The Unit of Analysis in Mathematics Education: Bridging the
Political-Technical Divide?1
Paul Ernest
University of Exeter
ABSTRACT
Mathematics education is a complex, multi-disciplinary field of study which treats a
wide range of diverse but interrelated areas. These include the nature of mathematics,
the learning of mathematics, its teaching, and the social context surrounding both the
discipline and applications of mathematics itself, and as well as its teaching and
learning. But research and researchers in mathematics education fall loosely into two
camps: On the one hand there is technical research, drawing on mathematics,
psychology and pedagogy, concerned with narrow questions about the teaching and
learning of mathematics. On the other hand there is political and social research drawing
on sociology and philosophy, addressing large scale problems of social consequence.
These two camps tend to draw on different theoretical underpinnings as well as having
different interests. Is it possible to find a shared theoretical element, a single unit of
analysis for mathematics education which provides a unified approach to both analysing
and explaining all of these diverse aspects? Can such a unit provide a bridge across the
technical-political divide? Before these questions are addressed the meaning of the term
„unit of analysis‟ is clarified. After distinguishing between methodological and
ontological senses, I propose units of analysis within each of the four listed subdomains
of mathematics education. Then drawing on Blunden's (2009, 2010) interdisciplinary
version of cultural historical activity theory I propose a single over-arching unit of
analysis, the collaborative project, for the whole field, thus potentially bridging the
technical-political divide.
Mathematics Education
What makes up the field of mathematics education as a research domain? A good starting
point is Joseph Schwab's (1961) four commonplaces of teaching. These include the subject
taught, the learner, the teacher, and the milieu of teaching, comprising the relationship of the
subject, and its teaching and learning aims, to society in general. Applying these four
commonplaces of teaching to the field of mathematics education results in the specification of
four domains of theory that mathematics education as a research field must accommodate.
These are:
1. An account of the nature of mathematics (philosophy of mathematics),
2. A theory of learning mathematics (psychology of learning mathematics),
3. A theory of teaching mathematics (mathematical pedagogy),
4. A theory of the social context of mathematics education, incorporating the
interpersonal, cultural, social, societal and political milieu of mathematics education in
its theoretical and practical forms (sociology of mathematics).
This is a tentative listing because there might well be further subdomains of mathematics
education beyond those identified here. However, many of the issues not explicitly
mentioned, such as the aims of teaching and learning of mathematics, the mathematics
curriculum and its assessment systems, and issues of diversity defined in terms of class,
1
The (first) definite article in this title is intended to designate the topic of discussion as specific singular area,
and is not intended to imply that the unit (or units) of analysis proposed are unique or constitute the only viable
constructs.
ability/achievement, gender, race and culture, can in my view be treated under the fourth
heading, the social context of mathematics education.
Mathematics education is a newly emergent field of study in social science research that is
strongly influenced or even shaped by adjacent disciplines and practices. Top-down,
mathematics and psychology were the original parent disciplines and continue to be very
influential. Bottom up, pedagogical practices, and in particular the various technologies used
in mathematics and its teaching, have also been very influential.2 Thus a good deal of research
is based on detailed mathematical or psychological analyses of the teaching or learning of
mathematics, or on detailed studies of the effects of technologies, including information and
communication technologies. Such research is in the main detailed and technical, focusing on
scientific studies of the teaching and learning of mathematics, that is on the psychology and
pedagogy of mathematics. Typical theories employed have been neo-Piagetian and
constructivist, as well as process-product experimental designs within the scientific research
paradigm. There have also been detailed constructivist studies of individual learners and
groups of students within the interpretative research paradigm. My point is that such studies
are largely detailed and technical and mostly only refer to social contexts as secondary aspects
of the research insofar as they impinge on the main focus of study. Some studies are now also
using Vygotskyan and „community of practice‟ theories, so there are greater possibilities of
incorporating social aspects of learning, but these are mostly local interpersonal aspects of the
social.
In contrast, an area of research that has been growing over the past two or three decades
comprises studies primarily directed at the social and political aspects of mathematics
education. This includes studies of gender, race and class in school mathematics. It also
includes the aims and goals of teaching mathematics. Looking critically at aims and indeed at
the discipline of mathematics itself brings in philosophy: philosophy of education and
philosophy of mathematics. Thus typically sociology and philosophy underpin social and
political research in mathematics education.
I want to stress this distinction between types of mathematics education research according to
the style and focus of the research. On the one hand I want to claim there is technical
research, concerned to discover and test details of the teaching and learning of mathematics.
On the other hand there is political and social research with a social interest. Typically this is
concerned with social critiques and follows a social justice agenda. This is a crude distinction
because sometimes detailed studies have strong social implications, and some social justice
motivated research must become closely detailed for its substantiation. Despite this boundary
blurring, an important distinction can be drawn between technical and (socio)political
research. This distinction is also paralleled between four other dichotomies (or end points of
2
Mathematics has a long tradition of calculation aids and technologies (abacus, Napier‟s bones, logarithm
tables, ready reckoners, slide rules, mechanical calculators, electronic calculators, computers) that have been
influential in teaching mathematics. Of course mathematical apparatus such as the compasses, divider and ruler
(or straight edge) have been in use since ancient times. In addition, for over a century specially designed
mathematics teaching apparatus or manipulatives of all sorts has been developed, including Montessori‟s golden
beads, Cuisenaire rods, Dienes‟ blocks, etc. (One of the two major mathematics teaching organisations in United
Kingdom, the Association of Teachers of Mathematics was originally called the Association for Teaching Aids
in Mathematics when formed in 1955.) The list of aids indicates the powerful impact of various technologies
used in mathematics and its teaching (without even mentioning symbolic technologies such as Hindu-Arabic
numerals, decimal place value notation, algebraic notation, geomentric figures, logical and proof notations,
statistical tables and tests, computer software, etc.) This illustrates the technological, tool-based roots of much of
mathematical pedagogy.
