Introduction to Theories in Mathematics Education

1. Introduction to Theories in Mathematics Education Analysing and making sense of learning and teaching mathematics

2. Overview • What is theory • Skemp • Teaching & Learning • Epistemology • Mind and Cognition • Environment and culture • Some theoretical perspectives • Rationalism • Associationism • Piaget • Vygotsky • Constructivism (& Interactionism) • Sociocultural theories • Social practice theory • Activity theory • Sources Barbara Jaworski MEC 2008 2

3. What is theory? • What kinds of things do we have theories about? • Whose theories are they? Barbara Jaworski MEC 2008 2

4. Naïve empiricism The literature Grand theory Local (middle range) theory Personal theory (BJ) Bryman p. 5-7 Barbara Jaworski MEC 2008 2

5. Skemp • Instrumental and relational understanding • First and second order concepts • Concept building • Abstraction process • Generalising from experience • Explanation and definition Barbara Jaworski MEC 2008 2

6. Pedagogy What kind of activity from me (as teacher) will enable my students to learn? Learning theories Forms of practice or activity Methods and materials Organisation and approaches Didactics What kind of tasks will enable my students to engage with mathematics and to understand mathematics? Using knowledge of mathematics Using knowledge of pedagogy Teaching Barbara Jaworski MEC 2008 2

7. Learning • What does it mean to learn? • To learn mathematics? • How do we learn? • How do we learn mathematics? • What forms of learning do we recognise? • Is mathematics special? Notice the questioning style above… WHY? Barbara Jaworski MEC 2008 2

8. Epistemology What assumptions do we make about knowledge? • Explicit or tacit (Polanyi, Schön) • External, objective, given by a higher authority • Human, constructed by the individual • Human, growth through activity, practice, culture Barbara Jaworski MEC 2008 2

9. Research Questions Why do some students avail themselves to mathematics support whilst others do not? Which new technologies are being used in mathematics lectures and how? Does expanding the ability to generate examples increase other areas of mathematical ability? Is there a need to improve the teaching of mathematics and to what extent? To what extent is learning mathematics a social activity? Barbara Jaworski MEC 2008 2

10. Break Barbara Jaworski MEC 2008 2

11. Mind and Cognition • What is the mind? • Where is the mind? (Cobb) • How can we characterise mind? Barbara Jaworski MEC 2008 2

12. Rationalism • Mind consists of innate components of knowledge that work in relation to each other – biologically configured (hard wired) • Learning is based on revealing to ourselves, through a process of logical deduction or rational discourse, the knowledge we already have, motivated by innate instincts or urges. • People are innately different in their qualities: • nourished by external experience, individual knowledge grows, rather like a plant grows, nourished by external nutrients. Barbara Jaworski MEC 2008 2

13. Thus, to promote learning we need to provide rich learning experiences to nurture growth, to enable what is inside to come out; to make the most of potential. • A.S. Neill; Maria Montessori; Noam Chomsky • we have mental organs governing what we can learn about language, mathematics, music, etc. ‘Language Acquisition Device’ (LAD) • Howard Gardner’s multiple intelligences geared to different ways of knowing and special aptitudes (see next slide) • Failure is due to lack of innate intelligence, or ability Barbara Jaworski MEC 2008 2

14. Intelligence type Capability and perception Linguistic words and language Logical-Mathematical logic and numbers Musical music, sound, rhythm Bodily-Kinesthetic  body movement control Spatial-Visual images and space Interpersonal other people's feelings Intrapersonal self-awareness Barbara Jaworski MEC 2008 2

15. Associationism (behaviourism) We learn by making associations and recording them in memory: linked entities such that the recall of one will lead to the recall of another. • Association of events that occur closely in time: repetition reinforces linkage; chains of association are formed. • Lock, Hume and Mill: all learning is by association • Difficulty of observing mental states led to focus on observable behaviour. Barbara Jaworski MEC 2008 2

16. Experimentation in conditioning of reflexes: frequency is significant, particularly the time lapse between action and reinforcemente.g. Pavlov, Thorndike, Skinner • Social behavioural conditioning: e.g. aversion or reward treatment. • Necessity for reinforcement to be linked to required behaviour: e.g. teacher providing ‘incentives’. Impossibility of conditioning for wider complexity of knowledge Barbara Jaworski MEC 2008 2

17. Piaget • Stage theory • Schema theory • Assimilation, accommodation, reflective abstraction • Clinical method Barbara Jaworski MEC 2008 2

18. Vygotsky • Social nature of learning • Importance of language • Internalisation • Spontaneous and scientific concepts • Importance of instruction • Zone of proximal development Barbara Jaworski MEC 2008 2

