1. Theory in mathematics education
A sketch of key ideas and concepts

2. Overview Constructivism Sociocultural theories Social practice theory
Naïve, Radical & Social
Communication and Intersubjectivity
Sociocultural theories
Vygotsky – general law of cultural development
Mediation
Spontaneous and scientific concepts
Instruction and ZPD
Social practice theory
Activity theory
Barbara Jaworski MEC

3. Constructivism von Glasersfeld, Steffe, Cobb, Confrey
Naïve constructivism
Radical constructivism
Making sense /Cognition
Individual cogniser/constructor
Clinical interviews/Classroom experiments
Social constructivism
Intersubjectivity
First and second-order models
Barbara Jaworski MEC

4. 2 principles of Radical Constructivism
Constructivism is a theory of knowledge with roots in philosophy, psychology, and cybernetics. It asserts two main principles whose application has far reaching consequences for the study of cognitive development and learning, as well as for the practice of teaching, psychotherapy and interpersonal management in general. The two principles are:
1. Knowledge is not passively received but actively built up by the cognising subject;
2a The function of cognition is adaptive, in the biological sense of the term, tending towards fit or viability;
2b Cognition serves the subjects’ organisation of the experiential world, not the discovery of ontological reality.
(von Glasersfeld 1987; 1990; Cited in Jaworski, 1994)
Barbara Jaworski MEC

5. Post epistemological
If experience is the only contact a knower can have with the world, there is no way of comparing the products of experience with the reality from which whatever messages we receive are supposed to emanate. The question, how veridical the acquired knowledge might be, can therefore not be answered. To answer it one would have to compare what one knows with what exists in the ‘real’ world – and to do that one would have to know what ‘exists’. The paradox then is to assess the truth of your knowledge before you come to know it. (Glasersfeld 1983)
Radical constructivism is thus radical because it breaks with convention and delivers a theory of knowledge in which knowledge does not reflect an ‘objective’ ontological reality, but exclusively an ordering and organisation of a world constituted by our experience (Glasersfeld 1984)
(Cited in Jaworski, 1994 p. 17)
Noddings (1990) suggests that Radical Constructivism is post-epistemological
Barbara Jaworski MEC

6. Communication
Teachers and students are viewed as active meaning makers who continually give contextually based meanings to each others’ words and actions as they interact. The mathematical structures that the teacher ‘sees out there’ are considered to be the product of his or her own conceptual activity. From this perspective mathematical structures are not perceived, intuited, or taken in but are constructed by reflectively abstracting from the reorganising sensorimotor and conceptual activity. They are inventions of the mind. Consequently the teacher who points to mathematical structures is consciously reflecting on mathematical objects that he or she had previously constructed. Because the teachers and students each construct their own meanings for words and events in the context of the on-going interaction, it is readily apparent why communication often breaks down, why teachers and students frequently talk past each other. The constructivist’s problem is to account for successful communication .
(Cobb, 1998 cited in Jaworski 1994)
Barbara Jaworski MEC

7. Intersubjectivity
Intersubjectivity is a state where each participant in a socially-ongoing interaction feels assured that others involved in then interaction think pretty much as he or she imagines they do. … Intersubjectivity is not a claim of identical thinking. … Rather it is a claim that each person sees no reason to believe others think differently than he or she presumes they do. Thompson (ref??)
First and second-order models
See also Jaworski, 1994, pp
Compare with intersubjectivity as shared meanings and values within social settings.
Barbara Jaworski MEC

8. The third principle: social constructivism
Recognition of the social constructing of knowledge through its negotiation and mediation with others:
The third principle derives from the sociology of knowledge, and acknowledges that reality is constructed intersubjectively, that is it is socially negotiated between significant others who are able to share meanings and social perspectives of a common lifeworld (Berger & Luckmann, 1966). This principle acknowledges the sociocultural and socioemotional contexts of learning, highlights the central role of language in learning, and identifies the learner as an interactive co-constructor of knowledge .
(Taylor & Campbell-Williams, 1993; cited in Jaworski, 1994, p. 24)
Barbara Jaworski MEC

9. Sociocultural theories
Wertsch, Cole, Rogoff, Lerman
Culture and cognition
Language & discourse
Community
Participation
Mediation
Situated cognition
Distributed cognition
Barbara Jaworski MEC

