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Comments on Why do students mix up addition and multiplication, in the Multivariable Chain Rule with all partial derivatives?

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Why do students mix up addition and multiplication, in the Multivariable Chain Rule with all partial derivatives?

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I have taught multivariate calculus for years. Annually, some student always swaps + and ×. They trump up

$\color{red}{\dfrac{\partial z}{\partial s} = (\dfrac{\partial z}{\partial x} + \dfrac{\partial x}{\partial s}) [\dfrac{\partial z}{\partial y} + \dfrac{\partial y}{\partial s}]}$

instead of this definition in James Stewart, Daniel Clegg, Saleem Watson. Calculus Early Transcendentals (2021 9 edn). Page 987.

This definition refers to “Theorem 1” scanned below.

Why ? What am I failing to teach? How can I teach this better? How can I forestall further Mix Ups?

multivariable-calculus
posted 5 months ago
5mo ago
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2 comment threads

It's not necessary to ask the same question three times, much less on the same day. (1 comment)
At least a student a year sounds like a very small proportion (1 comment)
At least a student a year sounds like a very small proportion
trichoplax‭ wrote 5 months ago
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I don't know how many students you teach per year, but if only a little over 1 per year makes this mistake, it sounds like the large majority understand correctly. Are you concerned because you consider this proportion to be unexpectedly high, or because you can't relate to making the mistake at all?

For someone who is memorising equations without understanding, mixing up multiplication and addition seems a likely mistake to make. For someone who understands why multiplication is used, the mix up seems much less likely. Are you concerned that some of your students are memorising without understanding?