ISSN 1463-6840
Proceedings of the British Society
for
Research into Learning
Mathematics
Volume 32 Number 3
Proceedings of the Day Conference held at the
University of Cambridge, Saturday 17th November 2012
These proceedings consist of short research reports which were written
for the joint NORME-BSRLM day conference on 17 November 2012.
The aim of the proceedings is to communicate to the research community
the collective research represented at BSRLM conferences, as quickly as
possible.
We hope that members will use the proceedings to give feedback to the
authors and that through discussion and debate we will develop an
energetic and critical research community. We particularly welcome
presentations and papers from new researchers.
Published by the British Society for Research into Learning Mathematics.
Individual papers © contributing authors 2012
Other materials © BSRLM 2012
All rights reserved. No part of this publication may be produced or
transmitted in any form or by any means, electronic or mechanical,
including photocopying, recording or any information storage retrieval
system, without prior permission in writing from the publishers.
Editor: C. Smith, Institute of Education, London University
ISSN 1463-6840
Informal Proceedings of the British Society for Research into
Learning Mathematics (BSRLM)
Volume 32 Number 3, November 2012
Proceedings of the Day Conference held at the University of
Cambridge on 17th November 2012
A functional taxonomy of multiple representations: A tool for analysing
Technological Pedagogical Content Knowledge
1
Hatice Akkoç and Mehmet Fatih Ozmantar
Marmara University & Gaziantep University
Coverage of topics during a mathematics pedagogy module for undergraduate
pre-service primary teachers
7
Yahya Al Zahrani and Keith Jones
University of Southampton
Rethinking partnership in initial teacher education and developing professional
identities for a new subject specialist team which includes a joint schooluniversity appointment: A case study in mathematics
13
Rosa Archer, Siân Morgan and Sue Pope
University of Manchester
Argumentative activity in different beginning algebra classes and topics
19
Michal Ayalon and Ruhama Even
Weizmann Institute of Science, Israel
Calculating: What can Year 5 children do now?
25
Alison Borthwick and Micky Harcourt-Heath
Relentless consistency: Analysing a mathematics prospective teacher education
course through Fullan’s six secrets of change
31
Laurinda Brown
University of Bristol, Graduate School of Education
Educational game Euro-Axio-Polis: Mathematics, economic crisis and
sustainability
a
a
a
Maria Chionidou-Moskofoglou , Georgia Liarakou , Efstathios Stefos , Zoi
Moskofogloub
a
University of the Aegean- Rhodes Greece, bUniversity College London
37
I thought I knew all about square roots
43
Cosette Crisan
Institute of Education, University of London
Developing a pedagogy for hybrid spaces in Initial Teacher Education courses 49
Sue Cronin and Denise Hardwick
Liverpool Hope University
From failure to functionality: a study of the experience of vocational students
with functional mathematics in Further Education
55
Diane Dalby
University of Nottingham, UK.
Investigating secondary mathematics trainee teachers’ knowledge of fractions 61
Paul Dickinson and Sue Hough
Manchester Metropolitan University
Teacher noticing as a growth indicator for mathematics teacher development 67
Ceneida Fernándeza, Alf Colesb, Laurinda Brownb
a
University of Alicante (Spain); bUniversity of Bristol
Teacher-student dialogue during one-to-one interactions in a post-16
mathematics classroom
73
Clarissa Grandi
Thurston Community College/University of Cambridge
Using scenes of dialogue about mathematics with adult numeracy learners –
what it might tell us.
79
Graham Griffiths
Institute of Education, University of London
Professional development in mathematics teacher education
85
Guðný Helga Gunnarsdóttir, Jónína Vala Kristinsdóttir and Guðbjörg Pálsdóttir
University of Iceland – School of Education
Engaging students with pre-recorded “live” reflections on problem-solving:
potential applications for “Livescribe” pen technology
91
Mike Hickman
Faculty of Education and Theology, York St John University
A student teacher’s recontextualisation of teaching mathematics using ICT
Norulhuda Ismail
Institute of Education, University of London
97
Mathematical competence framework : An aid to identifying understanding? 103
Barbara Jaworski
Loughborough University, Mathematics Education Centre
The role of justification in small group discussions on patterning.
109
Dr Cecilia Kilhamn
Faculty of Education, University of Gothenburg, Sweden
Social inequalities, meta-awareness and literacy in mathematics education
115
Bodil Kleve
Oslo and Akershus University College of Applied sciences
Stimulating an increase in the uptake of Further Mathematics through a
multifaceted approach : Evaluation of the Further Mathematics Support
Programme.
121
Stephen Lee and Jeff Searle
Mathematics in Education and Industry and Durham University
Exchange as a (the?) core idea in school mathematics
126
John Mason
University of Oxford and Open University
Exploring the notion ‘cultural affordance’ with regard to mathematics software
132
John Monaghan and John Mason
University of Leeds; University of Oxford and Open University
Doing the same mathematics? Exploring changes over time in students'
participation in mathematical discourse through responses to GCSE questions
138
Candia Morgana, Sarah Tanga, Anna Sfardb
a
Institute of Education, University of London, UK; bUniversity of Haifa, Israel
Vending machines: A modelling example
144
Peter Osmon
King’s College, London
Gendered styles of linguistic peer interaction and equity of participation in a
small group investigating mathematics
150
Anna-Maija Partanen and Raimo Kaasila
Åbo Akademi University and University of Oulu, Finland
Beauty as fit: An empirical study of mathematical proofs
156
Manya Raman
Umeå University
Making sense of fractions in different contexts
161
Frode Rønning
Sør-Trøndelag University College and Norwegian University of Science and Technology,
Trondheim, Norway
Developing statistical literacy with Year 9 students: A collaborative research
project
167
Dr Sashi Sharmaa, Phil Doyleb, Viney Shandilc and Semisi Talakia’atuc
a
The University of Waikato; bThe University of Auckland; and cMarcellin College
Feedback on feedback on one mathematics enhancement course
173
Jayne Stansfield
Graduate School of Education, University of Bristol, UK and Bath Spa University UK
Developing an online coding manual for The Knowledge Quartet: An
international project
179
Tracy L. Weston, Bodil Kleve, Tim Rowland
University of Alabama; Oslo and Akershus University College of Applied Sciences;
University of East Anglia/University of Cambridge
Preservice primary school teachers’ performance on rotation of points and
shapes
a
a
Zeynep Yildiz , Hasan Unal , A. Sukru Ozdemir
185
b
a
Department of Elementary Education, Faculty of Education, Yildiz Technical University;
Department of Elementary Education, Faculty of Education, Marmara University
b
Working Group Reports
Report from the Sustainability in Mathematics Education Working Group: Task
design
191
Nichola Clarkea, Maria Chionidou-Moskofogloub, Zoi Moskofoglouc, Alison Parrishd,
Anna-Maija Partanene
a
University of Nottingham, UK; bUniversity of the Aegean-Rhodes Greece; cUniversity
College, London; dWarwick University UK; eAbo Akademi University, Denmark.
Report of the Mathematics education and the analysis of language working
group
Alf Coles and Yvette Solomon
University of Bristol and Manchester Metropolitan University
196
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
A functional taxonomy of multiple representations: A tool for analysing
Technological Pedagogical Content Knowledge
Hatice Akkoç and Mehmet Fatih Ozmantar
Marmara University & Gaziantep University
This study investigates the development of prospective mathematics
teachers’ use of multiple representations during teaching in technologyrich environments. Forty prospective teachers took part in a teacher
preparation programme which aims to develop technological pedagogical
content knowledge (TPCK). As part of this programme, prospective
teachers participated in workshops during which the TPCK framework
was introduced focusing on function and derivative concepts. Various
components of TPCK were considered. This study investigates one
particular component of TPCK: knowledge of using multiple
representations of a particular topic with technology. The content we
focus on in this paper is the “concept of radian measure”. Two out of forty
prospective teachers introduced the concept of radian measure as part of
their micro-teaching activities. The data obtained from semi-structured
interviews, videos of prospective teachers' lessons, their lessons plans and
teaching notes was analysed to investigate prospective teachers'
knowledge of representations and of connections established among
representations using technological tools such as Cabri Geometry
software. We use the framework of “functional taxonomy of multiple
representations” which differentiates three main functions that multiple
representations serve in learning situations: to complement, constrain and
construct. We discuss the educational implications of the study in
designing and conducting teacher preparation programmes related to the
successful integration of technology in teaching mathematics.
Keywords: technological pedagogical content knowledge, multiple
representations, concept of radian measure, prospective mathematics
teachers.
Introduction
This study is part of a research project which aims to develop prospective
mathematics teachers’ Technological Pedagogical Content Knowledge (TPCK)
(Mishra and Koehler 2006). TPCK has been a useful framework for exploring what
teachers need to know or to develop for effective teaching of particular content. The
components of TPCK have been examined by only a few researchers. Among those,
Pierson (2001) and Niess (2005) used four components of PCK suggested by
Grossman (1990) to define the components of TPCK. In our research project, four of
these components were adopted from Grossman (1990). A component regarding
multiple representations was added as the fifth component of TPCK:
 Knowledge of using multiple representations of a particular topic with
technology
 Knowledge of students’ difficulties with a particular topic and addressing
them using technology
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 1
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012



Knowledge of instructional strategies and methods for a particular topic using
technology
Knowledge of curricular materials available for teaching a particular topic
using technology
Knowledge of assessment of a particular topic with technology
In the project, TPCK framework with its five components was used to design a
course for prospective mathematics teachers. Prospective teachers participated in the
activities concerning each component. This study focuses on a particular component
of TPCK, namely the knowledge of using multiple representations (MRs) with
technology. This paper aims to bring the content dimension into play focusing on the
concept of radian measure and investigates how two prospective mathematics teachers
integrate technology into their lessons to use multiple representations (MRs) of
radians.
Theoretical framework
To investigate prospective teachers’ development with regard to using MRs in
technology-rich environments, we use the framework of “functional taxonomy of
multiple representations” which differentiates three main functions that multiple
representations serve in learning situations: to complement, constrain and construct
(Ainsworth 1999). MRs might have complementary roles; that is, different
representations involve distinct yet complementary information or may support
different processes. MRs might also have constraining roles. Representations can
confine inferences, allowing one to constrain potential (mis)understandings stemming
from the use of another one. Finally, MRs might help students construct a deeper
understanding by providing cognitive linking of representations which might
eventually lead one to ‘see’ complex ideas in a new way and apply them more
effectively (Kaput 1989).
Ainsworth (1999) describes pedagogical functions that MRs serve as
mentioned above and proposes “systematic design principles”. She suggests
discouraging translation for complementary roles of MRs, to automate translation for
constraining interpretation and to scaffold translation for constructing a deeper
understanding. Although these principles are speculative, they provide a framework
towards the pedagogy of using MRs. This study investigates how prospective teachers
use MRs under this framework.
Methodology
Forty prospective mathematics teachers took part in the course. They were enrolled in
a teacher preparation programme (which will award them a certificate for teaching
mathematics in high school for students aged between 15 and 19) in a state university
in Istanbul. As part of this course, prospective teachers participated in workshops
during which the TPCK framework was introduced focusing on function and
derivative concepts. With regard to multiple representations, the workshops focused
on examples of MRs and how to make links between them with or without using
technology. Students' preferences for different representations which might be used
for different tasks were also discussed. The content we focus on in this paper is the
“concept of radian measure”. Prospective teachers prepared and conducted their own
workshops and discussed the issues of representing radians. After these workshops,
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 2
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
two out of forty prospective teachers (Gamze and Mutlu) introduced the concept of
radian measure as part of their micro-teaching activities. Both prospective teachers
were female and twenty-two years old. The data obtained from semi-structured
interviews, videos of prospective teachers' micro-teaching lessons (where they teach
to their peers in a computer lab), their lessons plans and teaching notes was analysed
to investigate prospective teachers' knowledge of representations and of connections
established among representations using technological tools such as Cabri software.
To do that, representations either drawn on the board or constructed using the
software were recorded. In addition to that, verbatim transcripts of micro-teaching
lessons and interviews were coded to reveal how prospective teachers link different
representations.
Findings
In this section, findings obtained from the data analysis will be presented in two subsections. Each sub-section is devoted to each prospective teacher’s lesson and how
they use MRs and links between them to teach concept of radian measure.
Findings regarding Gamze’s lesson
Gamze started her lesson by giving a brief history of angle. She then defined angle,
positive arc and negative arc. She assessed prior knowledge of unit circle by giving
various points and asking her peers to find whether they were on the unit circle or not.
After defining the angle of 1 degree verbally she explained it graphically on the
board. In other words, she used graphical representation for a complementary
purpose. She explained 1 radian in a similar way. She first explained it verbally as
follows:
1 radian is the angle which faces an arc equivalent to the length of a radius
(Gamze).
She then drew a graphical representation of any angle other than 1 radian and asked
her peers the following question:
How many radiuses are there in this arc? (Gamze).
She then asked her peers to find out the measure of the central angle facing the whole
circle using the Cabri Geometry software. Together with the class, she constructed a
circle and found that the measure of the central angle is 6.28 (which is nearly 2π)
radians (See Figure 1).
As can be observed from Gamze’s approach, she used MRs for constructing a
deeper understanding.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 3
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Figure 1 Graphical representation of 6.28 radians by Gamze using Cabri Geometry software
Another way she used such a translation is concerned with the algebraic
representation of radian which is arc-length divided by radius, L/R. She used Cabri
Geometry software to translate the algebraic representation of radian measure (L/R) to
graphical representation. She asked her peers to construct three circles and find the
measure of the angle as shown in Figure 2.
Figure 2 Graphical representation of L/R by Gamze using Cabri Geometry software
She used scaffolding to translate between MRs by asking questions as follows:
Measure the arcs on these three circles and radiuses of these circles. Are they the
same?...What I wonder is whether the ratio of arc and radius is the same?...Ratios
are the same. So it’s not dependent on the length of the arc. Radian is the ratio of
the length of an arc over the length of the radius. So it’s L/R (Gamze).
As can be seen from the excerpts and Figure 2 above, Gamze promoted an
understanding of the meaning of radian angle. In other words, she used graphical and
algebraic representations for constructing a deeper understanding.
Gamze’s reflections on her lesson also indicate that she used MRs for
constructing a deeper understanding:
Different representations are important for conceptual relationships. I tried to use
multiple representations to promote understanding and translations… I think
Cabri Geometry is very appropriate software to show the relationship between
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 4
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
radian and length of arc. I used Cabri Geometry to emphasise the arc. I wanted to
show the relations in a dynamic way (Gamze).
Findings regarding Mutlu’s lesson
Mutlu started her lesson by explaining the concepts of angle, directed angle and
directed arc on the board. Drawing a circle and a central angle on the circle, she asked
her peers the relationship between an angle and the length of the corresponding arc.
Why do we need radian as an angle measurement when we already have degree?
(Mutlu).
She first explained graphically what 1 radian is then expressed it verbally. In other
words, she used verbal and graphical representation for a complementary purpose
(See Figure 3).
Figure 3 Graphical representation of 6.28 radians by Mutlu using Cabri Geometry software
She then asked her peers to find out how many radians a round angle is:
If this is 1 radian, then how many radians is the whole circle? Let’s look at this
together… Yes, 6 radians and there is some left here. It is nearly 0,28… Is this
number familiar to you? Nearly 2π (Mutlu)
As can be seen from the excerpts and Figure 3 above, Mutlu, together with her
peers, constructed a graphical representation and discovered that there are 6 radiuses
and approximately 0.28 radiuses on a circumference of the circle. In that sense, she
used Cabri Geometry to construct a deeper understanding of the concept of radian
measure. In other words, she translated verbal representation to graphical
representation using Cabri Geometry software to construct a deeper understanding of
radian.
Mutlu’s reflections on her lesson indicate that she also used MRs for
constraining interpretation as well as the other two purposes of MRs. She mentioned
that radian is generally understood in terms of π (not as an arbitrary real number such
as 2). To constrain this interpretation, she said that she asked her peers to construct a
graphical representation using Cabri Geometry and found that a round angle is
approximately 6.28 radians.
Discussion and conclusion
The analysis of data indicated that both prospective teachers used MRs aiming at a
conceptual understanding of radian, i.e. for constructing a deeper understanding. They
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 5
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
also used MRs for complementary purposes while only Mutlu used MRs for
constraining purposes.
For complementary purpose, both prospective teachers defined concept of
radian measure verbally and explained it using graphical representation by drawing a
circle and an arc on it. For constructing a deeper understanding of the concept of
radian measure, both prospective teachers asked their peers to find out the measure of
the central angle facing the whole circle using the Cabri Geometry software (See
Figure 1). This requires a translation from verbal to graphical representation which is
not automatic but rather should be constructed using the software. The case for using
MRs for constraining purpose was observed only in Mutlu's lesson. She mentioned
that radian is generally understood in terms of π (not as a real number such as 2). To
constrain this interpretation, she asked her peers to construct a graphical
representation using Cabri Geometry and showed that a round angle is approximately
6.28 radians.
These observations let us draw two main conclusions. First any successful
preparation program for technology integration should provide opportunities for
participants to appreciate the contribution of MRs for an effective use of technology
aiming at a conceptual understanding. Several studies (such as that of Juersvich et al.
2009) suggest that the links among the MRs are not often established by teachers
during instruction. Second, functional taxonomy of MRs provides a theoretical lens to
analyse (prospective) teachers’ practice of technology integration regarding how MRs
can be effectively used in technology-rich environments.
Acknowledgement
This study is part of a project (project number 107K531) funded by TUBITAK (The
Scientific and Technological Research Council of Turkey).
References
Ainsworth, S. 1999. The functions of multiple representations. Computers &
Education 33: 131-152.
Grossman, P. L. 1990. The making of a teacher: Teacher knowledge and teacher
education. New York: Teachers College Press.
Juersivich, N., J. Garofalo, and V. Fraser 2009. Student Teachers’ Use of
Technology-Generated Representations: Exemplars and Rationales. Journal of
Technology and Teacher Education 17: 149-173.
Kaput, J. J. 1989. Linking representations in the symbol systems of algebra. In
Research Issues in the Learning and Teaching of Algebra, ed. S. Wagner and
C. Kieran, 167-194. Hillsdale, NJ: Erlbaum.
Mishra, P., and M. J. Koehler. 2006. Technological Pedagogical Content Knowledge:
A Framework for Teacher Knowledge. Teachers College Record 108: 1017–
1054.
Niess, M.L. 2005. Preparing teachers to teach science and mathematics with
technology: Developing a technology pedagogical content knowledge.
Teaching and Teacher Education 21: 509–523.
Pierson, M. E. 2001. Technology integration practice as a function of pedagogical
expertise. Journal of Research on Computing in Education 33: 413-429.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 6
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Coverage of topics during a mathematics pedagogy module for undergraduate
pre-service primary teachers
Yahya Al Zahrani and Keith Jones
University of Southampton
Recently, research on teacher preparation has begun examining the
opportunities to learn that pre-service teachers have of the different forms
of knowledge thought to be necessary for effective teaching. This paper
reports on one component of a wider study of undergraduate pre-service
specialist primary mathematics teacher preparation: the pre-service
teachers’ opportunities to learn about the primary school mathematics
curriculum during a final-year undergraduate module on mathematics
pedagogy (MPM). Using data from observations of the complete teaching
of this module at two university colleges in Saudi Arabia, the findings
indicate that while the pre-service teachers had some opportunity to learn
about teaching aspects of the primary school geometry curriculum, they
had little or no opportunity to learn about teaching topics related to the
algebra taught in the upper primary school years. The main reason for this
discrepancy was that while the MPM contained some sessions on primary
school geometry, there were no sessions explicitly related to primary
school algebra even though the current version of the relevant primary
school curriculum now includes some algebra for Grades 5 and 6 (pupils
aged 10-12).
Keywords: opportunity to learn, school mathematics curriculum,
geometry, algebra, pre-service primary mathematics teacher education
Introduction
For some time, research on teacher education in general, and on initial teacher
education in particular, has focused on the forms of knowledge that teachers need in
order to teach most effectively (see, for example, Rowland and Ruthven 2011). Such
forms of knowledge have commonly been categorised into ‘subject matter
knowledge’ and ‘pedagogical content knowledge’ (see, for example Petrou and
Goulding 2011). Here, ‘subject matter knowledge’ (SMK) is, in general, taken to refer
to the key facts, theories, models and concepts of mathematics, together with the
processes by which such theories and models are generated and established as valid.
Pedagogical content knowledge (PCK), in contrast to SMK, encompasses the
representations, examples and applications of mathematics that mathematics teachers
use in order to make mathematics comprehensible to students, together with the
strategies that such teachers use in order to overcome students’ difficulties in learning
mathematics. PCK also includes knowledge of the school curriculum.
Researchers have, more recently, begun examining the opportunities to learn
(OTL) that pre-service teachers have of these different forms of knowledge (see, for
example, Chapter 7 of Tatto et al. 2012). One major reason for this focus on OTL is
that pre-service mathematics teachers can experience difficulties in teaching primary
school mathematics even though they have completed relevant university-based
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 7
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
training; this being an example of what is generally referred to as the ‘theory-practice
gap’ (for more on this ‘gap’ see, for example Brouwer and Korthagen 2005).
Programmes for preparing primary mathematics teachers are diverse, globally.
In some countries, for example the UK and Germany, primary teachers are prepared
as generalists to teach all primary school subjects (though in the UK the training of
some specialist primary mathematics teachers is beginning in 2013). In contrast, in
some countries such as Thailand, Malaysia and Saudi Arabia, primary teachers are
prepared as specialists to teach mainly mathematics. Even so, there is a lack of studies
concerned with university-based teacher preparation curricula, with Stuart and Tatto
(2000, 493) commenting that “much less has been written on the professional
curriculum for teacher preparation”. This study is addressing this issue by analysing
the OTL aspects of the primary school curriculum during a Mathematics Pedagogy
Module (MPM) taken by undergraduate specialist primary pre-service mathematics
teachers in the first semester of their fourth year of study, immediately prior to
spending a semester in school.
Research into the design of mathematics teacher education programmes
A major study that is providing a global perspective on the design of initial teacher
preparation programmes is the Teacher Education and Development Study in
Mathematics (TEDS-M) being undertaken by Tatto and colleagues (see Tatto et al.
2008; 2012). TEDS-M is aiming to build a comprehensive picture of primary and
secondary mathematics teacher education around the world. The TEDS-M study has
three components: the first is examining teacher education policy, schooling, and
social contexts at the national level, the second is studying primary and lower
secondary mathematics teacher education routes, institutions, programmes, standards,
and expectations for teacher learning, while the third is determining the knowledge of
mathematics and related teaching of future primary and lower secondary school
mathematics teachers.
In analysing the characteristics of mathematics teacher preparation across the
17 countries participating in TEDS-M, Tatto et al. (2012) report a diversity of practice
in terms of institutional arrangements and regulatory systems. For example, Tatto et
al. (2012) show that initial teacher preparation programmes that focus on preparing
teachers to teach in lower and upper-secondary schools provide more opportunities to
learn mathematics in depth comparing to the programmes that prepare teachers to
teach at the primary level. This is likely to be because the overwhelming majority of
secondary school mathematics teaches are specialists, while this is not the case for
primary teachers of mathematics. In terms of opportunity to learn about the relevant
school mathematics curriculum, the TEDS-M results show that for future primary
mathematics teacher there is a high degree of variability across countries and
programme groups. Greater OTL was found in preparation programmes for specialist
primary mathematics teachers and for programmes preparing teachers to teach the
higher grade levels (see Tatto et al. 2012, 181).
Theoretical framework: opportunity to learn (OTL)
The TEDS-M study (see Tatto et al. 2008; Tatto et al. 2012) uses the concept of
opportunity to learn (OTL) in order to investigate pre-service teachers’ pedagogical
content knowledge of mathematics subject topics (such as number, geometry, algebra,
and data). The term OTL was first coined by Husen (1967):
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
“students have had the opportunity to study a particular topic or learn how to
solve a particular type of problem presented by the test …if they have not had
such an opportunity, they might in some cases transfer learning from related
topics to produce a solution but certainly their chance of responding correctly to
the test item would be reduced”. (Husen 1967, 162-163)
Carroll (1963, 727) is perhaps best known for taking up the term OTL as “time
allowed for learning” For this study, the notion of OTL in the TEDS-M (2008)
framework was used to examine the extent to which the content of the MPM provided
opportunity for the pre-service primary mathematics teachers to learn about the
primary school mathematics curriculum.
Methodology
The purpose of the study was to analyse the extent to which pre-service primary
mathematics teachers had opportunity to learn how to teach geometry and algebra as
specified in the relevant primary school mathematics curriculum. The study was
implemented in Saudi Arabia and focused on the mathematics pedagogy module
MPM that was taught during the second semester of the academic year 2011-2012 at
two university colleges.
Data was collected by observing university mathematics education lecturers
teaching the MPM at each of the two university colleges. To document each taught
session, an observation sheet was used. This observation sheet divided each session
into 12 parts, each lasting for ten minutes (1-10 minutes; 11-20 minutes, 21-30
minutes and so on). The role of the researcher-as-observer was to determine what type
of mathematical content was taught by the mathematics education university lecturers
every 10 minutes in each session of the MPM. The type of mathematical content was
based on the TEDS-M framework (Tatto et al. 2008).
The following categories were used:
Very heavy emphasis: if the lecturer focuses on topics related to the concepts:
Geometry, Algebra for 75%≤100% of the session time (= 91 ≤ 120 minutes)
Heavy emphasis: if the lecturer focuses on topics related to the concepts: Geometry,
Algebra for 50% < 75% of the session time to the concepts (= 61 ≤ 90 minutes)
Average emphasis: if the lecturer focuses on topics related to the concepts: Geometry,
Algebra for 25 % < 50% of the session time to the concepts (= (31≤ 60minutes)
Little emphasis: if the lecturer devotes less than of 25% of the sessions time to the
concepts (= ≤ 30 minutes)
As the study was conducted in Saudi Arabia, it is germane to know that the
primary mathematics school curriculum in Saudi Arabia is specified across six grades.
In each grade the curriculum emphasises different topics across the four mathematical
subject areas of Numbers, Algebra, Geometry, and Data. Table 1 shows a comparison
of the 2002 primary mathematics curriculum for Grades 1 to 6, compared with the
curriculum in 2012.
Grade
primary mathematical school topics 2002
1
(pupils
aged
6-7
Years)
Comparison and classification, numbers up
to 5, location and style, numbers up to 10,
numbers up to 20, combine. Additions
methods, subtraction, fractions.
primary mathematical school topics 2012
Comparison and classification, numbers up
to 5, location and style, numbers up to 10,
numbers up to 20, combine. Additions
methods, subtraction, measurement,
geometric shapes and fractions, money and
time.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
2
3
Numbers up to 100 patterns, combining
methods, methods of subtraction, data
representation and reading, collecting twodigit numbers, fractions, numbers until
1000, geometric shapes, measurement:
length, area, measurement: collection of 3digit numbers, subtraction of 3-digit
numbers
Numbers up to 100 patterns, combining
methods, methods of subtraction, data
representation and reading, collecting twodigit numbers, fractions, numbers until
1000, geometric shapes, measurement:
length, area, measurement: Capacity and
weight, collection of 3-digit numbers,
subtraction of 3-digit numbers
Addition, subtraction, multiplication 1,
multiplication 2. Division 1, division 2,
measurement, geometric shapes, display and
interpretation of data, fractures
Addition, subtraction, multiplication 1,
multiplication 2. Division 1, division 2,
measurement, geometric shapes, display and
interpretation of data, fractures
Addition and subtraction organize and
display data and interpretation, patterns and
algebra, multiplication in the number of
number one, multiplication in a two-digit
number. Divide by the number of number
one; identify geometric shapes and its
description. Measurement, fractions usual,
and decimal.
Addition and subtraction organize and
display data and interpretation, patterns and
algebra, multiplication in the number of
number one, multiplication in a two-digit
4
number. Divide by the number of number
one; identify geometric shapes and its
description, measurement, fractions and
decimals.
Addition, subtraction, multiplication,
Addition, subtraction, multiplication,
division, use of algebraic expressions for
division. Normal for instance, 2/3, 4/5.
example (3+x)-1=? x=2, functions and
Representation and representation of data,
equations such as 2x=6, fractions such as
5
denominators and complications, collect and 2/3, 4/5. Representation of data,
put fractions, geometric shapes,
denominators and complications. Geometric
measurement: perimeter, area and volume.
shapes, measurement such as perimeter, area
and volume.
Topics in algebra: functions and numerical
patterns such as 2, 4, 8, or 15, 10, 5, 0.
Operations on decimals, fractions normal
Statistics and graphical representations,
6
and decimal fractions, measurement: length,
operations on decimals, fractions and
(pupils capacity and mass. Normal fractions, ratio
decimals, measurement such as length,
aged and proportion, percentages and
capacity and mass. Ratio and proportion,
11-12 probabilities, Geometric , polygons,
percentages and probabilities, Geometry:
Years) measurement: perimeter, area and volume,
polygons, measurement: perimeter, area and
volume,
Table 1: the KSA primary mathematical school topics 2002/2012
Source: Obecan Education: 2002-2012 [changes by 2012 shown in italics]
As can be seen from Table 1, the main change in the primary mathematics
curriculum in 2012, compared with 2002, is the introduction of algebra topics for
pupils in Grades 5 and 6 (pupils aged 10-12).
Analysis and result
Table 2 shows, for each session of the MPM, the percentage of time devoted
to, and the degree of emphasis on, the two school mathematics subject areas of
primary geometry and algebra.
From Table 2, it can be seen that the pre-service primary mathematics teachers
had average (or below) opportunity to learn concepts related to geometric topics. In
contrast, in terms of OTL about algebra topics, there was no coverage at all.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 10
Sessions
N of the
Mathematics
Pedagogy
Module
1
sessions
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
degree of emphasis on school
geometry topics
*
VHE
(1)
-------
*
HE
(2)
-------
*
AE
(3)
-------
*
LE
(4)
0
24
0
14
0
degree of emphasis on school
algebra topics
%
*
VHE
(1)
-------
*
HE
(2)
-------
*
AE
(3)
-------
*
LE
(4)
26
24
18
10
22
%
x(1)
0
21.6
Classification
concept
2
y(1)
20
20
x(2)
0
15
2 Counting
concept
4
y(2)
11.6
8.3
The four
x(3)
0
18.3
3 operations
y(3)
0
0
8
6.6
(+, - , ×, ÷)
Fractions and
x(4)
---0
0
---4
3.3
4 operations on
------y(4)
0
0
6
5
them
Geometry
x(5)
--40
-33.3
---8
6.6
concept, e.g. a
----5
straight line, y(5)
56
46.6
0
0
--angles
Geometric
------x(6)
48
40
0
0
shapes and
6
their
------y(6)
54
45
6
5
properties
Geometric
------x(7)
58
48.3
12
10
7
models, e.g.
------1
38
331.6
10
8.3
cylinder, cube y(7)
x(8)
--50
-441.3
---8
6.6
8 Measurement
units1
y(8)
--40
-33.3
---0
0
Applications
-------0
14
11.6
of quantitative x(9)
9
and qualitative
------1
analyses of the y(9)
8
6.6
10
8.3
problems
* (1) VHE Very heavy emphasis (2) HE heavy emphasis (3) AE Average emphasis (4) LE little
emphasis
Table 2: emphasis on school mathematics topics during sessions at university colleges x and y
1
Discussion
Overall, the data indicate that while the pre-service teachers received average
opportunity to learn topics related to teaching the primary school geometry
curriculum, they had little or no opportunity to learn topics related to teaching primary
school algebra. A key reason for this discrepancy is that while there are some sessions
of the MPM related to primary school geometry, there are no sessions related to
primary school algebra. Even though the mathematics school curriculum in Saudi
Arabia has changed over the period 2002 to 2012, and now includes some algebra
topics for pupils in Grades 5 and 6, the MPM has not changed for more than 10 years
(according to the directory of undergraduate courses 2002-2012 at each of the
university colleges).
Conclusion
This study showed that there was average emphasis on some topics in school
geometry during the MPM. However, there was little or no emphasis on school
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 11
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
algebra. This confirms how there is variation of opportunity to learn in some topics
related to both subject areas of geometry and algebra.
What remains unclear is how to decide how much to emphasise topics such as
school geometry and algebra in a teacher preparation programme for pre-service
primary mathematics teachers. The implications of this are that more research is
needed on how much, and in what way, topics related to geometry, algebra or other
mathematical areas should be included in pre-service mathematics teachers
curriculum to match topics in primary mathematics school curriculum.
References
Brouwer, N., and F. Korthagen. 2005. Can teacher education make a difference?
American Educational Research Journal 42(1): 153-224.
Carroll, J. 1963. A model for school learning. Teachers College Record 64: 723-733.
Husen, T., ed. 1967. International study of achievement in mathematics: a
comparison of twelve countries. New York: John Wiley.
Obecan Education 2012. Mathematics and science. Riyadh: Obecan Education.
Petrou, M. and M. Goulding 2011.Conceptualising teachers’ mathematical knowledge
in teaching. In Mathematical knowledge in teaching, ed. T. Rowland and K.
Ruthven, 9-25. New York: Springer.
Rowland, T., and K. Ruthven, eds. 2011. Mathematical knowledge in teaching. New
York: Springer.
Stuart, J. S. and M. T. Tatto 2000. Designs for initial teacher preparation programs: an
international view. International Journal of Educational Research 33(5): 493514.
Tatto, M. T., J. Schwille, S. Senk, L. Ingvarson, R. Peck, and G. Rowley. 2008.
Teacher education and development study in mathematics (TEDS-M):
conceptual framework. East Lansing, MI: TEDS-M.
Tatto, M. T., J. Schwille, S. Senk, L. Ingvarson, G. Rowley, R. Peck, K. Bankov, M.
Rodriguez and M. Rackase. 2012. Policy, practice, and readiness to teach
primary and secondary mathematics in 17 countries. Amsterdam:
International Association for the Evaluation of Educational Achievement.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 12
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Rethinking partnership in initial teacher education and developing professional
identities for a new subject specialist team which includes a joint schooluniversity appointment: A case study in mathematics
Rosa Archer, Siân Morgan and Sue Pope
University of Manchester
In a time of rapid and extensive change in initial teacher education policy,
a new team of mathematics educators is establishing at the University of
Manchester. How does a new team of mathematics educators (some with
experience of other institutions) establish itself and ensure that previous
strengths and successes are maintained and developed? One member of
the team is a joint school-university appointment. What are the
affordances of a joint school-university appointment? What are the
personal challenges for the appointee and colleagues working with the
appointee – in school and in university? Evidence for the paper is through
personal reflective accounts, focus group discussions with school and
university colleagues, an anonymous questionnaire of student teachers and
their course outcomes. The outcomes of this early experience have
implications for the developing practice of the University of Manchester
PGCE mathematics team and the way in which university and school
based colleagues work together to optimise learning for beginning
teachers, as new models of ITE are adopted within a well-established
partnership. These implications may provide areas for consideration by
institutions rethinking partnership in initial teacher education.
Keywords: initial teacher education, partnership, secondary mathematics
Introduction
In the rapidly changing landscape of initial teacher education in England following
the change of government in 2010, the need to appoint a new team of mathematics
educators presented both challenges and opportunities. Alongside experienced
mathematics educators, the university worked with one of its partnership schools in
the vanguard of Teaching Schools to make a joint appointment. An experienced
teacher and former National Strategies consultant, the appointee brought
complementary strengths to the university tutor team.
Conscious that this was a novel situation, the team determined to investigate
the impact on student outcomes and their emerging professional identities. We
adopted a case study approach (Wellington 2000) using mixed methods: student
questionnaire and summative attainment data, focus group and one to one interviews
and tutors’ reflective diaries, with a view to providing a rich evidence base. The
principal aim was to ensure that the quality of the provision was maintained, whilst
enhancing learning of tutors, teachers/mentors and student teachers through the
opportunities of the new arrangements. Through exploring this novel context we
hoped to be able to identify priorities for our future development and provide a case
which others might find a valuable reference point when considering ways of
developing their initial teacher education provision.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 13
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
National Context
When the coalition government came to power in 2010 it set out its agenda for
education in England in the white paper The Importance of Teaching (Department for
Education (DfE) 2010). A principle aim was to put more teacher training into schools.
At the same time Ofsted (2010) reported that the best quality and value for money
initial teacher training (ITT) was higher education institution (HEI) led. HEI courses,
whatever their duration, have a substantial school element (24 weeks for secondary
and 18 weeks for primary) and guaranteed teaching experience in at least two
contrasting schools. As well as ensuring students meet expectations for teaching they
include academic study which results in a qualification which enables teachers to
work in other countries as well as England.
School based routes into teaching have existed for several years and the
aspiration to have all ITT in schools is not a new one. Anthea Millett as the chief
executive of the Teacher Training Agency, a non-departmental government body with
responsibility for teacher recruitment and training, (1995-1999) was also very keen on
moving ITT into schools. Despite preferential funding and a lighter touch inspection
regime, the graduate teacher training programme (GTP) and school based ITT
consortia (SCITTs) provides just one in five of all new teachers (Smithers and
Robinson 2011). Evidence from Ofsted (2010, 2012) is clear that the quality of school
based ITT is far more variable and it is less cost effective. Often schools take on
trainees with the hope of alleviating staffing shortages. Consequently trainees find
themselves with substantial teaching responsibilities and the emphasis is on survival,
as opposed to development as critically reflective practitioners with an understanding
of how young people learn and develop, particularly in the context of their specialist
area (primary or a subject at secondary).
In 2003, based on a scheme in USA (Teach for America), Teach First brought
200 graduates with firsts or upper class seconds into teaching for two years. In 2010
the numbers had increased to 500. Teach First participants have six weeks training
and then work in challenging inner city schools with high proportions of
disadvantaged youngsters that traditionally struggle to recruit and retain staff. Teach
First recruits high performing graduates who can become teachers of secondary
mathematics with just grade B at A level (Teach First 2012). There is on-going
support for Teach First trainees throughout the first year, but the assumption is that
subject specific development happens largely in school. This is unlikely to happen as
Teach First participants are likely to be working in schools with a shortage of subject
specialists, indeed subject specific pedagogy is identified by Ofsted as an area for
improvement (Teach First 2012). The current secretary of state is a strong proponent
of Teach First (DfE 2010) and further expansion of the scheme has recently been
announced (DfE 2012a).
For a mathematics PGCE course (the most popular route into teaching) half a
degree or equivalent in mathematics is usually required. Six- or nine-month subject
knowledge enhancement courses enable suitable candidates with an A level and
relatively little undergraduate experience to develop their knowledge and
understanding of their chosen specialism. In July 2012 the Secretary of State
announced that schools could employ whoever they wanted as teachers (DfE 2012b).
Whilst this has always been possible, schools have usually used the untrained
teacher/instructor pay scale rather than paying a qualified teacher salary. Teach First
and the July announcement seem to contradict the government’s espoused
commitment to teacher subject expertise. From September 2012, all student teachers
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 14
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
will have to pay £9000 fees for the PGCE course and there will be differentiated
bursaries dependent on degree classification. It is too early to say what impact these
changes will have, however many colleagues have expressed concerns (e.g.
Association for Language Learning (ALL) 2012).
The white paper (DfE 2010) also introduced Teaching Schools to be akin to
Teaching hospitals. Teaching Schools are expected to take a lead in both initial
teacher education and professional development for a group of local schools, typically
both primary and secondary. They are expected to work with at least one local
university ITT provider in developing their provision. School Direct has been
introduced as a new route into teaching, where aspirant teachers are recruited and
trained by the Teaching School and its partner schools, with variable levels of
university input. School Direct (salaried) replaces GTP (Teaching Agency 2012)
This new model, with schools taking a larger share of responsibility for ITT
creates the potential for a new type of professional who is both a teacher and an
academic, somebody who occupies a third space. A third space is a “territory between
academic and professional domains, which is colonised primarily by less bounded
forms of professional” (Whitchurch 2008, 377).
The particular context
The University of Manchester Postgraduate Certificate of Education (PGCE)
programme is well established and well regarded. It has consistently been graded
Outstanding since inspection of initial teacher education was introduced in 2002. The
programme has a strong partnership with a substantial core of schools and colleges
that have worked with the University over many years. Many mentors completed their
PGCE at the University. The entire mathematics course team was renewed during the
2011-12 academic year, following a long period of stable staffing. Alongside two
academics with experience in other universities, a joint university/school appointment
was made. The school is a long established partner with the University and was one of
the first Teaching Schools.
The overriding concern of the team was to ensure that all student teachers
were supported and challenged to be as successful as possible both in school and
academically and that standards were maintained. The PGCE requires students to
complete six Masters level assignments and four individual study packs.
All students have weekly meetings with their mentors, and termly school visits
by their tutors where the tutor, mentor and student teacher discuss progress and agree
targets, and termly tutorials. Tutors are expected to quality assure the school
placement and moderate mentors’ judgements about progress during these visits.
Students also receive support and feedback from tutors on their preparation for
school, files and academic assignments; when necessary additional school visits and
tutorials are provided.
The mathematics education tutors contribute to all aspects of the PGCE
programme, recruitment and selection, mentor training and update sessions. Tutors
also run seminar sessions for the education and professional studies strand of the
PGCE programme where students work in mixed subject groups. The University
expects all tutors to undertake research and scholarly activity. As in many other
higher education institutions (Pope and Mewborn 2009) any tutor who does not have
research qualifications is expected to complete a Masters or Doctorate as required.
Mentors are expected to be role models for the student teachers, demonstrating
and nurturing reflective practice. They are expected to have a weekly meeting with
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 15
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
their student teacher reviewing progress and negotiating targets. Usually the student
teacher works with at least one of the mentor’s classes. The mentor co-ordinates the
school experience of the student(s) within the department, and assesses their progress
at the end of each placement. The vast majority of students are placed individually.
Where students are placed in pairs or threes they each have a personal mentor.
The Teaching School has a very strong mathematics department involved in a
range of outreach and collaborative work with its family of schools, both primary and
secondary. The head of mathematics and the joint school/university appointee have
substantial experience in advisory and consultancy work. The joint school/university
appointee enabled a different model of placements to be explored. The school
department takes four student teachers at any one time and the appointee is mentor to
all four. A small number of schools offer contrasting placements to the Teaching
School which is a high performing selective girls’ school, enabling eight students to
participate during the year. The student teachers write a short application to be
involved, mindful of the need to be confident with mathematics up to A level.
The school/university joint appointee has both the university tutor role and
school mentor role for the student teachers alongside becoming a valued senior
member of the school.
The evidence
At the end of the academic year the team collated the PGCE student outcomes
including school grades and academic attainment, and course evaluations. Team
members undertook interviews with school based and university based colleagues and
identified key points from their reflective journals. The PGCE student data was
collected as a matter of course and students gave their permission for its use as part of
the case study. Colleagues based in school and university were invited to contribute to
the evidence base and volunteers from the university took part in a focus group
discussion, while school colleagues had one to one discussions with individual
members of the mathematics education team. The joint appointee was not involved in
the interviews. A semi-structured interview schedule was devised to help ensure that
the same themes were discussed with all interviewees but also to allow the
interviewer to probe where appropriate.
We investigated whether the tutor groups were equivalent in terms of prior
attainment and outcomes. Although the students who had worked with the joint
university/school appointee were slightly stronger academically and did particularly
well in school, there was no statistically significant difference in performance. All the
students who had worked with the joint appointee reported that there was coherence
between school and university expectations and were generally happy with the course
documentation. Other students were slightly less happy with the overall coherence
between school and university expectations.
The focus group of university tutors was very positive about the new team,
they felt the team was collegiate, pro-active, had high expectations and were very
committed. They recommended that team members established clearer boundaries
with students and were more confident and assertive in whole course discussions and
developments, as the team members had a great deal to contribute.
School based colleagues were excited by the opportunity to work more closely
with the university. They said that having more student teachers in the department had
been very rewarding. They observed that student teachers really benefitted from
having access to their mentor/tutor whilst in school. A major concern was that the role
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 16
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
and responsibilities of the joint appointee lacked clarity and colleagues tended to defer
everything to do with the student teachers to the joint appointee, even though the
department had a number of experienced mentors.
During the year there had been a Teaching Agency (TA) funded project on
lesson study (Lewis and Tsuchida 1988) involving the Teaching School and
University with a number of other partner schools. Teachers, mentors, tutors and
student teachers worked together to plan, teach, evaluate, refine, teach and evaluate
lessons designed to promote mathematical dialogue and argumentation. This was led
by the joint appointee working with a professor from the University. The outcomes
were shared at a conference involving colleagues from all the schools involved and all
the student teachers (AGGS 2012). The team plans to build on this experience to
enhance the effectiveness of the PGCE and its contribution to the professional
development of teachers in partnership schools.
The tutors found the year challenging: new cultures, documentation,
expectations and relationships. For the joint appointee a new school, moving into
initial teacher education and embarking on Masters level study was rewarding but
very demanding. Time spent travelling between school and university was a lost
opportunity for enculturation in either environment. Making space away from the
student teachers when in school was difficult. The biggest challenge was, as a mentor,
having to assess students on school practice whilst also being the tutor.
Conclusion
The year provided significant challenges for the tutors developing as a team in a wellestablished ITT provider. In addition the joint school/university appointee had to
negotiate a new school in which she had a senior appointment and embark on Masters
level study. There is no evidence to suggest that the student teachers had a less good
preparation despite the new team.
University and school colleagues were positive about the new team and the
new type of appointment. Benefits were perceived for the course as a whole and the
student teachers. The Teaching Agency funded lesson study pilot was particularly
successful, strengthening partnership with schools, enhancing the professional
development of all involved and informing future developments of the course.
Building on the experiences of the first year the team intend to
 incorporate collaborative planning, teaching, reflection and evaluation
into the programme for all students
 work more closely with a number of schools
 pro-actively involve mentors in the university elements of the course
 exploit synergies across the different emerging routes into teaching
 be more explicit with student teachers about why they are asked to do
what they do
 ensure that academic assignments are relevant to students’ personal
professional development.
The team will research the impact of lesson study on the development of the
student teachers, colleagues in partner schools and their own understanding of
effective pedagogy.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 17
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
References
Association for Language Learning (ALL). 2012. Statement from the Association for
Language Learning on ITE reform 12/12/2012. http://www.alllanguages.org.uk/uploads/files/Press%20Releases/Statement%20on%20ITE%
20reform%20from%20ALL.pdf
Altrincham Girls’ Grammar School (AGGS). 2012. Lesson study conference.
http://www.aggs.trafford.sch.uk/index.php/teaching-school/teaching-schoolnews/991-lesson-study-conference-enhancing-dialogue-and-questioning-inmathematics-classrooms
Day, C., B. Elliott and A. Kington. 2005. Reform, standards and teacher identity:
challenges of sustaining commitment. Teaching and teacher education 21:
563-577.
Day, C., P. Sammons, G. Stobart, A. Kington and Q. Gu. 2007. Teachers matter:
connecting lives, work and effectiveness. Maidenhead: Open University Press.
DfE. 2010. The importance of teaching. London: HMSO.
DfE. 2012a. Tripling number of top graduates recruited through Teach First Press
release 14/6/2012.
http://www.education.gov.uk/inthenews/inthenews/a00210309/triplingnumber-of-top-graduates-recruited-through-teach-first
DfE. 2012b. Academies to have the same freedom as free schools over teachers Press
release 27/7/2012.
http://www.education.gov.uk/inthenews/inthenews/a00212396/academies-tohave-same-freedom-as-free-schools-over-teachers
HE academy. http://www.heacademy.ac.uk/professional-recognition
Ofsted. 2010. Annual Report 2009/10. London: Ofsted.
Ofsted. 2012. Annual report 2010/11. London: Ofsted.
Lewis, C. and I. Tsuchida. 1988. A lesson is like a swiftly flowing river: how research
lessons improve Japanese education. American Educator Winter, 12-17 and
50-51.
Pope, S. and D.S. Mewborn. 2009. Becoming a teacher educator: perspectives from
the United Kingdom and the United States. In ICMI Study series 15: The
Professional Education and Development of Teachers, ed. R. Even and D.
Loewenberg Ball, 113-120. New York: Springer.
Smithers, A. and P. Robinson. 2011. The good teacher training guide 2011.
Buckingham: Buckingham University.
Teaching Agency. 2012. http://www.education.gov.uk/get-into-teaching/teachertraining-options/school-based-training/school-direct.aspx
Teach First. 2012.
http://graduates.teachfirst.org.uk/recruitment/requirements/teaching-subjectrequirements.html
Teach for America. http://www.teachforamerica.org/our-mission TDA 2011
http://www.tda.gov.uk/training-provider/itt/schooldirect.aspx (accessed
21/03/2012).
Wellington, J. 2000. Educational research, contemporary issues and practical
approaches. London: Continuum.
Whitchurch, C. 2008. Shifting identities and blurring boundaries: the emergence of
third space professionals in UK higher education. Higher Education Quarterly
62:377–396.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 18
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Argumentative activity in different beginning algebra classes and topics
Michal Ayalon and Ruhama Even
Weizmann Institute of Science, Israel
This study compares students’ opportunities to engage in argumentative
activity between two classes taught by the same teacher and across two
topics in beginning algebra: forming and investigating algebraic
expressions and equivalence of algebraic expressions. The study
comprises two case studies, in which each teacher taught two 7th grade
classes. All four classes used the same textbook. Analysis of classroom
videotapes revealed that the opportunities to engage in argumentative
activity related to forming and investigating algebraic expressions were
similar in each teacher's two classes. By contrast, substantial differences
were found between one teacher's classes with regard to the opportunities
to engage in argumentative activity related to equivalence of algebraic
expressions. The discussion highlights the contribution of the topic, the
teacher, and the class to shaping argumentative activity.
Keywords: argumentative activity, mathematics, topic, teacher, class,
deductive reasoning, inductive reasoning.
Background
In recent years, there has been a growing appreciation of the importance of
incorporating argumentation into school mathematics. First, because the principal
facets of argumentative activity – justifying claims, generating and justifying
conjectures, and evaluating arguments – are all essential components of doing,
communicating, and recording mathematics. In addition, accumulating research
suggests that participation in argumentative activities – which encourage students to
explore, confront, and justify different ideas and hypotheses – promotes mathematical
understanding (e.g., Yackel and Hanna 2003).
However, studies have shown that argumentation is not widely used in
mathematics classrooms (e.g., Hiebert et al. 2003). Research also shows that students
commonly use different kinds of justifications, which often depart from the norms of
the field (e.g., Harel and Sowder 2007). Specifically, research shows that deductive
reasoning is a source of great difficulties for students, and that students often have
difficulties in constructing arguments treating the general case (Harel and Sowder
2007). Instead, students often employ inductive reasoning, which is considered to be
the simplest and most pervasive form of everyday problem-solving activities (Nisbett
et al. 1983), and is often students' preferred way to form, test, and justify
mathematical conjectures (Harel and Sowder 2007).
Studies point to a variety of roles for the teacher in creating opportunities for
argumentation (e.g., Yackel 2002). An important role is encouraging students to take
an active part in the argumentative activity, e.g., prompting them to generate claims,
to provide justifications and to critically evaluate different arguments. Another
important role of the teacher involves responding to students’ arguments. Thus, for
example, the teacher plays a significant role in explicating students’ justifications to
emphasize the structure of the argument, and in supplying argumentative support that
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 19
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
was either omitted or left implicit. Moreover, as a representative of the mathematics
community, an important role of the teacher is to present to students what constitutes
acceptable mathematical arguments and to model particular ways of constructing and
confronting arguments.
However, how the interactions among the teacher, the class, and the
mathematical topic shape students’ opportunities to engage in argumentative activity
is not well understood. The study reported here examines this issue. For this purpose,
we use two case studies to compare students’ opportunities to engage in
argumentative activity between two classes taught by the same teacher, when learning
two beginning algebra topics: forming and investigating algebraic expressions and
equivalence of algebraic expression. Each topic requires a different kind of reasoning:
Work on forming and investigating algebraic expressions by using substitution of
numerical values into expressions mainly requires inductive reasoning. In contrast,
work on equivalence of algebraic expressions requires extensive use of deductive
reasoning. The specific research question examined is: How do (1) the contribution of
the teacher to the argumentative activity, (2) the contribution of the students to the
argumentative activity, and (3) the types of justifications, vary between two classes
taught by the same teacher using the same textbook and across two beginning algebra
topics – forming and investigating algebraic expressions, and equivalence of
algebraic expressions?
Methodology
Participants, setting, and textbook
Sarah taught two of the classes, S1 and S2, each in a different school. Rebecca taught
the other two classes, R1 and R2, each in a different school. Class work in Sarah’s
and Rebecca’s classes consisted almost entirely of work on tasks from the textbook.
The textbook used in the four classes was part of the Everybody Learns Mathematics
program (1995-2002). This study focuses on four central units: Two units deal with
forming and investigating algebraic expressions, mainly by substituting numerical
values into expressions as a means to develop a sense about their behaviour (e.g., task
1 in Figure 1). Work within these units largely requires inductive reasoning. Two
additional units focus on equivalence of algebraic expressions, dealing with
identifying, generating, and justifying the equivalence or non-equivalence of
expressions by employing several ideas, such as substituting numerical values into
expressions as a means to prove non-equivalence, substituting numerical values into
expressions as an inadequate means to prove equivalence, and expanding and
simplifying expressions as a means to maintain/prove equivalence (e.g., task 2 in
Figure 1). Work within this topic requires extensive use of deductive reasoning, i.e.,
proving equivalence and non-equivalence of expressions.
Data collection
The main data source was video and audiotapes of the teaching of the four units in
each of the four classes.
Data analysis
Detailed data analysis of the lessons included only the whole-class work. The videotaped lessons were transcribed and the argumentative activity in each class during the
whole-class work on each topic was then analysed.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
1) Consider the algebraic expression 4 – k:
Find a positive number and a negative number whose substitution yields a positive result.
Is there a positive number whose substitution yields a negative result? Demonstrate it.
Is there a negative number whose substitution yields a negative result? Explain why.
2) Find among the following pairs of expressions a pair in which the expressions are not
equivalent:
2∙m, m∙2
1∙m, m
m – 4, 4 – m
m + 4, 4 + m
For each of the remaining pairs, find a property that shows that the expressions are equivalent.
Figure 1. Examples of textbook tasks (abbreviated from Robinson and Taizi 1997).
The first step of analysis was to examine the teacher’s and the students’ utterances
according to their argumentative function within the whole-class work (e.g., claim,
request for claim, justification, request for justification). The second step was to
identify the teacher's and students' argumentative moves associated with each claim,
indicating them as an argumentative sequence. Two kinds of claims were the focus of
the analysis. One was related to generalizations of the behaviour of algebraic
expressions in the case of forming and investigating algebraic expressions (10 such
claims were found in each of the four classes). A second kind of claims was about
determining the equivalence of algebraic expressions in the case of equivalence of
algebraic expressions (13, 11, 33, 30 claims in S1, S2, R1, and R2 respectively). The
third step of the analysis involved classifying the types of justifications raised in the
argumentative sequences into one of two types: (1) justifications based on a general
mathematical rule, and (2) justifications based on a numerical example. We then
compared for each topic the two classes taught by each teacher on the contribution of
the teacher to the argumentative activity, the contribution of the students to the
argumentative activity, and the types of justifications suggested in class.
Argumentative activity in Sarah's classes
Analysis of classroom data revealed that the teacher’s contribution to the
argumentative activity, the students’ contribution to the argumentative activity, and
the types of justifications suggested in class were similar in Sarah’s two classes, for
each of the two mathematics topics.
 Sarah’s contribution. Sarah prompted her students to establish the claims
(generalization for the behaviour of algebraic expressions or determining the
equivalence of algebraic expressions). She was the one who usually provided
the justifications for the claims, supporting them with proof-related ideas on
which they are based.
 Students’ contribution. The students provided the claims.
 Types of justifications. Almost all of the justifications in both classes were
based on general mathematical rules.
The following episode from S1 class work on task 2 in Figure 1 illustrates the
recurrent argumentative sequence in Sarah’s two classes in the two topics. Sarah
pointed at the expressions 2∙m and m∙2:
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 21
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The
contributor
Utterance
The argumentative moves
Sarah
Are they equivalent?
Request for a claim
Student
Equivalent
Claim
Sarah
Right. These expressions are
equivalent. In order to prove
equivalence we have to use the
properties. Here it is the
commutative property. We have
multiplication so we are permitted
to replace the order and it will be
the same.
Justification (based on a general
mathematical rule)
+ the proof-related idea on which the
justification is based.
Argumentative activity in Rebecca's classes
As in Sarah’s classes, analysis of classroom data revealed that the teacher’s
contribution to the argumentative activity, the students’ contribution to the
argumentative activity, and the types of justifications suggested in class, were similar
in Rebecca’s two classes during the whole-class work on forming and investigating
algebraic expressions.
 Rebecca’s contribution. Unlike Sarah, in addition to prompting her students
to establish claims (generalization for the behaviour of algebraic expressions),
Rebecca also requested students to justify the claims and encouraged a
dialectical discourse among students, by asking for their opinion about a claim
raised in class. Her response to students’ arguments was approval.
 Students’ contribution. Rebecca’s students provided the claims, the
justifications, and collectively evaluated claims offered in class.
 Types of justifications. Almost all the justifications in both classes were based
on general mathematical rules.
The following episode from R2 class work on task 1 in Figure 1 illustrates the
recurrent argumentative sequence in both of Rebecca’s classes on this topic. After
substituting positive and negative numbers into the expression 4 – k, Rebecca asked
the class to generalize the outcomes produced by the substitutions, and a student
suggested a generalization. Rebecca asked for the other students’ opinion about it,
which led to students’ collective evaluation:
The
contributor
Utterance
The argumentative moves
Rebecca
Which numbers will give positive results?
Request for a claim
Student 1
Any number smaller than four
Claim
Rebecca
Did you hear what she said? Is she right?
Challenge for evaluation
Student 2
I don’t think she is right
Objection
Student 3
Because if she substitutes half…
Justification (for the opposition)
Student 4
If she substitutes half it will be okay
Opposition
Student 5
Minus one?
Justification (for the first opposition)
Rebecca
Substitute minus one here [points to the
algebraic expression]
Challenge for examination
Later on, the students accepted the initial generalization and justified it.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 22
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
In contrast to the previous results, in the case of equivalence of algebraic
expressions, while similarity was found in both classes in the teacher’s contribution to
the argumentative activity during the whole-class work, considerable differences were
found between the classes with regard to the contribution of the students to the
argumentative activity and the types of justifications suggested in class.
 Rebecca’s contribution. As in the previous case, in both her classes, Rebecca
encouraged her students to establish claims (determining the equivalence of
algebraic expressions), to justify claims, and to evaluate claims. Her response
to students’ justifications was approving the correct ones or encouraging a
different justification in cases of the incorrect ones, with no explicit distinction
between adequate and inadequate justifications.
 Students’ contribution. In both classes students provided the claims and the
justifications. However, whereas in R1 students’ arguments were frequently
challenged and evaluated by their peers, in R2, despite of Rebecca's
encouragement, no critical evaluation among students developed.
