Yesterday, I promised a rant about how textbooks are written in such a way as to encourage rote memorization of math rather than actually understanding what’s going on. This is not, of course, an original observation, but I noticed a particular example a few days ago that made me angry, and I wanted to get it off my chest. One of the more popular precalculus textbooks (Larson and Hostetler, I think but I can’t remember for sure), has approximately the following to say about the law of cosines (this is not an exact quote):

For any triangle with sides a, b, and c and opposite angles A, B, and C, the law of cosines is:

c^2 = a^2 + b^2 – 2ab cos(C)

The law of cosines is useful for solving a triangle when you know SAS (i.e. two sides and the angle between them).

After this there are a bunch of examples of how to use the law of cosines to solve triangles. This chapter contains: *no *explanation of the history of the law of cosines; *no* explanation of how the law of cosines is derived (here are six different proofs — a few of them are rather arcane (e.g. the one using the Ptolemy’s theorem), but none are very difficult); *no* intuitive explanation of why this formula is even close to reasonable (it’s basically the Pythagorean theorem with the -2ab cos(C) term included to account for the amount of error created by the fact that the triangle isn’t a right triangle); and *no* explanation of why you might *want* to “solve a triangle”.

The book is actually worse than this, because it actually lists *three* forms of the law of cosines:

c^2 = a^2 + b^2 – 2ab cos(C)

b^2 = a^2 + c^2 – 2ac cos(B)

a^2 = b^2 + c^2 – 2bc cos(A)

thus implying that making these substitutions is too difficult to expect students to accomplish on their own and thus they must memorize three forms of the law.

Is it any wonder students think math is difficult? And tedious? And useless? Under these circumstances, what’s to distinguish between c^2 = a^2 + b^2 – 2abcos(C) and c^2 = a^2 + b^2 + 2ab cos(C) and c = a + b – 2ab cos(C). What’s to keep you from thinking that someone just made up a bunch of random rules and stuck them in a book to torture you with? I know that’s not true, and most of my students know that’s not *really* true, but I don’t think many of them believe it. Where’s the excitement of discovery? Where’s the beauty? Where’s the satisfaction of figuring something out in order to solve a problem *you want to have the answer to*? Where’s the glory of *insight*?

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Great insight on this matter. As a once struggling student of math I can relate to the points you made about rote memorization and not having the slightest idea why this and that is as the books says.

I am not an angry math student. I do understand why, but I had to find it out myself – based on my interest. I guess a little research is good for the scientific mind as it teaches you that nothing gets served in the “real” world. But some things, namely those which have been known for literally thousands of years, shoudn’t need to be re-discovered by every student.

I could go on. But I’ll stop.

I have plenty of things to complain about with this book (and I dearly hope we switch to a new one when we do our next textbook adoption) but in my actual teaching of a Pre-Calculus class, I didn’t mind this section because

a.) I never would expect the students to memorize the formula. I don’t have it memorized.

b.) We work through the proof ourselves.

c.) We get the Pythagorean theorem intuition ourselves.

Both b.) and c.) could be potentially ruined by the book spelling things out.

As self-study, this chapter is clearly miserable.

Excellent thoughts. I just had a parent rave about FOIL method and how she love the FOIL method. I blinked. I never got math until I took my Montessori training. Algebra II was a really bad class for me in high school. I thought I should go get a refresher so I could speak intelligently with her. I realized the FOIL thing was just a formula which wasn’t understanding the principal. I couldn’t apply it to larger equations or odd situations. So I filed it back away and now smile politely.

EV