WHAT TEACHERS LEARN FROM CHILDREN’S
MATHEMATICAL ARGUMENTS IN DISCUSSION: MOVING TO
A NEW PEDAGOGICAL MODEL
Julie Ryan, Derek Kassem and Charles Sarland 
Liverpool John Moores University
Paper presented at the British Educational Research Association Annual Conference, Heriot-Watt
University, Edinburgh, 11-13 September 2003
Abstract
This paper is drawn from a study of teachers’ development of collaborative discussion in mathematics
classrooms where children’s misconceptions, errors and methods were used as the basis for group and
whole class discussion in year 5 classrooms in two contrasting schools in Liverpool. Research-based
materials used to begin, sustain and extend children’s discussion were presented to two teachers during a
university-based training session for the study of teachers’ understanding of children’s mathematical
arguments in discussion. How the teachers adapted or rejected these materials and strategies to their own
practice was tracked over a three-month period. This paper reports on one research question from the
study: how do teachers begin to use children’s mathematical discussion in their classrooms. The
teachers’ lessons were videotaped and these recordings were used across the three-month period to
prompt teacher reflection on what they were developing. The already highly successful practice of the
two teachers was securely embedded in the National Numeracy Strategy Framework and its model of
‘interactive whole class’ teaching. The introduction of children’s collaborative discussion was found by
the teachers to significantly challenge this model in terms of their pedagogy. Constraints of the NNS
pedagogical model were related in particular to time and the unitising of learning episodes. However, the
teachers endorsed the opportunity to explore new practice within a supported research environment
where they were able to take control of the agenda and develop their own practice rather than deliver the
practice of an outside agency.
Keywords: mathematical discussion; accounts of practice; errors and misconceptions;
teacher reflection.
Introduction
Collaborative discussion in England’s key stage 2 and 3 mathematics classrooms is a
rare occurrence. The ‘numeracy hour’ or Daily Maths Lesson prescribed by the
government’s National Numeracy Strategy (NNS) lays down a framework for ‘whole
class interactive teaching’ designed around a timed oral/mental starter, a main activity
with differentiated tasks for children of different ability and a closing plenary.
Teachers are judged by their proficiency in using this model by the government’s
inspection body Ofsted.
This delivery model, together with mandatory testing which establishes a school’s
public success, appears to have had a profound impact on the culture of the
mathematics classroom. We live in an era where education is cast narrowly as the
acquisition of competences and where teachers face the same conflicts, pressures and
demands behind many cycles of reform. Schon’s comment is still timely twenty years
on.
Practitioners are frequently embroiled in conflicts of values, goals, purposes, and interests.
Teachers are faced with pressures for increased efficiency in the context of contracting budgets,
demand that they rigorously ‘teach the basics’, exhortations to encourage creativity, build
citizenship, help students to examine their values. (Schon, 1983, p.17)

Note: The authors gratefully acknowledge the work of the two teachers identified here as
Debra and Kate and their year 5 children who made this study possible.
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The NNS aims to be a complete package: curriculum, assessment, lessons, tools,
artefacts and videos of exemplary practice. It is a rare teacher indeed who challenges
its content and style during a school inspection regime or as an expression of
professional discourse. While ‘interaction’ is promoted, it is found to be essentially
teacher-centred; teacher talk dominates. The whole class interaction exemplified in
NNS video materials supplied to schools does not show examples of interaction that go
beyond the teacher-centred Initiate-Respond-Evaluate/Feedback model (Dillon, 1988;
Mercer, 1996; Sinclair & Coulthard, 1975): significantly children are never seen to
explore a misconception, develop their argument or engage in conflict and reflection
which, from a constructivist perspective, are seen to be so productive for learning.
The NNS advises teachers to be aware of children’s misconceptions and errors
(DfEE, 1999) and to adopt a corrective approach. How to approach ‘errors’ more
productively however presents dilemmas: should errors be given centre stage in
classroom dialogue, should they be corrected with immediate ‘treatment’ before they
spread or can they be a source for enhancing mathematical understanding? (We use the
term ‘error’ to include children’s misconceptions, alternative methods and mistakes
that give some useful insight into their mathematical thinking.)
These were dilemmas we were keen to explore with classroom teachers who were
already proficient in the NNS model but who were willing to explore children’s
mathematical errors through discussion. How would they adapt their current teaching
in the light of research-based alternative suggestions for classroom practice? We were
aware that the adaptation might require a re-conceptualisation of the teaching and
learning frames (Schon, 1983) in the light of the NNS system.
Our plan was to explore a shift from current practice by twinning errors (content)
and discussion (method). The research we drew on stemmed from small group
discussions that children had had of their errors on mathematical test items in a
research environment (Ryan & Williams, 2000). The errors had been established from
large scale national testing (Williams & Ryan, 2000) and the likely arguments that
children used to back them had been organised into teaching planning tools (Ryan &
Williams, 2003). We were keen to study what teachers could make of these research
ideas.
Theoretical Perspectives
Teacher reflection on their practice was central to the study. Reflection on practice
(Grimmett & Erickson, 1988; Schon, 1983, 1987a, 1987b; Smyth, 1992) is widely seen
as an appropriate tool in the study of the professional development of teachers.
Accounts of practice produced through narrative inquiry have been regarded as a
means for teachers “to become better acquainted with their own story” (Conle, 2000,
p.51).
… researcher/teacher collaboration also came to be seen as an opportunity for professional
development for the teachers being studied in the research projects. To become better acquainted
with their own story was indeed interesting for busy teachers who had little time for reflective
writing themselves. (Conle, 2000, p.51)
Johnson (2002) however cautions researchers about the stability of accounts that
teachers give of their practice and proposes “that it is entirely possible that the same
teacher would produce a different version of her work, if … methods of accounting
were altered” (ibid, p.22).