 Types of justifications. In R1, all justifications relied on general rules –
simplifying and expanding algebraic expressions by using properties of real
numbers. In contrast, in R2, students repeatedly suggested substituting
numerical values into expressions to prove equivalence (a specific case of
supportive examples for universal statements as mathematically invalid).
The following episode from R2 class work on task 2 in Figure 1 illustrates the
recurrent argumentative sequence in R2. Rebecca pointed at the pair of expressions m
+ 4 and 4 + m written on the board:
The
contributor
Utterance
The argumentative moves
Rebecca
Are they equivalent?
Request for a claim
Student
Equivalent
Claim
Rebecca
How can I prove it?
Request for justification
Student
Because if you substitute 2 you get 6 in both
Justification
Rebecca
Okay. But maybe it is by coincidence?
Request for justification
Student
Substitute 3
justification
Rebecca
[Substituting 3 in both expressions and
obtaining 7 in both]. Do we have to
substitute more numbers in order to prove
that they are equivalent? What do you think?
Request for justification
Substitute 4
justification
Student
Discussion
One main finding was the identification of a typical approach to argumentation of
each teacher, as manifested in both her classes and during the teaching of both topics.
Sarah’s argumentation approach exposed students to mathematical arguments and
explicit ideas of proving, but it did not give the students a significant role in their
generation and evaluation. Rebecca’s approach to argumentation largely shifted to
students the responsibility for justifying and evaluating claims, but she seldom
discussed the arguments raised in class or offered an explicit distinction between
adequate and inadequate ones. While restricted to the cases of this study, this finding
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 23
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
of a “constant” teaching approach to argumentation that hardly changes even when
the situations change provides new information for research dealing with teaching
mathematics in general and in encouraging class argumentation in particular, and
requires further examination.
Still, another main finding was the identification of differences in the
opportunities for argumentative activity in the case of one of the teachers in the
teaching of one of the topics. The argumentative activity during work on forming and
investigating algebraic expressions was similar in Sarah’s two classes as well as in
Rebecca’s two classes. However, whereas the argumentative activity during work on
equivalence of algebraic expressions was similar in Sarah’s classes, there were
substantial differences during work on this topic between Rebecca’s classes. These
differences were expressed in different types of justifications provided by students in
each class and in the extent to which dialectical discourse developed in each class.
These differences can be related to the intersection of mathematical situations that
involve deductive reasoning, known to be difficult for students (Harel and Sowder
2007), and Rebecca’s approach to argumentation, which included students but hardly
acted on their contributions. In contrast, work associated with forming and
investigating algebraic expressions basically involves inductive reasoning, known to
be students’ usual preferred way to form and test mathematical conjectures (e.g.,
Harel and Sowder 2007). Consequently, it is possible that the use of inductive
reasoning suited students’ preferences. In Sarah’s case, however, her “non-inclusive”
approach apparently prevented the class and the mathematical topic from playing a
dominant role; thus they did not serve as a source of differences in neither of the
topics. These findings emphasize the need for further research into the role of the
mathematical topic – in addition to the teacher – and in particular inductive- and
deductive-related topics, in shaping the argumentative activity in class. Likewise, they
highlight the need to incorporate attention to another factor: the classroom.
References
Harel, G., and L. Sowder. 2007. Toward a comprehensive perspective on proof. In
Second handbook of research on mathematics teaching and
learning: A project of the National Council of Teachers of Mathematics, ed. F.
K. Lester, 805-842. Charlotte, NC: Information Age.
Hiebert, J., R. Gallimore, H. Garnier, K. B. Givvin, H. Hollingsworth, J. Jacobs, J.
Stigler. 2003. Teaching mathematics in seven countries: Results from the
TIMSS 1999 video study. Washington, DC: National Centre for Education
Statistics.
Nisbett, R., D. Krantz, C. Jepson, and Z. Kunda. 1983. The use of statistical heuristics
in everyday inductive reasoning. Psychological Review, 90: 339-363.
Robinson, N., and N. Taizi. 1997. On algebraic expressions 1. Rehovot, Israel:
Weizmann Institute of Science. (in Hebrew)
Yackel, E. 2002. What we can learn from analyzing the teacher's role in collective
argumentation. Journal of Mathematical Behavior, 21: 423-440.
Yackel, E., and G. Hanna. 2003. Reasoning and proof. In A research companion to
principles and standards for school mathematics, ed by J. Kilpatrick, W. G.
Martin, and D. Schifter, 227-236. Reston, VA: National Council of Teachers
of Mathematics.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 24
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Calculating: What can Year 5 children do now?
Alison Borthwick and Micky Harcourt-Heath
In 2006, 2008 and 2010 we collected and analysed answers from a Year 5
QCA test paper to explore the range of calculation strategies used by a
sample of approximately 1000 Year 5 children. Once again in 2012 we
have repeated this research using the same group of 22 schools. This
paper explores the findings from the 2012 data, including case studies. It
examines the range of strategies used by the children. We conclude by
considering if and how the use of particular calculation strategies has
impacted on the overall results and we ask if this shows greater clarity
about which strategies lead children to success.
Keywords: calculations, strategies, primary mathematics
Introduction
This Year 5 research emerged (Borthwick and Harcourt-Heath 2007) when the
National Numeracy Strategy (NNS) (DfEE 1999) was the main framework that
teachers used to support them in planning and delivering mathematics in the English
National Curriculum. Prior to the introduction of the NNS the mathematics
curriculum had focused more on the applications of mathematics and less on written
calculation strategies. However, the NNS placed more emphasis on arithmetic skills
and children were exposed to perhaps alternative methods for calculating than they
had been shown before (e.g. number lines and the grid method). The UK now has a
renewed Primary Framework for Mathematics (DfES 2006), which still includes this
emphasis on written calculation strategies. One of the main aims of both the original
and revised mathematics curriculum was to provide children with the “ability to
calculate accurately and efficiently, both mentally and with pencil and paper, drawing
on a range of calculation strategies” (DfEE 1999, 4). We have also retained this aim
as our benchmark when analysing the Year 5 data.
However, while our longitudinal study continues to follow the progress of
children’s success with written calculation strategies, other research shows that this
proficiency with calculations is not yet secure for many pupils. Howat (2006) reported
that children (aged 8 years old) were still failing in arithmetic because they were
unable to sufficiently understand that a ten in a two-digit number could be ten ones or
one ten. While there is a plethora of research that examines children’s progress and
understanding in specific calculation strategies (e.g. Anghileri 2001; Anghileri,
Beishuizen and van Putten 2002), our study is unique in that it involves large scale
data spanning across the last six years which looks at strategies for all four operations.
However, this paper concentrates only on the 2012 outcomes rather than the previous
data (Borthwick and Harcourt-Heath 2007; 2010).
Methodology and context
Data was collected from test papers completed by Year 5 children from 22 schools
throughout Norfolk. A range of primary and junior schools were selected. Responses
to four questions from each of the papers were analysed for their calculation
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
strategies. One question each for addition, subtraction, multiplication and division
was used.
Calculation
Question
Addition
546 + 423
Subtraction
317 – 180
Multiplication
56 x 24
Division
222 ÷ 3
Table 1. Questions from QCA Year 5 test paper
The four questions we selected were chosen as they had no context and
required children to perform a calculation, as opposed to less abstract problems that
involve children in some interpretation before a calculation can be carried out. The
categories used for analysis were determined by the National Numeracy Strategy
(DfEE 1999) and other research (e.g. Beishuizen 1999).
Findings and discussion
Each of the following sections looks at proportions of children using the range of
strategies for the four questions and includes examples of children’s work.
Addition
94% correct / 6% incorrect
546 + 423
Number
Correct
Number
Incorrect
Percentage
Correct
Percentage
Incorrect
Not attempted
Standard algorithm
430
10
98%
2%
Number Line
32
7
95%
5%
Partitioning
179
9
95%
5%
Expanded vertical
168
6
97%
3%
Answer only
114
22
84%
16%
Other
14
8
64%
36%
Totals
937
62
94%
6%
Table 2: Results from 999 children for addition question.
This question was by its very nature the least useful because it did not require
bridging through ten or one hundred. As a result, a number of different strategies
were identified.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 26
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Subtraction
69% correct / 31% incorrect
317 – 180
Number
Correct
Not attempted
Number
Incorrect
Percentage
Correct
Percentage
Incorrect
11
Standard Algorithm
– decomposition
106
71
60%
40%
Standard Algorithm
– equal addition
0
2
0%
100%
Number Line
484
70
87%
13%
Negative Number
13
5
72%
28%
Counting Up
20
65
24%
76%
Counting Back
16
1
94%
6%
Answer only
28
9
76%
24%
Other
24
74
24%
76%
Totals
691
308
69%
31%
Table 3: Results from 999 children for subtraction question.
Almost all children attempted to answer this question, with the number line
emerging as the most often selected and successful strategy (see Figure 2 below for an
example). However, those children who employed the counting up strategy but did
not record the number line were not as successful as those who drew it to aid their
thinking. Almost one fifth of children selected the standard algorithm but this was
much less successfully employed.
As illustrated in Figure 1 below, some children still demonstrate a lack of
understanding about subtraction by using partitioning inappropriately and incorrectly.
Figure 1
Figure 2
Multiplication
42% correct / 58% incorrect
56 x 24
Number
Correct
Number
Incorrect
Percentage
Correct
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Percentage
Incorrect
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Not attempted
109
Standard Algorithm
5
19
21%
79%
397
283
58%
42%
Expanded Vertical
7
7
50%
50%
Two partial products
only
0
70
Answer Only
0
14
0%
100%
Other
11
77
13%
87%
Totals
420
579
42%
58%
Grid Method
100%
Table 4: Results from 999 children for multiplication question.
Over two thirds of the children chose to use the grid method for completing
the multiplication calculation. We were surprised to note that this category had both
the highest number of correct (397) and the highest number of incorrect (283)
responses.
While Figure 3 below shows an appropriate grid structure, the presentation of
the multiples of tens numbers (e.g. 50 and 120) might cause us to question issues of
place value. It could be suggested that children had been taught to think when
multiplying 20 by 50 that you simply multiply 2 by 5 and add two zeros. The
particular example shown also demonstrates the impact of incorrect partial product
calculations on the overall answer. The second example, Figure 4, shows a typical
representation of the ‘two partial products’ category that more than 7% of the children
used.
Figure 3
Figure 4
Division
38% correct / 62% incorrect
222 ÷ 3
Number
Correct
Not attempted
Standard Algorithm
Number
Incorrect
Percentage
Correct
Percentage
Incorrect
151
48
38
56%
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44%
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Chunking Down
58
19
75%
25%
Chunking Up
83
47
64%
36%
Number Line
152
142
52%
48%
Answer Only
19
96
17%
83%
Other
23
123
16%
84%
Totals
383
616
38%
62%
Table 5: Results from 999 children for division question.
This calculation was the least well answered with 15% not even attempting it.
Although the number line was selected most often, a significant proportion of children
did not gain a correct answer through its use. Examination of the children’s responses
revealed that the underlying strategies are not secure, for example, children repeat the
subtraction of 3 but they seem not to be moving to the next stage where they are
subtracting multiple ‘chunks’ of 3. This leads to inefficiency and simple errors in
calculation.
Case study
Figure 5
Figure 6
The two examples shown in Figures 5 and 6 are taken from children in the same class.
95% of the 21 children in this group answered the division question correctly and they
all used the same ‘sharing’ strategy and similar layout. Whilst this could demonstrate
that the children had been taught to answer this algorithmically, closer analysis
showed that children had chosen different sized ‘chunks’. This would suggest that
while children have been taught this particular method, they have also been given the
associated mental skills and understanding to make it their own, even to the point
where the first child has used a negative chunk to readjust.
It is interesting to note that in the same school the parallel class employed a
range of division strategies with a lower 65% of children answering correctly.
Conclusion
The overarching aim of this paper was to report on the findings from the 2012 data
and examine the range of strategies used by the children. From this analysis it is clear
that the number line still seems to be under-utilised, despite the wealth of research
that reinforces the strength of this particular method (e.g. Beishuizen 1999; Anghileri,
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 29
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Beishuizen and van Putten 2002). While more informal strategies are being used,
according to this research, there is some evidence to suggest that these are being
sometimes taught algorithmically. This would suggest that teachers’ subject
knowledge is still weak, despite the fact that it is widely recognised that the
mathematical subject knowledge of teachers is an important factor in the teaching and
learning of mathematics (Williams 2008). Indeed it was Williams (2008) who
recommended that every school should have access to a maths specialist teacher
(MaST). This has not yet been fulfilled. Division remains the weakest calculation in
terms of success in the 2012 data and does seem to be as Watson and Mason (2012)
describe, for many children, ‘the odd one out’. While Watson and Mason talk of
children developing ‘coping strategies’ to ‘get away with it’ our research would show
that for many children, they simply do not even tackle this calculation. This research
tells us that there are some Year 5 children who are still not able to complete age
related calculation questions for all four rules. This continues to have implications for
schools with regard to the policies they adopt for calculations but also the importance
they place on other aspects of learning mathematics, such as representation (e.g.
Barmby et al. 2011).
References
Anghileri, J. 2001. Development of division strategies for Year 5 pupils in ten English
schools. British Educational Research Journal, 27 (1): 85-103.
----------- 2007. Developing number sense. London: Continuum.
Anghileri, J., M. Beishuizen and K. van Putten. 2002. From informal strategies to
structured procedures: Mind the gap! Educational Studies in Mathematics, 49
(2): 149-170.
Barmby, P., T. Harries, S. Higgins, and J. Suggate. 2009. The array representation and
primary children’s understanding and reasoning in multiplication. Educational
Studies in Mathematics, 70 (3): 217-41.
Beishuizen, M. 1999. The empty number line as a new model. In Issues in Teaching
Numeracy in Primary Schools, ed. I. Thompson. Buckingham: Open
University Press.
Borthwick, A. and M. Harcourt-Heath. 2007. Calculation strategies used by Year 5
children. Proceedings of the British Society for Research into Learning
Mathematics, 27 (1): 12-17.
---------- 2010. Calculating: What can Year 5 children do? Proceedings of the British
Society for Research into Learning Mathematics, 30 (3): 13-18.
Department for Education and Employment. 1999. Framework for teaching
Mathematics from Reception to Year 6. London: DfEE.
Department for Education and Skills. 2006. Primary framework for Literacy and
Mathematics. London: DfES.
Howat, H. 2006. Participation in elementary mathematics: an analysis of engagement,
attainment and intervention. Unpublished PhD thesis, University of Warwick.
Watson, A. and J. Mason. 2012. Division – the sleeping dragon. Mathematical
Teaching, 230: 27-29.
Williams, P. 2008. Independent review of mathematics teaching in early years
settings and primary schools. London: DfCSF.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 30
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Relentless consistency: Analysing a mathematics prospective teacher education
course through Fullan’s six secrets of change
Laurinda Brown
University of Bristol, Graduate School of Education
In the leadership of change literature, Michael Fullan’s work is influential.
He has developed theories about the process of working rather than the
content of that process. The work of a mathematics teacher educator could
be seen as leading change for a group of prospective teachers. This paper
aims to use Fullan’s ‘six secrets of change’ to analyse the structure of the
mathematics education aspects of the one-year University of Bristol Postgraduate Certificate of Education (PGCE) course, to gain insight into both
practices that illustrate Fullan’s ‘secrets’ and possible developments to the
course given aspects of the secrets not in evidence. Fullan’s idea of
‘relentless consistency’ seems to fit with the way the prospective teachers
evaluate strengths of the course.
Key words: mathematics education; leadership of change; mathematics
teacher education: relentless consistency.
Introduction
I first worked with a one-year PGCE group at the University of Bristol, Graduate
School of Education in 1990. In the UK, prospective secondary mathematics teachers
will have a degree in mathematics or a mathematics-related subject and apply to a
university education department for a one-year PGCE course either directly after
completing their degree or, later in life, after having worked in such careers as being
an actuary, engineering, ICT professional or even managing a pub or tree-felling!
Two of us work together running the PGCE course and we like to interview and offer
places to those students who contribute to the widest spread of age; experience; and
views and applications of mathematics as possible. We find that the multiplicity of
views and the fact that we, as tutors, do not believe that there is one way of teaching
mathematics lead to an energised learning environment where the interactions and
sharing between the group of prospective teachers is central. Their task, given to them
at the start of the year, is to become the teacher that is possible for them. The
importance of the group interactions is often commented on as part of our end-of-year
evaluations. Given that our prospective teachers already have their mathematicsrelated degrees, we do not need to teach them advanced mathematics as such. We do,
however, spend time in workshops where they transform their learning of
mathematics to extend the range of their possible offers to their pupils through
listening to and working with the different ways their fellow prospective teachers
have of solving mathematical problems or of presenting activities to their students.
However, we have not found a way of analysing the structure of the course to allow
us to gain a sense of why these ways of working provide the positive learning
experiences that taking the course seems to provide consistently over the years, and
what we are not doing that could potentially develop the course further. Although, of
course, there have been innovations on the course, often responding to feedback, it
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
has basically stayed the same structure for the last twenty years, throughout many
changes in the mathematics curriculum in schools.
Design of the Course
The PGCE mathematics course was designed on enactivist principles. A strapline that
I would use for this is ‘seeing more, seeing differently’. There is not space in this
paper to describe enactivist principles in detail but my colleague, Alf Coles, and I
have written about these most recently in a paper in ZDM (Brown and Coles 2011),
describing how practices of ‘deliberate analysis’ can be used by novices and experts.
Novices do not have to behave in different ways from experts when they learn. These
practices are used on the PGCE course at Bristol where Alf Coles and I now work
together. We are working to support the prospective teachers in extending their range
of practices and to do this they have to become aware of what they are not doing. We
have various strategies for this but in this paper, I want to illustrate how our own
learning can be exemplified by looking through a different perspective to support us
in seeing what is not there to develop our own practices as teacher educators.
Fullan’s Six Secrets of Change
After working with the Blair government in the UK to implement the National
Strategies for numeracy and literacy, Fullan applied his learning to the raising of
standards in literacy and numeracy in Ontario, Canada. The large-scale project
description can be found on the web and states that “Our goal is to have 75 per cent of
12-year-old students achieving a high standard of proficiency in reading, writing, and
mathematics” (Ministry of Education, Ontario, ‘Reach every student’ 2008, 5) over an
initial four years of implementation. Fullan’s learning, applied in the Ontario project,
was distilled in his book Six Secrets of Change (2008). The six secrets read like
sound-bites. They are statements related, crucially, to the process of working as
leaders of change rather than anything to do with the content of the change process.
So, the sound-bites do not mention literacy or numeracy, for instance. The six secrets
of change are: 1. Love your employees; 2. Connect peers with purpose; 3. Capacity
building prevails; 4. Learning is the work; 5. Transparency rules; and 6. Systems
learn. In what follows, for each of these secrets, there will be a paragraph explaining
some of the thinking and strategies suggested by Fullan. After these paragraphs,
sections of the PGCE syllabus and handbook will be discussed, seen through the
headings to give insight into the processes used. The use of any framework applied to
the familiar is only useful if it can see what we would not normally see. Where are the
gaps between the framework of Fullan’s Six Secrets of Change and what we currently
do that could shed light on where, perhaps, we could develop the course in the future.
Although the six secrets are given separate labels they need to be seen as inter-related
in that “the same action can enhance several secrets simultaneously” (Fullan 2008,
37). All six secrets, any one of which can support an aspect of a community, “in total
point to what is missing” (37).
1. Love your employees
We need to value teachers (employees) as much as the children and parents
(customers). Fullan quotes Barber and Mourshed, “the quality of the education system
cannot exceed the quality of its teachers” (2007, 23). So, “one of the ways you love
your employees is by creating the conditions for them to succeed” (25). How do you
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
support people to “find meaning, increased skill development and personal
satisfaction in making contributions that simultaneously fulfil their own goals and the
goals of the organization” (25)?
2. Connect peers with purpose
There is an acceptance that we learn through doing when this is related to a purpose,
such as implementing change through new materials; new behaviours/practices;
or/and new beliefs/understandings in a cyclical manner. Findings are focused at a
meta-level to the content of teaching and learning. The basic challenge here is, how
can teachers (or children in schools) take forward the agenda as their own?
3. Capacity building prevails
Initially, Fullan advises leaders to give descriptive, not judgemental, feedback,
building feedback into the system. Over time, the conversations can become more
open and are learning conversations, in that both parties are learning. In schools,
children might be feeding back what they have been doing on A3 sheets for class
discussion and in meetings of teachers, similarly, teachers might share how they did a
problem and discuss with each other.
4. Learning is the work
Fullan discusses the importance of ‘relentless consistency’ within the system, not to
dampen creativity but to allow the rethinking and redoing cycle that seems to be so
important. In his work with teachers, ‘snapshot views’ are used to support them
becoming aware of their own learning. The system supports the teachers in observing
themselves, “making a science of performance”.
5. Transparency rules
This is not “attempting to use the measurement tail to wag the performance dog” (93),
nor “measuring things that are not amenable to action” (94). So, transparency is
“openness about results” and “what practices are most strongly connected to
successful outcomes” (99). Therefore, in a non-punitive system, transparency rules
when it is combined with deep learning in context as opposed to league tables
(paraphrased 103).
6. Systems learn
Focus on developing many leaders working together, instead of relying on key
individuals. These leaders “approach complexity with a combination of humility and
faith that effectiveness can be maximized” (109). Secret 6 is the meta-secret because
it builds on secrets one to five. Guidelines for action for leaders are “Act and talk as if
you were in control and project confidence; take credit and some blame; talk about the
future; be specific about the few things that matter and keep repeating them” (Pfeffer
and Sutton 2006, 206, quoted in Fullan 2008, 119).
Discussion
Each year, during the summer between cohorts on the PGCE course, which starts in
September and finishes in June/July, our mathematics course handbook is updated.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Ofsted, one year, praised our handbook for being slim-line whilst describing exactly
what happens. Through changes in competences to standards; national curricula for
mathematics; strategies; and initiatives such as Every Child Matters, our handbook, at
its heart, remains unchanged. Why this is possible can be seen from this extract from
the introduction to the ‘programme and us’, at the start of the handbook (see next
section for a discussion):
There is not one way to teach mathematics. Schools use a variety of approaches
and we see the programme as allowing each individual student teacher:
to experience that variety by: working in at least two different schools with
different approaches supported by an Associate Tutor (AT) in each school;
sharing the impressions of others on the PGCE programme; and day visits to three
schools.
to discover how best to use themselves and their talents to teach mathematics
effectively to those children with whom they work, supported by sharing
perspectives on reading and research
to develop flexibility of approach in their classroom
to learn new skills.
You will find a range of age, work experience, technological skills, mathematical
interests and mathematical expertise within the group. The programme aims to
use the strengths of the group of student teachers in partnership with the PGCE
tutors and the Associate Tutors [mathematics department mentors in schools] to
support each other through:
working at issues of teaching and learning
doing mathematics together: at your own level to plug gaps in your knowledge,
e.g., find an applied mathematician to help you work at mechanics which you
have never done; tackling activities to see what the children might experience and
extend your appreciation of the range of possible approaches
sharing technical skills such as using computer equipment and packages and
developing academic writing
working in a variety of schools with different practices and comparing and
contrasting those with the experiences of others in the group.
Applying Fullan’s Framework
Fullan would argue that planning is important but better as a 5-page document (rather
than a thick manual) where the structures are built of doing and evaluating because
“you are more likely to behave your way into new ways of thinking than you are to
think your way into new ways of behaving” (2008). Our slim-line handbook inducts
the students into processes: they are going to be working in a group; doing
mathematics; sharing skills; comparing and contrasting experiences with others; and,
most importantly, discovering how best to use themselves to teach mathematics
effectively (so that children learn). So, these prospective teachers are being supported
in finding their own meaning (Secret 1), and we are creating conditions in which
evidence would say the majority of them thrive. In the first session of the year, we
share the purpose with them of the year being about finding the teacher they can
become (Secret 2). We continue throughout the year to check out where they are in
this task through tutorials and when we visit them in schools.
The handbook talks in terms of processes, not of the content of syllabuses or
of particular mathematics. The course is described at a meta-level. In the timetable for
the Autumn Term of the course, structures emerge. On Friday mornings there are
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
‘Groups’, where we split the cohort into two, effectively tutor groups. This sub-group
has the same tutor, who will visit them in school and work with them in reflecting
time on Friday mornings at the university. Monday mornings are workshops where
we work at some mathematical activities together as a class and then develop our
thinking on issues that arise. Similarly, there are patterns that emerge over the year,
for instance, when the prospective teachers arrive back from a period of school
practice, they sit in reflecting teams of three (in almost all cases!) to discuss their
developing practice using the details of their experiences to distil out issues. These
practices have the feel of ‘relentless consistency’ (Secret 4). The way the course
works is through these rethinking and redoing cycles.
During the group sessions on Friday, we are explicit about a way of working
where they share details of their practices and listen to others to extend their range of
possible strategies, not judge what someone else offers. Over time, the group learns to
trust this process and shares more openly in learning conversations (Secret 3). From
the details of practice arise intentions or issues, such as, how do we get children
sharing responses to an activity? The group often then develops a range of strategies
to tackle such an issue from both their observations of other teachers in the different
schools and their own teaching. So, the relentless consistency of these practices does
not dampen creativity but supports the prospective teachers in both seeing the
strategies they use as valuable to others, whilst also seeing more and differently in
that they are opened up to strategies they were not aware of that become possibilities
for future action for themselves. There is ‘deep learning in context’, not a league table
of the best to worst prospective teachers in the group (Secret 5). The sharing is in
relation to teaching strategies that support the children to learn effectively.
And the system learns (Secret 6). As leaders of the group, we keep repeating
the things that matter, e.g., “no right or wrong action, just what you did and reflecting
on it”, and there do not seem to be many of these statements. We talk about the
future, since there are communities of ex-PGCE teachers in the schools that we work
with in partnership. As our student teachers learn to learn about the children in their
classrooms as mathematics learners, we learn about the patterns related to becoming a
teacher of mathematics. The student teachers have the task of learning to teach
mathematics, however, we cannot do it for them. We do ‘project confidence’ (Secret
6), because experience tells us that what we do works, even when we do not answer
their requests for a lesson that will work tomorrow. We do not teach in their practice
schools. What we can do on the course is provide them with the conditions to
succeed.
What’s not there?
When I first read Fullan’s book, there was so much that I felt I was recognising and
images from our course were present for me. Here was another language I could use
to describe the background structures to the course. Although I have tried to give an
indication of how each secret could be illustrated, it is the case that for me the six are
inter-related. What also happened was that, in reflecting on the six, I became aware of
what was missing so we can further develop the course.
The mathematics PGCE course is not run in isolation from the whole PGCE
course and there are 4 points during the year, called Review Points, where the
prospective teachers and their ATs think about progress. This is not against the
standards, as such, because we have a course document where ‘pen portraits’ have
been written that describe, at each Review Point, what behaviours could be evidenced
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
for each of the (currently) 8 standards and from Very Good, through Good and
Satisfactory to Pass. In reading Secret 4, I initially considered that these Review
Points were the snapshot views. However, so much else from the secrets was being
contradicted through their use, including the judgements that somehow take away
from the individual student teacher’s sense of purpose.
As time went by, I realised that what would be a strong development for the
course would be if the prospective teachers had a mechanism through which they
could take ‘snapshot views’ to support them becoming aware of their own learning,
the system supporting them in ‘making a science of their own learning’.
Coincidentally, Alf Coles (2012) has an interest in using video for professional
development and the next connection was obvious. We are now in a process through
which I can imagine that prospective teachers in the future will use video-recordings
of their lessons over the year with snapshots illustrating their progress and awareness
of their learning. In the first year, we ran a research project where 7 of the group with
their ATS and the two UTs worked as a collaborative group to develop use of ICT and
supported the student teachers in taking a video showing progress in learning of the
children whilst using an ICT programme. During this academic year, we have now
built the same task into an assignment for everyone in the PGCE mathematics group.
We are looking long term for video recordings to become part of the culture of the
course as a learning tool for the student teachers’ progress. This is already beginning!
At a recent meeting of ATs, one AT talked, without our direction, about how he had
used video recordings in his department to support professional development after
working with them on our course as a student teacher. Learning is the work and this
positive feedback bodes well for the relentless consistency of their use in the future!
References
Barber, M. and M. Mourshed. 2007. How the world’s best-performing school systems
come out on top. London: McKinsey and Co.
Brown, L. and A. Coles. 2011. Developing expertise: how enactivism re-frames
mathematics teacher development. ZDM Mathematics Education 43, 861-873.
Coles, A. 2012. Using video for professional development: the role of the discussion
facilitator. Journal of Mathematics Teacher Education. DOI 10.1007/s10857012-9225-0.
Fullan, M. 2008. Six secrets of change: what the best leaders do to help their
organizations survive and thrive. San Francisco, CA: Jossey-Bass.
Pfeffer, J. and R. I. Sutton. 2006. Hard facts, dangerous half-truths and total
nonsense: profiting from evidence-based management. Boston: Harvard
Business School Press.
Ministry of Education, Ontario. 2008. Reach every student: Energizing Ontario
Education. http://www.edu.gov.on.ca/eng/document/energize/.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 36
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Educational game Euro-Axio-Polis: Mathematics, economic crisis and
sustainability
Maria Chionidou-Moskofogloua, Georgia Liarakoua, Efstathios Stefosa, Zoi
Moskofogloub
a
University of the Aegean- Rhodes Greece, bUniversity College London
A game called Euro-Axio-Polis was constructed by students of the
Aegean University aiming to promote teaching and learning on
mathematics and sustainability for 6th grade pupils. 40 students played
Euro-Axio-Polis and Monopoly to investigate differences between the two
games, and wrote five key words that characterized each game. Also 19
sixth grade pupils played the Euro-Axio-Polis game during students’
teaching practice and wrote five key words about the game. The research
results suggest that Monopoly reflects capitalist economic terms and
social values while Euro-Axio-Polis reflects social values associated with
sustainable development such as solidarity and equity. Pupils were more
likely than students to make reference to socio-political issues such as
parliament, education, democracy, elections and political power. As far as
mathematics is concerned, most students and half of 6th grade pupils
recall the mathematical concepts percentages and interest rates while they
played Euro-Axio-Polis.
Keywords: teacher training, cross thematic teaching approaches,
mathematics, sustainability and values
Overview
Research in mathematics education and training over the last decades has focused on
the enhancement of new pedagogical and epistemological approaches to learning and
teaching mathematics. However, the shift from the positivist paradigm about
mathematics as well as from the teacher-centered instructional models towards
constructivist and emancipatory ones is a challenging task for many mathematics
education students. It is difficult for them to understand, for example, how
mathematics is related to cultural issues and ideas that may affect peoples’ everyday
lives (Burton 2004; Chasapis 1996) and what is the meaning of contextual and
authentic learning in mathematics education (Lave and Wenger 1991). As is the case
for facilitation of pupils’ mathematical learning, trainee teachers may also learn better
by doing, i.e., designing and exploring new approaches to teaching and learning in a
collaborative context, developing some novel learning activities, experimenting and
reflecting critically on them, and transforming their own new mathematical discourse
as well as their instructional schemas. By doing so students are also beginning to learn
a) how to relate theory with practice in mathematics education (Jaworski 2006;
Sakonidis 2012; Rowland et al. 2012; Lerman, Murphy and Winbourne 2012); b) how
to deal with complexity while creating and managing meaningful and flexible
learning environment (Potari and Jaworski 2002); c) how to realize critical
epistemological concepts, such as Leone’s Burton “four epistemological challenges”
(2004) to mathematics education.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Having adopted most of the above dimensions of pedagogical philosophy, the
formal Curriculum Development Design for Mathematics, introduced by the Greek
Ministry of Education (Pedagogical Institute 2003), has given emphasis, among
others, on a) the implementation of some inter-disciplinary and cross-thematic
(Chionidou-Moskofoglou 2007) projects within all school subjects; b) the provision of
extra time for a “Flexible Zone” type of instruction during which open and innovative
approaches to learning and teaching can be applied by teachers; c) teachers’ efforts
for pedagogical enrichment of the school instructional context; d) pedagogical
instrumentalization of digital technology; e) exploring and introducing new
approaches to mathematics attainment evaluation. In an attempt to facilitate university
students’ professional empowerment towards this direction, an educational game has
been designed and evaluated as an alternative ecological approach to mathematical
instruction that makes connections between mathematics and socio-economic problem
solving. The idea of approaching mathematics as a socio- cultural product has been
introduced and a very promising area for this notion is this concept of sustainability
(Petocz and Reid 2003; Clarke 2012).
Sustainability has become a central notion in environmental policy discourse
over the last two decades. Trying to regulate the relations between human societies
and nature, sustainability is a complex and open concept that lends itself to many
different interpretations (Liarakou and Flogaiti 2007).
However, there is some consensus that sustainability brings together three
different axes: environment, economy and society. They constitute the pillars of
sustainability: three interdependent and overlapping systems, the proper functioning
of all three is a necessary condition for achieving sustainability. Environment refers to
the effective protection of nature and prudent use of natural resources. Economy
stems from the need to establish a prosperous and viable economic exchange which
takes into consideration the limits of economic growth and is based on a redefinition
of consumption levels. As far as society is concerned, human welfare and rights,
promotion of democratic and participatory systems and processes are among the
issues which play a key role in sustainability.
Description of Euro-Axio-Polis
The game Euro-Axio-Polis was constructed according to the above mentioned
constructivist theoretical background by 4th year university students in primary
education and the first author from March to June 2012. The objective of the game is
to teach the concept of ‘percentage’ and ‘interest rate’ to 6th grade pupils during their
teaching practice. Additional aims of Euro-Axio-Polis are: a) challenging the
prevailing function of mathematics as a means of reproduction of the dominant
ideology and the market economy, b) raising university students’ and pupils’
awareness of sustainability and c) contributing to the students’ social empowerment
and emancipation.
Euro-Axio-Polis rules have been designed to suit a classroom of
approximately 20 pupils. Duration of the game is 40-50 minutes. Players are divided
into five groups of four. Each team represents one of the 29 European Union countries
that have financial transactions with the European Central Bank (ECB). All teams
start from the STARTING POINT with one billion Euros in cash. Each country plays
throwing the dice and, depending on the number it gets, places the pawn on the
corresponding box. According to the options given in the particular box, the team
decides on its actions. In the frames with the mark “YOU DECIDE”, the country gets a
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
card and decides on its next actions depending on the given options – dilemmas (e.g.
What does your country choose? (A) to pay Doctors without Borders €170 million or
(B) to receive for its treasury €170 million by cutting pension funds for the health of
its citizens (See Figure 1). After the 50 minute duration, pupils vote which country is
the winner according to their own criteria: the winning country may be the one that
has raised the most money or has made the ‘best’ decisions, i.e. has the greatest
number of (A)s or (B)s as in Table 1 below.
PENSIONS REDUCE
YOU DECIDE
What does your country choose?
-To pay Doctors Without Borders 170 million €
(A) or
-To receive for its treasury 170 million € by
cutting pension funds for the health of its citizens
(B)
500.000.000 €
REDUCES IN INSURANCE FUNDS FOR
HEALTH
500 X 106 €
Figure 1 Example game cards and board
Sewage installation
Financial aid to Child’s Smile
(Α)
(Α)
(Β)
(Β)
(Α)
(Α)
(Α)
Sale of Hazardous electronic waste
Sale of state
land
Pensions reduce
Non-participation in the Olympics
Sale of works of Art
Construction of Nursing homes
Construction of sports centers
Maintenance and enhancement of
monuments and archeological
sites
Financial Support to Doctors
Without Borders
(Α)
Cuts in pension funds for public health
(Β)
Total
X?
(Β)
(Β)
(Β)
X?
Table 1: Grouping of decisions
The research process
The game was played and evaluated by 40 senior students (3 males and 37 females) of
the Department of Primary Education of the University of the Aegean and 19 pupils
of sixth grade of the Primary School. The aim was to investigate the differences
between the classic Monopoly game and Euro-Axio-polis. This assessment has been
part of an ongoing process which includes a variety of techniques.
The qualitative method was chosen as the research approach. Data collection
took place in the form of a written questionnaire, from which conceptual connections
made by university students and pupils emerged. The questionnaire included two
open-ended questions asking participants to write at least five words which, in their
opinion, characterise Monopoly game in its classic form and Euro-Axio-polis game
respectively. Having played the Euro-Axio-polis game, students and pupils answered
spontaneously and without having been affected by predefined concepts of the
researchers nor guided to specific answers. Data gathered from the above questions
were based on the written responses of participants. They were organized in thematic
areas, which appointed the classification of responses. Finally, qualitative data were
interpreted into quantitative data.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Results
University students’ responses were grouped into classes according to their semantic
content. Four categories were formed covering all the words that were recorded: a)
financial terms b) game components c) values and d) mathematical concepts.
Regarding the keywords that characterize the classic Monopoly, the category
with the highest frequency in students’ responses is the one of financial terms. This
class gathered a total of 51 reports while 12 different words were reported. The
following terms gathered the most reports: buying and selling (15), money (10),
investing (5), business (4), taxes (4), rent (4), bank (3) and shares (2). The second
category, with a total of 12 references, is related to the game’s components. The term
with the most references in this category is the prison/jail (5) to which players can be
sent. The second term is decision (2) the player should make about buying and selling.
The third category includes two values, competition (2) and individualism (2)
and a reference of the term ‘value’. Both values that were recorded are related to the
capitalist economic model, in which the game is supported. The low percentage
obtained by the fourth category of words ‘mathematical concepts’ is impressive
because it was mentioned by only 1 student.
In the second question, in which University students were asked to write down
words that characterise the Euro-Axio-Polis, the picture is different. The category that
collected most reports (41) is that of values. Nine different values were reported with
equity (21) in the first position; other recorded values were solidarity (4), charity (2),
respect (2), active participation (2) and fellowship (2), while justice and altruism had
only one report. In contrast with Monopoly, the reported values of Euro-Axio-Polis
reflect - to a bigger extent - values inherent in sustainability. These values are also
related to the economic realm, especially equity which refers to wealth distribution.
The category of economic conditions was in second place with 19 references.
In contrast to the same category of the classic Monopoly game in which the term
buying and selling dominated, most reports are compiled by the economic crisis
notion (6) while the remaining terms refer to various economic terms such as money
(5) and investment (3). The third category referring to the game’s component got 17
references, including European countries (4) and sustainability (3). Finally, the
category ‘Mathematical concepts’ had very few references in this question too: only
two students reported on percentages and interest rate during the game which may
means that mathematics was an invisible culture for the majority of the students.
6th grade pupils’ responses about Euro-Axio-polis were also grouped into
classes according to their semantic content. Beside the four categories that emerged
during the analysis of University students’ answers, two new ones were added: sociopolitical issues and emotions.
The category with the highest frequency (28) in pupils’ answers refers to the
game’s components. The following terms gathered most of the pupils’ references:
monopoly (9), countries (5), starting point (2), European Union (2) and decision (2).
The difference from the University students’ answers concerning this category is
evident: while pupils refer mostly to descriptive elements of Euro-Axio-polis,
University students reveal more qualitative ones (e.g. cooperation, dilemmas) and
stress the elements that differentiate the play of the two games. Concerning the
category of economic terms, the following ones gathered most of the students’
notions: economics (9), euro (3), income (2) and tax (2). While current economic
crisis terms ware included in game cards, pupils reported a variety of other words
related to the actual situation in Greece. Playing Euro-Axio-polis game brought to
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 40
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
their mind socio-political issues like politics (3), society (3), parliament (3),
education, democracy, elections, power, SYRIZA (Radical Left Coalition Party),
Merkel, Hitler.
The fourth category includes the following nine references related to
mathematics: mathematics (5), percentages (2), operations (1) and numbers (1). This
low frequency rate could be explained by the idea of mathematics becoming part of an
‘invisible culture’ in the pupil’s eye. Ranked in fifth place was another category
expressing pupils’ emotions affected by Euro-Axio-polis game. The game seeed to
create positive feelings in pupils since it was associated with joy (3), entertainment
(2), enthusiasm and interest. Finally, in contrast to students, few pupils associated
Euro-Axio-Polis game with values. The only value reported is charity, mentioned by
six pupils.
Conclusion
One of the objectives of the educational game Euro-Axio-Polis was to create an
authentic, mathematics instructional environment in which concepts such as
percentages and interest rates are embedded in some realistic and playful learning
activities while, at the same time, mathematics education is meaningfully integrated in
a realistic cultural context where the prevailing idea of mathematics as a politically
neutral instrument at the service of a dominant capitalistic values reproduction
ideology is under question. The latter is achieved when the players of this game are
confronted with dilemmas and decision making situations in which socio-cultural
issues, such as sustainability vs. economic investment and profit, as well as and value
judgment discourse, were involved.
The results of this survey conducted with university students, as well as with
6th grade pupils, are encouraging. While, in the case of Monopoly, economic
conditions and values that refer to the ascendant economic model dominate students’
discourse, this is not the case with Euro-Axio-Polis. Here values such as solidarity,
equity and social interdependence prevail, which are associated with sustainable
development. The game also evokes the actual crisis in Greece, which resembles the
current socio-political condition of other European countries too. University students
highlighted terms related to economic crisis, while pupils referred mostly to sociopolitical aspects of the crisis. Keywords used by participants to describe the
components of the game are very interesting. While in Monopoly terms describing the
game (e.g. prison, command) are reported, in Euro-Axio-polis qualitative
characteristics were brought up, such as cooperation between individuals and groups,
resolving ethical dilemmas etc.
Furthermore, another significant outcome which rose through students’
answers and may be worthwhile to be further researched, is the fact that only a few
students appreciate that mathematics can be learned within a sociopolitical context. It
seemed that mathematics in realistic contexts is an invisible culture (ChionidouMoskofoglou, Vitsilaki and Vasiliadis 2006) for the most of university students and
6th grade pupils.
Future research attempt should focus on the investigation of how Euro-AxioPolis embedded in school and lifelong learning curriculum and teaching approaches,
should support university students and school pupils in developing meaningful, not
isolated and functional mathematics in a socio-political context gaining happiness and
independence in mathematics (Smith 2011).
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 41
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
References
Burton, L. 2004. Mathematicians as enquirers. Berlin:Springer.
Chasapis, D. 1996. The reference frames of mathematical concepts in primary
education teaching and their ideological orientations. Proceedings of the 1st
Panhellenic Congress on Mathematics Education, 113-123.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
I thought I knew all about square roots
Cosette Crisan
Institute of Education, University of London
Following on from my observations of the inconsistencies and misuse of
the radical symbol amongst pupils, undergraduates, teachers and some
authors of school textbooks, I became interested in those decisions that
teachers take when confronted with inaccurate or ambiguous
representations of the square root concept and its associated symbol
notation. The impact that the ambiguous treatment of this mathematical
concept and its associated symbol notation has on a number of PGCE
students’ conceptual understanding and pedagogical affinity will be
discussed.
Keywords: square roots, ambiguous definition, textbooks
How it all started
My interest with this particular mathematical concept started a number of years ago,
just as I was embarking on teaching my Year 8 pupils about square roots. It was my
first year of teaching mathematics at secondary school level after having taught
various pure mathematics courses at university level for over ten years. I remember
glancing at the textbook the pupils were using and as I did so I was very surprised to
find a new symbol which I was not familiar with. The textbook introduced the symbol
± , according to which the notation  16 was understood to stand for the positive and
negative square root of 16. As I expected, my pupils found this new notation
confusing, especially after having studied the square root the previous year when the
textbook simply and clearly stated that “A square root is represented by the symbol
. For example, 16  4 and – 4” (Evans et al. 2008) (note and not or in the
definition above, introducing or indicating a further ambiguity about yet another
mathematical symbol, namely ± ).
As a mathematician, I felt uncomfortable with the situation. The square root
symbol , referred to as ‘the radical symbol’ is assigned to the positive square root
of any non-negative real number, since x 2  x for any real number x and thus its
value is always a non-negative real number. While I did not expect this level of rigour
in defining new concepts or symbols to Year 8 pupils (nor did I think that was
desirable at this level of pupils’ mathematical education), I was worried by the
textbook’s incorrect definition and use of a mathematical symbol together with the
lack of consistency and rigour in treating a mathematical concept..
In Crisan (2008, 2012) I identified the widespread misuse of the radical
symbol amongst the authors of a large number of school textbooks. Most of the many
teachers I talked to about the square root of a number did not seem to question the
textbook definition; but used it according to how it was introduced by the class
textbooks. It was not unusual for teachers to report to me that they taught pupils that
9  3 at KS3 and KS4 foundation level, while teaching pupils that 9  3 at KS4
higher level and KS5. Just a handful of teachers said that they were very keen to point
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
out the textbook inaccuracies to their pupils, teaching them to use the symbol
for
the positive square root value of a number only. They did so despite running into
difficulties at times, such as when confronted with examination marking schemes that
awarded marks for the negative values of a square root.
The study
While Ball and Phelps (2008) argue that teachers need to be able to make judgments
about the mathematical quality of instructional materials and modify them as
necessary, can we rest assured that users (teachers) of these resources are able to
identify inaccuracies and ambiguities and know what to do about ‘putting them right’
given for example, the constraints of the departmental practices or exam board
syllabus specifications?
For this reason I decided to carry out a small study involving prospective
teachers, students on a Post Graduate Certificate of Education (PGCE) course, and
present them with a number of mathematics questions to solve involving the square
root. The aim of this study was to explore the participants’ knowledge about the
square root and its associated symbol notation and to the decisions they take in the
planning for teaching when confronted with inaccurate definitions or ambiguous
representations of this concept held by other participants or present in the instructional
materials consulted. I was also interested to find their sources of conviction when
adopting a particular ‘definition’ of the concept and how they justify their choices.
In this study the eight secondary mathematics PGCE volunteers were engaged
in a number of mathematics and pedagogically specific tasks with the aim of gaining
access to their knowledge, views, beliefs and intended practices. The participants
were split into two groups according to their availabilities for group discussion (group
I – pseudonyms: Jan, Jemma, Jack and Joan; group II – pseudonyms: Billy, Barry,
Ben and Bea).
Data Collection
Participants were first given a piece of homework consisting of questions where the
concept of square root was likely to be employed. The mathematics questions were
designed so that they would bring to the surface the ambiguities and inconsistencies
of this concept and its associated symbol. The participants were then invited to talk to
each other about how they solved/answered the questions set. During the discussion,
implications for teaching about square roots arose naturally, either through the
participants’ reflection on how they had been taught the topic or how they would
teach the topic themselves. Immersion of the participants’ mathematical work in the
pedagogical space was taken further through another task, namely fictional pupils’
scenarios. The participants were asked to give written feedback to three fictional
pupils’ responses (Emma-KS3, Peter-KS4 and Lucy-KS5) characterised by a subtle
mathematical error in a question involving the square root, throwing further light on
the choices the participants made about treating this concept.
Discussion and findings
In the following I will report on some aspects of the participants’ approaches to
solving some of the questions set as homework, supporting their written and oral
explanations with data collected during the group discussions and some of their
written feedback to the fictional pupils’ scenarios where relevant.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The participants’ knowledge and understanding of the square root of a number
When discussing the answer to the question asking them to solve the equation x 2  16 ,
all the participants were in agreement that the solutions were x  4 . The solutions
were reached either by solving the equation by factorisation (one participant), by
using the graphical approach (two participants) or, in the most popular approach, by
‘taking the square root’ of both sides, the latter giving
x 2  16 hence x   4 since
16 equals ± 4 (five participants). Group I were happy with this above explanation
when given by Jan. A similar solution was put forward by Billy in group II, but he
changed his mind very soon after offering his explanation. He then quickly said:
Actually, strictly speaking that is not right, is it? Looking at it now, I would amend it
to say that x   16 since x 2   x and 16 equals 4. After this contribution, the
participants debated whether the answer when ‘taking the square root’ was either
positive or negative. Sometimes it could be +, sometimes it could be –, said Barry,
while Ben attempted to clarify this point by saying: It depends how you want to define
the root function. Billy interrupted abruptly to say: The root function is defined as two
numbers multiplied together to give the original number and so
x 2   x . However,
he then changed his mind to say that 16 should equal ± 4 , and so the equation
x 2  16 reduces to solving x  4 , an equation in a format unfamiliar to all
participants in group II.
The explanations put forward by Bill, Barry and Ben illustrate the two facets
of this ‘elementary procept’ (Gray and Tall 1994), an amalgam of a process (the
inverse of the square function) which produces a mathematical object (the square root
of a number) and a symbol which is used to represent either process or object (the
radical symbol notation). The radical symbol
is used for both a process and a
concept, giving thus rise to ambiguity.
Indeed, such ambiguity gave rise to a further interesting debate which took
place when solving another question asking them to give the answer to
following solutions were put forward:

9 2  81   9 ;

9 
2
1
2 2
(9 ) ,
which can then be taken forward by using the order of the
operations (brackets first)

1
2
2 2
9  (9 )
1
as (81) 2
 81  9 ;
, which if using the order of the operations (laws of indices)
2
can be taken further as 9

9 2 . The
1
2
 91  9 and finally,
92 = 9 since the square and square root cancel each other (given that the
square root and square functions are inverse of each other)
Despite the obvious equality 9 2  81 , all four explanations were regarded as
being valid and the participants in group I did not seem to be able to find any ‘fault’ in
the reasoning approaches presented above, as all explanations seemed to have a
logical, firm foundation. This is not an identity, but they can be equal, Jan then said.
The participants understood that this was ambiguous, and tried to ‘get to the bottom’
of this ambiguity. While doing so, they had a lengthy discussion about the differences
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
between a mapping and a function. Jack concluded that it is all to do with the fiddling
things like … between … functions and mappings, which I cannot quite put my finger
on why.
The participants in group II had a similar debate when comparing each other’s
answers to another question asking them to simplify
25y 2
. In the light of the earlier
discussion about solving the equation x 2  16 , they settled for the following
convention: for a variable y 2  y , while for a number 25  5 and so the solution of
the equation was 5 y , which ‘worked’ when these values were substituted back into
the equation. At this point, Ben summarised that perhaps in different contexts, the
square root could mean different things. He went on to say that if working in the
context of graphs and functions at KS5, one can consider only the positive value,
whereas when finding the square root of numbers, one could consider the + or –.
Both Billy and Barry illustrated this aspect with the formula for calculating the roots
of a quadratic equation, namely
 b  b 2  4ac
, justifying the presence of the ± as the
2a
result of calculating the square root of a number (the numerical value of b 2  4ac ).
When prompted to consider more carefully the quadratic formula, the participants
realized that in fact the ± becomes redundant in the formula.
Sources of conviction
During the group discussion, if conflicting or non-equivalent views of how to work
with the square root were encountered, the participants were invited to discuss, debate
and reach a consensus. Most of the participants’ sources of conviction, which they
used in order to justify their answers, were external in nature. The participants relied
on what they remembered from school or what they learned from the instructional
materials they consulted when doing the mathematics homework.
While consulting the materials available to them (textbooks, dictionary,
mathematics glossary, examination papers with marking schemes, web sites), the
participants commented on the inconsistencies in how the square root was presented.
For example, while browsing an A-level textbook (Pledger et. al., 2004), the
participants realised that according to the chapter on surds, 25 = 5 with no mention
of the  , while the following chapter on quadratic functions draws pupils attention to
the fact that 25 = + 5 or - 5 . The other instructional materials reviewed suggested
that 16  4 or - 4 , that 16  4 and -4 , that 16 = ±4 , introduced the new notation
±
16 standing for the positive and negative square root of 16, or gave pupils a choice,
namely that 16 is 4 most of the time, but that it could also be -4, depending on the
context of the problem to be solved. Quite annoyed by this, Billy thought that this was
abuse of language and notation at A-level and that mathematics should not be about
free choices. Billy went on to say that in his view this was the result of simplifying
things for the sake of our pupils. He explained how taking an easy route with Year 7
pupils when introduced to the positive and negative square root of 25 without a clear
distinction about the symbols in use is similar to the difficulties pupils have with the
incorrect (but widely accepted) way of reading -7-12 as ‘minus 7 minus 12’, leading
to difficulties in understanding the operation that needs to be performed.
During the group discussion Bea expressed her frustration with the fact that
her group were not making much progress in checking the rest of the homework
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
questions due to confusion over the definition of the square root. She shared with the
group that she was taught at school that the square root of a number is always a nonnegative number and as a result her answers to this question (and other similar to this
one) were non-negative numbers. In fact, she was confused by the polemic
surrounded the + or -: I cannot see what the problem is? x 2 = x for any x real
number; this is the definition of the square root, so why not use it? Bea explained that
taking the square root of both sides yields x = 16 , hence ± x = 4 , resulting in x = ± 4.
The definition presented by Bea created some uneasiness amongst the other
participants as they did not think it would be of much use since the square root is
introduced to pupils much earlier than the concept of modulus function, or function
for that matter. However, the participants in group II liked the clarity of this definition
and adhered to it. For example, Barry in group II gives the following feedback to one
fictional pupil scenario (Emma - KS3): However, by convention, we usually take
4 to just mean the positive root, i.e. 2, and he is consistent in the feedback to the
pupils.
In group I, the discussion led to the participants making a clear distinction
between the square root of a number and the square root of a square number written in
index form and evidence collected through their feedback to fictional pupils’
scenarios indicated that the participants were prepared to work with these two facets
of the square root concept even if it led to conflicting pedagogical decisions. In her
feedback, Jan tells Emma, the KS3 fictional pupil that 25  9  16  4 so when you
see
you must consider both the positive and the negative roots. However, in her
feedback to Peter, the KS5 pupil she explains that 72 can only equal 7, as this is
about the square root being the inverse process to squaring,
Discussion and some findings
The participants brought to the group discussion different knowledge and
understanding about the concept of square root of a number.
Strong held beliefs
With one exception, all the participants identified + 4 and - 4 as the square
roots of 16 and their written answers revealed that they used the radical symbol to
denote any of these square roots, i.e. 16   4 . This is how we were taught since very
little, said Jan and this explains why the participants (especially those in group I)
invested a lot of energy in defending this knowledge. The participants’ sources of
conviction were external in nature in most cases, recalling and reproducing definitions
they remembered from school or textbooks, while not claiming any ownership of the
square root concept. Initially, when encountering ambiguities in the questions they
were solving, the participants worked on the premise that their knowledge of square
roots was correct, i.e. 16  4 , as most of the participants were taught, hence they
looked elsewhere for resolving any issues they encountered instead of revisiting their
knowledge and understanding of the concept.
Competing Claims
However, the discomfort amongst the participants in group II caused by the
logical inconsistencies ( 9 2  81 ) motivated the participants to reconsider their
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
knowledge of this concept. They felt ready to alter it and to adhere to Bea’s definition
of the concept as it clearly was free of any ambiguities. Despite realizing that they
were not going to be able to introduce this definition at KS3 and 4 levels, the
participants were happy to present the use of the radical symbol to younger pupils as a
‘convention’ for positive square roots only, confident that they had a firm
mathematical foundation for this argument.
The participants in group I however could not reach a consensus and as a
result they accepted both facets of the square root. They were still not clear about the
underlying mathematics of the concept, but made some pedagogical decisions:
teaching pupils that 9   3 at KS3 and KS4 foundation level, while 9  3 at KS4
higher level and KS5, complying with the textbooks they consulted. Both definitions
were seen as valid and the participants’ feedback to pupils’ responses suggested that
the square root symbol was used differently for different year groups.
The use of instructional materials
It was important to expose the prospective teachers to situations where textbooks give
different but not equivalent or even ambiguous definitions of a mathematical concept.
Good textbooks providing accurate information are needed. This does not necessarily
mean that formal definitions should be introduced to the pupils, but authors of such
textbooks have to be very careful when less formal definitions are introduced, without
careful considerations for the implications for further learning
This study highlighted the need for prospective teachers to revisit their subject
knowledge and develop an appreciation of mathematics as a coherent discipline,
where different areas of mathematics are related and interconnected (square root
definition, functions, mappings, relationships, identities, symbol use were aspects
considered by the participants). It is this view and understanding of mathematics that
enable teachers to scrutinise the available instructional resources and to decide for
themselves on the appropriate pedagogical approaches and not rely on how they were
taught when at school or on the authority of textbooks or examination boards.
References
Ball, D. L.and G. Phelps. 2008. Content knowledge for teaching: What makes it
special? Journal of Teacher Education 59: 389-407.
Crisan C. 2008. Square roots: positive or negative. Mathematics Teaching, 209: 44.
Crisan C. 2011. What is the square root of sixteen? Is this the question? Mathematics
Teaching 230: 21-22.
Evans, K., K. Gordon, T. Senior and B. Speed. 2008. New mathematics
frameworking, Year 7, Pupils Book 3, Collins Education.
Gray, E. M. and D. O. Tall. 1994. Duality, ambiguity and flexibility: A proceptual
view of simple arithmetic. The Journal for Research in Mathematics
Education 26 (2): 115 -141.
TIMSS. 1995. Press Release June 10, 1997
http://timss.bc.edu/timss1995i/Presspop1.html Accessed on 08 April 2012.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 48
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Developing a pedagogy for hybrid spaces in Initial Teacher Education courses
Sue Cronin and Denise Hardwick
Liverpool Hope University
We share an emerging pedagogy for Initial Teacher Education (ITE)
mathematics tutors who are seeking new ways to work with student
teachers in what Zeichner (2010) defines as hybrid spaces. In terms of
Initial Teacher Education, hybrid spaces are those spaces which are
formed to “bring together school and university based teacher educators
and practitioners and academic knowledge in new ways to enhance the
learning of prospective teachers” (92). For the last three years the PGCE
secondary mathematics programme in the authors’ university has included
a Saturated Learning Project (SLP). This has involved taking all of the
secondary mathematics students into school one morning for each of ten
weeks to work with groups of pupils in a shared communal space,
supported by class teachers and university tutor. The project has now been
extended to the PGCE primary course with ten student teachers
specialising in mathematics. They also worked over a number of weeks
with a group of Y6 pupils. The experiences in such hybrid spaces enriched
and extended students’ practical and pedagogical knowledge by
facilitating understanding of theories about teaching and learning
mathematics in a real, shared context. This new pedagogical approach is
strengthening school-university partnership and improving learning
experiences for both student teachers and their pupils.
