On the role of the Parsimony Principle in AMT Boris Koichu Technion – Israel Institute of Technology October 16, 2007

The Principle of Parsimony (Ockham’s Razor) Classic formulation: “One should not make more assumptions than the minimum needed” = = “One should only add new assumptions when forced to do so by the evidence” Status: logical principle, heuristics Classic implications: modeling, software programming, keeping science on track by not allowing it to accept “wild speculations” 2

The Principle of Parsimony (modification) Modification: “One does not make more efforts than the minimum needed in achieving a goal” = = “One only makes more efforts when forced to do so by the evidence than the goal is not achieved with less efforts” Status: assumption, heuristics Implications: explanation of some phenomena in MT (AMT) 3

Outline of the talk Two examples and one non-example Suggestions about the role of the Principle of Parsimony (PP) in developing AMT An open question about PP 4

Example 1: Should a mathematical definition be minimal? Zaslavsky, O, & Shir, K. (2005). Students’ conceptions of a mathematical definition, JRME, 36(4), Minimal Definition: A square is a quadrilateral in which all sides are equal and one angle is 90 o Not Minimal Definition: A square is a quadrilateral in which all sides are equal and all angles are 90 o 5

A square is a quadrilateral in which all sides are equal and all angles are 90 o Erez: It’s correct, but it is not a definition. Yoav: It’s correct, and it is a definition. Erez: It has too many details. Yoav: Too many details, but it is still a definition. Omer: What do “too many details” has to do with that? Erez: Well…In fact…maybe it is. “We can see that Erez began rethinking the issue of minimality. As a result, later he was willing to consider a nonminimal statement as a definition” Zaslavsky & Shir (2005, p. 329) 6

There is a hot debate about the issue of minimality in the literature. Why? An (indirect) answer from the paper: Why minimal? Because of mathematical-logical considerations Why not minimal? Because of communicative considerations (example-based reasoning, clarity) 7

Explanation in terms of PP Why should a definition be minimal? Because of the classical PP: One should not make more assumptions than the minimum needed Why may a definition be not minimal? Because of the reformulated PP: One does not make more efforts than the minimum needed in achieving a (communicative) goal 8

Why did Erez rethink his position about minimality? Perhaps, because he encountered cognitive dissonance between two instantiations of the PP and tried to balance them. Cognitive dissonance means the ability of a person to simultaneously hold at least two opinions or beliefs that are logically or psychologically inconsistent (Festinger, 1957). 9

Example 2: Koichu & Berman (2005). When do gifted high school students use geometry to solve geometry problems? The Journal of Secondary Gifted Education, 16(4), EEC: = Efficiency vs. Elegancy Conflict Int.: How do you approach difficult geometry problems? Saul: Generally speaking, I try to understand what the fuzziest point in the problem is and then I apply my intuition to this point… If I don’t have any idea what to do, I just use different not nice methods. Int.: What do you mean? Saul: I use the special methods that I have learned, like complex numbers in geometry, or trigo…where there is no choice. Int.: Why do you think that these methods are not nice? Saul: They could be nice, but…[sighs, pause 5 seconds]. Int.: Which methods are "nice"? Saul: “Nice” is when I have a geometry problem and I solve it by means of classic geometry. 10 Saul

Angle problem: Let ABC be an acute angle triangle, AH is the longest altitude of the triangle and BM is a median, AH=BM. Prove that First step – Drawing (about 60 sec.) Silence (about 10 sec.) 11 Second step – "It would be better to compute everything by formulas.“ (about 5 min.) Alex: I’ve finished. It is not a difficult problem. Int.: Anyway, you wrote a lot, but there is a very short geometrical solution… Alex: Maybe, but in order to find a geometrical solution you usually spend much more time. Personally, I like geometrical solutions, but I look for them only if I cannot solve a problem by trigo.