2
continua) that occur in the literature. These concern the identity of mathematics education
academics, their academic interests, the underpinning methodological research paradigms,
and the underlying base disciplines:
1. The academic as functional vs. critical (Lerman et al., 2003)3
2. The technical vs. social justice orientation of academic interests (Ernest 2007)
3. Interpretative and scientific paradigm research paradigm vs. Critical-Theoretic
paradigm research, according to their interests (Habermas 1972)
4. The underlying disciplines: mathematics, psychology and pedagogy vs. philosophy
and sociology.4
These are contrasted in Table 1.
Table 1: Dichotomies paralleling the technical / political divide in research
Domain of dichotomy
Academic identity
Academic interest
Research methodology
(paradigm)
Base disciplines
Technical research
Functional
Technical
Interpretative, Scientific
Social/Political research
Critical
Social Justice
Critical
Mathematics, Psychology, Pedagogy
Philosophy, Sociology
By showing the domains of difference as dichotomies I am of course simplifying complex
relationships and continua. My aim is to demonstrate the validity and widespread presence of
the distinction, rather than claiming that the two positions are wholly disjoint, and that a
single fault line divides all of these domains of dichotomy in the same place.
Because of the underlying diversity between these two positions, mathematics education
might be said to suffer from a technical/ political divide in research. What I have labeled as
technical researchers can regard social and political research as too diffuse and as irrelevant to
the classroom the teaching and learning of mathematics; perhaps even a political distortion.
In contrast, social/political researchers can regard technical research as overly narrow and
detailed, and tacitly supportive of the political status quo through its failure to engage with
larger socio-political issues. Such differences are exacerbated by the lack of shared theories or
underlying disciplines of the two sides.
Is it possible to bridge this divide? A shared „super theory‟ might offer grounds – if not for
reconciliation – at least for using the same theoretical language. This might be the aim of
some ambitious projects such as that of Godino in his grand onto-semiotic model of
mathematics education (Godino, Batanero and Vicenc 2007), although I have yet to see it
applied to politically orientated research. Instead, what I offer here is something rather more
modest, a proposed unit of analysis as a shared theoretical basis or language of description.
Since mathematics education is by its very nature an interdisciplinary field of study the
question is, can a single unit of analysis be found to link the disparate domains that make up
the field? Asking for an overall unit of analysis for mathematics education is very ambitious,
3
In this research more complex distinction are drawn using 2 axes (1) the form of engagement (functional vs.
critical) and (2) positioning in overall activity (looking inwards or outwards) resulting in four academic „types‟.
4
Pedagogy is not a discipline in itself, nor does it rest on an underlying discipline – rather it rests on an applied
inter-disciplinary area. The lack of an underpinning discipline for pedagogy is reflected in the lack strong
theories in research in the area which, with the possible exception of the Theory of Didactic Situations
(Brousseau 1997). The area also often draws upon information and communication technologies, itself a recent
interdisciplinary development.
3
given that it is first necessary to show that each of the four elements of mathematics education
listed above can itself be based on an individual unit of analysis. Only when this challenge has
been met then is it necessary to show that there is a „super unit‟ of analysis that can serve as a
basis for all four of these element simultaneously. This is what I propose to do in this paper,
after I have first clarified what is meant by the term „unit of analysis‟.
The Unit of Analysis
Across the social sciences a number of authors have proposed or called for a unit of analysis.
For example, Demorest (1995) proposes the personal script as a unit of analysis for the study
of personality. More generally, in Wikipedia it is suggested that the unit of analysis is the
focus of any research study in the social sciences. “The unit of analysis is the major entity that
is being analyzed in the study. It is the 'what' or 'whom' that is being studied. In social science
research, typical units of analysis include individuals (most common), groups, social
organizations and social artifacts.” (Wikipedia 2010) This is widespread and similar
descriptions appear in research methodology texts, such as Trochim and Donnelly (2007). In
addition, the term unit of analysis is also is used in statistical methods and research
methodology to describe what is taken as a unit for statistical analysis, e.g., as in Knapp
(1977). Hopkins (1982), for example, discusses the methodological choice of group means
versus individual observations in a study without any concerns about the ontological
implications of such a choice. These are legitimate but restricted usages of the term unit of
analysis, and I shall term these them methodological. These usages do not correspond to the
central notion or usage I wish to focus on here.
What I wish to signify by unit of analysis goes beyond simply being the methodological focus
of a study. As well as serving as a methodological focus it is intended that a unit of analysis
should be a prototype or microcosm that represents the key relationships as well as the entities
of a study. I term this the ontological use or meaning of the term.5
This expanded idea is better captured by Eckert and McConnell-Ginet (2003) who introduced
the idea of community of practice to sociolinguistics as their unit of analysis in research on
language and gender. They chose community of practice as their unit because, as they put it,
they wanted a smaller unit than a social network. Their unit of analysis reflects the
relationships they regard as central to their field of study, and in particular the speaker‟s
agency. Co-membership of a community of practice is defined by them on three criteria:
mutual engagement, a jointly negotiated enterprise, and a shared repertoire. Thus the term
„unit of analysis‟ is employed by Eckert and McConnell-Ginet in a deeper way to embody the
prototypical unit in terms of both the entities studied and their relationships. As I shall
elaborate below, the choice of community of practice as a unit of analysis in this ontological
sense is widespread both across the social sciences in general, and in mathematics education
research as well.