Keywords: hybrid spaces, saturated learning, initial teacher education
Context and background of Initial Teacher Education
Initial teacher education (ITE) in England is at present more than at any other time in
its history a site of great contestation and change. The pace of political reform is
exponential and will force unparalleled and abrupt cultural and organisational changes
by universities and partner schools. The new UK coalition government’s drive to
shift the focus of control of teacher education into schools by reforming the current
system has significant and not yet fully understood implications for Higher Education
(McNamara and Menter 2011). Placing greater emphasis on the workplace and
employment based routes will require university initial teacher educators to reconsider
and reposition themselves within the field. Justifying critically the unique and
valuable learning spaces created for the beginning teacher by the university is an
important step forward towards a new vision of professional learning. This paper sets
forward the response of tutors at a particular university and how the development of
hybrid spaces (Zeichner 2010) may be part of a new pedagogy which offers additional
expansive learning experiences and new democratic ways of working with schools to
support student teachers and their professional development.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Origins of the Hybrid Space
Zeichner (2010) outlines hybrid spaces in ITE as those spaces which involve a
“rejection of binaries such as practitioner and academic knowledge and theory and
practice and involve the integration of what are often seen as competing discourses in
new ways” (2010, 92). Three years ago the secondary mathematics education tutor
introduced the Saturated Learning Project (SLP) as an enhancement to the exisiting
programme offered as part of the university taught course. The project was in keeping
with the vision of a closer, more democratic partnership with school mathematics
departments involved in the ITE programme. It was designed to allow an exploration
of new ways of working more closely with partner schools, using the cohort of PGCE
mathematics student teachers as co-enquirers. The project involved challenging
boundaries between the ‘academic’ learning situated in the university and the
‘professional’ learning situated in the school setting. The SLP created a new hybrid
space in which academic and professional practice were brought closer together by
moving one of the weekly university sessions into a partner school and involving the
mathematics department more closely in the content and purpose of the sessions.
The original SLP involved the entire cohort of secondary mathematics
students placed in a pilot school for a morning a week, working with the same two
groups of pupils for ten weeks. It contrasted as a learning experience with the
traditional ‘solo’ model used on the university PGCE secondary course, where a
trainee is placed on their own in a school with a supervising mentor. This is a model
which forms the basis for many secondary teacher training courses run by universities
and, as Bullough et al. (2002) note, one that has remained little changed for 50 years.
Placing all the student mathematics teachers in the one learning space presented a new
learning experience for not just the student teachers but for the university tutor, the
school teachers and colleagues.
The design of the SLP facilitated the formation of small communities of
enquiry (Senge 1990) as the school teachers, student teachers and university tutor
worked collectively with the same group of pupils. The experience provided a new
space to enrich and extend students’ practical and pedagogical knowledge by
facilitating understanding of theories about teaching and learning mathematics in an
authentic, shared context. The student teachers developed practices which were not
the same as those in their individual placement schools and thus the SLP afforded
knowledge of a different practical and pedagogical nature to reflect on and against.
In 2012 this saturated model was extended into a partner primary school. The
ten specialist mathematics PGCE primary students worked over a series of weeks with
a Y6 class who were preparing for the Key Stage 2 National Curriculum tests (NCTs)
and in addition worked with the mathematics coordinator to prepare a series of
enrichment activities for all year groups as part of a mathematics week.
Methodology
The project evolved as an action research project. Action research is characterised as a
form of:
self-reflective enquiry undertaken by participants in social (including
educational) situations in order to improve the rationality and justice of (a) their
own social or educational practices, (b) their understanding of these practices, and
(c) the situations in which the practices are carried out. It is most rationally
empowering when undertaken by participants collaboratively ... sometimes in
cooperation with outsiders. (Kemmis 1983: 34).
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
As university tutors, instigating the research project, we were both insider researchers
as well as cooperating outsiders. Although we were fully participatory in the research
project it could be argued that we were outside of the school community in which the
project was sited. There were advantages to this position as we had, as insiders, some
of what Coughlan (2001) refers to as “the pre-understanding from being an actor in
the processes being studied” (2001, 49), together with a degree of objectivity through
our external position as university tutors.
The original SLP projects took place in the first semester of the secondary
PGCE course and involved the secondary mathematics cohorts working with
examination groups of Y11 (16-year old) pupils on a one-to-one basis and small
groups of three or four Y7 pupils with two student teachers. In the second year all of
the learning took place in the school hall which provided a large physical space in
which all groups could work alongside each other. Before and after each session an
hour was set aside for preparation and anticipatory reflection (Van Manen 1995). The
morning ended with a further reflective hour when the student teachers initially
discussed their experiences and evaluated the successes and identified areas for
improvement in their own learning groups and then contributed comments and
reflections to the whole cohort.
The SLP project started at the beginning of the PGCE course so the tutor in
collaboration with the school’s head of department prepared the lessons. The student
teachers in their small working groups had to spend one hour before teaching
reviewing and adapting the materials. Initially the tutor felt it would have been of
greater benefit if the students had had the opportunity to co-construct and plan their
own lessons for their pupils. However the time constraints of the programme overall
and the very early stage of the student teachers’ own professional development meant
this was an unrealistic aim.
In retrospect this model offered some advantages to the student teachers and
the tutor as it ensured more effectively the delivery of a quality experience for the
pupils and provided a scaffolded experience in lesson planning and the development
of active learning strategies. The students concentrated their efforts on translating the
tutor’s and head of department’s plans and activities into engaging experiences for the
particular pupils they were working with. The subsequent discussions focussed on the
effectiveness of the learning experienced by their pupils.
Evaluation of the SLP
The research project evaluation had two different aspects; firstly the impact of the
project on the pupils’ learning and the school more generally, and secondly the impact
on the student teachers. As initial teacher educators we have a responsibility to
provide expansive learning experiences for our student teachers but ultimately these
must have a value in preparing them to teach i.e. have a resulting impact on pupil
learning. From the school’s perspective the partnership with ITE providers needs to
be seen as one that offers positive benefits for their school. Increasingly as schools are
focused on performance and accountability many hold a deficit model of beginning
teachers not seeing the possibilities of their positive contributions to their pupils early
on in their PGCE course.
The projects have been evaluated using largely qualitative methods involving
questionnaires, short interviews and focus group discussions. These have involved all
of the stakeholders: heads of department, coordinators and classroom teachers, a
sample of the pupils together with the student teachers. In all three years the
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secondary SLP has involved working with Key Stage 4 (15-16 years) examination
classes and groups of Y7 or Y8 pupils (11-13 years). Some quantitative data of the
KS4 pupils’ progress has also been collected.
The outcomes of the project have been extremely positive from all of the
participants’ perspectives. All of the schools involved have evaluated the project as
beneficial for their pupils and staff. The heads of department have viewed SLP as
contributing to their KS4 pupils’ achievement in public examinations. The expected
grades for pupils in the SLP pilot school were exceeded with 55% of the pupils
involved in the sessions achieving an examination grade higher than originally
predicted. The second year SLP project head of mathematics wrote a report which
included the following:
We believe with the explained support and personalised pathways put in place
through the SLP, these pupils are in a good position to build upon these
foundations and bring their attainment in line with National expectations of 3
levels [of progress] from KS2 to KS4. (Head of Department report to Governors,
2010)
The pupils’ views were collated using a questionnaire. A majority of the
responses have been encouraging with over 85% making positive comments. All
pupils who responded said they felt it was a good project. Across the three years of
the project the examination groups’ responses have been particularly positive, perhaps
understandably as they have appreciated the individual support afforded by the
student teachers as they approach their formal examinations. The Y7 and Y8 pupils’
responses have also been positive, their evaluations indicated after the individual
support, the enjoyable nature of the practical mathematical activities ranked second
for the thing they liked most about working with the extra student teachers.
The primary project has had similar positive responses from the school. The
coordinator was pleased that the NCT mathematics results increased from 47%
achieving level 4 and above in the previous year to 53%. In particular she felt the
impact had been most significant on those achieving level 5 and above which
increased from 18% to 28%. Although this increase has many contributory factors the
teachers’ perceptions were that the SLP had made a difference. The feedback from
primary pupils was also very positive. A concern for many of the student teachers is
that they will swamp the pupils and be overwhelming for them. However, as with the
secondary pupils’ responses the primary pupils were largely un-phased by the
additional adults and saw the benefits of greater attention and access to support and
guidance.
Impact on the beginning teachers
A more detailed evaluation of the student teachers’ opinions was obtained through
questionnaires and focus group discussions. Overall the student teachers found the
experience beneficial in several ways. They recognised the value of getting a close up
understanding of a learner over a series of weeks, as one put it: “it helps you get
inside the head of a learner”. They also found working closely with a peer extremely
beneficial. In particular they valued the peer support when they were not sure how to
support learner understanding of a concept. Many have also commented on the
benefits in listening to and watching the teaching styles of their peers.
During the SLP I was paired with another teacher and this proved an excellent
learning experience not only for the children but for the teachers. When a pupil
could not understand a concept being taught by one of us, we could look across
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
and ask for another interpretation. It highlighted that all teachers are different and
they have been taught differently. PGCE secondary student 2011
Not having to prepare the lesson materials was also viewed as a positive by
many of the students who felt it allowed them to concentrate on the pedagogical
aspects of the tasks and spend time considering the most effective ways to teach. It
also provided a dry run for activities which they transferred and adapted for their own
solo teaching.
The SLP has been identified by over 85% of all secondary student teachers
involved over the three years as having a significant positive impact on their practice,
as one student noted; “I have subsequently used some of the ideas in my own class
teaching and found them to be very successful.” The students also recognised that the
pupils had benefited from their support but most of their comments were about the
benefits to themselves rather than the pupils.
Although I stated that our mission was to help prepare the Y11 pupils for their
GCSE exam, it has to be recognised that the PGCE students benefited from the
SLP as much if not more than the pupils. (PGCE Secondary Student 2010)
Maths Week gave me a chance to experiment and take risks without the worry of
being observed which meant I produced much more creative lessons. (PGCE
Primary Student 2012)
Gave valuable experience of working in an inner city school with a lot of special
needs. (PGCE Primary 2012)
I remember being quite shocked by the inability of GCSE pupils to recall basic
number facts during the early SLP session. But the experience stood me in good
stead when I began to teach my own lessons as I was better prepared to deal with
such issues by the time my first school placement began. (PGCE Secondary
Student 2012)
Conclusion and discussion- what is afforded by the Hybrid space?
One of the original driving forces for the SLP was the awareness of how many
students move towards a privileging of school experience over university experiences,
often viewing the two aspects of their professional learning as separate, indeed
disparate. Allen (2009) argues student teachers re-orientate their practice as they
increase the time spent in school, giving agency to the school based practice over their
university experiences. This dichotomy of theory and practice is one initial teacher
educators need to challenge. The development of the student teacher over the PGCE
course should not be a process of displacement; with the student teachers substituting
theories about practice with all the situated practices of the placement school, but one
of critical integration. Edwards and Protheroe suggest that the student teachers’
knowledge is “heavily situated and that students are not acquiring new ways of
interpreting learning that are easily transferable” (2003, 227). This is supported by
Hobson et al (2008) who found a majority of student teachers viewed the university
component of their courses as least relevant. Indeed over half failed to see the links to
the authentic classroom setting.
By relocating the site of learning the university tutors were able to work with
the students in an authentic setting and to facilitate a greater level of connection
between theory and practice. The SLP affords more opportunity for the mathematics
tutor to offer different perspectives at possible sites for contestation in the school
context. Wilson (2005) notes that one of the dangers of the university – school model,
is that student teachers spend two thirds of their time in school where there may be
limited opportunity to discuss with anyone their emerging practice on a practical
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
level. She argues that this “may severely limit the novice teachers’ capacity to be
critically reflective of their own practice” (2005, 375). The SLP creates a reflective
space which allows time to consider some of these important practical issues
concerning developing mathematical understanding with peers, teachers and tutor.
References
Allen, J. 2009. Valuing practice over theory: How beginning teachers re-orient their
practice in the transition from the university to the workplace. Teaching and
Teacher Education 25: 647-654.
Bullough, R., J. Young, L. Erickson, J.R. Birrell, D.C. Clark, M.W. Egan, C.F.
Berrie, V. Halesand G. Smith. 2002. Rethinking field experiences: Partnership
teaching versus single placement teaching. Journal of Teacher Education
(53): 68-80.
Coghlan, D. 2001. Insider action research projects implications for practicing
managers. Management Learning 32: 49-60.
Hobson, A.J., A. Malderez, L. Tracey, M. Giannakaki, G. Pell and P.D. Tomlison.
2008. Student teachers’ experiences of initial teacher preparation in England:
core themes and variations. Research Papers in Education 23: 407-433.
Hopkins, D. 1985. A teacher's guide to classroom research. Philadelphia: Open
University Press.
Kemmis, S.1983. Action research. In International encyclopedia of education:
Research and studies, ed. T. Husen and T Postlewaite, 32-45. Oxford:
Pergamon.
McNamara, O. and I. Menter. 2011. 'Interesting times' in UK teacher education.
Research Intelligence (116): 9-10.
Reason, P., and H. Bradbury. 2008. Introduction. In The Sage handbook of action
research: Participatory inquiry and practice, ed. P. Reason and H. Bradbury,
1-10. Thousand Oaks, CA: Sage Publications
Senge, P. M. 1990. The fifth discipline: The art and practice of the learning
organization, London: Random House.
Van Manen, M. 1995. On the epistemology of reflective practice. Teachers and
Teaching: Theory and Practice 1: 33-50.
Wilson, E. 2005. Pedagogical startegies in initial education. Teachers and Teaching:
Theory and Practice 11: 359-378.
Zeichner, K. 2010. Rethinking the connections between campus courses and field
Experiences in college and university–based teacher education. Journal of
Teacher Education 61: 81-99.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 54
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
From failure to functionality: a study of the experience of vocational students
with functional mathematics in Further Education
Diane Dalby
University of Nottingham, UK.
Many students who undertake vocational courses in Further Education
colleges in England enter post-compulsory education as mathematical
‘failures’ at GCSE level but their experience in college has the potential to
change not just their attainment, but also their future attitude and
‘functionality’ with mathematics in employment and society. This paper
outlines the early stages of a mixed methods study to identify the main
influences on the student experience and their effects on the aspirational
trajectory from ‘failure’ to ‘functionality’.
Keywords: functional, mathematics, Further Education, vocational.
The context for the study
Many students who have not achieved grades A*-C at GCSE ( the standard English
and Welsh mathematics qualification taken at age 16) choose to undertake vocational
training post-16, often in a Further Education college, where they are often
recommended to improve these skills and may be expected to take a functional
mathematics qualification in addition to their vocational course. It is the experience of
these vocational students with functional mathematics that is the focus of this
research.
International comparisons of mathematical performance show England in a
relatively weak position (Organisation for Economic Co-operation and Development
(OECD) 2010) and adult numeracy levels have been a concern since this was
highlighted by the Moser Report (1999). Despite the Skills for Life Strategy
(Department for Education and Employment (DfEE) 2001), there has been little
significant change in adult numeracy skills (Department for Business, Innovation &
Skills (BIS) 2011) and recognition of the need to improve the mathematics skills of
the nation is not lacking in recent reports although the means of effecting the change
is still unclear. Evidence of the transmission of low numeracy across generations
(Parsons and Bynner 2005) and suggestions that school leavers with low levels of
mathematics may be disadvantaged economically as adults (Ananiadou, Jenkins, and
Wolf 2004) provide further reasons to improve the mathematical skills of young
adults, both for their own benefit and for the future of the next generation.
The mission for colleges is to transform these ‘failures’ into ‘successes’ within
the constraints of policy, funding and curriculum. The first stages of this research
indicate that this involves not just the challenge of getting students through an
examination but also a battle in the affective domain to change their established
attitudes to mathematics.
Research Aims
The research takes the form of a comparative study of the experience of vocational
students with functional mathematics in colleges with different staffing structures.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The aim is to identify the major influences and their effects on the learning of
functional mathematics.
Staffing structures for functional mathematics may be divided broadly into
two types: those with a centralised functional mathematics team and those with
functional mathematics teachers dispersed into the vocational departments. However,
between these two structural extremes lie a range of hybrid management structures
that combine some dispersed functions with a variable level of centralised control.
The three colleges engaged in the research use either a dispersed arrangement or a
more centralised hybrid arrangement but are such that comparisons between the
effects of centralisation or dispersion can be made.
The main research question of ‘What factors influence the experience of
functional mathematics for vocational students in Further Education?’ is followed by
the additional questions summarised below, but, from a social constructivist view, it is
the social interactions within these areas that are of particular interest.
 What effect do college structures and policies have on the student
experience?
 In what ways is functional mathematics relevant to students?
 What approaches to teaching functional mathematics are being used and
what effect do they have on student learning?
 What influence do the students’ prior experience and background have?
 What influence do the attitudes, beliefs and values of vocational and
mathematics staff have on the students’ experience of functional
mathematics?
Potential factors affecting the student experience
Structures, policies and systems
Organisational structures link people together, creating bonds or barriers and there are
both benefits and disadvantages in the different structures. For example, a dispersed
structure may facilitate a more integrated approach to functional mathematics,
resulting in greater relevance for students, but can isolate functional mathematics
specialists from their professional community leading to negative effects on teacher
attitudes.
College policies operate within the constraints of government policy and
funding but individual colleges do retain some freedom. Some may direct all students
on a particular vocational course to take functional mathematics but others may
exempt those with high grades in GCSE mathematics or even direct all students to a
different functional skill. Early indications from the research suggest that these
policies affect individual student attitudes depending on whether they perceive a need
to improve their mathematical skills or not.
The functional mathematics curriculum
The functional mathematics curriculum (Qualification and Curriculum Authority
(QCA) 2007) requires students to be able to make sense of situations, represent them,
analyse them, use appropriate mathematics, interpret results and communicate. This is
based on the assumption that learners need certain mathematical skills and abilities
“to gain the most out of life, learning and work.” (QCA 2007, 3). Early indications
from the research suggest that this concept of mathematics for real life and work has
been adopted by functional mathematics teachers. There is some ambiguity about
what skills people actually need (Roper, Threlfall, and Monaghan 2006) and whether
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
this is just knowledge with limited utility value (Ernest 2004) or a wider set of skills
that goes beyond basic numeracy (Hoyles et al. 2002). However, the view that
functionality involves problem-solving and communication and “requires more than
fluency with ‘the basics’” (Wake 2005, 6) is consistent with the views of functional
mathematics teachers at this stage in the research.
For students, how they relate to the functional mathematics curriculum is a
key issue. Disaffection or lack of interest in mathematics often stems from a failure to
see relevance (Nardi and Steward 2003) . This may be because mathematics does not
relate to their personal goals and interests (Ernest 2004) and is perceived to have no
practical usefulness, transferable process skills or professional exchange value (Sealey
and Noyes 2010). The emphasis in the curriculum on being able to apply mathematics
in a range of contexts (QCA 2007) restricts the opportunity to use a relevant context
and the problems of ‘transferability’ between the classroom and real life mathematics
(Lerman 1999; Nunes, Schliemann, and Carraher 1993) also present difficulties for
students.
Prior experience
The view that the nature of an individual is both socially constructed and emergent is
a useful starting point for a consideration of students in Further Education since they
bring with them a legacy from their previous lives but are still engaged in a learning
process that shapes their future.
Affective factors such as attitudes, beliefs and emotions, have been shown to
have an influence on the learning of mathematics (Hannula 2002; Zan et al. 2006) and
the concept of attitude as a set of emotions associated with the situation, combined
with a belief about the expected consequences and the relationship to the individual’s
personal values (Hannula 2002) is useful for this study. Affective and cognitive
structures are closely intertwined (Goldin et al. 2011) and there is some evidence of
this in early discussions with students. Both stable and rapidly changing affective
traits have been recognised (Goldin 2003) suggesting that although deep emotions
and beliefs may be resistant, there is some scope for change.
Social and cultural factors produce dispositions towards certain behaviours
that are often resistant and may adversely affect student attainment (Noyes 2009) or
performance in the classroom (Lubienski 2000). Not all these factors can be examined
in this research but initial discussions indicate that attitudes from the past are evident
in students and remain a significant influence on initial attitudes.
Vocational staff
The transition from school to college brings students into a new social environment
and learning community in which they adjust, establish their identity and adopt
behaviours that relate to the group norms.
In an organisation, the complex set of rules or traditions often referred to as
‘organisational culture’ (Deal and Kennedy 2000) reproduces patterns of thinking,
feeling and behaviour in a community. In a large and complex organisation there may
be several localised, departmental communities with different values and attitudes but
the vocational department is the main learning community for vocational students.
Attitudes and values are often transmitted implicitly and as students adjust to the
values and behaviours of the department, vocational teachers can become significant
influences. Their frequent social interactions with students may have more effect than
those of the functional mathematics teacher and differences between the vocational
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
and functional mathematics teams may only serve to reinforce beliefs that functional
mathematics is unrelated to their vocational world.
Functional mathematics teachers
Teachers of functional mathematics may be part of a mathematics team, a functional
skills team or a vocational team in the college structure. The combination of subjects
that they teach and the team or department they are affiliated to will affect their sense
of identity within their learning community and their approaches to teaching
functional mathematics. Early interviews indicate that functional mathematics
teachers have clear ideas about the concept of functional mathematics and how to
teach it but the external assessment and college performance measures do have an
impact.
Initial lesson observations indicate that teaching approaches are varied but it is
teachers’ beliefs about the value of functional mathematics and their ability to build
positive relationships with students that are emerging as significant. In the interviews
teachers frequently referred to their main challenges as: changing negative student
attitudes, persuading students that functional mathematics is relevant to them and
boosting the confidence of students who have already experienced failure.
Research methods
The range of and type of information to be gathered is wide and a mixed methods
approach is appropriate since both qualitative and quantitative data will be collected
and integrated at the analysis stage.
The main methods are: interviews with managers to gain an overview of
structures and policies; questionnaires for functional mathematics teachers on their
background, beliefs, teaching approaches and attitudes; interviews with functional
mathematics teachers to further explore these areas; lesson observations of certain
student groups; student focus groups to gain a student perspective on the lessons, their
beliefs about functional mathematics and their prior experience; questionnaires and
interviews with vocational staff to gain understanding of their beliefs and attitudes
towards functional mathematics. The central part of the research concerns the student
experience in the classroom and the triangulation of student perceptions, functional
mathematics teachers’ views and lesson observations by the researcher.
Some early indications are described in the following section, based on ten
individual interviews with functional mathematics teachers, 30 questionnaires, two
student focus group discussions and ten lesson observations, plus preliminary work
with five student groups, three teachers and one functional mathematics team.
Early emerging themes
The first indications reinforce the suggestion that less than a grade C in GCSE
mathematics is regarded as failure. Low attainment is strongly linked to negative
emotional responses and expectations of continuing failure. Comments such as
“We’re thick, therefore we’re doing functional maths”, “I’m never going to get it, I
feel so stupid” and “Always dreaded it, since school” illustrate the negativity,
assumptions and lack of confidence present in many students.
Students frequently stated a belief that functional mathematics had value and
the comment “You can’t get anywhere without maths” is typical. Their explanations
revealed that some see functional mathematics skills as useful tools for real life and
others acknowledge the exchange value of the qualification to access further training
or a career. However, their beliefs about the value of mathematics, coupled with their
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
own lack of success, seem to reinforce feelings of failure for some students yet
provide motivation for others.
Staff interviews show that teacher backgrounds vary widely and that few
would be seen as mathematics specialists in a school situation. All the staff
interviewed had had other careers before entering teaching and many had used
mathematics in that career.
The teachers interviewed all shared an enthusiasm for functional mathematics
and strongly believed that students needed these skills for life, even if they would not
actually use mathematics in a job. They made a clear distinction between
‘functionality’ and ‘traditional’ mathematics, referring to functional mathematics as
the application of mathematical skills in real life situations and the development of
transferable, problem-solving skills. They believed that functional mathematics has
value but this is more about the value of the skills students develop than the actual
qualification they achieve, which they feel has variable levels of acceptance amongst
employers and HE institutions. There was strong opinion that functional mathematics
is useful and that even students with high grades in GCSE mathematics benefit from a
functional mathematics course since it develops skills that are often lacking.
There are some indications from staff interviews that student attitudes to
mathematics and their attainment can change. In preliminary work, students at the end
of their course agreed that the teacher-student relationship in college was different to
school and they felt more positive about mathematics as a result. In the main study
student comments such as “If I’d had Pete as a teacher at school I’d have passed my
GCSE” and “I really enjoyed that lesson, considering I hate maths” suggest that new
relationships and environments can change attitudes but it may take time. As one
teacher commented when referring to the relevance of functional mathematics to real
life, “They don’t see it at first but in the end they do.”
Concluding comments
The transition from school to Further Education provides an opportunity for change in
students who may have previously experienced failure with mathematics. The early
indications of this research are that students bring a legacy from school but it is
possible to provide an environment in which student beliefs and attitudes can be reshaped, useful mathematical skills for the future can be developed and students can
gain the confidence to use them.
References
Ananiadou, K., A. Jenkins and A. Wolf. 2004. "Basic skills and workplace learning:
what do we actually know about their benefits?" Studies in Continuing
Education no. 26 (2):289-308.
BIS. 2011. Skills for life survey: Headline findings. London: Department for
Business, Innovation and Skills.
Deal, T. E., and A. Kennedy. 2000. Corporate cultures: The rites and rituals of
corporate life. Cambridge, Massachusets: Perseus Books.
DfEE. 2001. Skills for Life: The national strategy for improving adult literacy and
numeracy skills. London: HMSO.
Ernest, P. 2004. Relevance versus ytility. In International Perspectives on Learning
and Teaching Mathematics, ed. B. Clarke, D. M. Clarke, G. Emanuaelson, B.
Johansson, D. Lambin, F. Lester, A. Wallby and K .Wallby, 313-327.
Goteborg: National Center for Mathematics Education.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 59
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Goldin, G. 2003. "Affect, meta-affect, and mathematical belief structures." In Beliefs:
a hidden variable in mathematics education?, edited by G. Leder, E.
Pehkonen and G. Torner, 59-72. New York: Kluwer Academic Publishers.
Goldin, G.A., Y.M. Epstein, R.Y. Schorr and L.B. Warner. 2011. "Beliefs and
engagement structures: behind the affective dimension of mathematical
learning." ZDM The International Journal on Mathematics Education no. 43:
547-560.
Hannula, M.S. 2002. "Attitude towards mathematics: Emotions, expectations and
values." Educational Studies in Mathematics no. 49 (1): 25-46.
Hoyles, C., A. Wolf, S. Molyneux-Hodgson and P. Kent. 2002. Mathematical skills in
the workplace: final report to the Science Technology and Mathematics
Council. London: Institute of Education and the STM Council.
Lerman, S. 1999. Culturally situated knowledge and the problem of transfer in the
learning of mathematics. In Learning mathematics: From hierarchies to
networks, ed. L Burton, 93-107. London: Falmer.
Lubienski, S.T. 2000. "A clash of social class cultures? Students' experiences in a
discussion-intensive seventh-grade mathematics classroom." The Elementary
School Journal no. 100 (4):377-403.
Moser, Claus. 1999. Improving Literacy and Numeracy: A Fresh Start. London:
Department for Education and Employment.
Nardi, E., and S. Steward. 2003. "Is mathematics TIRED? A profile of quiet
disaffection in the secondary mathematics classroom." British Educational
Research Journal no. 29 (3):345-366.
Noyes, A. 2009. "Exploring social patterns of participation in university-entrance
level mathematics in England." Research in Mathematics Education no. 11
(2):167-183.
Nunes, T., A.D. Schliemann and D.W. Carraher. 1993. Street mathematics and school
mathematics. Cambridge: Cambridge University Press.
OECD. 2010. PISA 2009 Results: What students know and can do: Student
performance in reading, mathematics and science. Paris: OECD.
Parsons, S., and J. Bynner. 2005. Does numeracy matter more? London: National
Research and Development Centre for Adult Literacy and Numeracy.
QCA. 2007. Functional skills standards. London: Qualfications and Curriculum
Authority.
Roper, T., J. Threlfall and J. Monaghan. 2006. "Functional mathematics: What is it?"
Research in Mathematics Education no. 8 (1):89-98.
Sealey, P., and A. Noyes. 2010. "On the relevance of the mathematics curriculum to
young people." The Curriculum Journal no. 21 (3):239-253.
Wake, G. 2005. "Functional mathematics: More than “back to basics”." Nuffield
Review of 14-19 Education and Training. Aims, Learning and Curriculum
Series, Discussion Paper no. 17:1-11.
Zan, R., L. Brown, J. Evans, and M.S. Hannula. 2006. "Affect in mathematics
education: An introduction." Educational Studies in Mathematics no. 63
(2):113-121.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 60
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Investigating secondary mathematics trainee teachers’ knowledge of fractions
Paul Dickinson and Sue Hough
Manchester Metropolitan University
At Manchester Metropolitan University, approximately eighty students
each year qualify to become teachers of secondary mathematics. Of these,
roughly half do not have a mathematics degree, but have studied on a
Subject Knowledge Enhancement (SKE) course. This research study is
concerned not with the pure mathematical knowledge of such trainees, but
with the nature of their knowledge. Asking them relatively routine
questions on fractions showed almost all trainees reaching for a known
procedure to answer the questions. Furthermore, when asked how they
knew they were correct, most trainees used the procedure as the authority
for this. The trainees then studied the teaching of fractions, after which
they taught the topic in school. This paper focusses on the first part of the
study, which analyses the trainees’ own knowledge of fractions. A later
paper will report on the classroom work of the trainees.
Keywords: secondary; understanding; fractions; trainee teachers
Introduction
A 52 year-old policewoman was asked how she would work out
. She
smiled, wryly. ‘What did you say, a quarter plus a half?’ she says, writing the two
fractions with her index finger on the empty table in front of her.
‘It’s something to do with common denominator?’ she asks (gesturing a
horizontal line underneath her imaginary quarter and a half). ‘Then is it something
to do with cross multiplying’ (again gesturing this with her index finger pen on
the table).
‘Do you know the answer?’ I ask.
‘Oh yeah, it’s three-quarters.’
It would seem she knew this answer all along and yet her first preference was to
attempt to re-call a procedure which she had probably not used for over 30 years. We
have seen other evidence of this when working in classrooms, with trainee teachers,
and with adults. Why is it that so many people seem to elect to use a formal method
instead of their common sense intuitions?
Research focus
Although much has been written about trainee teachers’ subject knowledge (e.g.
Schulman 1986; Ball 1990; Brown et al. 1999; Goulding, Rowland and Barber
2002), much of it has focussed on primary teachers and/or on pedagogic subject
knowledge. With the advent of so many trainees now coming from SKE courses, it
seems pertinent to look at the nature of the subject knowledge of such trainees.
Consequently, our current research focuses on the two questions:
 To what extent is teacher trainees’ knowledge of fractions dominated by
procedural routines?
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
 What would be the issues in trainee teachers adopting a more conceptual
approach to the teaching of fractions
This paper deals specifically with the first of these, the issue of trainees’ subject
knowledge, and describes research undertaken at Manchester Metropolitan University
(MMU) in 2011.
Data Collection
We wanted an instrument that would enable us to measure the trainees’ knowledge of
fractions and in particular how procedural / conceptual this knowledge was. When
trainees are interviewed for the PGCE at MMU, they are usually asked the question,
‘work out
, followed by: ‘How do you know you are right?’ Our experiences of
this have led us to believe that for many trainees their answers are dominated by a
procedural knowledge of fractions. We chose to investigate this further by examining
the subject knowledge of 31 trainees who were part-way through a subject knowledge
enhancement (SKE) course. Using a test seemed the most appropriate method for
collecting our initial data, as it would enable us to quickly gather knowledge about the
whole cohort. We were aware of issues of validity and whether the test would actually
be able to measure what we intended it to measure (Mertler 2006). This led to us
designing a format of test which was in two parts, though the trainees were not
initially aware of this.
Initially, the trainees were asked to answer a series of questions on fractions. We
emphasised that we were not concerned with correct or incorrect answers but with
looking at the methods they had used. After this was completed, we asked the trainees
to look at each question again and say how they knew their answers were correct. We
suspected that the first responses would be dominated by procedural routines, and
hence the second question was introduced to expose other ways of thinking about
fractions. We were conscious that sometimes people feel under pressure to use a
known procedure because this is what they perceive to be the ‘correct method’. So the
second question gave each participant an opportunity to expose an alternative
strategy.
In designing the questions, we tried to ensure that they covered the range of
content knowledge normally expected in school level mathematics. We included
‘bare’ fractions questions and questions which were set in context, so that in the
analysis we might be able to see whether this had any impact on the initial methods
used. We also chose questions which had previously been used with pupils
(Dickinson and Eade 2005) and questions which had been used as part of continuing
professional development (CPD) for experienced teachers (Fosnot and Dolk 2002).
This was deliberate as it gave us an opportunity for comparison and also an
opportunity to establish the reliability of the questions.
Data Analysis
The test: strategies used to answer the questions
While questions covered all aspects of fractions, for this analysis we focus on the
three parts of the first question and the trainees’ responses to these. To work out
the trainees used a variety of approaches. Several converted
into a topheavy fraction and applied the rules for multiplying fractions, a few used the long
multiplication routine to find
, some chose to partition it into (3 lots of 14) +
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
( lot of 14) or alternatively (14 lots of 3) + (14 lots of
), some worked out
then
Fosnot and Dolk (2002) describe two categories of approach to problems of
this type: firstly the use of an algorithm or procedure and secondly the use of ‘number
sense.’ Applying ‘number sense’ involves having an awareness of a number of
strategies and making a case-by-case judgement about the best strategy to use.
Twenty out of the thirty-one trainees went for an algorithmic approach (either by
using a standard procedure to multiply fractions or by using long multiplication). Of
those using number sense (some form of partitioning) only three went for the
efficiency of doubling
and then multiplying by 7. The results would suggest a
strong leaning towards the use of algorithms / procedures. Responses to the second
question,
, were also procedural with all 31 trainees employing a version of the
strategy ‘top x top over bottom x bottom’.
The final part of this first question,
, also revealed a strong preference
for the standard procedure with 28 of the 31 trainees applying a version of ‘invert and
multiply’. In only 3 cases did the trainees appear to be using a ‘number sense’
approach by choosing to retain the division element of the question.
Use of algorithms versus ‘number sense’- some pros and cons
From the responses, trainees showed a clear preference for the use of procedures, as
perhaps was to be expected. Procedures are quick and efficient and (provided you
remember exactly what to do), can be easy to use and produce accurate answers.
There is something quite powerful about knowing, for example, that whenever one
sees a division of fractions question, all that needs to be done is to ‘invert and
multiply’.
Applying ‘number sense’ on the other hand requires having many strategies at
your disposal and deciding on which strategy to use (Fosnot and Dolk 2002). This
implies that the user needs to have a deeper understanding of number and of the
connections that exist within the world of numbers (for example, an understanding
that when multiplying two numbers together, the same result comes from halving the
first, doubling the second and then multiplying).
Where learners are able to apply number sense, then their methods have the potential
to be even quicker and more efficient than using a procedure. We saw this in the case
of
. Seeing this as
is potentially a lot quicker than converting to
fractions and multiplying. Seeing a calculation in an ‘easier numbers’ form can also
minimise the opportunity for calculation errors.
Possible reasons for the widespread use of procedures
This is the way they were taught at school
Perhaps the most influential factor is the way the trainees were taught at school
(Bramald, Hardman and Leat 1995) and whether the focus was on learning procedures
or on developing mathematical concepts. Much has been written about the tension
between these two approaches, for example Brown (1999, 3) refers to the “swings of
the pendulum” between approaches which stress the “accurate use of calculating
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
procedures” and those which favour developing “number sense”. Others (see
Thompson 1994, 1997; Sugarman 1997; Beishuizen 1999) discuss the issues relating
to the teaching of algorithms for number operations compared with the teaching of
mental strategies, informal written methods and a development of number sense.
Despite the heavy investment into research of this nature and indeed the emphasis
placed by the National Numeracy Strategy (NNS 1999) on the use of models such as
the empty number line, the overriding emphasis in UK textbooks is still to show a
procedure and then produce questions which practice that procedure (Haggarty and
Pepin 2001). Consequently, it is likely that, in whichever era the trainees were taught
their school mathematics, the goal will have been to have knowledge of the standard
written procedures. It is also possible that this may have been the only approach they
were taught.
The trainee’s awareness of non-procedural approaches to working with fractions is
limited.
One question to consider here is: ‘Were the trainees making a positive choice to use a
procedural approach over a number sense approach or did they not possess the facility
to try the questions any other way?’ Swan (2006, 16) refers to the fact that despite
spending many years practising techniques, it is possible to gain very little
“substantial understanding of the underlying concepts”. We suspect that many of the
trainees would have little conceptual (or ‘relational’ (Skemp 1976)) understanding of
why their procedures worked. The second element of the test, whereby trainees had to
say how they knew they were right was designed to expose some of the issues relating
to the type of understanding they possessed.
A discussion of the responses to the questions: How do you know you are right?
Having analysed the responses we were able to distinguish four main categories.
Categories 1 and 2 link closely to the responses one might expect from someone who
has predominantly an ‘instrumental understanding’ (Skemp 1976) of maths.
Categories 3 and 4 relate to having a ‘conceptual understanding’ of maths.
Category 1 – Uses the algorithm to justify the algorithm
This was a common occurrence across all the questions. One trainee stated ‘I know I
am right because I trust the method’; another said “I’ve always done it this way”.
Some repeated exactly the same calculations; some reversed the sum and then applied
another procedure. Several (see Figure 1) simply described the procedure they had
used as a justification for why they must be right. These trainees could be said to be
displaying ‘instrumental understanding’ as described by Skemp (1976) in that they
can apply the rule but their only justification for this rule is the rule itself.
Category 2 – Acknowledges that they don’t know why they are right
Many trainees were not able to offer any explanation as to how they knew they were
right. Unlike those in Category 1, these trainees seemed to recognise the limitations
of their algorithm as a means of giving credence to their answer. Several referred to
the fact that their method was ‘just a rule’; something they had learned at school and
never really questioned.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Figure 1: Trainee teachers use of algorithm to justify the use of an algorithm
A couple of the trainees said to us later that it was only since coming on the SKE
course that they had started to question the way in which they understood maths. This
is another indicator that for many the experience of learning maths at school may be
almost exclusively procedural.
Category 3 – Uses an alternative ‘sense making’ strategy
Having first used a procedure to answer the question, these trainees answered the
question again using a ‘number sense’ approach. Like those in category 2, the
algorithm provided them with little sense of whether they were right or not, but these
trainees looked for, and were able to find, an alternative way of looking at the
problem. Given the brevity and relative simplicity of many of these approaches, it
seems strange that more trainees did not adopt these strategies initially. It would
appear that sometimes people feel compelled to use the ‘standard method’, even when
more complicated, as this is what is believed to be ‘proper maths’.
Category 4 – Draws a picture
This strategy was rarely used despite the fact that classic early notions of fractions are
developed around pictures (Lamon 1999).Even when the question was set in the
context of “How many inches can be fitted into of an inch?”, only five out of 31
drew pictures. Two of these drew a circular diagram, showing no affinity with the
linear representation inferred by the context. When asked to find another way of
proving that
is not equal to , only 11% drew pictures and yet this is relatively
easy to see if you do draw a picture. It was clear from analysing these tests that:
1. Most trainees demonstrated a procedural (rather that conceptual) knowledge of
fractions
2. Most trainees appeared satisfied to use the authority of a procedure to justify a
procedure, although others did recognise the need to find other ways to justify
their procedures (even if they did not yet know what these were).
It is important to recognise that many of these trainees may not see a need to
teach fractions in any way other than how they learned them at school. Countering
this represents a huge challenge, as it involves changing people’s beliefs. According
to Swan (2006), central beliefs are often established young, firmly held onto, and are
incredibly difficult to change, particularly once one reaches adulthood. Working with
the trainees on these issues represents the second part of this research study.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
References
Ball, D. L. 1990. The mathematical understandings that prospective teachers bring to
teacher education. Elementary School Journal 90: 449–466.
Beishuizen, M. 1999. The empty number line as a new model. In Issues in Teaching
Numeracy in Primary Schools, ed. I. Thompson, 157-168. Suffolk: Open
University Press.
Bramald, R., F. Hardman, and D. Leat. 1995. Initial teacher trainees and their views
of teaching and learning. Teaching and Teacher Education 11(1): 23-31.
Brown, M. 1999. Swings of the Pendulum. In Issues in Teaching Numeracy in
Primary Schools, ed. I. Thompson, 3-16. Suffolk: Open University Press.
Brown, T., O. McNamara, L. Jones, and U. Hanley. 1999. Primary student teachers’
understanding of mathematics and its teaching. British Education Research
Journal 25: 299–322.
Dickinson, P. and F. Eade. 2005. Trialling Realistic Mathematics Education (RME) in
English secondary schools. Proceedings of the British Society for Research
into Learning Mathematics 25(3).
Fosnot, C.T. and M. Dolk, 2002. Young mathematicians at work: Constructing
fractions, decimals, and percents. Portsmouth: Heinemann.
Goulding, M., T. Rowland and P. Barber. 2002. Does it matter? Primary teacher
trainees' subject knowledge in mathematics. British Educational Research
Journal 28(5): 689-704
Haggarty, L. and B. Pepin, 2001. Mathematics textbooks and their use in English,
French and German classrooms: a way to understand teaching and learning
cultures. Zentralblatt für Didaktik der Mathematik: International Reviews on
Mathematical Education 33(5):158-175.
Lamon, S. J. 1999. Teaching fractions and ratios for understanding. Mahwah, NJ,
USA: Lawrence Erlbaum Associates.
Mertler, C. A. 2006. Action research: Teachers as researchers in the classroom.
London: Sage Productions.
National Numeracy Strategy. 1999. Sudbury: Department for Education
Schulman, L. S. 1986. Those who understand: Knowledge growth in teaching.
Educational Researcher 15(2): 4- 31.
Skemp, R. 1976. Relational understanding and instrumental understanding.
Mathematics Teaching, the journal of the Association of Teachers of
Mathematics 77:20-26
Sugarman, I. 1997. Teaching for strategies. In Teaching and learning early number,
ed. I. Thompson, 142-154. Buckingham: Open University Press.
Swan, M. 2006. Collaborative Learning in Mathematics. A Challenge to our beliefs
and practices. London: NRDC.
Thompson, I. 1994. Young children's idiosyncratic written algorithms for addition.
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From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 66
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Teacher noticing as a growth indicator for mathematics teacher development
Ceneida Fernándeza, Alf Colesb, Laurinda Brownb
a
University of Alicante (Spain); bUniversity of Bristol
In this paper, we report on our analysis of four transcripts of teacher
meetings that took place over the academic year 2011-12. These meetings
took place in the context of a project looking into tackling
underachievement in primary mathematics through a focus on creativity.
We bring the idea of growth indicators (Jacobs, Lamb and Philipp 2010)
within the framework of noticing (Mason 2002) in order to analyse shifts
in teacher discourse. There is evidence of growth but we conclude by
discussing the complexity of teacher change and problems with any set of
indicators.
Key words: noticing, primary school teachers, mathematics teacher
development, growth indicators.
Background
In this paper we report on our analysis of four transcripts of teacher meetings that
took place over the academic year 2011-12, in the context of a project aimed at
tackling underachievement in primary mathematics through creativity. The project is
a collaboration between the University of Bristol and the charity ‘5x5x5=creativity’
(5x5x5), it is funded over the period September 2011 to July 2013, in part by the
Rayne Foundation. For the purposes of the project, we were defining creativity within
mathematics to be indicated by students noticing patterns, asking their own questions,
making their own conjectures. In the first year, which we report on here, three
primary/infant schools in the South West region of the UK were involved. One
teacher from each of the three schools joined a project group that met five times over
the academic year. These were twilight meetings that generally lasted just over an
hour. Alf convened this group and, in between meetings, was able to visit the schools
to observe and then lead sessions with the teachers’ classes, with a focus on running
activities and class discussion in a way that allowed and supported student creativity.
Alf made on average 10 visits to each school. The focus of the meetings was on
teachers sharing the work they had been doing, including strategies for developing
creativity and for tackling underachievement. The ages of the focus classrooms were
year 2 (aged 6-7) in schools A and B and a mixed year 3-4 in school C.
Theoretical Framework
Noticing is an important skill for teachers. However, noticing effectively is
challeging. Although this skill has been conceptualized from different perspectives,
the common theme is how teachers process complex classroom events. Mason (2002)
considered noticing to be a fundamental element of expertise in teaching,
characterized by: (a) keeping and using a record, (b) developing sensitivities, (c)
recognizing choices, (d) preparing to notice at the right moment and, (e) validating
with others. On the other hand, van Es and Sherin (2002) considered that noticing
includes: (a) identifying noteworthy aspects of a classroom situation, (b) using
knowledge about the context to reason about classroom interactions, and (c) making
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
connections between specific classroom events and broader principles of teaching and
learning.
Recent studies have provided different contexts for the development of the
skill of noticing. For example, Coles (2012) proposed aspects within his role of
facilitating discussion of teaching videos. One of these aspects is moving to
interpretation: during a period of time constructing ‘accounts of’ (Mason 2002) what
was observed on a video clip, the task is to reconstruct the precise words or actions
and their chronology. In focusing on the detail of what was noticed or observed, it is
possible to then move to accounts for (interpretations of what occurred and why)
avoiding judgmental comments. Noticing is supported by having a period of time
describing the episode in all its detail and re-watching the clip when needed.
In this research, we are going to select a particular focus for noticing:
children’s mathematical thinking. In this context, Jacobs, Lamb and Philipp (2010)
conceptualize teachers’ competence in noticing as a set of three interrelated skills:
attending to children’s strategies, interpreting children’s understanding and deciding
how to respond on the basis of children’s understanding. Their findings also indicated
that this skill could be developed, providing growth indicators that can help
professional developers identify and celebrate shifts in teachers’ professional noticing
of children’s mathematical thinking Specifically (ibid, 196 (numbering added)).,
1. A shift from general strategy descriptions to descriptions that include the
mathematically important details.
2. A shift from general comments about teaching and learning to comments
specifically addressing the children’s understanding.
3. A shift from overgeneralizing children’s understandings to carefully linking
interpretations to specific details of the situation.
4. A shift from considering children only as a group to considering individual
children, both in terms of their understandings and what follow-up problems
will extend those understandings.
5. A shift from reasoning about next steps in the abstract to reasoning that
includes consideration of children’s existing understandings and anticipation
of their future strategies.
6. A shift from providing suggestions for next problems in general terms to
specific problems with careful attention to number selection.
For the purposes of this paper, we focus on the first four indicators as the last
two are linked to instructional decisions. In the meetings that we analyse, teachers are
reflecting on their work with their classes and so did not talk about ‘future strategies’
or ‘next problems’.
A teacher gives a general strategy description when he/she identifies a tool or
mentions that the problem was solved successfully but omits details of how the
problem was solved (indicator 1). If, later on, for example thinking about wholenumber operations, the same teacher comments how children counted, used tools or
drawings to represent quantities, or decomposed numbers to make them easier to
manipulate, we would see a shift into the consideration of ‘mathematically important
detail’ (indicator 1). Teachers may give general comments about teaching and
learning, such as, “I learned that it’s important to allow students to use different tools
to come up with mathematical problem solution” (Jacobs, Lamb and Philipp 2010,
186). If, afterwards, they make sense of the details of a student strategy and note how
these details reflected what the children did understand, for example recognizing the
ability to count by 2s or the ability to switch between counting by 2s and 1s we could
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
identify a shift into giving comments specifically addressing the children’s
understanding (indicator 2). A teacher overgeneralizes children’s understandings
when they go beyond the evidence provided. For instance, saying, “children
understand subtraction and addition — and which to choose when presented with a
problem…” (ibid, 186). This broad conclusion is difficult to justify on the basis of the
children’s performance on a single problem for which many may have used different
strategies. If, later on, teachers make sense of the details of a student strategy and note
how these details reflected what the children did understand in specific situations, we
would said that there is a shift into linking interpretations to specific details of the
situation (indicator 3). Finally, considering children as a group is another
characteristic of overgeneralising children’s understanding; a shift is indicated by
discussion of anything linked to individual understanding (indicator 4).
Recently, research has shown evidence of prospective teachers’ professional
noticing of children’s mathematical thinking development in relation to the
framework above. Fernández, Llinares and Valls (2012) show that participation in online debates supports this development in the specific domain of proportional
reasoning. Text produced by prospective teachers in on-line debates helped some of
the teachers attend to the mathematical elements of proportional and non-proportional
situations and link these elements with characteristics of students’ understandings. In
Fernández, Llinares, and Valls (2012) there was evidence of such shifts from general
strategy descriptions (before the participation in the on-line debate) to descriptions
that included the mathematically important details (after the participation). However,
more studies, focusing on the different contexts that could improve this skill, are
needed. Our objective in this paper is to analyze the discussions of in-service primary
school teachers who participated in the project introduced above. We were interested
to see if there was evidence of any shifts in relation to the first four indicators.
Data and analysis
In this research we are going to focus on two of the three in-service teachers: Sara and
Anna (pseudonyms). They are in-service teachers for the schools A and B,
respectively. School A is a rural primary school with high levels of mobility in the
student population. School B is an infant school in an urban area with high levels of
social deprivation. We have not considered the third teacher involved in the study or
school C, since in that school the teacher who was involved was swapped half way
through the year, so neither teacher was involved for the whole year.
The data we consider in this paper is the transcripts of the four meetings
between staff that were audiotaped. The first teacher meeting was not audio-recorded
to allow for an ethical discussion. Other data from the project that we have not
analysed includes lesson field notes and students’ work. For the analysis we three
researchers analysed individually the transcript of the first meeting looking for
evidence of the aforementioned shifts (Jacobs, Lamb and Philipp 2010). Then,
agreements and disagreements were discussed in an attempt to share the evidence for
shifts. Once we shared this evidence and came to an agreement, we applied these
filters to the rest of the meeting data.
Results
In this section, we present some evidence of the shifts in the two in-service teachers,
Sara and Anna. We begin by offering two sections of Sara talking, which were chosen
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
as they were the first comments she offered, in responding to the invitation to reflect
on the work she had been doing in her classroom.
Sara (meeting 2)
The first session, we looked at the place-value chart didn’t we before the last
meeting and I talked with Alf about some of my children used to work with
decimals. And also to consolidate multiplying and dividing by 10s, 100s and
1000s and in the first session we looked at the chart and noticed patterns and
then they had to make a journey, how could they get, going up and down the
columns and choose a number and take a journey and get back to the same
place, and so they could do a journey of two steps, they might take five
multiply by ten and divide by ten to get back, they could try and take it further.
By the end of that session some of them were getting more adventurous
because we’d shown them decimals on the other side of the chart.
Sara (meeting 3)
This is M’s from last year, he did similar kind of activities where they revisited
their work cut it out and made comments… Today they started with shapes.
It’s the investigation of how many sticks are you using. So he started to
comment about what he noticed and how he felt. So looking at what the
answers were and just showing he found it quite there and he found it easy but
he’s got all this other work about patterns.
We observe that there is a shift across these meetings from considering
children only as a group in meeting 2, to considering individual children in meeting 3
(indicator 4). Some evidence of this shift is when she says, in meeting 2, “they could
do a journey”, “some of them were getting more adventurous”. Later on, in meeting
3, she considers individual children, for example, she talks about the work of “M”.
We also see a shift from general comments about teaching and learning to comments
specifically addressing the children’s understanding (indicator 2). In meeting 2, Sara
says “we looked at the place-value chart… to consolidate multiplying and dividing by
10s, 100s and 1000s” “they could do a journey of two steps” (general teaching and
learning comments). In meeting 3, she says “he started to comment about he noticed
and how he felt…. He’s got all this other work about patterns” (she has addressed the
child’s understanding). In contrast, we do not see a shift in indicator 4 (from
overgeneralizing children’s understandings to carefully linking interpretations to
specific details of the situation). An example of generalizing children’s
understandings in meeting 3 is when she says, “So looking at what the answers were
and just showing he found it quite [ ] there and he found it easy but he’s got all this
other work about patterns”. Although she has addressed the child’s understanding,
she goes beyond the evidence provided: “he found it easy but he’s got all this other
work about patterns” (what Jacobs, Lamb and Philipp (2010) called limited evidence
of interpretation of children’s understanding). We see, across the two transcripts,
general strategy descriptions without mathematically important details (indicator 1).
In the next transcripts, focusing on Anna in meetings 2 and 3, we observe that
she talks on both occasions about individual children (indicator 4), this was a general
pattern across all meetings. In meeting 2 she talks about “M’s” progress and in
meeting 3, she continues talking about this child, we have selected these excepts
below for analysis, to see what has changed in how she talks about the same child.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Anna (meeting 2)
So, we’ve got this boy who actually I don’t know if you remember M on the
first session and he sat one of the first times when you came in when he copied
and he sat next to A who records really neatly. He didn’t know what was going
on but he copied how she recorded as in one number in each box. So, I was
he’s copied, he hasn’t done anything. But actually from that he’s recording his
own and recording in that way which is really nice. So here it was, they could
each choose, they chose their own number and practicing how many different
ways they could make that number using the Cuisenaire, so he picked up the
yellow. So we worked out what number that was and it was ‘five’. So, then he
started building his five wall and recording it and for him this is amazing. So,
he is knowing that it all equals five. He is beginning to see well he’s adding
them together even though it’s not in the 1 plus 2 plus 3.
Anna (meeting 3)
And then M. He tried this with Cuisenaire and realized he couldn’t really work
it out so he moved onto a hundred square when he was doing his finding out
about the five times table and so then spotted the pattern that he is going and
circling on the hundred square, so he could just carry it on. And that was the
first step in January of him being able to notice a pattern that he could then
use.
Anna has given comments addressing the children’s understanding, and does
not give general comments about teaching and learning (indicator 2). For example, in
meeting 2, she says “he picked up the yellow. So we worked out what number that
was and it was ‘five’. So, then he started building his five wall and recording it…he’s
adding them together even though it’s not in the 1 plus 2 plus 3”. And in meeting 3,
she says, “he tried this with Cuisenaire and realized he couldn’t really work it out so
he moved onto a hundred square when he was doing his finding out about the five
times table and so then spotted the pattern that he is going and circling on the
hundred square”. However we can observe a shift from overgeneralizing children’s
understanding in meeting 2 to linking interpretation to specific details of the situation
in meeting 3 (indicator 3). The evidence is that in meeting 2 she says “So, he is
knowing that it all equals five. He is beginning to see well he’s adding them together
even though it’s not in the 1 plus 2 plus 3”. Although there is attention paid here to
the children’s understanding, we read an overgeneralisation in the comment “he is
beginning to see well he’s adding”, which we do not read as something it is possible
to observe directly. In meeting 3, she says “And that was the first step in January of
him being able to notice a pattern that he could then use”. Here, in contrast to
meeting 2, the comment is a careful interpretation of specific details – M has noticed a
pattern that he was able to continue and this was the first time he had done this during
the year. In these two contrasting comments we see evidence of Anna considering
mathematically important details in both (indicator 1) although perhaps, as ever, there
are more mathematical issues that could be raised.
Discussion
At the BSRLM session in Cambridge we valued highly the comments we received
from participants at our session, where we asked people to use the framework of
growth indicators to analyse the transcripts above. The analysis above has been
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
informed by the discussion. However, in offering these transcripts we also wanted to
raise questions about the growth indicators themselves and this also came out of the
discussion at BSRLM. One participant pointed to a phrase of Anna’s from the second
meeting, that was not particularly relevant to the growth indicators, but which he felt
was strong evidence for change. Anna says, “So I was he’s copied, he hasn’t done
anything. But actually from that he’s recording his own and recording in that way
which is really nice.” In these comments, Anna is demonstrating an awareness of her
own learning. She is noticing that her ideas altered about the value of this student
copying a recording method from another student. This kind of noticing is not part of
the framework of growth indicators and yet, for the participant in the session, is a key
feature of teacher growth.
In the session we also discussed some underlying assumptions behind the
whole notion of ‘growth’. The word perhaps carries implications of a linear or unidirectional movement or some kind of ideal endpoint. In contrast, we bring to mind a
phrase of a 5x5x5 artist, Catherine Lamont who, when talking about positive changes
in some students in the context of her own work, stopped herself and commented:
“it’s not even a move forward, it’s a move.” In the transcripts of Sara and Anna,
above, we also see evidence of ‘moves’ without necessarily wanting to invoke a
direction or value judgment.
Acknowledgments
The research reported here has been funded by the Rayne Foundation and the University of Bristol.
Ceneida’s time has been financed in part by the Universidad de Alicante (Spain) under birth project
nºGRE10-10 and in part by the grant from Conselleria d’Educació, Formació i Ocupació de la
Generalitat Valenciana (BEST/2012/293).
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interpretations of classroom interactions. Journal of Technology and Teacher
Education 10: 571-596.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Teacher-student dialogue during one-to-one interactions in a post-16
mathematics classroom
Clarissa Grandi
Thurston Community College/University of Cambridge
Recent developments in mathematics education place an unprecedented
emphasis on the role of discourse in developing students’ conceptual
understanding, with a corresponding de-emphasis on the use of ‘telling’:
the stating of facts and demonstration of procedures. This action research
study investigated teacher-student dialogue during one-to-one interactions
in my post-16 mathematics classroom. The participants were four A-level
students. Data sources included clinical interviews, student feedback
interviews and an analytical log; and the data were coded using a
framework of scaffolding categories drawn largely from current research
literature. The findings suggest that, although I utilised more ‘telling’ than
‘questioning’ interventions, often these ‘telling’ actions served useful and
necessary functions. They also indicate that my scaffolding skills
developed as a result of the process of critical analysis; and that the
scaffolding strategies valued by my students were those that they felt best
promoted their independence. The study concludes by suggesting that
context is a crucially important factor in addressing the dilemma of
whether or not to tell.
Keywords: post-16 mathematics classroom, ‘dilemma of telling’, teacherstudent dialogue, scaffolding strategies.
Introduction
Current reforms in mathematics education, influenced by a social constructivist view
of learning, place dialogue at the heart of the development of conceptual
understanding and mathematical thinking skills. Teachers are now seen as ‘facilitators
of learning’ (Smith 1996; Lobato, Clarke and Ellis 2005) who manage discussion
within a student’s ZPD by employing suitable scaffolding and fading techniques
(Wood, Bruner and Ross 1976; Vygotsky 1978). Underlying these ideas is a strong
criticism of transmissive teaching styles, often referred to as ‘teaching by telling’.
However, there is very little in terms of specific guidance for teachers about how best
to achieve these reform aims (Chazan and Ball 1995; Smith 1996; Baxter and
Williams 2010). This has led to what Baxter and Williams describe as the “dilemma
of telling: how to facilitate students coming to certain understandings without directly
telling them what they need to know or do” (8). This has been a recurring dilemma in
my own practice at an English 13 – 18 comprehensive school.
Research Literature
Kyriacou and Issitt (2007) note that research on teacher-student dialogue in this
country is scant, especially so at the local level of one-to-one interaction. What
research there is into whole-class teaching generally reveals a prevalence of
transmissive ‘teaching by telling’, and little evidence of effective scaffolding that
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
might effect a handover to independence (Myhill and Warren 2005; Kyriacou and
Issitt 2007). Reasons proposed for the prevalence of the transmission model include
acknowledgement that scaffolding can be a difficult and uncomfortable task, carried
out in a pressured environment; and that teachers’ beliefs about the nature of
mathematics, as well as their own schooling, can affect their competence at
scaffolding (Schoenfeld 1992; Myhill and Warren 2005). When effective scaffolding
was observed, teachers were seen to hold back from telling, instead eliciting student
thinking through the use of probing questions, along with carefully tailored questions
and prompts that provided just enough guidance for breakthrough (Tanner and Jones
2000; Goos 2004; Cheeseman 2009; Ferguson and McDonough 2010).
