Making Sense of Mathematics
An approach to learning mathematics at Key Stage 4 based on RME
Background
The Freudenthal Institute, University of Utrecht was set up in 1971 in response to a
perceived need to improve the quality of mathematics teaching in Dutch schools. This
led to the development of a research strategy and to a theory of mathematics
pedagogy called Realistic Mathematics Education (RME). RME uses realistic
contexts to help pupils develop mathematically. Pupils engage with problems using
common sense/intuitions, collaboration with other pupils, well judged activities and
appropriate teacher and textbook interventions. This approach to teaching is used in
Dutch schools today and continues to be refined.
In 1991, the University of Wisconsin, funded by the USA’s National Science
Foundation and in collaboration with the Freudenthal Institute, started to develop the
Mathematics in Context (MiC) approach based on RME. The initial materials were
drafted by staff from Freudenthal Institute on the basis of 20 years of experience of
curriculum development. After revision by staff from the University of Wisconsin, the
materials were trialled, revised and re-trialled over a period of five years.
The first version of MiC was published in 1996/7 and it has undergone several
revisions since then. The teacher material, which supports the pupil books, provides a
comprehensive analysis of issues pertaining to the topic and provides the teacher with
insights into teaching and learning trajectories.
The research evidence from the USA suggests that this approach to teaching was
particularly effective in raising standards and influencing pupils’ and teachers’
attitudes1.
In 2004, the Gatsby Charitable Foundation agreed to fund a project, run by
Manchester Metropolitan University (MMU), based around trialling RME over a three
year period in English secondary schools. At that time, the Economic and Social
Research Council (ESRC) also agreed to fund research into how teachers’ beliefs and
behaviours change as a result of engagement in the project. Essentially the project was
conceived in terms of trialling the MiC materials through Key Stage 3 in a variety of
schools (in total over 1500 pupils and 35 teachers were involved in the project). The
project pupils fully utilised the MiC materials over a period of three years but with the
expectation that they cover the National Curriculum and take the end of Key Stage
examination. The aims of the project included exploring pupil development over time
and generating an understanding of the needs of teachers trained in different
paradigms. The outcomes from this project are summarised in Anghileri, J. (2009)2
and Hanley, U. et al (2007)3.
Making sense of Mathematics
July 2009
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Moving Forward
The Making Sense of Mathematics (MSM) project began in 2007 as an extension to
the work of the Gatsby funded Key Stage 3 project. Resources for the MSM project
were produced as a result of collaboration between the Freudenthal Institute and
MMU and currently consist of a series of 9 booklets designed to deliver the Key Stage
4 Foundation level curriculum. These booklets build upon the experiences gained
from the Gatsby project and take account of difficulties highlighted by the Key Stage
3 teachers such as the need for RME based materials which feature British contexts
and are more closely linked to UK assessment systems.
The MSM project has involved Foundation level classes from 6 schools in the first
cohort and 10 schools in the second cohort. MMU has supplied resources to these
schools and has provided ongoing support in the form of twilight training sessions and
school based observations. Feedback given by the teachers has been used to revise the
materials which are currently in their second version.
There have been several challenges related to working with Foundation pupils using
RME based pedagogy:
(i)
Foundation pupils have a repeated history of failure in mathematics;
consequently they lack confidence and are reluctant to expose their ideas.
Many display a range of well honed non-engagement tactics and teachers
have reported that it can take several months for the pupils to start to
interact in a meaningful way, which allows for the processes of listening to,
discussing with, and learning from others.
(ii)
Foundation pupils are well rehearsed in procedural ways of engaging with
mathematics, despite their lack of success with this approach to learning
and have long since lost a belief that mathematics could make sense to
them. It can take time for them to recognise that less formal ways of
approaching problems, more closely linked to their intuitions are indeed
valid and can be very fruitful.
There were similar issues during the Key Stage 3 project in terms of recognising the
need for it to be the pupils who are “doing the maths” rather than the teacher, but this
has been even more striking at Key Stage 4. It highlights the need for CPD which
enables teachers to create genuinely interactive classrooms even when working with
less inclined pupils.
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Key Findings from the MSM project to date:
Teachers
Working with MSM can influence the practice of teachers in the following ways.
(i)
Over time the teachers start to recognise the need to engage pupils in
genuine debate as they explain and listen to each other’s mathematical
thinking. In this pursuit the teachers begin to acquire a repertoire of
strategies to promote such engagement.
(ii)
They realise the power of context not only as a means of engaging pupils
in debate but also as a domain where pupils can make sense.
(iii)
They begin to ask different types of questions which prompt the pupils to
move between different mathematical representations and to visualise.
Examples of such questions are ‘Where can you see the 120 squares?
(showing a picture of an empty 15 by 8 rectangle), and ‘Where can you see
the diameter in this? (pointing to the circumference of a circle).
(iv)
They begin to question their beliefs about what is effective teaching and
comment on the dilemma caused by matching this approach with the
objective led climate in British classrooms.
(v)
They begin to use the MSM resources and approach with some of their
other classes.
Pupils
Working with MSM can influence the behaviour of pupils in the following ways.
(i)
Using contexts which are familiar to them can prompt reluctant pupils to
contribute and to articulate their thinking in a domain which appears less
obviously mathematical and therefore, to them, less threatening.
(ii)
Over time pupils begin to re-engage with mathematical thinking. They
start to show more confidence in describing their own methods and a
greater inclination to listen to others. In all areas of the curriculum it is
possible to see evidence of mathematical development.
(iii)
The early stages of analysis of Problem Solving data for MSM pupils from
Cohort 1 has revealed considerable progress over control pupils in their
approach to a fractions question and to an area question. MSM pupils
tended to access these questions by drawing pictures, by making sense, as
opposed to performing some routine on the numbers.