In modern times the concept of unit of analysis was first foregrounded by Vygotsky who
argued that psychology needs to be founded on such a unit. He rejected the stimulus-response
unit as proposed by behaviourists and first proposed word meaning as his unit. The idea is
that, just as in chemistry and physics in which all matter can be based on the atom or molecule
as a unit (ignoring sub-atomic particle physics that goes even deeper), so too any other
5
Evans (2011) argues that the methodological/ontological distinction is not as clear cut as I claim, and that what
I have termed methodological uses of the term unit of analysis have deeper ontological implications than just
statistical convenience. In some ways this parallels Quine‟s (1969) argument that the range of the variables
involved in any theory or language usage defines or enlarges the boundaries of its ontological commitments.
4
science or field of study ideally should be based on an irreducible elementary assemblage of
elements, its unit of analysis.6
Historically, a number of theorists have argued that a unit of analysis, or as it was
alternatively described, an explanatory cell, is needed in different fields of study. Goethe
(1996) described the need for an underlying basic cell form or Urphänomen (ur-phenomenon)
from which all more elaborated forms could be built up. Hegel (1822) argues that private
property plays an equivalent role. In the Philosophy of Right Hegel proposes that all the social
and political phenomena of the modern nation state grow out of the notion of private property,
which he calls „abstract right‟ – the cell or unit of analysis for what Hegel called „objective
spirit‟. Marx (1867) proposes the commodity relation (the exchange of commodities between
persons) as the unit of analysis of his political economics. His major work Das Kapital bases
its critique of capitalism and the related organization of society on this unit of analysis. More
recently the behaviourists, and in particular Pavlov, posited the conditioned reflex (stimulusresponse) as a unit of analysis.
However, ultimately it is the work of Vygotsky (1934) that raised widespread awareness of,
and interest in, the concept of a unit of analysis. As is well known, word meaning is stated to
be Vygotsky‟s (1934) unit of analysis in his seminal work Thought and Language (also
translated as Thinking and Speech in Vygotsky, 1987). However, scholars have argued that in
fact tool-mediated action serves as his later and more developed unit of analysis (Zinchenko
1985), and of course this latter serves as the basis for the unit of analysis of Activity Theory
(Leontyev 1978). Lerman (2000), drawing on Minick (1987) and Zinchenko (1985), offers a
nuanced account of the development of Vygotsky‟s thought. This includes the shift between
different units of analysis in his work, from word meaning in earlier work to socially
embedded tool-mediated action, in his later work.
Despite these changes, Vygotsky is clear about the essential aspects of a unit of analysis,
understood in what I term its ontological sense. The unit “designates a product of analysis that
possesses all the basic characteristics of the whole. The unit is a vital and irreducible part of
the whole.” Vygotsky (1987: 46, original emphasis). One of Vygotsky‟s lasting contributions
is the theoretical importance and prominence he accorded to the concept of a unit of analysis.
“The basic concept of using a unit of activity [analysis] that maintains the functions of the
larger system is one of Vygotsky‟s important contributions” (Rogoff 1998: 683). Rogoff goes
on to cite corroborations of this judgement in the work of Bakhtin, Cole, Leontyev, Wertsch
and Zinchenko.
Building on the work of Vygotsky, and further developing his concepts, Blunden (2009,
2010) argues that if a unit of analysis is to respect Vygotskyan ideas it needs to be a
prototypical entity within a study that serves to represent the key relationships involved,
possibly in simplified form. In order to fulfil this function it needs to have the following
properties.
1. It should be a singular and indivisible entity (not a collection or combination) so that it
cannot be broken down into simpler parts that can perform the same function.
2. It should exhibit the central properties of a class of more developed phenomena. As the
most primitive of its type, it should be able to serve as a prototype or basic form for the
area it represents
6
The molecule is an irreducible unit – not because it does not have parts - but because further analysis into
elements or particles results in the loss of its characteristics, especially its chemical properties.
5
3. It should itself be an existent phenomenon that can be located within the realm of
entities with which the subject area is concerned.
A number of authors have proposed different units of analysis corresponding to these notions,
that is, defined ontological units of analysis. Radzikhovskii (1991) offers an account of
dialogue as a unit of analysis of consciousness. Lave and Wenger (1991) identify the informal
community of practice as their unit of analysis, as does Wenger (1998) in his subsequent
work. Likewise, Brown and Duguid (2001) take the community of practice as a unifying unit
of analysis for understanding knowledge in the corporation or firm. Thus the idea of unit of
analysis, with the deeper ontological meaning, is widely used in the social sciences.
My intention in this paper is to apply this idea to mathematics education. In each of the
following sections I present a unit of analysis for the four commonplace-based domains of
mathematics education listed above. I then go on to suggest that Blunden‟s (2009, 2010) own
proposed unit of analysis (the collaborative human project) for an interdisciplinary theory of
activity can serve as an overall unifying unit of analysis for mathematics education.
1. THE NATURE OF MATHEMATICS, THE PHILOSOPHY OF MATHEMATICS
Traditional philosophy of mathematics has proposed a number of objective elements of
mathematics as serving a function similar to a unit of analysis. These elements extend, in
order of increasing complexity, from the mathematical concept, term (or expression),
proposition or sentence, proof, to the mathematical theory. Probably the most common item
used equivalently to a unit of analysis for mathematics is the proposition or sentence, since
this is the smallest unit with a full epistemological function, namely expressing a claim or
assertion that can be characterised as true or false.
A major problem with any of the putative units of analysis in this list including the sentence is
that they objectify and mystify mathematical knowledge and its components, expressing it in
a self-subsistent form that has no essential relation, as many persons see it, with human
beings. Philosophical positions that follow this line put mathematical knowledge beyond the
reach of human making or shaping. Of course there are good arguments for objectifying
mathematical expressions and knowledge, and there is a long and honourable tradition in
epistemology and the philosophy of mathematics that proposes such views and indeed such a
philosophy. I will not rehearse the arguments that I and others have offered elsewhere in
critique of what may be termed absolutist philosophies of mathematics (Ernest 1991, 1998,
Hersh 1997). However, from a humanistic perspective, such an objectification appears to be
an unjustified mystification. Although I do not expect this assertion to convince absolutists, I
shall assume this perspective and develop the ideas here in line with my social constructivist
philosophy presented in preliminary form in Ernest (1991) and in developed form in Ernest
(1998). On this basis, what I propose to do here is to adopt as a unit of analysis the underlying
idea of persons in conversation which I have previously used as an epistemological unit
(Ernest 1998).