But is achieving effective teacher-student dialogue in mathematics teaching as
simple as striving to eliminate an ingrained habit of telling? Chazan and Ball (1995)
propose that a blanket exhortation to avoid telling is inadequate because it ignores the
importance of context. Lobato, Clarke and Ellis (2005) point out that many kinds of
telling perform useful functions in the development of conceptual understanding, and
can thus be reconciled with a constructivist viewpoint. These two sets of researchers,
along with Baxter and Williams (2010), suggest that it is important to gain further
understanding of the function of teacher actions through analysis of the intentions
behind their scaffolding decisions.
Research Questions
Having decided that the aim of my research was to improve the quality of the teacherpupil dialogue in my A-level classroom through a process of critical reflection, I
formalised the following research questions:
RQ1: What does a critical analysis of the form and function of my utterances reveal
about the nature of my scaffolding strategies?
RQ2: Can the form and function of my scaffolding interventions be changed as a
result of investigation on my part?
RQ3: What does student feedback reveal about what students valued about the
scaffolding strategies I employed?
Research Design and Participants
My formalised RQs, with their emphasis on reflective action, arose out of an
interpretivist viewpoint and led quite naturally to the use of an action research
methodology. The small scale of my study and the time constraints placed upon it,
restricted the number of action research cycles to two. After outlining my research
aims to the 12 students in my Year 13 core maths group, six male students
volunteered to take part. As a small sample was sufficient for the introspective, indepth nature of my study, I used purposive sampling to select four participants.
Data Collection Tools
Clinical Interviews
In order to answer RQ1 and RQ2 I decided that audio recording would provide the
clearest data set. I also decided that it might be best to record myself interacting with
a single student in a one-to-one situation outside of the bustle of the classroom – ‘in
vitro’ rather than ‘in vivo’. I therefore opted to use a clinical, task-based interview, of
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
the type closely associated with Piaget’s work, in which the interviewer’s responses
are contingent on the subject’s reactions to the task (Rowland 2000). This means of
eliciting student thinking by contingent prompting and probing is a similar discourse
model to that involved in the type of local level on-the-fly scaffolding (van Lier 1996;
Brush and Saye 2002) that I wished to develop in my own practice, and therefore
seemed to provide a rich means of analysing my performance. Interviews took place
during those lessons when an adjoining classroom was vacant for interview use, with
both classroom doors remaining open. In order to maintain further links with a
familiar setting, I used questions from the A-level textbook, selecting two for each
cycle of intervention: questions that were sufficiently challenging for the participants
to require assistance. Transcription, including paralinguistic messages (pauses,
interruptions and heavily stressed words), was carried out promptly to minimise data
loss.
Student Feedback Interviews
In order to answer RQ3, participants were interviewed immediately after their clinical
interview, using the same recording method. The following open questions were
devised to enable the participant to reply without restriction, and to allow me to probe
more deeply or clear up misunderstandings if these arose:
Q1
Did you find any aspect of the teacher input helpful?
Q2
Was there anything that wasn’t helpful?
Q3
Is there anything that might have been more helpful for me to do?
Q4
Is there anything you would like to add?
Analytical Log
In order to carry out the process of critical reflection inherent in the two action
research cycles, I used an analytical log in which to record my evaluation of the
clinical interviews. I also recorded the thoughts, feelings and insights that arose
during the process of analysing the interview transcripts. As a result, the log had a
narrative quality more characteristic of a journal of reflection. In this way I hoped to
bring my own subjectivity to bear on the analytical process, and to unearth the
intentionality behind my utterances.
Data Analysis
On one level I wanted to identify the form of my dialogic interventions. I therefore
colour-coded the text of the transcript using green font for questioning and red font
for telling. However, following Lobato et al (2005), I also wanted to identify the
function of my utterances. I began with 6 categories borrowed from Anghileri (2006),
but it soon became clear that my coding framework needed to be more fluid, and I
ended with a total of 12 categories, a mixture of predetermined and emergent codes:
Checking, Confirming, Convention, Demonstrating, Directing, Explaining, Focusing,
Funnelling, Parallel modelling, Probing, Prompting and Rephrasing. Each category
was colour-coded, and the transcript was then colour-highlighted accordingly.
Transcripts from the student feedback interviews were coded in the first
instance according to the participant’s perception of the ‘helpfulness’ or otherwise of
a particular scaffolding intervention. The above function codes were then applied.
The analytical log was coded according to whether I had criticised or approved
each scaffolding intervention, and both form and function codes were applied.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Findings
RQ1: What did a critical analysis of the form and function of my utterances reveal
to me about the nature of my scaffolding strategies?
Analysis of the form of my scaffolding interactions in the first cycle suggested that I
overwhelmingly relied on telling (113 out of 170, with the remaining 57 coded as
questions). However, analysis of the function of those interactions revealed that a
large proportion of the telling actions were simple confirmations of the rightness or
wrongness of student ideas. With confirming excluded, the most common telling
categories were explaining conceptual content; demonstrating a procedure; directing
by providing instructions, advice or suggestions; and outlining a convention.
Analysis of the ‘critical’ content of my analytical log revealed that I was
dissatisfied with instances where I employed telling to demonstrate, direct, explain or
funnel, and where I used questioning to funnel. In cases where a student was unable to
recall a procedure, I felt that parallel modelling would have been a more useful
strategy than demonstrating using the question itself. In the cases where I was critical
of my explaining interventions, I felt that it would have been more beneficial to have
assisted the student with probing and prompting guidance. I also noted that there was
a controlling element to my directing, sometimes due to lack of confidence. With
regard to the funnelling instances, I reflected that I seemed to be hurrying the student
towards the answer instead of allowing him more time to respond to my questioning.
Analysis of the ‘approving’ content of my analytical log revealed that I was
more satisfied with instances where I employed telling to confirm, discuss convention,
and parallel model, and when I used questioning to probe and prompt. I felt that
confirming was a necessary part of my scaffolding strategy. I also felt that ‘telling to
share a convention’ was the only way to impart arbitrary mathematical knowledge,
and hence was a necessary intervention. I approved of one instance in which I
directed the student on how to set out his work, as I felt this also involved the sharing
of a conventional norm. I noted that probing questions revealed student thinking, and,
in the case of one individual, elicited his longest responses. And finally I reflected that
prompting questions enabled the student to work through problems more
independently, whilst also allowing for the possibility of internalisation for future
independent use.
RQ2: Can the form and function of my scaffolding interventions be changed as a
result of investigation on my part?
Analysis of the form of my scaffolding utterances in the second cycle of clinical
interviews revealed that I used a greater proportion of questioning interventions than
pre-investigation (telling accounted for 79 out of 134 coded utterances, with the
remaining 55 coded as questions). There were some notable changes in the function of
my scaffolding interventions that may have resulted from my investigation. I
demonstrated and explained a good deal less, having been critical of my use of those
interventions previously. I parallel modelled more often, and also probed more often
and more directly. The final observed change was that I was now utilising indirect
prompts – a form of fading – which I had not done in the first cycle of clinical
interviews.
Analysis of the critical content of my second cycle analytical log revealed that
I was dissatisfied with instances where I used questioning to focus, funnel, probe and
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
prompt. A common theme underlay these criticisms: the observation that I was not
giving the students sufficient time to think. Finally I noted that my lack of confidence
with using an untried method had caused me to intervene and change the way the
student was approaching a particular question.
Analysis of the approving content of my analytical log revealed that I was
satisfied with many more of my scaffolding interventions that I had been previously:
specifically instances where I employed telling to discuss convention, direct (when
procedural content was involved), focus, parallel model and probe; and where I used
questioning to focus, parallel model, probe and prompt. I was also pleased with my
use of indirect fading prompts.
RQ3: What did student feedback reveal about what students valued about the
scaffolding strategies I employed?
Analysis of the feedback interview responses from the first cycle of clinical
interviews revealed that discussing a conventional norm, explaining and prompting
were valued strategies. And interestingly, one student made the suggestion that
parallel modelling would have helped him more – exactly mirroring the conclusion I
had reached myself.
Analysis of student responses from the second cycle of clinical interviews
revealed that prompting, parallel modelling and confirming were valued scaffolding
strategies. One student also suggested that more use of demonstrating would have
helped him, specifically the use of diagrams to enable him to visualise the situation
more easily.
Conclusion
What has emerged from the analysis of my utterances is that the situation is far more
complex than the widespread notion, cited in Baxter and Williams, that “teachers
should not lecture, demonstrate or ‘tell’” (2010, 8). My findings are consistent with
Chazan and Ball’s (1995) argument that context is all-important, and is a crucial
consideration in the management of the dilemma of telling. This discovery, coupled
with the realisation that I had, indeed, been able to develop my scaffolding skills – to
tell more selectively and question more skilfully – has made me a more confident
practitioner; and my coding framework continues to serve as a useful reflective tool.
Such is the paucity of research into teacher-student interactions (Kyriacou and
Issitt 2007), particularly at secondary level, that there is abundant scope for teacherresearchers to undertake studies into ‘on the fly’ teacher-student interactions in their
classrooms. In this regard, the coding framework I have devised may prove a useful
tool to others wishing to examine and develop their scaffolding strategies. The impact
of classroom pressures on scaffolding strategies – something that policy makers often
seem to overlook – is a further topic that may be of interest to the teacher-researcher.
References
Anghileri, J. 2006. Scaffolding practices that enhance mathematics learning. Journal
of Mathematics Teacher Education 9: 33-52.
Baxter, J.A., and S. Williams. 2010. Social and analytic scaffolding in middle school
mathematics: managing the dilemma of telling. Journal of Mathematics
Teacher Education 13: 7-26.
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Brush, T., and J. Saye. 2002. A summary of research exploring hard and soft
scaffolding for teachers and students using a multimedia supported learning
environment. Journal of Interactive Online Learning 1 (2): 1-12.
Chazan, D., and D. Ball. 1995. Beyond exhortations not to tell: The teacher’s role in
discussion-intensive mathematics classes. NCRTL Craft Paper 95 (2): 1-26.
Cheeseman, J. 2009. Challenging mathematical conversations. In Crossing divides:
Proceedings of the 32nd annual conference of the Mathematics Education
Research Group of Australasia, ed. R. Hunter, B. Bicknell, and T. Burgess,
Vol. 1. Palmerston North, NZ: MERGA.
Ferguson, S. and A. McDonough. 2010. The impact of two teachers’ use of specific
scaffolding practices on low-attaining upper primary students. In Shaping the
future of mathematics education: Proceedings of the 33rd annual conference
of the Mathematics Education Research Group of Australasia, ed. L. Sparrow,
B. Kissane, and C. Hurst. Fremantle, Western Australia: John Curtin College
of the Arts.
Goos, M. 2004. Learning mathematics in a classroom community of inquiry. Journal
of Research in Mathematics Education 35 (4): 258-291.
Kyriacou, C. and J. Issitt. 2007. Teacher-pupil dialogue in mathematics lessons.
Proceedings of the British Society for Research into Learning Mathematics 27
(3): 61-65.
Lobato, J., D. Clarke, and A.B. Ellis. 2005. Initiating and eliciting in teaching: A
reformulation of telling. Journal for Research in Mathematics Education 36
(2): 101-136.
Myhill, D. and P. Warren. 2005. Scaffolds or straitjackets? Critical moments in
classroom discourse. Educational Review 57 (1): 55-69.
Rowland, T. 2000. The Pragmatics of Mathematics Education: Vagueness in
Mathematical Discourse. London: Falmer Press.
Schoenfeld, A.H. 1992. Learning to think mathematically: Problem solving,
metacognition, and sense-making in mathematics. In Handbook for Research
on Mathematics Teaching and Learning, ed. D. Grouws, 334-370. New York:
MacMillan.
Smith, J.P. 1996. Efficacy and teaching mathematics by telling: A Challenge for
Reform. Journal for Research in Mathematics Education 27 (4): 387-402.
Tanner, H., and S. Jones. 2000. Scaffolding for success: reflective discourse and the
effective teaching of mathematical thinking skills. In Research in
Mathematics Education Volume. 2, ed. T. Rowland and C. Morgan, 19-32.
London: British Society for Research into Learning Mathematics.
van Lier, L. 1996. Interaction in the language curriculum: Awareness, autonomy and
authenticity. Harlow: Longman.
Vygotsky, L. S. 1978. Mind in society: The development of higher psychological
processes. Cambridge, Massachusetts: Harvard University Press.
Wood, D., J.S. Bruner, and G. Ross. 1976. The role of tutoring in problem solving.
Journal of Child Psychology and Psychiatry 17: 89-100.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Using scenes of dialogue about mathematics with adult numeracy learners –
what it might tell us.
Graham Griffiths
Institute of Education, University of London
The study concerns the use of prepared dialogue scenes involving
mathematics with groups of adult learners. It is intended to consider how
we might characterise discussion following the reading of scenes of
dialogue. The article outlines some examples of scenes and the response
from the use of these in an early exploratory phase with some adult
learners intending to become teaching assistants. A discussion of the
scenes and responses leads to some conclusion about the characteristics of
more appropriate scenes for the main study.
Keywords: adult, numeracy, dialogue, intervention, discussion
Introduction
The study concerns the use of prepared dialogue scenes involving mathematics with
groups of adult learners. It is intended to answer the following question:
How might we characterise the discussion following reading of scenes of
dialogue?
The idea for this work came from two broad areas, one of which concerns
learner-learner interactions and the other concerns the use of participants in
verbalising the words of others.
A few years ago, I was involved in a project in which discussion of
mathematical concepts by learners was a key part of the learning intervention. What
interested me was that learner-learner interactions were at times rather minimal. The
reports from the sessions contained very few records of learner-learner interactions. A
search around learner interactions in the literature produced more teacher-learner
interactions than learner-learner. Indeed most of these were concerned with school
learners rather than adults, the area which was of most interest to me.
A second influence for me began in an observation that I made when
attending a research seminar. I had noted that in one session participants were asked
to read out the parts of dialogue that were collected in the course of the research. I
noted that this appeared to be an effective way of presenting the information. The
most obvious aspect of this was that a change in voice appeared to produce a positive
difference in delivery with participants actively engaged rather than passive
observers. From this, I started to use this approach in my teacher training by asking
participants to read the dialogue (and at times narrator aspects) when
investigating various literature. In particular, this appeared to work effectively when
looking at the work of Jean Lave and others (Lave, Murtaugh and de la Rocha 1984)
with adults in the supermarket and with the dialogue scenes written by Lakatos (1976)
in Proofs and Refutations (more below).
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
This study is an investigation into the use of such dialogue scenes as a learning
activity within the classroom. Some proposed scenes have been collected (and
constructed) and used in an exploratory stage with adult learners.
Who are the participants?
It is perhaps useful to spend some time exploring the context of this activity and the
participants before considering the scenes and their use.
This work will be undertaken with adult numeracy learners. Each of these
three words ‘adult’, ‘numeracy’ and ‘learners’ deserves some consideration. The
meaning of ‘adult’ is not necessarily straight forward. There are a number of points
that could count as the beginning of adulthood: ages 16, 17, 18, 21 and 25 all have
cultural or legal significance. Those writers such as Knowles (1990), who are
interested in adult learning, and the notion of androgogy, have argued that the context
of people’s lives play a significant role in learning. It is fairly clear that all the ages
noted above are likely to contain individuals with a range of life stories and histories.
The study proposed will run in a further education college with learners who have
self-selected to join programmes and who could be any age from 16 upwards although
it is most likely that the vast majority will be in their 20s. These are individuals who
have not had entirely successful experiences with education in the past but who now
are looking for a second chance (see Swain et al. 2005).
Next comes the word ‘numeracy’. A contentious word with a range of
meanings and connotations and used to contrast with the word ‘mathematics’. In the
primary National Numeracy Strategy (Department for Education and Employment
(DfEE) 1999) it was argued that the subject of study was mathematics and that there
was an intent to develop ‘numerate’ students. In the world of adult education,
‘numeracy’ has been used to connote the relationship that mathematics has with the
context in which it works. In the proposed study, the word will be a description of the
learners. That is, the learners that have chosen courses that come under the funding
streams for ‘adult numeracy’. The subject under study may be called mathematics or
numeracy and, while the relationship between the learners and the subject will be
important, as will the words that they use, in the proposed research the term
‘mathematics’ will be used for the subject of study and ‘numeracy’ for the learner.
And now the third of those words - used quite a lot in the preceding
paragraphs - ‘learners’. It has become the norm to use this word in the postcompulsory sector. This has been introduced to enable an overarching term for all
those involved in learning in the sector, which may include workplace learning where
individuals feel that the term ‘student’ is not a good description. Nevertheless, the
participants in this study will be adult learners in a further education college and
therefore might equally be described as students.
Examples of dialogue scenes
The following are examples of the type of dialogue that might be used. These scenes
have been trialled in an exploratory stage of the work with a small group of three
volunteer adult numeracy learners. These learners were studying adult numeracy in
order to qualify as school teaching assistants The scenes were read and discussed in
one separate session rather than being embedded in normal classroom activity and,
thus, may only be indicative of their intended use.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
A scene about reverse percentages
This scene was developed as part of a CPD package for teachers along with associated
training resources (Swan 2005). The scene is used here to illustrate what may be
achieved although the content is not the most appropriate to the target group of this
study. The discussion of such ‘reverse percentage’ problems will prove difficult to
most participant learners. The discussion in the scene follows information stating that
fares had increased by 20% in one month then later reduced by 20%, and a character
proposing that the fares are “back to what they were” before the increase.
Harriet ; that’s wrong, because … they went up by 20%, say you had £100 that’s
5 , no 10.
Andy ; yes, £10 so its 90 quid, no 20% so that’s £80. 20% of 100 is 80, … no, 20.
Harriet : five twenties are in a hundred.
Dan; say the fare was 100 and it went up by 20%, that’s 120.
Sara: then it went back down, so that’s the same.
Harriet : no, because 20% of £120 is more than 20% of £100. It will go down by
more so it will be less. Are you with me?
Andy: Would it go down by more?
Harriet; Yes because 20% of 120 is more than 20% of 100.
Andy: What is 20% of 120?
Dan: 96…
Harriet: It will go down more so it will be less than 100.
Dan: it will go down to 96.
(Swan 2005: 28)
The scene is useful here as it contains some clear mathematical ideas, namely
the calculation of percentages, concerns the discussion of the solution to a problem
involving the mathematical idea and uses a range of formal and informal language.
A scene from Season 1 Episode 8 of The Wire
The next scene shows how a child (Sarah) is having difficulties with calculations of
her school homework, and finds it easier to understand a related problem in the
context of drug sales. Sarah discusses her difficulties with Wallace, a drug dealer who
is looking after her and asks for help with a text book problem.
W: This one here? A bus travelling on Central Avenue begins its route by picking
up 8 passengers, at the next stop it picks up 4 more and an additional 2 at the 3 rd
stop while discharging 1. The next to last stop, 3 passengers get off the bus while
another 2 get on. How many passengers are still on the bus when the last stop is
reached? …. Just do it in your head. [tosses book away]
After a discussion with a third character about a deal the scene returns to Sarah’s problem. .
S: Eight?
W: Damn, Sara. Look. You work in the ground stash, you got twenty tall pinks,
two picks come out for you and ask for two each, another one cops 3, then Bodie
hands you up 10 more, but some white guy rolls up in a car, waves you down a
piece for 8. How many vials you got left?
S: [thinks for a bit] fi’teen
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W: how … you able to keep The Count but you not able to do the book problem
then.
S: The Count be wrong, they [cut] you up.
(Simon and Burns 2002)
This interchange – while fictional – can be seen as illustrative of what is
sometimes called the ‘transfer problem’ (Evans 1999). Following the work of those
investigating ‘real world’ mathematics such as Nunes, Carraher and Schliemann
(1993) and Lave, Murtaugh and de la Rocha (1984) it has been noted that moving
between contexts is either difficult or impossible. This scene involves some
mathematics – addition and subtraction of two digit whole numbers - at relatively low
levels of the curriculum combined with a great deal of informal language.
A scene discussing the point of mathematical study
This is an example of a self-produced scene developed to raise discussion about the
point of studying mathematics.
Teacher : mathematics helps us to understand how to build bridges, send
submarines to the bottom of the ocean and rockets to the moon.
Jo(e) : didn’t the millennium bridge have to be closed down because they hadn’t
worked out that it would wobble?
Toni/y : and didn’t NASA mess up with metres and yards and lost a satellite.
Sam: and I’m not going to build bridges or send people to the moon anyway.
Teacher : aren’t there other subjects that you do where you might not use it
straight away.
Alex : yeah, I think this is interesting
Many teachers will recognise the questions raised by learners about the
purpose of learning mathematics. This scene may help to raise this issue with learners
and involves mostly everyday language although it does not contain any mathematical
calculations.
Some analysis of the scenes
I take a social constructivist view of learning and situate the study within those that
interpret discourse. I am interested in the interactions that follow the reading of given
scenes of dialogue that involve mathematics. Sfard (2008) proposes a structure that
sees mathematical discourse through four properties: (1) word use, (2) visual
mediators, (3) narrative and (4) routines. Other notions such as Engeström’s model for
activity theory (e.g. Engeström 1999) offer ways of interpreting language use in
relation to the backgrounds and experience of the participants concerned.
To illustrate some issues in the choice of scenes I will outline some examples
that occurred within the exploratory study of word use within the scenes. This
exploratory study allowed some consideration of the scenes’ appropriateness for the
main study. In particular, I am interested in the ways in which the learners use
language to discuss the scenes. For example the extent to which they repeat the
language used within the scenes in their discussions is noted.
The following extract comes from the discussion by learners and researcher of
the CPD percentage scene. Words and phrases of interest have been italicised.
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Learner 1: It’s telling you that it is going up 20% and then coming back down
again.
Researcher: So does that take it back to the same level, or different?
Learner 1: Um, near enough the same level?
Researcher: what do you think?
Learner 2: I think that it is almost the same level but the argument of Harriet …
£120 20% and £100 20% will be different.
Learner 1 uses some of the more everyday language from the scene, when
discussing the scene ‘going up’ and ‘coming back down’. This learner also adopts the
language of the researcher in response to the ‘level’ which is not included in the
scene. Learner 2 directly uses the text to answer the question although the learner is
still hedging about the resolution to the problem.
The following interchange relates to the use of the drug scene.
Learner 1 - The child is used to the second calculation … it’s in its everyday life.
The first bit probably doesn’t happen very often. But the second part is probably
like us going to the shops and buying bread every day.
Researcher - a teacher the other day … said that you can’t talk about drugs with a
class
Learner 2 – no not really
Learner 3 – you’re dealing with adults, you can talk about anything with adults
This discussion shows a possible problem in using this scene in the study. The
learners have shown that they draw some meaning from the text but there appears to
be less opportunity for developing discussion from a mathematical angle. This may be
because the mathematical concepts involved – addition and subtraction of integers –
are well understood by these learners. It is possible that for other learners the ‘word
problem’ aspect may produce some interesting discussion.
The following quotation is from one learner responding to the third scene.
It will not come easily in our minds that constructing a bridge needs mathematics
… to build an effective and sound quality bridge that will last for a number of
years.
And it will be building bridges between your mental ability as well, … yes some
people believe maths is difficult … and if they think maths is difficult I want to
build a bridge where they can have fun and at the same time learn real maths.
The learner has taken language from the text – ‘the bridge’ – and used it as a
metaphor for her own views. This use of language here does exemplify the type of
discussion that was intended during the intervention. The difficulty is that while this is
an important discussion about the appreciation of mathematics the scene does not
provide an opportunity for much discussion about particular mathematical concepts.
Overall, from the use of these three scenes, some criteria for scene selection is
emerging. Scenes should involve:
(a) a discussion of a mathematical problem;
(b) an appropriate level of mathematical content; and
(c) a range of everyday and technical vocabulary.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Conclusion
The use of such dialogue scenes has potential as a learning intervention. If we see the
use of language as a key to learning then the recording of learner-learner interchanges
about the scenes should bring insight into learning for the education community.
The three scenes discussed all show that there is a potential for the
development of mathematical thinking through the discussion of such scenes. The
difficulty is in the selection and/or construction of scenes that address appropriate
mathematical concepts while allowing for an appreciation of mathematics at the same
time. Nevertheless, criteria are emerging that will help the choice and construction of
appropriate scenes.
References
DfEE. 1999. The national numeracy strategy. London: DfEE.
Engeström, Y. 1999. Innovative learning in work teams: Analysing cycles of
knowledge creation in practice. In Perspectives on Activity theory, ed. Y.
Engeström R. Miettinen and R. Punamäki, 377-406. Cambridge: Cambridge
University Press.
Evans, J. 1999. Building bridges: Reflections on the problem of transfer of learning in
mathematics. Educational Studies in Mathematics 39, 23-44.
Knowles, M. S. 1990. The adult learner: A neglected species. Houston: Gulf
Publishing Company.
Lakatos, I. 1976. Proofs and refutations. Cambridge: Cambridge University Press.
Lave, J., M. Murtaugh, and O. de la Rocha. 1984. The dialectic of arithmetic in
grocery shopping. In Everyday Cognition: Its Development in Social Context,
eds. B. Rogof, and J. Lave, 67-94. London: Harvard University Press:.
Nunes T., D. W. Carraher and A. D. Schliemann. 1993. Street mathematics and
school mathematics. Cambridge: Cambridge University Press.
Sfard, A. 2008. Thinking as communicating. Cambridge: Cambridge University Press.
Simon, D., and E. Burns. 2002. The Wire Season 1 Episode 8 Lessons. Unpublished
script.
Swain, J., E. Baker, D. Holder, B. Newmarch, and D. Coben. 2005. ‘Beyond the daily
application’: making numeracy teaching meaningful to adult learners,
London: NRDC
Swan, M. 2005. Improving learning in mathematics: challenges and strategies.
London : DfES Standards Unit (available online at
https://www.ncetm.org.uk/public/files/224/improving_learning_in_mathemati
cs.pdf accessed 03.02.13)
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Professional development in mathematics teacher education
Guðný Helga Gunnarsdóttir, Jónína Vala Kristinsdóttir and Guðbjörg Pálsdóttir
University of Iceland – School of Education
Icelandic student teachers’ professional development starts at the onset of
their initial teacher education. We have studied our teaching as teacher
educators with a focus on the development of learning communities and
reflective practices that are considered important elements of effective
professional development. Our studies have given us some guidelines to
work with and strengthened our beliefs on the importance of collaboration
and discussions.
Keywords: teacher education; learning-community; professional
development
Introduction
Professional development is an important part of teacher education for both student
teachers and teacher educators. Professional development is a life long process. In our
teacher educational program the student teachers are supposed to develop their
professional theory from the onset of their studies and are introduced to various ways
to develop professionally. Teacher education for compulsory schools (6–16 year old
students) in Iceland has in recent years been undergoing radical changes. It has
changed from being a three-year bachelor program to a more research based five-year
master degree (300 credits). Student teachers who want to become mathematics
teachers take a 120-credit specialisation in mathematics and mathematics education
both at bachelors and masters level.
The authors of this paper have taught different mathematics education courses
for more than 20 years and have taken part in developing the teacher education
program in cooperation with colleagues. During this period of change we have studied
our teaching as teacher educators with focus on the development of learning
communities and reflective practices (Guðjónsdóttir and Kristinsdóttir 2011;
Gunnarsdóttir and Pálsdóttir 2011). In our mathematics education courses students
have been introduced to various ways to collaborate and develop professionally. They
have used lesson study, collaborative lesson planning and co-teaching. They have also
worked on group assignments on important issues in mathematics teaching and
learning as well as assignments that challenge them to develop their own professional
perspective and identity. Our aim is also to introduce professional learning strategies
to our students that they can use when they enter the teaching profession. In this paper
we will report on our ongoing research on our teaching and development of the
mathematics teacher education program.
Effective professional development
Several researchers have pointed out some principles for effective professional
development by synthesizing results from various research and development projects
(Borko 2004; Desimone 2009; Loucks-Horsley et al. 2010; Wei et al. 2009)
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Wei et al. (2009) define effective professional development as development
that leads to improved knowledge and instruction by the teachers and improved
student learning. They draw on research from both the US and elsewhere that links
student learning to teacher development. They put forward four main principles for
designing professional learning:
 Professional development should be intensive, ongoing, and connected
to practice.
 Professional development should focus on student learning and address
the teaching of specific curriculum content.
 Professional development should align with school improvement
priorities and goals.
 Professional development should build strong working relationships
among teachers. (Darling-Hammond et al. 2009)
They also indicate that other factors like school-based coaching and mentoring
and induction programs for new teachers are important and likely to increase the
effectiveness of teachers. Intensive professional development rooted in practice is also
most likely to change teaching practices and lead to increased student learning.
According to Loucks-Horsley et al. (2010) effective professional development
is designed to address students learning goals and needs. It is driven by images of
effective classroom learning and teaching and gives teachers opportunities to develop
both their content and pedagogical content knowledge and inquire into their practice.
It is research based and implies active learning for teachers in learning communities
with their colleagues and other experts. It is a lifelong process, linked to other parts of
the school system and should be continuously under evaluation.
Professional learning communities seem to play an important role in
supporting teachers in continuously improving their teaching and sustaining their
professional learning (Fernandez 2002; Loucks-Horsley et al. 2010). Lesson study is
referred to as an example of a professional development strategy that has many of the
aspects that characterize effective professional development. Lesson study enhances
teachers’ knowledge and quality teaching, it develops leadership capacity and the
building of professional learning communities (Loucks-Horsley et al. 2010).
According to Desimone (2009) there is a consensus among researchers on the
main critical features of professional development that can be linked with changes in
teachers practice and knowledge and to some degree in student learning. She points
out five main features. These are focus on content, active learning, coherence,
duration and collective participation. According to Desimone there is strong evidence
that focus on content and how students learn that content, in professional
development, can be linked to teacher development and to some extent to student
learning. Active learning where teachers engage in various activities like
observations, reviewing of student work and discussions is also an important feature.
Collective participation and duration are equally important. Teachers need time to
work with, reflect on and try out new ideas and they need to do this in a learning
community with others dealing with the same issues. The critical features Desimone
points out seem to capture the core in principles for effective professional
development both Darling- Hammond et al. (2009) and Loucks-Horsley et al. (2010)
present. They also have much in common with what Borko et al. (2011) claim to be
the shared view of many teacher educators on professional development. According to
this view professional development for teachers should be a collective endeavour, it
should be about the work of teaching and the learning opportunities should be situated
within the teachers practice.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Some examples from our research
In the design of our mathematics education courses we have introduced our student
teachers to various ways of collaborating and building learning communities during
their studies by:
 giving students good possibilities for developing a professional
language and collaboration competency,
 creating opportunities for student teachers to focus on students’
mathematical learning,
 introducing to them effective professional learning strategies.
We have developed our ways in teaching based on our studies of our own
teaching. In these studies we have gathered data from oral and written assignments,
interviews, course materials, recordings of evaluation meetings and our course notes.
We analyse and categorise the data by emerging themes and reflect on them together
with the intension to improve our practice. We will here give some examples from our
studies on lesson study, reflective diaries, and student teachers’ reflection on their
own learning and on our reflective practice.
Lesson study
In lesson study a group of 15 student teachers planned one lesson in grade 8–10 and
the lesson was taught twice. The focus of the data analysis was on how the process
affected the student teachers. Four themes emerged from the data; Professional
language, collaborative competence, pupils learning and mathematical content.
The data shows that the student teachers developed their competencies in
using professional language. When describing their ideas and asking into each other’s
ideas they discussed thoroughly and went into depth and therefore needed theoretical
concepts both from mathematics and mathematics education. It was also evident that
they started to refer to the theories and literature they all had studied to make their
ideas clearer and to give them more weight. They made an effort to develop ideas
together. The lesson study process requires collaborative competence. The whole
group has to discuss and come to a conclusion. The student teachers experienced a
learning community when they created a lesson plan together and took joint
responsibility for the lesson. They experienced taking decisions and thinking together.
Through their practice with lesson study they felt how important conversations were.
The student teachers started with discussing the teaching approach and wanted to
build a lesson that the pupils would find interesting and fun to participate in. They
discussed ideas they thought the pupils would like and ended with making a game.
Based on their notes from observing the pupils learning in the lesson during the first
round of teaching they focused on the flow of the lesson. They discussed and wrote
about the connection between teaching and learning. The lesson was taught in 3–4
different schools at a time so the student teachers could also compare and discuss how
the lesson developed differently in different schools even though they all had the
same teaching plan. They were telling stories about pupils that gained understanding
and made some discoveries. When deciding on the mathematical content the student
teachers chose to work with prime numbers and composite numbers. During the
planning process they refined their own understanding of the content. They discussed
how prime numbers were related to other content in number theory and other fields of
mathematics. They also discussed what it implies to teach prime numbers. The student
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
teachers found lesson study a positive experience. They were more conscious about
how complex mathematics teaching is and the advantages of planning it together.
Reflective diary
In most of our courses one of the assignments is to write a reflective diary. As an
example all participants read the book Connecting Mathematical Ideas by Boaler and
Humphreys (2005). In choosing the books the main concern was that its content was
on teaching and learning from both practical and theoretical points of view. The
student teachers discussed each chapter of the book in small groups and wrote
together a reflective diary based on their discussions. We emphasised that they
reflected on the text and connected what they read to their experience from their own
learning, studies of theories and teaching practice. Recently we conducted an
interview study with five new mathematics teachers. They referred to the reflective
diary and the group discussions as an experience that has been helpful to them in their
practice. The content of the books became so familiar to them that they often referred
to them in their discussions with their colleagues. They have kept the books and
brought them with them to their schools. From this study we have learned that the
discussion of a text is just as important as the reading.
Student teachers’ reflections on their own learning
We urge our students to reflect on their mathematics learning in the teacher education
programs as well as their former learning in school. When they study new research on
pupils’ ways of learning mathematics and different approaches to mathematics
teaching they get inspired to teach their pupils in a way that gives all pupils an
opportunity for meaningful mathematics learning. Reflecting on their experience as
mathematics learners and relating to their studies, three student teachers wrote:
When we went to school the teacher described the procedures for calculating
numbers, the traditional algorithm. Then we practiced the algorithms individually.
We never worked together or even discussed our procedures. We cannot
remember that we ever explored relationships between numbers or used any
mathematical models. The focus was on memorizing and rote learning and the
problems were without context. (Anna, Hanna & Sigga February 2010)
It seems to be so deeply rooted in our culture that this is the way we learn
mathematics that when the student teachers look back this is what they recall.
According to their experience Icelandic classrooms seem to resemble classrooms in
other western countries as described by Stiegler and Hiebert (1999, 2004).
When asking the student teachers about other things they did at school they
remember having played games and explored together into many fields in science,
arts and crafts, where they used mathematics as a tool, measured, calculated,
transformed, reasoned, etc. but they never thought of this experience as mathematics
learning. They remember to have been active learners but stereotypes of mathematics
learners as passive receivers are the images they give of their own learning.
In order to help our students develop their understanding of their own way of
learning mathematics we emphasise that they reflect on their own thinking while
solving mathematical tasks. According to Mason (2009) learning mathematics can be
supported by providing opportunities for learners to manipulate familiar objects. The
aim is to get a sense of relationships that are instances of important properties such as
mathematical concepts and facts. Through doing, talking and attempting to record,
they can work towards articulating those concepts and facts. Important things happen
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
when learners try to reconstruct for themselves, in their own words and in
conventional terms, what they are coming to understand. In order to support learning,
it helps to sensitise yourself to learners’ struggles, and the best way to do this is to
challenge yourself mathematically by placing yourself in a similar situation and
become a learner again.
Björn one of our student teachers wrote in his final remarks on the end of term
reflective assignment:
Finally I want to add that this assignment has been very helpful. It is important to
take time to think what one has been doing during the winter. I have discovered
things about myself, that I of course had some vague ideas about, but are
important to write down because then they somehow become more real. What I
have discussed here does not only relate to my mathematics learning but gives a
good picture of me as a person. Therefore my reflection on my way of studying
mathematics has helped me to understand my way of learning not only
mathematics but in general. (Björn May 2006)
The assignment was individual but Björn was writing about his reflections on
learning in a community with his fellow student teachers. Exploring mathematics with
others, doing, talking and reflecting together as well as discussing what they had read
about research on children’s mathematics learning helped him reflect on how
mathematics learning gradually became meaningful to him.
Our professional development
Our collaboration has helped us see from a broad range of views how our student
teachers are learning, and in so doing we believe that we have managed to respond to
them in a more professional way. By discussing our responses to them and helping
each other understand their learning we have opened up a forum and encouraged them
to critically reflect on their classroom practice in the light of research. By giving the
student teachers access to research on children’s mathematical development, their
capacity to evaluate students’ learning, through analysis of their engagement in
authentic mathematical problems, has been enhanced. We have experienced that when
the student teachers investigate mathematics their confidence in solving problems
increases. Additionally, their understanding of how pupils use diverse ways to solve
mathematical problems expands.
The transformation from theory to practice does not proceed automatically.
Teacher educators can create learning communities for student teachers and should be
responsible for supporting them in teaching mathematics in inclusive schools. As
teacher educators we have the desire to identify approaches to teacher education to
ensure that teachers meet the demand to develop relative to the complexity in
mathematics teaching.
The complexity of teaching about teaching is embedded in the nature of
teaching itself and demands a sophisticated understanding of practice (Loughran
2007). In analysing the development of our own teaching we have found that theories
and research findings have affected our ways of thinking about mathematics teaching
and learning. We have found it rewarding to build our work on research in
mathematics education. Gradually we have realized how important it is for teacher
educators to understand that pedagogy of teacher education must go beyond the
transmission of information about teaching. Student teachers need not only to
concentrate on learning what is being taught, but also the way in which that teaching
is conducted. Teaching is a complex process that cannot be learned once and for all
and it is important in teacher education to open the doors to research in the field.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Writing about our research enhances our understanding of how our
collaboration has grown to develop a community of inquiry where we reflect on our
work together. It has also affected the learning community that we have developed
along with the student teachers.
References
Boaler , J., and C. Humphreys. 2005. Connecting mathematical ideas: Middle school
video cases to support teaching and learning. Portsmouth, NH; Heinemann.
Borko, H. 2004. Professional Development and Teacher Learning: Mapping the
Terrain. Educational Researcher, 33(8): 3–15.
Borko, H., K. Koellner, J. Jacobs, and N. Seago, 2011. Using video representations of
teaching in practice-based professional development programs. ZDM, 43(1):
175–187.
Darling-Hammond, L., R. C. Wei, A. Andree, N. Richardson and S. Orphanos. 2009.
Professional learning in the learning profession: A status report on teacher
development in the United States and abroad. Dallas, TX: National Staff
Development Council.
Desimone, L. M. 2009. Improving impact studies of teachers professional
development: Toward better conceptualizations and measures. Educational
Researcher, 38(3): 181–199.
Fernandez, C. 2002. Learning from Japanese Approaches to Professional
Development: The Case of Lesson Study. Journal of Teacher Education,
53(5): 393–405.
Guðjónsdóttir, H., and J. V. Kristinsdóttir. 2011. Team teaching about mathematics
for all: Collaborative self-study. In What counts in teaching mathematics, ed.
S. Schuck and P. Pereira, 29–44. Dordrecht: Springer.
Gunnarsdóttir, G. H., and G. Pálsdóttir. 2011. Lesson study in teacher education: A
tool to establish a learning community. In Proceedings of the Seventh
Congress of the European Society for Research in Mathematics Education, ed.
M. Pytlak, E. Swoboda ans T. Rowland, 2660–2669. Rzeszów: University of
Rzeszów.
Loucks-Horsley, S., K. E. Stiles, S. Mundry, P. W. Hewson, and N. Love. 2010.
Designing Professional Development for Teachers of Science and
Mathematics (3rd edition.). Thousand Oaks, CA: Corwin.
Loughran, J. 2007. Enacting a pedagogy of teacher education. In Enacting a pedagogy
of teacher education, ed. T. Russell and J. Loughran, 1–15. New York/Oxon:
Routledge.
Mason, J. 2009. Learning from listening to yourself. In Listening counts: Listening to
young learners of mathematics, ed. J. Houssart and J. Mason, 157–170. Stoke:
Trentham Books.
Stiegler, J. W., and J. Hiebert. 1999. The teaching gap: Best ideas from the world’s
teachers for improving education in the classroom. New York: Free Press.
Stiegler, J. W., and J. Hiebert. 2004. Improving mathematics teaching. Educational
Leadership 61(5): 12–17.
Wei, R. C., L. Darling-Hammond, A. Andree, N. Richardson and S. Orphanos. 2009.
Professional learning in the learning profession: A status report on teacher
professional development in the United States and abroad. Technical report.
Dallas, TX: National Staff Development Council.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 90
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Engaging students with pre-recorded “live” reflections on problem-solving:
potential applications for “Livescribe” pen technology
Mike Hickman
Faculty of Education and Theology, York St John University
Building on the author's PhD work with part time postgraduate (PGCE)
primary student teachers, this paper considers the potential application of
Livescribe pen technology to facilitate/support reflection on collaborative
mathematical problem solving, allowing opportunities for participants to
engage in ‘live’ reflection on their ‘free’ problem solving performance in
order to elicit reasoning/effective strategies and thereby inform their
future practice. With recorded (group) thinking aloud, followed and
supplemented by a stimulated recall/task-based interview opportunity and
associated problem solving/talk framework, participants are encouraged to
articulate their problem solving strategies, experiences and understanding
with the benefit of potentially reduced influence from the researcher. The
risk of think-aloud protocols impacting negatively on problem solving
performance is arguably reduced by the use of a technology that allows
the ‘replay’ of participants’ workings/jottings alongside their verbal
contributions.
Keywords: digital audio; thinking aloud; primary; PGCE; problem
solving; stimulated recall.
Introduction
As discussed in Hickman (2011), the overarching focus of this project is on the ways
in which digital audio can support student teachers’ learning and levels of confidence
in teaching primary mathematics (specifically problem solving) to their own pupils.
To this end, and utilising a think-aloud protocol (T-AP) informed by the work of
Ericsson and Simon (1993), their verbal contributions during collaborative problem
solving activities (taken from Primary National Strategies materials) are recorded
using digital audio recorders with the recordings subsequently played back to them in
stimulated recall interviews (SRIs). The SRIs allow opportunities for participants to
reflect upon the different types of verbal contributions made (in line with Mercer’s
(1995) talk framework i.e. identifying ‘exploratory’ and ‘cumulative’ contributions
and considering their impact upon the group’s ‘success’ at solving the given problem)
as they ‘relive’/replay their original work. They also arguably provide opportunities
for the student teachers to identify effective problem solving strategies to take forward
into their classroom practice, although this is not a major consideration of the current
iteration of this work (which is not directly concerned with following the participants
into the classroom as it considers student teachers’ perceptions of their levels of
confidence in teaching primary problem solving).
Both T-AP and SRI allow opportunities for participants to reflect upon their
mathematical problem solving performance – the former during a task ‘in the
moment’ in a self-directed fashion; the latter at some point afterwards, although with
the caveat that, as recommended by Fox-Turnball (2009, 206) it “should occur as
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
soon as possible after the task is completed”. Both Ericsson and Simon (1993) and
Robertson (2001) comment upon the different ‘levels’/types of thinking aloud that are
possible, with greater amounts of verbalisation potentially causing greater disruption
to mathematical thinking and performance. This provides a rationale for avoiding
excessive verbalisation during the task and exploring opportunities for post-task
reflection via SRI. The Livescribe pen (detailed below) has supported the combination
of both methodologies to allow for verbalisation of the strategies employed during a
group problem solving event to be ‘revisited’ within SRI and even potentially built
upon, with some new learning arguably taking place as a result of this ‘live’ reflection
on pre-recorded work. Such a concentration on student teachers’ verbal contributions
fits well with Duval’s (2006, 104) point that “research [of this kind]…must be based
on what students do really by themselves, on their productions, on their voices” and in
the case of this project, the students’ reflection affords the potential for them to learn
about their own learning (in a metacognitive sense) from their own voices.
In this way, the SRI provides the opportunity to identify and/or reconsider
participants’ “knowledge” of their problem solving strategies. This process was, in
part, influenced and informed by Goldin’s (1997, 41) task based interviews which
allow researchers “to observe and draw inferences from mathematical behavior”. This
four stage exploration begins with ‘free’ problem solving “with sufficient time… [for
response]… and only non-directive follow-up questions” and culminates in
“exploratory (metacognitive) questions (e.g., “Do you think you could explain how
you thought about the problem?”)” (45). The initial recording with think-aloud
protocol and Livescribe pen affords the opportunity for ‘free’ problem solving;
Goldin’s (1997) succeeding stages are evident within the SRI that follows.
Livescribe pens
The brand name Livescribe refers to a digital pen with built-in digital audio recorder
and camera which tracks the user’s writing across special proprietary paper, recording
both the marks made and any speech/utterances produced during the writing –
providing the user has remembered to press ‘record’, of course!. While the pen can be
used as an ordinary pen on regular paper, any writing produced will not be attached to
audio recordings made (indeed, the pen is not able to record sound without ‘tapping’
the record icon on the proprietary paper so its usefulness with ordinary paper is
singularly reduced) and replay will not be possible.
Recordings can be listened back to in one of two ways: either by tapping any
written word on the note paper to listen back to the exact word/s said when the word
was being written or by connecting the pen, via USB, to a computer and using the
‘Livescribe desktop’ to upload the contents of the pen for replay on screen. Having
done this, it is then possible to ‘play back’ an entire page of work (or more) with
associated audio. The writing appears, in real time, on a virtual version of the original
paper on the computer screen (with the sound coming through the more powerful
computer speakers instead of the very small speaker contained in the pen). Either one
of these methods would be able to facilitate recall, should it be required. In the pilot
work for this project, it quickly became clear that the latter approach, when working
on group problem solving and therefore group recall, was preferable as the whole
group could more easily both see and hear the replay of their work. Within the SRI,
the group are able to hear their original spoken contributions and consider how these
relate to specific written symbols and working; this has, indeed, prompted some
interesting and useful observations:
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
‘I had three columns written on the paper and, like XXX was saying, that the
image on the top of the sheet, wasn’t it, with the three…?’
‘Yeah, and so I’ve gone…right…nine…that would that…whatever…and that,
whatever…and then we went systematically down and then it occurred to me,
why don’t we start with nine?’ (Stimulated recall transcript)
The above extract from a recall session illustrates the way in which the
recording informed the response given – the second respondent identified their first
contribution of “nine” and their resultant attempt to work “systematically” through the
problem from the ‘animated jottings’ provided by Livescribe. If students had listened
back solely to their verbal contributions, the strategy employed to attack this
particular problem (‘make as many three digit numbers as possible with 25 beads on
one abacus’) would not have been clear – indeed, this particular individual’s one and
only verbal contribution at this point in the recording was the word “nine”. Even with
the T-AP employed strongly encouraging participants to explain their reasoning, such
was the enthusiasm of the group (with overlapping speech and much in the way of
‘unfinished’ thought) that the student had been unable to add to their statement,
‘swept away’ by contributions from their peers.
‘So it comes down to what you can’t see…in the audio…there was an attempt to
try to verbalise this, I know…I tried to tell someone, can you just say out
loud…?’(Stimulated recall transcript)
As seen above, participants attempted to ‘hold each other to task’ by
indicating where things needed to be (more effectively) verbalised; they were also
able to indirectly reflect upon the inadequacies of audio recording/T-AP alone as a
method for capturing their contributions (and this will, indeed, be used to inform
further iterations of the work). Within the Livescribe SRIs, the audio supported notes
often provided evidence of exploratory contributions that would not otherwise have
been evident (such as proposing nine as an appropriate, systematic place to begin
when addressing the problem above). To an extent, then, it could be argued that this
‘makes up for’ and even potentially enhances the quality of the participants’ original
mathematical discussion.
So, we’ve got all the combinations of 9. 8 and 7…so you’ve got 3 9s in each, 9
appears three times in each column, 8 appears twice…
And that must be because [of] the number bonds in 25…something to do with
number bonds… (‘Beads’ Digital Audio transcript)
In the extract above, participants have already gone some way towards solving
the ‘beads’ problem just eight minutes into the recording; that they continued their
discussion for a further eight minutes illustrates their level of uncertainty. The
Livescribe-supported SRI allowed them to revisit this and actually confirm their
original thoughts.
I’m really desperately trying to think ‘cos you knew reading through this...why
did we make it so difficult? It’s not difficult, is it? [Murmur of agreement]
(Stimulated recall transcript)
In some respects, the ‘live’ reflection within the SRI enables a ‘second go’ at
thinking aloud and, as will be further discussed below, this additional layer of
thinking aloud brings aspects to participants’ attention that were not evident when
first tackling the problem.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Contrasting digital audio recorders and Livescribe pens
As indicated above, initial pilot work involved ‘traditional’ digital audio recorders,
albeit supported by transcripts of the participants’ problem solving work and their
original jottings within the original SRI. This led, as will be discussed below, to some
confusion amongst the students in identifying which of the written jottings matched
with their verbal comments and, indeed, one problem which quickly became apparent
was that the discussion in the SRI became over-focused on these more ‘technical’
issues.
Other technologies such as tablet computers may provide some of the
advantages of the Livescribe pen in that they can allow the recording of audio in
conjunction with written notes (also affording the playback of such recordings
alongside jottings), as indicated by Weibel et al. (2011). However, there are
advantages to paper-based working that fit well with this project and its postgraduate
student teacher participants: paper is “portable, cheap and robust” and it is “much
more convenient to scan through a book than to browse a digital document” (Weibel
et al. 2011, 258). This, in part, informs the use of Livescribe over tablet computers or
other similar technologies in this work. In addition, it is arguably equally beneficial to
employ a technology that requires less in the way of formal briefing or training, given
the relative simplicity of the pens compared to other technologies, when working with
student teachers with varied levels of ICT experience and whose confidence in and
contribution to problem solving tasks is the primary concern of the work. Although
the intention of the project was always to utilise both the T-AP and SRI
methodologies (which, of course, could have stood alone as independent methods for
capturing data on domain specific problem solving), participants were able to identify
their exploratory comments more effectively within the Livescribe supported SRI than
those produced by digital audio alone. It is also clear that the import of their original
exploratory statements had not been recognised via the T-AP alone due to their
listening to each other’s contributions (as would be expected in group problem
solving opportunities), the level of concentration required on own verbal contributions
and, indeed, their awareness of being recorded in the first place.
Beyond just ‘missing’ exploratory contributions made in the original problem
solving event, participants had also missed connections with previous problems
encountered and successfully solved. The framework proposed in Hickman (2011) is
informed by Mercer (1995), Hošpesová and Novotná (2009) and Seal’s (2006)
identification of the importance of exploratory talk. For the purposes of this project,
‘exploratory’ contributions have been split up into those that restate the problem by
using analogy to clarify it to other members of the group and those that restate it in
mathematical form (i.e. identifying operations required that were not explicitly stated
in the original question). Both categories are arguably assisted by an appreciation of
‘similar’ problems (i.e. a problem is seen to be ‘like’ another that had previously been
encountered); with the T-AP alone, however, contributions of this kind were not much
in evidence. For example, in the SRI (but not in the T-AP) of the abacus problem,
participants noted that they had, in fact, been presented with a problem similar to one
that had previously been encountered (indeed, the problem had been chosen for this
reason) – they had simply failed to notice this on first encounter. Watching their
tabulation of the Beads problem on the screen in consort with their verbal offerings
had made this clearer to them. Therefore, the Livescribe supported SRI afforded
students the opportunity to make connections, from their original contributions and
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
working, that had not been explicitly identified in the original problem solving
session.
One contention of this project is, then, that Livescribe pen technology is better
able than the more ‘traditional’ digital audio recorders to afford participants the
opportunity to revisit their actions in the moment whilst standing outside the moment.
The ‘replaying’ of their written notes alongside (essentially ‘in time with’) their
original verbal contributions potentially also provides a stronger prompt for recall and
ultimately reflection than the more typically employed (in SRI) video and audio
technologies which do not so strongly connect the written with the spoken.
Conclusion
The ‘unfinished’ nature of much of the participants’ verbalised thought in the T-AP
recordings produced to date was caused in part by the attempt to reduce the impact of
the protocol on their problem solving performance by limiting the amount of thinking
aloud that actually had to be articulated. This reflected Ericsson and Simon’s (1993)
comments about the degree to which more ‘extreme’ verbalisations of thoughts,
moves and actions will ultimately impact on the mathematics. A higher level of
verbalisation could potentially have reduced the number of ‘unfinished’ thoughts but
ultimately would not have prevented other issues impacting upon these recordings,
including interruptions from other members of the group that inadvertently cut short
their peers’ speech. It is, therefore, perhaps unsurprising that such thought was not
always effectively built upon (in Mercer’s (1995) ‘cumulative’ sense) within the
original problem solving opportunities. Participants’ attention was split between the
demands of the problem set, the need to solve this with/alongside their colleagues
(which, of course, is not itself without problems due to the risks of ‘exposing’
mathematical uncertainties in front of peers) and the ‘artificial’ situation of being
audio recorded/having to think aloud whilst engaging in this work. We suggest that
the Livescribe-supported SRI reduced the impact of these factors by providing a
valuable opportunity for them to clarify what had originally been propounded both by
themselves and their peers and revisit their learning in a way that fits well with
Polya’s (1957) concept of ‘looking back’ at problem solving work. It allows them to
concern themselves less with the think-aloud protocol during the original recording,
even omit details that would be seriously ‘missed’ if recorded by conventional digital
audio recorders, as a combination of spoken and written material is employed within
the SRI to prompt their recall. Indeed, their SRI contributions to date indicate that the
technology allows them to identify for themselves especially productive/beneficial
contributions made, that may not have been recognised as such by observers, and this
again fits well with the four stage exploration of Goldin’s (1997) task-based
interviews. Some refinement to the T-AP utilised to underpin this work is almost
certainly required to address some of the issues encountered by groups such as
unintentional interruptions. More also arguably needs to be done to effectively
‘capture’ participants’ resource use whilst solving problems (although an effective TAP that ensured participants articulated clearly their choice of appropriate resources
and thinking behind this would prevent this being a major problem). However, the
Livescribe technology itself has shown some promise in prompting productive
responses that encourage deeper exploration and even exploratory talk (Mercer, 1995)
and this, when enhanced further, may be of significant benefit to future classroom
practitioners.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
References
Duval, R. 2006. A cognitive analysis of problems of comprehension in a learning of
mathematics. Educational Studies in Mathematics, 61: 103-131.
Ericsson, K.A., and H.A. Simon. 1993. Protocol analysis: Verbal reports as data.
Cambridge, MA: MIT Press.
Fox-Turnball, W. 2009. Stimulated recall using autophotography - A method for
investigating technology education. Proceedings of the Pupil's Attitudes
toward Technology Conference (PATT-22), 24-28 Aug 2009:204-217.
http://www.iteaconnect.org/Conference/PATT/PATT22/FoxTurnbull.pdf.
Accessed 21.12.12
Goldin, G.A. 1997. Observing mathematical problem solving through task-based
interviews. Journal for Research in Mathematics Education, 9: 40-177.
Hickman, M. 2011. A talk framework for primary problem solving. Informal
Proceedings of the British Society for Research into Learning Mathematics,
31 (3). http://www.bsrlm.org.uk/IPs/ip31-3/BSRLM-IP-31-3-13.pdf Accessed
20.12.12
Hošpesová, A., and J. Novotná. 2009. The process of problem solving in school
teaching. In Proceedings of the 33rd Conference of the International Group
for the Psychology of Mathematics Education 3, eds. M.Tzekaki, M.
Kaldrimidou and H. Sakonidis, 193-200. Thessaloniki, Greece: PME.
Mercer, N. 1995. The guided construction of knowledge: Talk amongst teachers and
learners. Clevedon: Multilingual Matters.
Polya, G. 1957. How to solve it: A new aspect of mathematic method. New York:
Doubleday Anchor Books.
Robertson, S.I. 2001. Problem solving. Hove: Psychology Press.
Seal, C. 2006. How can we encourage pupil dialogue in collaborative group work?
(Summary produced for the National Teacher Research Panel Conference,
2006). http://www.edupa.uva.es/schemesofwork/ntrp/lib/pdf/seal.pdf
Accessed 20.12.12
Weibel, N., A. Fouse, E. Hutchins and J.D. Hollan. 2011. Supporting an integrated
paper-digital workflow for observational research. In Proceedings of the 16th
International Conference on Intelligent User Interfaces: 257-266.
http://adamfouse.com/pdfs/weibel-iui-2011.pdf. Accessed 20.12.12.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 96
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
A student teacher’s recontextualisation of teaching mathematics using ICT
Norulhuda Ismail
Institute of Education, University of London
In university mathematics education courses, messages about the
pedagogy and content of teaching mathematics are conveyed to student
teachers. During the teaching practicum, mentor teachers also have their
own set of messages about mathematics teaching. My research
investigates the messages conveyed to student teachers and the ways
student teachers acknowledge these messages and incorporate them into
their teaching using the notion of recontextualisation. The use of
information and communication technology (ICT) in teaching
mathematics is generally viewed positively in the university and by
mentor teachers. In this paper I share some data and analysis of the
messages about ICT, and how one student teacher recontextualises these
messages into his own teaching of mathematics.
Keywords: ICT, student teachers, recontextualisation
Introduction
For student teachers, the teaching practicum is a difficult experience as they try to
select ‘approved’ methods of teaching set by the university discourse. A study by Goh
and Matthews (2011) on Malaysian student teachers’ journal writing revealed their
frustrations in choosing the appropriate methodology and techniques for teaching.
They are also worried that they are unable to answer students’ questions. The
problems faced by Malaysian student teachers could be due to the design of teacher
training programs in Malaysia. Lee (2004) has described some of the weaknesses of
teacher training programs in Malaysia which focus mostly on general pedagogical
knowledge such as time on task, questioning techniques and preparing lesson plans
and not on actual methods for teaching subjects. Furthermore, the presence of a
mentor teacher who may not have aligned views about the appropriate approaches in
teaching mathematics may make the practicum an even more confusing experience
for many student teachers. Lee (2004) highlighted that mentor teachers are not being
prepared to provide effective supervision to help student teachers develop their
practices in teaching.
This scenario has led me to develop a research project to investigate the
various messages about teaching mathematics provided by a teacher training program
in Malaysia. I am also investigating how student teachers acknowledge and apply
these messages during the teaching practicum in consideration of the messages about
teaching mathematics of their mentor teachers. One of the themes that has arisen from
the data is the emphasis on use of information and communication technology (ICT)
in teaching mathematics. I will present in this paper analysis on the program’s
messages about use of ICT in teaching mathematics. Finally, I will show how one
student teacher acknowledges and applies this message in his teaching, with respect to
his mentor teacher’s interest.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Theoretical framework
I am using the notion of recontextualisation (Bernstein 2000) to conceptualise the
transformation of messages about teaching mathematics from the teacher training
program to the setting of the mathematics classroom. Recontextualisation selectively
relocates, and refocuses a discourse and relates it to other aspects to focus onto
another principle. Recontextualisation will occur as student teachers adjust the
messages about teaching mathematics to the new site, subject to the conditions of the
social and political relations of the new site (Thomas 2003). This means that, in
recontextualisation, student teachers will select from their experiences concerning
teaching mathematics in the teacher training program and refocus it to the principle of
teaching the class at hand. For example, when planning a lesson, a student teacher has
to select ICT tools to incorporate into their lesson. In the teacher training program
setting, they have experienced in the teacher training program setting guidelines on
appropriate ICT tools and how to use these tools in teaching mathematics. They select
from this experience, relocate and refocus the guidelines, while also considering the
requirements of the mentor teacher.
