ORIGINAL ARTICLE

Theories in practice: mathematics teaching and mathematicsteacher education

Stephen Lerman

Accepted: 2 May 2013 / Published online: 10 May 2013

� FIZ Karlsruhe 2013

Abstract Whilst research on the teaching of mathematics

and the preparation of teachers of mathematics has been of

major concern in our field for some decades, one can see a

proliferation of such studies and of theories in relation to

that work in recent years. This article is a reaction to the

other papers in this special issue but I attempt, at the same

time, to offer a different perspective. I examine first the

theories of learning that are either explicitly or implicitly

presented, noting the need for such theories in relation to

teacher learning, separating them into: socio-cultural the-

ories; Piagetian theory; and learning from practice. I go on

to discuss the role of social and individual perspectives in

authors’ approach. In the final section I consider the nature

of the knowledge labelled as mathematical knowledge for

teaching (MKT). I suggest that there is an implied telos

about ‘good teaching’ in much of our research and that per-

haps the challenge is to study what happens in practice and

offer multiple stories of that practice in the spirit of ‘‘wild

profusion’’ (Lather in Getting lost: Feminist efforts towards a

double(d) science. SUNY Press, New York, 2007).

1 Introduction

The expectations on authors contributing to this special

issue as set by the editors are to address at least one of what

seem in general to be separate lines of inquiry (areas of

study) within the field of mathematics teaching and teacher

education research: teacher knowledge; beliefs and affect;

and identity. Authors are expected to clarify their position

in relation to whether they work with a social theory or an

individual one, and also where they stand on the relations

between theory and practice. Finally they are asked to take

into account the impact of their theoretical and research

focuses on instruction and student learning. The set of

papers here indeed fulfill the expectations. They are all the

work of established and creative researchers in the field of

mathematics teaching and teacher education and have

responded to some, or in one or two cases to all, of the

challenges set by the editors of this special issue. I am

pleased to have been invited to write this response given

the quality of the contributions and because of my on-going

engagement with research in this area since my doctoral

dissertation work almost 30 years ago.

My response will mirror the special issue goals to a large

extent. In the first section I will discuss the theories of

teacher learning that are taken up in these papers, as I

consider this to be absolutely key to discussions about

mathematics teaching and mathematics teacher education

in particular, and of what is understood in these papers by

‘teacher knowledge’. The authors’ positions in relation to

the three lines of inquiry will be addressed in that section.

The following section will examine the researchers’ posi-

tions on whether they work with a social or an individual

perspective. Subsequently I will discuss perspectives on

research in mathematics teaching and teacher education,

taking up the relationship between theory and practice.

2 Teacher learning and teacher knowledge

In a paper on teacher learning (Lerman 1997) I argued that

much research talks in terms of teacher development or

teacher change, rather than learning, and I suggested that

perhaps this was a consequence of a lack of theories for this

S. Lerman (&)

Department of Education, London South Bank University,

London, UK

e-mail: lermans@lsbu.ac.uk

123

ZDM Mathematics Education (2013) 45:623–631

DOI 10.1007/s11858-013-0510-x

field. Most of our work at that time had been on students’

learning and we had strong and well-developed theories,

from Piaget, Vygotsky, and others. However we had not

addressed how adults learn and in particular how teachers

learn. Teaching is very different from most fields of

inquiry, having a face towards practice and a face towards

theory. Whilst we draw on psychology, for example, we

recontextualise from that field into the concerns we face in

education, and our concern is with how research can affect

and perhaps improve practice. The process of selection in

recontextualisation is always value laden. Furthermore a

discussion of theories of teacher learning cannot be

divorced from the nature of what constitutes ‘teacher

knowledge’. If we think in terms of mathematical knowl-

edge we must note notions such as internationally accepted

and agreed conventions, certainty, applications, and so on.

Is mathematical knowledge for teaching (MKT) or the

other equivalent names of the same kind? I think we have

to agree that it is not. It does not carry the same universality

or certainty or universal agreement as mathematical

knowledge. Perhaps one of the challenges for us as a field

is that we deal with the certainty of mathematical knowl-

edge at the same time as the uncertainty of pedagogic

knowledge and perhaps tacitly expect the latter to be more

like the former. I will return to this issue in my final

remarks.