The idea of persons in conversation as epistemological unit in the social constructivist
philosophy of mathematics rests on a synthesis of the philosophical ideas of Wittgenstein
(1952) and Lakatos (1976). Wittgenstein bases his philosophy on the notions of language
games and forms of life. He proposes that humans share forms of life that are social activities
with shared purposes. Within these activities humans operate with language games that are
6
part of, and embedded in, these forms of life and are the semiotic means of working towards,
and coordinating cooperation in achieving the shared purposes and goals. Persons in
conversation with the implicit social context of such activities acknowledged thus
corresponds to Wittgenstein‟s basic ideas of language games in forms of life.
Lakatos (1976) argues that the development of mathematical knowledge, including its
warranting, takes place through proofs and refutations, following a Logic of Mathematical
Discovery. In this process or Logic, based on Hegel‟s Dialectic, different speakers/authors
take turns in proposing and critiquing mathematical concepts, propositions, proofs and
theories. This too can therefore be characterised as a conversation between persons, although
Lakatos would not be happy of the foregrounding of the social dimension of knowledge
construction.
Thus the idea of conversation between persons as a unit of analysis for the philosophy of
mathematics builds on the foundations of both Wittgenstein‟s and Lakatos‟ work, and in my
account in Ernest (1998) also sits on Vygotsky‟s psychology. As a unit of analysis
conversation can be represented as a triple or diagrammatically as a triangle with the vertices
corresponding to Speaker/Proposer, Listener/Critic, and Text (semiotic expression of
mathematics, both formal/written and informal/spoken/otherwise represented). The roles of
Speaker and Listener alternate in conversation, and both roles are internalized within a single
person (following Vygotsky‟s theories, but also in accordance with Mead 1964). Indeed I
doubt that a person can be a competent mathematician without such internalization of both
these roles, as one needs to propose and construct new, putative mathematical knowledge, and
then critique and edit it oneself, in order to shape it in accessible form so that it can be
responded to, although not necessarily accepted, by others. Thus a mathematical author
incorporates criticisms and refutations that public presentation will inevitably bring, insofar as
she can anticipate them, in mathematical texts. Furthermore, in any true conversation the
participants need to alternate between the roles of speaker/listener and proposer/critic.
Figure 1: Conversation as the Unit of Analysis for Mathematics
Mathematical
Text
Speaker /
Proposer
Listener /
Critic
7
Note that by speaker I also include writer, the utterer of the text in some mode or other.
Likewise the listener also incorporates the reader of the text.
Thus the unit of analysis for mathematics can be represented as in Figure 1. In this figure, the
arrow between speaker and listener indicates the back and forth flow of conversation,
mediated by the text. The arrow from the speaker to the text shows the relationship of the
author to the text indicating its production and revision, and the reverse direction arrow shows
the author‟s reading of her own text, either in original or modified form. The arrow between
the text and the reader indicates similar bidirectional relations. And as indicated above, the
roles of speaker and listener are alternated/switched.
What Figure 1 shows is the path of a single cycle in a conversation, for conversations involve
a repeated cycling or spiralling as speakers utter or propose a text, followed by listeners
responding to this text and modifying it or uttering their own responsive text. This back and
forth between speaker and listener each modifying the text (sometimes purely by appending
their own response) continues repeatedly with other persons participating. The number of
persons involved can be anything greater or equal to two (or one, in an internalized
conversation), and indeed the identities of the speaker, listener and text can and often will
change during the process of conversation. Furthermore, conversations can be extended
indefinitely over time and space through repeated face to face meetings, or through
information and communication technology mediated communications such as written texts,
letters, telephone conversations, electronic exchange of documents, and so on (Ernest 1994a).
Conversation as a unit of analysis for mathematics shows the mechanism or communicational
means whereby mathematical knowledge productions are created, reformatted, questioned,
modified, warranted and transmitted. This unit of analysis emphasizes that mathematical
knowledge representations are always materially present in some form and are an organic and
evolving part of human culture. However, the material presence of texts or other knowledge
representations does not mean that knowledge can be turned into a material product. Only in
the presence of a reader or interpreter does a text become meaningful. Furthermore a person
can only act as a reader or interpreter after enculturation, that is after an extensive social
apprenticeship in language use. So the social aspect of conversation cannot be meaningfully
stripped away to reveal a new objectification of knowledge. If it were so, conversation could
not serve as an epistemological unit, let alone a unit of analysis.
Although the insight that conversation works as a unit of analysis may be controversial in the
philosophy of mathematics, beyond in the social sciences it is more commonplace.
Collingwood (1939) proposes the epistemological project of substituting a 'logic of question
and answer' for a logic of propositions. In Gadamer‟s (1965: 333) view the quest for
knowledge "contains within itself the original meaning of conversation and the structure of
question and answer". And lastly, according to Harré (1990: 117) “in providing an ontology ...
we must take the individual speaker locked into a pair with an interlocutor as the
conversational unit.”
2.
LEARNING MATHEMATICS
There has been a great deal of attention given to the theories of learning mathematics over the
past two decades (see, e.g., Steffe et al. 1995, Sriraman 2010) with different forms of
constructivism much discussed and criticized. Various forms of individual or radical
8
constructivism seem to be based on a binary relationship between the learner or subject and its
environment or object. In such models the learner actively interacts with the environment both
selecting what it samples from the environment and actively interpreting its experiences of the
environment and regularities that occur within them to build up a growing knowledge base
and interpretative framework. This model has strengths that derive from its recognition of 1.