10. Vygotsky
Human learning presupposes a special social nature and a process by which children grow into the intellectural life of those around them. (1978, p.88)
Every function of a child’s cultural development appears twice: first, on the social level, and later, on the individual level; first between people (interpsychological), and then inside the child (intrapsychological). This applies equally to voluntary attention, to logical memory, and to the formation of concepts. All the higher functions originate as actual relations between human individuals.
(1978, p. 57. Emphasis in original.)
Barbara Jaworski MEC

11. Mediation
Vygotsky emphasized the importance in social participation, or action, of the mediation of tools or signs.
Wertsch (1991), following “the tradition of the theory of activity proposed by A. N. Leont’ev”, refers to “goal-directed action” and writes, “human action typically employs ‘mediational means’ such as tools and language”.
As social beings we use tools (physical) or signs (intellectual) to enable us to achieve the objects or goals of our activity.
One of the most important tools, or signs, is language. We use language both to express meaning and to make sense of the activity in which we engage.
Barbara Jaworski MEC

12. Based on Vygotsky’s model of a complex mediated act
MEDIATING ARTEFACTS
SUBJECT
OBJECT
OUTCOME
Based on Vygotsky’s model of a complex mediated act
Barbara Jaworski MEC

13. individual(s)-acting-with-mediational-means
Wertsch (1991) emphasises 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)’” (p. 12).
Leont’ev writes, “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 ).
Wertsch contrasts two ideas: rather than seeing mental functioning as deriving from participation in social life, he suggests that
the specific structures and processes of intramental processing can be traced to their genetic precursors on the intermental plane (p. 27).
In other words, individual mental thought processes and functioning have their origins fundamentally in social interaction.
Barbara Jaworski MEC

14. Spontaneous and scientific concepts
Spontaneous concepts
Learning through participation in social endeavour
Scientific concepts
Needing pedagogical mediation for their appropriation (Schmittau, 2003, p.226)
Vygotsky 1986, pp
Barbara Jaworski MEC

15. Zone of proximal development
ZPD is ‘an account of how the more competent assist the young and less competent to reach the higher ground from which to reflect more abstractly about the nature of things (Bruner, 1985).
The ZPD is the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers. (Vygotsky, 1978, pp. 84-7)
Thus the notion of a [ZPD] enables us to propound a new formula, namely that the only ‘good learning’ is that which is in advance of development (p.89).
Bruner: scaffolding, vicarious consciousness
Barbara Jaworski MEC

16. Social practice theory
Lave & Wenger, Wenger, Adler
Situated cognition
Community of practice
Knowledge in practice
Learning in practice
Belonging and becoming
Participation
Legitimate peripheral participation
Reification
Barbara Jaworski MEC

17. Community of practice
The term ‘community’ designates a group of people identifiable by who they are in terms of how they relate to each other, their common activities and ways of thinking, beliefs and values. Activities are likely to be explicit, whereas ways of thinking, beliefs and values are more implicit.
Wenger (1998, p. 5) describes community as “a way of talking about the social configurations in which our enterprises are defined as worth pursuing and our participation is recognisable as competence”.
In a learning community, “learning involves transformation of participation in collaborative endeavour” (Rogoff, 1996, p. 388).
The concept of practice connotes doing, but not just doing in and of itself. It is doing in a historical and social context that gives structure and meaning to what we do. In this sense practice is always social practice (Wenger, 1998, p.47).
Mathematics as a social practice (Hemmi)
Barbara Jaworski MEC

18. Legitimate peripheral participation
L&W 1991
The newcomer joining the practice and being (in a legitimate way) on its periphery
Masters, or old-stagers, being the ones who define the practice and perpetuate it.
Newcomers working towards centrality in the practice.
Barbara Jaworski MEC

19. Participation and reification
According to Wenger in a community of practice we have
Mutual engagement: doing things together, relationships, social complexity, community maintenance
Joint enterprise: mutual accountability, interpretations, rhythms, local response
Shared repertoire: stories, artefacts, tools, actions, discourses, historical events, concepts
(1998, p. 72ff)
Learning involves participation and reification – “dual modes of existence through time” (p. 87).
Barbara Jaworski MEC

