Is coloring equations good practice or bad?

Question

I've noticed that though there is a way to color text or equations, in mathjax, though very few users use it. On the otherhand, those who do, use it regularly.

To me, coloring seems a nice way to remove the barrier that digital or printed text entails - in a class, a professor might show with their hands physically how something is moved from one step to another, (for example how certain terms might cancel, or how some terms might be rearranged to yield a more fruitful representation) whereas this is difficult to implement in text.

Coloring often removes this restriction. Similarly colored parts of the equation might suggest some relation between the two, like simplification, or approximation of one by the other. Another use might be to highlight a part of a proof or derivation which deserves special attention, or has some things to note about, later in the post. (As opposed to tagging equations, color can be used to target specific expressions, instead of a whole equation)

Keeping in mind all this, is it bad practice to use color in answers? Especially equations?

Answer 1

(I'm a new user so I don't have anything to contribute as far as norms here go, but I'm really interested in how to effectively convey information in an accessible way!)

Like others have said, the most obvious downside is colorblind people and people using screen readers. Colorblind people will miss information and so you should make sure the color doesn't convey anything essential, like it does in the Fourier Transform jaska posted.

Adding color may make it harder for people with screen readers to parse the rest of the equation as well. I know some people with screen readers read the TeX source, and using a lot of color will make a complete mess of something that would otherwise be comprehensible. I don't know how the screenreaders that have math support render color, but if it's including it, it will probably make it harder to understand.

To some extent, you'd run into the same screenreader issue with bold and italics. From a google it looks like the ones that have math support do read it, but it would be a less frequent intrusion than some uses of color. If the TeX source is being read, I imagine \mathbf would be less disruptive than \color{#c00}.

But using color can really help people with learning disabilities or who are just daunted by big equations. You can do things with it that you just can't do with bolding, italics, or underlining, and I think it's often much easier to make sense of in an equation.

The other potential downside is that color controls your reader's attention, and you'll send them towards wrong things if you don't use it in a thoughtful way.

After looking at some of Bill Dubuque's posts, I wanted to point out some examples of things I consider really effective uses of color. Like this:

This is a special case of telescopy. For as below we can write the RHS as a product of its term ratios $$\rm\ g(n)\ =\ \frac{g(n)}{\color{#c00}{g(n-1)}}\ \frac{\color{#c00}{g(n-1)}}{\color{#0a0}{g(n-2)}}\ \frac{\color{#0a0}{g(n-2)}}{\cdots }\ \cdots\ \frac{\cdots}{\color{brown}{g(3)}}\ \frac{\color{brown}{g(3)}}{\color{blue}{g(2)}}\ \frac{\color{blue}{g(2)}}{1}$$

Or this:

[Q: Is there an elementary proof that $\sum \limits_{k=1}^n \frac1k$ is never an integer?]

Hint $\ $ Since there is a unique denominator $\rm\:\color{#C00} {2^K}\:$ having maximal power of $2,\,$ upon multiplying all terms through by $\rm\:2^{K-1}$ one deduces the contradiction that $\rm\ 1/2\, =\, c/d \;$ with $\rm\: d \:$ odd, $ $ e.g.

$$\begin{eqnarray} & &\rm\ \ \ \ \color{green}{m} &=&\ \ 1 &+& \frac{1}{2} &+& \frac{1}{3} &+&\, \color{#C00}{\frac{1}{4}} &+& \frac{1}{5} &+& \frac{1}{6} &+& \frac{1}{7} \\ &\Rightarrow\ &\rm\ \ \color{green}{2m} &=&\ \ 2 &+&\ 1 &+& \frac{2}{3} &+&\, \color{#C00}{\frac{1}{2}} &+& \frac{2}{5} &+& \frac{1}{3} &+& \frac{2}{7}^\phantom{M^M}\\ &\Rightarrow\ & -\color{#C00}{\frac{1}{2}}\ \ &=&\ \ 2 &+&\ 1 &+& \frac{2}{3} &-&\rm \color{green}{2m} &+& \frac{2}{5} &+& \frac{1}{3} &+& \frac{2}{7}^\phantom{M^M} \end{eqnarray}$$

The prior sum has all odd denominators so reduces to a fraction with odd denominator $\rm\,d\, |\, 3\cdot 5\cdot 7$.