The experiential basis of all knowledge, 2. The importance of interpretation and the active
construction of meaning, and 3. the learner and subject as a growing being, whose
interpretative framework is, at least partially, self-constructed. However, the weakness of this
model is that it describes equally well the growth of knowledge of an amoeba, a shark or a
higher primate. It omits the fact that human beings are fundamentally social creatures whose
cultural inheritance (most notably language) natures and shapes the way that they (we)
interact with the world and each other. Thus, for example, Bottino and Chiappini (2002) argue
that the individual student in interaction with tools (e.g. computer software) is an inadequate
unit of analysis because of the stripping away of the social context.
Sfard (1998) distinguishes between two metaphors for learning: the acquisition and the
participation metaphors. Constructivism in its individualistic forms described above falls
under the acquisition metaphor because it reifies knowledge as something to be gained and
acquired, and thus as something that can be transmitted or delivered. But this perspective
makes the category mistake of thinking that when learners demonstrate competence or
abilities they „have‟ or acquire something, rather than they are just able to do something. As in
traditional absolutist accounts of the nature of mathematics, from the acquisitionist
perspective learning is seen as based on mathematical knowledge as externalized objects such
as concepts, rules or truths (propositions or theorems) and methods, that can be acquired or
owned by learners. These objects would be the unit of analysis from this perspective.
In contrast Sfard‟s (1998) participation metaphor foregrounds the fact that knowing is
manifested in doing, and it also requires recognition of the constitutive and ineliminable
social dimension. Doing is action in the real world that we all inhabit, not in some fictional
space of reified concepts and knowledge. It is learned socially by participation in relationships
or human communities, even if only fleetingly. In these social contexts one learns to use a
range of cultural tools including the full range of communicative modes and technologies.
Some key examples of these in mathematics are gestures, signs, spoken and written language,
mathematical notations, pens, books, electronic calculators, computers, and the World Wide
Web.
With regard to the learning of mathematics a number of authors agree that the isolated
individual is not an adequate unit of analysis. Cobb (2000) argues that the mainstream
characterisations of the individual as the unit of analysis within cognitive psychology have
been delegitimized by situated accounts of intelligence and learning. Likewise, as mentioned
above, Bottino and Chiappini (2002) argue that the individual student alone in interaction
with tools (e.g. computer software) is an inadequate unit of analysis. Lerman (2000) is critical
of any unit of analysis based on the isolated individual and proposes the person-in-practice or
mind-in-society as unit of analysis. Perhaps playfully he goes on to extend this unit of
analysis to the rather Byzantine ideas of person-in-practice-in-person or mind-in-society-inmind. Irrespective of this formulation his main point is that the social dimension is an integral
part of both self and learning that no adequate unit of analysis can ignore or factor this out.
Lave and Wenger‟s (1991) „community of practice‟ idea has been widely promoted as a unit
of analysis for learning (e.g., Boylan 2007, Graven & Lerman 2003). Its adoption is not
9
without dissenters, such as Boaler (2000b) who argues that it does not adequately cover
formal learning situations, such as schools and universities.
In keeping with the thrust of these developments in my quest for a unit of analysis for human
learning I adopt a Vygotskyan or neo-Vygotskyan social constructivist theory, This sees an
essential role for other humans (i.e., participation) in all but the most rudimentary (preverbal)
human learning. Even there, gestures and touch play a key role from day one (birth), and
represent a primitive form of conversation. Vygotsky sees learning as taking place primarily
in the learner‟s Zone of Proximal Development. This is the cognitive space in which a more
capable other, a relative expert (no pun intended), helps a less capable learner master some
tool-based action in pursuit of a goal. Such tools can include language, eating utensils, toys,
algebraic symbolism, computers, cars, etc., in other words, all the tools, materials and
information and communication technology artefacts that humans have developed over that
past few hundred thousand years. Thus the concept of tool or cultural artefact is taken in the
broadest sense from material tools such as a stick or hammer or jet plane, to conceptual and
semiotic tools and artefacts such as language, texts, performed plays, paintings, ideas and
theories. Tool-mediated action is the unit of analysis for this theory of learning. It is
represented as a triple with the Learner / novice in one position, the teacher / expert in
another, and the action as the third apex (with the mediating tool/cultural artefact within it,
sometimes implicitly). This triple as the unit of analysis for learning theory is represented as
in Figure 2.
Figure 2: Tool-Mediated Action as the Unit of Analysis for Learning Theory
Action mediated
by Cultural Tool
Teacher / Expert
Learner / Novice
In Figure 2 the arrow between teacher and learner indicates the back and forth flow of actions,
mediated by the cultural tool (including language). The arrow from the teacher to the tool
shows this agent‟s active use of this tool, while the reverse arrow indicates that teacher is also
attending to and interpreting the tool in action. The arrow between the tool and the learner
indicates a similar bidirectional relationship. This could also be described as a conversation,
and evidently resembles Figure 1 (Ernest 1994b).
10
According to this model, learner mastery of some action using a mediating tool or cultural
artefact is the prototypical learning activity and takes place within the learner‟s ZPD.
Typically, as shown in the unit of analysis illustrated in Figure 2, this involves a more capable
other guiding the learner. Learning on one‟s own can be understood as a simplified version of
this triad where the social element is implicit in the tool/cultural artefact. A learner reading a
book or solving some mathematical problems on her own can only do this because of earlier
work within such a triad, with guided use of a tool, under the supervision of a more capable
other. It could also be interpreted in terms of the learner having internalized the role of the
teacher.
