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On the role of the Parsimony Principle
in AMT
Boris KoichuTechnion – 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”
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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)
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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
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Example 1: Should a mathematical definition be minimal?Zaslavsky, O, & Shir, K. (2005). Students’ conceptions of a mathematical
definition, JRME, 36(4), 317-346.
• Minimal Definition: A square is a quadrilateral in which all sides are equal and one angle is 90o
• Not Minimal Definition: A square is a quadrilateral in which all sides are equal and all angles are 90o
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A square is a quadrilateral in which all sides are equal and all angles are 90o
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)
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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
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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).
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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), 168-179.
EEC: = Efficiency vs. Elegancy ConflictInt.: 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.
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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.)
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60B
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 PPWhy 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.
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Why do some gifted students experience EEC?
Perhaps, because they encounter cognitive dissonance between two instantiations of the PP and try balancing them.
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Non-example:What is the greatest number?
(anecdotic evidence collected in a car)
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Dalia: 20B: 21Dalia: 25B: 26Dalia: 20…10B: You mean 30?Dalia: Yea, 30B: 31Dalia: 37B: 38Dalia: 30…10… 40B: 41Dalia: But dad, don’t tell numbers in order!
B: Why not?Dalia: Because there are greater numbers…B: OK… 45Dalia: 59B: 63Dalia: 1000B: Wow! 1005Dalia: 1009B: 1010Dalia: 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. -1Dalia: OK. -0B: You know, 0 is a special number, -0=0Dalia: OK. Let’s play again “the greatest number”. From the beginning. 1030.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.
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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. 275-282, 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).
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An attempt of theorizing: PP and APOS theory
Action
Process
Scheme
Object
Procept
Balancing different fo
rms 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?
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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)
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A concluding question:
Is assuming PP as a force that drives learning beneficial with respect to the
above criteria?
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