The authors in this collection all engage with theories of

learning but in different ways. I have separated the authors’

approaches into three sub-sections: socio-cultural theories;

Piagetian theory; and learning from practice. The papers

are not equally distributed between these, with the majority

being in socio-cultural theory, perhaps because there is

such a range of theories under that umbrella, or perhaps

because the notion of identity, common amongst them, is

so elusive. There is also some overlap as many refer to the

connections between theory and practice. Nevertheless,

grouping the papers in this way offers a useful perspective,

I believe.

2.1 Sociocultural theories

Sociocultural theories start from the assumption that tea-

cher learning is about changing participation in social

practices. In this collection the authors draw on activity

theory (Potari 2013); zone theory (Goos 2013); identity

development in social contexts (Gellert, Espinoza and

Barbe 2013); patterns of participation (Skott 2013); and

shared participation in renewal (Brown, Heywood, Solo-

mon and Zagorianakos 2013).

Potari sees the adoption of an informed stance on using

research as a tool for inquiry as potentially generating

‘professional learning’ amongst teachers. I take the term to

refer to the two faces of educational research that I

mentioned above, towards practice and towards theories.

Based on a study of secondary mathematics teachers and

recent graduates of mathematics all undertaking a master’s

degree, Potari shows how the different practices in which

they engaged, with research through readings and with

practice through reports on their teaching, could be ana-

lysed through third generation activity theory. Activity

theory enables a reading of activity that highlights con-

tradictions and identifies which elements in the activity, be

they the rules or the community or others, are generating

these contradictions for the participants, leading to devel-

opmental cycles of activity. The analysis showed, accord-

ing to Potari, that the inquiry process generated a deeper

‘‘understanding of their students’ thinking’’ amongst the

teachers, who started from practice when engaging with

research, and that the novices saw the research as some-

thing to be tested and verified by their practice in schools.

Whilst one study cannot be expected to stake a strong

claim for any particular approach, as a community we do

need to worry about sustainability when activities take

place through participation in University courses. The

discussion with the teachers near the end of the paper

highlights this concern. Third generation activity theory

has been shown here to be useful for the researchers to

identify the motivators of shifts and challenges but many of

Engestrom’s studies have engaged the participants in

explicit use of the theory too, though this does not appear

to have been the programme of the researchers. I wonder

whether Leont’ev’s second generation theory might be

useful here, as participants might find the levels of opera-

tion, action and activity useful in seeing how they them-

selves engage with the activity of planning, teaching and

reflecting (Jaworski et al. 2013).

Goos has carried out substantial research in recent years

extending Vygotsky’s zone of proximal development

(ZPD) by drawing on Valsiner’s additional two zones, of

free movement (ZFM) and of promoted action (ZPA). She

argues that productive tensions can emerge ‘‘between

teachers’ thinking, action, and professional environments’’

and that these ‘‘tensions can become opportunities for

teacher change’’. Goos presents a convincing demonstra-

tion, in the case of Brian, of someone whose reading and

discussions with others led him to change aspects of his

practice that Goos characterises in terms of his ZFM and

ZPA. It seems that Brian saw these changes as advancing

his knowledge and understanding and benefiting the stu-

dents through their learning of mathematical concepts and

not of memorising, and forgetting, techniques.

What is important here, for me at least, is that Goos

supplies a material indication of what teacher learning

means in terms of changes in the zones, a practice-based

source of evidence of learning, as described by both the

teacher and the researcher. I use the term ‘material’ in the

624 S. Lerman

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sense that Vygotsky used it, a Marxist idea that conveys the

connection between activity and learning, the goal orien-

tation of all learning, and its observability. I should add that

there is no evidence in the paper that Brian has access to

the language of the zones. It would be interesting to know

if he has acquired that language. Goos also writes in terms

of Brian’s past being brought through to his present, and

his present being projected forward to his future. This

approach draws also on a materialist analysis that identifies

change and defines learning in those terms (see also Meira

and Lerman 2009) and I appreciate greatly the perspective

the author offers. Although some aspects of the environ-

ment in which Brian, and all teachers, work are mentioned

here, a stronger sociological element to the analysis is

perhaps also needed here I think. The ZFM and ZPA

analysis enables a conversation regarding the teacher’s

choices and actions but the implications for students of

differing social groups needs also to be accounted for,

given the focus on addressing practice in this special issue,

and for that we need sociological theories (see e.g. Cooper

and Dunne 2000).

