Math Needs Better PR

I was recently reading a book on Greenfoot, a Java-based GUI intended for teaching programming to high schoolers and college underclassman. In the “Newton’s Lab” project, the writer assuaged the reader who might be leery of the mathematics in that particular project. Remember, the reader was told: Programming can do a variety of things, including both mathematics and creative tasks.

This is a sadly typical dichotomy, with the implication being that mathematics isn’t creative. Never mind that everything involving computers is inherently mathematical, particularly the things we tend to think of as “creative works”. Every piece of music, every work of art, every recording of a visual artist… it all has to be converted into a string of digits via predictable mathematical formulas, and then reconverted into a pseudo-analog presentation for the viewer. Even the font you’re reading this text in is the result of several fields of mathematics.

My perspective is broader than that, though. Mathematics is seen as boring by many people because it’s presented in a boring fashion. There’s not enough “ooo neat!” coming from the people who talk about it. I think one of the reasons why Vi Hart is so popular is because she sounds genuinely excited about mathematics. I don’t think it’s a coincidence that she first trained in art and then became attracted to mathematics, or that her father is a mathematical artist in his own right.

I’ve been thinking about this PR issue for a while. I wrote a post recently on what is apparently generally called the “Power of a Point” theorem. It has a few names, both in itself and for specific subtheorems. Here is the theorem:

Take a piece of paper. Draw a dot somewhere. Draw a circle somewhere. Any line you draw through that dot will touch the circle twice, once, or not at all. Ignore the not at all. If it touches twice, multiply the distance between the point and each time it touches the circle, and call that result “Bob”. Why “Bob”? Why not? Anyway, if it touches once, square the distance between the dot and that point, and call that result “Bob”. Bob will always be the same value for the same dot and the same circle. Move the dot around, and Bob changes. Move the circle around, or make it bigger or smaller, and Bob changes. But for a particular dot and a particular circle, Bob will be the same regardless of the line you draw.

Now, that’s cool. That’s the sort of thing an excited mathematics teacher should be able to shout “Lemme show ya somethin'” over. We need more moments like that.

Here’s how a fairly typical High School Geometry book decided to explain the theorem. First, it broke it into two theorems, depending on whether the point is inside or outside the circle. Then it offered these theorems: “If two chords intersect in a circle, then the products of the lengths of the chord segments are equal. If two secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.” (Source: Glencoe Geometry, 2012 CCSS edition, pages 750 and 752.)

I suppose it’s mathematically accurate, although not quite rigorous enough (it’s not the products of the lengths of any two chord segments; it’s the products of the lengths of the chord segments on the same secant). Splitting it into two theorems misses an opportunity to generalize, and makes the second theorem more difficult to understand. In my opinion, by selecting two specific secants, the text misses the point that it’s any line from the point to the circle.

There’s another theorem, presented separately, that the two tangents between a point and a circle will always be the same length. The Bob’s Your Secant theorem renders this separate theorem redundant. Presenting the theorem that the two tangents are always equal first is pedagogically useful, but then, when the *ahem* “Secant Segments Theorem” is then presented, that’s a good time to bring up the earlier one. “Remember when you learned that the two tangents are the same length? Well, it turns out, there’s a general pattern between the segment lengths between a point and a circle, and here it is.”

There are numerous books at the bookstore and Amazon that attempt to sell mathematics to the unconvinced, with varying degrees of success. These are usually marketed to adults who find themselves needing to muck through mathematics because they never quite learned it right in the first place, and to teen sufferers of the modern mathematics textbook. Why can’t we cut out the middleman, and make high school textbooks interesting and engaging in the first place?

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