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Barbara Jaworski & Simon Goodchild
Agder University College, Norway
A developmental research project, Learning Communities in Mathematics (LCM)
bases its activity on the theoretical concepts of inquiry and community. It seeks to
create knowledge and improve practice in the learning and teaching of mathematics
through developing inquiry communities between teachers in schools and
didacticians in a university setting. Analysis of data requires a recognition of the
complexity of socially embedded factors, and we draw on activity theory to address
complexity and deal with issues and tensions related to learning within the project.
This theoretical paper presents our early thinking in analysing inquiry community
within an activity theory frame.
We present a theoretical paper related to a research project, LCM1 (Learning
Communities in Mathematics) introduced in previous papers. The theoretical basis of
our project is community of inquiry which is addressed in Jaworski (2004, 2005 and
in press). In Goodchild & Jaworski (2005), we introduced an activity theory frame
for analysing data within the project; here we provide a more detailed theorisation.
Teachers and didacticians each bring specialised knowledge to developing teaching,
and hence learning, of mathematics. Together we can use, and explore the use of this
knowledge in order to improve the mathematical learning experiences of students in
classrooms and to know more about the creation of good opportunities for learning.
The words “together” and “explore” adumbrate the concept of inquiry community
(Wells, 1999). Fundamentally, inquiry and exploration are about questioning: asking
and seeking to answer questions. Together, we ask and seek to answer questions to
enable us to know more about mathematics teaching and learning. Moreover, the
asking of questions is a developmental tool in drawing students, teachers and
didacticians into a deeper awareness of their own actions, motives and goals
(Jaworski, 1994; Mason, 2001).
Thus we engage in developmental research: research which both studies the
developmental process and, simultaneously, promotes development through
engagement and questioning. We recognise engagement in inquiry activity within a
well defined community as a significant means of coming to know. Thus, a major aim
of our critical questioning approach is to take us, as a community, deeper into
knowing mathematics teaching and learning. Not only are research questions defined
and explored (through suitable data collection and analysis), but the whole research
process is subject to question and exploration. We look critically at our research
activity while engaging in and with it (Chaiklin, 1993).
2006. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds.). Proceedings 30th Conference of the
International Group for the Psychology of Mathematics Education, Vol. 3, pp. 353-360. Prague: PME. 3 - 353
Jaworski & Goodchild
Teachers in the project belong to particular school communities within the school
system, functioning within educational norms in society, and a political framework.
Everyday factors such as curriculum, school timetables, responsibilities of teachers,
time and energy afford and constrain what is possible (Engeström, 1999). We would
not describe such communities as communities of inquiry, although they might be
termed communities of practice (Wenger, 1998)2. They are groups of people
dedicated to specific activity with established ways of thinking and doing which
may be questioned but are most often taken for granted in the everyday momentum.
Didacticians belong to a university community in which research activity is a norm
involving familiarity with inquiry at formal levels. As teacher educators,
didacticians are expected to question and theorise teaching in schools. However,
questioning or theorising their own activity (of teaching and research) may not be a
community norm. Thus, a desire to generate communities of inquiry within the
project requires a serious addressing of the activity and goals of these various
communities and a searching for ways of generating the kinds of thinking and
coming to know that we expect from inquiry activity (Cochran Smith & Lytle,
1999; Wells, 1999)
The LCM project was designed by didacticians who sought funding and had initial
responsibility for the project. Schools and teachers were recruited after funding had
been secured (Jaworski, 2005). Initial activity was motivated by the need to
establish a project community and to start to understand jointly what inquiry might
mean within the project. Didacticians have designed activity to create opportunity to
work with teachers, to ask questions and to see common purposes in using inquiry
approaches that bring both groups closer in thinking about and improving
mathematics teaching and learning. We design workshops, and tasks for workshops,
through which parallel design activity can start to take place in schools. This design
process is generative and transformative (Kelly, 2003). We use tasks necessitating
inquiry to generate inquiry activity through which a joint community, with common
goals can emerge.