This last point brings me to the paper by Gellert, Espi-

noza and Barbe, in which they address directly the teacher

in social context in its widest sense. Gellert et al. see

identity development as a critique of studies of the idea that

teachers’ practices are an externalisation of their internal

states of knowledge and beliefs. They set out to show,

through the story of a specific teacher, how identity

develops in social contexts and they elaborate what is to be

understood as those contexts and what they produce in

terms of identity change. The authors address the call from

the editors of the special issue to be aware of which of the

three lines of inquiry they are addressing, knowledge,

beliefs or identity. As for many of the papers in this

excellent collection Gellert et al. do not want to separate

them, and they provide a table that summarises the

dynamics, in terms of mode and telos, of each of these lines

of inquiry. I find this table to be very fruitful in enabling

readers to see what each calls for in terms of approach and

in terms of research. Too often, I feel, the telos of an

assumption in this area is left unexamined. ‘Knowledge’

suggests a body of stuff that has to be acquired and as I

have argued above, mathematical knowledge may be the

paradigm for all sorts of fields of inquiry, including Pia-

get’s, but it may not be appropriate for teachers’ knowl-

edge of the teaching of mathematics. Gellert et al. call this

telos ‘sufficiency’. Similarly the identification of changing

teachers’ beliefs has a telos of ‘properness’.

Decades of study, including my own doctoral thesis

(Lerman 1986), assumed that teachers need to change their

practices without recognising that there was an implied and

value laden direction of that change (see also Lerman

2002). Identity has a telos of ‘fitting’ to a situation, which

opens up the analysis to the social context, powerfully

described by Stephen Ball’s notion of performativity, taken

from Baudrillard, but also potentially to ‘‘resistance and

subversion’’. Of course, those contexts can also include

inspiring in-service courses, not just the constraints of

political demands, often fuelled by the desire to emulate

the performance of other countries in international com-

parisons. The analysis of Claudia’s teaching and how she

sees herself in times of change is a creative demonstration

of how essential is the wide angle as well as the close-up

zooms of the researcher’s lens. ‘Identity’ is an over-used

concept, often taken in rather simplistic ways from

Wenger’s 1998 book. I believe more work needs to be done

by authors on how they take the concept into their research

and Gellert et al.’s embedding of their analysis in the lit-

erature of Black, Mendick and Solomon, in Bernstein and

in Stephen Ball is done well.

A final concern: the authors say, ‘‘In times of reform, it

is essentially the identity of teachers that is at stake’’. In

this sense the term ‘reform’, so ubiquitous in education

across the world, and certainly for us in mathematics

education, needs some examination. Re-form carries a

notion of going back to a preferred past or the construction

of a new form that is seen to be better than a past. All

visions of education, whether of the whole system or spe-

cifically of mathematics (or other subjects) education, are

value laden, and those values are often implicit. Examining

those values, what drives them, what they are reacting

against, can support an analysis of who might gain and who

might lose under that reform.

Skott notes the acquisition/participation duality and

rather than try for complementarity calls on an examination

of patterns of participation (PoP) in the range of social

practices with which the teacher engages as a way of

understanding teachers’ practices and the potential for

change. As in Gellert et al.’s paper Skott recognises the

need to account for the teacher’s negotiation across

sometimes conflicting practices and his PoP framework

aims to do that. He sees PoP as an approach that explicitly

moves on from, whilst not rejecting, the study of beliefs as

these are still seen, he says, as ‘‘a main obstacle to edu-

cational change’’. Skott argues that the notion of beliefs is

still confused with and across ‘‘knowledge, conceptions,

emotions and values’’, and sees them, drawing on Sfard, as

reified social experiences. Skott makes an important shift in

focusing on classroom practices as emergent and hence the

need to study how teachers’ practices change across time.

This dynamic process is studied through eliciting the tea-

cher’s interpretations of what she does in classrooms and

how that relates to her prior engagement in other social

practices.