19. 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 2

20. Sociocultural theories Wertsch, Cole, Rogoff, Lerman • Culture and cognition • Language & discourse • Community • Participation • Mediation • Situated cognition • Distributed cognition Barbara Jaworski MEC 2008 2

21. Social practice theory Lave & Wenger, Wenger, Adler • Situated cognition • e.g., Brown et al, Greeno, Carraher & Schleiman, Kirchner & Whitson • Knowledge in practice • Learning in practice • Participation • Legitimate peripheral participation • Reification • Community of practice • Belonging and becoming Barbara Jaworski MEC 2008 2

22. Activity theory Leontev, Davidov, Engeström, Mellin Olsen • Activity • Mediation • Tools and signs, artefacts • Vygotsky’s mediational triangle (next slide) • Expanded mediational triangle (next next slide) Barbara Jaworski MEC 2008 2

23. MEDIATING ARTEFACTS SUBJECT OBJECT OUTCOME Based on Vygotsky’s model of acomplex mediatedact Barbara Jaworski MEC 2008 2

24. MEDIATING ARTEFACTS OBJECT SUBJECT OUTCOME RULES COMMUNITY DIVISION OF LABOUR Engeström’s ’complex model of an activity system’ Barbara Jaworski MEC 2008 2

25. Sources 1 • Wood: general introduction to Piaget, Vygotsky and Bruner • Jaworski: Explaining constructivism in terms of individual learning and development • Ernest: providing an epistemological framework for talking about theory • Sierpinska and Lerman: relating theory of knowledge and theories of learning Barbara Jaworski MEC 2008 2

26. Piaget Skemp Von Glasersfeld Steffe/Confrey/ Cobb  Bauersfeld/Voigt Ernest  Jaworski  Vygotsky Bruner  Cole/Wertsch/ Engeström Lerman Bartolini Bussi Steinbring/Seeger Mellin-Olsen Sources 2 Barbara Jaworski MEC 2008 2

27. Some References Adler, J. (2000). Social Practice Theory and Mathematics Teacher Education. Nordic Studies in Mathematics Education. 8, 3, 31-54. Ayer A. J. & Winch R. (Eds) (1958) British Empirical Philosophers: Locke, Berkeley, Hume, Reid andJ.S.Mill. London: Routledge and Kegan Paul 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. Davis, B., Maher, C., & Noddings, N. (1990). Constructivist Views of The Teaching and Learning of Mathematics. Journal for Research in Mathematics Education. Monograph 4. NJ: National Council of Teachers of Mathematics. [Chapters from von Glasersfeld; Noddings; Cobb, Wood & Yackel; Confrey; Steffe) Ernest, P. (1991). The Philosophy of Mathematics Education. London: Falmer. Goodchild, S. (2001). Students’ Goals: a case study of activity in a mathematics classroom. Bergen, Norway: Caspar Forlag. Goos, M., Galbraith, P., & Rensahw, P. (1999). Establishing a community of practice in a secondary mathematics classroom. In L. Burton (Ed.), Learning mathematics: from hierarchies to networks. London: Falmer Press. Barbara Jaworski MEC 2008 2

28. Some refs continued Jaworski, B. (1988). Is versus seeing as: constructivism and the mathematics classroom. In D. Pimm (Ed.), Mathematics Teachers and Children. London: Hodder and Stoughton. Jaworski, B. (1994). Investigating Mathematics Teaaching: A constructivist enquiry. London: Falmer Press.Jaworski, B. (2002). Social constructivism in mathematics learning and teaching. In L. Haggarty (Ed.), Teaching Mathematics in Secondary Schools. London: Routledge. Kirshner, D. & Whitson, J. A. (Eds) (1997). Situated Cognition: Social, Semiotic and Psychological Perspectives. London: Lawrence Erlbaum Associates. Lerman, S. (1996). Intersubjectivity in Mathematics Learning: A challenge to the radical constructivist paradigm. Journal for Research in Mathematics Education, 27(2), 133-150. Piaget, J. (1950). The Psychology of Intelligence. London: Routledge and Kegan Paul. Polanyi, M. (1958). Personal Knowledge. London: Routledge and Kegan Paul. Schön, D. (1987). Educating the reflective practitioner. Oxford: Jossey Bass. Steffe, L. P. & Thompson, P. W. (2000). Interaction or intersubjectivity. A reply to Lerman. Journal for Research in Mathematics Education, 31(2), 191-209. Vygotsky, L. (1978). Mind in Society. London: Harvard University Press. Barbara Jaworski MEC 2008 2