Workshops have encouraged all of us to do mathematics together, to inquire in
tackling mathematical problems, to raise questions about learning and teaching and
to start to think and plan for the classroom. In schools, teacher teams, with
didactician support, follow up experiences from workshops to explore possibilities
for inquiry activity in classrooms, engaging themselves in inquiry through their
design of tasks for students. These words express, simply, aspects of the project
design and of its implementation but they underestimate the complexity of the
process and the problematic nature of interpreting project goals into the realities of
engagement in the project (Goodchild & Jaworski, 2005; Jaworski, 2005). We see
tackling issues and tensions as forming the essence of our learning: at a practical
level, for the project to make progress; and, at a theoretical level, to conceptualise
their role in our learning development, both theoretical and practical. It is here that
we are exploring the use of activity theory as an analytical framework and toolbox.
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Key concepts and terms
Activity theory develops from the work of Vygotsky, particularly his arguments that
cognition arises through the internalisation of external operations that occur in
sociocultural contexts (Vygotsky, 1978). In identifying an intermediate link in the
stimulus-response process, Vygotsky proposed the notion of a “complex mediated
act” which “permits humans … to control their behaviour from the outside. The use
of signs leads humans to a specific structure of behaviour that breaks away from
biological development and creates new forms of a culturally-based psychological
process” (1978, p. 40, italics in original). Through consideration of sociocultural
artefacts that mediate between stimulus and response, the idea of a complex mediated
act has been developed further. For example, following “the tradition of the theory of
activity proposed by A. N. Leont’ev”, Wertsch refers to “goal-directed action” and
writes, “human action typically employs ‘mediational means’ such as tools and
language”. He goes on to emphasise that “the relationship between action and
mediational means is so fundamental that it is more appropriate, when referring to the
agent involved, to speak of ‘individual(s)-acting-with-mediational-means’ than to
speak simply of ‘individual(s)’” (1991, p. 12). A. N. Leont’ev makes the following
point, “in a society, humans do not simply find external conditions to which they
must adapt their activity. Rather these social conditions bear with them the motives
and goals of their activity, its means and modes. In a word, society produces the
activity of the individuals it forms” (1979, pp. 47-48). So, according to Wertsch (p.
27), rather than “the idea that mental functioning in the individual derives from
participation in social life”, “the specific structures and processes of intramental
processing can be traced to their genetic precursors on the intermental plane”.
The key idea for us here is that human activity is motivated within the sociocultural
and historical processes of human life and comprises (mediated) goal-directed action.
According to Leont’ev, “Activity is the non-additive, molar unit of life … it is not a
reaction, or aggregate of reactions, but a system with its own structure, its own
internal transformations, and its own development” (p. 46). He proposed a three
tiered explanation of activity. First, human activity is always energised by a motive.
Second, the basic components of human activity are the actions that translate activity
motive into reality, where each action is subordinated to a conscious goal. Activity
can be seen as comprising actions relating to associated goals. Thirdly, operations are
the means by which an action is carried out, and are associated with the conditions
under which actions take place. Leont’ev’s three tiers or levels can be summarised as:
activity ÅÆ motive; actions ÅÆ goals; operations ÅÆ conditions.
Leont’ev writes emphatically about the movement of the elements between the
‘levels’ within an activity system: activity can become actions and actions develop
into activity, goals become motives and vice-versa, similarly with operationsconditions. The crucial differences seem to be: first, goals are conscious, if the
motive of activity becomes conscious it becomes a motive-goal; second, motive is
about an energizing force for the activity and the actions, it is not something that is
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attained but rather drives the activity forward; on the other hand goals are results that
can be achieved. Leont’ev writes “The basic ‘components’ of various human
activities are the actions that translate them into reality, We call a process an action
when it is subordinated to the idea of achieving a result, i.e. a process that is
subordinated to a conscious goal” (pp. 59- 60).