As in other papers, Skott extends his analysis from

beliefs to deal also with knowledge and identity. The study

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of one teacher, Anna, provides exemplars for Skott of the

methodology of PoP, which involves the researcher’s

interpretations, consisting of quite strong inferences, of

these beliefs, knowledge and identity as they emerge across

time in teachers’ accounts of themselves and their actions

and of their learning. Perhaps the implications of the

interpretive stance and therefore for the interpretation that

is offered in his approach are not sufficiently recognised. I

consider, though, that the emergent approach, and the

search for methods that attempt to allow for dynamic

explanations of teachers’ changing actions over time, is a

most important move. In the spirit of the meta-analysis

called for in the expectations for this special issue it would

be interesting to see Skott’s response to the mode of

dynamic and telos of beliefs research proposed by Gellert

et al.

The paper by Brown, Heywood, Solomon and Zago-

rianakos brings the emergence and dynamic process to the

heart of their account of teacher knowledge and identity

and indeed of mathematical knowledge. The examples

given of a group of student teachers are very useful in

helping readers engage with the challenging theoretical

framework which draws mainly on Badiou, whose Lac-

anian inspired account of subjectivity leads the authors to

argue that, as a consequence of different pedagogical

attitudes, different mathematical objects can emerge in

shared situations, to the extent that sharing can take place.

They argue that we are in new times and prior approa-

ches, in which the teacher takes a privileged position in

relation to what students need to acquire and designs

ways of enabling their students to acquire that knowledge

and can assess its acquisition, are no longer adequate.

Their lovely example of students moving around in

relation to each other in order to experience from ‘inside’

the motion of the earth, moon and sun is not just about

learning about conics: ‘‘an object must be in the world for

it to exist’’.

In this sense an ontology of mathematical objects is

emergent as are the people engaged in experiencing the

creation of that world. ‘‘In becoming teachers they are

participating in the becoming of mathematics’’. For the

teacher and the students education is a perpetual renewal.

It is not clear to me how what we might describe as

Popper’s world 3 objects, the established and accepted

mathematical knowledge, appear and interact in the

vision of experience of mathematics provided in the

paper, but that might be my lack of imagination, coming

reasonably fresh to Brown et al.’s ideas. I have to add,

given the stranglehold of Government on teaching in the

UK, I worry for the frustration of these teachers as they

imagine what their own classrooms could be like, fol-

lowing their creative and imaginative course at the

University.

2.2 Piagetian theory

Simon (2013), whose contributions over many years to

perspectives on teaching, and to what constitutes con-

structivist teaching in particular, lead the field, acknowl-

edges the need for a theory of teacher learning explicitly. In

particular he points out how the reform revolution has in

fact left most classrooms and teachers largely unaffected

and we need to understand why that happens in order to

conceive of ways to change it. That involves understanding

how the learning process functions such that, when teach-

ers are confronted with new information, through research

texts or policy texts, or other means, they may articulate a

new understanding but their practice does not change.

Simon proposes the Piagetian contrast of assimilation and

accommodation as potentially offering a way of envisaging

change, or its lack, in teachers’ perspectives on mathe-

matical knowledge and on its acquisition by students. He

emphasises that the theoretical construct of assimilatory

structures is a researcher’s construction, not a claim of the

existence of such structures. As compared to other expla-

nations of the resistance of teachers’ practices to change I

find this very convincing.

Simon then demonstrates how a perception-based per-

spective can be seen to be a major assimilatory structure.

The researchers interpreted Ivy’s practice to be one

whereby she assumed students perceived the world in the

same way that she did, in particular in relation to how the

base-10 blocks activity was seen to be showing the same

concepts as the division algorithm representation. The

difference between Ivy’s perception-based perspective and

the researchers’ conception-based perspective highlights

how Ivy’s students had no way of seeing the parallels.

Simon’s caveat must be repeated here: these are research-

ers’ explanations or analyses and are not intended to claim

any actuality in Ivy’s cognitive structures, since the con-

structivist hypotheses about learning prevent the statement

of anything in the observed person’s conceptual frame-

work; one can only say what the observer/researcher con-

jectures. One should note the contrast with the materialist

one offered by the ZPD, ZFM and ZPA in the Goos paper.

