processes of abstraction in context the nested epistemic actions model tommy dreyfus rina...

PROCESSES OF ABSTRACTION IN CONTEXT

THE NESTED EPISTEMIC ACTIONS MODEL

Tommy DreyfusRina HershkowitzBaruch Schwarz

ISF Jerusalem workshop on Guided Construction of Knowledge in Classrooms

February 5, 2007

February 5, 2007 Nested epistemic actions 2

ABSTRACTION

AS AN ACTIVITY

ACTIONSEPISTEMIC

R B C

NESTED

INTERACTING

social

curr

icul

ar

hist

oric

al

learningcontext


An example The 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

7 13

9 15

3 9

5 11


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.


HaNe’s justification of the DPP

1) Did we already prove it?

2) Extending the distributive law

3) Hesitation

4) The difference 12 becomes significant

5) Closure


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.


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


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


THE EPISTEMIC ACTIONS

• Actions used in processes of construction of knowledge

• Observability

• Data-based theory-building led to– Recognizing– Building-with– Constructing

• RBC


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...


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)


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)


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 C

C1


1

2

3

4

5

6

7

89

10

11

12

13

14

15

16

C2

C2

C1

C1

C3

C4

C3

InteractingParallel

Constructions

For example:BranchingCombining

Related to context,Specifically tothe way computer tools are used


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)


Consolidation (cont.)

• Characteristics of consolidation– Awareness– Confidence– Flexibility

• Conducive for consolidation– Problem solving– Reflection– Further constructing


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


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


Thank you for listening!

[email protected]

processes of abstraction in context the nested epistemic actions model tommy dreyfus rina...

Documents