## Sunday, October 14, 2012

### More Fun With Statistics

The other day I wrote about a homework problem in first-year Stats that, according to my thinking, illustrates much that is wrong with the way math is taught. A few days ago I met my student again, and she had more homework. Let's have a look at it.

The assignment begins with the following general instructions:

It's good to show your work. But I find it disturbing that the professor assumes that a formula must be part of your "|work". What if you figure out the problem by pure logic? How do you reduce that to a formula? I certainly can't.

What's going on here is that the professor is revealing her true philosophy of math education: learning math consists of memorizing a bunch of formulas and learning how to recognize which formula to use on which problem. Her instructions don't make much sense otherwise. And the problem she gave last week about the woman picking random shoes out of her closet...well, the whole problem doesn't make any sense unless you interpret it as a case of "match-the-numbers-the-correct-formula".

Today's assignment includes more of the same. To be sure, most of the problems are OK, but a few of them are objectionable. I have issues with the following three:

1a). An industrial plant will randomly select six machines from an assembly line containing 20 machines for a quality control check. How many ways are there to do this?

1b). An ice cream shop offers an end of season special Fall deal for a fixed cost, where you can choose two of twenty flavors, two of eight toppings, and one of four cones. How many different ice cream treat combinations are available?

4. On a 15-item true-false test, where a true item is as likely to appear as a false item, what is the probability of getting 10 true items on the test?

It's actually the middle problem that I find most objectionable, so let's set that aside for the time being. My complaints about the other two items are perhaps mere quibbles, but I find them to be revealing of an unhealthy attitude. Let's talk about the industrial plant first.

The statistics of quality control is an important and practical issue of which there is much to be learned. My problem with this question is that it has nothing to do with the quality control. It is true that there are so-and-so-many ways of picking six machines out of twenty, but I cannot for the life of me imagine any practical situation where you would care how many such choices there are. The beauty of a good real-world math problem is that it illustrates an interesting connection between math and the real world: and this question, while at first pretending to deal with quality control, turns out in the end to be nothing more than a case of plug-the-numbers-into-the formula.

My issue with the third question is similar, but in this case it is the pointless awkwardness of the question which stands out. There is a very important and interesting question which could be asked here, and in fact it is answered with the same formula which is intended for this item. But the professor goes into great contortions to avoid asking the interesting, practical question. Instead she asks an artificial and contrived question which is of no general interest but happens to be answerable by using the same formula as the interesting question.

In case you haven't figured it out yet, here is the practical and interesting question which the professor fails to ask:

"On a true-false test with 15 questions, what is your chance of getting ten out of fifteen by simply guessing?"

If you can go back and re-read the professor's question I think you can see how she asks something theoretical and esoteric that really misses the point that ought to be made.

By the way, it should also be noted that what is significant here is not really the chance of scoring exactly ten out of fifteen, but the chance of scoring ten or better, which is a somewhat more invovled calculation. These are the things that are important in math, and ought to be talked about. But they become irrelevant if your purpose is simply to plug-the-correct-numbers-into-the-formulas.

Which brings us to the middle question, the one about the ice-cream stand. Let's take this one up when we return.