We could go on to say that the students have to learn,

which they can only do for themselves, what representa-

tions are intended to say, what they are intended to rep-

resent. They are not obvious, though teachers may think

they are. Students are not going to have the same under-

standing of concepts and representations unless suitable

activities have been designed to enable them to construct

appropriate meanings. In other languages of learning one

would say: students are not going to have the same

understanding of concepts and representations as the tea-

cher unless they have been inducted into the accepted

meanings of mathematics. This induction process has been

626 S. Lerman

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interpreted in various social forms including: communities

of practice; zones of proximal development; the pedagogic

device; or Wittgenstein’s description of understanding as

being inducted into how concepts are used and being able

to go beyond. In Piagetian theory the individual comes to

understand through reflective abstraction the process of

reflection by the individual on their mental operations. For

Piaget ‘‘it [reflective abstraction] alone supports and ani-

mates the immense edifice of logico-mathematical con-

struction’’ (Piaget, 1980, p. 92).

To return to the teacher, how is Ivy to perceive the

difference between the perception-based and conception-

based perceptions, which if achieved would constitute

teacher learning and by extension student learning? Being

consistent in the theory of learning: what would trigger

accommodation; will sufficient experiences of mental

operations lead to reflective abstraction? The adequacy of

the extraction of the notion of assimilation from the Pi-

agetian model depends, for me, in regards to the concerns

of this special issue, on this question: Is teacher knowledge

of the same form as the immense edifice of logico-math-

ematical construction? Applying the notion of reflective

abstraction to how children develop multiplication is only

the same as how teachers develop new ways of seeing how

they might bring students to learn, that is, to teach, if the

body of knowledge about teaching mathematics exhibits

the same kinds of structures as mathematics itself: cer-

tainty, replicability, generality and the like. I don’t think

this is the case.

2.3 Learning from practice

The paper by Van Zoest and Thames (2013) calls for

studying teachers’ characteristics, their term for the three

lines of inquiry, through studying how these are realised in

practice because, they argue, all studies on teaching need to

be about the improvement of practice; that has to be the

rationale for the field of mathematics education research,

especially research on mathematics teaching and teacher

education. They examine four studies in a search to

exemplify their practice-based approach and then attempt

to look across three of the four to encapsulate how these

studies contribute to understanding and improving practice

whilst the fourth, although valuable in itself, does not as it

‘‘stops short of explicitly addressing the dynamics of

instruction’’.

Van Zoest and Thames start from the assumption that

teaching and teachers matter in terms of the effect on

student achievement but that the field does not ‘‘adequately

understand how and why’’. They use a notion from Ball

and Forzani, that of research in education, taken to mean

that instruction is the key variable. This requires a focus on

all three vertices of the familiar triangle of teachers,

content and students. Separating the three lines of inquiry

similarly weakens the likelihood of research having an

effect on practice. Their concern is with how these char-

acteristics impact on student learning and they are looking

to how practice-based approaches can generate research

questions that ‘‘would support the improvement of

practice’’.

It is important to the authors, as signalled in their title,

that researchers achieve coherence between research on

teacher characteristics and practice. Coherence is here

understood as researchers building on each other’s research

in the field of teacher characteristics in a concerted effort to

develop theory about teachers’ practices that can lead to

successful learning. They highlight the likely obstacles to

such coherence by examining the limitations of the sepa-

rate studies of teacher characteristics: that beliefs research

does not help overcome teachers’ resistance to changing

their beliefs; that the body of professional knowledge is not

yet sufficiently developed; and that identity research is too

distinct from the other two to support a coherent research

orientation. Their proposal is that the development of

practice-based approaches can provide this coherence. I am

convinced that ethnographic studies of teachers and

teaching and student learning are the appropriate starting

point for research on mathematics teaching and teacher

education, but I am not convinced that greater coherence

can emerge. I am not overly worried about this as I am

sceptical about the possibility that this body of professional

knowledge can be sufficiently developed as to constitute an

accepted body of knowledge. Researchers working on

practice-based approaches bring their values about good

teaching and need to make these values explicit. Of course,

these will vary across different researchers and so coher-

ence is not likely to be achieved, but those studies can still

serve as valuable insights into classrooms and teachers and

learning. I wonder, also, where teachers’ learning from

their practice might play a role in Van Zoest and Thames’s

work. I refer for example to Doerr’s extensive study (see

e.g. Doerr and Lerman 2010). Finally, whilst I welcome the

central focus on one of the faces in our research field, that

of practice, theories of what one is looking at both afford

and constrain what one sees.