Methodology
The data collection consisted of observing and video recording sessions from courses
in the university setting that focused on developing student teachers’ knowledge about
teaching of mathematics. The courses are Methods for teaching mathematics,
Microteaching and Laboratory in mathematics education.
I am using a critical discourse analysis approach to draw out the messages
about teaching mathematics from the university setting. To do this, I focus on the
objects (what counts as an ICT tool) and value statements regarding the objects. I also
focus on suggested methods in using the tools and value statements regarding the
methods. In the end, I may develop an observational scheme that will assist in helping
to identify student teachers’ recontextualisation of these messages.
In the school setting, six student teachers and their mentors were participants
for this research. The mentor teachers were interviewed and their messages about
teaching mathematics were drawn out. The student teachers were observed three times
each and interviewed at least once. In considering student teacher’s
recontextualisation of using ICT, I focus on two aspects. First I focus on student
teachers knowing the messages about using ICT in the interviews. This concerns their
acknowledgement of the messages, and the ways in which student teachers position
themselves in the acknowledgement. Then, I focus on student teachers acting out the
messages of using ICT in their mathematics teaching, focusing on the functions of the
tools and the similarities and differences in the ways they use ICT from the university
setting and the mentor’s advice.
Data analysis of messages from university setting
In the analysis regarding the messages about mathematics teaching, I focus on the
objects demonstrated to the student teachers, and value statement regarding the
objects. The table is a portion of a transcript from a Microteaching class, where a
lecturer was giving her initial comments on a student teacher’s lesson regarding her
use of a ministry approved ICT tool in teaching matrices. In the text, the object talked
about is technology and the value statement about the use of the official ministry tool
is that it saves time for teachers. The message given here is that the use of official
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
ministry tool is viewed as sufficient (good) because teachers can save time in teaching
preparation as they do not have to build their own tools.
Table 1: data analysis extract
Text
analysis
Ok. Let’s look at her strengths.
One thing is she used technology.
That is very good. you can apply that
Object: technology
Evaluation of use of ICT: very good/
more than sufficient.
You don’t have to do it yourself, so there is Further explanation of why it’s good
no need to waste time.
because it saves time.
Messages about teaching mathematics using ICT in university setting
The three courses, Methods for teaching mathematics, Microteaching and Laboratory
in mathematics education, held distinct messages according to the objectives of each
activity, therefore portraying different evaluations of use of ICT in teaching
mathematics. The Methods class focused on the ICT objects, emphasising the novelty
and the quality of each tool displayed. Therefore the values conveyed focused on the
ICT tool itself. The Microteaching class was about applying ICT tool into teaching.
Here, the comments focused on the teacher’s ability to find a balance between the role
of the teacher and the role of the ICT. Finally, the Laboratory class aim was to
develop technological pedagogical content knowledge among student teachers and the
activity was to demonstrate by allowing student teachers experience learning school
level mathematics using ICT based tools. Table two summarises the messages
conveyed about teaching mathematics using ICT tools and value statements regarding
the use of ICT.
Table 2: messages conveyed about use of ICT in teaching mathematics
1a. What objects
are demonstrated
or displayed to
student teachers?
Methods for
teaching math
Tool 1: A song
about the rules of
exponent.
Tool 2: An
energetic song
about the difference
of ‘log’ use in
everyday life and in
mathematics.
Tool 3: A static
powerpoint with no
animations.
Tool 4: A
PowerPoint about
vectors with
animation and very
colourful.
Microteaching
Tool 1: a ministry
approved tool
which has narration
and animation
about the
introduction to
matrices. It has the
starter set that
showed items
ordered in rows and
columns. It had
activities where
students or the
teacher can key in
the answer. Had use
of matrices in other
fields.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 99
Laboratory in
math education
Tool: MSWLogo.
Students learn to
program the turtle
to move around the
playground.
Through the
activity, the
students construct
knowledge about
polygons.
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
1b. What values
are placed on the
objects?
Tool 1: the song is
satisfactory. But,
the mathematical
notations need to be
correct.
Tool 3: the static
PowerPoint is usual
and unauthentic.
Tool 4: the
PowerPoint with
animations is
satisfactory.
2a. What is the
suggested method
in using these
tools?
2b. What values
are placed on the
suggested
methods?
Tool 2: Must
explain the
difference between
use of certain terms
in life and in
mathematics.
Teachers have to
make this clear in
the lesson.
Tool 4: students
should use this
PowerPoint which
has animations and
colours as an
example in creating
their own
PowerPoint.
Tool 2:
differentiating the
use of terms in
mathematics and in
life helps students
to understand better
the language of
mathematics.
Tool 4: colourful
and animated
powerpoints is
motivating to
students.
Tool 1: the ICT tool
was interesting.
However, it was
only interesting for
the first two
minutes because
students may not be
able to concentrate
on the display for
long. It was good
because it assisted
in the teaching and
learning process
where the ‘teacher’
could use the tool
for several
segments of the
lesson.
The ministry tool
was played for the
starter set which
explained examples
of matrices in
everyday life.
Tool is open
source, free, and
compatible with
many operating
systems.
Introduction of tool
using tutorial.
Giving direct/basic
instructions.
Math activity:
Whole class
problem solving
In the closure, a
activity about
mini activity was
creating polygons.
conducted where
Student teachers
students had to
construct
individually solve
knowledge of
some matrices
interior angles
problems in
while learning to
cryptography.
program the turtle
to create polygons,
guided by the
lecturer.
The use of ICT is
Students can
viewed positively.
construct
However, there
knowledge about
needs to be a
mathematics
balance between
through
the use of ICT and
technological
pedagogical
based activities.
strategies. Teaching However, teachers
needs to be student need to closely
centred where
guide students so
students are active, that the students
have group
construct the
activities.
knowledge
intended.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Saiful’s mentor teacher’s messages about teaching using ICT
Saiful is a student teacher participant who took the same university courses prior to
his teacher training semester. The rest of the paper depicts Saiful’s mentor teacher’s
messages and Saiful’s recontextualisation of teaching mathematics using ICT. Saiful’s
mentor teacher is highly interested in seeing him incorporate the use of ICT especially
in developing and using interesting and animated PowerPoint. This view seems to be
aligned with tool 4 in the Methods class, where the use of multimedia based ICT
incorporates amusing and motivational elements into the classroom.
Saiful’s recontextualisation of teaching mathematics using ICT
Saiful Knowing the Messages about teaching using ICT
During the interview, Saiful clearly identifies his mentor teacher’s interests in using
PowerPoint to teach mathematics as it includes creative elements.
My mentor teacher, she likes fun activities, such as PowerPoint, she likes use of
teaching aids which are very creative, so the teaching does not seem too
traditional bound.
In using ICT, Saiful states that he does not use PowerPoint much because he
views its use is to be limited. Saiful appears to align himself with the laboratory class,
where the use of mathematics applications allows students to experiment with
mathematical objects. He explains how he used a mathematics application in class to
teach straight lines.
I think mathematics applications are a lot better than PowerPoint. I used it for
form four students teaching straight lines. The students can key in the gradients
and see how a small and big gradient looks like. It’s online.
Furthermore, Saiful views that the use of PowerPoint is limited for teaching
mathematics because this requires both practical work and understanding. This
suggests that he views PowerPoints as only useful for displaying notes such as tool 3
in methods class.
I think that use of PowerPoint is useful in teaching mathematics. However, there
is a limit. When compared with other subjects that require more reading,
mathematics is an understanding and practical subject. So, these two aspects need
to be considered when using PowerPoint.
Saiful acting out the messages about teaching using ICT
In one of the lessons I observed, Saiful’s mentor teacher was also there to observe
him. Saiful had taken into consideration his mentor teacher’s preferences by
compiling several ICT based tools which he uses throughout the lesson. For the
starter, pictures of an obese and underweight man were displayed and students had to
guess the topic which was mass. In the exchange about the pictures, the term weight
was used consistently. However, the students were able to guess the topic name
correctly which is mass because Saiful had asked the students to open the textbook to
the topic page before the class began. Despite this, Saiful did not differentiate
between the different use of the terms ‘weight’ and ‘mass’ in mathematics and in life.
This was an element emphasised in Methods class where use of terms in life and in
mathematics should be differentiated by the teacher.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The avatar selected stated the definitions of mass, but it was exactly as the
content in the textbook. However, there was an element of amusement as the students
laughed when Saiful played the avatar. Their response seems to be aligned with
Methods class and the mentor teacher’s view that ICT should give an element of
amusement for students.
The notes displayed were static and exactly the same as in the textbook. This
is one of the concerns of Microteaching class, that the use of ICT can make the class
still appear dull. In Microteaching class, it is advised that group activity must be
included to overcome this. However, Saiful was unable to conduct the group activity
because he did not prepare the weighing tools beforehand. To compensate, Saiful
asked the students to guess the weight of selected objects by calling them out to him.
Conclusion
This research focuses on the ways student teachers recontextualise the messages about
teaching mathematics using ICT from the university setting and the mentor teacher.
During the interview, Saiful aligns himself with the Laboratory class, where he says
he prefers using technological tools to teach because it allows students to experiment
with mathematics knowledge. However, the class observed did not include any
elements of experimentation. This observation seem to show a mismatch between his
own verbal preference about having a lot of interesting activities, to his actions where
the tools displayed were just resources for the content and did not provide interesting
activities for students to conduct. Saiful also attempted to apply group work as
advised in Microteaching class, but the tools were not prepared beforehand, so the
group activity was unable to be carried out successfully.
No officials from the university were present during this observation.
Therefore it is possible the criteria selected for this lesson was dominantly from the
expectations of the mentor teacher because she was there to observe him. The analysis
show that although a student teacher is clearly aware of the interests of his mentor
teacher in seeing him teach using PowerPoint, his own values about mathematics
learning being practical and his beliefs that PowerPoint is limited for teaching
mathematics means that he does not entirely fulfil the mentor teacher’s interest. To
compensate, Saiful compiled a set of tools to support his teaching. However, there is
rigidness in his selection as the avatar and the notes were clear repetition from the
textbooks and did not provide much variety to the lesson.
References
Bernstein, B. 2000. Pedagogy, symbolic control and identity: Theory, research,
critique: Oxford : Rowman & Littlefield
Goh, P.S. and B. Matthews. 2011. Listening to the concerns of student teachers in
Malaysia during teaching practice. Australian Journal of Teacher Education
36, no 3: 92-103.
Lee, M. 2004. Malaysian teacher education into the new century In Reform of teacher
education in the Asia-pacific in the new millennium: Trends and challenges,
eds Cheng, Y, Chow, K and Mok, M, 81-91: Springer Netherlands.
Thomas, P. 2003. The recontextualization of management: A discourse-based
approach to analysing the development of management thinking. Journal of
Management Studies 40, no 4: 775-801.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 102
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Mathematical competence framework : An aid to identifying understanding?
Barbara Jaworski
Loughborough University, Mathematics Education Centre
Research into the teaching of mathematics to engineering students to
promote their conceptual understanding (Jaworski and Matthews 2011)
has shown the problematic nature of planning for and identifying
understanding. I review the project briefly and introduce the idea of
competencies from the Danish project, KOM (translated as Competencies
and Mathematical Learning). Through the medium of designing inquirybased tasks for students and use of the competency framework for
analysis of tasks, I consider the relevance of such a competency-based
analysis and its usefulness (or otherwise) for recognising student
understanding. This leads to important questions for further research of a
developmental nature.
Keywords: mathematical competency; engineering students, inquiry-based
teaching, sociocultural setting, developmental research.
Mathematics for Engineering students:
In this paper I discuss a research project which aimed to study the design and teaching
mathematics in ways which enable students’ conceptual learning and understanding of
mathematics for flexible use in engineering contexts. The project was fundamentally
about teaching: in particular, how teaching relates to learning with understanding.
The project, ESUM, Engineering Students Understanding Mathematics,
involved an innovation in teaching and learning. It was a developmental research
project; that is, it involved research that both studies development and contributes to
that development. It focused centrally on INQUIRY – inquiry in mathematics and
learning mathematics and inquiry in the teaching process. It attracted support from the
Royal Academy of Engineering through the UK HE-STEM programme. Funding
supported a researcher to work with the teaching team and paid for a literature review.
Research questions included

How can we enable engineering students’ more conceptual understanding of
mathematics?
o What teaching approach (and why)?
DESIGN
o What means of perceiving students’ outcomes?
APPROACH
o What outcomes?
EVALUATION
A sociocultural frame
In seeking mathematical understanding, we were interested not only in cognitive
processes, but in the whole context and culture in which we are active. (Schmittau
2003; Vygotsky 1978; Wertsch 1991). This included the following principles:
 All learning is social; knowledge grows in the social domain within which
individual knowledge is formed.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012


Learning takes place through participation in social and cultural worlds
mediated by social and cultural tools.
Scientific concepts grow through pedagogical mediation.
Thus we were interested in
 mathematical meanings, relating to an established body of mathematical
knowledge;
 perspectives of students and teachers, relating to learning mathematics;
 institutional dynamics and constraints influencing perspectives on learning and
teaching;
 worlds (cultures) in and beyond the institutional setting creating parameters and
boundaries for engaging in learning and teaching.
The institutional setting (pre-innovation)
A university three-year BSc for first year students in Materials Engineering included a
one year module in mathematics. The ESUM project studied the first semester of this
module. Students were fresh from school still with perspectives from their school
culture. The module was allocated two lectures and one tutorial per week (each 50
minutes). The university encouraged use of a Virtual Learning Environment –
LEARN – for communication, holding notes and resources. Assessment was by exam
(60%) and 8 computer based tests (40%). Teaching was largely traditional in style
with perceived instrumental approaches to mathematics (Artigue, Batanero and Kent
2007; Hiebert 1986; Skemp 1977).
The teaching-research team (co-learners)
The teaching team of three experienced teachers, two having extensive experience of
teaching engineering students, had responsibility for interpretation of curriculum,
design of innovation and teaching approach, design of questions/tasks/group project
(with the help of PhD students). One member (the lecturer) taught the module.
The research team, of four people, included the teaching team plus a research
officer (paid for with the HE-STEM funding). Together they designed research which
included

Research in practice (insider research))

Research on practice (outsider research) (Bassey 1995)
Learning through inquiry – a developmental research methodology & innovation
The inquiry approach aimed to promote learning through an inquiry community in
mathematics AND in mathematics teaching). A Community of Inquiry (CoI) was
seen to be based on processes of participation and reification as described by Wenger
(1998) in a Community of Practice (CoP). It embraced the principles that addressing
inquiry-based questions challenges existing ideas and engages students in meaningmaking more deeply; it motivates ‘wanting to know’, encouraging asking one’s own
questions, and looking critically at outcomes; it enables development of a critical
sense through critical alignment (Jaworski 2006).
A developmental research process involved linked forms of research:
Insider research – cyclic approach: Insiders, teachers who are also researchers,
engage in cycles of activity involving design (of tasks), work in practice (act/teach &
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 104
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
observe), reflect on and analyse what has been done, feedback to further planning,
disseminate to others in the field.
Outsider research – data and analysis: Involving research into processes and
practices from the outside – taking out data and analysing it through a rigorous
research process including design documentation, student surveys, observation of
practice (audio recording), interviews; analysis relevant to the kinds of data; overall
activity theory analysis; dissemination and publication (Jaworski 2003).
The innovation involved a modification to teaching, with implications for
learning mathematics. It included use of inquiry-based questions in lectures and
particularly in tutorials; a GeoGebra environment for demonstration in lectures and
for student exploratory use in tutorials in relation to inquiry-based questions; small
group activity: students in groups of three or four working on tasks in tutorials,
discussing solutions together and with the lecturer; a small group (assessed) project:
tasks given by the lecturer for exploration by students in a group with requirement to
submit a group project for assessment; and changes to assessment to include the
assessed project. Figures 1, 2 and 3 show examples of tasks which were designed and
used in the ESUM innovation:
Think about what we mean by a function
and write down two examples. Try to
make them different examples.
1. Open question/task in a lecture
In the topic area of real valued functions
of one variable
Consider the function f(x) = x2 + 2x (x is
real)
a) Give an equation of a line that
intersects the graph of this function:
(i) Twice (ii) Once (iii) Never
(Adapted from Pilzer et al. 2003, 7)
b) If we have the function f(x) = ax2+bx+c
what can you say about lines which
intersect this function twice?
3. Tutorial task – for small group work
c) Write down equations for three straight
lines and draw them in GeoGebra
d) Find a (quadratic) function such that
the graph of the function cuts one of your
lines twice, one of them only once, and the
third not at all and show the result in
GeoGebra.
e) Repeat for three different lines (what
does it mean to be different?)
2. A lecture and tutorial task
Findings from the ESUM project indicated important differences between
perceptions towards mathematical learning, the value of inquiry processes and use of
GeoGebra of those designing and delivering teaching (the teaching team) and those
experiencing the teaching and learning from it (the students). For details see Jaworski
and Matthews (2011), Jaworski, Robinson, Matthews and Croft (2012). Here I focus
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 105
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
on our desire to improve students’ conceptual understandings of mathematics
(compared with previous cohorts) which proved elusive in the ESUM analysis.
What does it mean to understand and how can we recognise understanding?
It was clear in observation of teaching sessions the extent to which students engaged
with mathematics and their degrees of conceptualization. This was pleasing in many
respects (for those teaching), however, due to being very local and specific, it did not
reveal general characteristics or provide objective insight to the nature of conceptual
understanding. Examination and test scores showed improvement on previous
cohorts, but this was not indicative of the quality of understanding. Thus we sought
an alternative approach to discerning understanding.
We became aware that the mathematics working group of the European
Society for Engineering Education (SEFI) was promoting a set of competencies
deriving from the work of the Danish KOM Project (e.g., Niss 2003; SEFI 2011) for
the design of mathematics teaching for engineering students. We decided to look
critically at what these might offer. The SEFI document quotes Niss (2003, 183) as
follows:
Possessing mathematical competence means having knowledge of, understanding,
doing and using mathematics and having a well-founded opinion about it, in a
variety of situations and contexts where mathematics plays or can play a role.
A mathematical competency is a distinct major constituent in mathematical
competence
Eight competencies have been identified as follows. See SEFI (2011) and
Niss (2003) for a detailed breakdown of what each competency includes.
The ability to ask and answer questions
in and with mathematics
The ability to deal with mathematical language
and tools
1. Thinking mathematically
5. Representing mathematical entities
2. Reasoning mathematically
6. Handling mathematical symbols and formalism
3. Posing and solving mathematical
problems
7. Communicating in, with and about
mathematics
4. Modelling mathematically
8. Making use of aids and tools
We began by using these competencies to analyse some of our tasks. For
example in Task 2a above, given in a lecture in which students had to work on the
task in their seats talking with their neighbours, we analysed as follows:
 The function is easy to sketch – it is easy to see lines which cross it in the
three conditions [5]
 Students have to talk to each other [7]
 They have to think about equations for their lines [1] [3] [6]
 They start to reason about the differences between the lines [2]
 They have to give feedback to the lecturer and others in the cohort [2] [7]
We see further analysis of Task 2b-e:
b) generalising from (a) [1, 2, 7]
d) tackling an open-ended problem
[1, 2, 3, 7, 8]
c) Inventing own mathematical objects and using a
technological tool [1, 2, 5, 6, 7, 8]
e) Addressing mathematical generality
[1, 2, 3, 5, 6, 7]
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 106
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
From this example, we believe that our tasks are broadly in line with the stated
competencies. There seems to be a region of synergy between the competencies and
goals of inquiry- based learning. The tasks designed for the latter seem to fulfil the
former. Our next challenge is to try to recognize student understanding in relation to
the competencies. The Danish team has suggested three dimensions for specifying
and measuring progress:



Degree of coverage: The extent to which the person masters the characteristic aspects
of a competency.
Radius of action: The contexts and situations in which a person can activate a
competency.
Technical level: How conceptually and technically advanced the entities and tools are
with which the person can activate the competence.
The SEFI Mathematics working group is in the process of specifying what
such dimensions can look like in relation to the mathematical curriculum for
engineering students. With regard to ESUM, we ask how our data might allow us to
address the three dimensions in order to discern what competencies students
gained/achieved. In fact, existing data is not adequate: it was not collected for this
purpose, so we ask what data we would need to collect; for example we can record
data and analyse it from events such as:




In lectures: we can ask further questions and encourage students to respond (we
recognise that not all can/will do so).
In tutorials: we can visit groups, talk with them about their current thinking, probe
and challenge appropriately (of course, we cannot be with all groups all of the time).
Assessment: in tests or exams, we can design suitable questions and analyse students’
responses (which may or may not reveal understanding).
Assessment through group project with written report: we can look critically for
evidence of understanding (we also need to consider who has done most of the work)
Analysis of the data would allow us to seek evidence of competencies having been
addressed. For this to be effective for successive groups of students we need a
systematic process which can be achieved quickly and efficiently which requires
assessment instruments to be developed to have accord with competency statements.
We can see above some of the constraints to this process, and recognise that cultural
issues revealed through ESUM will also present challenges (see Jaworski et al. 2012)
Questions for further research using a competency framework
From the above, we see a use of the competencies in evaluating design of tasks and a
potential development of instruments for a systematic use of competencies against the
three dimensions to measure student progress. The latter needs further consideration.
A third potentially valuable use of competencies would be in providing a formative
presence, for example, in creating opportunities for students to achieve competency
and as a tool for teachers in working with students to achieve competency. With
respect to this third area of consideration we ask: what are the elements of creating the
sociocultural setting in which the desired mathematical practices and ways of being
are nurtured as central to participation? ESUM identified students as having an
essentially strategic focus towards their studies; thus we might also ask: are there
ways in which students can develop an awareness of competency as a means of
changing the nature of their strategic focus?
Within our sociocultural frame in which we consider teaching for learning
mathematics in relation to a cohort of students rather than in terms of individuals – in
which we have to take seriously systemic and cultural factors as revealed by ESUM –
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 107
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
we see the above questions as motivating for further research of a developmental
nature. The questions are challenging for the design of both teaching and research:
teaching is seen as a research process through which teachers and students can come
closer in their understandings of what it means to learn mathematics effectively for
engineering contexts. We hope to pursue these questions and invite others who are
interested to join us in this endeavour.
References
Artigue, M., C. Batanero and P. Kent. 2007. Mathematics thinking and learning at
post secondary level. In Second handbook of research on mathematics
teaching and learning, ed. F. Lester, 1011-1050. Charlotte, NC: Information
Age Publishing.
Bassey, M. 1995. Creating education through research. Edinburgh: British
Educational Research Association.
Hiebert, J. 1986. Conceptual and procedural knowledge: The case of mathematics.
Hillslade, NJ: Erlbaum.
Jaworski, B. 2003. Research practice into/influencing mathematics teaching and
learning development: Towards a theoretical framework based on co-learning
partnerships., Educational Studies in Mathematics 54: 249-282.
——— 2006. Theory and practice in mathematics teaching development: Critical
inquiry as a mode of learning in teaching. Journal of Mathematics Teacher
Education 9: 187-211.
Jaworski, B., and J. Matthews. 2011. Developing teaching of mathematics to first year
engineering students. Teaching Mathematics and Its Applications 30: 178185.
Jaworski, B., C. Robinson, J. Matthews, and A. C. Croft. 2012. An activity theory
analysis of teaching goals versus student epistemological positions.
International Journal of Technology in Mathematics Education 19: 147-52.
Niss, M. 2003. The Danish “KOM” project and possible consequences for teacher
education. In Educating for the future: Proceedings of an international
symposium on mathematics teacher education, ed. R. Straesser, G. Brandell, B
Grevholm and O. Hellenius, 179-190. Gothenburg, Sweden: NCM,
Gothenburg University.
SEFI (European Society for Engineering Education). Draft, 2011. A framework for
mathematics curricula in engineering education: A report of the mathematics
working group. SEFI (European Society for Engineering Education).
Pilzer, S., M. Robinson, D. Lomen, D., Flath, D., Hughes Hallet, B. Lahme, J. Morris,
W. McCallum, and J Thrash, J. 2003. ConcepTests to accompany calculus,
Third Edition. Hoboken NJ: John Wiley & Son.
Schmittau, J. 2003. Cultural-historical theory and mathematics education. In
Vygotsky’s educational theory in cultural context, ed. A. Kozulin, B. Gindis,
V. S. Ageyev and S. M. Miller, 225-245. Cambridge: Cambridge University
Press.
Skemp, R. 1976. Relational understanding and instrumental understanding.
Mathematics Teaching 77: 20-26.
Vygotsky, L. 1978. Mind in society. Cambridge, MA: Harvard University Press.
Wenger, E. 1998. Communities of practice. Cambridge: Cambridge University Press.
Wertsch, J. V. 1991. Voices of the mind: A sociocultural approach to mediated
action. Cambridge, MA.: Harvard University Press.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 108
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The role of justification in small group discussions on patterning.
Dr Cecilia Kilhamn
Faculty of Education, University of Gothenburg, Sweden
Swedish students have not been successful in solving geometrical pattern
tasks in the TIMSS study and as a result it has been introduced as explicit
core content in the National Syllabus (Lgr11) for grades 1-6. Analysis of
video recordings of three student groups working with a task taken from
TIMSS07 showed that students’ initial approach to the task was often
unsuccessful. In this situation it was then a call for justification that led
them on, for example through questioning why a solution was correct or
what the answer meant. The call for justification came from the teacher,
from other students or from a student’s wish to understand. An
implication of this study is that students would benefit from incorporating
justification as an essential part of their problem solving process.
Key words: algebra, patterns, justification, video data, TIMSS tasks,
problem solving
Introduction
Generalizations and patterns are often highlighted as key ideas in mathematics,
essential parts of early algebra and fundamental to algebraic reasoning (e.g. Cai and
Knuth 2011). Working with problems of detecting and/or generating patterns,
describing a term and its position in a sequence, is an approach to algebra as
generalization aimed at enhancing students’ insight into detecting sameness and
differences, making distinctions, repeating, ordering, classifying and labeling (Mason
1996, 83). Lee (1996, 106) writes: “As an introduction to algebra, an entry into the
culture, I think a generalizing approach is grounded historically, philosophically, and
psychologically and has proven its merits pedagogically wherever it has been tried.”
Although patterning has been acknowledged in school curricula in many
countries it has not been a prominent part of Swedish school textbooks or of
classroom practices. As a result of declining results on TIMSS tests, particularly
concerning algebra, patterning was explicitly introduced as core content for grades 16 in the Swedish National Curriculum Lgr11 (Skolverket 2011). In the rather short
mathematics syllabus part of the curriculum (pages 59-63 cover the whole syllabus for
mathematics in grades 1 through 9) patterning is mentioned twice, as core content for
grades 1-3 as well as grades 4-6, in the short, and quite general phrase: “How simple
patterns in number sequences and simple geometrical forms can be constructed,
described and expressed.” Currently patterning problems are making their way into
textbooks, and teachers are starting to do patterning in their classes. A general
question to ask in this situation is if teachers understand what students are supposed to
learn by doing patterning tasks. Will simply exposing students to patterning activities
result in better understanding of algebra, more qualified algebraic reasoning, higher
problem solving skills, and eventually show up as better results on future TIMSS
achievement tests? This was suggested in the official TIMSS report from the TIMSS
2007 test, which commented on the poor result on task M05_03 (see figure 1) with the
words “Since additive changes […] are not considered particularly difficult to encode,
the difficulties are probably due to lack of exposure to patterning” (Skolverket 2008,
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 109
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gestures becoming part of a linguistic repertoire that helps them notice and articulate
specific aspects of a pattern. In such a view on learning it is not possible to separate
the levels of perception, verbalizing and expressing a pattern since all these aspects of
patterning contribute to the linguistic repertoire that affords development of algebraic
thinking.
Patterning activities in textbooks are commonly structured to help students
generate patterns by asking them to continue the pattern, describe the next figure,
describe the pattern and a create a formula for the nth figure. The TIMSS task above
asks students to make use of a pattern they first need to detect, which makes it a
problem solving task without the scaffold of a guided step-by-step procedure. It is
thus more of a true problem to solve than a didactically designed patterning task. Lee
(1996) addresses the problem-solving issue of patterning using the term ‘perceptual
agility’ to describe the ability to see several patterns in a sequence of figures or
numbers and judging which patterns are useful. Such agility is closely related to
justification and argumentation. Through justification a pattern will be validated and
argumentation will support or refute the usefulness of different patterns.
From 1969 to 1994 Swedish curricula focused more on instruction about
problem solving and learning for problem solving than learning through problem
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 110
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
solving (Wyndhamn, Riesbeck and Schoultz 2000). Problem solving in the Nordic
countries during this time has been described as ‘applied problem solving’
(Zimmermann, 2001). The National Syllabus (Lgr 11) treats problem solving as both
core content and as an ability students should develop. Today there is a movement in
Sweden to work with ‘rich problems’ emphasising open problems, different solutions
and use of multiple representations (e.g. Haglund, Hedrén and Taflin 2010).
What value does problem solving and generalizing activities in school
mathematics have beyond learning the specific mathematics content embedded in the
problem? Zimmermann (2001, 57) lists some possible goals and characteristics of
problem-oriented mathematics instruction, such as “possibilities for the invention of
conjectures and their critical discussion, including refutations and proofs”,
“connecting thinking” and “opportunities for communication”. These goals bear many
similarities with the list of purposes for justification in school mathematics expressed
by 12 middle school teachers in a more recent study by Staples, Bartlo and Thanheiser
(2012). Justification, according to these teachers, promotes conceptual understanding
and fosters mathematical skills and dispositions. One teacher says:
Justification pushes students beyond a procedure to a deeper understanding of the
math. In order to justify their thinking, they have to justify not only the hows, but
get to the whys of what they’re doing. (454)
In classrooms the demand to ‘explain your thinking’ is often met by a verbalising of
the procedure, whereas a demand to ‘justify your solution’ could perhaps help student
develop mathematical reasoning and argumentation.
Method
Using a larger set of video data collected within the project VIDEOMAT (Kilhamn
and Röj-Lindberg 2012) this study is an analysis of small group discussions when
solving the TIMSS task presented above. This paper reports on 3 groups of students
from 2 different grade 6 classes working on the problem. The task was given by the
researchers following four lessons of introduction to algebra planned individually by
each teacher as part of the normal curriculum. The aim of the intervention was to
study how these students worked on the problems without specific instruction but
within the context of introductory algebra. The teachers handed out and gave
instructions to the group activity in slightly different ways for example concerning
whether they expected individual or group documentation. To encourage discussion
the final request in the original task to ‘Show the calculation’ was replaced by the
question ‘How do you know?’ The group discussions were video recorded and the
videos were then viewed many times by the author of this paper as well as other
members of the VIDEOMAT research team. Essential parts of the interactions were
transcribed. The analysis focused on students’ initial and subsequent strategies,
particularly changes of strategy, as well as the nature and effect of any teacher
intervention. The student groups spent 9 –14 minutes working on the problem.
Results and discussion
In this section the three groups, here called Alpha, Beta and Gamma, are presented
one at a time with analytical comments.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 111
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Group Alpha (S1T1-SG2)
This group of four girls only has one paper to write on and most of the time girl C
takes charge of the paper and does most of the talking. Initially A suggests drawing, B
wants to work it out with numbers, and C starts drawing squares. However, C keeps
losing track of how many she has drawn and the paper is too small. Her drawing is not
systematic. After 2 minutes they all start discussing number facts in search of some
factors for which the product is 73. They are off task for a few minutes discussing the
nine times table. After five minutes B raises her hand to attract the teacher’s attention
and starts a new drawing, this time completing one square at a time with three lines
for each. C counts as B draws, nodding her head for every count and making them
very clearly in threes: 23 24 25 ,…, 32 33 34. Then the drawing gets too small, and C
takes the paper and continues the drawing coming up with ideas of how to fit more
squares in. After 54 sticks the row of squares reaches the edge of the paper and B
takes over starting a new row of squares. After 11 minutes both A and B raise their
hands and the teacher finally joins them.
Teacher:
well then what have you done?
C:
we’re not done, because it’s too tiresome to draw all 73
B:
well we don’t know… Can’t you like, take 73 times 4?
Teacher:
what does that give you?
B:
I don’t know…
[A gestures that she has an idea, the teacher directs attention to her]
A:
is it, can’t you take, like, 73 divided by 3 minus 1. Because here,
it’s 3 otherwise and so if you divide those 3 and then you just take
away this first one here
Teacher:
yes, why did you think of that?
In the episode the teacher does not evaluate B’s suggestion but asks her to clarify. Girl
A, who has mainly participated as an observer, suddenly finds room to give a
suggestion that shows her perception of the addition of threes and the extra one,
possibly through the simultaneous drawing, nodding and counting of the others. The
teacher evaluates the solution and leaves, and at the end of 14 minutes the girls hand
in a paper with the solution 73/3 – 1, which is not quite correct. There is a slight call
for justification by the teacher helping the girls to get past the perceptual level and
beginning to verbalise the pattern, but the group never gets to the correct expression.
Group Beta (S3-SG1)
This group of two girls (B and D) and two boys (A and C) takes 9 minutes to solve
the problem. Like Alpha, they start by suggesting number facts (73·4, 73/4) and then
begin drawing squares, each one on their own paper. A, B and D spend the following
7 minutes drawing. A makes many drawings, rubbing them all out and starting over
several times. First she loses track of how many she has drawn. Then she gets
different total amounts (28 and 24). She checks both answers by multiplying 28·4 and
24·4. Neither of the products is 73 so they are slightly at a loss. C has suggested an
equation but keeps seeking a multiple of 4. When dividing 13/4 he realises that
12/4=3. C exclaims that 24 squares “feels right.” B starts a new drawing this time
systematically one square at a time counting 1234, 567, 8910…They hear from
another group that the answer is 24, but they are still not satisfied. After 7 minutes C
calls the teacher’s attention.
C:
Teacher:
look we counted how many we could do and it is 24.
why? [B looks up]
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
C:
B:
T:
because we counted, how many, of these there are for 73
or, you do 72 divided by no 73 divided by 3, and then minus 1.
that is very difficult to calculate. In which order should you take
minus and divided by? Why do you take minus 1?
In this episode, the students themselves want to find a justification, seek evaluation
from the teacher who answers by requesting a justification that goes beyond their
drawing. B who has been totally engrossed in her drawing suddenly looks up and
finds that she has seen the pattern and verbalises it. Again the teacher questions her
answer, asking her to explain why she takes minus 1 so that she can work out in
which order the subtraction and division need to be done. When this is resolved, they
compare ‘solution by drawing’ with ‘solution by the general expression’.
Group Gamma (S3-SG2)
The third group is a dysfunctional group with four members. Girl A works on the task
for 12 minutes, at times joined by boy B. C and D are mostly off task or trying to
copy what A is writing but never contributing with ideas. As in both the other groups
an initial strategy is to use number facts (13·4). Then A suggests an equation and
writes 3x=73, showing that she has perceived the pattern of multiples of three. She
calculates 73/3=24,3333… and the others copy her answer. They seem to have
finished when the teacher comes past. When seeing their answer, she questions the
result asking if it will not be an even number of squares, or a strange sort of square at
the end, one is not closed. Girl A laughs, and the teacher points to the picture asking
“what about the first square?” She leaves them to try again. Peers from another group
come by, telling them to divide 72 by 3 instead of 73 by 3. Girl A contemplates this,
wondering why. After some time she exclaims: “wait, you take it away to use at the
end don’t you!” She has now perceived the pattern and expressed it verbally. She
finally expresses the answer as “3x=72 and then +1”, and explains her solution to B.
In this episode the call for justification comes from the teacher when she evaluates the
initial solution and questions its validity. Also A herself feels a need for justification
once she knows the correct solution, and she is not satisfied until she can express
clearly why she has to take off an extra stick to add at the end.
Conclusions
A summary of the analysis shows that initial, but unsatisfactory, strategies were
similar in all groups. These were: using number facts, drawing squares, or writing an
equation. In the process of drawing systematically, in combination with gestures
and/or counting in 3’s, the pattern was perceived, but not readily verbalised. In each
group there was a turning point initiated by a call for justification of their first effort.
This call for justification came from the teacher or from students themselves. In
groups Alpha and Beta the teacher played a role of changing the pattern of
participation slightly, thus giving new ideas opportunity to surface. In group Alpha
the students finished when the teacher no longer asked for a justification, and
therefore the problem was not fully solved on the symbolisation level, whereas group
Beta were asked to justify also the order of operation and the reason for subtracting 1.
Undoubtedly, a massive exposure to similar additive pattern tasks would
probably result in better average achievement, but it might not make students better
equipped to solve slightly different problems. These findings suggest that students
would benefit from engaging in problem solving where the justification of a solution
becomes an essential part of the process. In addition to Polya’s four stages of problem
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
solving (Polya, 1990/1945), a fifth stage of justification and argumentation could be
added. The teacher’s role in small group problem solving activities is not so much to
guide students in a step-by-step procedure, or to evaluate their solutions, but rather to
ask them a) to justify what they are doing, b) to create opportunities for new ideas to
come forward and c) to expect valid mathematical argumentation.
Acknowledgements
This article is a result of research funded by the Joint Committee for Nordic Research
Councils for the Humanities and the Social Sciences (NOS-HS). The author is a
member of LinCS, a national centre of excellence for research on Learning,
Interaction and Mediated Communication funded by the Swedish Research Council.
References
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Kilhamn, C. and A-S. Röj-Lindberg. (in press). Seeking hidden dimensions of algebra
teaching through video analysis. In Nordic research in mathematics education,
past, present and future. ed. B. Grevholm. Oslo: Cappelen Damm.
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Approaches to algebra: Perspectives for research and teaching. eds. N.
Bednarz, C. Kieran and L. Lee. 87-106. Dortrecht: Kluwer Academic
Publishers.
Mason, J. 1996. Expressing generality and the roots of algebra. In Approaches to
algebra. Perspectives for research and teaching. eds N. Bednarz, C. Kieran
and L. Lee. 65-86. Dortrecht: Kluwer Academic Publishers.
Polya, G. 1990/1945. How to solve it. London: the Penguin Group.
Radford, L. 2001. The historical origins of algebraic thinking. In Perspectives on
school algebra. eds R. Sutherland, T. Rojano, A. Bell and R. Lins. 13-36.
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naturvetenskap i ett internationellt perspektiv. Rapport 323. Stockholm:
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leisure-time centre 2011. Stockholm: Skolverket.
Staples, M., J. Bartlo and E. Thanheiser. 2012. Justification as a teaching and learning
practice: its (potential) multifacted role in middle grades mathematics
classrooms. Mathematical behaviour, 31(4): 447-462.
Wyndhamn, J., E. Riesbeck and J. Schoultz. 2000. Problemlösning som metafor och
praktik.. Linköping: Linköpings Universitet.
Zimmermann, B. 2001. On some issues on mathematical problem solving from a
European perspective. In Problem solving around the world. Proceedings of
the topic study group 11 at ICME-9 in Japan 2000. ed E. Pehkonen. Turku:
University of Turku.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 114
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Social inequalities, meta-awareness and literacy in mathematics education
Bodil Kleve
Oslo and Akershus University College of Applied sciences
In this paper I take as a starting point social inequalities and pupils’
different learning possibilities as a result of their social background, and
consider mathematics on three levels: The level of Discourse, which
primarily encompasses cultural relations and communities of meanings in
school; the level of genre which concerns recognizable common cultural
texts and the frames of reference which support their understanding, and
finally, the level of paradigmatic and syntagmatic modes of thought
which are necessary for learning within mathematics. My argument is that
in order to decrease the school’s reinforcement of social inequalities,
teaching should be based on meta-awareness rather than acquisition
through pupils’ activities.
Keywords: mathematical discourse; genre; modes of thought.
Introduction
In 2006 a new curriculum reform, The Knowledge Promotion (LK06) was introduced
in Norway. The overall goal for this new curriculum was to raise the knowledge level
for all pupils in school and to change the school so that the impact of family
background on pupils’ school results should be less. In Norway, education is a
democratic right and social background should no longer be a reason for lack of
education. Yet, despite the democratization which has taken place, social inequalities
are increasing within the Norwegian educational system, as in many parts of the
world: educated parents foster educated children (Bakken 2004; Bourdieu 1995;
Zevenbergen 2001).
In taking the increasing social inequalities as a starting point I suggest that a
higher meta-awareness of both language and modes of thought will increase all
pupils’ possibilities for learning. The focus will be on pupils who are characterized as
previously low attaining in the school discourse. My argument is based on Bruner
(1986) and on other theorists who have developed his theories further. One of
Bruner’s main arguments is that we learn through the use of language and being
aware of the learning situation. The challenges will be addressed by taking a literacy
perspective which recognizes that mathematics as a school subject draws on a range
of discourses. Olson (1994) emphasizes that school subjects belong to different
textual communities, and to master a school subject is to develop the ability to
manipulate different texts:
To be literate it is not enough to know the words; one must learn how to
participate in the discourse of some textual community. And that implies knowing
which texts are important, how they are to be read and interpreted, and how they
are to be applied in talk and action. (273)
Gee (2003) emphasizes the difference between acquisition and learning,
reminding us that what many pupils already have acquired before they start school,
others have to actively learn. This is a problem which has been neglected in many
pedagogical reforms. Teaching which is mainly based on acquisition through pupils’
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
activities and not on meta-linguistic awareness will reinforce the differences which
are already there. Thus school can be looked upon as a reinforcement of social
inequalities.
Meta-linguistic awareness and literacy competence characterize the “winners”
in the Norwegian school (Bakken 2004), as in other countries. Many ‘weak’ pupils
find it difficult to distinguish everyday language and school language and these pupils
will also have difficulties in mathematics in their meeting with the new and strange in
the subject (Zevenbergen 2001).
The main purpose of this paper is to discuss pupils’ learning possibilities in
mathematics from a literacy perspective. The argumentation will take place on three
theoretical levels: I explore mathematics on the level of discourse, then I turn to the
level of genre, and thirdly I examine the implications of Bruner’s (1986) concept of
‘modes of thought’ in terms of ways of thinking and reasoning in the subject. First,
however, I start by discussing the impact of social inequalities for pupils’ learning,
and the role of their prior understanding about ‘the meaning’ of typical classroom
activities, that is, of playing the school game (Olson 2003).
Literacy and primary and secondary discourses
Pupils start school with different prior understandings about its activities and goals.
They have different experiences with books, literature and calculation, and different
affinities in relation to letters and numbers. These prior understandings, which
encompass experiences, language, habits, affinities and feelings, constitute what Gee
(2003) calls their “primary Discourse”.
The primary Discourse is a ‘value Discourse’ and is part of different networks
of meanings. It may, or may not, support school activities. Some pupils feel
comfortable at school because of a match with their primary Discourse, while for
others school may be more or less foreign. This is a challenge in a learning context.
School is more or less about constant meetings with new and different thinking and
texts, what Gee calls “secondary Discourses”. Ideally, the purpose of schooling is to
encourage openness to unfamiliar and new secondary Discourses.
Zevenbergen (2001) focuses on the potential difficulties pupils will meet in
mathematics classrooms. Like Gee, she emphasizes that pupils enter school and
mathematics classrooms with different social backgrounds and correspondingly
different language backgrounds. Drawing on Bourdieu, she argues that some pupils
are “predisposed” (47) to learn mathematics, not because of innate abilities but rather
because of their family habitus. These pupils are better equipped to cope with the
mathematical culture and to “position themselves more favorably in the eyes of their
teachers” (47). For others the opposite will happen, and success will be more elusive.
This initial habitus is also recognizable in Norwegian classrooms (Penne 2006).
In this paper, sociocultural differences in mathematics classrooms in Norway
are recognised. According to Gee (2003), literacy for pupils is a question of mastering
secondary Discourses. Pupils meet them in school and in mathematics. A precondition
is meta-awareness in the learning process incorporating contextual understanding and
interpretation.
Discourses, genres and modes of thought- three levels in the teaching/learning
process
In order to discuss the challenges sociocultural differences play for mathematics
teaching and learning, I consider mathematics on three levels: The level of Discourse,
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
which primarily encompasses cultural relations and communities of meanings in
school; the level of genre which concerns recognizable common cultural texts and the
frames of reference which support their understanding, and finally, the level of
paradigmatic and syntagmatic modes of thought (Bruner, 1986) which are necessary
for learning within mathematics. Developed by Olson as “modes of apprehension” in
the school context, these are “the frames of reference in terms of which children and
adults formulate their experience, the major modes in which they define the
discourses or disciplines that are the concern of schooling” (2003, 156). Thus one
learns to reason or think as a mathematician.
The level of discourse
A Discourse is a kind of ‘community of meaning’, of ways of thinking to understand
the world or a part of the world. Discourse gives meaning, a feeling of inclusion and
identity, for example in the profession of teaching. Within a Discourse, some frames
may be obvious while others are in motion, formulated by Gee (2001) as follows:
We can think of Discourses as identity kits. It's almost as if you get a tool kit full
of specific devices (i.e. ways with words, deeds, thoughts, values, actions,
interactions, objects, tools, and technologies) in terms of which you can enact
specific activities associated with that identity. (720)
Mathematics teachers are located within a Discourse or “identity kit” as is the
textbook in the subject. To mathematics teachers the Discourse is creating an implicit
world of knowledge or experience. However, from some pupils’ point of view, what
is obvious to teachers may not be certain. Some have a background providing them
with access towards unfamiliar Discourses or secondary Discourses, but others will
not recognize these without support from the teacher. Thus Solomon (2009)
emphasizes the teacher’s role in supporting mathematical literacy, and I agree that
this can only be facilitated through intervention from the teacher which makes rules,
language and nature of arguments in the subject more explicit. The only way pupils
can become party to what is frequently implicit knowledge is through awareness of
mathematics as a secondary Discourse.
According to Dowling (2001) formal mathematics is often projected onto a
practical task for the less able pupils, for example shopping, in the public domain. As
Walkerdine (1988) pointed out, the use of numbers in shopping is not the same as
studying number relationships in mathematics in the esoteric domain. Thus
mathematics presented in an everyday discourse may be embedded in practical tasks
and ‘less able’ pupils will not gain the desired access to the subject. As a result pupils’
predispositions for mathematics, or lack of such, will be reinforced at school.
Similarly, Kleve (2007) reported that perceived low-attaining pupils were
taught mathematics differently from high-attaining pupils in Norway. Low-attaining
pupils were confronted with more rote learning and focus on methods and procedures,
in comparison with pupils who were perceived to be more able. Furthermore the lowattaining pupils were not challenged in the same way to make connections between
different areas of mathematics.
The genre level and pupils’ prior understanding
Although there is much discussion in the literature about the relationship between
discourse and genre, I will adhere to Hyland’s (2003) definition of genre as follows:
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Genre refers to abstract, socially recognised ways of using language. It is based on
the assumptions that the features of a similar group of texts depend on the social
context of their creation and use, and that those features can be described in a way
that relates a text to others like it and to the choices and constraints acting on text
producers. Genres, then, are the effects of the action of individual social agents
acting both within the bounds of their history and the constraints of particular
contexts, and with a knowledge of existing generic types. (21)
Although a variety of genres are expressed in our curriculum, and teachers
themselves draw on these genres, research suggests that genres are rarely made clear
for pupils, who may lack the same control of genre. The challenge for teachers is to be
explicit about their use of genres and teach genres explicitly. Successful pupils come
to school with sufficient pre-understanding. Less successful pupils need the teacher’s
assistance to understand the implicit rules of genre in the subject (Solomon 2009). For
Hyland (2003), an approach which is sensitive to genre offers
… the most effective means for learners to both access and critique cultural and
linguistic resources … The provision of a rhetorical understanding of texts and a
metalanguage to analyze them allows students to see texts as artifacts that can be
explicitly questioned, compared, and deconstructed, so revealing the assumptions
and ideologies that underlie them. (125)
Prior understanding opens up the text’s meaning as linked to a cultural
community of meaning. It is the same issue in mathematics. Solomon (2009)
emphasizes the importance of awareness of genre in all subjects, also in mathematics.
Despite not being evident in mathematics classrooms, a wide range of genres are
being used. Graphs, for example are means of communicating information and
express meaning. Also mathematical definitions, proofs, equations, algorithms and
statistical tables are considered as expressions of genre. In the mathematical part of
the curriculum in Norway (Kunnskapsdepartementet 2006) these are integrated as
competence aims, which encompass a variety of genres in line with the description
presented by Marks and Mousley (1990):
In solving problems, writing reports, explaining theorems and carrying out other
mathematical tasks, we use a variety of genres...Events are recounted (narrative
genre), methods described (procedural genre), the nature of individual things and
classes of things explicated (description and report genres), judgments outlined
(explanatory genre), and arguments developed (expository genre). (119)
Genres may be discursively expressed, but they will always be more than this.
On one hand they represent different textual traditions. On the other hand genres are
part of successful pupils’ prior understanding; they are frames for understanding,
necessary for academic development and may be used as interpretive lenses (Bruner
1986; Feldman and Kalmar 1996). Many pupils need a specific prior understanding to
decode the genre signs necessary for a relevant interpretation of the text (CochranSmith 1994).
The research reviewed here demonstrates the importance for pupils to gain
awareness of mathematics discourse as well as learning about genre in the subject.
Discourse and genres make mathematics what it is.
Awareness of different modes of thought in mathematics
As a last point, awareness of different modes of thought as a prerequisite for learning
is discussed. Suggesting that it is necessary, but not sufficient, to work on the level of
discourse and genre, and building on Bruner’s (1986) distinction between
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
paradigmatic and syntagmatic modes of thought, I argue that working with
mathematics requires both modes of thought, or ‘modes of apprehension’.
For Bruner, the paradigmatic mode of thought is linked to a scientific way of
thinking that requires arguments based on decontextualized generalizations and
explanations (as in the case of mathematics). It requires the acknowledgement of an
unchangeable, permanent, abstract system. The syntagmatic mode of thought is
primarily narrative and requires hermeneutical ways of reasoning, and as such
contextualized interpretations. Bruner (1986) writes:
Let me begin by setting out my argument as baldly as possible, better to examine
its basis and its consequences. It is this. There are two modes of cognitive
functioning, two modes of thought, each providing distinct ways of ordering
experience, of constructing reality. The two (though complementary) are
irreducible to one another. Efforts to reduce one mode to the other or to ignore
one at the expense of the other inevitably fail to capture the rich diversity of
thought. (11)
The syntagmatic mode communicates an ‘experienced’ world, and is more or
less subjectively based and therefore cannot communicate absolute truth but, rather,
verisimilitude. We therefore have to interpret within contexts, within which parts can
be explained in the light of wholes and vice versa. In communicating and thinking in
the syntagmatic mode, the narrative structure is the most pervasive cognitive schema
(Bruner 1986). For Bruner it is unrealistic to suppose that the two modes can be
separated and that we can choose the one over the other.
Although, as Mason and Johnston-Wilder (2004) point out, people deal with
generalizations and abstractions all the time, in mathematics generalizations are
expressed in a succinct notation from which further conclusions, particular or general,
may be drawn: “Mathematics deals with relationships per se, and so context is of the
least importance; hence the prevalence of abstractions in mathematics” (132, my
emphasis). Oatley (1996) refers to Bruner’s ‘two modes of thought’ claiming that
objects expressed in the narrative or syntagmatic mode slips easier into the mind
whereas the mind is more resistant to objects expressed in the paradigmatic mode. He
refers to how Newton’s third law can be explained either narratively (syntagmatic) or
with a mathematical equation (paradigmatic mode of thought). He thus emphasizes
the need for both modes of thought in physics.
Meta-awareness in the learning process, why is it so important?
In this paper I have argued for meta-awareness for all pupils. The starting point was
social inequalities and pupils’ different learning possibilities as a result of their social
background, which forms their primary Discourse. Meta-awareness and literacy
competence characterize the winners in school. However, meta-awareness should not
be reserved for those whose social background, or ‘value Discourse’ supports school
activities. To decrease the school’s reinforcement of social inequalities, teaching
should be based on meta-awareness rather than acquisition through pupils’ activities.
My argument has been on three levels; discourse, genre and modes of thought. On the
level of Discourse, I have argued that the only way pupils can become party to
implicit knowledge is through awareness of mathematics as a secondary discourse.
The teacher plays a crucial role in this work. Also, it is important that the ‘less able’
pupils not only should be presented mathematics in an everyday discourse, because
then they will not gain the desired access to the subject. On the level of genre, it is
important for the teachers to be explicit about genres and to help pupils establish
sufficient pre-understanding. Finally, the argument has been that both modes of
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
thought, paradigmatic and syntagmatic are necessary for all pupils in the mathematics
learning process.
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Cochran-Smith, M. 1994. The making of a reader. New Jersey: Ablex Publishing.
Dowling, P. 2001. Reading mathematics texts. In Issues in mathematics teaching, ed.
P. Gates, 180-196. London: RoutledgeFalmer.
Feldman, C. F., and D. A. Kalmar. 1996. “Autobiography and fiction as modes of
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the process writing mistake again? Language and Education no. 4 (2):117135.
Mason, J. and S. Johnston-Wilder. 2004. Fundamental constructs in mathematics
education. London: RoutledgeFalmer.
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reading and writing. Cambridge: Cambridge University Press.
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selvforståelse og identitet gjennom språk og tekster. (Dr. polit), Dr. polit
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Stimulating an increase in the uptake of Further Mathematics through a
multifaceted approach : Evaluation of the Further Mathematics Support
Programme.
Stephen Lee and Jeff Searle
Mathematics in Education and Industry and Durham University
Over recent years there has been a marked increase in the number of
students studying A-level Further Mathematics in England. In 2012
12,688 students sat the qualification, with the numbers having more than
doubled from 5,627 in 2005 (Joint Council for Qualifications figures). The
increase has been evident despite the common perception that Further
Mathematics is a difficult subject.
The work of Mathematics in Education and Industry’s (MEI) governmentfunded Further Mathematics Support Programme (FMSP) has been highly
influential in stimulating this increase through not only enabling all
students who wish to study Further Mathematics to have access to tuition,
but also through supporting teachers and students in schools and colleges
in a variety of ways.
An external evaluation of the FMSP has been undertaken by the Centre
for Evaluation and Monitoring at Durham University. This paper reports
on aspects of the evaluation and how these relate to the multifaceted
approach taken by the FMSP to increase participation in Further
Mathematics, including: innovative tuition models, enrichment events,
extensive provision for teachers to undertake professional development
and also an insight into direct attempts by the FMSP to engage with
schools and colleges who have not traditionally offered the subject.
Keywords: Further Mathematics, evaluation, tuition, continued
professional development.
Introduction
This paper assumes some familiarity with the UK education system. In brief, most 16
year old students sit formal examinations in subjects including mathematics, known as
GCSE or level 2 qualifications. At this stage, academic-pathway students choose to
specialise in three or four subjects. Those who wish to continue their study of
mathematics to level 3 (advanced level) take A-level Mathematics and in addition
they can take A-level Further Mathematics. Both advanced courses in mathematics
are available at advanced subsidiary (AS) level, usually a one year course, or the full
A-level (A2) which is usually a two year course. Many students choose to take the
full A-level in mathematics and the AS-level in Further Mathematics,
Those who study A-level Further Mathematics are exposed to additional and
new material beyond that found in A-level Mathematics. Several topics met in A-level
Mathematics are developed further such as integration and differentiation, but some
completely new topics are studied in A-level Further Mathematics, such as complex
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
numbers and matrices. The study of applications of mathematics in mechanics,
statistics and decision mathematics met in A-level Mathematics can also be extended
in Further Mathematics, as well as the topics in pure mathematics.
Background to the FMSP
The services currently offered by the FMSP to students and teachers have evolved
over time. Input from external evaluations has played a part in this evolutionary path.
What has now become the FMSP started from a suggestion by a practising
teacher prior to the year 2000 who was worried about the decline of Further
Mathematics. He suggested that MEI should determine if anything could be done to
stem the decline. Subsequently, MEI responded by initiating a pilot project having
successfully obtained funding from the Gatsby Charitable Foundation. Details of the
pilot project entitled ‘Enabling Access to Further Mathematics’, including how it was
structured, can be seen in Stripp (2002).
The pilot project was deemed to be a success and was highlighted in the report
Making Mathematics Count by Professor Adrian Smith on post-14 mathematics
(2004). Searle (2010, 2008) gave some discussion of the concerns expressed around
2000 by academics in mathematics and other STEM subjects as to the lack of
readiness of terms of knowledge and fluency in mathematics seen in applicants to
degree level courses. Subsequently, in 2004, MEI received funding from the
Department for Education and Skills to enable their pilot to be rolled out nationwide
in England, with the project becoming known as the Further Mathematics Network
(FMN). The basic structure of the FMN was locally based management teams
supported and directed by a national central team. The activities of the locally based
management teams led to increasing engagement with many schools and colleges and
their students and teachers, and the number of students taking Further Mathematics
began to grow again. The number of students who took the full A-level in Further
Mathematics increased during the lifetime of the FMN from about 5000 to over
9000(Stripp 2007; Searle 2008).
In 2009 a new contract was awarded for a national Further Mathematics
Support Programme (FMSP), which was based on the FMN. MEI won the
competitive tender to manage the project centrally (Stripp 2010). As well as the
central team of MEI staff, in 2012, the locally based management of the FMSP is
through 30 Area Coordinators, who are employed by schools and universities and
Local Authorities. The Area Coordinators are now the primary facilitators of day-today engagement with schools and colleges and their students and teachers.
FMSP’s multi-faceted approach to increasing the uptake of Further
Mathematics
The primary goal of the FMSP is to give every student who can benefit from studying
Further Mathematics the opportunity to do so. In order to achieve this, a multi-faceted
approach has been developed. This approach involves a number of strands of activity,
including:
 Innovative tuition models in Further Mathematics
 Enrichment events which aim to inspire students
 A range of opportunities for teachers to undertake professional
development
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012

Direct attempts to engage with schools and colleges who have not
traditionally offered the subject
Student tuition
A vital aspect of the national FMSP is its flexibility to meet the particular needs of
students and teachers at a local level. For example student tuition by the FMSP is
provided in a number of ways, including:
 Face-to-face tuition (very small classes and/or involving school
consortia)
 Live Online Tuition (LOT)
 A mixture of the two – Live Interactive Lectures for FM (LIL FM)
To support students preparing for examinations in both A-level Mathematics
and Further Mathematics the FMSP offers a revision programme that is also flexible
in that it involves online revision events and/or face-to-face events. Student
participation in the recent live online events was quite large. Thousands of students,
along with a number of teachers, accessed the sessions. The live sessions are recorded
and many more thousands of students and teachers viewed the recordings when they
were made available after a live session had ended.
