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. 13 May 2008 Auckland, NZ: LOGOS 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

3. 13 May 2008 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

4. 13 May 2008 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, …

5. 13 May 2008 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

6. 13 May 2008 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

7. 13 May 2008 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

8. 13 May 2008 Auckland, NZ: LOGOS 8 Recognizing (a previous construct) The 're-cognition' of previously encountered mental constructs that are inherent in a given mathematical situation

9. 13 May 2008 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

10. 13 May 2008 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

11. 13 May 2008 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

12. 13 May 2008 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

13. 13 May 2008 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.

14. 13 May 2008 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

15. 13 May 2008 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)

16. 13 May 2008 Auckland, NZ: LOGOS 16 The role of context

17. 13 May 2008 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.

18. 13 May 2008 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.

19. 13 May 2008 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)

20. 13 May 2008 Auckland, NZ: LOGOS 20 Thank you! tommyd@post.tau.ac.il