The teacher accounts of practice revealed in conversation with the
researchers/interviewers (and in whole research group conversation) are produced
within a frame of communicative action – the framing talk of the interviewer’s
comments and questions – which, as Johnson (2002) warns, does not necessarily
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produce a definitive account. The conversational strategy as a method of enquiry is
limited and can encourage teachers to produce a particular version of practice
determined by the relationship between the interlocutors (Young, 1986 as cited in
Johnson, 2002). However, in this study there appeared to be some stability across
interviews and teacher-to-teacher conversation where accounts were confirmed or
consolidated.
‘Knowing-in-action’ (Schon, 1983), or ‘tacit knowing’ (Polanyi, 1967), is found not
only in everyday behaviour but especially in professional behaviour – that intriguing
knowing that is sometimes more than you can put into words, that knowing that
constitutes the artistry of a profession. Teachers, as with all skilled practitioners, have
much of their knowledge and expertise embedded in their practice; their knowing is in
their doing.
Knowing-in-action however can have some limitations. Practice can become
repetitive and routine and consequently lead to boredom, burn-out or ‘over-learning’
where a practitioner has learned to be “selectively inattentive to phenomena that do not
fit the categories of (their) knowing-in-action” (Schon, 1983, p.61) or where they are
drawn into patterns of behaviour that they can no longer correct. Reflection-in-practice
is seen as an antidote to professional over-learning (ibid.) – a provocation for new
learning. Reflection on reflection-in-action is “an intellectual business, and it does
require verbalization and symbolization” (Schon, 1987a). It involves making tacit
knowledge explicit.
… (B)oth ordinary people and professional practitioners often think about what they are doing,
sometimes even while doing it. Stimulated by surprise, they turn thought back on action and on
the knowing which is implicit in action. (Schon, 1983,p.50)
Roth (2003) attempts to theorise reflection on teaching in general and the potential
role of video as a medium in particular. He uses a “phenomenological perspective to
articulate a number of issues that frame the opportunities and constraints of video as a
tool that mediates reflection” (Roth, 2003).
First, each moment of praxis (lived experience of situated action) is marked by its intentionalhorizontal character. Second, each moment of praxis is marked by a particular rhythm and tempo
with which it unfolds, that is, by its characteristic temporality. Together, the two dimensions also
contribute to different Selves that teachers experience while teaching and reflecting on teaching.
(Roth, 2003, p.5)
The intention ‘in the moment’ of teaching and the intention in reflection ‘of the
moment’ are constrained differently, most dramatically in terms of time or horizon.
The first intention is to successfully complete the lesson, while the second intention
may be to consider ‘how could I do this differently?’, or simply, ‘what was I doing
here?’ where intention is constructed, or articulated, after the event. The video episode
becomes a tool which can be used again and again for different ends: what did I do,
why did I do it that way, should I be doing it differently? The temporal advantage
affords the switching between “different Selves” in reflection.
We address teacher knowledge in this study from a socio-cultural perspective;
knowledge conditioned by available tools, the environment of action and the language
of articulation. We draw attention to these conditions mediating teacher reflection on
their action. The video episode in particular can be seen as a boundary object
mediating reflection on practice. The reflection on action, mediated also by the
researchers’ interventions and lines of enquiry (eg. questions, comments and
suggestions), is revealed as a developing social practice of collaboration.
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The Study
The research study was focussed on the questions (a) how do primary teachers use
research-based materials to organise classroom discussion, (b) what persuades children
to change their minds in group and whole class discussion, (c) what do teachers attend
to in reflection of children’s arguments in discussion and (d) what impact does it have
on their practice. The materials and strategies already developed to begin, sustain and
extend children’s discussion were to be presented to two teachers during a universitybased training session. How teachers adapted these materials and strategies to their
own practice was to be tracked over a three-month period.
The ‘ready-made’ materials however proved not to be central to the teachers’ initial
interests – they were keen to first demonstrate their current practice and then to
develop their ideas of what mathematical discussion could be in terms of their own
securely-based NNS practice. The original aims were therefore re-conceived as the
collaborative nature of the research unfolded.
The research questions were fine-tuned to become (a) how do teachers develop
mathematical discussion in their classrooms, (b) what do they learn from such
discussions, (c) what materials and strategies are seen to be productive, and (d) what
impact does collaborative discussion have on teaching practice. This paper addresses
the first question.
The two year 5 primary teachers in Liverpool worked in different schools which
catered for different socio-economic communities. The two teachers, Kate and Debra,
in this study were nominated because they were considered to be successful year 5
mathematics teachers who have recently gained Advanced Skills Teacher status after
only four and five years teaching experience respectively. They were both in ‘Beacon’
schools which are charged with transferring good practice to other schools. Debra won
her AST in mathematics and is a Leading Maths Teacher for on the exemplary
teaching of mathematics; Kate won her AST for the teaching of Spanish and also
advises on exemplary teaching practice. These two teachers were experienced in
demonstrating their practice to visitors and were comfortable about the videotaping of
their lessons. They were also experienced in evaluating other teachers’ practice and
offering professional development support. They were not experienced however in
terms of reflection on their own practice with outsiders.
The three researchers in the study were university educators with former school
teaching backgrounds. One researcher had been involved in the development of the
discussion tools and therefore had a vision of what could work, another researcher had
a strongly developed democratic approach to development and therefore adopted a
more open-ended start to the study, the third researcher came with a non-mathematical
background and a preference for exploration of language and action research. These
different perspectives affected the direction of the study.
