It’s Mathemagic!

I’m often rather disturbed by how mathematics is taught in school. Take, for example, the way I was taught the following (important) equation:

$latex int_a^b f(x) ,mathrm dx = F(b) – F(a)$

Those of you familiar with calculus will immediately recognize what this means. Those of you who aren’t should know that the above set of squiggly lines means something very important in the upper levels of confusing math.

When I first was taught the equation I was taught it by being exposed to it exactly as I just exposed it to you: an equation. There were some words with it too, but they made just as much sense as the equation did at the time. A few moments later the teacher explained what the equation means: one can find the definite integral of an equation using its antiderivatives.

But we were never taught why that’s the case.

With the diabolical mind I have I set out to figure out just why this worked, and with the help of some prior knowledge (I’m nerdy) I figured out just why the antiderivate of an equation might help me find its definite integral. But nobody else in the class got the benefit of my epiphany — I wasn’t the one teaching it.

It has been my fervent belief for a very long time that the understanding is as crucial as the knowledge. I made it through years of school without studying (at all) through this method: you don’t need to remember how the fundamental theorem of calculus was written if you remember why it works. The rest of it comes naturally when you understand the math. You can solve problems you’ve never seen before if you understand the fundamentals — but if you memorized the equations and how to solve them, you’ll be baffled by new types of problems.

Too often I see students trying to study by memorizing formulas and equations for hours on end — and then failing their exams. Why? Because however readily they may be able to recite $latex x – x_0 = V_0t + frac{1}{2}at^2$, they still don’t know what it means or how to use it.

7 Comments

  1. I agree completely. My physics prof the other day told us that we should memorize the physics formulas… why? So we could do better on the MCATs.

    *sigh*

    I guess he wants us to understand them most importantly, but if you can’t do that, memorization is good enough.

    Interestingly, my problem with physics has been the opposite. I understand concepts well enough, but applying them is harder. I can’t easily memorize equations and I don’t have the skills or time to derive them during an exam. I don’t have the greatest math skills… what’s one to do?

  2. I think physics takes memorization and understanding. You need to know the equations and how to use them. You just have to be snappy in figuring out which equations will get you from where you are to what you want. That takes understanding of the relationships between variables.

  3. I think part of the problem in introductory physics is that a whole bunch of equations get thrown at the students, and they don’t realize how they are all related. When I was TA-ing most students were horrified at my suggestion that they derive an equation. I know that deriving an equation is not a guarantee that one knows what it means, but I think that if you can’t, you probably don’t have a much of clue about how to use it.

  4. $latex F(t) = 6x$
    $latex f(t) = frac{d}{dx}F(t) = 6$
    Let’s suppose f(t) is a function giving the velocities of some object, and F(t) represents its position at any given time t. Suppose I’m asked to find the distance traveled by the object from t=2 to t=4. That would be
    $latex int_2^4 f(t) dt$
    correct? (The integral of velocity is position — if you find the area under a velocity graph, you’ll end up with how far the object went. Try it.)

    With this problem we can see that the answer is 12. But if the integral was harder to do, we’d use the antiderivative of f(t) (F(t)) to solve. The distance traveled by the object is its final position minus its initial position, and F(t) tells us the object’s position — meaning that F(4) – F(2) is the distance traveled.

    So,
    $latex int_2^4 f(t) dt = F(4) – F(2)$

    Make sense?

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