Figure Skating and Math Textbooks

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This odd coupling of topics links two separate frustrations which seem to come from the same core problem.

For figure skating the problem is one particular feature of the new scoring system—the extra points given to jumps in the second half of a program.  There’s no question that those are harder to do—anything is harder to do when you’re dead tired.  The question is why that extra stamina should be rewarded as an achievement!  In the last Olympics, those extra points were primarily an opportunity to game the system.  What was missing in artistic achievement could be made up in pure stamina.  The winner of the woman’s gold medal had all of her jumps in the second half of the program.

The fundamental problem is that from within a system, people are tempted to put a value on anything hard, regardless of whether or not it means something to the outside world.  People who have spent their careers doing and teaching figure skating have a tendency to think that any hard jumps are achievements.

Having just had reason to work my way through a high school algebra textbook, I have to say (based on an admittedly small sample) that the textbook publishers have that problem in spades.  The book was filled with minutia that need to be memorized for exam questions.  Getting through all of that is certainly hard, but it’s not the stuff that will make kids confident and successful in mathematics.  And that certainly won’t make them like it or remember it.  (One more reason it’s tough to be a teacher!)

It seems to me that high school math has a pretty good story to tell.  Core algebra is based on a really brilliant idea, that an answer you are looking for can be handled just like a number.   That lets you create a simple kind of manual computer.  If you can write down the problem, then the manual computer can solve it.  Sort of like doing your taxes with turbo-tax.  The core ideas are simple and don’t need to become a challenge.  From there it’s straightforward to do linear systems, which are part of the vocabulary in science, engineering, finance, or just about anything else.  “Vocabulary” is actually the right word.  For most students, mathematics is a language in which they need to gain confidence and fluency.

Then with quadratic equations you hit a wall.  All the techniques you just learned fail.  But you’re saved by another really good idea you would (probably) never have come up with yourself.  You do a few exercises with that idea, working toward an answer that is basically all you need to know—the quadratic formula.  Any further messing around with higher-order polynomials and rational functions is only supplementary, examples of what you can do with other kinds of problems you might run into.  No more rules to learn.

Going beyond that you have topics with really exciting applications.  Conic sections are geometrically interesting and tied to the whole story of the Copernican revolution, Kepler’s laws (“equal areas swept out in equal time”), and Newton’s gravitation.  (You don’t need calculus to tell the story.)  Sequences and series cover ideas behind every financial analysis students will see.  Permutations and combinations are important for probability and statistics and introduce the first non-trivial example of a group.  Modular arithmetic is a useful feature of every programming language.   Trigonometry is important but can become unnecessarily confusing.   Sine and cosine functions are important parts of the vocabulary, but you can get lost forever in trigonometric identities and formulas.

Mathematics shouldn’t be a drag.  There are few core principles and very little actually to memorize.  You get to see some remarkable solutions to tough problems.  One famous mathematician said that mathematics should be “like floating down a river on your back.”  My father-in-law put it differently: “Mathematics is for people too lazy to do real work.”

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