Enrichment
The FMSP also offers enrichment events. These events aim to inspire students in
mathematics both at Key Stage 4 when they are studying for GCSE and also whilst
they are studying at advanced level. There are a number of enrichment opportunities
offered by the FMSP:
 Year 10 Team Mathematics Challenge (for students aged 14/15).
There are over 50 regional events, involving over 1000 schools and over
4000 students.
 Senior Team Mathematics Challenge in collaboration with the United
Kingdom Mathematics Trust (for students aged 16/17).
There are over 50 regional events, involving over 1000 schools and
colleges and over 4000 students. There is also a national final.
 In 2012/13 60 enrichment events for Key Stage 4 students (aged 15/16)
are taking place.
These events enable students to meet new ideas in mathematics and its
applications, as well as being given challenging problems to solve.
 Other one day events for various age groups including themes such as
‘Maths Works’ and ‘Taking Maths Further’.
Professional development
The FMSP has developed a variety of opportunities for teachers to undertake
professional development in the teaching of advanced mathematics. These
opportunities include:
 Face-to-face events
 Live Online Professional Development (LOPD) courses
 Extended 15 month professional development courses (Teaching Further
Mathematics (TFM) Teaching Advanced Mathematics (TAM))
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Many teachers take up these opportunities; during the academic year 2011-12
there were over 1000 teacher days of participation in professional development
offered by the FMSP.
Direct engagement
As might be expected with any national project there are a number of schools and
colleges who, for whatever reason, don’t engage with it. The FMSP has made direct
attempts to engage with schools and colleges who have not traditionally offered
Further Mathematics, some of which have led to the school or college now offering
Further Mathematics. Specific events like the Access to Further Mathematics
conferences for senior school leaders and teachers have acted to inform and advise
those unsure of the benefits, to them and their students, of having Further
Mathematics in their post 16 curriculum offer. These events too have resulted in
some schools and colleges now offering Further Mathematics.
Evaluating the FMSP
The Centre for Evaluation and Monitoring (CEM) at Durham University has
conducted external and extensive evaluation of the FMSP since its inception in 2009,
and of the FMN before then. To date, there have been three reports on the FMN (two
interim and one final) and three reports on the FMSP (one interim and two end of
Phase
reports,
see:
www.furthermaths.org.uk/fmnetwork_impact.php).
A
comprehensive review of many of the activities of the FMSP highlighted in the
previous section has been included in these evaluation reports. A large numbers of
interviews and surveys were conducted; teachers, students, event participants,
stakeholders and Area Coordinators were all involved. The evaluators also observed a
range of events first hand.
As well as receiving direct feedback on the activities of the FMSP as above,
the evaluators at CEM also reviewed student take up and achievement data year-onyear in AS and A-level Mathematics and Further Mathematics. Data on the 2009 and
2012 entries can be seen in Table 1. The percentage change in entries between the two
years is also displayed, as is a comparison between 2005 and 2012, which is the
lifetime to date of MEI’s Further Mathematics project.
Table 1 AS/A-level Mathematics and Further Mathematics certifications in 2009 and 2012 (Source:
Joint Council for Qualifications)
2009
2012
12688
2009-2012
percentage
change
26%
2005-2012
percentage
change
125%
A-level
Further
Mathematics
AS-level
Further
Mathematics
10073
12710
20370
60%
324%
A-level
Mathematics
AS-level
66552
78951
19%
64%
95408
139585
46%
46%
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Mathematics
The complete evaluation can be seen on the FMSP website (
www.furthermaths.org.uk/fmnetwork_impact.php). Included within the evaluation
reports was a summary that stated:
The FMSP is an effective and successful organisation, evidenced by the growth in
student numbers and the positive feedback from teachers when interviewed from
the perspective of a range of activities. Searle (2012, 31)
It went on to say:
The work of the FMSP is highly valued by students, teachers and more generally
by stakeholders, and this work should continue. Searle (2012, 31)
In summary
A brief overview of the strategies employed by the FMSP to enable any student who
could benefit from studying Further Mathematics to do so has been provided.
Support for students and teachers has been at the heart of the success of the
FMSP. Student support includes tutoring, enriching, and inspiring students in
mathematics. Teacher support includes professional development, advice, guidance
and information in developing Further Mathematics in their school or college. It is
predominantly the actions and enthusiasm of the Area Coordinators to meet needs and
demands in their local area that has now enabled many more students to study Further
Mathematics and teachers to teach Further Mathematics in a way that engages and
motivates students.
References
Searle, J. 2008. Evaluation of the Further Mathematics Network. In Improving
Educational Outcomes Conference, Durham University.
Searle, J. 2010. Investigating the impact of the Further Mathematics Network.
Proceedings of the British Society for Research into Learning Mathematics,
30 (1): 207-214.
Searle, J. 2012. Evaluation of the Further Mathematics Support Programme 20092012 - Summary Report: August 2012
www.furthermaths.org.uk/fmnetwork_impact.php
Smith, A. 2004. Making Mathematics Count: The report of Professor Adrian Smith’s
Inquiry into post-14 mathematics education. London: DfES.
Stripp, C. 2002. Enabling access to Further Mathematics. MSOR Connections, 2 (4):
19-22.
Stripp, C. 2007. The Further Mathematics Network. MSOR Connections, 7 (2): 31-35.
Stripp, C. 2010. The end of the Further Mathematics Network and the start of the new
Further Mathematics Support Programme. MSOR Connections, 10 (2): 35-40.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 125
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Exchange as a (the?) core idea in school mathematics
John Mason
University of Oxford and Open University
I propose that exchange is a core idea underlying much of school
mathematics. Alerted by young children struggling with the difference
between coins as objects and coins as having value, I began to explore the
action of exchanging one thing for another. If exchange is augmented to
include substitution then it shows up everywhere, from counting to
algebra, from money to currency, from ratio to algorithms and Turing
machines.
Introduction
The phenomenon of interest is children in years 2, 3 and 4 who when shown some
play-coins and asked “how much is there?” respond by counting the number of
objects rather than adding their total value. Primary teachers have been quick to tell
me that young children do not get to use coins in the way they themselves did when
young, because of credit cards etc.. Nevertheless there is an important awareness
which underpins not only mathematics but ordinary life, in which things have value(s)
and sometimes you are expected to attend to the quantity and sometimes to the value.
I began the session therefore with the observation that prior to the act of
counting, which requires coordinating the physical action of pointing with the verbal
act of reciting a memorised cultural poem, there is the physical action of exchanging
one thing for another, repeatedly. Thus
Task 0: I have a pile of red counters (all the same size) and you have yellow counters.
Exchange each of my red counters for a yellow counter until all the reds are gone.
What mathematical action is involved?
At the heart of this action is the awareness of one-to-one relationship. Here I am using
awareness in the sense of Gattegno (1987; see also Young and Messum 2011) to
mean ‘that which enables action’. However, the action of exchange depends on
discerning and distinguishing both the entity-ness of individual counters, the colours
of the counters, and distinguishing the red counters from each other without being
concerned about minor imperfections in the colouring or the shape. It also requires
some fine motor control, and sufficiently focused attention to complete all the
exchanges, repeating the exchange action over and over. Finally, there is an
expectation that repetition of the act of exchange is not simply a repeated physical act,
but is accompanied by some sort of growing sense of the act of exchanging ‘one thing
for another’.
In this and the following tasks my question is about the mathematical action,
but this question is for the teacher not the child!
Tasks involving such an exchange can be set in many different contexts,
changing the red counters to other objects. Also there can be a practice of lining up
the reds and the yellows as the exchange takes place. Someone commented that the
language of this task might be demanding for young children; however here I am
concerned with the mathematical awarenesses. I leave to primary experts how to
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phrase such tasks. I am confident that children will quickly learn what exchange
means through being immersed in such tasks.
At some point these exchanges become related to the act of counting (uttering
items from the verbal ‘poem’ consisting of number names) so that cardinality
becomes available as a focus of attention in exchange tasks.
Task 1: I have a pile of red counters. I exchange each of them for 3 yellow counters.
What mathematical action is involved?
The underlying awareness is what we (later) call multiplication. As Dave Hewitt
observed, if you attend to the exchange, you experience scaling (one to three); if you
attend to the growing pile of yellows you experience repeated addition. These are two
vital aspects of multiplication, but scaling gets overlooked when children are led to
believe that ‘multiplication is repeated addition’ rather than that ‘repeated addition is
one form of multiplication’. Note that engaging in one or two similar exchanges is
preliminary to but not the same as internalising a deep sense of exchange, and
different again from becoming consciously aware of the generality that is being
instantiated: any number of red counters, exchanging them for some specified number
of yellow counters. So far so good.
Task 2: I have a pile of red counters. I exchange 5 reds for 1 yellow and do this until I
can make no more exchanges. What mathematical action is involved?
With some encouragement people responded with terms such as ratio, division and
division with remainder. At some point in this sequence one could invite children to
exchange, say 5 small red counters for 1 large red counter. The notion of ‘value’
arises from context (a large red is ‘worth’ 5 small counters) as a subsidiary but
important awareness. Note however that the relative sizes of coins do not indicate
their relative value. Thus it is vital when attending to size to vary whether the larger
counter is worth more or less. This can be augmented by having large objects worth
the same or less than smaller objects when engaging in play-shops and other exchange
activities.
The task can be augmented by inviting children to explore what numbers of
red counters, once exchanged, end up with only yellow counters, or with exactly 1
yellow counter.
Now things get a bit tricky.
Task 3: I have a pile of red counters. I exchange 5 reds for 2 yellows and do this until
I can do no more exchanges. What mathematical action is involved?
Different ways of attending to the action might lead to different awarenesses. For
example, there is a doubling and a dividing by 5. If there is a remainder then the left
over reds are ‘worth’ 2/5ths of a yellow, so perhaps what is going on is multiplication
by 2/5, or multiplying by 2 and dividing by 5. However:
Task 4: I have a pile of red counters. I exchange 1 red for 2 browns, and 5 browns for
1 yellow. What mathematical action is involved?
I have a pile of red counters. I exchange 5 reds for 1 green, and 1 green for 2
yellows. What mathematical action is involved?
Essentially, the result can sometimes depend on order: if you double first and then
divide by 5 you may have some brown left over; if you divide by 5 first you may have
some red counters left that cannot be exchanged. For example, starting with 18 reds,
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
the first exchange rules end up with 7 yellows and 1 brown, while the second
exchange ends up with 6 yellows and 3 reds. How are these to be reconciled?
It depends on having absorbed the notion and language of value, so that seeing
each red as worth 2 browns, 3 reds are worth 6 browns which is ‘the same as’ 1 extra
yellow and 1 red remaining. Thus the exchanges can be reconciled, but, I suspect,
only after having encountered and developed some fluency with the language of
‘value’. This in turn can be supported by being immersed in many simpler exchanges
in many different contexts over a considerable period of time. Helen Williams
(workshop at ATM Easter 2012) has videotape of children engaged in a variety of
exchange tasks in different contexts, ending up with an auction in which it seems that
at least some of the children haven’t really grasped what bidding is about!
It is worth noting that Valerie Walkerdine (1988) challenged the practice of
using unrealistic values for pretend objects when trying to get children to work with
tasks.
I then went on to provide evidence that exchange, often in the form of
substitution, pervades school mathematics. A slight difference between these notions
is that for some people exchanges are reversible, while substitutions may not be.
Barter and Exchange
Barter has taken place long before and well after the introduction of money. For
example, there are amazingly complete records of exchanges in the town of Prato
(now a part of Florence) over two hundred years (Marshall 1999, 72-73). Here are
three instances:
Task 5 : I will exchange 3 of my sheep for 5 of your geese; I know I can exchange 7
geese for a colt …
As a baker I will exchange 12 loaves of bread for use of your horse for a day
As more and more people became merchants, it was necessary to educate sons into
the mechanics of barter. The renaissance painter Piero de la Francesca (1412-1492)
was asked by his patron to write a book for young men to learn arithmetic and in it
there are tasks such as
Task 6: Two men want to barter. One has cloth, the other wool. The piece of cloth is
worth 15 ducats. He puts it up for barter at 20 and 1/3 in ready money. A cento of
wool is worth 7 ducats. What price for barter so that neither is cheated?
I had to be helped to see that what it means is that the barter price is 20 but that 1/3
must be in cash (this at a time when coins were scarce). The solution provided
involves dividing 56 by 5:
Treat the 1/3 ready money as 1/3 of 20, that is 20/3. Reduce both the original and
the barter prices by the amount of ready money: 15 – 20/3 and 20 – 20/3, namely
25/3 and 40/3. The ratio of these gives the ‘inflation’ proportion required, namely
40/25 = 8/5 (and involves a division by 25/3). Then the wool merchant should
barter at 7 x 8/5 = 56/5 ducats, and this agrees with the answer given by
Francesca.
Adolescents sometimes like collecting ‘cards’ showing football players or the like,
and they engage in swaps such as “You can have any three from this pile in exchange
for … (two specific cards)”.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Functions
Whenever there is an input-output relationship there is a form of exchange, or at least
substitution going on: I exchange an x for the value associated with x. This applies
(without any formal language using x or f) to look-up tables such as timetables,
vending machines and so on. Often the exchange is one way: it cannot be reversed, or
cannot be reversed uniquely.
It is well known that the awareness which underpins functions and the
language of f(x) for functions remains mysterious for many students. Graphs of
functions are literally the coordination of input with output, which can be ‘seen’ as a
form of exchange.
Attention directed to functions, as elsewhere in algebra, often focuses on the
mechanics of manipulating symbols. Thus substitution into functions to find the
function values presents obstacles to students who have no mental images with which
to make sense of the act of substitution; perhaps exchange could provide that enactive
foundation. A plausible conjecture might be that with extensive experience of
exchange, and having integrated the discourse of exchange into their vocabulary,
students might not find the notion of function so abstract.
Task 7: If a configuration of n identical hexagons forms a shape with 4n + 2 edges on
its perimeter, how many edges will be made by 3n + 2 such hexagons in the
corresponding shape?
The multiple use of n is an obstacle for many, when all that is signalled is exchanging
each n in the formula 4n + 2 with the expression 3n + 2. Imagine the tension for
Scandinavian countries in which the pronunciation of the words for ‘one’ and the
letter ‘n’ are very hard to distinguish!
Patterns and Relationships
Task 8: You are shown the first three terms of a
sequence of black and white pictures, each
generated from the previous by means of the same
rule. How many little squares will there be in the
nth picture and what will be the proportion of
black squares?
What might be interesting in this task is to catch yourself looking for and trying to
articulate a relationship, which is presumably what people mean by ‘pattern spotting’.
If the relationship is ‘the same’ between each successive pair, then there is a property
which is being instantiated, and that will serve to generate pictures farther along in the
sequence.
In the session there was little time so I
directed attention to the way in which each
picture after the first is used to generate the next
picture. Then I showed the second sequence so
that participants could rehearse that particular
way of looking.
The underlying perception is that each
square is replaced with a 3 by 3 square, coloured according to a specific and invariant
rule. A great deal of ‘pattern spotting’ that is currently used to stimulate pre-algebraic
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
thinking involves substitution of something for something else, and expressing that
relationship as a property.
Newton’s Principle
Newton formulated an awareness that people develop spontaneously through
enaction, even if they do not articulate it the way Newton did: if you have a collection
of masses, then you can treat the system as a single mass (the sum of the masses)
concentrated at the centre of mass of the system. Statics as a part of mechanics
depends on this observation. But there are some slightly counter-intuitive aspects!
Task 9: Where is the centre of mass of three equal masses placed at the vertices of a
triangle, or four at the vertices of a quadrilateral?
Where is the centre of mass of three rods forming the edges of a triangle, or
four rods being the edges of a quadrilateral?
Where is the centre of mass of a uniform sheet of triangular (quadrilateral)
material?
It turns out that for a triangle two of these must coincide but the third only coincides
for special triangles, while for quadrilaterals, all three are in general slightly different.
(Thanks to an ATM workshop led by Jayne Stansfield for reminding me of this.)
Number Necklaces
I tried to show an animation from the internet (Von Worley 2012) which displays n
circles distributed around a circle in such a way as to display all the factors of n. My
question was going to be “what can one do with this?” and whether participants saw it
as involving substitution in the way that I do. Here are some sample frames:
Frames for 9, 14, 20 and 30
Actions
Whenever a mathematical investigation proceeds by locating and working with
actions that preserve some property in the objects acted upon, there is a form of
exchange going on. Any configuration can be replaced by the result of one of the
actions. Mathematical attention then focuses on the actions and how they are related.
For example, the inverse relation between addition and subtraction is a relation
between the actions of ‘adding n’ (for some n) and ‘subtracting n’, and likewise for
multiplication and division, exponentials and logarithms, differentiation and
integration. Furthermore, the properties of arithmetic (commutativity, associativity,
distributivity) that provide the properties for manipulating algebra, are relationships
between actions, and can again be seen as a form of exchange.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Statistics
Newton’s principle alerted me to the fact that every ‘statistic’ is a summary of a set of
data, and as such it stands for or (re)presents the original data. We exchange the mass
of data for statistical information such as the mean, median or mode, but also upon
occasion the maximum and the minimum.
The Whole of Algorithmic Mathematics
My final example involves Markov Sequences related to Post Productions. For
example
Task 10: Given a sequence of symbols such as $AAAAAAAA$BBBBB$, you are
permitted to replace any occurrence of A$B by AA$ and any occurrence of $$ by $.
What mathematical action is being enacted by carrying out all possible replacements,
over and over?
Interpreting $AAAAAAAA$ as a presentation of 8, and similarly for the Bs, the
replacement rule effectively calculates the sum of two numbers. Now construct a
similar replacement rule that will subtract two numbers. A little thought coupled with
appreciation of the previous example leads to the rules A$B is to be replaced by $,
and $$ is to be replaced by $. Finding a way to multiply and divide is rather trickier
but can be done. Furthermore, the action of any Turing machine can be presented by
replacement rules like these, so that exchange lies at the heart of all algorithmic
mathematics.
Summary
As with all mathematical topics, what matters is not the specific exercises or tasks, but
provoking students to be aware of the generality being instantiated.
Exchange certainly lies at the heart of the awarenesses that underpin counting
and basic arithmetic. It seems that in the form of substitution it underpins much of
school mathematics. The examples of exchange presented here were meant to
illustrate the pervasiveness of exchange in school mathematics, and are certainly not
exhaustive. Might it be the case that real appreciation of and familiarity with
exchange in the early years could provide the foundation for many more students to
find mathematical thinking both attractive and understandable? Yet to be considered
is whether geometrical thinking involves exchange in any substantive way.
References
Gattegno, C. 1987. The Science of Education Part I: theoretical considerations. New
York: Educational Solutions.
Marshall, R. 1999 The Local Merchants of Prato: small entrepreneurs in the late
medieval economy. Baltimore: Johns Hopkins University Press.
Von Worley, S. 2012. Dance, Factor, Dance. http://www.datapointed.net, accessed
Dec 2012.
Walkerdine, V. 1988. The Mastery Of Reason. London: Routledge & Kegan Paul.
Young, R. and P. Messum. 2011. How we learn and how we should be taught: An
introduction to the work of Caleb Gattegno. London: Duo Flamina.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 131
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Exploring the notion ‘cultural affordance’ with regard to mathematics software
John Monaghan and John Mason
University of Leeds; University of Oxford and Open University
About 10 years ago the Gibsons’ notion of ‘affordance’ was extended to
cultural objectives underlying designed computer systems. Chiappini
(2012) extends this idea to mathematics software. We critically, but
respectfully, review these extensions – does ‘cultural affordance’ add
anything new to valuations of software for doing mathematics?
Keywords: affordances, constraints, culture, mathematics, software
Introduction
This paper is an exploration of the construct ‘cultural affordance’ (CA), and whether
it adds anything to what we might, as mathematics educators, call ‘rich software (SW)
environments’. In the opening sections the first author outlines the genesis and
development of the construct ‘affordance’, culminating in a recent extension applied
to evaluating a SW system designed for learning/doing algebra. Then each author
critically considers these developments of the construct ‘affordance’. The paper ends
with an overview and matters/questions for further consideration.
The development of the construct ‘affordance’
E and J Gibson developed the constructs ‘affordances’, ‘constraints’ and
‘attunements’ over three decades, from the 1950s. A succinct account is:
The affordances of the environment are what it offers the animal, what it provides
or furnishes, either for good or ill … It implies the complementarity of the animal
and the environment. … If a terrestrial surface is nearly horizontal … nearly flat
… and sufficiently extended (relative to the size of the animal) and if its substance
is rigid (relative to the weight of the animal), then the surface affords support.
(Gibson 1979, 127)
Note that the Gibsons’ affordances are very basic things – knives have edges
that afford slicing. Norman (1988) equates affordances with perceived affordances,
which is not the Gibsons’ view – their affordances exist whether we perceive them or
not. Norman (1999) corrects his earlier ‘mistake’ and rants on about the misuse of the
term:
it is wrong to claim that the design of a graphical object on a screen “affords
clicking.” Sure, you can click on the object but you can click anywhere. Yes, the
object provides a target and it helps the user to know where to click and maybe
even what to expect in return, but those aren’t affordances, those are conventions
and feedback (ibid, 40)
The construct is widely used in mathematics education; Watson (2007), for
example, examines tasks and questions that afford participation in mathematics
classrooms. This, to us, is a legitimate application of the construct – it considers what
the environment (equipped with social norms) provides the animal (student) with
appropriate attunements.
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Turner & Turner (2002) take the construct much further, to what they call
‘ergonomic affordances’ and ‘cultural affordances’ which they introduce as:
affordances for embodied action are peculiarly central to effective interaction with
people and objects in a technologically mediated environment. In the real world,
embodied action recognises the constraints of our physical bodies … embodiment
allows us to use a wealth of non-verbal mechanisms and to make assumptions
about the perceptual resources and scope for action of other embodied beings (93)
A cultural affordance (CA) is a feature or set of features which arises from the
making, using or modifying of the artefact and in doing so endowing it with the
values of culture from which it arises. Unlike simple affordances or those which
arise from embodiment, CAs can only be recognised (in an extreme sense) by a
member of the culture which created it. CAs are exploited with the artefact in use
and will change if the artefact is put to a different use. (94)
It should be noted that the Turners are not mathematics educators. They design
and evaluate collaborative virtual environments (CVE). The project behind the theory
they develop is important in terms of critical safety-training simulations in maritime
and offshore work practices. Ergonomic cultural affordances may be important in
safety-training SW, but does this importance extend to the culture of mathematics?
Giampaolo Chiappini is a mathematics educator and Chiappini (2012) applies
the Turners’ constructs to his software Alnuset designed for high school algebra. He
starts by considering ergonomic affordances, e.g. the representation of algebraic
variables on the line through sliding points associated to letters that can be dragged
along the line with the mouse, and he lists a number of other ergonomic affordances.
We agree that sliders in mathematics SW systems can provide an ‘ergonomic
affordance’, because they afford interaction between the users’ bodily movement and
the system’s graphical/symbolic representation.
Chiappini then turns his attention to CAs:
The ergonomic affordances of … Alnuset are not sufficient in themselves to allow
students to master the meaning, values and principles of the cultural domain
which has inspired the creation of these ergonomic affordances … it is only
through an activity that features which emerge from .. [Alnuset] … can be
transformed into cultural affordances and can assume the values of the culture
from which they arise. (138)
To address how these meanings and values may be acquired by students he
turns to activity theory, focusing on “every human activity can be characterized by
contradictions” (138). He adopts Engeström’s notion of the cycle of expansive
learning where the evolution of activity goes through a number of phases.
The first phase … the assignment of an open problem on an important issue of
algebra learning and concerning an obstacle of an epistemological nature. …
Typically a conflict emerges in terms of an unexpected representative event as a
reaction of the system software to the student action that appears surprising to
their consciousness. …
In the second phase … students are requested to face tasks that broaden
problematic areas of the knowledge in question. … Tasks in this phase are
designed in order to exploit the visuo-spatial and deictic ergonomic affordance of
the algebraic line to allow students to explore the conditions, causes and
explicative mechanisms of conflicts …
In the third phase the use of the algebraic line is integrated with the axiomatic
algebraic model incorporated into the AlNuSet algebraic manipulator…. In this
phase the teacher encourages both the establishment of the algebraic axiomatic
model in the student’s practice and the development of meta-cognitive processes
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
involved in the re-configuration in symbolic terms of the algebraic meanings
expressed beforehand in visuo-spatial and deictic terms.
In the fourth phase [the teacher fosters] … a full awareness by students of the
developed knowledge through the comparison with the memory of their
knowledge before the beginning of the cycle. (139)
John Monaghan’s reaction
Chiappini’s approach (and software, not described here – see his paper) is interesting,
as is the evolution of the construct ‘affordance’. Of the two constructs, ‘ergonomic
affordance’ appears relatively unproblematic compared to ‘cultural affordance’. But
there is something appealing to the construct ‘cultural affordance’ with regard to the
culture of mathematics and software used in mathematics learning and teaching.
Spreadsheets are amongst the most widely ‘mathematical’ software in schools.
Although they were designed for finance, not mathematics instruction, the quasi
algebra, B2=2*A2+1, is ‘cell arithmetic’ and can only support the development of
some cultural aspects of algebra. Spreadsheets afford ‘filling down’. This ergonomic
affordance can be appropriated by mathematics teachers to solve equations by a
decimal search. For example, to solve x3=100, we can fill down single digits in one
column, fill down corresponding cubes in an adjacent column and see a solution
between 4 and 5; we can then fill down between 4 and 5 in steps of 0.1 and see a
solution between 4.6 and 4.7, etc. Such an appropriation is a cultural act on the part of
the mathematics teacher (for one aspect of mathematics). It may be that the construct
‘cultural affordance’ permits us to hone in on affordances of software system that
support (or do not support) aspects of mathematics that we wish to promote.
Chiappini brings in activity theory (AT) and Engeström’s version of AT in
particular – are these essential? Regarding AT, I think ‘yes and no’. ‘No’, I’m sure
that someone who is not particularly drawn to AT could put an interpretation on the
transformation of ergonomic affordances into ‘real’ mathematical understandings by
students which is not based on AT. ‘Yes’ in as much as AT does highlight that
learning mathematics is a cultural process (would someone who is not drawn to AT
even have an interest in the phrase ‘CA’?) Now, with regard to Engeström’s version
of AT, I first note that there are a number of forms of AT. Engeström’s version is a
‘systems’ approach and it could (and has) been argued (see LaCroix 2012) that it is
too big to capture the nitty-gritty details of students’ actions in doing and learning
mathematics. Personally I suspect the AT approach of Luis Radford (see LaCroix,
2012, again, for details), who looks at nitty-gritty student details and pays close
attention to gestures (which could be called ergonomic actions), may be a more
suitable AT approach to looking at how the affordances of a mathematics SW system
can assume, in the words of the Turners (above), “the values of the culture from
which they arise”.
Further to this, there appears to be a certain ‘Italian flavour’ to Chiappini’s
version of the Engeström approach. By this I mean, it is certainly an Engeström-based
approach but I have detected that an Italian way of sequencing learning and teaching
involves initially presenting students with tasks that are beyond their technical powers
and then attending to technical matters in intermediate lessons before returning to a
form of the original task; and this is basically what happens in the four phases above.
Now there is much to laud in this approach to sequencing learning and teaching but
there are other means as well. There is thus a sense in which Chiappini may be
‘prescribing’ rather than ‘describing’ students’ actions; I do not see anything wrong in
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
prescribing as long as we realise that what students do may differ to what we want
them to do.
Finally it needs to be asked whether the construct ‘affordance’ has been
stretched too far from “affordances are very basic things, knives have edges that
afford slicing”? To “assume the values of the culture from which they arise” requires
an elaborate peopled environment with specific tasks and sub-tasks “designed in order
to exploit the visuo-spatial and deictic ergonomic affordance” of a specific artifact
(Alnuset). Now I do not see anything ‘wrong’ with this elaboration, I just wonder
whether the Gibsons would recognize the animal-environment relation in this account.
John Mason’s responses
The notion of ‘cultural affordances’ has an immediate appeal for me, at least until it
runs up against the Gibsons’ insistence that they be independent of people, time and
place. It seems reasonable, even indisputable that immersion in a culture provides
access to cultural tools, whose attunements then afford specific actions. For example,
recognizing the possibility of studying a set of objects by studying actions acting on
those objects (in the way that, for example, analysis studies the reals, rationals and
complex numbers by studying various families of functions, or Klein’s approach to
geometry as the study of groups of permissible actions).
It seems to me that the CA construct with respect to software is a portmanteau
for the enculturation of one or more people into (some aspects of) a culture enjoyed
by the author of the software. It is the finer grained analysis of that enculturation
which is of interest to me, and I suspect to most mathematics educators.
I find the notions of affordances, constraints and attunements powerful
triggers to direct attention to important aspects of tasks generally, and software in
particular, but only by seeing them as evolving and developing during activity.
Affordances perceived at the beginning are usually a subset of the affordances
recognized later. Trying to encompass all of the affordances as basic, absolute
affordances fails to take into account the user and their evolving attunements. For
example, in spreadsheets, ‘fill down’ offers both ergonomic and cultural affordance.
Treating affordances as the union of all possible basic affordances in all possible
situations takes away the power of the framework for identifying what is possible inthe-moment, moment-by-moment.
One approach to a finer grained analysis of enculturation into, exploitation and
evolution of affordances could be through activity theory as used by Engstrom or
Radford. Another could be through abstraction of Bruner’s trio of modes of
(re)presentation Enactive–Iconic–Symbolic.
Take for example one of my own applets: Tangent Power (Mason 2012).
Define the tangent power of a point P with respect to a function f to be the number of
tangents through P to the graph of f. Some initial questions might be: What tangent
powers are possible and where are the points with a given tangent power? What are
the greatest and the least possible tangent powers, and where would these points they
be found? These are outer tasks (Tahta 1988) to initiate activity. An inner task (not to
be made explicit until after work on the task) is encountering inflection points as the
places where the first derivative changes direction.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
I usually start with a
quintic, displayed by pressing
the Particular Button at the top
left hand side of the screen.
At first sight there are a
mass of buttons and a
prominent graph.
What the buttons do will
only emerge through trial or
through watching someone
drive the software.
Following Turner & Turner and Chiappini
Usability (affordances): Fine-motor coordination required; reasonable
eyesight; focusing on some parts of the screen while ignoring others.
Ergonomic affordances: There appear to be buttons that could be pressed, but
even recognizing these requires some cultural capital; you can drag a point along an
axis which drives a tangent to the curve; you can display boundary regions; you can
create a point with a number-label to use as a label for the tangent-power of a region.
(Local) Cultural Affordances: Unless you have been told or shown, it is not
evident that to change a number (such as Poly Degree or New Number) you clickand-hold while typing in an appropriate number.
(Global) Cultural Affordances: Opportunity to explore various particular
instances of a phenomenon, and to generalise to a broad class of functions;
opportunity to challenge your sense of what happens to tangents at points with large
(in absolute value) x-coordinate; opportunity to challenge assumptions (concept
images) about tangents and whether they can cut or be tangent to a curve ‘elsewhere’,
or even cross the curve at a point of tangency; encounter geometric implications of a
first derivative having an extremal value.
Constraints: Polynomials of degree up to 7 whose variation fits on the screen.
Attunements: Users need to have some familiarity with graphs of functions
and tangents; in order to concentrate on the mathematical relationships and properties,
it is necessary to develop some facility with the use of the buttons etc.
Following Bruner
Enactive Affordances: Dragging (and animating) a point to animate a line through a
point or a tangent to the curve; dragging red points alters the curve; extension permits
a line at a fixed angle to the tangent; display of curve enveloped by those lines.
Iconic Affordances: Stabilising the image of particular polynomial, with a
(possible) sense of generality through the possibility of dragging red points to change
the polynomial; stabilising a particular position for point P and a particular line
through P; allowing line through P to be varied; stabilising image of a tangent;
animating the tangent; providing instances of enveloped curve when lines are at a
fixed angle to the tangent, for making conjectures.
Symbolic Affordances: Not really present;
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Overview
Seeing CAs as ‘features arising from making, using and modifying artefacts’ draws
attention to how conditions make some things more likely, and other things unlikely
or even impossible, so that both affordances and constraints contribute to creative
potential. Moreover, the affordance expected may not be the affordance experienced.
Seeing CAs as ‘internalised’, ‘condensed’, ‘reified’, ‘semantic contractions’
offers a psychological contribution, because what I make of and do with an artefact
can be slightly different to what you make of and do with it. For example many
people do not seem to make use of the affordance of styles in MSWord, yet it is
present, but perhaps not perceived, or if perceived, not acted upon for various
idiosyncratic as well as social reasons. Each user of a cultural artefact brings to it their
own propensities, stressings and ignorings, and intentions, so the artefact itself, as an
object in the material world may look the same, yet in conjunction with a person in a
social setting may bring to mind different possibilities.
Might not the adjective ‘cultural’ mislead attention away from the personal,
the psyche of the individual (their awareness, enactive potential and affective states)
as an important component? The cultural and indeed the historical play a role in the
genesis, but the condensation-contraction is likely to be personal. If the person’s state
were dominantly social, wouldn’t everyone in the class give the same response to a
teacher’s probe, or at least the group would agree on a response?
References
Chiappini, G. 2012. The transformation of ergonomic affordances into cultural
affordances: The case of the Alnuset system. International Journal for
Technology in Mathematics Education, 19(4):135-140.
Gibson, J.J. 1979. The ecological approach to visual perception. Boston: Houghton
Mifflin.
LaCroix, L.N. 2012. Mathematics learning through the lenses of cultural historical
activity theory and the theory of knowledge objectification. CERME7, WG16,
http://www.cerme7.univ.rzeszow.pl/WG/16/CERME7_WG16_%20LaCroix.pdf
Mason, J. 2012. Tangent Power. Applet available at
mcs.open.ac.uk/jhm3/Presentations/Presentations%202012
Mason J. (in press). Interactions Between Teacher, Student, Software and
Mathematics: getting a purchase on learning with technology. In The
Mathematics Teacher in the Digital Era: An International Perspective on
Technology Focused Professional Development, ed A. Clark-Wilson, O. and
N. Sinclair. Springer.
Norman, D.A. 1988. The psychology of everyday things. New York: Basic Books.
Norman, D.A. 1999. Affordances, conventions, and design. Interactions, May-June:
38-42.
Tahta, D. 1981. Some thoughts arising from the new Nicolet films. Mathematics
Teaching, 94:25-29.
Turner, P. & Turner, S. 2002. An affordance-based framework for CVE evaluation.
People and Computers XVII – The Proceedings of the Joint HCI-UPA
Conference, 89-104. London: Springer.
Watson, A. 2007. The nature of participation afforded by tasks, questions and prompts
in mathematics classrooms. Research in Mathematics Education: Papers of
the BSRLM 9:111-126.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 137
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Doing the same mathematics? Exploring changes over time in students'
participation in mathematical discourse through responses to GCSE questions
Candia Morgana, Sarah Tanga, Anna Sfardb
a
Institute of Education, University of London, UK; bUniversity of Haifa, Israel
The project “The Evolution of the Discourse of School Mathematics” uses
the lens of GCSE examinations to investigate changes over the last three
decades in what is expected of students in England. We have identified
differences in the discursive features of examination questions through
this period and now seek to investigate how these differences may have
affected the nature of student participation in mathematics discourse.
Students have been tested using questions varying in characteristics
typical of different points in time. We discuss the design of the test, and
present some preliminary results.
Keywords: assessment; examination; mathematical discourse
Introduction
During the past three decades or so in England there have been a number of changes
in curriculum and assessment policy and government interventions in pedagogy and
assessment practices. These changes form the background to our study, which seeks
to investigate how school mathematics has changed over the period.1 Rather than
focusing on the documents and policies that seek to regulate the curriculum, we try to
gain insight into the curriculum that students actually experience and the nature of the
mathematical discourse in which they are expected to learn to participate. We take
GCSE examinations as our window onto these expectations because of the welldocumented relationship between high-stakes examinations, curriculum and pedagogy
(e.g. Broadfoot 1996)
The study is framed by a theoretical assumption that understands doing
mathematics as participating in mathematical forms of discourse (Sfard 2008). Hence
we focus analytically on the discourse of examination texts and of student responses.
Phase 1 of the project has involved the development of an analytic framework,
described in (Tang, Morgan, and Sfard 2012), and analysis of a sample of
examination papers. We have no space here for the full details, but present below
some key findings that highlight differences found between examinations set at
different dates. The main focus of the present paper is Phase 2 of the project, in which
we investigate how students respond to examination questions that have differing
discursive characteristics. We have constructed and administered two versions of a
test, enabling us to compare student responses to ‘parallel’ questions. Below, we
describe the design of the test and present some results, raising questions about the
nature of the mathematical activity involved in examination success.
Phase 1: Analysis of examination questions
Our analysis of the changing discourse of examinations has made use of a sample of
Higher Tier question papers from two of the three English examination boards. The
1
The project “The Evolution of the Discourse of School Mathematics through the Lens of GCSE
Examinations” is funded by the ESRC grant reference: RES-062-23-2880
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
sample includes the papers for summer examinations taken in each of 8 years. The
years range from 1980 (pre-GCSE) to 2011, chosen to capture major changes in
curriculum and examination policy and practice within this time frame. The analysis
focuses on how mathematics and mathematical activity are construed and on the role
of the student-examinee in this. Here we summarise some of the differences found in
the analysis. Fuller details of some of these areas of the analysis have been and will be
reported elsewhere (Morgan and Tang 2012; Morgan, Tang, and Sfard 2011; Tang,
Morgan, and Sfard 2012).
Human agency in mathematical and non-mathematical processes
In considering the nature of mathematical activity construed in the examination texts,
we ask to what extent mathematical processes are presented as being performed by
human agents. Across the whole sample, agency in mathematical processes is
overwhelmingly obscured. The means by which this is done has changed from passive
voice “a tangent has been drawn” to use of relational statements “line AB is a
tangent”. A reduction in the use of passive voice constructions has been a deliberate
change made by the examination boards, following advice that passive voice lowers
reading comprehension. The corresponding increase in relational statements,
however, may further increase alienation, as the process itself (in this case “drawing”)
is now absent. While human agents are thus largely absent from mathematical
processes, they are to be found as actors in everyday practices in contextualised
questions.
Contextualisation
The proportion of contextualised questions rose substantially in the first few years of
GCSE, falling back in more recent years. Throughout the period, the majority of
contextualized questions demand little engagement with the context itself. In the most
recent years in our sample (2010 and 2011) we have coded approximately 40% of all
contextualized questions as “ritual”, that is, of a standard form widely used as
‘exercises’ in the classroom (Nyabanyaba 2002). This compares to just 8% of
contextualized questions coded as “ritual” in 1980.
Grammatical, logical and task complexity
The most recent examinations overwhelmingly use simple one-clause sentences. This
is accompanied by a marked decrease in the use of conjunction “and” and
implications (“hence”, “then”, “therefore”). Again, the reduction in grammatical
complexity follows an explicit policy of attempting to avoid linguistic characteristics
known to reduce reading comprehension. However, it is also relevant to ask whether,
by avoiding the complexity of sentences with dependent clauses or clauses joined by
conjunctions and implications, engagement with some important aspects of the logic
of mathematics are also avoided.
We have also considered the complexity of the mathematical activity expected
of students by considering the “grain size” (defined as the number of decisions
required to achieve a solution) of tasks. The analysis of this factor has not yet been
completed, though preliminary results suggest that, while the majority of tasks are of
grain size one or two, the proportion of tasks with higher grain size has decreased. In
1987, 16% of tasks involved three or more decisions, while only 8% of tasks had this
level of complexity in 2011.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Framing of student response
This part of the analysis concerns the degree of flexibility for students in producing
and presenting their answer. In 1980 and 1987, students had a separate answer book,
providing no guidance about the extent or shape of the expected response. Later
examination papers provide space for answering on the same page as the question
and, in many cases, gaps to fill in and lines to place the answer. There is variation
between and within years in how the form of student answers is defined. Methods of
framing include:
explicit statement of methods (e.g. “Use algebra to …”). In recent years there
is a tendency to ask students to “write down …” or to “calculate …” rather than
simply to “find …”. The use of imperatives (e.g. “Write down the amount …”) rather
than questions (e.g. “How many …?”) also constrains possible approaches to finding
an answer.
formatting answers. While in some cases a simple space or line is given for
students to write their final answer, in others, the format of the answer is strongly
determined. For example, the answer line for a question involving simultaneous
equations might be given as “x= ……, y= ……”. In recent years, the units of the
answer are commonly included on the answer line (e.g. “…… kg”).
Phase 2: Testing students
In Phase 2 of the project, we ask what differences the discursive characteristics of an
examination question make to the mathematics students engage in when answering. In
order to investigate this, we have designed two versions of a test with ‘parallel’
questions involving characteristics typical of examinations set at different dates. In
each case, an original question was included on one version of the test, while the other
version of the test included a ‘contrived’ adaptation of the question, making use of
discursive characteristics found in questions on a similar topic in another year. The
questions were distributed to ensure that each test contained four original questions
and four ‘contrived’ questions, four with ‘early’ discursive characteristics (1980 –
1995) and 4 with ‘late’ characteristics (1999-2011). This test has been administered to
a sample of 158 Year 10 students from six classes in four London schools (all entered
for Higher Tier GCSE). Half the students were assigned to each version of the test. In
the next sections we present the design and results of two questions that gave rise to
some striking differences in the responses to the two versions.
proportion – the ‘Election’ question
In table 1 we present the two versions of the question on proportion, summarising
some of main differences structuring our design of the ‘contrived’ question. Table 2
then shows some of the differences in student responses to the two versions, focusing
on the occurrence of some different correct strategies.
Table 1: Two versions of the proportion question
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
‘contrived’ question, based on 1999
original 1987 question
1. In the 1983 General Election, 650 Members of Parliament were elected. Shortly before 1. Before the 2010 General Election, an opinion poll asked voters which party they
the election, an opinion poll indicated these voting intentions:
intended to vote for.
Conservative 38%,
Labour
32%.
The results of the opinion poll were:
Conservative 38%
If Members of Parliament had been elected in the same proportions as the poll
results, find how many M.P's of parties other than Conservative or Labour would
have been elected.
Labour
32%
a) Write down what percentage of voters said they would vote for a party other than
Conservative or Labour.
………………………………..
[1]
650 Members of Parliament were elected.
The proportions of MPs elected for each party were the same as the poll results.
Answer ……………………………….. [2]
b) Calculate how many MPs of parties other than Conservative or Labour were elected.
………………………………..
[1]






increased human presence in (non-mathematical) processes: “ voting intentions” vs. “voters said
they would vote for …”
decreased grammatical complexity: two temporal phrases (in reverse order of time!) vs. one; “in
the same proportions” (qualifying phrase) vs. “the proportions were the same” (independent single
clause sentence)
decreased logical complexity: “If MPs had been elected, […] would have been elected”
(conditional structure) vs. “The proportions were …” (statement of fact)
decreased grain size: 1x3 vs. 1 + 1x2
1
increased explicitness of instructions: “Find” vs. “Write down”; “Calculate”
increased emphasis: use of space to separate points; bold to highlight negation
1
Table 2: Some results for the proportion question
1987
‘new’
52%
81%
calculate 30% of 650
38%
72%
calculate 70% of 650 and subtract
15%
4%
calculate 32% and 38% and subtract both from 650
23%
9%
fully correct answers
strategy
Unsurprisingly a high proportion of those doing the ‘new’ version of this
question have calculated 30% of 650 directly. The structure of the question, divided
into two explicit sub-tasks, suggests this approach. Although the overall success rate
for the original version is substantially lower, the proportions choosing to use a
correct strategy are relatively close (76% vs. 85%). It may be that those attempting the
more complex strategies have made more errors in calculation; our analysis has not
yet addressed this issue.
percentage change – the ‘Car’ question
In table 3, we present the two versions of the question on proportion. Table 4 then
shows some of the differences in student responses to the two versions, focusing on
the extent to which the context of the question is taken into account.
Table 3: Two versions of the percentage change question
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
original 2010 question
‘contrived’ question, based on 1980
Arwen buys a car for £4000
The value of the car depreciates by 10% each year.
Work out the value of the car after two years.
The value of a car depreciates by 10% each year. If it
cost £4000 originally, what would be its value after
two years?
£ ...................................




decreased human presence: The introduction of “Arwen” in the original 2010 question suggests
that the question is about an everyday practice, whereas the contrived question is in what Dowling
(1998) calls the expressive domain: clearly school mathematics, not everyday, even though
expressed in non-specialised vocabulary.
increased grammatical and logical complexity: simple sentences vs. conditional two-clause
sentence demanding hypothetical reasoning
decreased explicitness of instructions: explicit “Work out” vs. question seeking information
decreased specification of form of answer: space for working delimited by line for answer; units
given vs. open space
Table 4: some results for the percentage change question
2010
‘old’
63%
48%
full sentence answer
e.g. “The value after two years would be £3240”
0%
14%
£ sign used in answer
16%
62%
£ sign used in working but not consistently
20%
13%
£ sign used consistently throughout working
9%
18%
fully correct answers
Contextualisation
Again, the ‘new’ (original 2010) version of the question has a higher success
rate. We have not yet investigated the strategies used but wish to draw attention to
differences in how students located their responses in relation to the contextualisation
of the question. In presenting their answer, 14% of those doing the ‘old’ version (11
students) wrote a full sentence, relating the numerical result to the value of the car.
This was not done by any of those answering the 2010 version. We assume that the
printed answer line with the £ sign frames students’ response so that there is no
perceived need (or space) for other means of signalling the answer (although 16% still
wrote their answer with a £ sign elsewhere on the page). Less easy to explain is the
use of the £ sign in the working. While similar proportions used it at least once in
their working, twice as many of those doing the ‘old’ style question used it
consistently throughout.
Discussion
We have chosen to look at results for two questions differing substantially in their
success rates between the ‘old’ and ‘new’ versions. However, our main interest is not
in levels of difficulty but in the nature of the mathematical activity that students
engage in when responding to questions with different discursive characteristics.
Following Bezemer and Kress (2008), we ask what is gained and what is lost when
the discourse changes.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The analyses offered here focus on students’ choice of strategy and the
contextualisation of their responses. In the ‘Election’ question, splitting the task into
sub-tasks seems to have directed students towards using a more efficient strategy (and
perhaps thereby achieving greater success). However, making a decision about
strategy is in itself an important mathematical activity, involving students in
exercising agency as mathematical thinkers. In the ‘Car’ question, in spite of the
apparent attempt to make the context more ‘relevant’ by introducing human activity,
the tight framing of the answer space seems to reduce the extent to which students
engage with the context, not only in presenting their final answer but also throughout
their working. In both cases, students’ mathematical activity appears to be affected by
subtle changes in the discursive characteristics of the questions. Examination boards
have made some of these changes deliberately to increase student access and to
prevent “language getting in the way of the mathematics”. Our analysis suggests,
however, that “the mathematics”, which may appear the same, is itself changed for
some students.
This analysis of student responses has allowed us to form conjectures about
which features of the questions prompt particular types of response. In the next phase
of the project we intend to interview students who took these tests to probe more
deeply into the ways they participate in mathematical discourse as they read and
respond to questions with different discursive characteristics.
References
Bezemer, J. and G. Kress. 2008. Writing in multimodal texts: A social semiotic
account of designs for learning. Written Communication, 25 2: 166-195.
Broadfoot, P. M. 1996. Education, Assessment and Society. Buckingham: Open
University Press.
Dowling, P. 1998. The Sociology of Mathematics Education. London: Falmer
Morgan, C. and S. Tang. 2012. Studying changes in school mathematics over time
through the lens of examinations: The case of student positioning. In
Proceedings of the 36th Conference of the International Group for the
Psychology of Mathematics Education ed. T. Y. Tso. (Vol. 3: 241-248).
Taipei, Taiwan: PME.
Morgan, C., S. Tang, and A. Sfard. 2011. Grammatical structure and mathematical
activity: comparing examination questions. Proceedings of the British Society
for Research into the Learning of Mathematics, 31 3. Retrieved from
http://www.bsrlm.org.uk/IPs/ip31-3/BSRLM-IP-31-3-20.pdf
Nyabanyaba, T. 2002. Examining Examination: The ordinary level (O level)
mathematics examination in Lesotho and the impact of recent trends on
Basotho students' epistemological access. Unpublished PhD dissertation.
University of the Witwatersrand, Johannesburg, South Africa.
Sfard, A. 2008. Thinking as Communicating: Human development, the growth of
discourses, and mathematizing. Cambridge: Cambridge University Press.
Tang, S., C. Morgan and A. Sfard. 2012. Investigating the evolution of school
mathematics through the lens of examinations: developing an analytical
framework. Paper presented at the 12th International Congress on
Mathematical Education, Topic Study Group 28 on Language and
Mathematics, Seoul, Korea.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 143
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Vending machines: A modelling example
Peter Osmon
King’s College, London
Throughout the last century the mathematics of the continuum
underpinned the science and technology of the developed world. Today’s
developed world is increasingly dominated by the artefacts and processes
of information technology and it is discrete mathematics that underpins
this technology. A finite state machine description of the behaviour of
vending machines, in the form of state transition diagrams and state
transition tables, is used as an example to demonstrate that modelling
numerous artefacts of today’s everyday world would be within reach of
many 15-19 year old learners if the curriculum were to give more
emphasis to discrete mathematics
Keywords: mathematics applications, modelling, discrete mathematics,
finite state machines, state transition diagram, state transition table.
Introduction
This presentation is one of a series where the overall aim is to make the case for an
updated curriculum- one with less emphasis on the continuum and more on discrete
mathematics. The argument for this change is essentially that while during the
nineteenth and twentieth centuries continuum mathematics underpinned the science
and technology of the developed world, now in the twenty-first century our
civilisation is becoming IT-dominated and the mathematics that underpins it is
discrete. Moreover mathematics performs this underpinning role through modelling
and this is what applied mathematics should mean in the curriculum.
To elaborate this: looking for patterns and building models with them is how
we understand the world around us. Mathematics is the science of patterns, and so can
help with model-making and hence with our understanding of the world. Applied
mathematics is model-making and using in the context of either the everyday world or
some professional discipline such as science or engineering. But learners in school
have limited knowledge of (a) mathematics (b) application domains (principally
science and their everyday world), and these limitations narrow the range of models
they can hope to appreciate.
However, with respect to their limited mathematics knowledge, quite a lot of
the relevant discrete mathematics (sets, relations, logic, events, algorithms, sequential
machines) needed for understanding the behaviour of typical artefacts and processes
of our everyday twentieth-century world could, with a reformed curriculum, be within
reach of learners aged 15-19. In today’s mathematics curriculum, discrete
mathematics does not emerge as a distinct branch of the subject until university. The
topic Finite State Machines (FSMs) is an example of this accessible mathematics. See
for example Rosen (2007, 796-798).
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
An example of model-making in today’s everyday world
Vending machines of all kinds are part of our everyday environment. If we ignore the
detail of what a particular instance dispenses, it is clear that they exhibit similar
behaviours: there is a common behaviour pattern or at least a common family of
patterns. In what follows I aim to show that describing these behaviour patterns is
potentially within the reach of school mathematics. Finite State Machines (FSMs) is
the particular discrete mathematics topic needed for describing vending machine
behaviour. Figure 1 outlines the process of modeling the behaviour of a vending
machine by designing an appropriate FSM.
Before getting further into the example I should introduce the term “state”.
State is an intuitive concept that helps us understand the behaviour of entities, usually
systems, over time. Thus we speak of the state of the weather, of the economy, of
London’s transport network, of our health. “State” can be described mathematically.
A familiar example is the parabolic trajectory of a projectile subject to vertical
acceleration due to gravity. Its state at any moment is described by values of its
position and velocity variables (x,y,vx,vy). The projectile has a continuum of states.
Note that I have chosen not to use mathematical subscripts, considering instead that
the abbreviated state name vx is more appropriate for learners.
Now consider the behaviour of vending machines- these familiar entities have
sets of discrete states- rather than a continuum like the projectile. This kind of
behaviour is characteristic of the artefacts and systems in the IT dominated world in
which we all now live.
Mathematics
Finite State Machine
Make a model
(Understand)
Application
Behaviour of a vending machine
Figure 1. Designing an FSM to model the behaviour of a vending machine.
FSM notation: State Transition Diagrams (STDs)
An STD is a bubble and arrow diagram that describes the behaviour of an entity over
time. Bubbles represent states which are given names. The entity has a finite set of
states S = {S0, S1, S2, etc}. By convention, S0 is the initial state. When speaking
generally we call the current-state Scs. State names are written in the upper half of the
bubble. Characteristic of a state is its set of outputs O. A state’s output is written in
the lower half of the bubble.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Arrows show the possible transitions from Scs to a next-state Sns. A state has
a set of Inputs I = {I1, I2, etc} to choose which of several possible transitions occurs.
An arrow is labelled with the particular input which selects that transition. That is
there is a next-state function: Scs x I = Sns. (Understanding FSM models may be
helped by assuming that States have duration and Transitions are instantaneous- the
mathematics has nothing to say about such matters.)
Figure 2 aims to clarify STD notation. It shows a state bubble and transition
arrows into the state, from possible previous-states, and out of the state, into the
various possible next-states.
Possible transitions
into State S
STD notation
bubble (state) and
arrows (transitions)
State name: S
Output
during state: O
I1
I2
I3
Possible transitions out of State S
Input value selects one
Figure 2. STD notation: a state bubble- containing the State name and Outputs during that state- and
possible Transitions in and out of that state with the Input values that select them.
FSM application example: drinks vending machine
Now, to be specific, consider drinks vending machines as our everyday example, and
first consider a very simple machine that accepts 50p coins and offers a choice of two
drinks: Cola or Orange.
This machine has four states, as follows.
S0. Initial state: Waiting
Output: “Insert 50p coin”
Input of coin- causes transition toS1. Choosing drink
Output: “Cola/Orange”
Choice of Input buttons- causes a transition to either-
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
S2. Dispensing Cola or S3. Dispensing Orange
Output: “Please wait”
Input: dispensing finished- causes transition toS4. Drink is ready
Output: “Take drink”
Input: removal of drink- causes transition back to S0
From this information we can construct the STD for this simple machine. It looks like
the diagram in Figure 3.
STD for
Simple drinks
Vending machine
S0
Insert
coin
coin inserted
S0. Waiting
S1. Offering choice
S2. Dispensing Cola
S3. Dispensing Orange
S4. Drink is ready
S1
Drink
taken
Choose
Cola
Orange
S2
S3
Please
wait
Please
wait
Finished
Finished
S4
Take
drink
Figure 3. STD for the simple drinks vending machine
A State Transition Tables (STT) is an alternative notation for describing FSM
behaviour. The Table below describes the STT for the simple drinks machine. As the
table demonstrates, a STT is actually two tables: the output table and the next-state
table, corresponding to the machine’s output function and next-state function
respectively.
Current-state
S0 Waiting
S1 Offering choice
S2 Dispensing Cola
S3 Dispensing Orange
S4 Drink is ready
Output
Insert coin
Choose
Please wait
Please wait
Take drink
Input/Next-state
Coin-inserted/S1
Cola/S3 Orange/S4
Finished/S4
Finished/S4
Drink-taken/S0
STTs are more compact than STDs: they can describe, on a single page,
behaviour with more states and transitions. While it is generally harder to comprehend
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
behaviour from a tabular description, drawing large and complicated STDs can
become tedious, even with the aid of special software.
A more elaborate vending machine: more functionality and more states
Now consider a drinks vending machine with more functionality: drinks still cost 50p
each, but this machine accepts 5p, 10p, 20p, and 50p coins, and gives change. Further,
the machine offers a choice of five hot drinks - tea, coffee, strong-coffee, mocha,
chocolate - and also offers choices of additives - unsweetened/sugar/double-sugar and
black/milk/double-milk. When it comes to describing its behaviour with an FSM, this
means not just more states but more complicated connections- quite a lot to get one’s
head round. How to proceed?
Our vending machine problem provides an opportunity to introduce the
following two heuristics which are helpful in many problem solving situations (Polya
1945):
(A) Divide-and-conquer
“Factorise” the problem into parts: a Payment part and a Drinks-and-AdditivesChoices part.
(B) Easier-problem-first (applies to both parts):
Payment part: let the complications in progressively. We have already considered a
50p coin only machine, so now accept a range of coins- but no change given, and
then, at the next stage, give change.
Choices part: let the complications in progressively. We have already considered a
binary choice machine, now provide a five-way choice of drinks, and finally introduce
two levels of additives choice.
Following in the tradition of mathematics textbooks, the task of working out a
description of the more elaborate vending machine- as either a STD or a STT- is left
to the reader.
Some other applications of finite state machines
Vending machines in today’s world dispense a great variety of goods and services
besides drinks- perhaps the most common is the automatic teller machine (ATM) or
“hole in the wall” outside banks that dispenses money, or one’s bank account
information, in response to input information supplied by a magnetic strip on one’s
debit card, supplemented by choices input by keypad.
ATMs differ from most vending machines in that they are not self-contained
within a cabinet- not localised- but rely on electronic communications with a,
generally remotely located, bank database. Perhaps this also behaves as a finite state
machine. Communication between a pair of FSMs- the output from one providing the
input for the other and vice versa- is a more challenging modelling problem.
Vending machines of various kinds are by no means the only applications of
finite state machines in our everyday world. Other examples include control of traffic
lights, lifts, and traditional combination locks. In a rather different context- word
processing- FSMs can perform syntax checking. A problem example that
demonstrates this application, while not taking too long to work out, is devising an
FSM that checks that every left-hand bracket in a sentence or mathematical
expression containing nested pairs of brackets has a matching right-hand one.
My experience presenting a range of such examples to first year
undergraduates, who have no more than GCSE level mathematics, is that they get
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
quite interested in the problems and can manage to solve them; also, that this is a class
of problems that seem well-suited to collaborative group working.
Conclusions
This paper is about meaningful teaching of applied mathematics to 15-19 year olds. I
have argued that this means teaching modelling and that, to make models, learners
need knowledge of some application domain- and in practice this has to be either
science or else the everyday world- together with the relevant mathematics.
During the nineteenth and twentieth centuries, continuum mathematics
(calculus especially) underpinned the classical physics base of much nineteenth and
twentieth century industry as well as many everyday artefacts. Now, in the twentyfirst century, it is becoming apparent that today’s business and industry as well as
today’s everyday artefacts and processes, rest on information technology- which in
turn is underpinned by discrete mathematics (sets, logic, relations, algorithms, events,
sequential machines) rather than the formerly dominant continuum mathematics. But
in today’s mathematics curriculum, discrete mathematics does not emerge as a distinct
branch of the subject until university.
In this paper, by focusing on a particular topic in discrete mathematics, namely
finite state machines, I have sought to demonstrate that, were the curriculum to be
reformed to give appropriate recognition to the contemporary importance of discrete
mathematics, there are many familiar everyday artefacts and processes that would
then be accessible to younger learners to model.