Exemplifying AT terms and concepts in the LCM Project
Here we exemplify briefly the concepts and terms above with reference to examples
from the LCM project. These examples (rows in the table below) are deeply related
to each other and so could be considered elements of one complex activity system.
We separate them artificially to show elements of the three levels (the columns).
Exemplification is an oversimplification, but serves the purpose of clarifying how we
see these terms and concepts fitting our project and serving as a basis for analysis. In
each case actions and operations are only examples of many possibilities.
Activity (System) & Motive
Developmental research,
whose motive is to study
developmental processes and,
simultaneously, promote
development in the learning
and teaching of mathematics.
A school, whose motive is to
educate pupils.
The LCM project as
community of inquiry, with
motive to provide the
environment and modes of
action for teaching
development to be realised.
Actions & Goals
Asking researchable questions,
collecting and analysing data
leading to findings or
outcomes related to new
knowledge and/or practice.
Operations & Conditions
Making methodological
decisions related to
principled and effective
ways of collecting and
analysing data to address
research questions.
Organising teaching groups
Choosing topics and
and designing lessons to
planning classroom tasks
promote learning according to according to the school’s
the declared curriculum.
approaches to addressing the
curriculum. (Planlegg et
Creating opportunities for
Teachers and didacticians
working together and engaging working in groups in
in inquiry to achieve a working workshops on mathematical
community with practical
problems to exemplify
knowledge of inquiry
inquiry processes and
develop common
Mediation in goal directed action
Within an activity system, goal-directed actions are mediated by tools and signs as
represented by the basic mediational triangle deriving from Vygotsky and developed
further by Leont’ev (Figure 1) (e.g., Engeström, 1999; Vygotsky, 1978). Here we see
the human subject or group seeking to achieve a goal or object, mediated by some
tool or sign where the nature of the tool or sign is deeply embedded in the activity. In
recognition of this deep embeddedness, Engeström expanded the basic triangle to the
“complex model of an activity system” (1999, p. 31) to recognise mediation by or
through community, rules of activity and division of labour within the activity
system. Each of the connections within the expanded triangle indicates possible
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mediational means within the system. The double arrows throughout indicate
dialectical dependencies between the elements of a system.
Based on Vygotsky’s model of a complex mediated
Engeström’s ’complex model of an activity
Figure 1
Figure 2
Within the LCM project we engage in research that seeks to promote development in
teaching and learning mathematics and to study that development. We believe this
can be achieved through the development of an inquiry community comprising
teachers and didacticians. This is the energizing force of the project, it is the motive
for the activity of the project. The motive provides a rationale for the activity, and an
incentive for the actions that comprise the activity. The actions are ‘energized’ by the
motive but they are directed towards achieving some conscious goals, achievable
results that will arise from the actions. For example, we want to achieve a sense of
community, so we organise workshops and within those workshops opportunities for
teachers and didacticians to meet together, work together and discuss together.
Individual didacticians also spend time on their own seeking out mathematical
activities for the workshops; in this action the goal is to find tasks that show potential
to be of use. The result of this time spent is an ‘oppgave’3; neither the action nor the
‘oppgave’ is the central purpose of the project, but the motivation for the action of
finding the tasks is the same ‘energizing force’ (motive) of the project.
In the project activity system a number of mediational means are available to support
or enable the actions that comprise the activity. For Vygotsky the main tool was
language and the project seeks to develop a language of inquiry within the
community. It has been emphasised that the asking of questions is fundamental to
inquiry. In this respect the questions are an important tool or artefact within the
envisaged activity system. The workshops, tasks, research literature, meetings of
teachers in school or didacticians at the college also have a role as ‘mediating
artefacts’. The rules of the activity system, now, include rules underpinning rigorous
research and rules governing teachers’ and students’ work in school, such as
following the national curriculum. However, through the project we anticipate that
our understanding of the developmental research paradigm will grow and as teachers
engage to a greater degree in teaching characterised by inquiry processes it is
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possible that their interpretation of the curriculum ‘rules’ may change. At the outset
of the project, there were a number of separate communities, the community of
didacticians and a number of school communities, each pursuing their own activity
largely independent of the others. In coming together within the project it was
recognised by each community that we can learn together and develop our practice.