Barwell (2013), in common with Brown et al. and

Gellert et al., sees his approach, that of discursive psy-

chology, as an explicit move away from the view that

teachers’ practices are the realisation of their mental con-

tents. Some other authors in this collection hint at a similar

perspective. Barwell goes further, and suggests that

research on the three lines of inquiry, or teacher charac-

teristics as Van Zoest and Thames describe them, have a

scientific, or we might say scientistic, orientation (Lather

2007), with associated expectations on truth. In line with

the philosophy underlying discursive psychology, he

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argues that all we have is teachers’ or others’ utterances

and an observer’s account of teachers’ practices. This is not

due to the simplistic argument that one does not have

access to what is in a teacher’s head when she is teaching,

being interviewed, answering a questionnaire or keeping a

diary. It is stronger than that and arises from the recogni-

tion that all research methods are productive of the data

that emerge. Someone else dong the interview, or carrying

out the interview in another location or on another day will

elicit different data. To expect universal truth to arise from

educational research, from this perspective, is to be

adopting a metaphysical approach that has been strongly

challenged in recent decades, although it could be argued

that this challenge has had insufficient impact on mathe-

matics education research, though more on educational

research in general (e.g. Popkewitz 2000; Scheurich 1997).

Barwell also presents a strong case in opposition to the

understanding of the nature of representations that is

widely held in our field. ‘‘Part of the process of pedagogical

reasoning involves teachers finding ways to represent the

knowledge represented in their minds so that students can

form their own representations’’. Simon also pins down

what the research community in general subscribes to in

relation to representation: ‘‘Ivy demonstrates that teachers

who assimilate the new tools into a perception-based per-

spective may view these tools as transparent representa-

tions of abstract mathematical relationships’’. In his

critique of Shulman’s and Ball et al.’s use of the idea of

representations Barwell does not quite go so far as to say

that the foundational assumption is that there is an essence

to mathematical knowledge, an absolute, abstract, defined

collection of objects and all external and internal repre-

sentations are partial images of aspects of those objects.

The postmodern argument would be that there is no

essence, but the collection of representations is all there is,

though ‘all’ can be very substantial and difficult. This is

how I read his argument, however.

This paper leaves the reader wondering what next,

though. Barwell’s approach shifts us into epistemology and

away from ontology and gives us a way to read classroom

interactions in a highly nuanced way. I agree, and he adds

to the argument I have been developing above that calls for

starting from ethnography. But we need to be able to

articulate what these analyses say about and for teaching,

even if we leave behind the expectation that what they say

will contribute to an accepted body of knowledge about

mathematics teaching and teacher education.

Schoenfeld’s paper (2013) is quite different from the

others in this collection. Its aim is to produce, from the

decades of experience, experimentation and writing that

precede this work, and supported by substantial research

grants, ‘‘a workable, theoretically grounded scheme for

classroom observations’’. He draws constantly on research

findings in constructing the scheme, and its programme is

to ‘‘construct a model of an individual’s decision-making

that is entirely consistent with the individual’s behaviour

on a moment-by-moment basis’’. This huge and ambitious

programme is developed in stages by successive refine-

ments as the researchers seek to make the scheme workable

and consistent, and both necessary and sufficient. The

paper takes us through the stages of development of the

scheme, to the current model. That the scheme is called

TRU (Teaching for Robust Understanding of mathematics)

is entirely appropriate to the goals and to the expectation of

the outcome. There is no space in the paper for an elabo-

ration of the team’s and author’s philosophy in relation to

what is being produced, there is too much to report on here;

I expect this could be the subject of another paper. I wel-

come, however, the three observations that appear in the

discussion and speak to researchers who might work with

this model: that different constructs in the scheme may

appear to play more or less central roles, depending on

focus; that research should always involve a to-and-fro

between theory and data; and that a multiplicity of methods

and perspectives should be brought to bear on ‘what

counts’. From this side of the Atlantic, though, I fear that

schemes such as this, in spite of the purposes of the

researchers and developers of the scheme, will be used to

test teachers and become a tool for control. The experience

and wider vision of the developers, to develop a scheme for

classroom observation, can easily get lost in prescriptions

and measures to pass or fail teachers. That should not stop

the research and development work as presented here,

however.