Methodology
A case study methodology was adopted for the project given that contextual conditions
were central to the study (Yin, 2003). We hoped to study the teachers’ descriptions of
their practice, their beliefs about children’s learning, their reflections on changing
practice, the nature of researcher interventions and the use of materials already
developed for classroom discussion. A two-case design gave us the opportunity to
contrast styles of development and to consider teacher-to-teacher interaction at key
stages of reflection. Importantly, the two-case design afforded opportunities to
consider the impact of different contexts and the possibilities of generalisability of
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findings in order to inform the next step: how should the research materials be adapted
or further supported to make them more likely to be useful to practitioners for
professional development?
The selection of teachers recognised by their schools as highly competent
practitioners was deliberate. The two teachers had significant responsibility for
dissemination of practice within their school regions and were highly knowledgeable
of the mechanisms that were likely to support successful change in classroom practice.
They were also more likely to delineate the boundaries of development.
The three-month period available for the study restricted the focus to early
development only. The structure was three half-day meetings of the five participants at
the university, videotaping of three lessons in the teachers’ schools and paired or whole
group researcher-teacher reflection on each lesson.
The unit of analysis was accounts of practice as revealed in interview and prompted
by selected videotaped segments of the teachers’ lessons. The choice of videotaped
episodes for reflection was in the hands of the teachers. Attention was paid to the role
of the ‘interviewer’ and the framing of questions and comments so that a researcher’s
own ideology or preconceptions were either subordinated or made explicit.
Videotaping protocols were also outlined to secure common terms of reference.
The teachers were given their videotapes after each lesson so that they could watch
them privately and decide what parts could be used for reflection with the researchers.
Two cameras were used in each taping session: one taking a fixed wide view in whole
class interaction and then following the teacher and the other capturing independent
student group interactions.
The researchers drew up the agendas for the meetings (with focussed questions) and
made transcripts of the meetings from audiotapes. The transcripts form the basis of the
analysis below. One of the teachers was also available after the conclusion of the
project to analyse one of the transcripts and her comments are built into the analyses
below.
Analysis of First Meeting: Baseline
In the first meeting, the researchers outlined the aims of the project and sought the
teachers’ expectations. The researchers were to present ideas but the teachers were to
decide what they wanted to do – in this sense there was an open agenda. The teachers
described their current practice and reactions to errors and misconceptions in the
classroom. Video snippets of NNS materials, where teachers responded to children’s
errors in the classroom, were also discussed in order to uncover aspects of currently
promoted practice or innovation.
Expectations of the project
Debra and Kate had considerable responsibility for the professional evaluation and
development of their school colleagues and also demonstrated expert practice to other
teachers. However, they had little opportunity to ‘view’ their own practice and they
regarded this positively.
I do think it will be quite interesting to see what I do in the classroom. You never see what you
do, you see what you’ve done, but you don’t know how you’ve done it. (Debra, I, 20:20)
I agree. (It will be) interesting for professional development. We watch other teachers a lot and
we’re very observant and see how to develop them and their teaching, but watching yourself – we
don’t have the opportunity very often. (Kate, I, 22:10)
There was an initial belief that the researchers were evaluating the teachers’ practice
but the researchers reinforced the idea of description rather than judgement (R1, I,
21:35). The researchers stated that the teachers should feel free to develop their own
ideas for discussion in the classroom for this research project. This was unusual in their
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experience of professional development. Sometime later in the project Debra said that
she had been hoping to discover the researchers’ hidden agenda.
‘Cos then I was thinking, what do they really want me to do with it. So you start thinking, right, I
can do what I want, but what do they want to see out of this lesson. (Debra, 5, 029)
…in the end, (I said) like ‘forget what they want me to do!’ (Debra, 5, 035)
This proved to be a significant decision – she was able to control both content and style
within her own time frame. She appeared to be enthusiastic about developing ideas
with an outsider and other teachers in her school expressed interest in what she was
doing.
Nature of discussion
The complex question of what constitutes classroom discussion proved to be a critical
focus of the first meeting. In this sense the study proved to start more speculatively
than anticipated – there was a need to ‘slow down’ the study so that ownership was
explicit for all five stakeholders. This had a significant impact on the direction of the
project.
It was clear at this stage that ‘collaborative classroom discussion’ was a step too far
without a careful mapping of where the teachers were now and where they could
possibly go. Debra initially spoke of classroom ‘talk’ and questioned whether
information-giving constituted discussion.
(It will be) interesting to see what I class as mathematical discussion and what you class as
mathematical discussion. I think sometimes we’re always talking in the classroom but whether
you class it as discussion or just informative … it’s finding the difference between the two.
(Debra, I, 23:20)
Kate spoke of ‘interaction’ between teacher and pupils and then between pupils in
group tasks. Her comments were framed as questions as she tried to make sense of the
researchers’ agenda.
Basically you’re looking at our input, and how our teaching would then address certain
misconceptions of the children and … our interaction with them…and then you’d be looking at
their discussion in their group tasks? … And from that we analyse the video and see how we can
promote better and more constructive discussions? (Kate, I, 37:12)
The second stage of the meeting attempted to establish what classroom discussion
already took place in the teachers’ classrooms. A videotaped segment of a small group
of children engaged in collaborative discussion of their errors was shown and
discussed. Several questions were posed: how can discussion be used to support pupil
learning, how would you plan and organise a mathematical discussion and what
misconceptions or errors might arise in the topic you will be teaching?
Current practice and impediments to discussion
Both teachers stated that they “adhered to the numeracy strategy… generally I do stick
to that routine” (eg. Kate, I, 40:19). Kate had a “high ability set” and reported that
within the set she had a “very talented group of mathematicians… three children who
are on individual education plans because they are gifted and talented at maths”. These
children were given a lot of problem solving tasks to do. Kate reported “doing more
discussion (with them) than the rest of the class. The rest of the class needs to get a
handle on the basics, doing calculations, more focussed written tasks” (Kate, I, 41:04).
