PROCESSES OF ABSTRACTION IN CONTEXT THE NESTED EPISTEMIC ACTIONS MODEL

1. PROCESSES OF ABSTRACTION IN CONTEXTTHE NESTED EPISTEMIC ACTIONS MODEL Tommy DreyfusRina HershkowitzBaruch Schwarz ISF Jerusalem workshop on Guided Construction of Knowledge in Classrooms February 5, 2007

2. ABSTRACTION AS AN ACTIVITY historical curricular EPISTEMIC ACTIONS R B C NESTED INTERACTING learningcontext social Nested epistemic actions

3. An exampleThe task: All questions refer to the following two rectangles: • Find as many properties as you can • Compare the sums of the diagonals (DSP) • Will the property always hold? • Compare the products of the diagonals (DPP) • Will the property always hold? • Justify your claims Nested epistemic actions

4. The pair HaNe (Grade 7) • Had used letters • Had used (a+b)c=ac+bc • Had not used algebra as a tool for justification • Stated the DSP with no urge to justify • Stated the DPP • Raised the issue of justification • Justified the DSP! • Came back to justify DPP when asked to do so. Nested epistemic actions

5. HaNe’s justification of the DPP • Did we already prove it? • Extending the distributive law • Hesitation • The difference 12 becomes significant • Closure Nested epistemic actions

6. The second segment: (x+6) (x+2) • Ha133 And this [thinks] ... • Ne134 It's impossible to do the distributive law here. Wait, one can do ... • Ha135 This is 6x. • Ne136 Then this is 6x times x and 6x times 2. • Ha137 Wait, first, no ... • Ne138 Yes. • Ha139 No because this is x plus 6, this is not 6x, it's different. Wait. First one does x; then it's xx plus 2x, and here 6x plus 24. Then ... • In140 Not 24, why 24? • Ha141 Ah, 12. Nested epistemic actions

7. Algebra as a tool for justification • Together, the five segments constitute a proof of DPP • The girls used algebra as a tool for this proof • This is a new construct for them: Algebra can be used as a tool to prove general claims • Within this process, they have used the extended distributive law, another new construct for them • Their knowledge about algebra has (potentially) been restructured, made more profound Nested epistemic actions

8. ABSTRACTION AS AN ACTIVITY • The new construct arises during mathematical activity • The motivation for the new construct arises out of a mathematical need for it • The thinking mode is mathematical • Vertical mathematization (Freudenthal school) • Abstraction is an activity of vertically reorganizing previous mathematical constructs into a new mathematical construct • Abstraction is not objective and universal • It depends on context Nested epistemic actions

9. THE EPISTEMIC ACTIONS • Actions used in processes of construction of knowledge • Observability • Data-based theory-building led to • Recognizing • Building-with • Constructing • RBC Nested epistemic actions

10. Recognizing The 're-cognition' of previously encountered mental structures that are inherent in a given mathematical situation. • Ha133 And this [thinks] ... • Ne134 It's impossible to do the distributive law here. Wait, one can do ... • Ha135 This is 6x. • Ne136 This is 6x times x and 6x times 2. • Ha137 Wait, first, no ... • Ne138 Yes. • Ha139 No because this is x plus 6, this is not 6x... Nested epistemic actions

11. Building-With The combination of elements of mental structures in order to achieve a given goal. • Ne136 This is 6x times x and 6x times 2. • Ha137 Wait, first, no ... • Ne138 Yes. • Ha139 No because this is x plus 6, this is not 6x, it's different. Wait. First one does x; then it's xx plus 2x, and here 6x plus 24. Goals: • solving a problem [computing (x+6)(x+2)] • justifying a statement (DPP) Nested epistemic actions

12. Constructing • ‘Cognizing’ novel constructs • Vertical reorganization of knowledge • Objects to be constructed include: Methods, strategies, concepts • Process may be slow or sudden, short or long • For example: • The extended distributive law (short) • Algebra as a tool for justification (intermediate/long) Nested epistemic actions

13. C1 C Dynamic Nesting • In processes of abstraction, the epistemic actions are dynamically nested. • R nested in B: • Ha139 First one does x; then it's xx plus 2x, and here 6x plus 24. • R, B nested in C • Possibly lower level C nested in C. • C1 - Constructing the extended distributivelaw -consists of R and B actions • C - Algebra as a tool – consists of C1andmany R and B-actions Nested epistemic actions

14. InteractingParallelConstructions For example: Branching Combining Related to context, Specifically to the way computer tools are used Nested epistemic actions

15. Consolidation • Constructing • First instance • Awareness? • Fragility of knowledge • Sequences of activities • Countably infinite sets (Tsamir and Dreyfus) • The probability project (Hershkowitz, Hadas, Schwarz, Dreyfus) • Algebra/exponential growth (Tabach, Hershkowitz, Schwarz) Nested epistemic actions

16. Consolidation (cont.) • Characteristics of consolidation • Awareness • Confidence • Flexibility • Conducive for consolidation • Problem solving • Reflection • Further constructing Nested epistemic actions

17. In conclusion The genesis of an abstraction in three stages • The mathematical need for a new construct • Constructing – first time emergence • Consolidating – an indefinite process The analytical power of the model • Combining constructions and justification Nested epistemic actions

18. Some uses of the model • Dooley – Multiplication / Whole class interaction • Bikner-Ahsbahs – Fractions / Interest • Hershkowitz Schwarz and Dreyfus – Algebra / Dyadic interaction • Ron, Dreyfus and Hershkowitz – Probability / Partially correct knowledge constructs • Ozmantar and Monaghan – Function transformations / Consolidation • Williams – Calculus / Creativity • Kidron and Dreyfus – Dynamic processes, chaos, bifurcations / Solitary learner • Stehlikova – Abstract algebra / Solitary learner Nested epistemic actions

19. Thank you for listening! tommyd@post.tau.ac.il Nested epistemic actions