3.
TEACHING MATHEMATICS
The teaching of mathematics is a complex affair that can be represented relatively simply by a
triad as unit of analysis. The three elements are two asymmetric partners, the teacher and
student, plus the mathematical task used for instruction. This task is presented as a text using
one or several of a variety of means and modes of representation. Thus the proposed unit of
analysis is very similar to that for learning mathematics shown in Figure 2.
In keeping with modern semiotics I want to understand a text as a simple or compound sign
that can be represented as a selection or combination of spoken words, gestures, objects,
inscriptions using paper, chalkboard or computer displays, as well as recorded or moving
images. Mathematical texts can vary from, on the one hand, printed documents that utilize a
very restricted and formalized symbolic code in advanced mathematics teaching, to, on the
other hand, multimedia and multi-modal texts, such are used in kindergarten arithmetic. The
latter can include a selection of verbal sounds and spoken words, repetitive bodily
movements, arrays of sweets, pebbles, counters, various objects including specially designed
structural apparatus, sets of marks, icons, pictures, written language including number words,
symbolic numerals, and so on.
Texts figure in the teaching of mathematics in two ways, as texts authored or presented by the
teacher to the student, and by responsive texts constructed by the student and directed back to
the teacher. The first of these is primarily the task used in mathematics teaching. Such a task
has the following properties (based on Ernest 2008):
1.
It is an activity that is externally imposed or directed by a person or persons in
power (the teacher) representing and on behalf of a social institution (e.g., the
school).
2.
It is subject to the judgement of the persons in power (the teacher) as to when and
whether it is successfully completed.
3.
It is a purposeful and directional activity that requires human actions and work in
the striving to achieve its goal or goals.
4.
It requires learner acceptance of the imposed goal, explicitly or tacitly, in order for
the learner to consciously work towards achieving it.
5.
It requires and consists of working with texts: both reading and writing texts in
attempting to achieve the task goal.
6.
It carries with it set of assumptions about what to attend to and what to ignore
among the available meanings (Gerofsky 1996).
11
Figure 3: The Teacher-Student-Task Triple as the Unit of Analysis for Mathematics Teaching
Task presented
via text
Teacher
Learner
There is a very large literature on educational tasks in general and on the role and variety of
mathematical tasks used in the teaching and learning of mathematics. For example Brousseau
(1997) proposes the well known theory of didactical situations concerned with the nature of
mathematical tasks and their relationship with the social context of the teaching of
mathematics.
Figure 3 shows the Teacher-Student-Task triple as the unit of analysis for mathematical
teaching. Teacher communication with student(s) is mediated by texts presenting mathematics
teaching/learning tasks (and by student responses).
For present purposes it is enough to point out the overall structure is that in which a teacher
communicates a mathematical task to a student, that is directs a student‟s attention to a text
embodying a mathematical task, and possibly directs the student‟s attention beyond it to its
mathematical or social context. This is the instructional direction of communication (left to
right in Figure 3). However, as discussed above, and indicated by the double headed arrows,
such traffic is not intended to be unidirectional only. Students construct their own texts in
response to teacher texts and tasks, and then communicate these to the teacher in one or more
of the available modes, including oral, written on paper, written on chalkboard or electronic
whiteboard, by electronic transmission, etc. They may even pose problems themselves and
submit them to the teacher for approval. This is the responsive or assessment direction of
communication (right to left in Figure 3). The teacher responds to such student texts by
assessing them and feeding back further responses in the instructional direction, providing
either informal or formal assessment responses to the student, and possibly also providing
responses to others too, given the broad and varied functions of assessment.
The triple as illustrated in Figure 3 is not a controversial proposal as a unit of analysis for
mathematics teaching. Indeed similar diagrams are widespread in the literature. More likely
the criticism is that this triple is so rudimentary, so simplistic, that it adds little to the
understanding of the teaching and learning of mathematics. However, its emphasis on the
12
inescapably mediated nature of the teaching/learning of mathematics, the centrality of
textually presented tasks, and the essentially bi-directional direction of teacher-student
interaction are of value because they are not always evident in accounts of mathematics
teaching. Beyond these aspects, the unit of analysis is presented here as one that can be
subsumed by a still more general unit, which is the overall purpose of this paper.
4.
THEORY OF SOCIETY OR THE SOCIAL CONTEXT
A theory or model of the social context of mathematics education, one that fulfils all of the
theoretical needs of the area, is not something easily found. On the one had it needs to
accommodate the macro-social context, including the cultural, social, societal and political
milieu of mathematics education in its theoretical and practical forms. On the other hand it
must accommodate the micro-social, interpersonal, interactional aspects including the social
construction of agency, self and identity. In addition there must be space in it for knowledge,
especially mathematical knowledge.
A number of theories have either been used in mathematics education or appear to have
potential for such use. Some accommodate the macro-social but seem to have less potential
for the micro-social, such as:
 Marx‟s (1867) ideas of society as economic base and superstructure (and its
development by Gramsci, Althusser and others),
 Pierre Bourdieu‟s notions of reproduction and Habitus (Bourdieu and Passeron 1977)
 Basil Bernstein‟s (1996) ideas of the social role of knowledge.
Some theories are better on the micro-social but may have less potential for accommodating
the macro-social or knowledge, such as:
 The Family as a model for society and the state, first found in Aristotle (1995)
 Freire‟s (1972) ideas of education as emancipatory,
None of these above approaches provides a rich enough framework to encompass potentially
all of teachers, students, school mathematical knowledge, tools, texts, institutions, roles, rules,
aims, power, inequalities, and economic activity. Such models need to treat both
social/structural inequalities in society and the formation of identity in individuals.
One family of theories that has potential to accommodate both of the two realms are various
versions of Activity Theory.