The teachers thought that moving from current familiar practice might be a problem
for the children. The socio-mathematical norm of ‘being correct’ or ‘giving the teacher
the right answer’ in mathematics lessons was recognised but Debra was already
counteracting it by encouraging ‘helping each other’ and ‘giving reasons for answers’.
Well my lot don’t like to be wrong, so if you are discussing something, or saying what will you
do, they don’t like to say what they would do, in case it’s not what you want. (Debra, I, 53:40)
Her strategy to break this down was to group her children in mixed ability groups
knowing that they “actually do a lot of discussion from helping each other” (Debra, I,
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54:12). She had been encouraging them to give reasons for their answers from the
beginning of the year: “…initially they were just giving answers, but now they say,
well I’ve done this because…” (Debra, I, 54:20).
Kate’s class was the top set of year 5 and she set lower, middle and higher ability
groups within the class. She reported that:
The less able in the top set are more reluctant to discuss their methods and to volunteer the
answers and they will need more thinking time. So I’ve tried to do that more where you discuss it
with a partner and then put your hand up to give them more confidence. (Kate, I, 56:21)
Debra and Kate already had strategies in place to encourage collaboration between
children and to foster discussion in their classrooms. Debra used mixed ability groups
within her class and Kate used partners for the children to discuss their ideas in order
to foster public confidence.
It appeared that opinions or beliefs about ability (and setting) might influence
decisions about which children are suited or ready for discussion and how children
could be grouped for productive interaction. Kate and Debra appeared to have different
opinions at this stage.
Current strategies: errors, misconceptions, mistakes and misunderstandings
In the early stages of the meeting the five participants had found a shared meaning for
the terms ‘misconceptions’ and ‘errors’. When Debra encountered misconceptions and
misunderstandings she would “go back through it and we’ll identify where the actual
problem is and we’ll sort it out” (Debra, I, 65:15, italics added). Kate reported that she
would “stop the whole class and … go through it again …teach it in a different way”
(Kate, I, 66:00). A corrective strategy, led by the teacher, appeared to be the norm.
Strategies were also described as “taking a step back and moving it forwards again
so they understand” (Debra, I, 67:45, italics added) and “think it themselves … rather
than somebody having to identify (it) for them” (Debra, I, 68:40). Kate described how
her children did not make links to previous lessons. Her strategy was to “go back to the
number line” and to “go back again” to what a recurring decimal meant. These were
unfolding strategies.
Summary: Baseline
In the first meeting, the five participants began to negotiate shared meanings for the
research project. Shared meanings for key terms (eg. misconceptions, discussion) and
practice (eg. implementing classroom discussion, whose agenda prevails) were
explored but not finalised. These remained as ideas in progress.
Noticeably, the materials already developed, and presented to the two teachers as
possible resources, were not immediately taken up (materials such as: starter problems,
strategies for classroom organisation, transcripts of children’s discussions). The two
teachers were keen to develop the idea of classroom discussion using their existing
resources. The project shifted direction to being more evolutionary than originally
anticipated.
The original research question moved from ‘how do teachers use research-based
materials to organise class discussion’ to ‘how do teachers begin to use children’s
mathematical discussion in their classrooms’. It would be in their practice that their
‘definition’ of mathematical discussion would begin to emerge (ie. knowing-in-action).
Analysis: First Lesson
The teachers suggested that their first videotaped lesson should exemplify their current
practice so that it could be used as a baseline for their development of discussion in
their classroom.
Debra and Kate separately joined two of the researchers in an after school meeting
to discuss moments that they had identified for reflection from their videotapes. The
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five participants then met for a joint meeting to share reflections on the first lesson, to
review the project and plan the next step.
Debra’s and Kate’s intention in the moment of teaching and their intention in
reflection of the moment demonstrate Roth’s (2003) notion of switching between
“different Selves”. The videotaped episodes afforded a means to express “their own
story” (Conle, 2000). The affordance of videotaping also gave the teachers
opportunities to see usually unseen actions that indicated subtle individual behaviours
during whole class interaction when the teacher is normally dependent on explicit
social cues.
Case 1: Debra
Debra planned her first lesson using the NNS model: starter, main activity, plenary.
She had also planned to have “some element of discussion” in the lesson (Debra, 2,
25:55). The focus of her lesson was the comparison of fractions with different
denominators. The main activity for each group had been differentiated by ability and
took the form of a worksheet followed by differentiated consolidation or extension
tasks.
She said her usual practice was characterised by an emphasis on teacher-to-student
questioning, and it was this area of her practice that she became quite critical of on
reflection. “What I was trying to do there...”, but “I was leading them to the answer I
wanted” or, “I got side-tracked by ...”, so “I need to...”. This descriptive cycle was a
feature of her initial reflections.
So what I was trying to do there was see what they came up with and then try to move them on.
(Debra, 2, 05:03)
... because they’d say something, I’d move on from what they’d said, and get side-tracked
(Debra, 2, 05:40)
There were several tensions that Debra described: between her lesson
focus/objective and wanting to respond to children’s responses; between time planned
and time taken; between “coverage of the strand” and getting side-tracked by children.
... sometimes I look at my objectives for my mental starter, sometimes I just think, ‘well I only
covered, did three questions from that’, and something has sidetracked me, and the child said
something, and I said ‘yes, that’s brilliant’, and then gone on with that. ... which because of the
Strategy, you know it is wrong, but at the same time maybe it’s not ... (Debra, 2, 7:50)
... you sometimes do go off (track), and it can be a very valuable thing that the children learn at
the end of it, but you think, ‘I haven’t done my job properly because I’ve not covered what I’ve
written down I’m going to do. (Debra, 2, 11:00)
Debra used questioning in whole class interactions to seek out the reasoning and
logic behind children’s answers. “Basically I wanted to see what their reasoning
(was)”, “I was interested to see what kind of logic he’d applied” (Debra, 2, 23:53), and
“I do get them to explain an awful lot of times” (Debra, 2, 24:50).
