Can someone give me a crash course on trigonometry? Please?
I have a math competition next weekend, and I’m ridiculously unprepared, meaning that I’m doing the equivalent of cramming right now, only a week in advance. What fun.
For some reason, I thought that signing up for a test that included trigonometry and something called “analytical geometry” was a good idea, when I have very limited knowledge of trig, and geometry is not my forte. However, there’s no going back, so I might as well learn some extra math so I don’t look like a complete idiot.
While working through some practice problems, I couldn’t help but notice how many problems I couldn’t solve simply because I didn’t know a specific formula or identity, mostly trigonometric identities. In a test that specifically covers trigonometry, I guess that’s considered prerequisite knowledge, but it made me wonder to what extent formulas played a role in problem solving.
Obviously, pure formula memorization won’t do a bit of good in competitions if they can’t be applied properly, but in the case that they are, they’re super effective! (Sorry, couldn’t help making the reference) Formulas often save a considerable amount of time, which is a valuable commodity with the constraints of most competitions, but I don’t see them as an accurate gauge of problem solving ability.
The topic of Chinese math education ties in slightly with this as well because of their notorious memorization heavy curriculum and emphasis on standardized testing.
However, both my parents, having taking the infamous gaokao, insist that this test determining their college placement was not based off pure memorization, but rather complex application of concepts. (“If it was all formulas, then everyone would do well.”) They described their experience preparing as, roughly translated, “attacking a sea of problems”. By solving thousands of problems and applying formulas in different ways, both in math and science, the formulas naturally became ingrained into their minds, and they never made any intentional effort to memorize formulas.
Then again, this was decades ago, back when the gaokao was designed to reward whoever worked the hardest. I’m not sure how much has changed since then with the new parastructure of test prep companies. However, I believe that this outlines the fundamental difference between math education (and perhaps education in general) in China and America.
In Chinese, there’s a concept of “dead” and “live” when it comes to problems. A “dead” problem has been solved over and over before and only requires a set process to solve. This is what most of American math is. Outlining every step of how to solve a problem, and simply attaching different numbers to create “different” questions.
On the other hand, “live” problems require some problem solving skills. The process isn’t given to you– you have to figure it out yourself using a combination of “dead” skills. This requires extremely familiarity with the concepts and forumlas in order to know which ones to use and how to use them. I find this to be what makes math fun.
Of course, “live” problems become “dead” once solved over and over again, but that newly acquired”dead” skill simply becomes another tactic to use in solving more “live” problems. (Something tells me that modern Chinese math has become a slaughter of live problems without enough replacements to keep math interesting.)
So far, math competitions have been the best source of “live” problems that I can find, and although I am nowhere near the best, I still compete for the joy of the “aha” moment when everything in a problem clicks after bouts of desperation, as well as the the FML moments where I forget how to add correctly.
Whew, that was a tangent. (hehe, get it? tangent? trig? sigh.) Back to competition prep. My parents handed me this little pocketbook as reference material in studying, although they rarely used one themselves because they, you know, already had everything pounded into their heads.
Although I hadn’t read much Chinese lately, much less math-related Chinese, I figured this would be a good resource. Flipping through the pages, I saw some familiar concepts from Geometry and Algebra II, which was nice and comforting. Then I remembered that the book was written for middle schoolers. Argh.
A couple things in particular:
Can we just take a moment to appreciate how much nicer SLRs make mundane books look?
Anyways, I eventually found what I needed. The page of trig identities.
How I’m actually going to end up using this material is still unknown. All I know is that I’ve spent an inordinate amount of time writing this post instead of hardcore mathing, leaving me in an even more unprepared state. Oh NaBloPoMo, what have you done to me.