 Lave and Wenger‟s (1991) model of a community of practice,
 Engeström‟s (1987) extension of Activity Theory,
 Blunden‟s (2009) interdisciplinary version of Cultural Historical Activity Theory.
Lave and Wenger‟s (1991) model of a community of practice suggests a unit of analysis
consisting of participants in the practice (both novices/apprentices and experts/masters) and
the practice itself. The practice is organized around a purposive activity. Although it provides
a good way to conceptualize productive practices, it has not been elaborated to encompass
large scale social divisions (although Wenger 1998 is a possible exception to this claim). As
mentioned above, it has also been criticised for not adequately covering formal learning
situations (such as schools and universities) as opposed to informal workplace contexts
(Boaler 2000b). However, Blunden‟s (2009) Cultural Historical Activity Theory extends some
its key elements so I shall consider this instead.
13
Engeström‟s (1987) version of Activity Theory with its well known set of nested triangles
(Figure 4), represents social relationships and related elements. Engeström (1987) claims that
in order to analyze composite phenomena like social interactions and interpersonal
relationships, one needs a new unit of analysis. He argues that his theory is “a strong
candidate for such a unit of analysis in the concept of object-oriented, collective, and
culturally mediated human activity, or activity system” (Engeström & Miettinen 1999: 9).
Figure 4 Engeström’s dialectical triangle
Figure 4 shows the dialectical triangle developed by Engeström (1987), used as a model for
analysing activity systems. The elements of such a system include the object, subject,
mediating artefacts (signs and tools), rules, community, and division of labour. These lead to
the outcome of the system. The internal tensions and contradictions in such a system are the
driving force behind change and development (represented in Fig. 4 by three internal double
headed arrows, each originating from an apex). These contradictory forces are accentuated by
the continuous transformations that take place within and between the components of this
system and between the hierarchical levels of activity, action and operations (Engeström
1999, Leontyev 1978).
This is not the place here to elaborate Engeström‟s (1987) version of Activity Theory, but it is
clear from its elements that it is potentially a good candidate for accommodating the social
context of mathematics education. It includes relationships between individual (subject),
object, outcome, community, social rules, division of labour. From these it is a short step to
the environment, instruments of production, production itself, distribution, exchange and
consumption. The upper triangle of subject, object, and instruments or tools has been shown a
capable of modelling the development of individuals and identity within second generation
Activity Theory7 (Alvarez and Del Rio 1999, Gee 2001). However, as Blunden (2009) argues,
such a complex as is illustrated in Figure 4, with all of the elements listed, even with
redundancies or reducible elements eliminated, can hardly be regarded as a simple unit of
analysis. It is a macro-model, rather than a unit of analysis, from which elements and
relationships are selected to be foregrounded. Furthermore, many of these elements are
secondary or derived from the fundamental unit.
... attempts to incorporate „supra-individual‟ aspects of society such as social division of
labor, norms and rules, systems of production and distribution, and so on, fail to provide
7
Núñez (2009), drawing on Engeström (2001), distinguishes between three generations of Activity Theory
based on the work of: 1. Vygotsky, 2. Leontyev, 3. Engeström, each with their own unit of analysis.
14
the basis for a unit of analysis. The fact is that these societal phenomena exist for the
individual only through (1) the use of artifacts which originate and carry culturally
determined meaning from outside the immediate setting of their use, and (2) the
regularity of expectations and experiences of interaction with other individuals.
(Blunden 2009: 19)
Because of this irreducible complexity I reject the use of Engeström‟s (1987) version of
Activity Theory as a unit of analysis for the social context of mathematics education. This
in no way casts any shadow on its utility, but simply on its ability to serve the function
presently sought.
Blunden (2009) in his version of Cultural Historical Activity Theory addresses the issue of
finding a unit of analysis directly. He fixes upon „project collaboration‟ – the interaction
between two or more persons in pursuit of a common objective (p. 1) and the “artefactmediated collaboration of individuals in common project” (op. cit. p.19) as his unit of
analysis. He goes on to elaborate this notion further.
A project‟ differs from „an activity‟ understood à la Leontyev, as a system of actions
directed towards a given socially defined object in several respects. Firstly, a project
includes the individuals and all the artefacts and norms and rules indigenous to that
project. A project is always directed towards some ideal. Projects need to be understood
as historically articulated, and individual projects carry forward projects that may have a
long history. In this sense the idea of project is subject-centered rather than objectcentered. (op. cit. p.19)
From this idea Blunden is able to derive the secondary notions so prominent in Engeström‟s
(1987) account. “Notions of social norms, division of labor, markets, and so on, must
therefore be derived from their foundation in the artifact-mediated collaboration of
individuals in common projects or „project collaboration‟.
“To be clear, „project
collaboration‟ is not something different from activity, but simply a unit of activity, a unit of
joint mediated activity.” (Blunden 2009: 19).
Now the question arises, to what extent can the notion of project collaboration serve as a unit
of analysis for the social context of mathematics education? Can it accommodate different
social groups, inequalities, hierarchies, curriculum contestation, competing and rival groups
with different ideologies as well as small scale phenomena such as learning careers and
identity projects?
Blunden (2009, 2010) broadens and deepens his notions in ways that make such coverage
plausible. He notes that projects include conflict as well as cooperation. He sees cooperation
(pursuing the same end using some division of labour, whether natural or artificial) and
conflict (pursuit of mutually exclusive states of affairs) as special, limiting cases of
collaboration. He also remarks on other important limiting cases of collaboration concerning
„ownership‟ of the project: solidarity, where one subject voluntarily subordinates themself to
the other‟s ends, and cooption, where one subject subsumes another under their own project.