3 Social and/or individual perspectives

In a chapter published in 2000 (Lerman 2000) I argued that

there has been a turn in perspectives in education in general

and in mathematics education too, towards a social orien-

tation. It seems, however, that there is a common misun-

derstanding of the social turn, that it addresses the social

context of teachers, for example, but if one wants to

examine what teachers actually do or know, researchers

need to take a cognitive focus, by which is meant an

individual cognitive focus. There may also be a suggestion

that the analyses would be complementary.

However, as I wrote in a recent paper (Lerman 2013,

p. 40):

Whilst Vygotsky undoubtedly took a more cognitive

focus than many believe, the key point of his work

and that of his followers is that the origins of cog-

nition are, for them, cultural-historical in contrast to

the Piagetian-inspired individual origin. It makes

628 S. Lerman

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little sense then, to distinguish between the cognition

of the individual and the socio-cultural context.

Knowledge is first on the intersubjective plane and

then subsequently on the intrasubjective plane. The

unit of analysis has to include, therefore, the subject,

the situation, the task, the activity, the teacher and the

students, if not beyond the classroom as well, for

these constitute the intersubjective plane. For

research purposes we may choose a particular focus

but this merely means the other elements of the

intersubjective, cultural-historical context are out of

focus, or backgrounded, not absent (Lerman, 1998).

Developments in theory over the last few decades,

from Wittgenstein to postmodernism, have led to

social science research in general and much mathe-

matics education research in particular taking a social

turn (Lerman, 2000). The socio-cultural is too often

defined as social class, ethnicity and political issues

and then separated from what matters, learning. I am

arguing that this makes no sense in Vygotskian

theory.

Although the editors called for authors to clarify their

position in relation to whether they work with a social

theory or an individual one, few are explicit about their

positions. Potari argues that activity theory ‘‘can offer us a

tool to relate individual and social perspectives in research

focusing on mathematics teacher development’’. Simon

identifies two complementary ways of studying changes in

teaching: a social perspective examining the context of the

teacher’s work; and a cognitive perspective ‘‘focusing on

teachers’ conceptions, including conceptions of mathe-

matics, teaching and learning’’, and he chooses to focus, in

his paper in the collection, on the latter. Skott emphasises

his use of aspects of social theories and Goos confirms her

commitment to sociocultural research in the sense I have

described above in the lengthy quotation. Brown et al.,

Schoenfeld, Barwell, and Van Zoest and Thames do not

make any overt commitment on their position, though

readers can infer where they stand. Gellert et al. draw on

sociology of education, specifically Basil Bernstein’s work.

Their main argument is that ‘‘Formation of professional

identity has to do with developing a fitting to a situation, of

relating individually centered social processes to the

socially constructed institutionalized world’’. Again,

although no explicit position is stated in the position of

Gellert et al. it is clear where they stand.

Given the historical influence of cognitive theories

drawn from individualistic perspectives on our field I think

it is important for researchers to make clear how they

conceive of the relationship between the social and indi-

vidual as this gives the reader a sense of how they approach

the process of learning, whether of students or, as in this

case, of teachers. Units of analysis emerge also from

considerations of the relationship between the social and

the individual. Person-in-practice or second order models

of a student’s or a teacher’s model, for example, are quite

different units of analysis in use in our field, and the out-

comes in terms of conjectures about knowledge and

understanding are quite different too.