The final item in Engeström’s model is ‘division of labour’. Inevitably in the
envisaged activity system didacticians, teachers and students will have distinct roles
that engage them in different tasks and actions. As teachers increasingly recognise
and value their own research potential and didacticians participate in school and
classroom, we anticipate that the division or labour will change. Thus we see, and
expect to see, developments in the activity system as we engage in it.
The LCM project emerges from a vision of an activity system whose motive is to
engage, collaboratively, didacticians, teachers and students in developing and
researching the teaching and learning of mathematics through processes of inquiry.
At the time the project was proposed, this activity system did not exist, nor did it
exist when the proposal began to be implemented. Now, halfway through the initial
funding period, an activity system exists but we question the extent to which it
fulfils what was envisioned. The vision is of a coherent community of co-learners
taking roles relevant to the nature of their participation with responsibility as
partners within the project. For example teachers and didacticians might be both
insider and outsider researchers: insiders, as they seek to explore and develop their
own practice and outsiders as they explore characteristics in their students’ learning
and understanding of mathematics, or in the activity of their co-participants
(Jaworski, 2005).
The words above point not only to possible divergence between original goals and
current activity, but to the transformative nature of the process in which we engage
and the problematic nature of what we experience. In promoting development of
inquiry communities we are motivated by theoretically warranted visions of
transformation in mathematics teaching and learning. In the reality of project
implementation, we recognize people, relationships, existing systems, ways of being
and thinking and obstacles to change. Every event emerging in research embodies a
complex story (e.g., Goodchild & Jaworski, 2005). We learn about the development
of inquiry communities as teachers and didacticians act together and embrace
notions of inquiry. We have talked about using inquiry as a tool leading to
development of inquiry as a way of being (Jaworski, in press). We see ‘inquiry as a
tool’ in many circumstances within the project. For example, tasks designed for
workshops promote joint asking of questions and associated exploration. Tasks,
questions and exploration are tools mediating activity. Use of these tools involves
goal directed actions comprising condition-related operations. However, ‘inquiry as
a way of being’ is a motive-goal (i.e., where the motive becomes conscious) of
activity, rather than an outcome from it. Achieving an inquiry stance is ongoing and
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Our developmental process is a struggle with developing thinking related to
intransigencies in everyday activity. While workshop activity might prove
inspirational in illuminating concepts of inquiry, and promote associated actions in
school activity, the actions and emergent thinking contend with the demands of
school activity, and established ways of thinking about it. Reports at a workshop of
actions deriving from the previous workshop, reveal activity and thinking that both
indicates elements of progress, and reveals limitations and constraints in vision and
practice. As the project progresses, our analysis of data both charts the nature of
development and reveals the problematic nature of that development. We are
submerged in the complexity of relationships, interactions, demands from established
communities with their deeply embedded ways of thinking, and our ongoing
struggles with changing thinking and practice according to theoretical principles. We
both recognize our situation as a complexity of activity systems, and draw on
analytical frameworks in AT to navigate the complexity.
The AT triangle offers a unit of analysis for all levels of the project. Its value lies in
the possibility of exploring the mediating elements and the dialectical relationships
between elements. As the project is intended as developmental research the concern
is to engineer, monitor and research changes within the activity system. Engeström
suggests that contradictions and tensions take a central role as sources of change and
development and thus the model can be used as both development and research tool
in that it draws attention to those points where contradictions or tensions exists,
whether these be within the elements or the dialectical relationships between the
elements and thus prompts “a search for solutions … (that) reaches its peak when a
new model for the activity is designed and implemented” Engeström (1999, p. 34).
Engeström refers to this process as “the expansive cycle” (ibid. p. 33).