Reference
Rosen, K. H. 2007. Discrete mathematics and its applications. Sixth edition. New
York: McGraw-Hill,
Polya, G. 1945. How to solve it. Princeton: Princeton University Press
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Gendered styles of linguistic peer interaction and equity of participation in a
small group investigating mathematics
Anna-Maija Partanen and Raimo Kaasila
Åbo Akademi University and University of Oulu, Finland
In a teaching experiment with two Finnish upper secondary classes, the
basics of calculus were studied using an investigative approach and a
small-group setting. As part of the ethnographic teacher research, the
different styles of talking of the girls and boys in four groups were
analyzed through application of the concept of sociolinguistic subcultures.
This paper focuses on the interactions in one of the groups where two girls
and a boy discuss mathematics. We show how the linguistic strategies
typical of these boys prohibited the full potential of the contributions of
the girls to be utilized in the collective construction of meaning in the
group. Promoting democratic discussions in small groups may need
attention in terms of gendered ways of interacting.
Keywords: small groups, sociolinguistics, gender, mathematics education
Introduction
Small-group activities are widely used as a method of studying mathematics,
especially in problem-solving and inquiry approaches. Normally, they are found to
promote students’ mathematical learning, although research on the use of small-group
discussions in instruction has also revealed differentiated possibilities for student
participation in the group activities (Good, Mulryan and McCaslin 1992, Bennett et
al. 2010). If the democratic discussion of ideas constructed by all the students in a
group is prohibited, much of the potential of the working method is lost.
The first author, Partanen, conducted a teaching experiment with two of her
upper secondary classes in Finland, in which students investigated mathematics in
friendship groups of three to four. Partanen (2007) analysed the different
sociolinguistic subcultures (Maltz and Borker 1982) in four of the small groups and
found differences in the styles of talking of the girls and boys. In this paper we use the
earlier analysis to focus on the interactions within one of these groups containing two
girls and a boy. The aim of this paper is to investigate how the styles of talking of the
girls and boys were enacted in the discussions of this focus group.
Theoretical framework
Equity of participation in small-group discussions
Although research reviews on the use of small groups in instruction show that group
discussions promote students’ learning and acquisition of high order skills, they also
point to the observation that the quality of collaboration and interaction varies from
group to group, and that democratic and high quality interactions do not appear
naturally (Good, Mulryan and McCaslin 1992, Bennett et al. 2010). Differentiated
opportunities of participation for students in small groups in mathematics instruction
have been observed, for example, as a function of achievement (Rozenholz 1985) and
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
gender (Lindow, Wilkinson and Peterson 1985). Esmond (2009) also showed how the
type of tasks given to students influenced equity of participation in the small-group
activity.
Bennet et al. (2010) reported differences in interactional styles according to
gender in small-group discussions. All-male groups confronted differences in their
individual predictions and explanations, whilst all-female groups searched for
common features of their predictions and tried to avoid conflict. Mixed groups
interacted in a more constrained way, and it can be argued that the best of all-male
and all-female group interactions was lost in them (Bennet et al. 2010).
Sociolinguistic subcultures
Maltz and Borker (1982) write in their classic paper about different styles of talking
of American women and men in friendly conversations. They argue that girls learn to
do three things with words: 1) to create and maintain relationships of closeness and
equality, 2) to criticize others in acceptable ways, and 3) to interpret, accurately, the
speech of other girls. On the other hand, boys use speech in three major ways: 1) to
assert one’s position of dominance, 2) to attract and maintain an audience, and 3) to
assert oneself when other speakers have the floor (Maltz and Borker, 1982).
Four small groups as a context
The focus group of this paper is one of the four small groups studied in a teaching
experiment established by the first author (Partanen 2011). Partanen (2007) described
the different styles of talking of the girls and boys in the four small groups.
In the peer interaction of the groups studied, the girls invited and encouraged
others to speak, and they acknowledged what the others said more than the boys. For
example, the girls expressed proactive utterances that required (and received) a
response, and they used tag questions. They also gave more positive minimal
responses. The girls gave more space for the others to express their ideas than boys,
for example, by phrasing propositions that were meant to enhance the mathematical
discussion as questions or in conditional form. These features of the girls’ talk can be
interpreted as trying to avoid giving the impression of mathematical authority and also
recognizing the speech rights of others, which both contribute to building
relationships of equality (Partanen 2007).
The boys in the four small groups were more assertive than the girls. They
interrupted each other more often, and they had disputes, boasting, name calling,
jeering, and mocking. They also gave more orders to each other than the girls. In line
with Maltz and Borker (1982), the boys seemed to be very often in the process of
posturing and counter-posturing (Partanen 2007).
Methodology
The experimental courses in the term 2001/2002 were established for the dissertation
of the first author (Partanen 2011). She aimed at developing her own practice of using
the investigative small-group approach in teaching upper secondary mathematics in
her school Lyseonpuiston lukio in Finland. The project can be seen as teacher
research (Cochran-Smith and Lytle 1999). The research question for this paper is as
follows: how did the different sociolinguistic subcultures of the girls and boys in the
four small groups show up in the discussions of the focus group?
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The experimental classes consisted of 31 and 28 second-year students,
approximately 17 years old. They worked in friendship groups of three to four, and
almost all of the groups were single-sex groups. The course was one of the
compulsory courses for high level mathematics. Instead of teaching the important
concepts of calculus, limits, and derivatives directly, the teacher gave the students
questions and problems to be discussed and solved together. After the small-group
sessions, the ideas of the students were discussed and summarized, and the teacher
tried to connect her further teaching to the experiences of the students. The data for
this paper consists of six recorded discussions in one focus group that consists of two
girls, Anni and Jenni, and a boy, Veikko. The earlier analysis of the sociolinguistic
subcultures in the four small groups (Partanen 2007) showed that Veikko used
strategies of talking typical of both the boys and girls.
The way of analyzing data was close to that used in microethnographic
analysis of interaction (Erickson 1992). Transcribed discussions in the small group
were divided into episodes according to the themes. The episodes were then analyzed
in chronological order. For each episode, the group participation structure was
described. After this description, the teacher made conjectures of the typical
participation structure in the group. When she was looking at the next episode, she
revised and developed the conjectures. In this way, a holistic picture of the typical
interactions in the group developed in her writings. She continued revising the
conjectures until she felt certain satisfaction with the description. Finally, the typical
interactions in the small group were examined in the context of the sociolinguistic
subcultures analyzed in the interactions of the four small groups (Partanen 2007).
Results
Through the following two episodes, we are going to illustrate how the ways of
talking typical of the boys that were also used by Veikko prohibited the full potential
of the two girls to be utilized in the collective meaning-making processes of the small
group. Prior to the episode, the class had measured some position-time values for a
glider on an air track and fitted a simple quadratic function to the data. For the smallgroup session, the students were given questions about the meaning of the gradient of
chord and the instantaneous velocity. In episode 1, the students are considering the
meaning of the gradient of the chord (f(z) – f(1))/(z – 1) to the position-time graph.
Overlapping of speech is shown in the transcription.
Episode 1
31 Anni:
So, what do they mean? (looks at the previous two pages of her
notebook) Because this is time and that’s distance (points to the axis
in her calculator).
32 Jenni:
So, how do we draw it?
(takes her calculator)
33 Veikko: What does the gradient of the chord mean, then? (looks at Anni
triumphantly) Because it is time [indistinct].
34 Anni:
(does not notice the expression on Veikko’s face) Is it something like
an average, something like that? … I don’t know.
35 Anni:
(Jenni is following the discussion between Anni and Veikko)
But, isn’t it,
36 Veikko:
When time goes on
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
isn’t it average
Look, here, because this is
time (points to his notebook). Then, then, well x is, yes, here it meets
that (points to the graph in Anni’s calculator). The [indistinct] average
value or something like that.
39 Anni:
Um.
40 Veikko: Well, how was it, then?
41 Anni:
(points with her pen to the screen of the calculator). So that if it is the
average value, then the steeper it is, the longer the time is. For the
average thing.
37 Anni:
38 Veikko:
It seems that Anni was close to constructing an important idea: that the
gradient of the chord is the average velocity. Yet, Veikko interrupts her and, by doing
so, transforms the meaning of what Anni was saying. Anni gives up and returns to the
previously discussed idea that the longer the time interval is the steeper the
corresponding chord. Most probably, a learning opportunity for all the students was
destroyed. A few times, it happened that Anni was expressing a promising idea, and
Veikko prohibited it from being expressed so that a learning opportunity was lost.
Normally, Anni did not persist with her idea, like the boys in the other groups
sometimes did.
At the beginning of the data, it was typical of the participation structure in this
group that Veikko and Anni collaborated, trying to achieve a consensus about the
topic being discussed. Jenni either followed the discussions or worked alone with her
calculator. When she rarely expressed herself, she spoke timidly with a low voice.
Although the students listened to each other in their conversations, it was harder for
Veikko than for the girls.
After the first four small group sessions, Veikko had to be absent from a few
lessons. When he returned, the first topic was about constructing methods for finding
the equations of a tangent and a normal to a curve at a particular x-value. It was a year
ago when Veikko had studied the equations of lines, but the girls had attended the
course during the previous period, just a few weeks before. Jenni had the notes from
that course with her, and she seemed to have knowledge about the important methods
and formulae. The typical participation structure of this group changed when Jenni
had her chance to participate in the working of the small group.
The group had succeeded in finding out the equations of the tangent and
normal to the graph of a third order polynomial function. They were beginning to
write a summary about their investigation. Jenni asked Anni to write the summary on
a transparency. After a short and friendly debate, Anni accepted the task.
Episode 2
20 Veikko: Let’s first write that here. Firstly, we need to substitute this (points to
his notebook). Don’t write yet, but let’s discuss this. (Anni and Jenni
give a short laugh.) We should first substitute that x by minus one here
in the original expression to get the y-value. Then, we need the x.
21 Jenni:
No, but, that’s the gradient, I mean. (points to Anni’s notebook).
22 Veikko: Yeah, no, but, so, so that if we substitute that, and we’ll get the
gradient.
23 Jenni:
No.
24 Veikko: No, but, well, here we don’t need. (The girls laugh. Anni is holding her
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25 Jenni:
26 Veikko:
27
28
29
30
Jenni:
Veikko:
Jenni:
Anni:
31
32
33
34
Jenni:
Veikko:
Jenni:
Veikko:
35 Anni:
36 Veikko:
37 Anni:
38 Veikko:
39 Anni:
40 Jenni:
head by her hands.) Do we need to substitute this here? Yes.
Yes. To get the y. But this is it (points to Anni’s notebook).
And then we need to find the x. Then we
need to differentiate the original expression, to get the gradient of
Yes.
tangent.
Exactly (gives a short laugh). And, after that.
So, we shall first put it (takes her
pen). Shall I write that x = -1 is substituted in the equation, in that?
Yes.
Or, should we make it general? Or, just for this task?
Can we make it general?
So that if you first substitute x in this original equation (points to
Anni’s notebook).
No but, shall we
write that
we get
the equation of the, the tangent (points at a place in her
notebook). And then, let’s write that the gradient can be found by
substituting the
Differentiated.
Differentiated, yes.
Yes.
In this episode, Jenni is playing a much more active role than earlier. She
participates in organizing the group work (the debate before the episode). She
discusses with Veikko about the meaning of their results and she supports Anni’s
suggestions. Although, at the end of the data, there were episodes where Jenni was
not quite this active, she followed with attention the discussions between Anni and
Veikko and, every now and then, participated in them. We interpret these occurrences
so that Veikko’s assertiveness and willingness to take and hold the floor in the
discussions of the group excluded Jenni from participating in the collaboration.
Discussion
In the first episode discussed, Veikko interrupted Anni and thus prohibited her from
expressing what seemed to be a very promising idea. Anni did not persist with her
point of view. The second episode shows how Jenni participated in the small-group
activity much more after Veikko’s absence during his temporary confusion.
We see these episodes as examples of how the ways of talking typical of boys
(Partanen 2007) produced obstacles for the two girls in the group to participate in the
collective meaning-making processes when they were communicating in ways typical
of the girls. For developing the use of small-group discussions in mathematics
instruction, it is important to search for ways of establishing democratic participation.
If multi-vocal contributions of all the participants can be utilized, the group activity
will be enriched. One aspect that may lead to inequality in participation is the
different sociolinguistic subcultures of girls and boys (Maltz and Borker 1982).
Some researchers in science education have identified notable differences in
interactional styles according to gender (Bennet et al. 2010). Our analysis,
furthermore, shows how the differences in the styles of talking may influence the
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
possibilities for students to participate in the small-group discussions. The two
episodes also exemplify possible consequences of this influence for the collective
processes of meaning construction. Training is recommended for students and
teachers in the skills required for handling and participating in group discussions
(Bennet et al. 2010). In mathematics education research, the work of Cobb and Yackel
(1996) on social and sociomathematical norms has potential for contributing to
resolving the problem. However, the challenge still remains for future research and
developmental work, firstly, of identifying the important factors that contribute to
inequalities in the possibilities for participation and, secondly, of developing ways of
overcoming those problems.
References
Bennett, J., S. Hogarth, F. Lubben, B. Campbell, and A. Robinson. 2010. Talking
science: the research evidence on the use of small group discussions in science
teaching. International Journal of Science Education, 32: 69–95.
Cobb, P., and E. Yackel. 1996. Constructivist, emergent and sociocultural
perspectives in the context of developmental research. Educational
Psychologist, 31(3/4): 175–190.
Cochran-Smith, M., and S. Lytle. 1999. The teacher research movement: a decade
later. Educational Researcher, 28(7): 15–25.
Erickson, F. 1992. Ethnographic microanalysis of interaction. In The handbook of
qualitative research in education, eds. M. D. Lecompte, W. L. Millroy, and J.
Preissle, 201–225. San Diego: Academic Press Inc.
Esmonde, I. 2009. Mathematics learning in groups: analyzing equity in two
cooperative activity structures. The Journal of the Learning Sciences, 18: 247–
284.
Good, T. L., C. Mulryan, and M. McCaslin. 1992. Grouping for instruction in
mathematics: a call for programmatic research on small-group processes. In
Handbook of research on mathematics teaching and learning, ed. D. Grows,
165–196. New York: MacMillan.
Lindow, J., L. Wilkinson, and P. Peterson. 1985. Antecedents and consequences of
school-age children’s verbal disagreements during small-group learning.
Journal of Educational Psychology, 77: 658–667.
Maltz, D., and R. Borker. 1982. A cultural approach to male-female
miscommunication. In Language and social identity, ed. J. Gumperz, 196–
216. Cambridge: Cambridge University Press.
Partanen, A-M. 2007. Styles of linguistic peer interaction of girls and boys in four
small groups investigating mathematics. Journal of Philosophy of
Mathematics Education, 21(2).
http://people.exeter.ac.uk/PErnest/pome21/index.htm (accessed December 27,
2012)
Partanen, A-M. 2011. Challenging the school mathematics culture, an investigative
small-group approach – Ethnographic teacher research on social and
sociomathematical norms. PhD diss., University of Lapland, Finland.
Rosenholtz, S. 1985. Effective schools: interpreting the evidence. American Journal
of Education, 93: 352–388.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 155
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Beauty as fit: An empirical study of mathematical proofs
Manya Raman
Umeå University
Beauty has been discussed since ancient times, but discussions of beauty
within mathematics education are relatively limited. This lack of
discussion is surprising given the importance of beauty within the practice
of mathematics. This study explores one particular metaphor of beauty,
that of beauty as fit, as a way to distinguish between proofs that are
considered beautiful and those that are not. Several examples are
examined, supported by empirical data of mathematicians and
mathematics educators who judged and ranked different proofs in a
seminar on mathematical beauty.
Keywords: beauty, fit, proof, mathematician, mathematics
Introduction
The idea of beauty as fit is an ancient one. It was touted by the Stoics, who defined
beauty as “that which has fit proportion and alluring color.” (Cicero, as quoted in
Tatarkiewicz 1972) and the Pythagoreans who claimed, “order and proportion are
beautiful and fitting” (Aristotle, as quoted in Tatarkiewicz 1972). The metaphor
persists to modern times. Beardsley described one essential characteristic of aesthetic
experience to be “a feeling that things are working or have worked themselves out
fittingly” (Beardsley 1982).
This metaphor of beauty as fit can be found not only in the arts, but also in
mathematics, as Hardy famously asserted, “The mathematician’s patterns, like the
painter’s or the poet’s must be beautiful; the ideas like the colours or the words must
fit together in a harmonious way” (Hardy 1967). And Sinclair (2002) discusses some
of the ways that her sense of fit guided her in the process of discovering a proof of
Napoleon’s theorem. That fit can be used productively as a metaphor seems clear, but
we still know little about what fit means in mathematics, whether it has different
connotations in different contexts, and why the notion of fit might have anything to do
with beauty, or aesthetic preference more generally.
The focus on aesthetics in mathematics education is not new. In the 1970s
Papert (1978) suggested that mathematical thinking consists of three processes:
cognitive, affective, and aesthetic. At the time of his writing, only the first of these
was a serious area of research. Now, there has been substantial process made also on
the second. Aesthetics remains under-researched, despite an attempt in the 1980s to
jump-start the field (Dreyfus and Eisenberg 1986). In recent years there has been a bit
of activity in the field, mostly due to Nathalie Sinclair. One purpose of this paper,
part of a larger project conducted in cooperation with Lars-Daniel Öhman, a
mathematician at Umeå University, is to try to contribute to the momentum generated
from her work.
One of Sinclair’s main contributions (see e.g. Sinclair 2004) has been to shift
the focus away from judgements of mathematical objects (such as proofs) towards a
more holistic account of the role that aesthetics can play mathematical practice, three
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of which she identifies as generative, motivational, and evaluative. While this is an
important shift, the paper here concerns only the evaluative aspect, or in particular
what makes a particular proof beautiful. The reason for this focus is that it seems
difficult to move forward with any serious research program on aesthetics without
nailing down what is meant by mathematical beauty in the first place.
This brings us back to the metaphor of beauty as fit. Fit is by no means the
only metaphor used for beauty. Others include unity, perfection, moderation, and
metaphor itself (see Tatarkiewicz 1972 for an excellent account of the history of these
and other accounts of beauty). Yet, the metaphor seems productive, in ways we will
discuss below, and working mathematicians refer to it when making judgements about
the beauty of mathematical proofs. In other words the metaphor is persistent enough
that it seems important to try to understand exactly what it means.
The goal of this paper is to try to clarify at least some of the roles of fit in the
context of mathematical proof. We will discuss below, though two examples of
proofs: (i) what does it mean for a proof to fit a theorem, and (ii) what different types
of fit a beautiful proof might possess. The analysis of proofs is supported by data
collected in a year-long seminar on mathematical beauty, attended by mathematicians
and mathematics educators who provided their own subjective judgements about the
aesthetic values of the different proofs.
Examples
Pythagorean theorem
Let c be the length of the hypotenuse of a right triangle T0, and let a, b be the
lengths of the remaining two sides. Then the sum of the areas of the squares
constructed on sides a and b of T0 equals the area of the square constructed on
the hypotenuse, or symbolically a2 + b2 = c2.
The first example we will consider is the Pythagorean Theorem. This is a familiar
theorem, for which most mathematicians will know many different proofs, and most
likely have a favourite. Below we present one proof, from Euclid VI. 31, that is fairly
familiar and among those that the mathematicians in our seminar preferred, and a
second proof which might be new for many people and which proved to be less
popular. We begin with the proof from Euclid.
b
c
d
a
Figure 1
First proof. Consider Figure 1. The line d is perpendicular to c, and intersects the
vertex of the triangle. Let T1 be the right triangle with hypotenuse a and side d,
and let T2 be the right triangle with hypotenuse b and side d. Clearly, by the
principle of conservation of area, the sum of the areas of T1 and T2 equals that of
T0. We can, of course, consider these three triangles as being constructed on either
side of the original triangle. Also, by standard congruences (two shared angles),
all the triangles T0, T1 and T2 are congruent. Scaling a figure F in the plane by a
linear factor k changes the area of F by a factor k2. Therefore, if the theorem holds
for any set of congruent plane figures constructed on either side of the original
triangle, it holds for all such sets of congruent plane figures. As observed above,
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
the theorem holds for congruent right-angled triangles, and therefore holds for any
set of congruent figures, in particular, squares.
The missing algebra establishing that it is indeed the equation a2 + b2 = c2 that
follows from the scaling considerations can be presented in the following manner:
The linear scaling factor from T1 to T2 is b/a, from T2 to T0 is c/b and so on. If we let
Si be the area of Ti, for i = 0, 1, 2, it follows that S0 = S1 + S2 = (a/c)2 S0 + (b/c)2 S0,
from which we get c2 = a2 + b2 by cancelling S0 and multiplying through by c2.
Second proof. Suppose we have the subtraction formulas for sine and cosine:
(1)
cos(α − β) = cos(α) cos(β) + sin(α) sin(β)
(2)
sin(α − β) = sin(α) cos(β) − cos(α) sin(β).
Suppose that α is the angle opposite to side a, and β is the side opposite to
side b, and without loss of generality that 0 < α ≤ β < 90◦ . We now have cos(β)
= cos(α − (β − α)) = cos(α) cos(β − α) + sin(α) sin(β − α) = cos(α)(cos(α) cos(β)
+ sin(α) sin(β)) + sin(α)(sin(α) cos(β) − cos(α) sin(β)) = (cos2(α) + sin2(α))
cos(β), from which it follows that cos2(α) + sin2(α) = 1, since cos(β) is the
ratio between one leg and the hypotenuse of a right triangle, and as such is
never zero. The theorem now follows from the definitions of sine and cosine
and scaling.
In our seminar, these two proofs were presented along with six other proofs
of the theorem. Members of the seminar, both mathematicians and mathematics
educators were asked to rank the proofs from those they liked best to least, and to
write a word to describe what they thought of the proofs (e.g. beautiful, nice, slick).
The reason for posing the task as such was to separate the issues of preference and
beauty: one might like a proof for other reasons than its aesthetic appeal, and there
are words similar to beauty (like elegance) that might have a distinctly different
connotation.
The pilot data show that proof 1 above was preferred to proof 2 for all the
mathematicians and one of the mathematics educators. The words used to describe
the first proof included, “simple”, “beautiful”, and “conceptually correct”, while the
words used to describe the second proof included “ugly”, “clever”, and
“unnatural”. One of the mathematics educators also preferred proof 1, but the reasons
given by the other two for preferring proof 2 was that it was easier for them to
follow, having just seen the area argument for the first time and not grasping it
entirely.
The point of this first example is to distinguish a proof that our
mathematicians agreed fits a theorem (the first proof) from the one that does not (the
second). The mathematicians suggested that the reason the first one fits is that it gets
directly to what the Pythagorean theorem is about. With a very simple algebraic
calculation one can check that the sum of the similar squares behaves the same way as
the sum of the similar triangles, which conveniently both lie on the three sides of the
triangles and also make up the interior (so one can easily see that the sum of the first
two is the same as the second.) The proof is both economical – it doesn’t involve
outside information, as the second proof using trigonometry does – and it is
transparent– once you see the idea of the proof you immediately see why the
conclusion of the theorem follows. In contrast, the second proof, while neat or
perhaps new to the reader, involves extraneous information: the Pythagorean
relationship falls out of the calculation, but one does not have a sense of why the
theorem holds. The proof appears like a trick, a set of algebraic manipulations which
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give you the result while keeping you in the dark. This proof, while having some
aesthetic merits, was not considered by our mathematicians to be beautiful.
Pick’s theorem
Let A be the area of a lattice polygon, let I be the number of interior lattice
points, and let B be the number of boundary lattice points, including vertices.
Then A = I + B/2 − 1.
The second example we will consider is Pick’s theorem, which gives a simple
formula for calculating the area of a lattice polygon, that is a polygon constructed
on a grid of evenly spaced points. The theorem, first proven by Georg Alexander
Pick in 1899, is a classic result of geometry. An interior (lattice) point is a point of the
lattice that is properly contained in the polygon, and a boundary (lattice) point is a
point of the lattice that lies on the boundary of the polygon. We will assume two facts
as lemmas, first that it is always possible to triangulate a polygon (see Figure 2 as
an example), and the second that each of the elementary triangles has area ½.
There are many proofs of this theorem, but the one below is considered to be among
the most beautiful (see Raman and Öhman (2011) for another beautiful proof).
Figure 2: A triangulated lattice polygon
Proof sketch. For space reasons we sketch the proof below and refer to Aigner and
Ziegler (2009) for details. The idea of this proof is to conceive of the triangulated
lattice polygon as a polyhedron, with each triangle as a face, and the outer area
(outside of the boundary of the polygon) as a face. We can count the number of
edges in two different ways: 3 N = 2 eint + ebd. where N is the number of triangles,
eint is the number of interior edges, and ebd is the number of boundary edges. Note
that we are overcounting the edges on both sides, but by the same amount, namely
the number of edges that are shared by neighbouring triangles.
Next, we apply Euler’s formula, V + F - E = 2, where V = number of vertices, F =
number of faces, and E = number of edges. For our polyhedron, V = I + B, F = N
+ 1, and E = eint + ebd. Using substitution and algebra, one can now arrive at the
formula A = I + B/2 -1.
The point of this proof is to show that proofs can “ fit” in at least two
different ways. Proof 1 of the Pythagorean theorem, above, fits the theorem in a
way which we will label as “internal fit”, meaning that the proof directly
illuminates what the theorem is about, providing a sense of why the theorem is true.
The proof of Pick’s theorem above fits in a way we will label as “ external fit”,
meaning that the proof derives its beauty from the way it is connected to a family
of other theorems, in this case the theorems that can be proven using Euler’s
theorem (and this particular case is a surprising application of that theorem.) This
kind of fit does not convey meaning or explanation, per se, but links the theorem,
through the proof, to a class of theorems not previously thought to be related.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Discussion
We now return to the question of fit: what does it mean for a particular proof to fit a
particular theorem? Our analysis above indicates that there are at least two types of
fit— internal fit and external fit— that can both potentially give rise to the sensation
of beauty. We are far from proving that the relationship always holds, that is to say
that fit and beauty are always coupled. In fact, looking at the empirical data we see
that some people prefer proofs that we claim do not have fit. The reason for this
mismatch might arise from the fact that judgements of mathematical beauty must be
linked to understanding. If one does not understand a particular proof, one cannot
judge it as beautiful. So it is difficult to say exactly how fit and beauty relate, except
to say there seems to be some correlation among people with a particular level of
understanding.
Another potential lesson from this short exploration is that we should
distinguish between (1) a proof having a particular sort of fit to a theorem; and (2)
whether a particular person can see the fit. The first feature could be objective while
the second one is subjective. These two features are often confused, giving rise to the
knee jerk “beauty is in the eye of the beholder” type attitude. Making a distinction
between whether a proof is beautiful and whether a person can grasp that beauty can
help explain phenomena such as why mathematicians judge different proofs to be
beautiful, or why mathematicians and non-mathematicians do the same, without
drawing a necessary conclusion that mathematical beauty is subjective. Moreover, the
metaphor of ‘fit’ suggests a more objective view of beauty might be warranted—
whether a proof is appreciated as beautiful is a subjective claim, but whether a proof
fits a theorem, which relies more on the nature of the proof than our perception of it,
is a more objective one.
References
Aigner, M., and G. Ziegler. 2009. Proofs from THE BOOK (4th ed.). Berlin, New
York: Springer-Verlag.
Beardsley, M. C. 1982. The aesthetic point of view. Selected essays. Ithaca: Cornell
University Press.
Dreyfus, T., and T. Eisenberg. 1986. On the aesthetics of mathematical thought. For
the Learning of Mathematics, 6(1): 2-10.
Hardy, G. H. 1967/1999. A mathematician’s apology. New York: Cambridge
University Press.
Papert, S. 1978. The mathematical unconscious. In On aesthetics and science, ed. J.
Wechsler, 105-120. Boston: Birkh•auser.
Raman, M., and L.-D. •Öhman. 2011. Two beautiful proofs of Pick’s Theorem.
Proceedings of Seventh Conference of European Research in Mathematics
Education. Rzeszow, Poland. Feb. 9-13.
Raman, M., and L.-D. Öhman. (In prep). Beauty as fit: A mathematical case-study.
Sinclair, N. 2002. The kissing triangles: The aesthetics of mathematical discovery.
International Journal of Computers for Mathematical Learning 7: 45-63.
Sinclair, N. 2004. The roles of the aesthetic in mathematical inquiry. Mathematical
Thinking and Learning, 6(3): 261-284.
Tatarkiewicz, W. 1972. The great theory of beauty and its decline. The Journal of
Aesthetics and Art Criticism, 31(2): 165-18.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 160
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Making sense of fractions in different contexts
Frode Rønning
Sør-Trøndelag University College and Norwegian University of Science and Technology,
Trondheim, Norway
This presentation is based on a study of 20 pupils, aged 9-10, in a
Norwegian primary school. The pupils were exposed to two, rather
different, classroom situations and in both situations the concept of
fraction was central. The pupils were given tasks and in order to
accomplish these tasks it was necessary to make sense of fractions in
some way. An interesting observation is how the presence of different
mediating artefacts influences the pupils’ meaning making.
Keywords: Fractions, semiotic representations, mediating artefacts.
The classroom episodes
The first episode takes place in the pupils’ regular classroom, which is quite large and
holds an area furnished as a kitchen at one end. There are 20 pupils in the class and
the pupils come in groups of five to the kitchen area to do a particular task, making
batter for waffles. This task involves measuring out a number of ingredients (milk,
flour, butter) and in this paper I am particularly interested in what happens when the
pupils measure out 15 decilitres of milk. The milk comes in boxes marked “1/4 liter2”,
and the pupils have measuring beakers available that can take 1 litre. The beakers are
transparent, with a scale marked “1 dl, 2 dl, …. 9 dl, 1 lit” from bottom to top.
The second episode takes place some time later. In this episode the pupils
receive a task sheet with drawings of red and blue milk boxes of equal size and with
the information that a blue box contains 1/3 litre of milk and a red box contains 1/4
litre. Here the standard fractional notation with a horizontal bar is used. In this text I
will use the fraction notation a/b to save space. On the task sheet the following four
situations are described: A: Three blue boxes, B: Four blue boxes, C: Four red boxes,
and D: Three red boxes. The following questions are given:
 Which box, red or blue, contains most milk?
 Which situation, A, B, C or D, represents the largest quantity of milk?
 And which situation represents the smallest quantity of milk?
 Are there any situations with the same amount of milk?
 How many decilitres of milk are there in situation D?
 You need 15 decilitres of milk and you have boxes containing 1/4 litre, hence
red boxes. How many boxes do you need?
The pupils have only pencil and paper available and no concrete material. The
20 pupils are grouped in the same way as in the first episode and each group leaves
the rest of the class to join me in an adjacent room to work with the task for about 30
minutes.
Both episodes were video recorded and the video footage constitutes the most
important data for the analysis. Video recordings have been transcribed, first in
Norwegian, and later parts of the transcriptions have been translated into English. In
addition there exist notes and drawings made by the pupils in the second episode.
2
Norwegian spelling of litre
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The most important research questions for the study are: how do the children
make sense of fractions given with different representations, and, in what ways will
mediating artefacts influence the children’s sense making of fractions?
Theoretical framework
The study reported on in this paper is concerned with pupils applying and developing
mathematical knowledge in different settings, which calls for a stance that knowledge
is situated (Lave and Wenger 1991). Closely related to this is also the idea that the
knowledge depends on the sociocultural artefacts that mediate between stimulus and
response (Wertsch 1991). I will use the term artefacts to denote both physical tools,
such as measuring devices that are used in the described situations, and psychological
tools, such as language and signs. All the artefacts involved are considered as cultural
tools, containing both psychological and physical aspects (Säljö 2005/2006, 28).
My analysis of the pupils’ work in the two situations rests heavily on semiotic
theory. The concept of sign is fundamental, and according to Peirce
[a] sign is a thing which serves to convey knowledge of some other thing, which it
is said to stand for or represent. This thing is called the object of the sign; the idea
in the mind that the sign excites, which is a mental sign of the same object, is
called the interpretant of the sign. (Peirce 1998, 13, emphasis in original)
A sign has two functions, a semiotic function; “something that stands for
something else”, and an epistemological function, indicating “possibilities with which
the signs are endowed as means of knowing the objects of knowledge” (Steinbring
2006, 134). All mathematical objects are abstract but, despite this, mathematical
concepts and the signs representing them are used to refer also to real life situations.
A sign or symbol can therefore be thought of as representing a mathematical concept
as well as a concrete object or reference context. This is visualised in The
Epistemological Triangle (Steinbring 2006, 135) shown in Figure 1 below.
Object/reference
context
Sign/symbol
Concept
Figure 1: The epistemological triangle
The relations in the Epistemological Triangle are largely conventional and in a
learning process these relations are in development. In a given situation meaning is
created through mediation between the sign/symbol and object/reference context. This
means that the system is continuously in development based on interaction between
pupil/s and teacher. According to Steinbring “[t]he links between the corners of the
epistemological triangle are not defined explicitly and invariably, they rather form a
mutually supported, balanced system” (Steinbring 1997, 52).
Analysis of the two episodes
Although the two episodes can be said to deal with largely the same mathematical
topic they are very different in their context. Even though the first episode takes place
in a mathematics lesson it is very close to an everyday context. Both the task itself
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
and the artefacts that are used are of a nature that the children will recognise from
their daily life experiences. The purpose of the task is also of a very practical
character. The pupils were supposed to make the batter for the waffles and later in the
day they were going to cook the waffles and eat them together with other pupils at the
school. In the second situation the task is of a nature which could be said to be typical
for a school task. Although it makes reference to daily life artefacts (milk boxes) the
milk boxes are only imaginary and it had no practical consequences whether the task
was correctly solved or not.
Making sense of the symbol 1/4 liter
In the first episode most pupils noticed the text “1/4 liter” on the milk boxes and they
started to discuss the meaning of this sign. Several suggestions were offered for the
meaning of the sign. Here are some examples: “One four litres”, “One comma3 four
litres”, “four and a half litres”, “one and a half litres”. Some of the suggestions are
combined with common sense such as when the teacher challenges the proposal that it
is 4.5 litres in one box the pupils suggest that it must be decilitres, because, as one
pupil says, “it isn’t even half a litre”. In one of the groups Jessica suggests that one
box contains “one comma four” (i.e. 1.4) litres and James follows up by suggesting
that it will be 2.8 if they take two boxes. If they had relied on counting in this way
they would not have obtained the desired amount of milk but Ellie points to the fact
that there is an empty measuring beaker on the table which they can use. Jessica had
not seen this in the first place, but being made aware of it she and James start pouring
milk into the measuring beaker. Now the scale of the measuring beaker takes over the
role as a sign connecting the amount of milk to the boxes (reference context). This
new sign renders the original sign 1/4 liter obsolete and the pupils no longer have any
need to make sense of this sign. On the video one can see that the pupils follow
closely the level of milk rising in the measuring beaker when they pour in the fourth
box and they show no sign of making a connection between the fact that they have
used four boxes and that the scale shows 1 litre. Jessica says that “we need to have
five more decilitres”. Now they pour the milk into the bowl with the flour and fetch
another box of milk. Jessica pours in the content of the box into the now empty
measuring beaker and says “three decilitres” while looking and pointing at the scale.
During this process I ask how much there is in one box. Jessica looks at the sign 1/4
liter and says “one comma four litres”.
The measuring task has now been completed without ever making correct
sense of the sign 1/4 liter. When I ask them how many boxes they have used they
present the answer “six” which is obtained by counting the empty boxes. The excerpt
below shows how Jessica works only in the realm of decilitres and the number of
boxes only comes in because it is explicitly asked for by me, not because it is
necessary to complete the task.
3
Jessica:
Three, four, five six. Have you thrown away any?
Ellie:
Me. No.
Jessica:
OK, then we have used six.
Frode:
Six, to get 15 decilitres?
Jessica:
Yes, we had 10 before, and then we took five now.
In Norwegian “comma” is used for the decimal point, so “one comma four” would be 1.4.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
My interpretation of this episode is that during the process the measuring
beaker has been introduced as a new sign, replacing the sign given in fractional
notation, and the mediation of the concept 15 dl takes place between the scale of the
measuring beaker and the milk boxes instead of between the sign 1/4 liter and the
milk boxes. The effect of the measuring beaker can be seen also in the reasoning of
Joseph and Thomas, who were urged by the teacher not to use the measuring beaker.
Joseph:
Ohh. A quarter of a litre, that is … a quarter … ten decilitres is
one litre. We have to have three of these then, then it will be. Five
of these I think … no not five. How much should we, Thomas, if
we take three of these, no four, then it is one litre and we want
fifteen decilitres, and that is, and ten decilitres that is one litre.
But how many more than four do we have to take then?
Thomas:
Then we have to take four, and then we have to take … two
Joseph:
Then we have two, and ten decilitres here. And then it is fifteen.
The excerpt above shows that 1/4 is replaced by “a quarter” and that “four plus
two boxes” will equal one and a half litre. Without the measuring beaker the sign 1/4
is given meaning in order to solve the task.
Which box, red or blue, contains most milk?
This is the first question on the sheet given to the pupils in the second episode. Here
the reference context is taken to be the pictures of the coloured, equally sized, boxes
and the sign is given in the standard fraction notation, such as in Figure 2.
1
3
Volume of one box
Figure 2: The epistemological triangle for the volume
To compare the volume of the red box to the volume of the blue box the only
available representation is the symbol given as a fraction. The reference context gives
no information about the relative size of the boxes. The pupils soon agree that 1/3 >
1/4 and to justify their argument they create a new reference context in terms of a
rectangle divided in strips. An example is shown in Figure 3. The drawings are not
made to match the actual situation but I interpret that the drawings are meant to show
that when m > n, 1/m < 1/n. This interpretation is supported by a statement from one
of the pupils saying, “the larger the number below, the smaller is the actual part”.
Figure 3: Comparing fractions
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 164
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
How many boxes to get 15 dl?
In Episode 1 the pupils relied on the measuring beaker to get the correct amount of
milk, except Joseph and Thomas who were encouraged to manage without it. In
Episode 2 the option of using a measuring beaker, or any other physical artefact, is
not there, so the pupils have to rely on making sense of the signs. I have previously
described how the group with Jessica and James completely depend on the scale of
the measuring beaker to get 15 dl in Episode 1 and that they answer the question
about how many boxes they have used by just counting the empty boxes. Below is
part of the dialogue when the same group solves the problem of getting 15 dl in
Episode 2.
Jessica:
Five, ten, fifteen. It will be three
Ellie:
So it is three
Frode:
OK, five, ten, fifteen. That is three
Ellie:
It is just like in D, one, two, three
Jessica:
Three boxes, it will be three boxes
James:
Three boxes, no, we should have fifteen
Ellie:
We are not supposed to have fifteen boxes, but fifteen decilitres
Jessica:
Yes, and each box takes two and a half decilitres
Emily:
Couldn’t we…
Jessica:
I did not understand this
Ellie:
Me neither
Emily:
Two comma five, two comma five, that is five, and then we have
five three times in fifteen, and then it is two for each, so it is six
Ellie:
OK, but then I did not understand anything
…
Emily:
Every five is two boxes, so it is three, therefore six.
After some initial confusion Emily comes up with a solution by converting 2
times 2.5 to 5 and then 3 times 5 to 15. Then she finds the number of boxes, 6, by
taking 2 times 3. Compared to the solution by Joseph and Thomas presented before
there are similarities but also differences. Both solutions entail building up the total
amount using a multiplicative procedure but in different ways. Joseph and Thomas
find that 4 boxes equal 1 litre and that they need 2 more to get 15 dl = 1.5 litre. This
reasoning was repeated by Joseph in Episode 2 when he said about Situation C (where
there are 4 boxes of 1/4 l): “C is one litre, which is ten decilitres. Then I need half of
C again, and that is two and therefore it is six. Two plus four is six.” Both solutions
involve two steps, where the first step establishes a relation between a number of
boxes and a number of decilitres that is easy to handle further to get 15. Presented in a
table the two solutions can be illustrated as shown in Tables 1 and 2 below.
Boxes
Dl
4
10
4+(half of)4= 15=10+(half of)10=
4+2=6
10+5
Table 1: Joseph’s solution
Boxes
dl
2
5
3x2=6 15=3x5
Table 2: Emily’s solution
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
The solutions shown in Tables 1 and 2 end with the answer six boxes as a
result of a process where the symbol “1/4 liter” plays a crucial role. Joseph, in
accordance with his reasoning in Episode 1, connects 1/4 with “a quarter” and “four
quarters equal one whole (litre)”. Emily links decimal notation to fractional notation,
2.5 dl = 1/4 l, and then she uses 5 dl as a starting point to count 5-10-15. In Episode 1
the presence of the measuring beaker made the interpretation of the symbol 1/4 liter
redundant. Instead of 1/4 liter being the sign that mediates between boxes and
decilitres the scale of the measuring beaker was used as the mediating artefact. The
scale functions as an indexical sign (Peirce 1998) that has a real connection to the
object that it represents, namely the milk in the measuring beaker.
Further analysis of the two situations described in this paper can be found in
Rønning (2010 and in press).
References
Lave, J., and E. Wenger. 1991. Situated learning: Legitimate peripheral participation.
Cambridge: Cambridge University Press.
Peirce, C. S. 1998. The essential Peirce. Selected philosophical writings. Vol. 2
(1893-1913). (Edited by the Peirce Edition Project). Bloomington, IN: Indiana
University Press.
Rønning, F. 2010. Tensions between an everyday solution and a school solution to a
measuring problem. In Proceedings of the Sixth Congress of the European
Society for Research in Mathematics Education. January 28th - February 1st
2009, Lyon, France, eds. V. Durand-Guerrier, S. Soury-Lavergne, and F.
Arzarello, 1013-1022. Lyon: INRP.
––– in press. Making sense of fractions given with different semiotic representations.
Paper to be presented at the Eighth Congress of the European Society for
Research in Mathematics Education, February 2013.
Säljö, R. 2006. Læring og kulturelle redskaper. Om læreprosesser og den kollektive
hukommelsen (S. Moen, Trans.) [Learning and cultural tools. On processes of
learning and collective memory]. Oslo: Cappelen Akademisk Forlag. (Original
work published 2005).
Steinbring, H. 1997. Epistemological investigation of classroom interaction in
elementary mathematics teaching. Educational Studies in Mathematics, 32:
49-92.
––– 2006. What makes a sign a mathematical sign? An epistemological perspective
on mathematical interaction. Educational Studies in Mathematics, 61: 133162.
Wertsch, J. V. 1991. Voices of the mind. Cambridge, MA: Harvard University Press.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 166
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Developing statistical literacy with Year 9 students: A collaborative research
project
Dr Sashi Sharmaa, Phil Doyleb, Viney Shandilc and Semisi Talakia’atuc
a
The University of Waikato; bThe University of Auckland; and cMarcellin College
Despite statistical literacy being relatively new in statistics education
research, it needs special attention as attempts are being made to enhance
the teaching, learning and assessing of this strand. It is important that
teachers are aware of the challenges of teaching and assessing of this
literacy. In this collaborative research study, two cycles of teaching
experiments were carried out in two year 9 classes. The data set consisted
of audio and video-recordings of classroom sessions, copies of students’
written work, audio recorded interviews conducted with students, and
field notes of the classroom sessions. The results shed light on tools and
techniques which the research team used to help students develop critical
statistical literacy skills. The findings have implications for teaching and
further research.
Key words: statistical literacy, high school students, collaborative research,
teaching experiments, relevant contexts, data based arguments
Introduction
Every day, people all over the world are bombarded with a complex array of numbers
and statistics (Budgett and Pfannkuch 2010; Gal 2004;Paulos, 2001; Schield, 2010).
For example, statistics of opinion polling, business, employment and health regularly
appear in the news media and research reports. According to a number of educators
(e.g. Best 2001; Gal 2004; Paulos 2001), people without statistical literacy may be
misled or have difficulty in interpreting and critically evaluating such messages. Best
(2001) writes that consumers need to understand that statistics is a social construct
and that people debating social problems may chose statistics selectively and present
them to support their point of view. For example, gun-control advocates may be more
likely to report the number of children killed by guns, whereas opponents of guncontrol may prefer to count citizens who use guns to defend themselves from attack.
However, people often choose to rely on an author’s interpretation and seem not to
engage adequately with such information.
The importance of statistics in everyday life and workplace have led to calls
for an increased attention to statistical literacy in the mathematics curriculum
(Ministry of Education 2007; Schield 2010; Shaughnessy 2007; Watson 2006).
Schield (2010) argues that one of the most important goals for teaching statistics in
schools is to prepare students to deal with the statistical information that increasingly
impacts on their lives. More specifically, critical stance (Gal 2004) - the ability to take
and evaluative stance with respect to statistical flaws and biases contained in media,
marketing and financial reports - is of vital importance in the quest for statistical
literacy. In New Zealand, Begg et al. (2004) have called for a greater emphasis to be
placed on statistical literacy in the curriculum so that students can become active and
critical citizens. The use of the term statistical literacy is much more explicit in the
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
new curriculum document with the addition of statistical literacy achievement
objectives (Ministry of Education 2007).
Additionally, schools are being asked to prepare students to be flexible
thinkers, lifelong learners, and to manage complexities of an uncertain world
(Ministry of Education 2007). They need to think for themselves when faced with
contradictory information from diverse sources and contexts (Gal 2004; Paul 2011).
Watson (2003) stated that in this century decision making for all citizens is likely to
be made based on the critical thinking skills derived from the statistical literacy
strand.
Gal (2004) sees statistical literacy as the need for students to be able to
interpret results from studies and reports and to be able to pose critical and reflective
questions about those reports. Gal would like students to come away from a statistical
literacy class with an ability to evaluate statements from reports and ask a set of
questions such as: Where did the data come from? What kind of study is it?
According to Watson , statistical literacy is the “ meeting point of the chance and data
curriculum and the everyday world, where encounters involve unrehearsed contexts
and spontaneous decision-making based on the ability to apply statistical tools,
general contextual knowledge, and critical literacy skills” (2006, 11). Clearly, the type
of statistical literacy that Gal (2004) and Watson (2006) propose is different from just
being able to read and evaluate data and graphs. Aspects of Gal’s notion of statistical
literacy have been incorporated in the New Zealand Curriculum.
It is interesting to see terms like statistical thinking and statistical literacy in
the revised curriculum document as well as notions of critical thinking in the key
competencies and descriptions of effective pedagogy (Ministry of Education 2007).
However, many of the theories and developments of statistics education are still very
new. It is not clear how many teachers are aware of the theories and developments in
statistics education and how many teachers understand teaching as inquiry and the
implications of research in their classrooms. For instance, there may be a
match/mismatch between the stance taken by the current curriculum towards
statistical literacy and what teachers understand of statistical literacy (Doyle 2008).
Research in New Zealand and overseas (Garfield and Ben-Zvi 2008; Hill,
Rowan and Ball 2005; Hunter 2010) has consistently acknowledged the importance of
the teacher in student learning. A key theme of Effective Pedagogy in
Mathematics/Pāngarau Best Evidence Synthesis Iteration [BES] is that “quality
teaching is not simply a matter of ‘knowing your subject’ or ‘being born a teacher”
(Anthony and Walshaw 2007, 4). There is a need for quality professional
development in secondary schools to make sure that good practice occurs in as many
classrooms as possible. Teachers need to become fully conversant with the theory to
participate in the research process. A design research approach allows researchers and
teachers to work together within a research process in which researchers and teachers
work together to explore student learning.
The research approach
According to Bakker (2004), statistical ideas need to be developed slowly and
systematically using carefully designed sequences of activities in appropriate learning
environments, which challenge students to explore, conjecture and evaluate their
reasoning. One way to develop these sequences of activities is through a researchand-development process called design research (Cobb 2000). Design research is
cyclic with action and critical reflection taking place in turn. There are benefits for
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
teachers and researchers undertaking such research. In this type of collaborative
research the teacher is involved in the whole process and takes part in posing
questions, collecting data, drawing conclusions and writing reports.
Research design and data collection methods
The following inter-related research questions guided our study:
 How can we support students to develop statistical literacy within a data
evaluation environment?
 How can we develop a classroom culture where students learn to make and
support statistical arguments based on data in response to a question of interest to
them?
 What learning activities and tools can be used in the classroom to develop
students’ statistical critical thinking skills?
Preparation for the teaching experiment
This phase consisted of literature review (statistical literacy, teaching experiment) and
the first attempt at reformulating a teaching sequence. Then, the research team
proposed a sequence of ideas, skills, knowledge and attitudes that they hoped students
would construct as they participate in activities. The team planned activities to help
move students along a path towards the desired learning goals. As part of the
activities, students evaluated statistical investigations or activities undertaken by
others including data collection methods, choice of measures and validity of findings
(Ministry of Education 2007). The team envisioned how dialogue and statistical
activity would unfold as a result of planned classroom activities.
The teaching took place in regular classrooms and as part of mathematics
teaching. The teaching activities were spread over up to two weeks to suit the school
schedule. The research team was involved in designing, teaching, observing and
evaluating sequences of activities. There were two cycles of teaching experiments.
The goal was to improve the design by checking and revising conjectures about the
trajectory of learning for both the classroom community and the individual students.
Students’ thinking and understanding was given a central place in the design and
implementation of teaching eight lesson in each cycle. The research team performed a
retrospective analysis after each lesson to reflect on and redirect the learning. In
addition the team performed analysis of the unit after an entire teaching experiment
has been completed. The continually changing knowledge of the research team
created continual change in the learning sequence.
Data Collection
The data set consisted of video-recordings of classroom sessions conducted during the
design experiment, copies of all the students’ written work, audio recorded miniinterviews conducted with students, and field notes of the classroom sessions. Semistructured interviews were also conducted while the design experiment was in
progress, with six groups of three students. These interviews were scheduled after
class sessions and focused on students’ interpretation of classroom events with a
particular emphasis on the identities they were developing as consumers of statistics. .
Each teacher-researcher kept a logbook of specific events that took place during the
data collection period. The team was engaged in conscious reflection and evaluation
of situations as they unfolded.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Data Analysis
The research team read the transcripts, watched the videotapes, and formulated
conjectures on students’ learning on the basis of episodes identified in the transcripts
and video. The generated conjectures were tested against other episodes and the rest
of the collected data.
Results
Statistical literacy is more than the ability to do calculations and read tables and
graphs. Our findings show that students are actually quite good at this. Students were
able to interpret and critically evaluate statistical information and data related
arguments. Additionally, they were able to discuss and communicate their
understanding and opinions to others. Students can be exposed to critical questions in
statistics as reflected in the following student quote:
The simplest question I want to ask is how they got the information. Now that we
have talked about statistic … and now that we probably understand a bit about
statistics, I would want to ask how they got the information
We noticed that literacy skills are critical in the development of statistical
literacy. Students were required to communicate their opinions clearly orally and in
writing. Students in the class were of different language abilities and needed to
interact in order to improve the group’s statistical communication. This presented
various demands on students’ literacy skills as indicated in the following student
quote:
Because usually, like in normal maths, we don’t use literacy … like we use
addition, subtraction but we actually have some kind of literacy for the things we
do in statistics.
The classroom discourse was important for statistical literacy. Most of our
classroom activities included group and whole class discussion of the data. This
typically involved a small group activity in which the students worked on problems
together and then reported back to the whole class. The two teachers took time to
remind the students how to work in groups (e.g. how to agree and disagree and how to
present to the class). Our results show that students can be taught how to question and
challenge in respectful ways as part of classroom discourse. Students found group
work useful:
When you are working alone you just get one point of view and when you are
working in a group you get different perspectives of other ideas … how other
people are thinking, learning in class
Context is an important component of statistical literacy. Our findings show
that students need exposure to both familiar and unfamiliar contexts. Engagement
with context helps students develop higher order thinking skills. However, our results
show that some contextual knowledge may be a barrier for some students. This is
revealed in the journal entry below:
My students found the language used in the Hans Rosling video difficult to
understand. I had to show the clip a couple of times. Some students even
questioned why I was using this clip..
Teachers were able to address this in two ways. The first was to start from
familiar contexts before moving to unfamiliar contexts. The other was to use contexts
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
of interest to the students. This involved handing over some of the control and
planning of lessons to students.
Teachers had an important role in the construction of a purposeful classroom
environment. Teachers needed to guide the pedagogical setting so statistically
relevant aspects were discussed
Limitations and implications for teaching and research
The limitations of design research can relate to technical and human aspects. On the
technical side, the recording devices used in the study may not have captured
everything that was said by the students and the teachers. On the human side,
interview data may be subjective, hence has limitations associated with reliability.
Students’ views, during interviews in particular, may have been influenced by our
unequal relationship. Their teachers assessed their work, so during the interviews,
students may have said things they thought we wanted to hear. Another human
limitation relates to researcher prejudices and biases. Since we were both the
practitioners and the researchers, data collection and analysis could have been
affected by our predispositions and partiality. Major implications for practice and
research that can be drawn from this study are discussed below.
We envision statistical literacy going beyond calculations. It is more than the
ability to do calculations and read tables and graphs. Students should be able to
interpret and critically evaluate statistical information and data related arguments.
Additionally, they should be able to discuss and communicate their understanding and
opinions to others. This has potential consequences in how the teaching of statistical
literacy might be altered for greater effectiveness. For example, ample class time
should be spent on discussion and reflection rather than presentation of information.
As well as statistical knowledge, literacy knowledge and skills are important
for statistical literacy. Since all statistical messages are conveyed through written or
oral text the understanding of statistical messages requires the activation of various
literacy skills. Additionally, students are required to communicate their opinions
clearly orally or in writing so others can judge the validity of their arguments. These
present various demands on students’ literacy skills. Teachers need to help students
access information.
We believe that the nature of the learning environment and classroom culture
are major contributors to success for students, and teachers need to put a high priority
on building a classroom climate that positively engages all students. Students need to
understand the importance of sharing their opinions in order to advance their
statistical ideas. It would be valuable for teachers to help students reflect on the
purposes of explaining and justifying their thinking to others
The ability to interpret and critically evaluate reports that contain statistical
elements is paramount in our information laden society. Teachers need to give
students some basic foundations for critiquing and evaluating statistically based
information that they encounter in daily life. We assume that students can be taught
these reasoning skills through using media articles as a springboard into learning
about how to evaluate these reports. Consequently they will become familiar with a
list of worry questions and apply them to real life examples without prompting,
consistent with Gal (2004).
Groth (2007) argues that the relationship between educational research and
teaching practice has often been stormy because researchers are often interested in
theoretical aspects and general questions whereas teachers are usually interested in
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
solving problems related to situations that arise in the classroom on daily basis. We
believe that a partnership between schools and universities can help strengthen cyclic
flow of information.
References
Anthony, A, and M. Walshaw. 2007. Effective pedagogy in mathematics/Pangarau
best evidence synthesis iteration [BES] Wellington: Ministry of Education
Bakker, A. 2004. Reasoning about shape as a pattern in variability. Statistics
Education Research Journal, 3, no. 2: 64-83.
Budgett, S. and M. Pfannkuch. 2010. Using media reports to promote statistical
Literacy for non-quantitative major. In Proceedings of the 8th International
Conference on the Teaching of Statistics, ed. C. Reading, Ljubljana, Solvenia:
International Statistical Institute and International. Association for Statistical
Education. Available www.stat.auckland.ac.nz/~iase/publications.php [© 2010
ISI/IASE]
Begg, A. M. Pfannkuch. M. Camden. P. Hughes. A. Noble. and C. Wild. 2004. The
school statistics curriculum: statistics and probability education literature
review. Auckland: Auckland Uniservices Ltd, University of Auckland.
Cobb, P. 2000. Conducting teaching experiments in collaboration with teachers, In
Handbook of research design in mathematics and science, ed. A. Kelly and R.
Lesh, 307-333. Mahwah, NJ: Lawrence Erlbaum.
Doyle, P. 2008. Developing statistical literacy with students and teachers in the
secondary mathematics classroom, Unpublished masters thesis. Waikato
University, Hamilton, New Zealand.
Gal, I. 2004. Statistical literacy: Meanings, components, responsibilities. In The
challenge of developing statistical literacy, reasoning and thinking, ed. J. B.
Garfield and D. Ben-Zvi, 47-78. Dordrecht, The Netherlands: Kluwer.
Groth, R. E. 2007. Reflections on a research-inspired lesson about the fairness of dice.
Mathematics Teaching in the Middle school 13: 237-243.
Garfield, J. B. and D. Ben-Zvi. 2008. Preparing school teachers to develop students’
statistical reasoning. In Proceedings of the ICMI Study 18 and 2008 IASE
Roundtable Conference, ed. C. Batanero, G. Burrill C. Reading and A.
Rossman. Joint ICMI/IASE Study: Teaching Statistics in School Mathematics,
Challenges for Teaching and Teacher Education.
Hill, H., B. Rowan. and D. Ball. 2005) Effects of teachers’ mathematical knowledge
for teaching on student achievement. American Educational Research Journal
42: 371-406.
Hunter, R. 2010. Changing roles and identities in the construction of a mathematical
community of inquiry. Journal of Mathematics Teacher Education 13, 397409.
Ministry of Education. 2007. The New Zealand curriculum. Wellington: Learning
Media.
Paul. R. 2011. Critical thinking: How to prepare students for a rapidly changing
world. The Critical Thinking Foundation: USA.
Paulos, J. A. 2001. Innumeracy: Mathematical illiteracy and its consequences. New
York: Hill and Wang.
Watson, J. M. 2006. Statistical literacy at school: Growth and goals Mahwah, NJ:
Lawrence Erlbaum.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 172
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Feedback on feedback on one mathematics enhancement course
Jayne Stansfield
Graduate School of Education, University of Bristol, UK and Bath Spa University UK
This paper reports on changes in students’ perceptions of assessment
during a Mathematics Subject Knowledge Enhancement Course (MEC).
Students’ views were gathered pre- and post-MEC via an open-question
questionnaire with semi-structured interviews for some. Pre- and postMEC understanding of mathematics features highly in the students’ sense
of progress, but few had experienced feedback prior to the MEC. PostMEC feedback is viewed as the most useful aspect aiding their sense of
progress.
Keywords: summative assessment; formative assessment; feedback;
understanding
Introduction
The Mathematics Subject Knowledge Enhancement Course (MEC) is designed for
post-graduates whose degrees contain insufficient mathematical subject knowledge
for direct entry to Initial Teacher Training (ITT). It aims to provide students with deep
understanding of mathematical concepts and their inter-connectedness, as opposed to
surface or rote learning.
Understanding cannot be represented by any single or simple model (Pirie
1988). Several models exist such as Skemp’s (1979) schema, in which isolated
concepts become more connected as understanding takes place. That connections are
an important part of understanding is backed up by Mousley (2004), whose literature
review demonstrated that development of understanding is focused on ‘connected
knowing’. Hence I am using the following succinct summary as a working definition
of understanding.
A mathematical idea or procedure or fact is understood if it is part of an internal
network. More specifically, the mathematics is understood if its mental
representation is part of a network of representations. The degree of
understanding is determined by the number and the strength of the connections.
(Hiebert and Carpenter 1992, 67)
Moreover, Hiebert and Carpenter (1992) point out that a variety of tasks are needed in
order to avoid an individual task being done by rote with no understanding.