4 Researching mathematics teaching and teacher

education

The special issue editors hoped that the papers would

‘‘initiate a meta-discussion about the relationships between

the three lines of inquiry in order to take the general field

forward, and investigate the possibilities—or lack

thereof—for more coherent approaches to the field.’’ I

suppose one of the roles of the reactor to this set of papers

is to add to the initiation of a meta-discussion. It is cer-

tainly not easy to set up a meta-discussion within a set of

papers reporting on each author or authors’ research

(though see, e.g., Educational Researcher 38(7), Oct 2009)

but I will try to facilitate it by commenting on the rela-

tionship between theories and practice in the sub-field of

mathematics teaching and teacher education, as this reflects

back also on the three lines of inquiry. I will also comment

upon the extent to which the papers take the field forward.

As I have signalled above, theories about practice both

constrain and afford what one sees and how one interprets

what one sees. The papers by many of the authors indicate

clearly how they see their theory or theories informing their

research. The process of recontextualisation that always

takes place in moving concepts from one practice, where

they have perhaps originated, to another is subject to values

and ideals. In addition, as the term implies, it is also about

the new context and how it is perceived by the person

stepping into that context, as Ensor (2001) shows in rela-

tion to pre-service students and their move into schools and

Gellert et al. show in this collection in relation to Claudia. I

would suggest that these papers indicate the need to work

at one’s theoretical perspective. The lens through which a

researcher organises their research, reads the data, revisits

theory and interprets the findings is critical, and without

such work the values of the researcher are hidden but never

absent, of course. One learns from the scrutiny and critique

of colleagues.

Elsewhere I have expressed concerns about issues of

complementarity, incommensurability and design science

(Lerman 2010) and I hope I have made clear the need for

work on theory. Skott is explicit about not trying to con-

struct a complementarity across the acquisition/participa-

tion dichotomy and Simon similarly for the social/

individual dichotomy (though I have expressed

Theories in practice 629

123

reservations above); Brown et al. take the philosophy of

Badiou and recontextualise it for their purposes in an

educational setting; Barwell remains in the domain of

analysis of text where critical psychology works; Potari,

Goos, and Gellert et al., explicate and then apply theories

to their studies; and Van Zoest and Thames are eclectic

about theory, their concern being a focus directly on

practice. As I have said, Schoenfeld’s paper is rather dif-

ferent in relation to theory and practice. Perhaps what one

can take from all these papers concerning the three lines of

inquiry is that they need theory, whether the aim is to bring

them together or to make clear the different approaches

called for in their study.

There is a case for saying that teachers can and do learn

about their teaching and much research points to it (e.g.

Doerr and Lerman 2010). As Simon signals in his discus-

sion of assimilation and accommodation, it may well take

place only when something brings about a disequilibrium

for the teacher, such as a critical incident, a comment from

an observer, or from engagement with new ideas from

others, perhaps through further study or participation in a

research project. Except perhaps for lesson study, there is a

worrying unsustainability of investigation by teachers of

their teaching beyond engagement in research projects or

higher degrees. The challenges of the context of education

in so many countries around the world, contexts which are

productive of teacher identities as teachers seek survival in

ever-changing regimes of prescription, goes a long way to

explaining the resilience of assimilatory structures.

The hope that these papers and the meta-discussion it

generates will ‘take the general field forward’ begs the

question ‘where is forward’? What is the telos implied by

any author’s use of the idea of progress? I feel sure that

research to understand the work of the teacher, ethnog-

raphies of the classroom, as seen here in many of these

papers, through suitable theoretical lenses articulated by

the researchers, is ‘the way forward’ but in a local sense. I

am not optimistic about a global way forward, as will be

obvious from my comments throughout this response

paper. Contexts, values, cultural traditions and so on are so

varied across the world that acceptability of knowledge of

mathematics education is not the same as acceptability of

groups, rings and fields. Human meaning-making and

action are evolving, emerging, dynamic; this theme comes

through explicitly, if in different ways, in Skott, in Gellert

et al. and in Brown et al., and implicitly in the others. As

Kanes and Lerman (2008, p. 343) suggest, ethnography in

research studies in mathematics education, whether of

school students or of teachers, ‘‘is to trace the actual tra-

jectories around learning within communities of practice

and drawing on this to speculate on hypothetical trajecto-

ries in more general settings.’’ That is perhaps as far as we

can go in fulfilling the expectations of the special issue, but

it is still quite a challenge.

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