Development of Mathematical and Physical Reasoning Abilities
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Transcript of Development of Mathematical and Physical Reasoning Abilities
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Development of Mathematical and Physical Reasoning Abilities
Jay McClelland
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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?
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Rule-like behavior and deviationsTorque-difference effectGradual change in sensitivity to distance if measured on a continuous scaleDifferences in readiness to progress from targetted experiences
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Current Interests
• Numerosity and counting• Understanding of fractions• Geometry & trigonomety
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cos(20-90)
sin(20) -sin(20) cos(20) -cos(20)
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The Probes
func(±k+Δ)func = sin or cossign = +k or -kΔ = -180, -90, 0, 90, or 180order = ±k+Δ or Δ±kk = random angle {10,20,30,40,50,60,70,80}Each type of probe appeared once in each block
of 40 trials
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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
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sin(90–x) = cos(x)
All Students Take Calculus
How often did you ______ ?
NeverRarely Sometimes OftenAlways
• use rules or formulas• visualize a right triangle• visualize the sine and
cosine functions as waves
• visualize a unit circle• use a mnemonic• other
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Self Report Results
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Accuracy by Reported Circle Use
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cos(-40+0)
sin(40) -sin(40) cos(40) -cos(40)
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sin(-x+0) and cos(-x+0)by reported circle use
sin
cos
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cos(70)
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cos(–70+0)
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Effect of Unit Circle Lesson byPre-Lesson Performance
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Effect of Unit Circle Lesson vs. Rule Lesson
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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