He argues that collaboration not only provides a starting point for science, but is also
normative and thus provides a reference point for ethics. Blunden analyses three broad types
of cooperation: hierarchy, exchange and collaboration, which between them potentially
accommodate all forms of social organisation. He draws on John Steiner‟s (2000) categories
15
of different types of collaboration to show the range of projects and activities that can be
described or elaborated from his basic unit.
Social movements, nations, religious communities all constitute themselves as projects.
The pursuit of an art or profession, is also constituted as a project, with practitioners
striving to perfect the art, each generation standing on the shoulders of the generation
before. (Blunden 2009: 24)
The outcome is a very promising unit, within Blunden‟s particular take on Cultural-Historical
Activity Theory, on which to base an analysis of the leading social phenomena, in general,
and the social context of mathematics education, in particular.
So much for it as a potential unit for macro-social phenomena. What about micro-social
phenomena such as individual identity? A number of authors regard Activity Theory as a
valuable theoretical tool for treating this area, e.g., Alvarez and Del Rio (1999), Côte and
Levine (2002), Gee (2001), Rogers (2007) and Wenger (1998). Blunden sees such phenomena
as accommodated within his own version of Activity Theory.
When concepts first appear, they constitute projects, but in time, they become
objectified and merge into the fabric of social life, the language and culture generally.
Once a concept has become objectified, it ceases to have an independent life, but
participates as an aspect of all subsequent projects. Some concepts however, not yet
objectified, retain vitality, and constitute living, self-conscious projects. Consciousness
is therefore constituted by participation in a multiplicity of different projects and
activity organized around a multiplicity of different more or less independent concepts,
which represent the sediment of past projects. (Blunden 2009: 25)
Although not based on this particular formulation, within the mathematics education research
community a growing body of research on identity and individual agency discusses or utilises
Activity Theory as its basis, including Black et al. (2010), Boaler and Greeno (2000), Graven
and Lerman (2003), Grootenboer et al. (2006), Williams, J. (n. d.). Thus the prospects appear
optimistic for Blunden‟s (1009, 2010) project collaboration to serve as a unit of analysis for
mathematics education.
Collaborative Projects as the Unit of Analysis for Mathematics Education
Figure 5 offers a representation of the collaborative project as unit of analysis. It comprises
the individuals engaged in the project shown as participants 1 and 2. It also comprises the
project itself with its goals and direction towards some object, and the tools and artefacts
including norms, rules and roles indigenous to that project.
In Figure 5 the arrow between the participants indicates cooperative activity, encompassed by
the project itself. This provides both the direction of the activity as well as the means of
accomplishing the project or at least the means of engaging in actions subsumed into the
project.
Does the collaborative project constitute a potential Unit of Analysis in the ontological sense?
To answer this it must be seen to satisfy the three criteria proposed earlier in the paper. First
of all, it is minimal in that cannot be broken down into simpler parts that can perform the
same function. A project cannot exist without the agency of its participants, and it cannot be a
16
collaborative project without more than one participants (except in the internalized ways
described above). Second, a collaborative project is not an abstraction, but an existent
phenomenon, for every instantiation is constituted by the historically situated activities of real
persons. Third, as demonstrated in the previous section, in outline at least, collaborative
projects exhibit the central properties of a class of more developed phenomena on both the
macro-social and micro-social planes. It only remains to show that it is a prototype or basic
form for the subareas of mathematics education discussed above.
Figure 5: The Collaborative Project as Unit of Analysis for Social Phenomena
Collaborative
Project
Participant 1
Paricipant 2
Without labouring the point it is pretty evident from the form of the Figures 1, 2 and 3 that
these can be subsumed into the collaborative project form as represented by Figure 5. Each
has been represented as a triangle with persons or persons-in-roles as represented by the two
base vertices. In each figure human activities or the tools resulting from such activities have
been represented as the apex. Whether this is text exchanged and co-constructed as in Figure
1 representing the conversational nature of mathematics, or as in Fig. 2, during the learning of
mathematics, whether it represents actions mediated by the cultural tools in Figure 3, during
the teaching of mathematics, these are all fundamentally analogous or inter-translatable.8
What the collaborative project form adds, as well as generalizing them all, is to emphasize the
goals and directedness of all of these fundamental units of activity. The creation of
mathematics, and its learning and teaching, all involve persons in relationships with texts and
cultural tools as mediating means that are ordered by the goals and purposes of the activities,
the projects.
8
In this brief paper I somewhat gloss over the differences between the apex elements in Figures 1 to 3 (be they
human activities or tools). Thus my argument only demonstrates the plausibility of the unification of the various
units of analysis under the single collaborative project unit, and needs to be more extensively argued and
elaborated to be fully convincing.
17
Thus I have come to the end of my quest. I have demonstrated that it is plausible that the
collaborative project, at least potentially, can serve as a unit of analysis for mathematics
education. Furthermore, it does so in the ontological sense distinguished above. What this
shows is the perhaps surprising fact that that same unit can be applied not only to the teaching
and learning of mathematics, but also to the nature and philosophy of mathematics and to the
social context of mathematics education too: not only for technical research, but also forpolitical and social research. The value of this is that it leads to an interdisciplinary
unification. It links the elements of mathematics education that are drawn from philosophy,
psychology, pedagogy and sociology. In doing so it offers a bridge across the technicalpolitical divide. It presents a single theoretical underpinning that allows the two camps of
technical and social-political researchers to communicate through a shared language. No
longer need these two camps be separated by the scale or mode of description of their
concerns.
However, it must be acknowledged that this proposed unification does not come without
presuppositions and indeed costs. It is persuasive only if a social view of teaching, learning
and mathematics, and in particular an Activity Theoretical account is adopted. But the form of
Activity Theory used here is not ideologically narrow, and it is open to expansion and further
elaboration to better theorize the phenomena involved. So a theoretical straitjacket is not part
of the cost, even if a social perspective is required for this solution.
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