HackerHow can you tell a correct answer from understanding?
Ask two people to explain their reasoning, then change the presentation and ask them to reconstruct the answer. Peer instruction and geometric intuition offer different ways to expose gaps.
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What does explaining an answer to a peer reveal?
Collaborative learning gains a concrete mechanism when students have to explain their reasoning to one another. Agreement on an answer alone tells the teacher less.
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most great learning happens in groups the collaboration is the stuff of growththe students who had the right answer pretty quickly convinced those who did notWhy do experts miss the step a beginner needs?
Familiarity can hide prerequisites. Compare the techniques each speaker uses to expose the gap between what a presenter assumes and what a learner can reconstruct.
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they will hallucinate that there's material in your presentation that isn't thereyou're no longer aware of the conceptual difficulties of beginning learnerHide a corner instead of asking for three rules
Put 2, 3 and 8 at the corners of a triangle: 2 cubed equals 8. Hide the 8 to ask for a power, the 3 to ask for a logarithm, or the 2 to ask for a root. Ask the learner to name the missing quantity before calculating. Someone who knows the calculator buttons but not which question they answer now has a way to express the confusion.
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The symbol as a whole represents the value that should go in the missing corner.A tougher transfer test is to rewrite a probability puzzle without changing its sampling rule. In the sphere puzzle, choose lines through the center, then flip fair coins to choose their endpoints. Now one can count choices instead of computing the areas of changing spherical triangles. Why do these steps still produce uniformly random points? “Use coins when you see a sphere” misses the part of the explanation that matters.
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That first line passes through the sphere at two points, so flip a coin to decide which of those two points is going to be p1.Does getting the answer mean understanding the concept?
Concept questions in physics and geometric intuition in linear algebra reveal gaps that procedural success can hide. Neither approach rules out calculation practice.
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good problem solving does not necessarily mean understandingwe almost always get computers to handle that half,Try it on your own example
Change a familiar exercise so the usual steps cannot simply be copied. What explanation would you listen for?