On completion of the course, I am required to report on students’ readiness to
progress to their ITT course. Since the inception of the MEC, I have been determined
that assessment occurring throughout the course should support the students’ learning
and understanding and give a sense of progress.
The assessment regime is based on Black and Wiliam’s (1998) guidance for
using assessment to focus on learning. A wide range of tools are used such as
researching and presenting a topic of their own choice; traditional tests; writing their
own test and sitting one written by a peer; posters; investigations; and others. Initially,
all tasks are used formatively i.e., to aid student learning. Work is returned to students
with tutor comments. Students write their response to this and then use a criterionreferenced grid to grade their work. It is only after this process that tutors give grades.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Summative, formative assessment and feedback
Black and Wiliam’s (1998) opinion that formative assessment impacts positively on
student learning is backed up by Higgins, Hartley and Skelton (2002) who describe
how assessment methods influence the quality of learning, saying, moreover, that
formative feedback resulting from assessment can lead to deep rather than surface
learning and that a positive impact occurs when students ‘connect’ with the feedback.
Deep learning is not an automatic consequence of feedback. According to Struyven,
Dochy and Janssens, “students’ perceptions about assessment and their approaches to
learning are strongly related” (2005, 336) and “a surface approach to learning is easily
induced, whereas promoting the deep approach seems to be more problematic”. So, it
is likely that some feedback methods will be more effective than others. Murtagh and
Baker’s (2009) analysis of feedback delivery methods, “revealed explicitly that the
students much welcome all of the feedback strategies that are employed across the
programme” (2009, 24), whilst one-one tutorials were the most highly rated method.
Orsmond, Merry and Reiling (2004) caution that, since the student learning
experience is shaped by assessment, the tutor feedback and student learning should be
inseparable. If they become separated the formative aspect is lost. Bailey and Garner
(2010) highlight some difficulties with feedback such as it can be opaque; its purposes
can be ambivalent; and practices vary between tutors. MEC tutors have been working
on these issues over several years with assessment planned as an integral part of the
course and moderation to ensure consistent standards of feedback.
Ideally, we would use comment-only marking as suggested by Black and
Wiliam (1998) but, given that summative assessment is required by the institution, the
choice seems to be either to use the formative task in a summative manner or to have
formative tasks and separate summative tasks, duplicating effort to find out nothing
new. Indeed, Newton (2007) argues that there is no difference between the two forms
of assessment, only the use to which the results are put. Taras thinks that the only
difference between summative and formative assessment is timing, arguing that all
assessment is in fact summative of the learning to that point and “formative
assessment is in fact summative assessment plus feedback which is used by the
learner” (2005, 466). In Newton’s terms, our assessment regime has been designed to
prioritise formative over summative assessment, moreover “summative judgements
could be derived from an aggregation of judgements made for formative purposes.”
(2007, 154).
Some tasks on the MEC are accepted by the students without complaint
whilst, anecdotally, one in particular is often seen as not ‘valid’. This task involves
writing a test for peers. Each test is taken by one other member of the cohort, i.e.,
each student sits a different test. They are assessed on how fully their own test covers
the topic and also on their ability to answer the peer’s test. In my opinion, this task is
the most valid, i.e., assessing that which it is intended to assess, of those used but less
reliable, i.e., less likely to give exactly the same result if repeated. Perhaps their use of
‘valid’ could be substituted with ‘fair’. I think this process is fair because we feedback
based on what we see individuals have done, need to do and could do. Yet some
students perceive unfairness, perhaps because we use the results from this task to form
a summative statement, and because as Taras says, this “requires reliability (of grades
or classification) to take precedence over validity (of assessment)” (2005, 474).
Hence, I decided to investigate what perceptions students hold of assessment,
in particular what their perceptions are at the beginning and end of the MEC, not just
of fairness but more generally.
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Research Design/Method
I planned this naturalistic investigation as a preliminary study to attempt to identify
what views students hold. I wanted to minimise the effect of my own and others’
opinions in order to hear students’ views as clearly as possible.
With 19 students starting the course, interviewing all was not practical. An
open-question questionnaire was used to elicit views from all. This was repeated at
the end of the course, slightly amended to ask about their time on the MEC.
Additionally, permission was requested for use of students’ reflective logs;
assessment feedback; circle-time discussions; interviews at the end of the course; and
possible future interviews.
16 students completed the course of whom 10 gave permission for all of the
above; 1 refused use of anything; and the rest gave varying permissions. Of the 10
students who gave full permissions, 5 were interviewed using naturalistic/semistructured interviews in order to allow opinions to be expressed freely (Gray 2009;
Cohen, Manion and Morrison 2011). I attempted to choose these to be representative
of the cohort in terms of gender and prior qualifications but, ultimately, the choice
was made pragmatically based on who was available and would be easily accessible
in the future for follow-up interviews.
The pre-course answers were analysed using a generalised form of thematic
analysis based on Rapley (2011). Member checking (Cohen, Manion and Morrison
2011) was performed by asking all students to code their own responses under these
themes and my coding adjusted as a result. Post-course answers were then analysed
using the same themes.
Data analysis
My focus here is on the two questions I have analysed at this point: Q1 “Describe how
you knew how well you were doing in mathematics.” and Q3 “What do you think is
most useful for you to know how well you are doing in mathematics?” The number of
responses coded under each theme is shown in table 1 below.
Theme
Q1
Q3
Correct Answers
8
2
Marks
7
2
Easy/ability
3
0
Enjoyment
1
0
Understanding
8
5
Teachers
4
4
Confidence
3
2
Comparison with others
2
0
Self-help?
1
6
Reliance on method?
2
1
Lack of fear (student insisted this 1
0
is not any of above)
Table 1: Pre MEC - Prevalence of themes from Q1&3 of questionnaire
The table appears to show an apparent mismatch between how they knew and
how they find it most useful to know. Understanding features highly in both lists. But
correct marks and answers, which are highest in how they actually knew, are replaced
by self-help when considering what they found most useful, raising several questions
that I wish to explore in more detail, for instance who decided if the answers are
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
correct? The switch to self-help, when thinking about what is most useful, may imply,
perhaps, the teachers rather than the students? One student coded “High marks in
exams” as ‘Correct answers’ whilst I had coded it as ‘Marks’. It may be that the two
categories are in fact one or need to be split in some different way. Only one student
mentioned “feedback” which they coded as ‘Teachers’.
Post-course results
13 students completed the post-course questionnaire. See table 2 below.
Theme
Q1
Q3
Correct Answers
2
0
Marks
6
4
Easy/ability
0
0
Enjoyment
0
0
Understanding
7
4
Teachers
1
2
Confidence
3
0
Comparison with others
1
0
Self-help?
1
0
Reliance on method?
0
0
Lack of fear
0
0
Feedback
4
3
Table 2: Post MEC - Prevalence of themes from Q1&3 of questionnaire
Table 1 included responses from 17 students therefore direct numerical
comparison with Table 2 is difficult, however, there are several things that I noticed
immediately. ‘Correct answers’ appears to be far less important for how they knew
how well they were doing, although ‘marks’ and ‘understanding’ remain important.
However, a new theme of feedback was needed in order to code several responses
e.g., Assessment feedback, “being regularly assessed and getting assessment
feedback”, whilst four themes were not mentioned at all.
Responses to ‘what is most useful?’ are coded under four themes only: marks;
understanding; feedback; and teachers. ‘Correct answers’ is no longer a frequently
occurring theme, perhaps this indicates that marks and correct answers are either
somehow different and dependence on correct answers has decreased, or that
perceptions of correct answers has changed. ‘Feedback’ and ‘teachers’ need further
exploration since ‘feedback’ is given by teachers and the only response in the precourse data that mentioned ‘feedback’ was coded as ‘teachers’ by the participant.
Interviews
In order to try to understand what has changed, the 5 interview transcripts were
inspected alongside the rest of the data on a student-by-student basis. I include a brief
overview of 2 students below.
Student A
In the pre-MEC questionnaire, ‘A’ talked about fear; “teacher scared the living
daylights out of me” but also of her enjoyment of doing questions if she could
understand them. She continued to demonstrate this dichotomy during the MEC. Her
reflective logs mentioned understanding frequently. For example, “I just wanted to
understand why…. woke up in the morning understanding”. References to fear also
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
occurred frequently e.g., “Graphs bring me out in a cold sweat. I feel fearful before
I’ve even read the question.” Moreover, she raised questions for herself to follow up,
e.g., “I’m not really sure what triangle numbers are all about, yet. What’s their
relevance?” Fear and real interest are sitting (uncomfortably?) side by side.
The other theme repeatedly coming out of her log was the amount of time she
was spending coupled with a sense of failure, e.g., commenting on the pace of
lessons, “I understand and can apply suvat equations when I work on the questions at
home, but in class I feel panicky” or “I’ve spent hours and hours working on this and I
feel very deflated”. She continued to work with determination and later reported, “….
I’ve been able to complete the papers…” and I found no more mention of fear.
In the post-MEC questionnaire, ‘A’ described how she, “Didn’t find ‘exam’
results useful at all in school” perhaps implying that now she does? In interviews, she
explained that examinations feel like they are testing knowledge but other types of
assessment task are more than that. ‘A’ particularly liked tasks that could be taken
home and worked on in her own time. I surmised perhaps as a result of the effect of
time pressure but actually ‘A’ saw these as a continuation of the learning process
(“Every single one I took home I learnt so much more than learning it for an exam”)
and therefore useful.
‘A’ compared how she knew how well she was doing prior to the MEC as
only from “end of year exams” and “getting the ‘right’ answer in class”, but after the
MEC as “understanding. Understood links with other areas. Assessments.” She also
said the “staged assessments” were the most useful way for her to know this.
Although she does not use the word feedback, in my opinion this is implied because
she engaged thoroughly with the feedback process, e.g., “I am happy that I fully
understand the concepts of straight line graphs. I can see where I didn’t use precise
terminology in part 2 and understand my errors.”
Student K
‘K’ said that correct answers, marks and understanding were important pre-MEC,
describing learning as a feat of memory, “I definitely remember at school cramming
before an exam and going in and it’s all just in that short-term memory pull it all out
onto the page bang and the examiner says end of the exam pick your bag up and you
walk out and you can’t even remember the questions you’ve answered”. Post-MEC he
considered understanding and feedback as important. Talking about MEC assessments
he said “where you know really having to justify everything from first principles but
then actually having reflected on it I say well this is great”
In his reflective log, he talked about feedback from tutors as important but also
the ability to explain to others, as he saw this as essential for a future in teaching. He
also enjoyed the assessment task and found the feedback useful e.g., “I enjoyed this
piece of work as it helped me check my understanding”.
Conclusion
The MEC students’ responses indicate their views on how they know how well they
are doing have altered from a reliance on correct answers and marks to feedback,
although it is not clear in what ways correct answers and marks overlap or differ.
Connection with the feedback is evident. I would argue that subsequent work
frequently demonstrates that feedback has been acted on, although evidence of this is
not given here. Questionnaires and interview responses describe feedback as useful
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
with assessment tasks being enjoyable and part of the learning process, indicating a
close link is present between the formative feedback and their learning.
Further discussion is needed with this set of participants to clarify boundaries
between the themes in order to enable more accurate coding when working with
future cohorts. In particular, it is not clear whether ‘correct answers’ and ‘marks’ are
the same or in what way they differ. Nor is it apparent who decides the answers are
correct (students, teachers, book etc.). Where the coding ‘teachers’ has been used
what is the nature of reliance on the teacher and does this include marks and written
feedback? Understanding featured highly both pre- and post-MEC. As yet I have
made no attempt to analyse what students mean by understanding. Their perception of
what it means to understand may be very different to mine. This would be a valuable
future investigation.
References
Bailey, R., and M. Garner. 2010. Is the feedback in higher education assessment
worth the paper it is written on? Teachers' reflections on their practices.
Teaching in Higher Education 15:187-98.
Black, P., and D. Wiliam. 1998. Inside the black box: Raising standards through
classroom assessment. London: King's College.
Cohen, L., L. Manion and K. Morrison. 2011. Research Methods in Education (7th
Ed.) Oxford: Routledge.
Gray, D. 2009. Doing Research in the Real World. London: Sage.
Hiebert, J. and T.P. Carpenter. 1992. Learning and teaching with understanding. In
Handbook of Research on Mathematics Teaching and Learning, ed. D.A.
Grouws, 65-97. New York: Macmillan.
Higgins, R., P. Hartley and A. Skelton. 2002. The conscientious consumer:
Reconsidering the role of assessment feedback in student learning. Studies in
Higher Education 27: 53-64.
Mousley, J. 2004. An aspect of mathematical understanding: The notion of
"connected knowing" Proceedings of the 28th Conference of the International
Group for the Psychology of Mathematics 2004: 377-84.
Murtagh, L., and L. Baker. 2009. Feedback to feed forward: Student response to
tutors' written comments on assignments. Practitioner Research in Higher
Education 3: 20-28.
Newton, P. 2007. Clarifying the purposes of educational assessment. Assessment in
Education 14: 149-70.
Orsmond, P., S. Merry and K. Reiling. 2000. The use of student derived marking
criteria in peer and self-assessment. Assessment and Evaluation in Higher
Education 25: 21-38.
Pirie, S. 1988. Understanding: Instrumental, relational, formal, intuitive... How can
we know? For the Learning of Mathematics 9(3): 7-11.
Rapley, T. 2011. Some Pragmatics of Qualitative Data Analysis.In Qualitative
Research ed. D. Silverman, 273-290. London: Sage.
Skemp, R. 1979. Intelligence, learning, and action. Chichester: John Wiley &Sons,
Struyven, K., F. Dochy and S. Janssens. 2005. Students' perception of evaluation and
assessment in higher education: a review. Assessment and Evaluation in
Higher Education 30: 325-41.
Taras, M. 2005. Assessment - summative and formative - some theoretical reflections.
British Journal of Educational Studies, 53: 466-78.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 178
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Developing an online coding manual for The Knowledge Quartet: An
international project
Tracy L. Weston, Bodil Kleve, Tim Rowland
University of Alabama; Oslo and Akershus University College of Applied Sciences;
University of East Anglia/University of Cambridge
This paper provides a brief overview of the work to date of an
international research team that has worked together since Fall 2011. The
team members are mathematics educators and researchers who use the
Knowledge Quartet (Rowland et al. 2009) in their work as researchers as a
framework by which to observe, code, comment on and/or evaluate
primary and secondary mathematics teaching across various countries,
curricula, and approaches to mathematics teaching. The countries
represented on the team include the UK, Norway, Ireland, Italy, Cyprus,
Turkey and the United States. The team has developed a Knowledge
Quartet coding manual for researchers which is freely available for other
researchers to use. This is a collection of primary and secondary vignettes
that exemplify each of the 21 Knowledge Quartet (KQ) codes, with
classroom episodes and commentaries provided for each code. This work
provides increased clarity on what each of the KQ dimensions ‘look like’
in a classroom setting, and is helpful to researchers interested in analysing
classroom teaching using the KQ. This paper provides an overview of the
Knowledge Quartet, describes the working methods of the team and offers
examples of classroom vignettes that exemplify two of the codes as an
indication of what can be found on the coding manual website
(www.knowledgequartet.org).
Keywords: mathematical knowledge in teaching; classroom observations;
data analysis; primary mathematics teaching; secondary mathematics
teaching.
Background
Beginning in 2011 an international team of researchers began working collaboratively
to develop a coding manual to support researchers interested in using the Knowledge
Quartet (Rowland et al. 2009) in data analysis. The Knowledge Quartet (KQ) is an
empirically grounded theory of knowledge for teaching in which the distinction
between different kinds of mathematical knowledge is of lesser significance than the
classification of the situations in which mathematical knowledge surfaces in teaching
(see Rowland, 2008). It can be considered what Ball and Bass would call a “practicebased theory of knowledge for teaching” (2003, 5). Based on empirical grounded
theory and an iterative process of grouping similar codes, four dimensions exist on the
KQ framework which are depicted in Figure 1.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Figure 1: The relationships of the four dimensions that comprise The Knowledge Quartet (Rowland,
Turner, Thwaites & Huckstep, 2009).
The KQ identifies three categories of situations in which teachers’
mathematics-related knowledge is revealed in the classroom: transformation,
connection, and contingency (Rowland, Huckstep and Thwaites 2005). Foundation,
which comprises a teacher’s mathematical content knowledge and theoretical
knowledge of mathematics teaching and learning, supports each of these categories of
situations. Transformation is the category most similar to Shulman’s
conceptualization of pedagogical content knowledge, that is, how a teacher takes
his/her own content knowledge and transforms it into ways that are accessible and
pedagogically powerful to pupils. This category pays special attention to the teacher’s
use of representations, examples, explanations, and analogies. A second dimension is
connection, which is whether a teacher makes instructional decisions with an
awareness of connections across the domain of mathematics (that mathematics is not,
after all, a subject that contains discrete topics) and an ability to sequence experiences
for pupils, anticipate what pupils will likely find ‘hard’ or ‘easy’ and understand
typical misconceptions in a given topic. Since not all aspects of a lesson can be
planned for ahead of time, contingency is the dimension that focuses on how a teacher
must think on his/her feet in unplanned and unexpected moments, such as to respond
to pupils’ statements, answers, and questions. Within each of the four dimensions
there exist four to eight codes which identify specific aspects of mathematics teaching
to consider in planning, reflection, and evaluation.
To date, the majority of writing about the Knowledge Quartet has been
focused on describing the framework (Rowland et al. 2009) and its origin (Rowland
2008) and has been written to support teacher development of mathematical
knowledge in teaching (MKiT). In recent years team members have been using the
KQ as a tool to support focused reflection on the application of teacher knowledge of
mathematics subject-matter and didactics in mathematics teaching (Corcoran 2007;
Kleve 2009; Rowland and Turner 2009; Turner 2009) and working with early-career
teachers, pre-service teachers and their school-based mentors, and with universitybased mathematics teacher educators, in applying the KQ to the development of
mathematics teaching. Through these interactions we have seen that participants often
conceptualise one or more of the dimensions of the KQ in ways that differ from the
understandings shared within the research team which conducted the classroombased research leading to its development and conceptualisation. Therefore, we have
seen that the framework is open to interpretive risks and mis-appropriation.
Furthermore, the majority of the writings have focused on explaining the essence of
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
each of the four dimensions rather than identifying definitions for each of the
underlying codes. These considerations are supported by Ruthven (2011):
Essentially, the Knowledge Quartet provides a repertoire of ideal types that
provide a heuristic to guide attention to, and analysis of, mathematical
knowledge-in-use within teaching. However, whereas the basic codes of the
taxonomy are clearly grounded in prototypical teaching actions, their grouping to
form a more discursive set of superordinate categories – Foundation,
Transformation, Connection and Contingency – appears to risk introducing too
great an interpretative flexibility unless these categories remain firmly anchored
in grounded exemplars of the subordinate codes” (85, emphasis added).
Beyond his categorization of generic and content specific aspects of teacher
knowledge, Shulman (1986) also identified ataxonomy for the forms in which
knowledge might be represented, including propositional knowledge, case knowledge,
and strategic knowledge. Case knowledge contains salient instances of theoretical
constructs in order to illuminate them, and a subcategory of this domain is the use of
prototypes. It is within case knowledge that we situate the project at hand.
Project aim
Compared to previous work, this project focused on researchers (not teachers) and
expanded KQ use into secondary grades and across countries and curriculum. The aim
of the project was to assist researchers interested in analysing classroom teaching
using the Knowledge Quartet by providing comprehensive coverage to ‘grounded
exemplars’ of the 21 contributory codes from primary and secondary classrooms. An
international team of 15 researchers was assembled. All team members were familiar
with the KQ and used it in their own research as a framework by which to observe,
code, comment on and/or evaluate primary and secondary mathematics teaching
across various countries, curricula, and approaches to teaching. The team includes
representatives from the UK, Norway, Ireland, Italy, Cyprus, Turkey and the United
States.4
Project methods
In Autumn 2011 team members individually examined their data and identified
available codes that they could contribute to the project. A template was developed in
which the scenario of how the episode unfolded was captured. Often this included
excerpts of transcripts and/or photographs from the lesson. Then a commentary was
written, which analyzed the excerpt and explained why it is representative of the
particular code and why it is a strong example. In January 2012 each team member
submitted scenarios and commentary for at least three codes from his/her data to offer
as especially strong, paradigmatic exemplars. Drafts of each scenario were written by
individual team members remotely and shared via Dropbox. In February 2012,
scenarios were assigned to each team member to review for agreement of the code
with the scenario and improvement of the commentary. In March 2012, 12 team
4
Tim Rowland, University of East Anglia/University of Cambridge, UK; Tracy Weston, University of
Alabama, US; Anne Thwaites, University of Cambridge, UK; Fay Turner, University of Cambridge,
UK; Bodil Kleve, Oslo and Akershus University College of Applied Sciences, NO; Dolores Corcoran,
St. Patick’s University, IE; Ray Huntley, Brunel University, UK; Gwen Inson, Brunel University, UK;
Marilena Petrou, Cyprus/UK; Ove Gunnar Drageset, University of Tromsoe, NO; Nicola Bretscher,
Kings College London, UK; Mona Nosrati, University of Cambridge, UK/NO; Marco Bardelli, IT;
Semiha Kula, Dokuz Eylül University, TR; Esra Bukova Guzel, Dokuz Eylül University, TR.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
members met for the Knowledge Quartet Coding Manual Conference at the
University of Cambridge. Groups of three team members evaluated and revised each
scenario and commentary. Throughout the spring and summer, individuals again
worked remotely to revise scenarios based on conference feedback.
To date, 55 total scenarios and commentary have been written. These
scenarios and commentary combine to form a ‘KQ coding manual’ for researchers to
use. This is a collection of primary and secondary vignettes that exemplify each of the
21 KQ codes, with classroom episodes and commentaries provided for each code. The
collection of codes and commentary is freely available online at
www.knowledgequartet.org. The website provides an overview of the Knowledge
Quartet and its four dimensions as well as the work to-date of the international team’s
scenarios and commentaries describing mathematics teaching across multiple
countries, topics, and pupil ages. Additional scenarios and commentaries continue to
be added to the website.
Sample scenarios
In order to exemplify our work we will present two scenarios which illustrate two of
the codes. First we present an example of the code Responding to students’ ideas
(RSI), a code within the Contingency dimension. Second, we present an example of
the code Decisions about sequencing within the Connection dimension.
Responding to students’ ideas
The following scenario from a lesson that took place in 2002 (Rowland 2010) is
offered here as a kind of prototype of the RSI code. Jason was reviewing elementary
fraction concepts with a Year 3 (pupil age 7–8) class. The pupils each had a small
oblong whiteboard and a dry-wipe pen. Jason asked them to ‘split’ their individual
whiteboards into two. Most of the children predictably drew a line through the centre
of the oblong, parallel to one of the sides, but one boy, Elliot, drew a diagonal line.
Jason praised him for his originality, and then asked the class to split their boards
‘into four’. Again, most children drew two lines parallel to the sides, but Elliot drew
the two diagonals. Jason’s response was to bring Elliot’s solution to the attention of
the class, but to leave them to decide whether it is correct.
This scenario is interesting mathematically, and not so ‘elementary’ in the
context of the Year 3 curriculum. Responding to Elliot’s solution, either by teacher
exposition, or in interaction with the class, makes demands on Jason’s content
knowledge, both Subject Matter Knowledge (SMK) and Pedagogical Content
Knowledge (PCK), in three significant respects. Jason has to decide whether the noncongruent parts of Elliot’s board are equal, but also what notions of ‘equal’ will be
meaningful to his 7–8 year-old students, and what kinds of legitimate mathematical
arguments about area will be accessible to them.
Decisions about sequencing
Connection as a dimension in the Knowledge Quartet is “concerned with the decisions
about sequencing and connectivity” (Rowland, Turner, Thwaites, & Huckstep, 2009,
36). One of the codes within this dimension is Decisions about Sequencing (DS),
which is concerned with “introduc(ing) ideas and strategies in an appropriately
progressive order” (37). We suggest that in the scenario described below, the
sequence of the exercises was consciously done by the teachers. The teacher had
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
prepared four different exercises to be done in whole class before the pupils were told
to work individually with tasks from the textbook. The lesson objective was to learn
to calculate with improper fractions. Placing this lesson in the Connection dimension
of the KQ and coding it as Decision about Sequencing is based on the progression of
the exercises discussed below.
The first exercise 6/8+5/8 was presented with both illustrations and numbers.
To work out this exercise pupils calculated with numbers, converting improper
fractions to mixed number which was illustrated on the figure by pulling shaded
pieces from one rectangle to fill up the other on the smart board. In this exercise it
was possible to get the correct answer 1 3/8 by counting shaded pieces on the
illustration. The second exercise was presented without numbers. The teacher had
shaded 5/8 of one circle and 4/8 of another circle and pupils were asked how large a
part of the first was shaded and then of the second before they worked out the answer.
The answer, 9/8, was converted to 1 1/8 which was illustrated on the figure. This
time, it was not possible to pull the pieces. The teacher erased from one circle and
filled up the other. When starting the third exercise 3/5+3/5 and 7/10+5/10, the
teacher said, “let us try without illustrations”. This suggests that he consciously
wanted the pupils to calculate the sum of two fractions which added up to an improper
fraction, without having illustrations as mediating tool.
The fourth exercise was different from the first three in several ways. It was
illustrated with two circles, each divided in quarters. All quarters were shaded and the
calculation 2-1/4 was written below. This time the illustrations were not on two sides
of an equal sign, the calculation was subtraction, and it started with a whole number.
The exercise for the pupils was both to illustrate how much to erase from the figure
and also to work it out with numbers.
Hans’ choices of illustrations and numbers / only illustrations / only numbers
reflected a progression in the lesson. However, the fourth exercise required a
subtraction and thus introduced an added complexity. In this example subtraction was
thought of as “take away”, but could also be comparison. Thus there was a leap, or a
missing link. It might have been preferable here to have an exercise that was adding,
with one of the numbers as a mixed number and the other as a fraction. Also, whether
the exercises chosen were appropriate for developing a solid concept of improper
fractions may be discussed. In all exercises the fractions were presented as part of a
whole. According to research, fractions as part of a whole is inconsistent with the
existence of improper fractions and possibilities for obtaining a well-developed
concept of fractions are limited if one focuses on fractions as part of a whole (Kleve,
2009).
Discussion
Both of the proceeding classroom vignettes are offered as exemplars of a given KQ
code (RSI and DS, respectively). We readily acknowledge there may be ‘room for
improvement’ and indeed have identified some possible instructional decisions to this
end. It was not uncommon in our work for scenarios to seem strong exemplars of one
KQ code, and simultaneously lacking in another. Other scenarios were considered
strong examples of multiple KQ codes, and in these instances the team worked toward
a consensus of which KQ code seemed ‘best’ exemplified by the scenario.
An underlying question during this project was whether any adjustments
would need to be made to the Knowledge Quartet when applied to secondary grades.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Although the content involved is different in upper grades, it was not necessary to add
or remove any codes to capture effective mathematics teaching to pupils beyond the
primary grades. The majority of the scenarios on the website are from primary grades
(which is helpful in that the mathematics do not get to be so difficult as to burden the
reader trying to sort out the mathematics instead of thinking about the code), and
approximately one-quarter of the codes are from secondary grades and will be helpful
to researchers interested in using the KQ to analyze secondary teaching.
The team continues to collect and write scenarios, with the near-term goal of
having at least three scenarios per each of the 21 KQ codes. We encourage the use
and sharing of the www.knowledgequartet.org website as this work provides increased
clarity on what each of the KQ codes ‘look like’ in a classroom setting and is helpful
to researchers interested in analyzing classroom teaching using the KQ across a wide
range of countries, contexts, and pupil ages.
References
Ball, D. L., and H. Bass. 2003. Towards a practice-based theory of mathematical
knowledge for teaching. In Proceedings of the 2002 Annual Meeting of the
Canadian Mathematics Education Study Group, ed. B. Davis and E. Simmt, 314. Edmontson, AB: CMESG/GCEDM.
Kleve, B. 2009. Aspects of a teacher's mathematical knowledge on a lesson on
fractions. Proceedings of the British Society for Research into Learning
Mathematics 29(3): 67–72.
Rowland, T., F. Turner, A. Thwaites and P. Huckstep. 2009. Developing primary
mathematics teaching, Reflecting on practice with the Knowledge Quartet.
London: Sage.
Rowland, T. 2008. Researching teachers’ mathematics disciplinary knowledge. In
International handbook of mathematics teacher education: Vol.1. Knowledge
and beliefs in mathematics teaching and teaching development, ed. P. Sullivan
and T. Wood, 273-298. Rotterdam, The Netherlands: Sense Publishers.
Rowland, T., P. Huckstep and A. Thwaites. 2005. Elementary teachers’ mathematics
subject knowledge: the knowledge quartet and the case of Naomi. Journal of
Mathematics Teacher Education, 8:255-281.
Ruthven, K. 2011. Conceptualising mathematical knowledge in teaching. In
Mathematical Knowledge in Teaching, ed. T. Rowland and K. Ruthven, 83-96.
New York: Springer.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Preservice primary school teachers’ performance on rotation of points and
shapes
Zeynep Yildiza, Hasan Unala, A. Sukru Ozdemirb
a
Department of Elementary Education, Faculty of Education, Yildiz Technical University;
Department of Elementary Education, Faculty of Education, Marmara University
b
In this study, the purpose was to reveal thinking styles and different points
of view of pre-service primary school teachers about the concept of
“rotation” in mathematics. The study was conducted with undergraduate
students who are studying in the department of primary school teacher
education. The subject of “rotation” in this study has two sub-topics which
are rotation of points around a point and rotation of shapes about a point
in a coordinate plane. A test about rotation was applied to 44 students and
then interviews were made with 5 students. Results of the study include an
analysis of correct and incorrect answers of students.
Keywords: preschool mathematics teachers, education, rotation
Overview
According to Pehlivan (2008), knowledge, skills, attitudes and habits gained at the
primary level are highly influential on individuals’ later lives. Accordingly, it is stated
that the importance of classroom teachers who undertake a major part of individuals’
education at this level and their qualities cannot be ignored.
In Turkey, in order to respond to social upheaval, education systems and
accordingly teacher training institutions are in the process of reconstruction (Alkan
2000; cited in Karaca 2008). These configurations also bring the necessity for
teachers who have to find a balance between modernism and traditional methods to
have some qualifications. These qualifications can be classified as in the following
(Delors et al. 1996, cited in Karaca 2008).
 Teachers should constantly be investigative in order to help students to
construct knowledge personally.
 Teachers should constantly keep students’ individual thinking capacities
awake during learning by going beyond their own disciplines.
 Teachers should commit themselves to educate students in accordance with
objectives.
 Teachers should teach students how to learn and which cognitive tools can
help them to get more useful outcomes.
 Teachers should be able to use new information-communication technologies
which are developing rapidly and which increase the quality of teaching
process.
According to Pırasa (2009), knowledge of teaching mathematics has components
which are still discussed and which are evolving. This knowledge which was seen as
unary and which was recently considered as if it consisted only of subject knowledge,
was redeveloped together with the completion of the definition of the subject
education. At first, in classroom teacher training programs, student teachers’
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knowledge of teaching mathematics was developed by providing two different
knowledge types: professional knowledge (about teaching) and subject knowledge
(about mathematics). Later, the program was reorganised to add a course concerned
with teaching mathematics, requiring an extensive and comprehensive editing of the
original program content (Pırasa 2009). The content of the teacher education
programmes in Turkey was changed with the new curriculum which was put into
practice in 2005-2006, taking constructivist learning theory as a base (Ministry of
Education, 2005). According to Selim (2009) constructive learning theory is a
teaching and learning approach which is based on relating new knowledge to the
existing knowledge of individuals. In constructivist learning, when the learners
process the knowledge which is obtained through observation, experience or transfer
from external sources in their minds, then this information becomes meaningful. In
this study, we analyse the performance of student teachers on rotation tasks typical of
this new constructivist curriculum.
The ‘rotation’ topic includes sub-topics such as rotating points around points
or rotating figures around points. With this research, it is intended to understand preservice primary school teachers’ thinking skills and different viewpoints about the
subject.
Methodology
A ‘scanning model’ was used in the study. The scanning model is an approach which
aims to define a present or past case as it exists today (Arlı and Nazik 2001).
Qualitative and quantitative data collection instruments were used in this research. For
collecting quantitative data, a test about rotation was conducted by researchers. In
order to collect qualitative data, some students were interviewed and detailed
questions were asked about their solutions.
Study Group
This research was carried out with 44 students who are studying in the department of
primary school teacher education in a state university in Istanbul from Marmara
region.
Implementation
The test about “rotating a point or figures around a specific point in the coordinate
system” was applied to students. Students were asked to complete this test in one
lesson. During this time, they were asked both to solve the questions and to write
explanations about their solutions. A semi-circular protractor was given to students
while they were solving questions. After this test, face-to-face interviews with five
students were conducted to get detailed information about their thinking while solving
the test questions. Five students were determined according to their papers. Especially
students who made interesting solutions were chosen. In these interviews, students
were asked how they made their solutions and how they thought while they were
making solutions.
Data Collection Instruments
The test which was used during research was created by the researchers and included
six questions. Each question was expressed and asked on a coordinate plane. Enough
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
space next to questions was provided to students for making calculations and writing
explanations about the solutions. Questions included the following processes:
 Rotating a point around the origin.
 Rotating a point around a point.
 Rotating figures around a point inside them.
 Rotating figures around a point outside them.
 Rotating figures around a point on the figure.
Findings
The first question was to rotate a point given in a coordinate system by 90 degrees
around an origin. When the answers were analysed it was seen that students mostly
found the correct solutions by using ‘using a protractor’ and ‘counting squares’
techniques. Students who got wrong solutions had mistakes mostly because they
thought they must take symmetry. So, they drew the 90 degree arc incorrectly.
The second question was to rotate a rectangle by 180 degrees around a point
on that rectangle. When the answers analysed it was seen that all the students with
correct answers rotated the corner points of the figure around the asked point which is
one of the corners. Then, they formed the rotated image rectangle by combining
newly formed points accordingly. The following mistakes were frequently made by
the students who got wrong solutions. Some students specified the required point in
the figure and then drew a 180 degree arc starting from that point. They drew the
rectangle with one corner at the end point of the arc that they had drawn. While
solving the same question, some other students rotated the figure apparently randomly
or intuitively. Some other students rotated the figure around the origin or reflected in
the y-axis.
The third question was to rotate a point by 270 degrees around a point
different from the origin. When the answers were analysed it was seen that students
mostly used ‘drawing by using a protractor’ and ‘drawing a 270 degree arc’ methods.
In drawing by using a protractor; firstly they drew a line segment by combining point
and reference point. Then they drew another line segment of the same length to form a
270 degree angle whose corner point was a reference point. They placed the image
point on the other end of this line segment. Some students answered this question
correctly by drawing a 270 degree circular arc. For this, they drew a 270 degree arc
centered at the reference point and with one end at the point to be rotated (the object).
They placed the image point on the other end of the arc. Nearly half of the students’
wrong answers also tried to solve this question by drawing a 270 degree arc.
However, they specified the centre of this arc incorrectly, and so their solutions were
also incorrect. The remaining incorrect answers did not pay attention to the distance
between the object point and the reference point. That is, the distance between the
object point and the reference point and the distance between the image point and the
reference point were not drawn equally.
The fourth question was to rotate a point 180 degrees around a point that is
different from the origin. When the answers were analysed it was seen that most of
the students who gave correct answers to this question found their solution either by
using a protractor or by taking symmetry. When the solutions of the students with
incorrect answers were analysed, drawing mistakes can be seen originating from the
misuse of protractor or not drawing equally the lengths which should be equal
between object point-reference point and image point–reference point. Besides,
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students thought that they needed to get symmetry of the point for rotating 180 degree
operation, but they used one of the horizontal or vertical axes of the reference point as
a reflective symmetry axis.
The fifth question was to rotate a point 90 degree around a point outside a
triangular figure. When the questions were analysed it was seen that this question has
the lowest percentage of correct answers. Four students, who found the correct answer
for this question, rotated the corner points of the figure by applying the rotation rules
correctly and made their drawings according to this. Approximately one third of the
students who gave incorrect answers tried to solve the question by reflecting the
figure in a vertical axis through the reference point. Some students specified a point in
the triangle; they rotated this point around reference point at desired amount and
direction. Later on, they drew the triangle ‘by eye’ so that the point would stay inside.
Also some students generally rotated the figure 90 degree inferentially by taking one
edge of the figure into consideration and completing the triangle by eye.
The sixth question was to rotate a figure 270 degree around a point inside
itself. When the answers were analysed it was seen that students mostly rotated the
figure around the reference point by specifying the corner points of the figure, and
then they drew the rotated form of the figure according to this. Besides, there were
also students who got the correct figure by rotating the rectangle 90 degree three
times. When the incorrect solutions were analysed, it was seen that most of them had
attempted rotation but drew the rotated figure incorrectly. In addition to this, students
who reflected or translated the figure were also identified. Some students tried to find
solution by drawing a 270 degree arc from the reference point. And three of the
students made operations such as rotating the figure 3-dimensionally.
In the research, the data which was collected after implementation through
face to face interviews with students were evaluated. In these interviews, students
who made incorrect solutions stated that they do not have enough information about
using a protractor. They showed this as the reason of their mistakes in most of the
questions. Especially they had difficulties in deciding which point (the object point
which will be rotated or the reference point) to place at the
centre of protractor.
While some students were making drawings, they
did not paid necessary attention to the equality of lengths
which should be equal. When the reason of this was asked,
they showed the involvement of millimeters. They stated
the difficulty of making millimetric measurements. Figure
1 gives an example of this kind of solution.
Figure 1: A student’s solutions
Some of the students expressed the fact about their
without equality of lengths
drawings that instead of using a protractor or making
mathematical drawings they made mental and imaginary
rotations.
Figure 2: Students’ solutions using symmetry of the shape or point
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
While rotating the point around the origin, there were students who got symmetry
according to x-axis as seen in first solution above. When students were asked to
explain their solutions, they told that they got symmetry according to x-axis but they
could not explain the reason of this. In second solutions above, it was seen that
symmetry of a figure, which was asked to rotate 180 degree around F point, was
determined according to y-axis. In third solution above, similarly the symmetry of the
triangle which was asked to rotate according to point A was determined according to
y-axis.
Figure 3: Students’ solutions using symmetry of the shape or point
The above solutions related to 180, 270 and again 270 degree rotations
respectively. In the all three solutions above in Figure 3, an arc was drawn for the
desired angle with a protractor centred at a random point. The students did not recall
which point they used as a centre of rotation and they could not make any explanation
about this situation. While the centre of the arc which was drawn should be the
reference point (point that will be rotated around) they did not pay any attention to
that point.
Figure 4 is an example of the solutions of some students
who tried to make rotation as 3-dimensional. For the
reason of this, they expressed that they thought the
figure as 3-dimensional like a book or a notebook but
not two dimensional.
Figure 4: Student’s solution about thinking 3-dimensionally
Figure 5: Student’s solution
about taking symmetry
according to origin
One of the incorrect responses to rotate a figure 180
degree around a point inside that figure is seen in
Figure 5. When this student was asked about how he or
she found this solution, he or she stated that since 180
degree rotation was asked, he or she thought that it was
necessary to take symmetry. For this reason, the student
made his or her drawing by taking the symmetry of the
figure and the point inside that figure, according to
origin.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Conclusion
When the solutions of pre-service teachers were searched generally, it is possible to
make the following comments.
Students who made right solutions have enough basic knowledge about the subject. In
interviews these students gave answers with self-assurance. When their solutions were
analysed, their drawings show how they reached their answer.
In addition to that there are also students who reached the correct solution by
different methods. When these different methods analysed, we can make this
comment. Knowing how to use protractor, and doing millimetric drawings may be
evidence of that they learned this subject as conceptual when they are in primary
school. Although they haven’t learned this subject again in the middle school, they
can make connections between other mathematics subjects which they learned
afterward. This can be a result of that they constructed their knowledge strongly.
For the incorrect solutions, the general result after interviews was that these
students don’t have enough knowledge about the subject and they have some
deficiency about basic concepts. Not knowing how to use protractor may be an
indicator of that situation.
Our results show the thinking and misconceptions of some pre-service
teachers. We consider that determining this kind of conceptual deficiency amongst so
many student teachers is important. By adding this kind of self-assessment and
discussion of misconceptions to teacher education programs, quality of teacher
education may be increased.
References
Alkan, C. 2000. Meslek ve Öğretmenlik Mesleği, In Öğretmenlik Mesleğine Giriş. ed.
V. Sönmez. Ankara: Anı Yayınları
Arlı, M. ve Nazik, M.H. 2001. Bilimsel Araştırmaya Giriş, Gazi Kitapevi, Ankara.
Karaca, E. 2008. Eğitimde Kalite Arayışları Ve Eğitim Fakültelerinin Yeniden
Yapılandırılması, Dumlupınar Üniversitesi, Sosyal Bilimler Dergisi, Sayı 21
Ministry of Education/Milli Eğitim Bakanlığı. 2005. İlköğretim Matematik Dersi
Öğretim Programı ve Kılavuzu, Ankara: MEB Yayınları
Pırasa, N. 2009. Sınıf Öğretmeni Adaylarının Matematik Öğretimiyle İlgili
Bilgilerinin Değişim Sürecinin İncelenmesi, Karadeniz Teknik Üniversitesi,
Doktora Tezi
Pehlivan, K. B. 2008. Sınıf Öğretmeni Adaylarının Sosyo-kültürel Özellikleri ve
Öğretmenlik Mesleğine Yönelik Tutumları Üzerine Bir Çalışma, Mersin
Üniversitesi Eğitim Fakültesi Dergisi, Cilt 4, Sayı 2, Aralık 2008, ss. 151-168.
Selim, Y. 2009. Matematik Öğretmen Adaylarının Bilgisayar Destekli Olarak
Hazırladıkları Öğretim Materyalinin Niteliği ile Matematik ve Öğretmenlik
Meslek Bilgileri Arasındaki İlişkilerin İncelenmesi, Atatürk Üniversitesi,
Doktora tezi
Delors, J. and the Task Force on Education for the Twenty-first Century 1996.
Learning: The Treasure Within. Paris: UNESCO.
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Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Working Group Reports
Report from the Sustainability in Mathematics Education Working Group: Task
design
Nichola Clarkea, Maria Chionidou-Moskofogloub, Zoi Moskofoglouc, Alison Parrishd,
Anna-Maija Partanene
a
University of Nottingham, UK; bUniversity of the Aegean-Rhodes Greece; cUniversity
College, London; dWarwick University UK; eAbo Akademi University, Denmark.
The Sustainability in Mathematics Education Working Group discusses
research on how to integrate learning about climate change and
sustainable living with the learning of mathematics. In the third group
meeting, participants from Denmark, Greece and the UK focused on the
design of cross-curricular tasks for the simultaneous learning of
mathematics and sustainability issues. We drew on examples of task
design experiences from Greece and the UK.
Keywords: sustainability, climate change, mathematics, teaching, learning,
task design, curriculum, critical mathematics education, practice, systems,
social justice, local.
Introduction
In this third meeting of the Sustainability Working Group, we explored some of the
particular features of designing tasks for learning mathematics whilst also learning
about climate change and sustainable living. We considered some of the ways in
which producing cross-curricular tasks might raise particular issues for a task
designer. We took advantage of the three national contexts represented in the meeting
(Denmark, Greece, UK) to consider the ways in which local context can inform what
and how students learn about sustainable living, climate change, and mathematics.
Our discussions in this session stemmed from elaboration of two examples of tasks
designed for learning about sustainability issues whilst learning the mathematics
outlined below. The initial foci for discussion were drawn from the research agenda
set in the first working group (Clarke 2012):
(1) To design high quality materials for use in teaching sustainability in
mathematics lessons.
(2) To evaluate materials for learning about sustainability in mathematics
lessons.
The Euro-Axio-Polis Game
Chionidou-Moskofolou, Moskofolou, Liarakou and Stefos (2011) have designed a
game with fourth year education students engaged in initial teacher education practice
at the University of the Aegean. The game is called Euro-Axio-Polis. This is a board
game aimed at 6th grade students, in which pupils make spending decisions in small
groups, stimulating whole-class discussion about mathematical calculation techniques
and the effect of repeated percentage changes, but also stimulating argument about
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difficult social choices. Designed with education students, this project also helps
beginning teachers embed teaching about sustainability in their practices.
The mathematical content of the task includes calculating percentage changes,
using interest rates, repeated percentage change, and manipulation of large numbers in
a money context. The mathematical practices in which students engage include
attending to others’ actions, turn taking, understanding others’ calculations, and
justifying and explaining their own calculations. The sustainability context of the
game is making difficult economic choices given scarce resources, and choosing
between actions with different ecological and economic impacts. The pupils are
learning about the types of choices faced in their political context, and become aware
that social choices are based not just in economic constraints, but are based in values.
Clarke’s virtual water tasks
Clarke (working with a science teacher and environmentalist, Michael Sparks) is
developing a sequence of tasks on the measure of virtual water. This is a concept
developed by Allan (2011) to make sense of the amount of embedded water used in
production of goods and services. Clarke works with schools engaged in sustainable
living projects to help them embed sustainability in classroom curricula. The
sustainability focus of these whole-school actions is food security and urban growing.
Clarke also works to promote mathematical learning about sustainability in out-ofschool contexts.
Clarke claims that if students are to learn how to live sustainable lives in
response to constraints imposed by climate change, they need good understanding of
scaling relationships (Wake 2011). Scaling is especially import because students need
to notice that small personal changes they as individuals might choose to make, if
adopted by large numbers of people, could have large impact. Scalings work
(potentially) across time, from individual at a time to cumulative individual action,
and also from individual to group actions. Students need also to gain a sense of the
relative effect of the different scalable actions they might engage with. Sustainability
contexts thus afford important learning opportunities on proportional reasoning.
The tasks on virtual water also afford learning about measures and quantification.
Often, the measures used in work on sustainability are complex, involving averages,
and combinations of averages. These defined measures need to be explicated as part
of students’ learning. Clarke uses the context of virtual water to develop students’
understanding of devising and critiquing measures.
The virtual water context also affords work on sustainability: understanding
the effects of scalable actions on the amount of actual and embedded water used, and
the way water is imported and exported in virtual form. This helps students to make
sense of the impact of choices they make about water use. Students also learn to
engage in sustainable practices, by identifying resource use and considering
alternative choices of action, based on their use of their measures and their own
personal values.
Clarke linked her work to some research questions raised in the report of the
first working group (Clarke 2012):
What mathematics do students need to understand and be able to use if they
are to understand everyday life choices about sustainable living?
How can students best learn the mathematics they require to understand
everyday choices linked to sustainability issues?
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What values and principles inform our choices of what mathematics to teach and
how to teach it to students, to help them learn about living sustainably?
Which parts of the mathematics and science of sustainability and climate change
are accessible to students, at what stages of learning?
She argues that much of the science and mathematics of climate change is
complicated: basic principles such as the greenhouse effect can be explained using
school mathematics and science, but communicating in more depth requires greater
mathematical and scientific sophistication. However, if the reality of climate change
is accepted (and this is the current scientific consensus) there are different choices to
be made between actions. The mathematics about those choices can appear
simultaneously too simple (scaling) and too complex (large and very small numbers;
complex measures) for use in schools. Nevertheless, students need to be taught to
engage with this type of value-laden estimation task, and to use and critique such
measures as a central part of learning to live sustainable lives.
Discussion
We spent some time discussing the role of values in the tasks presented to students. In
both Chionidou-Moskofolou et al.’s game and in Clarke and Sparks’s scaling tasks,
students are presented with choices between possible actions. Their mathematical
work shows different resources use consequences of choices, with different economic
and social impacts. In neither case did the authors intend that the choice on offer was
false, and students were to be constrained to a particular form of (environmentally
preferable) action. The designers were trying to avoid ‘moral’ overtones in those
choices, by giving real options linked to different viable sets of values, raising
awareness of resource use rather than condemning one set of choices.
The task designers were thus not intending that students learned the ‘correct’
choice to make. Rather, part of what students are being offered is the opportunity to
engage in public discussion of a plurality of values that inform different choices, and
the practice of justifying choices and values to others. What blurs the issue is the need
to offer particular alternative forms of action in the tasks: those choices are clearly
informed by the designers’ own values and their intentions to produce constructive
argument.
The group agreed on the importance of developing negotiation and
argumentation skills, given the complexity of many of the environmental choices to
be considered. For example, although a food might have relatively low virtual water
content, its embedded energy costs might be high. Making judgments about
incommensurable resources requires consideration of values, not just mathematics
(Brown and Barwell 2011 discussed in Clarke 2012).
The group discussion also explored the different educational contexts within
which tasks were being offered. Partanen described the Danish schools context, in
which there is a requirement that students address green issues in school learning.
Parrish and Clarke described the shifting educational and political contexts of the UK,
and differences between school subjects. This raised research questions: Why do
some teachers choose to teach about sustainability in the context of mathematics
lessons? Why do others choose not to? What impact do national and school policies
have on teacher choices?
Chionidou-Moskofoglou and Moskofoglou developed discussion of local
context. They outlined the relevance of the Euro-Axio-Polis Game to the Greek
national context. The game engages students in thinking about large-scale economic
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decisions. These might be argued to lack realism since all the numbers are initially
rounded to the nearest hundred thousand Euros, the percentages are neat integer
values, and school children do not really make these types of economic choices.
However, Chionidou-Moskofoglou et al.’s research findings show that students were
nevertheless highly engaged with the game and many argued passionately, perhaps as
if the decisions were real for them (Chionidou-Moskofoglou et al. 2011). The hard
economic decisions being made in Greek politics have impact on students’ lives,
through cuts, unemployment, and national debt. This perhaps makes national
economic decision-making a “live question” (Peirce 1877) for Greek children.
Similarly, Clarke described how children in Lancashire and Hong Kong
schools who are engaged in growing projects raised questions about planting
arrangements, water use and crop yields. Those students are raising what for them are
“live questions”. Since those students raise issues there is no need to make
assumptions or guess what (all) young people want, in an attempt to design a task that
is relevant or real, because the crop production work of the group of students provides
them with relevant mathematics in a truly shared context. That context works for
those students, but would not necessarily work for others; the same is true for
economic decision making in the Greek students’ context. We thus began to develop a
sense of the importance of localism in task design for teaching about mathematics and
sustainability, perhaps reminiscent of Lave’s discussion of the difference between
abstraction and generality (Lave 1988). However, this also raises the problem of
which mathematics can be made a “live question” for which students.
Conclusions
Exploring issues drawing on three different national contexts allowed us to share
experiences of working on sustainability in mathematics lessons. We opened up
possibilities for tasks, task designing, and research on the efficacy of tasks, and
considered working together to explore topics and share the load of designing tasks,
identifying potential difficulties for teachers. We considered the value of large-scale
contexts and how to attend to the importance of different locally-live issues whilst
drawing on (abstract) mathematics. We agreed to continue our discussion online, and
a literature review was suggested as a topic for the next working group meeting in the
UK.
Many thanks to all participating colleagues for their lively, interesting
contributions.
References
Allan, J.A. 2011. Virtual water: Tackling the threat to our planet’s most valuable
resource. London: I.B. Tauris.
Brown, T. and Barwell, R. 2011. Mathematics and climate change. In Psychology of
Mathematics Education Newsletter, February/March 2011, pages 6-8.
Clarke, N. 2012. Report of the first sustainability in mathematics education working
group. In Informal Proceedings of the British Society for Research into
Learning Mathematics, ed. C. Smith, 32(1) March 2012. Available online:
http://www.bsrlm.org.uk/IPs/ip32-1/BSRLM-IP-32-1-03.pdf Accessed 2.1.13.
Chionidou-Moskofolou, M., Z. Moskofolou, G. Liarakou, and E. Stefos. 2011.
Lecture on Euro-Axio-Polis presented at BLOD, Athens 2011. Available
online: http://www.blod.gr/lectures/Pages/viewlecture.aspx?LectureID=523
Accessed 12.12.12.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 194
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Lave, J. 1988. Cognition in practice: Mind, mathematics and culture in everyday life.
Cambridge: Cambridge University Press.
Peirce, C.S. 1877. The fixation of belief. Available online:
http://www.peirce.org/writings/p107.html Accessed 2.1.13.
Wake, G. 2011. Modelling in an integrated mathematics and science curriculum:
Bridging the divide. Paper presented at the 7th Congress of the European
Society for Research in Mathematics Education, University of Rzeszów,
Poland.
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 195
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
Report of the Mathematics education and the analysis of language working
group
Alf Coles and Yvette Solomon
University of Bristol and Manchester Metropolitan University
In this paper, we report on the discussion and issues raised at the working
group session at the day conference in Cambridge.
Key words: language, gesture, analysis, mathematics education.
Brief history of the working group
At Cambridge there was the third meeting of the re-formed Mathematics Education
and the Analysis of Language Working Group. In the first meeting (November 2011)
we worked with conversation analysis and linguistic ethnography approaches to
analyzing data. This was followed in March 2012 by a session that triangulated
conversation analysis and a multi-modal approach (see Farsani 2012). The aims of the
group are to share and develop approaches to the analysis of classroom talk. We aim
to dwell in the detail of how we work with language in our own mathematics
education research. In this session we worked on some data collected by Yvette
during a teacher-training course on which she taught in the UK, asking the questions
of how we understand transcript data and what do different transcription methods
allow or constrain?
Context of data
We offered the working group three different transcriptions of the same event, in
which some prospective teachers were performing (with their bodies) a demonstration
of how the earth moves around the sun. The prospective teachers’ task was to explain
why we have seasons. They were modelling this with M moving around D and
spinning, whilst leaning her body towards and away from D, who was holding a torch
(the sun). The most pared down transcript (that we offered first) is below. If you were
not at the group meeting, you might want to try to make sense of this data and
consider what you bring to your sense making.
Transcript 1
A: well this is [‘] summer
A: This is in winter
D: Right.
D: Northern hemisphere’s in winter.
D: the southern hemisphere’s in summer
A: Earth’s moving this way
D: Mandy’s moving anticlockwise round me.
D: but she spins round clockwise as well
A: yeah
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 196
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
A: Stop!
A: Now this is summer
A: and this is winter when you’re at the furthest point aren’t you
A: we had erm summer and a cold winter as well
M: shall I finish off? [laughing]
A: and there you go
In the session, we then offered the transcript below, with more details about
gestures and movement. You might want to consider what, for you, is the same of
different about engaging with the following transcript compared to the one above.
Transcript 2
A: well this is [‘] summer [Ashley moves towards Mandy and supports her with
one foot underneath Mandy’s raised foot, indicating that the foot end of Mandy is
summer. Mandy is leaning back at an angle, Danielle is holding the torch. Mandy
starts to wobble, Ashley holds on to her]
A: This is in winter [at the same time Mandy taps her own head, Ashley gestures
towards Mandy’s head]
D: Right.
D: [Lifts hand in air to indicate upwards] Northern hemisphere’s in winter.
D: [Lowers hand to point at Mandy’s foot] the southern hemisphere’s in summer
[??]
A: Earth’s moving [‘] this way [Ashley and Danielle both gesture to indicate orbit
in slow anticlockwise sweeping circular movement with their lower arms]
D: Mandy’s moving anticlockwise round me.
[At ‘round me’ Mandy simultaneously raises her left hand and gestures fast and
tight anti-clockwise rotation from the wrist]
D: but she spins round clockwise as well [Danielle sweeps her hand down and fast
to turn her lower arm movement into a clockwise rotation]
A: yeah
[Mandy starts to spin round clockwise (according to her own body, ie she moves
to the right as she turns) and orbit anticlockwise at the same time. Ashley holds
on to her to keep her balance (Mandy is on one foot all the time)]
A: Stop! [Mandy has completed a 180degree orbit. Danielle has turned round and
is shining the torch on to Mandy, who has her back turned]
A: [moves to hold Mandy and demonstrate]
A: [voice for audience] Now this is summer [Mandy pats the back of her own
head]
A: and this is winter [both point to Mandy’s raised foot] when you’re at the
furthest point aren’t you [non-public teacher-to-child voiced question directed at
M]
[.. indistinct brief dialogue between A and D]
A: we had erm summer and a cold winter as well [?indistinct, transcription may
not be faithful]
M: shall I finish off? [laughing] [ completes orbit]
A: and there you go
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 197
Smith, C. (Ed.) Proceedings of the British Society for Research into Learning Mathematics 32(3) November 2012
We then had a third transcript, not reproduced here, which included snapshots
from a video and, in the session, we also showed participants the short video clip
itself, on which this transcript is based.
Discussion
We now summarise some of the issues that arose from discussion of the task of
engaging with the data above.
There were several comments about an initial preference for working with the
most pared down transcript. Engaging with transcript 1 first, forced an attempt to
reconstruct the events and work out what they meant. The effect of then being offered
transcript 2 was described, by several people, as a shift from working something out
yourself to then being told. It was as though the ‘authorial’ voice of the researcher
was much more present in transcript 2. It was clear there were interpretations in
transcript 2, for example, “non-public teacher-to-child voiced question”. On
reflection, of course, it was recognised that the authorial voice and interpretation of
the researcher is just as present in transcript 1 but perhaps not as visible. This became
apparent when we watched the video clip. There were a lot of other voices and noises
on the clip and the intentions and interests of the researcher suddenly became
relevant. We needed to know the context of why Yvette had transcribed what she had
– and this context was in fact an interest in embodied understandings of mathematics.
What is left out of the data we present each other is often not alluded to in research
reports.
Another preference for transcript 1 was that the relationships between speakers
and turns in the dialogue were more apparent than when all the detail was added. One
sense that came across from the discussion was that, as researchers, we need to see the
data in its “fullest” form, i.e., in this case the video, but that to actually work on the
job of analysis a pared down transcript was easier. Exactly this issue is raised in
Powell, Francisco and Maher (2003, 412), do we use tapes as data or transcripts as
data? A preference for transcripts with more non-verbal detail was also expressed and
it was noted that even in the transcript with the images, there was not an attempt to
convey tones of voice, or pauses and timings and these “vocal” aspects of talk can be
important in our interpretations, particularly if we want to be able to tell how
“hedged” contributions are, i.e., how much they are expressed in ways that
communicate a lack of certainty.
One participant reflected on how she had been constrained to use audio rather
than video recordings of lessons, in order to comply with ethical concerns expressed
by some students, but that she had ultimately valued this constraint and the way it
made her focus on aspects of talk only.
We hope the group will continue at the next BSRLM meeting and we invite
anyone to contact Alf if they have some data/issues they would like to share.
References
Farsani, D. 2012. Mathematics Education and the Analysis of Language Working
Group Report: Making multimodal mathematical meaning. Proceedings of the
British Society for Research into Learning Mathematics, 32(1): 19-24
Powell, A., J. Francisco and C. Maher. 2003. An analytical model for studying the
development of learners’ mathematical ideas and reasoning using videotape
data. Journal of Mathematical Behaviour, 22: 405-435
From Informal Proceedings 32-3 (BSRLM) available at bsrlm.org.uk © the author - 198