Reflection on particular episodes then became reflection on her general teaching
style. Debra described her usual whole class teaching pattern as initially teachercentred but then focussed on children’s public responses.
If it's something new, I'll go through, I'll explain a couple of times and then I'll get the children to
talk me through and I'll scribe and they have to tell me exactly what to do. (Debra, 2, 28:50)
The responses were generally describing steps in a computation. The researchers
prompted ‘could this dialogue lead to classroom discussion?’
...if there was a two-way, not just a question and an answer, but questions and answers coming
from both sides rather than just me saying 'what would you do now, what would you do next' but
when they actually say, 'I've done that bit but what do I...' so then they start throwing questions
back ... (Debra, 2, 29:50)
It was this movement towards Debra’s re-definition of classroom discussion that
became the focus of the reflection with the two researchers. At this stage her definition
was centred on sustained teacher-student dialogue where student reasoning was
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articulated. Debra also thought that this type of interaction sometimes destabilised the
children.
They'll give you the answer and I'll say, ‘explain how you got it’, and they'll immediately,
immediately think they're wrong, and I'll say, ‘no the answer's right, but how did you get there,
the processes you go through are as important as actually getting the right answer’. But it's funny
the way, immediately, if you say something like ‘how did you get that answer’, initially last
September they'd just go, ‘oh I must be wrong’. (Debra, 2, 30:55)
Debra especially wanted to see an excerpt from the second camera tape where she
thought a group of children were discussing their main task. She had not seen this
second tape but the children involved had told her after the lesson that even though it
might appear that they had not done much work, they had in fact been working quite
hard. The three girls had become stuck on a problem on their worksheet where they
were to compare 3/5 and 7/8. They could no longer strictly follow the procedures that
had been explained initially where one denominator was a factor of the other.
Yes, it's fifths and eighths but, um, it was too early on for them to have a go at that one, because
they'd all done the first one fine and then they'd, it’s fifths and eighths, and it'd really thrown
them but then the others were like say you'd have fifths and fifteenths or quarters and, um,
sixteenths, so it was more obvious what you had to (do). (Debra, 2, 36:30)
One of the girls, Betty, was suggesting that their problem was, “how many fives go
into eight?” Her friends were initially non-responsive. Betty persisted, suggesting “one
point three” but the group was still clearly puzzled. It appeared that Betty’s
perseverance had created the climate for different ideas. Debra explained what the girls
were doing and how they revealed their arguments.
She looks at the example, and says, 'well, look, you've doubled the quarters, and got eighths' and
then she says, 'double, oh, maybe we have to double the fifths' ... and that's what she gets, and she
gets tenths, and then Gina picks up and says 'no, you've doubled the quarters because we wanted
eighths, and you've timesed the top and bottom by two, but we want to see how many fives go, to
multiply five by, to go into eight', so they've not got all the way, but they understand what they're
doing. (Debra, 2, 40:00)
Debra was clearly excited by the children’s interactions, whereas she had been
initially critical of the design of her worksheet that presented this question too early.
This is fabulous seeing this now! ... I just think it's really interesting, because obviously you don't
see... It's good that she hasn't given up actually, she doesn't give up, she constantly struggles with
it... You can feel them thinking, can't you! ... They're just kind of bouncing ideas ... (Debra, 2,
43;00-48:00)
The children were keen to justify the time they had spent on the question.
... then afterwards Jill came up to me ... and said ...’we were trying that one for ages’, I think she
was saying though, ‘I haven't done much work because I was trying that one for ages’ and
insinuating it's all on the video, you know, so like proof that they were actually having a go, it
was just that one question. So she was quite conscious you know of how long they'd spent on it
and how much work they'd done trying to work it out. (Debra, 2, 50:00)
Initially Debra had suggested that the problem created by this question could have
been fixed by re-ordering the content of the lesson (fraction equivalence practice
followed by fraction comparison). One of the researchers however suggested another
framing of the episode.
I noticed that you thought that that was a critical, a criticism of the worksheet, that the question
came in as number two, but maybe it wasn’t, maybe that’s one that actually generates the conflict
and therefore discussion and therefore the sorting out of ideas. That’s me imposing ...
(Researcher 1, 2, 51:00)
Debra also identified “underlying misconceptions in there that need to be worked
out” (Debra, 2, 52:00). She decided to use the video episode not only as a prompt for
the next lesson where the misconception was to be sorted out, but also as modelling
what she wanted from the other children in terms of ‘discussion’.
Debra was beginning to build her own model for discussion: (a) exchange ideas, (b)
find differences, and (c) ‘sort it out’. A key condition for discussion was a conflict of
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ideas or a problematic. At this stage too she was formulating organisational strategies
for grouping children, guidelines for interaction, and reporting back to the whole class
that would support the ‘sorting out’.
Case 2: Kate
Kate also planned her first lesson using the NNS model: starter, main activities,
plenary. The main activity for each group had been differentiated by ability and ranged
from practical activity, to worksheet to problem solving. Her role was explaining the
tasks and then assisting a chosen group through questioning.
Kate identified how she used a particular group of children as a target group but “I
hadn’t realised that I do tend to rely on them a lot” (Kate, 3, 11:12) and “I didn’t realise
it was so apparent” (Kate, 3, 37:54). She identified this group as “the middle ability
group” (Kate, 3, 38:54) within this top-set class.
