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How can I explain to an Algebra I student why he should use lowercase letters when naming his variables (i.e. $q = $ number of quarters $vs.$ $Q = $ number of quarters? I am not interested in the historical answer given in this post: Why are variables lowercased? I would instead like a simple example and/or rational reason that a rather stubborn 8th grade student would understand.

Thanks all!

N.B. Maybe I am the one who is in the wrong here but I have been told that lowercase letters are better.

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    $\begingroup$ Does it actually matter? $\endgroup$
    – user61527
    May 1, 2014 at 20:20
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    $\begingroup$ You only have a certain amount of political capital with the 8th grader. Is this really what you want to spend it on? $\endgroup$
    – MJD
    May 1, 2014 at 20:23
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    $\begingroup$ I think the answer is that it doesn't matter at all from a rational point of view. You (or your student) can choose whatever letter you want to represent the number of quarters, as long as you state clearly what you mean. That said, it would be unkind to your readers to choose, say, $Q$ for the number of years and $Y$ for the number of quarters. And it's traditional to use $i,j,k,l,m,n$ only to denote integers; not following these traditions will make your mathematical writing a little harder to understand. But there's nothing logically or mathematically necessary about the traditions. $\endgroup$
    – Jack Lee
    May 1, 2014 at 20:30
  • $\begingroup$ It doesn't matter, but one reason you could give to support your reasoning is that oftentimes capital letters are used to denote some type of constant (e.g. G, R). $\endgroup$
    – Grid
    May 1, 2014 at 20:37
  • $\begingroup$ It really doesn’t matter. In a lot of situations there is a link between upper and lower case e.g. $F(x)$ and $f(x)$. I think the main reason standard variables are lower case is because upper case variables are often assigned as constants or have a specific meaning $\endgroup$
    – Henry Lee
    Oct 15, 2018 at 12:33

2 Answers 2

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In my opinion, you should do exactly the opposite.

Teach them that variable names are absolutely arbitrary. You can pick uppercase letters, lowercase letters, little icons, or whatever. The only important thing is that two variables with the same name, or picture, or whatever refers to the same quantity.

It really bugs me that math teaching revolves so much about the syntax of things, instead of focusing on what it actually means. I don't deny that doing math requires some sort of formalism - but it should always be made clear that the formalism is just a convenient way to communicate mathematical ideas, and that what's actually important is the idea behind it.

Once you've established that notation is about communication of ideas, not about their content, it follows (and I think they'll agree) that deviating from universally agreed upon notation is usually a bad idea. But not because the math somehow becomes wrong, but simply because it makes it harder to get your point across. It's like inventing your own, private language, and wondering why nobody understands you.

Though I must say that in the case of variables, saying that they should always be given lowercase letters as names is a gross oversimplication. A variable representing a matrix, for example, will usually be given an uppercase latter. The same often goes for points in geometry. So for variables, what we really do is we try to partition the available letters in such a way that similar things can be recognized as such. Sometimes that means using letters that are close together in the alphabet. Sometimes that means using both upper-case and lower-case letters, because we want to emphasize that $f$ and $F$, respectively $g$ and $G$ are somehow related. What should be taught, I think, is to pick sensible variable names, i.e. ones that make the intention clear, not any particular variable names (which, in a given context, might not be sensible at all).

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  • $\begingroup$ Although beyond the scope of 8th grade Algebra, we also use uppercase and lowercase to indicate state functions, path functions, extrinsic and intrinsic variables, and others but the use is inconsistent. $\endgroup$
    – mchid
    Mar 7, 2020 at 20:57
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While the points below do not apply to the 8th grade, it is preferable to teach the habits of good style early on.

A principle of easily read mathematical typography is that symbols with the same syntactic properties, particularly with regard to concatenation, should belong to the same typographical class (where "class" is loosely defined for convenience in particular applications).

For example, if $\pmb x,\pmb y\in\Bbb R^n$ and $A,B\in\Bbb R^{n\times n}$, then formulae such as $$z=\pmb x^\top\! A\pmb y\pmb y^\top\!B\pmb x+\pmb y^\top\!\pmb y\quad\text{and}\quad Z=\pmb x\pmb y^\top\! A\pmb y\pmb x^\top+B $$are easily read; while, in the same formulae with the typography randomized to $$Z=x^T A Y Y^Tbx+Y^TY\quad\text{and}\quad z=xY^T AYx^T+b,$$where distinction by capitalization and font is dropped, it is harder to see what's going on (albeit easier to type!).

If you are only dealing with real-valued quantities—not operators, sets and spaces, or vectors and matrices (as ab0ve)—then keeping to lower case is less important, especially if you need lots of different variables and constants. Even then, it's better to avoid capitals, or restrict their use to symbols that play a particular role.

In physics, engineering, and other areas where mathematics applies, different priorities hold, and many symbols are capitalized by long-established convention.

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