development of mathematical and physical reasoning abilities
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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? - PowerPoint PPT PresentationTRANSCRIPT
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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