She unravelled her knowing-in-action in some detail as she described how she
gauged their responses to questions to guide the direction of her whole class teaching.
Kate also started to make her idea of discussion explicit.
My idea of discussion in maths is that children are explaining to me the strategies they were
using, and in doing so helping the rest of the class. (Kate, 3, 13:20)
I do tend to do that quite a lot (asking children to explain their mental processes). (Kate, 3,
14:32)
But they do argue with each other as well, and they do listen to each other quite well, this group,
and they will contest ideas…(Kate, 3, 15:08)
(Discussion can happen) between children in problem solving activities or for reassurance when
they need to test out their answers on somebody and explain to each other… It would be by them
looking at each other’s work and saying ‘that’s not right because…’ and then challenging each
other. (Kate, 7, 3:04)
One researcher asked, “how is it that teachers seem to generate discussion?” (R3, 3,
20:30). Kate and Debra both agreed that there was a need for careful choice of a
problem, or “creating a difficulty” (Kate, 3, 21:20). Kate said that in her current
questioning practice she tried to give children “time to think” (Kate, 3, 25:40) by not
immediately responding to an answer and that the children were becoming used to this
technique.
Teacher reflection in a group situation was also seen by Kate to be valuable.
It’s interesting for me … the things that I hadn’t noticed, even watching back myself … and I
thought… I’d go, ‘ooh, that’s interesting, look what I do, look at that’, and notice things about
myself, and I did pick up things like that, but watching it with (the researchers) they pointed out
things that I hadn’t even noticed. (Kate, 3, 36:52)
… I’m getting more things from it now discussing it as a group and it’s not like I thought it
would be, I’m feeling far more comfortable with it. (Kate, 3, 37:40)
Much of the joint reflection was spent watching Kate’s first lesson and the early
focus was on her interactions with the class. Debra asked whether a moment had been
captured where the children were interacting without the teacher.
…Is there a part where you’ve just got the children, and not the, ‘cos we found that when we got
rid of me out of the equation, we found quite an interesting discussion going on … (Debra, 3,
48:50)
I don’t think so, it was problem solving ... (Kate, 3, 48:55)
Kate used questioning extensively and sought children’s reasoning, especially in the
small group situations of the lesson main activity. She encouraged children to listen to
each other but found it more likely to occur with the top set who were usually given
problem-solving tasks.
… what’s going on there, is that they are suggesting things to each other … (they) are more used
to problem solving and they will explain their reason, and they justify and convince … (Kate, 3,
52:45)
But they’ve been given it to solve as a group task, and they’re looking to achieve that aim
together … they just want to get the right answer, and they don’t mind who comes up with it …
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the worst thing for them is when you get to the end of the lesson and they’ve not solved it …
(Kate, 3, 57:40)
Kate’s notion of discussion here is separated from ‘problem-solving’ yet she
identifies terms like reason, justification and convincing. She felt that this was territory
for her top group at this stage. The socio-mathematical norm was ‘finding the right
answers by the end of the lesson’.
Analysis: A Discussion Lesson
Kate and Debra planned their second lessons with discussion as the focus. Their ideas
were still evolving after the first meeting, the first lesson reflection and the second
meeting.
Case 1: Debra
Debra asked for a planning meeting to test out her ideas before her second lesson. “I’d
like to give them a misconception and see where they go with it” (Debra, 4, p.1). Her
SAT’s analysis had shown that decimals and reading scales were areas where her year
5 children had problems. She was open to suggestions. The researcher then offered
some of the research materials that had been developed for the project (items,
discussion prompts, worksheets, organisational strategies) and alternative ways of
using them.
Debra was keen to provide a structure for the children “…because having not done
anything (like this) … I think they’ll need some kind of structure” (Debra, 4, p.2). She
recognised that the children’s normal motivation was to get the right answer and that
this culture would need to be shifted carefully.
… we could have some kind of recording system of their own as they go through, so they can
say, ‘initially I thought this, and Jack thought this, and such-and-such said, but we said…and then
we changed our mind’. So they don’t just say, ‘this is the right one’. (Debra, 4, p.3)
She was keen to model the process for the children first in a whole class situation.
The video snippet from the previous lesson showed three girls discussing a problem so
she decided to use that to also model what she wanted.
I don’t want to leap in too much, that’s the only thing, but if we could give them an example of
the kind of thing that I wanted… give them something that is wrong…and I’ll say, ‘is it right,
why isn’t it right’. And so they start to do discussion … (Debra, 4, p.4)
Her agenda was to shift the socio-mathematical norms in terms of what constitutes
‘doing mathematics’.
… I’m looking for … what’s the mathematical thinking … so children can verbalise what’s going
on up here, what they’re thinking, rather than just doing it, which a lot of them do, just do things.
And to start to questions themselves, why am I doing it, is it right, what else can I draw on, the
supports … so they become independent thinkers. (Debra, 4, p6)
She reiterated her earlier concerns with the NNS teaching model where coverage
was a priority.
I think your main concern is going to be time and you’ve got to get through that much, you know,
do you have time to stop and talk about it ... But it’s realising the benefits of what the children
gain from using discussion in the classroom. (Debra, 4, p.7)
In her review of the project transcripts Debra again mentioned “how we are
continually obsessed with covering the strategy and time limitations” and that this has
a profound influence on the possibilities of discussion in the classroom (Debra, 8,
6:55).
Debra was looking for a strategy to get the children to unravel the thinking behind a
misconception. The researcher suggested:
…you could say, ‘there is a wrong answer, now why did the person think this, because there is a
good reason behind it. How can you get into the head of someone who was convinced that this
was the right answer?’ (R1, 4, p.8)
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Debra decided to put the children in the personal role of ‘teacher’ to unravel the
understanding behind the incorrect response.
