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 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
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.)