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What does this symbol mean?

Posted by on Mar 25, 2019 in How To Think Like A Mathematician, Mathematical thinking, Mathematics education | 0 comments

Another useful question is to ask students is “What does this symbol mean?”. Mathematics uses a lot of symbols and even a small symbol may be representing a very complicated concept. And, as was observed yesterday, the same symbol may be representing different concepts! In particular, the zeros in
\lambda _1 v_1 + \lambda _2 v_2 + \dots \lambda _n v_n = 0 \implies \lambda _1= \lambda _2 =\dots = \lambda _n=0
are different. One is a vector, one a scalar.

More generally, consider the symbols in the equation below taken from a course on fluid dynamics:
\rho \frac{D\bf{u}}{Dt} = -\nabla p + \nabla \cdot \boldsymbol{\tau} + \rho \bf{g} .
Asking the students what the symbols mean can be enlightening as to what they are having trouble with. The \nabla behaves differently when combined with \cdot (the dot). Do students know that? Some symbols represent vectors, some scalars. It’s hard to know whether students grasp this until I ask what the symbols mean to them.

Even asking a student to read out such an expression can be interesting. I have met students who are unsure of how to pronounce the Greek alphabet and will refer to — admittedly with some embarrassment — any symbol they don’t know as “that funny symbol”.

In conclusion, it is important to ask the students to describe what the symbols mean.

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