Or, ‘if you were the teacher, what would you say to make the child realise’, you know. (Debra, 4,
p.8)
Debra reflected on her first discussion lesson in a meeting with one of the researchers
five days after the event (transcript 5) and also in the final group meeting (transcript 6).
She said that she had been very apprehensive before the lesson.
… I was very apprehensive … I had no concept of how the lesson was going to go. I knew what I
wanted to do, I knew the structure of what I wanted to do, but I had no idea of actually how the
lesson was running and how the children would respond. (Debra, 5, 009)
Debra felt that her discussion lesson was very different to her normal practice using
the NNS framework.
(This is very different) to anything I’ve done … the actual set up of it was very different to what
I’d done…actually taking away the National Numeracy Strategy, and just going with
mathematical discussion … (Debra, 5, 020)
She identified what was essentially different.
… originally I saw it as a very separate lesson, but when I look back on it, I thought it’s actually
not. The only thing that was different was, at no point did I say to the children, ‘right to
consolidate your understanding I want you to do this activity’… it was all children just talking to
each other and thinking out loud, and showing each other what they understood just through
talking … (Debra, 5, 043)
Debra had considered the possible social interactions in this first lesson. She had
organised the children into new groups, deliberately putting them with at least one
friend who they “could work with, if you got stuck you could discuss problems with…
you feel comfortable working with…” (Debra, 5, 120). She had decided that she
would move from group to group and refrain from too much intervention “letting their
discussion go the way it would” (Debra, 5, 135).
She had used the research-based materials – discussion prompts, group response
sheets, and group reporting – but had added to them and personalised them in creative
ways. Discussion was modelled using a video snippet of children from the previous
lesson discussing a ‘problem’. The new problem for the discussion groups was to
unravel an error that a fictitious child had made when ordering decimals. The
children’s role was to be ‘teacher detectives’: “if you were the teacher, what would you
do to help this child understand?” (Debra, 5, 244). One child in each group was
assigned as scribe, to “keep everybody together or to have an idea of when things were
floundering” (Debra, 6, 12:00). The whole group reported their work in the plenary.
The children were told to give themselves personal thinking time, to jot ideas down on
their own individual sheet before starting group discussion. They were also told when
to start organising their group report.
Debra reported that she had many surprises and some shocks during the lesson. For
example, one girl who was usually quiet and hid in maths responded enthusiastically in
discussion (Debra, 5, 154), a usually articulate boy did not take over his group (Debra,
5, 184), and some groups had not worked out the ‘correct’ answer before unravelling
the misconception (Debra, 5, 370).
Case 2: Kate
Kate planned her discussion lesson around the structure of the NNS framework but
described it as using “a more limited (teacher) input”. She also re-mixed the groups
from same ability groups to mixed ability groups. She used problem-solving and set up
a competition between groups.
Kate reported that moving from setted groups to mixed ability groups had been
problematic. The socio-cultural norm had been disturbed.
And I had a problem with the high ability children. (They) didn’t want the low ability children in
their group because they thought they wouldn’t get through the problems as quickly, um, they
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were spoiling their group. And then I had the middle and low ability complaining they didn’t
have enough high ability children in their group. (Kate, 6, 24:55)
She was surprised that generally the high ability children did not support other
children in their group. She had hoped that they would take on a teacher’s role.
I don’t think the mixed ability groups worked in the way that I wanted, but the high ability
children weren’t supporting the others enough in their understanding, and they were trying to get
through the problems rather than help each other. (Kate, 6, 27:10)
However, there were some groups that had worked well.
There was one little girl who took real control of the group … she managed to engineer it so that
everyone was having a say (Kate, 6, 27:54)
The setted groups that had been established for the year had served to form strong
friendship bonds and Kate felt that the re-mix had proved generally unsuccessful. She
also felt that the competition had not engendered the right atmosphere. She decided to
revert to the setted groups for the next lesson.
After the lesson Kate discussed the problems of social interaction with the children
and confronted them with the need for groups to work cooperatively. Some of the
children articulated what role they had taken.
And they recognised that themselves … (one boy said) well I tend to just do the work and tell
people what to do. (Kate, 6, 36:00)
Kate reported that there was a change for her in moving from the NNS to a
discussion lesson but that this was not a bad thing.
I did feel I had to change the way I taught to fit the project but I didn’t think that was necessarily
a bad thing … it was at odds with my conception of the NNS as a numeracy hour should be
taught … and promoting discussion didn’t really lend itself to that structure … It is more
developing a skill rather than developing mathematical concepts that I could teach the children. I
didn’t feel I was actually teaching the children … I felt it was more them coming to their own
understanding of how to solve the problems … I thought they lent themselves to it better… think
they coped very well with it. (Kate, 9, 7:04)
She decided that the problem-solving context had not been targetted to present a
difficulty which would promote a discussion (Kate, 6, 38:44). She felt that
differentiated tasks would be more appropriate (Kate, 6, 33:40).
Kate’s confidence in using discussion was growing and she was prepared to
experiment with the social organisation of her setted groups.
I would use it more regularly now. I am more confident with using discussion in maths now, I
would say. But I would still experiment more with different groups and whether it’s better to do
mixed ability or paired work or whether it’s better to keep in the group work but in their own
ability. (Kate, 9, 5:15)
She also raised the issue of teacher intervention when children’s discussion stalled
or was going in the wrong direction. She and the children were aware of time
constraints of lessons.
… It’s knowing when to intervene … when the discussion needs to be steered in another
direction … we were all conscious of the time and the need to re-focus in a way. (Kate, 6, 42:00)
She identified the need for children to practice different roles within their groups.
… with practice maybe they could take on that role, one person in the group could say, ‘take a
step back, and say hang on a minute, why don’t we try it this way?’. But it’s them recognising
that as well. That would come from practice. (Kate, 6, 43:50)
Kate decided that the children needed a structure for the recording of their group
work.
