Guershon HAREL

Two Fundamental Questions: A DNR Perspective Guershon Harel University of California, San Diego [email_address] http:// www.math.ucsd.edu/~harel

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[object Object],Instructional Principles Duality Necessity Repeated Reasoning

The N ecessity Principle For students to learn the mathematics we intend to teach them, they must have a need for it, where ‘need’ refers to intellectual need , not social or economic need. D N R

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[object Object],10 10 30 70 70 ×30≠1500 100 300 700 1500

[object Object],2/3 100 ÷2/3=150 2 700÷2/3=1050 1050 ×2≠1500 100 300 700 1500

[object Object],√ 2 100 ÷ √2 300÷(100÷√2)=3×√2 700÷√2 (3×√2) × (700÷√2) ≠1500 100 300 700 1500

[object Object],[object Object],[object Object],[object Object],[object Object],t 100 ÷ t 300÷(100÷t)=3t 700÷t 3t×(700÷t)=2100 ≠1500 100 300 700 1500

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t A ÷ t (A+200)÷(A÷t)= (A+200)t/A (A+600)÷t A A+200 A+600 A+1400

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[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],Repeated Reasoning Necessitating the concept of variable

Students and Teachers’ Conceptions of Proof: Selected Results ,[object Object],[object Object],[object Object]

Teaching Actions with Limited Effect ,[object Object],[object Object],[object Object]

Why showing the limitations inherent in the use of examples is not effective? ,[object Object],[object Object],[object Object]

Necessitating the Concept of Diagonalization

Guershon HAREL

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Guershon HAREL