The color completely commands your attention. When I look at the equation in the first example, I don't read it left to right. The first thing I see is the pattern, and then I read out from there. In the second example, I'm trying to match the colors up before I've really read the text.

It works in Bill's posts because understanding how the colored things are connected is central to understanding the post—the color directs your attention right where it needs to go, and you quickly see what he's trying to show you. But if it didn't draw you to the right place, or if wasn't clear what it was supposed to show, you'd be distracted and confused before you even fully read the text. If you're not positive that color will lead your reader in the right direction, it's probably best to avoid it.

Another use might be to highlight a part of a proof or derivation which deserves special attention, or has some things to note about, later in the post.

I think the latter would be a mistake. The reader should understand why there's color as they're reading the proof. It's too prominent to expect a reader to just keep going until you see fit to explain.

Answer 2

(I've never used Meta before -- I noted there's no answer yet, but a lot of comments, so I hope I'm not doing something wrong)

To first give an example of what I would consider good coloring; I like the explanation of Fourier Transform, on betterexplained and technically copied from altdevblogaday):

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To be fair, he could have made it a little easier for colorblind people (like myself - and that's something else you need to keep in mind). And therein lies the issue. While this is one of the best examples of a color-coded equation I can think of, there are immediate "bad" consequences:

• Hard for colorblind people to distinguish
• Still somewhat reduces readability
• Doesn't emphasize if a part of the equation is "more important" than another

Anyways, it's hard to tell when an equation would be better with color, and that's totally up to your discretion when you post an answer. However, I usually would recommend to err on the side of "no color". As mentioned in the comments, colors can be distracting, and should be used to emphasize the part of the equation you're looking at, rather than splitting it into pieces.

I think that it's just as easy to go ahead and explain with words, since you're likely to do that anyways. Pretty colors are no substitute for a lucid explanation.

A good example of readable (but lacking in content) labeling below.

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Answer 3

Sometimes coloring is the best way to show how one step relates to the next or how different parts of one line are related, usually in a telescoping manner.

$$\sum_{n=1}^\infty\left(\frac1n-\frac1{n+1}\right)=\left(\frac11-\color{red}{\frac12}\right)+\left(\color{red}{\frac12}-\color{blue}{\frac13}\right)+\left(\color{blue}{\frac13}+\dots\right)$$

I think that using colors like this helps students understand telescoping series very well. Simply stating why a telescoping series works the way it does may not snap very fast for some students.

Another example is to help with equating parts, as Euler did to solve the Basel Problem.

$$\frac{\sin(x)}x=\dots-\color{green}{\frac1{\pi^2}\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}\dots\right)}\color{red}{x^2}\dots=1-\color{green}{\frac16}\color{red}{x^2}+\frac1{120}x^4-\dots$$

$$\implies\frac1{\pi^2}\left(\frac1{1^2}+\frac1{2^2}+\frac1{3^2}\dots\right)=\frac16$$

Students again may have difficulty seeing that you can simply set those two equal if you don't point out that they have the same $x$ term.

Also, colors are much more visual, and if you ask around on the matheducator's SE site, you'll find that visuals can be extremely helpful to students.

Of course, there are some setbacks, like the fact that staring at colors could put strain on your eyes, but I think it may be worth it, depending on the problem and who you are trying to present your answer to.

Answer 4

A BBC News article Revision techniques - the good, the OK and the useless, mentioned by the recent Guardian's article The science of revision: nine ways pupils can revise for exams more effectively, might be related. Using highlighters (colours) would be bad as it would break up ideas into components and therefore break down the 'chain of thought'. People would be more likely to work things out - make brain connections - when everything is 'normal'. Therefore, it would make sense for equations and such to be left as they normally are and not to link them colour-wise.

Following on from this research, I infer that ANY method of isolation or categorisation of terms / ideas would be, in fact, contributing to the 'breakdown of chain of thought' and therefore not beneficial. This is just another idea, on top of the obvious colour-blindness issues.