Theory in mathematics education

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

2. Overview • Constructivism • 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 2008 4

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 2008 4

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 2008 4

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 2008 4

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 2008 4

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.208-212 • Compare with intersubjectivity as shared meanings and values within social settings. Barbara Jaworski MEC 2008 4

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 2008 4

9. Sociocultural theories Wertsch, Cole, Rogoff, Lerman • Culture and cognition • Language & discourse • Community • Participation • Mediation • Situated cognition • Distributed cognition Barbara Jaworski MEC 2008 4

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 2008 4

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 2008 4

12. MEDIATING ARTEFACTS SUBJECT OBJECT OUTCOME Based on Vygotsky’s model of acomplex mediatedact Barbara Jaworski MEC 2008 4

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. 47-48). • 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 2008 4

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 146-209 Barbara Jaworski MEC 2008 4

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 2008 4

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 2008 4

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 2008 4

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 2008 4

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 2008 4

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 2008 4

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 2008 4

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 2008 4

23. Barbara Jaworski MEC 2008 4

24. MEDIATING ARTEFACTS OBJECT SUBJECT OUTCOME RULES COMMUNITY DIVISION OF LABOUR Engeström’s ’complex model of an activity system’ Barbara Jaworski MEC 2008 4

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. 19-38). 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 the30th Conference of the International Group for the Psychology of Mathematics Education, Charles University, Prague,v. 3, pp.353-360.Lave, J.: 1996 ‘Teaching as Learning, in Practice’, in Mind Culture and Activity, 3(3), pp. 149-164. 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. 37-71). 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.388-414. 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.225-245. 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 2008 4