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New answers tagged intuition

1

Modular arythmetic works just like a clock I've seen several students expect "half-hours" making an appearance in $\mathbb{Z}_{12}$. This leads them to be confused about statements like $5^{-1}=5$ rather than $5^{-1}=0:12$

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A point (say, in $\mathbb{R}^n$) is a vector. Vectors and points are really no different. They are both $n$-tuples in $\mathbb{R}^n$. The difference between two points (in $\mathbb{R}^n$) is a vector, but a vector has no fixed position. Points are positions in space. Vectors are displacements. It makes no sense to add two points, but it does make sense to ...

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For a much lower-level topic, consider explaining to beginning algebra students why "like terms" can be combined. On a few occasions, I have resorted to reasoning with students that adding algebraic expressions is like adding quantities with units. [Our curriculum begins with units and geometry before algebra, so this is usually safe ground in my class.] If ...

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I shall post my humble and incomplete list of bad explanations I've given or heard over the years: A function is continuous if you can draw its graph without lifting your pencil As you mentioned this is bad, but, depending on the level of the student, it can be a reason for big or small misunderstandings. At a high school level, this simply ignores the ...

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I think there is nothing wrong with reasoning using differentials/infinitesimals in introductory calculus and physics classes provided the nature of the reasoning is made clear and explicit. More precisely, my contention is that in general such uses serve to motivate definitions rather than to hide hard-to-formalize details of proofs, and it is operationally ...

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