It is easy to be discouraged when plans do not result in envisioned outcomes.
Particularly in the activity in schools, there have been many obstacles to realisation of
teams of teachers working in inquiry modes. The developmental nature of the project
is that we work with the perceived obstacles and through this work relationships
develop and forms of activity emerge that could not have been predicted in the
original design. We work with what we have, and rethink according to theoretical
principles and emerging reality. Periodically, in the cycles of activity and thinking, a
recognisably new way of acting and thinking emerges, and becomes new activity. We
see this as an expansive cycle. Its importance for the project is twofold. Firstly it is
manifested in a build up of tension with transformative power in that it promotes a
new wave of activity with clear actions and goals. Secondly, and possibly most
powerfully, it creates new learning in which we gain new insights to theoretical
realisation in social and historical complexity.
Chaiklin, S. (1993). Understanding the social scientific practice of understanding practice.
In S. Chaiklin & J. Lave (Eds.), Understanding practice (pp. 377-401). Cambridge:
Cambridge University Press.
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Cochran Smith, M. & Lytle, S. L. (1999). Relationships of knowledge and practice: teacher
learning in communities’. In A. Iran-Nejad & P. D. Pearson (Eds.), Review of Research
in Education (pp. 249-305). Washington: American Educational Research Association.
Engeström, Y. (1999). Activity theory and individual and social transformation. In Y.
Engeström, R. Miettinen & R-L Punamäki (Eds.), Perspectives on activity theory (pp. 1938). Cambridge: Cambridge University Press.
Goodchild, S. & Jaworski, B. (2005). Identifying contradictions in a teaching and learning
development project. In H. L. Chick & J. L. Vincent, (Eds.), Proceedings of the 29th
Conference of the International Group for the Psychology of Mathematics Education (pp.
41-47). Melbourne, Australia: University of Melbourne.
Jaworski, B. (1994). Investigating mathematics teaching. London: Falmer Press.
Jaworski, B. (2004). Insiders and outsiders in mathematics teaching development: The
design and study of classroom activity. In O. Macnamara & R. Barwell (Eds.), Research
in Mathematics Education: Papers of the British Society for Research into Learning
Mathematics: Vol 6 (pp. 3-22). London: BSRLM.
Jaworski, B. (2005). Learning communities in mathematics: Creating an inquiry community
between teachers and didacticians. In R. Barwell and A. Noyes (Eds.), Research in
Mathematics Education: Papers of the British Society for Research into Learning
Mathematics, Vol. 7 (pp. 101-119). London: BSRLM
Jaworski B. (in press). Theory and practice in mathematics teaching development: Critical
inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2).
Kelly, A. E. (2003). Research as design. Educational Researcher, 32 (1), 3-4
Leont’ev, A. N. (1979). The problem of activity in psychology. In J. V. Wertsch (Ed.), The
concept of activity in Soviet psychology (pp. 37-71). New York: M. E. Sharpe.
Mason, J. (2001). Researching your own classroom practice. London: Routledge Falmer.
Vygotsky, L. S. (1978). Mind in society. Cambridge MA: Harvard University Press.
Wertsch, J. V. (1991). Voices of the mind. Cambridge MA: Harvard University Press.
Wells, G. (1999). Dialogic inquiry. Cambridge: Cambridge University Press.
Wenger, E. (1998). Communities of practice. Cambridge: Cambridge University Press.
The LCM Project is supported within the KUL Programme (Knowledge, Education and Learning)
of the Norwegian Research Council. Project number 157949/S20.
We have demonstrated a fundamental difference between community of practice and community
of inquiry (e.g., Jaworski, in press). Space does not allow articulation here.
Key Norwegian concepts and terms have entered into our vocabulary and are difficult to replace
simply in English. Planlegg et opplegg refers to teachers’ planning of tasks for the classroom, and
their resulting lesson plans. Tasks and all their related feature are called oppgaver.
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