Creativity, Mathematics, and What to Teach

This week’s post is about the role that creativity plays in mathematics and the role mathematics plays in creativity. The post started as an angry rant, but Occupy Math wants to limit the amount of time spent shaking his fist at the sky, so the rant part of this post will be brief.

A political scientist named Andrew Hacker has published a book entitled The Math Myth and Other STEM Delusions. The thesis of this book is that too many students drop out of high school because we are teaching them math beyond arithmetic, which they don’t need. I thank Abigail Ann Young for pointing out this controversy. Others have already taken a good shot at refuting Dr. Hacker’s pack of arrant nonsense in places like the Atlantic and Slate. I think they did a pretty good job explaining why he’s horribly, tragically wrong and a quick search will net you hundreds more items on this debacle, er, debate. Here is a cartoon about possible dark motives for teaching only arithmetic, with thanks to Elizabeth Knowles for pointing it out.

Dr. Hacker says that abstract math damages students and drives them out of school. Occupy Math’s biggest objection to this idea is that it’s obviously completely wrong. Math didn’t use to drive people out of school in Canada and the U.S. and it still doesn’t in Russia, Japan, China, Korea, and a bunch of other countries. Solving the problem of students dropping out of high school by banning math (at least math harder than arithmetic) is like solving the problem of freeway congestion by destroying all the on-ramps. Yeah, you did solve the traffic problem, but now no-one can get anywhere.

To accept Hacker’s thesis, you must believe that North Americans are degenerate, lesser descendants of a wiser race and intrinsically inferior to much of the rest of the world.

Occupy Math finds this notion not only wrong, but an example of surrendering to despair and abandoning the benefits of mathematics for imaginary gains. The motto of Occupy Math is that math is the right of all free people. Knowing math helps people to stay free and strong. Hacker recommends herding most of the students in and then welding the door on the dungeon of ignorance shut!

Suppose that kids say that hard math classes drove them out of high school or that you check their grades and attendance and the worst numbers are in math class. Does this mean math is the problem? Let’s use our imagination and examine a few alternative hypotheses:

      • Math classes are often taught by people who lack subject mastery or fear the subject. Poor math instruction is a likely culprit.
      • We, in North America and parts of Europe, have a pervasive cultural belief that math is too hard for people to learn, that being bad at math is okay, that normal people cannot do math. Math is an acceptable excuse for failure even if the real problems are with family, depression, or one of the other maladies that high school students endure.
      • Math education is subject to idiotic fads and the people designing math education are often neither involved in its teaching nor skilled with math. In other words bad curriculum design, rather than mathematics itself, may cause this problem.
      • Funding for education has been a bottom priority in many places with math behind only music and art for being on the chopping block. The tantrum-throwing Governor of Kansas, for example, may have created a situation where Kansas K-12 schools may not open next fall. Inadequate funding is definitely a contributing factor to declining student performance.

From the point of view of a teacher on the ground in first year university courses, Occupy Math is sure that implementing the reforms outlined by Dr. Hacker would act to impoverish students both intellectually and financially, deny them access to fascinating majors, and help reduce our society to a state including something very like serfdom. Worst of all,

Students gifted in math would very likely never notice that gift in Hacker’s world.

Occupy Math will end the rant portion of this post by quoting from one of his favorite novels, A Wizard of Earthsea by the incomparable Ursula K. LeGuin: “For to keep dark the mind of the mageborn, that is a dangerous thing.”

Let us now turn to the connections between math and creativity. Consider the notion that a person with mathematical training has a wider variety of mental tools than a person without the training. If you have more mental tools, your options for understanding a situation or solving a problem are larger. Occupy Math frequently says that the most powerful technique in mathematics is to keep changing your point of view until the problem becomes easy. Let’s look at an example, in the form of a puzzle.

gridpuzzle

The goal of this puzzle is to move from the starting square to the ending square, moving between squares that share a face. You may visit a square at most one time, and your score is the total of the numbers in all the squares you visit. My colleague Richard Hoshino brought this problem to the CMESG conference session on problem solving. He is currently at Quest University and you may enjoy his novel The Math Olympian. Spoiler warning!

A best-possible solution (there are several):

solution

If you play with the puzzle a little, you will find the best you can do is a total of 98 (the total of all the numbers is 100). For some reason you have to skip a square and, since you have to start with 1, skipping a square with a 2 in it gives you the best score. A normal person might then say “why?” A mathematician would say “prove you have the best score!” (maybe without the exclamation point). Consider the following picture:

grid

This picture takes the basic grid of the puzzle and colors it like a checkerboard. Since we move between squares that share a face, we must move between squares that are assigned different colors. On the other hand “start” and “finish” are the same color. A path that visited all the squares would have to alternate colors and so, if it visits 16 squares, begin and end on different colors. That means no path beginning at “start” and ending at “finish” can go to all the squares. The solution given with a score of 98 skips only one square – the best possible number of squares to skip – and that square has the lowest possible value (given that we must use 1). It is now self-evident that the highest possible score is 98 and we know “why” (mathspeak: we have proved 98 is the largest possible score).

Occupy Math likes the checkerboard viewpoint because it provides a very short solution to the problem. One of the math-education researchers at the CMESG conference said that “using the checkerboard coloring is an advanced concept, and so not a simple solution.” Occupy Math does not disagree, “short” and “simple” can be quite different, but he does think that:

Coming up with the checkerboard viewpoint is an act of creativity!

This is also a good example of coming up with a point of view that makes the problem (proving that 98 is the highest possible score) much easier. In general, training in math expands your mental viewpoint, increasing your creativity. Conversely, making progress in math is much easier if you are at least somewhat creative. Many human activities are called “left brain” or “right brain”. While this is a terrific oversimplification, mathematics is both a left and a right brain activity. Arithmetic is mostly left brain – so teaching students only arithmetic would let them miss the creativity boost that mathematics generates and, with it, most of the benefits of studying mathematics.

Occupy Math mostly solves problems other people , biologists, engineers, computer scientists, and others bring to him. For that kind of work, creativity is at a premium, meaning that the statements in this post about creativity are real-world and application-tested. There are many other ways that math inspires and enables creativity. As long-time readers know, for example, Occupy Math knows fractals. The fractals in the linked post can be clicked on to get more fractals, because too much is never enough. The trig flowers are another example.

Mathematics is endless and fascinating, a playground for creative minds, and a greater human achievement than the Sphinx, the Colossus of Rhodes, or sliced bread. Do any readers have stories or examples of mathematical creativity? Do you know a cool puzzle? Have you or any of your students come up with something interesting, worth sharing? Please share by commenting or tweeting!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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