Colourised Equations As An Aid To Mathematics

[Colin McAllister]

Yesterday, during a discussion about cognition, a colleague informed me that Physicist Richard Feynman perceived colours when he saw equations. "When I see equations," he once said "I see the letters in colors - I don't know why." (Ref 1). This ability is a form of synesthesia by which there is cross-over from one sense to another. Daniel Tammet, an amazing savant, also has this gift. He "sees numbers as shapes, colors, and textures, and performs extraordinary calculations in his head", as he describes in his book, Born on a Blue Day. This suggested to me that colourised equations would be a good aid to learning for some people. I took the well known roots of the quadratic equation:

and use blue to identify irrational parts and red to identify negative terms. The mapping is not clear cut, because one term is plus-or-minus, and it is only the result of the square root that is irrational.

This image was created by entering the LaTeX formula:
\frac{\uc{red}{-b}\pm{\uc{blue}{\sqrt[]{(b^2 \uc{red}{- 4*a*c})}}}}{2a}
into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/. The LaTeX command \uc{red}{-b} defines that the term -b is to be coloured red.

The benefits of colouring may not be apparent for a single equation. I suggest that you try it the next time you need to present a long and tedious algebraic evaluation. The low-tech approach would be to use coloured highlighting pens to mark up a printed copy of the equations.

For the benefit anyone who is unable to view the images, the equation shown by the above images is:
(-b +/- sqrt(b^2 - 4*a*c))/2a
The negative terms -b and -4*a*c are coloured red and the sqrt() function, which may be irrational is coloured blue. The square root may also be an imaginary number, if it is the square root of a negative number.

Ref 1: Sparks of Genius, By Robert Scott Root-Bernstein. The page containing Feynman's description may be viewed in Google Books.

[Bradford Hansen-Smith]

This makes sense to me.
When I color a model it is usually a way to code certain elements, to track design levels in complex models or in movement systems where changes are difficult to follow. Color coding helps to isolate information, while keeping the context in tact, or that would be difficult to see if not colored. There is a tradition of using color to map geometry models, games, and puzzles, as well as many other areas of mapping information, much can be seen extensively on the web.

No reason mathematical symbols should not be colored, it may well help students see functions and understand relationships that without some kind of visual differentiation is difficult.
Color in nature is informational as well as pleasing to the complexities of interrelated forms and systems we see.

[Bob Mathews]

Colored equations is great, and I believe it really does focus student attention where you need it -- if used properly. I used colored equations myself when I was still teaching, but one thing that bears mentioning is that we need to watch out if color is our only means of distinguishing between parts of an equation. Studies into color blindness provide varying numbers, but several point to as many as 5% of the adult males having some form of color blindness. Chances are, if a member of your audience is color blind, you'll never know it. Therefore, we need to consider this in advance.

Wikipedia's article on color blindness is helpful. An article by Geetesh Bajaj on Indezine is also helpful. The latter provides hints for making your PowerPoint slides "color-blind friendly", but the techniques presented there apply to other means of presentation too, and certainly apply to mathematics.

[Bob Mathews]

Bob Mathews said:

Colored equations is great...

Actually I meant to say "Coloring equations is great...".

[Colin McAllister]

Another application of colouring is to present each digit in its standard resistor colour code. The pattern of repeating decimals for certain fractions becomes much more memorable. For example 264/999:

This was generated from the LaTeX code:

\pagecolor{gray}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}/\uc{white}9\uc{white}9\uc{white}9 = 0.\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}\uc{red}{2}\uc{blue}{6}\uc{yellow}{4}...

using into the equation renderer at: http://www.hamline.edu/~arundquist/equationeditor/.

[Colin McAllister]

Bob, thank you for the link. I ran the 264/999 image through the http://vischeck.com/ colour vision simulator and here is what the resistor colour coded number may look like to some people:
Original image:

Deuteranope (a form of red/green color deficit), 2 and 4 are similar.

Protanope (another form of red/green color deficit), 2 and 4 are similar.

Tritanope (a blue/yellow deficit- very rare), 4 and 9 are similar.

[Colleen Young]

I have used 'colourful agebra' for some years now! Students agree it is helpful

It works particularly well for introducing simplifying expressions