Understanding Abstraction in Mathematics Education: An Analytical Framework

1. Abstraction in Context -an introduction Tommy Dreyfus, Tel Aviv University, Israel MERGA 31, Brisbane, AUS June 30, 2008 Research supported by the Israel Science Foundation under grants 973/02 and 1166/05

2. The complexity of (research in) mathematics education • Even a seemingly simple event in a mathematics classroom is a complex issue • Different researchers have different interests and think in different theoretical frameworks about such events • Their focus may be (some but not all of) cognitive, social, cultural, affective, beliefs, design, learning environment, … • As researchers, we have to be aware that we always deal with some aspects of a problem or situation and ignore others Auckland, NZ: LOGOS

3. The focus • Historically: • A curriculum development program • Design-research-design cycles • ‘Rich’ activities • What remains (is consolidated)? • The focus is on cognitive processes, especially abstraction, emergence of new knowledge constructs • The learning environment is considered as context within which these processes take place • We propose a framework that allows us to analyse such processes at the micro-level Auckland, NZ: LOGOS

4. Abstraction in Context (AiC) • Approach developed over the past ten years with Rina Hershkowitz, Baruch Schwarz and others • Abstraction is a process of interweaving earlier constructs and leading to a construct that is new for the learner • Abstraction is an activity of vertical [Freudenthal, Treffers & Goffree] reorganisation of knowledge, within mathematics and by mathematical means • Vygotsky, Davydov, … Auckland, NZ: LOGOS

5. Abstraction in Context • Processes of abstraction take place in context • learning context (classroom, available tools incl ICT) • historical (prior experience and learning) • social context (peers, teacher) • curricular (task sequence) • More on context below, if time permits Auckland, NZ: LOGOS

6. The nested epistemic actions model of abstraction in context • This is the name of our tool for analysis • The name expresses that • epistemic actions form the main tool of analysis • epistemic actions are dynamically nested • we attribute great importance to context Auckland, NZ: LOGOS

7. Epistemic actions • Epistemic Actions are observable mental actions by means of which knowledge is constructed (Pontecorvo & Girardet, 1993) • We found the following three epistemic actions useful for the analysis of processes of abstraction: • Recognizing • Building-With • Constructing Auckland, NZ: LOGOS

8. Recognizing (a previous construct) • The 're-cognition' of previously encountered mental constructs that are inherent in a given mathematical situation Auckland, NZ: LOGOS

9. Building-with (previous constructs) • The combination of mental constructs in order to achieve a given goal • Goals: • solving a problem • understanding and explaining a situation • reflecting on a process Auckland, NZ: LOGOS

10. Constructing (a new construct) • ‘Cognizing’ novel constructs • Assembling and integrating previous constructs by vertical mathematization to produce a new construct • Constructs include • Methods • Concepts • Strategies • Process may be slow or sudden Auckland, NZ: LOGOS

11. Dynamic Nesting • In processes of abstraction, the epistemic actions are dynamically nested: • R-actions are nested in B-actions: you cannot build-with a construct unless you have first recognized it • Similarly, R-actions and B-actions are always nested in C-actions; in fact, C-actions consist of (alternating) R and B actions • C-actions at different levels may be nested in each other since I may need a certain construct in order to reach another one Auckland, NZ: LOGOS

12. The genesis of an abstraction • Processes of abstraction have three stages • The need for a new construct • The emergence of a new construct • The Consolidation of the new construct • The second stage is the central one, and so far I have mainly related to this stage • I will now briefly relate to the other two stages Auckland, NZ: LOGOS

13. Stage 1: The need for a new construct • This need is inherent in the design but it is relative to the context: • The student population • Their prior knowledge and experience • Available tools such as computer tools • Habits of collaboration • Our research, so far, has concentrated on the second and third stages of processes of abstraction; we have taken the need for granted – provided by the instructional design. We plan research on the first stage in the near future. Auckland, NZ: LOGOS

14. Stage 3: Consolidation • Consolidation is a long-term process • Consolidation is likely to occur during problem-solving and reflection activities • Consolidation contributes to awareness of one’s use of the constructs and to flexible problem solving Auckland, NZ: LOGOS

15. Mechanisms of consolidation • The analysis of the work of students in sequences of activities over several lessons has allowed us to identify several mechanism of consolidation • The most interesting of these is the consolidation of a previous construct during the process of constructing a further one, with the earlier one serving as an element in constructing the new one • For the other mechanisms, as well as for example, I refer to the literature (Schwarz, Hershkowitz & Dreyfus, 2008) Auckland, NZ: LOGOS

16. The role of context Auckland, NZ: LOGOS

17. Context • Computer tools may be a component of the context. In a recent paper, we analyzed the influence of a computer tool on construction of knowledge (Kidron & Dreyfus, 2008). More research in this direction is planned. • Another important aspect of context is social context. For example, in MERJ (Hershkowitz et al. 2007), we analyzed the social construction of knowledge by student groups in classrooms. Auckland, NZ: LOGOS

18. Social context • Hershkowitz et al. investigated processes by which two groups of individual students (three students each) construct shared knowledge and consolidate it. • We identified an interactive flow of knowledge from one student to the others, in the group, until they reach a shared knowledge– a common basis of knowledge, which allowed them to continue together the constructing of further knowledge in the same topic. Auckland, NZ: LOGOS

19. Sample topics (student age/authors) of published AiC-based studies • Rate of change as a function (14/HDS) • Algebra as a tool for justification (12/DHS) • The power of a countably infinite set (16/TD) • Elementary probability concepts (13/RDH, …) • Function transformation (17/OM) • Bifurcations in a dynamical system (adult/DK) • Limits (adult/K) • Finite arithmetic structures (adult/S) Auckland, NZ: LOGOS

20. Thank you! tommyd@post.tau.ac.il Auckland, NZ: LOGOS