(iv)
Pupils recognise that this approach is different, they comment ‘It doesn’t
feel like you are doing maths’, ‘It’s like you are doing geography….lots of
subjects’. Likewise some pupils who enjoy working procedurally through
exercises find it difficult to adjust to a more exposed way of learning.
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Implications for developing MSM at Higher Level
In designing and writing the materials for Higher level pupils, the following issues
will be explored in detail.
(i)
One of the concerns raised about the RME approach is the facility it has
for enabling the learner to ‘vertically mathematise’ (to have the ability to
move from informal or contextualised mathematics to more formal or
abstract mathematics). This is pertinent to achieving success at higher level
where pupils are required to work effectively and efficiently with formal
mathematics across a variety of representations. In the previous projects
we saw evidence that pupils were vertically mathematising . For example
many pupils started to recognise the power of the ratio table model for
solving problems from a wide range of mathematical topics. What models
and learning trajectories are helpful to enable vertical mathematisation at
Higher level?
(ii)
The use of context: - What contexts are helpful at Higher level to enable
pupils to make sense? Is it necessary to return to real life contexts or can
these contexts come from the world of mathematics? How do we design
materials which prompt pupils to shift between the levels of abstraction
and to recognise the power of returning to a previous level as a means of
making sense of the formal?
(iii)
The use of models: - Some extremely powerful models emerged at
Foundation level including the number bar, the ratio table and the use of
squares for area. What are the significant models at Higher level, those that
will be effective in promoting higher order thinking?
(iv)
Higher level pupils have usually experienced a good deal of success
through learning mathematics in a procedural way; many will not see the
need to change their approach. What strategies can the teacher employ to
promote real engagement with mathematical thinking and re-focus pupils
on exploring underlying structures as well as learning how to ‘do’?
(v)
An emerging thread from the MSM project is the use of questions which
prompt pupils to look for one mathematical representation within another.
What strategies will be effective at Higher level as a means of enabling
pupils to make links across the mathematical domains?
In order to address these issues there is a need for:
(i)
Consultation with the Freudenthal Institute. In conjunction with MMU
they devised the outline for the Foundation scheme and wrote the first few
booklets. They have vast experience writing across all levels including
higher ability.
(ii)
A field study which closely examines the implementation of some sample
material at Higher level prior to writing the whole scheme. This will take
place in 09/10 working closely with 4 schools that have previous
experience using RME. The field study is to include:
(a) Consultation with experienced RME teachers;
(b) Observation in RME classrooms;
(c) Discussion with a range of higher level pupils.
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Implications for adopting MSM across the UK
The long term vision is that courses like MSM will be adopted across the UK
throughout the various Key Stages. The following are some of the issues that would
need to be considered before this could happen.
(i)
The need for ongoing CPD over a substantial period of time was
highlighted in both the Key Stage 3 and MSM projects. This is a radically
different approach and teachers need guidance about how to act and what
to expect both in terms of their own development and the development of
their pupils. Given the current ‘rarely cover’ situation in schools CPD
would need to take the form of twilight sessions and/or paid holiday
INSET days.
(ii)
It should also be noted that time is needed for the practitioners to acquire
the appropriate skills and to develop a level of understanding of RME
which enables them to make their own choices about how to act.
(iii)
Online support is needed as a means of updating schools, supplementing
resources and providing a helpline service to address individual queries.
Several teachers in the Key Stage 3 project experienced doubt around the
November of the first term. Support from the Curriculum developers and
from experienced RME practitioners is particularly useful at this time.
(iv)
An emerging factor from the MSM project is the recognition that teachers
need to develop a repertoire of interactive teaching skills alongside the
RME pedagogy. The training programme must address this need.
(v)
The use of classroom video has been extremely productive as a means of
enabling teachers to analyse their practice and to share effective strategies.
This would need development into a video package which includes lesson
excerpts with commentary.
(vi)
The RME approach provides a strikingly different route through the
mathematical content of UK national schemes. In recognition of this it will
be necessary to outline trajectories through the mathematics curriculum
which take as much account of the journey as the outcomes. This would
help to relieve teachers from the pressure they feel to teach to a formal
end-point even when this is not appropriate.
(vii)
Further classroom based research would be needed to consider the possible
adoption of this approach at Key Stages 1 and 2.
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Conclusion
There are many lessons to be learnt from other countries where RME has been
introduced successfully.
These include the need to develop and trial scaffolding materials which take account
of research in RME and teachers’ needs, and to provide high quality CPD to ensure
that teachers have a deep understanding of the pedagogy they are using. Meeting these
requirements forms a major part of this project.
There are, however, other issues that are specific to the UK, particularly those
associated with the assessment regime and the culture within which education is
taking place. It would be quite unrealistic to expect to achieve success simply by
importing a package from another country. It is essential for RME to be presented in a
form that is fit for purpose in the UK. This, too, is being addressed in this project.
References
1.
Romberg, T. A. (2001). Designing middle-school mathematics materials using
problems set in context to help students progress from informal to formal
mathematical reasoning. Madison: University of Wisconsin-Madison.
<http://66.102.9.104/search?q=cache:ASzZRTZ2lUJ:www.wcer.wisc.edu/ncisla/publications/articles/MiCChapter.pdf+designin
g+middle+school+mathematics&hl=en&ie=UTF-8>
2.
Anghileri, J (2009) Gatsby Project: Mathematics in Context: Summary
Evaluation Report. (Provided as a separate document)
3.
Hanley, U. and Darby, S. (2007) Working with curriculum innovation: teacher
identity and the development of viable practice, Research in Mathematics
Education, vol 8, pages 53-66.
Making sense of Mathematics
July 2009