Jach's personal blog

(Largely containing a mind-dump to myselves: past, present, and future)
Current favorite quote: "Supposedly smart people are weirdly ignorant of Bayes' Rule." William B Vogt, 2010

Circus Math Considered Harmful

(I always wanted to use that phrase in a title.)

There's a problem with math education. I don't mean a specific field of math, but math in general. At every level of schooling, the problem rears its head. The problem even extends into other disciplines where the math is meant to be a tool rather than the subject.

This isn't a new observation. The problem is recognized by a lot of people, who attack it from different directions. Students claim they don't like "math". How can we get more students to like math? Is it really math they dislike? Or is it just "math"? Teachers claim students are doing worse on tests, so while they're making the tests easier in order to keep their jobs and funding, they ask how can we get students to master more complicated material? Is it really that the material is so hard, or are unrealistic expectations being made, or are artificial barriers being created to buff up the apparent difficulty?

Is memorization important? Memorization has certainly helped me appear more competent in math than I am. But it doesn't really help me solve problems that much better--perhaps if you can memorize the underlying concepts that would be sufficient. For example, memorizing that derivatives represent change and if you can find where a change hits a zero point you can find extrema will help you solve a large number of problems. Much more than just memorizing that the derivative of $$x^n$$ is $$n*x^(n-1)$$.

The biggest problem I see in math education and math application outside of specific math courses is that of circus math. It's math used completely for the purpose of show. Professors are hugely guilty of this--what is a lecture but a circus act with the secondary goal of transmitting knowledge? You'll notice that if the act is performed poorly, with the speaker stuttering or "umming" or going too fast or too slow, it doesn't matter how valuable the intended knowledge transfer could have been. The focus is on the poor performance, not the information. Besides, everyone knows if you really want just the information you can either read it in a book, derive it yourself, or watch an online lecture where you can pause, rewind, back-reference, and whatever you please to gain a fuller picture of the matter on your own time.

I don't want to suggest there isn't a purpose to show, I love Feynman lectures precisely because of the character he brings to them. It's why I can watch them multiple times, along with other video recordings of his thoughts and personality. But it's just out-of-place when it comes to doing math. It goes beyond busywork, it's pure show and not very good show. I shall give two examples. The first is general and may contain unfamiliar terms, the second should hopefully be more obvious for even a high schooler.

In the midst of solving some physics problem, you have produced a transfer function $$G(s)$$ and an input function $$X(s)$$ in Laplace-space. You need $$y(t)$$ to continue with your particular physics problem, so you first need to find $$Y(s) = G(s) * X(s)$$. Okay, that was easy. Now it turns out that we need to do an inverse Laplace transform to get $$y(t)$$. But the product of G and X isn't in a form where the inverse Laplace transform is obvious or is in our tables! So we waste time and algebraically manipulate it until it does. Maybe this involves the technique of partial fractions, maybe it involves the technique of completing the square, but we mess with it until it looks nicer and then we do the inverse Laplace transform. This "messing with it" is circus math. It's not necessary, and not for fun, though there are times when circus math is indeed fun. What you should have done was ask your calculator, or Matlab, or Wolfram Alpha, or some other open source alternative, to do the inverse Laplace transform for you and get back to whatever it was you were doing that needed $$y(t)$$ in the first place.

My second example is a "problem" from "economics". You are asked to choose between two plans, A or B, both of which have a 5% interest rate involved for the gains. They have the following cash flows:

Year A B
0 -$2000 -$2800
1 +800 +1100
2 +800 +1100
3 +800 +1100

One way to "solve" this "problem" is to use "Present Worth Analysis." But go ahead and take an intuitive guess which plan seems more attractive; that is, which plan gives you a higher net benefit.

If you chose plan B, you're right! It seems obvious, doesn't it? The interest rate is the same, the time period is the same, it seems intuitive that because $800*3 < $1100*3 and because 1100*3 - 800*3 = 900 > 800, two facts you can calculate easily and immediately, plan B seems better. The actual "math":

Net present worth of A is 800 * (P/A, 5%, 3) - 2000
Net present worth of B is 1100 * (P/A, 5%, 3) - 2800.

What's that crap in the parens? Well the book the problem comes from seems to sort of realize the real math is circus math, so it uses that notation to say "Hey, just go look up the number in the table at the back of the book!" The formula behind it is $$(P/A, i, n) = \frac{(1+i)^n - 1}{i(1 + i)^n}$$. Where does the formula come from? Have fun finding out! (It's not too difficult to derive, but.)

The problem itself is a joke problem, a sleight of hand used for a further demonstration of the circus math. The assumptions involved seem almost untenable for the real world. Maybe there is a place where the assumptions hold and the problem becomes useful, though. Of course this is a mild problem, there are many others with far less tenable assumptions.

With these two somewhat weird examples I hope I've hinted at the full illustration that circus math represents. Any time a professor chooses to work out a math problem by hand instead of use a computer program, when the working out of the math problem by hand isn't the subject of the course, that professor is just doing circus math. That professor is wasting time and obfuscating the discussion. If we really want students to know how to do things by hand like our computer-less ancestors did and be able to synthesize new ideas and prove new theorems with the help of a computer and solve complicated problems that use the computer in the end anyway, well okay, we can try and do that I suppose but I doubt it will work. It hasn't so far. My guess is because parents seem to think their kids ought to be able to do everything they can do and more, but this forces us to assume that our computer-less ancestors were retarded and weren't using their full intelligence to handle those problems.

I'd at least like to see the label "an exercise in circus math for" attached to courses such as differential equations, where the entire premise of the course is to solve simple toy math problems in various ways before realizing that any real-world problem of interest is going to require so many complications that you'll have to use a computer...

Maybe that's what makes the American college system so "great". The fact that if you want to actually do anything, you have to figure it out on your own. It seems like this stagnates progress, though. Perhaps this will be the subject of another post. Until then, I'll conclude by repeating the introduction.

Circus math, the idea of doing math for the sake of doing math, especially by hand, is harmful. It makes students hate math as an entire field when they've only experienced it in its worst usage. It makes other students hate classes outside the math department that nevertheless use circus math to pad their lectures, to reduce the potential utility of the material in its real-world professional-setting form, and to have students mimic the performance of so that they have something to grade. It stifles progress because instead of learning how to solve complicated problems with the aid of a computer, students learn how to solve toy problems by hand and forget it when the semester is over.

Posted on 2012-03-14 by Jach

Tags: math, philosophy, rant


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