I'm sure you've gotten used to all of the excuses I come up with for long period of inactivity (which are becoming more common, unfortunately). But I'll just get directly to my random thought of now.

In March of this year, an article was published in Keith Devlin's column of the Mathematics Association of America by Paul Lockhart. Now, I know Lockhart is an experienced mathematician and mathematics teacher and I'm but a lowly 12th grader in high school. However, one thing he notes is that "The only people who understand what is going on are the ones most often blamed and least often heard: the students." (3). So, I use this to justify my comment on this article.

First, I totally agree with Lockhart that math education today is dismal. At least, if not moreso than science education. People generally enjoy something if it's one of two things: useful, or interesting. Learning to decorate is useful, so people will willingly learn it in order to better themselves. And black holes and the Large Hadron Collider are just so friggin' cool that people love 'em. Unfortunately, present math classes are neither. They're far too based on rote memorization and don't really present the underlying concepts at least until another grade. In this way, I agree that the curriculum is flawed.

However, I don't quite agree with Lockhart's solution to it. This may just be the anti-Twainian acadmelitist in me, but while allowing students to pursue their own questions is fun, there is only so much that can be learned. It took who knows how long for mankind to come up with the concept of 0 (even in the Mayan civilization). And it wasn't until the 17th century that people started understanding negative numbers. Now, I'm not saying that he expects kids these days to figure out 3000 years worth of mathematics in 12 years. But saying that having a lesson plan "insures that your lesson will be planned, and therefore false" does a disservice. Lesson plans are useful and SHOULD be used to keep everyone in the class on the same page (that doesn't necessarily imply that lesson plans ARE doing this now). You might have one student pondering what it means to take 6 away from 3, but most others might still not have thought of that question. Lesson plans allow all of the students to have a similar knowledge base.

Now, what do I WISH could be done? Well, I believe that learning the concepts and context behind the mathematics is the best way to teach it. I love the way Lockhart was able to not only say, but elegantly show how the area of a triangle is 1/2bh. This provides those "Aha!" moments that are fundamental to understanding a concept. Now, I entirely support setting time aside in class to show the diagram and asking students "Now, how can I definitively find the area of this triangle?" and setting up a discussion. This way, you keep a balance between rote memorization and pure, yet extremely difficult creativity. In fact, there is this mathematics teacher at my school, who really pursues the learning of concepts. I take the example of how he introduced his geometry class to the Pythagorean Theorem (I'm not a primary source by the way, I never actually had him for a class, and am going by what I've heard). He makes sure people understand the physical basis of the Pythagorean theorem, the way the Greeks originally understood it. They didn't quite have the algebraic concept of "squaring" the number, that's very abstract. But they understood that if you have a right triangle, and if you take a physical square with the length of one leg, and another square with the length of the other leg, if you cut those squares and arrange them correctly, they make a square equal in area to a square with a side length of the hypotenuse. Here's a graphical rendition of what I just said:

They were able to prove that (in this case), a^2+c^2=b^2. That teacher made sure that the students knew that basis, and then proceeded to prove the theorem five different ways (you can find some here) I only WISH my geometry teacher taught that to me. But alas, most students thought that it was way to overboard. Anyway, I believe THAT kind of teaching is what would be best for students (or for me at least).

So, what I want to leave is that while Math education is pretty flawed, going to a free-for-all lesson plan of pure imaginative creativity isn't quite the solution. In painting, there's still a standard for learning forms and perspective and all of the other terminology. Although, like painting, math is an art. It is also not an unstructured one; it follows certain rules and in most cases, those rules are best taught than derived from scratch. Though the concepts behind them should be developed rock-solid.

(By the way, although I have never had that teacher for a math class, I do independently study proofs and logic with him, something that's greatly missing in math classes today)

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