20. Belonging to a community of practice
Identity in engagement, imagination and alignment (Wenger, 1998)
Normal desirable states (Brown & McIntyre, 1993)
Critical alignment (Jaworski 2006)
Barbara Jaworski MEC

21. Activity theory Leontev, Davidov, Engeström, Mellin Olsen Activity
Mediation
Tools and signs, artefacts
Vygotsky’s mediational triangle
Leont’ev’s Motives and goals
Expanded mediational triangle
Barbara Jaworski MEC

22. Motives and goals
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  motive; actions  goals; operations  conditions.
Barbara Jaworski MEC

23. Activity (System) & Motive Actions & Goals Operations & Conditions
Developmental research, whose motive is to study developmental processes and, simultaneously, promote development in the learning and teaching of mathematics.
Asking researchable questions, collecting and analysing data leading to findings or outcomes related to new knowledge and/or practice.
Making methodological decisions related to principled and effective ways of collecting and analysing data to address research questions.
The LCM project as community of inquiry, with motive to provide the environment and modes of action for teaching development to be realised.
Creating opportunities for working together and engaging in inquiry to achieve a working community with practical knowledge of inquiry processes.
Teachers and didacticians working in groups in workshops on mathematical problems to exemplify inquiry processes and develop common understandings.
Working together on mathematics as a common area of interest and purpose in order to learn more about inquiry processes and their application in classrooms
Designing mathematical tasks that will be accessible across a spectrum of mathematical experience and relatable to teaching at a range of levels
Assembling a set of resources (e.g., books, software, ideas) and possibilities for translating these into usable forms in the workshop environment and beyond
Barbara Jaworski MEC

24. Engeström’s ’complex model of an activity system’
MEDIATING ARTEFACTS
SUBJECT
OBJECT
OUTCOME
RULES
COMMUNITY
DIVISION OF LABOUR
Engeström’s ’complex model of an activity system’
Barbara Jaworski MEC

25. References
Brown, S. & McIntyre, D.: 1993, Making Sense of Teaching, Open University Press, Buckingham
Bruner, J. S. (1985). Vygotsky: A historical and conceptual perspective. In J. V. Wertsch (Ed.) Culture Communication and Cognition: Vygotskian Perspectives. Cambridge: Cambridge University Press.
Engeström, Y. (1999). Activity theory and individual and social transformation. In Y. Engeström, R. Miettinen & R-L Punamäki (Eds.), Perspectives on activity theory (pp ). Cambridge: Cambridge University Press.
Jaworski, B. (1994). Investigating mathematics teaching. London: Falmer Press.
Jaworski B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2) pp. xx-xx
Jaworski, B., & Goodchild, S.: 2006, ‘Inquiry community in an activity theory frame’, in Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education, Charles University, Prague,v. 3, pp Lave, J.: 1996 ‘Teaching as Learning, in Practice’, in Mind Culture and Activity, 3(3), pp. 
Lave, J. & Wenger, E.: 1991, Situated Learning: Legitimate Peripheral Participation. Cambridge University Press, Cambridge, MA.
Leont’ev, A. N. (1979). The problem of activity in psychology. In J. V. Wertsch (Ed.), The concept of activity in Soviet psychology (pp ). New York: M. E. Sharpe.
Rogoff, B., Matusov, E. and White, C.: 1996, ‘Models of Teaching and Learning: Participation in a community of learners’, in D. R. Olson and N. Torrance (eds.), The Handbook of Education and Human Development, Blackwell, Oxford, pp
Schmittau, J.: 2003: ‘Cultural-Historical Theory and Mathematics Education’, in A. Kozulin, B. Gindis, V. S. Ageyev and S. M. Miller, (eds.), Vygotsky’s Educational Theory in Cultural Context, Cambridge University Press, Cambridge, pp
Thompson, P. (2002). Didactic objects and didactic models in radical constructivism. In K. Gravemeier, R. Lehrer, B. van Oers, & L Verschaffel (Eds)., Symbolizing. Modeling and Tool Use in Mathematics Education. Dordrecht, The Netherlands: Kluwer.
Vygotsky, L. S.: 1978, Mind in society, Harvard University Press, Cambridge MA.
Vygotsky, L. S. (1986) Thought and Language. Cambridge, MA: MIT Press.
Barbara Jaworski MEC