Explanation in terms of PP Why do gifted students like geometrical solutions more than algebraic? Because of the classic PP: One should use no more mathematical tools than the minimum needed. Why do gifted students solve geometry problems algebraically anyway? Because of the reformulated PP: One should not make more (intellectual) effort than the minimum needed. 12

Why do some gifted students experience EEC? Perhaps, because they encounter cognitive dissonance between two instantiations of the PP and try balancing them. 13

Non-example: What is the greatest number? (anecdotic evidence collected in a car) 14 Dalia: 20 B:21 Dalia:25 B:26 Dalia: 20…10 B:You mean 30? Dalia:Yea, 30 B:31 Dalia: 37 B:38 Dalia: 30…10… 40 B: 41 Dalia: But dad, don’t tell numbers in order! B:Why not? Dalia:Because there are greater numbers… B:OK… 45 Dalia:59 B:63 Dalia: 1000 B:Wow! 1005 Dalia: 1009 B: 1010 Dalia: Let’s play “the smallest number” B:OK. Dalia: 0. You know, when you count, you count 0, 1, 2, 3, 4… B:You’re right, but there are even smaller numbers. -1 Dalia:OK. -0 B:You know, 0 is a special number, -0=0 Dalia: OK. Let’s play again “the greatest number”. From the beginning B:1033. How do you think, who can win the game? Dalia: I don’t know. Let’s play another game.

What is the point? Different instantiations of PP (logical- mathematical, communicative, competitive and more) act simultaneously. Balancing between different instantiations of PP seems to be a force driving learning. It can be argued that PP has origins in brain (evolutionary psychology), not only in mind. 15

An attempt of theorizing: PP and APOS theory Action Process Object Scheme Dubinsky, E. McDonald, M. (2001). APOS: A constructivist theory of learning in undergraduate mathematics education research, in D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study, pp , Kluwer.

An attempt of theorizing: PP and APOS theory Action : A transformation of objects perceived by the individual as external. Process : Object : Scheme : Dubinsky, E. McDonald, M. (2001).

An attempt of theorizing: PP and APOS theory Action : A transformation of objects perceived by the individual as external. Process : When an action is repeated and the individual reflects upon it, he or she can make an internal mental construction called a process which the individual can think of as performing the same kind of action. Object : Scheme : Dubinsky, E. McDonald, M. (2001).

An attempt of theorizing: PP and APOS theory Action : A transformation of objects perceived by the individual as external. Process : When an action is repeated and the individual reflects upon it, he or she can make an internal mental construction called a process which the individual can think of as performing the same kind of action. Object : is constructed from a process when the individual becomes aware of the process as a totality and realizes that transformations can act on it. Scheme : Dubinsky, E. McDonald, M. (2001).

An attempt of theorizing: PP and APOS theory Action : A transformation of objects perceived by the individual as external. Process : When an action is repeated and the individual reflects upon it, he or she can make an internal mental construction called a process which the individual can think of as performing the same kind of action. Object : is constructed from a process when the individual becomes aware of the process as a totality and realizes that transformations can act on it. Scheme : an individual’s collection of actions, processes, objects, and other schemas which are linked by some general principles to form a framework in the individual’s mind that may be brought to bear upon a problem situation. Dubinsky, E. McDonald, M. (2001).

21 An attempt of theorizing: PP and APOS theory Action Process Scheme Object Procept Balancing different forms of PP

A concluding question: Does incorporation of PP in the model(s) of developing AMT fit PP? In other words, is considering PP forced by the evidence and thus justified? 22

Functions of theory in math education Models and theories in mathematics education can: · support prediction, · have explanatory power, · be applicable to a broad range of phenomena, · help organize one’s thinking about complex, interrelated phenomena, · serve as a tool for analyzing data, and · provide a language for communication of ideas about learning that go beyond superficial descriptions. Schoenfeld (1999); Dubinsky and McDonald (2001) 23

A concluding question: Is assuming PP as a force that drives learning beneficial with respect to the above criteria? 24

Let’s think about it… Thank you! 25