Perhaps I didn’t make it explicit enough as to how they should record their workings. (Kate, 9,
5:15)
I hope they were learning from each other ways of recording … the top people can record their
work … and they can learn that it is OK to make a mistake as well. And to jot their ideas down
and if it’s wrong, it’s wrong. (Kate, 9, 6:09)
Like Debra, Kate was attempting to shift the socio-mathematical norms within her
class. She had also identified the need to develop strategies, either for herself or for the
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groups, for deciding what to do when an impasse was apparently reached, resonating
with problem-solving strategies for ‘what do you do when you are stuck?’
Conclusions and Discussion
The research project became an investigation of what mathematical discussion could
be: a study of two teachers defining ‘mathematical discussion’ of misconceptions and
errors in their classroom alongside researchers. An important finding is that the
original research agenda needed to be ‘slowed’ down so that the teachers could start
from a secure position of existing practice and adapt it confidently to their own vision.
Researcher/teacher collaboration, as in Conle’s (2000) research projects, was seen
to be a productive model for professional development. The researchers as well as the
busy teachers were becoming acquainted with ‘their own story’.
The speculative beginnings of the research project served to highlight the need for
teachers to control their own development in a creative way. The research project
presented a somewhat open agenda to the teachers and they reported that this was a
refreshing and unusual approach in terms of their professional control. Their
experience was of complete packages of exemplary practice that they were to
disseminate to their colleagues: “this is what you should do!” (Debra, 8, 6:32).
The teachers’ accounts of and for their practice reveal explicitly the impact of the
culture of the National Numeracy Strategy. Significantly time is an overwhelming
constraint underlying the imperatives of coverage of curriculum or the need to ‘get
through’ objectives. These imperatives were reported repeatedly by the teachers and
presented some tensions for them in the development of collaborative discussion.
The teachers reported that they were sometimes seeking explanations and
justifications from the children in whole class interaction, yet the time imperative
requires that you do not get side-tracked from your objectives by the children’s
responses. “(B)ecause of the Strategy, you know it is wrong, but at the same maybe it’s
not” indicates a conflict between being a strategist and being a teacher.
The time structure of the NNS appeared to provide a strong framing of practice and
was seen to be an initial impediment to discussion.
I would say that my understanding was, there was a definite ten, twenty, twenty, ten structure and
promoting discussion didn’t really lend itself to that structure. (Kate, 9, 7:07)
It may be that ‘time’ signifies structure, which in turn infers a pedagogy that must
fit the time/structure. The NNS may be understood perhaps to have packaged
‘pedagogy’: getting through objectives, covering the strand, using consolidating
activities, not getting sidetracked, knowing what children are learning. Kate suffered
some anxiety about whether she was ‘teaching’ when group discussion was happening
– she reported feeling at odds with how a NNS “numeracy hour should be taught”.
There are auditing and accountability influences here.
Similarly for children the socio-mathematical norms of the NNS were prevalent: get
the right answer, finish tasks by the end of the lesson, and record your work. Some
children felt the need to explain why they had made seemingly little progress on a
worksheet by saying “we were trying that one for ages” – they had spent considerable
time discussing an early problematic question with no written record of productive
activity, yet their discussion had been persistent and had demonstrated considerable
and critical thought.
The NNS presented a strong and secure frame for the teachers’ practice and it was
one they had mastered with considerable public recognition. The shift from delivery of
curriculum to placing children’s ideas centre stage through collaborative discussion
was reported as a considerable pedagogical jump by the two teachers in this study –
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they often mentioned how different the practice seemed to be. The compatibility of the
new with the established practice was not (yet) achieved.
Socio-cultural norms relating to within-class grouping appeared to have an impact
on the possibilities for discussion. Some children, used to being in the ‘top’ ability
group within their class, were resistant to mixed ability grouping for discussion.
Friendship bonds and imperatives to succeed within the lesson time frame were strong.
This was identified by the teacher and she has plans to change this behaviour.
The need to close discussion or complete the lesson within the ‘hour’ also reflects a
unitising of teaching and learning episodes within the NNS in an accountability culture
where assessment is a driving force and the checking off of objectives monitors
learning.
A critical breakthrough moment for development of discussion was recognising
existing practices of children’s discussion within their classroom. Debra’s delight in
watching her children’s persistence with a problem, their error, and their sharing of
ideas was palpable. This video snippet proved to be the inspiration for her development
of children’s collaborative discussion of errors.
The teachers in this study identified these tensions associated with the NNS lesson.
In different ways they were stretching the boundaries of their current practice and
productively compared their ideas and strategies with each other. They were
enthusiastic and reported satisfaction in terms of their own professional development.
They are very keen to continue with the project and to involve a larger group of
teachers in several schools.
Both teachers were confident that the children were learning a new skill and that
they learned more about the children’s understanding within the new frame. However,
Debra noted that the current culture of “hard evidence and proof” (Debra, 8, 10:23) for
learning would need to be confronted if a discussion methodology was to be widely
accepted. This may reflect the need for official sanction or imprimatur for teaching
methods or it may reflect a sensible scientific stance when considerable time and effort
is required to imbed innovative practice. Assessment too is a critical issue: How does
the teacher know what the child has learned from a group discussion? These are ideas
for future consideration.
References
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Transcripts
Quotations throughout are coded as (speaker, transcript no., time/counter/page)
1: First whole group meeting
2. Debra , R1 and R3
3. Second whole group meeting
4. Debra and R1
5. Debra and R1
6. Third whole group meeting
7. Kate, R2 and R3
8. Debra’s written comments on transcript 2
9. Kate and R2
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