The design of mathematical notation

The above page is from an 1847 edition of Euclid made by Oliver Byrne.

I bring it up to show that the way we present mathematics is a cultural construct. There are alternate methods. Perhaps (as is sometimes the case in the above volume) they are not always clear, but when something in mathematical notation design is confusing to new students it should be looked at with a skeptical eye and we should be willing to break new territory: new fonts, new interactivity, new colors.

In The Design of Everyday Things Donald Norman argues that many things attributed to “human error” are really design error.


It’s labelled as PUSH, it’s the fault of the user if they pull instead, right?


Just as if a student does the calculation on the right, it must be entirely their fault, right?

Brent Yorgey calls this “quite possibly the most horribly chosen mathematical notation in the history of the world” and writes:

I would like to find whoever made up this notation for inverse trig functions and make them pay $10 to every student who has ever been confused by it. The only problem is they probably don’t have A GAZILLION DOLLARS.

When discovering a design that is inherently confusing, the proper response is to change the design.

I. An actual textbook example

If m < n, the line y = 0 is a horizontal asymptote.

If m = n, the line y = \frac{a_m}{a_n} is a horizontal asymptote.

If m > n, the line has no horizontal asymptote. The graph’s end behavior is the same as the graph of y = \frac{a_m}{b_n}x^{m-n}

The letters m and n aren’t the two most visually and verbally distinguishable choices, especially in small fonts. The first time I taught this it was directly from the textbook version, and students were wildly baffled; I consider it the fault of the design, not the students.

One improvement is to change the letters:

Now, at least, eyes won’t strain and blur just to tell the difference between the letters. More experimentally, words can stand for variables (just as in computer programming):

It’s much easier for students to think of TOP < BOTTOM, TOP = BOTTOM, and TOP > BOTTOM instead of m < n, m = n, and m > n.

There is even some precedent in logic for words (in an all caps font) standing in for the logic symbols.

II. On color

Subtle changes in color can affect clarity of a text.

While textbooks have introduced color almost universally, the changes in color have not touched the mathematical notation itself. This is understandable when pencil and paper still predominate, but if the layout is done with computers there’s no reason even the most extreme color changes can’t be made . . .

. . . just as editors for programmers use syntax highlighting.

III. Interactive design

Text design no longer needs to be static. Consider the example above, rendered any way the user so chooses:

The settings could be consistent, as in the syntax highlighting example, or it could be a design where the equation starts “plain” but the user can click to modify the colors of specific parts: all constants, all exponents, all operations.

IV. What is possible with design?

Here’s a new rendition of Alice in Wonderland for the iPad:

When will mathematical texts reach the same level?

(To be continued in Textbooks of the Future.)


13 Responses

  1. I have to agree with Brent Yorgey about sin^-1(x). Followed by the convergent evolution wherein 2(x) means 2*x, but f(x) does not mean f*x. Most unfortunate.

    • sin^-1(x) is easily resolvable by just using arcsin.

      I’m not sure how to fix the f(x) issue though. Maybe use f[x] for functions? That would require a lot of changing.

      • Personally, I’d go with f→(x) as the notation for “f of x”. It can also be typed as f->(x) if you don’t want to resort to characters not on a normal computer keyboard.

      • Actually a reply to Doug S, but the reply depth has been reached. Using the “goes to” arrow for functions is a really bad idea, since it already has a much more mnemonic meaning in limits.

  2. […] of the Exterior Angle Theorem June 25, 2010 This isn’t the point of Jason’s post (although he makes a great point about confusing notation), but I had forgotten a different proof […]

  3. I love this post. I always change the variables in the book to “make sense”. Then I start with a story and words, then numbers, THEN notation. And, I always use different colors while illustrating. But, I have to make my worksheets in black and white. So, I have my kids hi-light certain things in different color hi-lighers (big packs can be found cheap at Walmart).

    This year colored, erasable pencils are on the supply list and my worksheets will be more “empty”, like graphic organizers, so that the students can write all of the notes in the various colors.

    I think I am going to make Purple (Violet) for Variables, Coefficients are Green (Cyan), Exponents are Blue (like high in the sky). I hope that the colors will make it easier for me to see errors too when walking around.

    • Thanks! I know teachers have been doing all of these for years, but it’s always been at the teacher level; I’ve never seen this kind of presentation in a textbook.

  4. Carnival of Mathematics #67…

    In The design of mathematical notation at The Number Warrior, Jason Dyer considers mathematical notation as a design issue and gives a series of examples of design that hinders, rather than aides, understanding….

  5. If Brent really wants to bill somebody for Sin^-1 as a symbol for Arcsine, he should blame John Herschel, son of the great British Astronomer/mathematician William.

    See more at

  6. Hi Jason. You have 120% agreement from me! We complain so much about students not “getting” it, when we should be determining where the confusion really lies.

    You may be interested in my Towards more Meaningful Math Notation.

  7. […] Watch the football-based equations, Q*Bert binomial theorem videos, and the math notation examples for inspiration.  The blog connects well to other sites, through Math Teachers at Play […]

  8. […] is a test case for my mission to reform the design of mathematical notation. The original proof was a very clear one, from Niven’s Numbers: Rational and Irrational; I […]

  9. […] theoretical questions, games, lesson plans, strategies, and more. Notable recent posts include The Design of Mathematical Notation, Nonlinear Lesson Design (Giant Ants of Doom), and Is “One, Two, Many” a […]

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