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Development of Mathematical and Physical Reasoning Abilities

Development of Mathematical and Physical Reasoning Abilities. Jay McClelland. Questions. How do we acquire concepts we don’t already have? How do we acquire representations of physical variables and of its importance in reasoning? Why does the ability to reason about things develop so slowly?

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Development of Mathematical and Physical Reasoning Abilities

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  1. Development of Mathematical and Physical Reasoning Abilities Jay McClelland

  2. Questions • How do we acquire concepts we don’t already have? • How do we acquire representations of physical variables and of its importance in reasoning? • Why does the ability to reason about things develop so slowly? • What makes someone ready to learn, and someone else unready to learn?

  3. Rule-like behavior and deviations Torque-difference effect Gradual change in sensitivity to distance if measured on a continuous scale Differences in readiness to progress from targetted experiences

  4. Current Interests • Numerosity and counting • Understanding of fractions • Geometry & trigonomety

  5. cos(20-90) sin(20) -sin(20) cos(20) -cos(20)

  6. The Probes func(±k+Δ) func = sin or cos sign = +k or -k Δ = -180, -90, 0, 90, or 180 order = ±k+Δ or Δ±k k = random angle {10,20,30,40,50,60,70,80} Each type of probe appeared once in each block of 40 trials

  7. A Sufficient Set of Rules • sin(x±180) = -sin(x) • cos(x±180) = -cos(x) • sin(-x) = -sin(x) • cos(-x) = cos(x) • sin(90-x)=cos(x) • plus some very simple algebra

  8. How often did you ______ ? sin(90–x) = cos(x) • use rules or formulas • visualize a right triangle • visualize the sine and cosine functions as waves • visualize a unit circle • use a mnemonic • other All Students Take Calculus Never Rarely Sometimes Often Always

  9. Self Report Results

  10. Accuracy by Reported Circle Use

  11. cos(-40+0) sin(40) -sin(40) cos(40) -cos(40)

  12. sin(-x+0) and cos(-x+0)by reported circle use sin cos

  13. cos(70)

  14. cos(–70+0)

  15. Effect of Unit Circle Lesson byPre-Lesson Performance

  16. Effect of Unit Circle Lesson vs. Rule Lesson

  17. What is thinking? What are Symbols? • Perhaps thinking is not always symbolic after all – not even mathematical thinking • Perhaps symbols are devices that evoke non-symbolic representations in the mind • 25 • cos(-70) • And maybe that’s what language comprehension and some other forms of thought are about as well

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