Right Wing Nation

Years ago when I was in my first year as a PhD student, one of the courses I had to take was phonology. Now, you need to understand that there are two types of linguists (formalists, at any rate): Syntax and sematics people, and phonology and phonetics people. The two groups are mutually exclusive: Syntacticians typically find what they do to be intuitive, while they find phonology to be incredibly difficult, and vice versa. I am one of the former.

On the first day of phonology class, the professor had us play Wheel of Fortune. He put a series of underlines separated by spaces on the board, and asked us to solve the puzzle. It looked like this:

_ _ _     _ _ _     _ _     _ _ _     _ _ _

I solved it in very little time, and he asked me to explain how I solved it. I don’t remember what it was (I am still in contact with a friend who was in that class and remembers), but it went something like this.

“The lengths of the words suggest that it is a complex noun phrase. The lengths of the first two words suggest that it is a noun phrase, a definite article followed by a noun. The lengths of the last three words suggest that it is a modifying prepositional phrase, since the third word is only two letters, and prepositions tend to be very short, and again, the next word is three letters, suggesting another definite article; that would make the last three words a preposition, an article (most likely), and a noun. Going on that assumption, the first and the fourth words are almost surely defiite articles (the). I asked for T and H for that reason, and got two more Ts and one more H than I had anticipated. I asked for N because the fourth word could be “in” or “on” (”of” is a possibility, and so is “by,” and probably several other prepositions, but I have twice as much probability of being right with the first because both end in N). Alternatively, it could be a clause, particularly if IS is the verb, but the first and last two word pairs are still almost sure to be noun phrases, so if the complex noun phrase strategy didn’t work, I would then have switched to the clause strategy . . . ”

The puzzle then looked like this:

T H _     _ _ T     _ N     T H _     H _ T

And I guessed it (The Cat in the Hat). The phonology professor — recall that this was a phonology class — began with, “I believe what you meant to say was . . .” and then went on to describe a very similar analytical process to solving it, but by relying on phonological, and not syntactical, cues.

Thus my hell in phonology class began, and on the very first day. It was a very long semester. But I’m getting off track.

Syntactical v. phonological isn’t relevant. Analytical is. Whether you are like the phonology professor or me, you will arrive at the solution by picking it apart analytically. To generalize, syntactical and phonological are analogous to “learning styles,” while analytical thought is independent of them. It is something that must be learned, and should be (but never is) taught. And I would argue that any academic field which does not require analytical thought cannot be called academic even in the loosest of definitions. Analytical thought is the fundamental tool for approaching any kind of problem, and it is lack of analytical thought that causes students the most difficulty.

Consider: Everything, from reading the problem to setting up and modeling it, to solving it, requires analytical thought. In fact, the whole is an analytical thought process. My biggest problem with the “discovery learning” proponents is that analytical thought is never developed. And, indeed, why would it be, if you take a “holistic,” “student centered” approach, and never teach students how to pick something apart into meaningful units?

As an example, let’s take business stats. They cover the spectrum, from simple descriptive stats to ANOVA, linear regression, chi-square, t-tests, and simulations. Yet, while many students find stats in general to be difficult, many of those are just as confounded by very simple stats problems as they are more complex ones.

Many business stats problems are simple data distribution problems. Let’s take the following example:

Kelleytread Tires produces a particular model tire whose tread wear life is normally distributed with a mean of 39,000 miles and a standard deviation of 5,300 miles. The manufacturer wishes to provide a guaranteed tread life for this model which would be exceeded by 98% of all tires. What tread life would meet this requirement?

Let’s pick this apart. What do we know? We know from the text of the problem that:

• Tread wear life is normally distributed
• Mean tread wear life is 39000
• Stdev of tread wear life is 5300

And what do we need to know? We need to know what guaranteed tread life would be exceeded by 98% of all tires.

Let’s go back to what we know. The information given us has to be relevant in some way to finding the solution. And indeed, it is. Imagine the tread wear life for, oh, several thousand tires plotted in a histogram. What does it look like? It tells us in the problem.

It’s a normal distribution. A bell curve.

Visualize a bell curve. Now visualize a line that cuts off the bottom 2%, because we want a tread life that will exceed the 2% cutoff. Like this:

Before we do this, of course, we have covered normal distributions and their properties, as well as means, standard deviations, and descriptive stats in general. We have also covered two things crucial to solving this problem: z-scores, or a value on a normal distribution expressed in standard deviations, and an incredibly useful Excel function, NORMSINV.

NORMSINV takes a probability on a normal distribution curve and returns its z-score. We can then turn the z-score into a regular number, because, remember, the z-score is the regular number expressed in standard deviations. Oh, one other thing. The z-score for the mean is zero. Values above the mean are expressed in positive standard deviations, and values below, in negative standard deviations. The z-score, then, is the distance of a number from the mean expressed in standard deviations.

The syntax is NORMSINV is:

=NORMSINV(probability) Simple, eh?

So we need to set up our problem in Excel, first entering the data given (mean and stdev). Now, we have two probabilities: The right area, or the 98%, and the left area, 2%. So we’ll enter those next. Then, we’ll add a label, z, for our NORMSINV function (because it will return the z-score), and another label for the value, our answer.

The NORMSINV function takes our probability (2%, named P_left_area), and returns that value expressed as the number of standard devations from the mean. Because the probability is below the mean, the z-score is negative.

All that’s left to do is turn our z-score into a number. It’s the number in standard deviations, so to turn it back into the number, we multiply by the standard deviation. Then we add it to the mean:

So a tread life of 28,115 miles would be exceeded by 98% of all tires.

Provided you have the necessary knowledge of normal distributions and their properties, the NORMSIV function, and z-scores, this is a problem that shouldn’t take students more than three or four minutes to solve. There are only two calculations, one a function with only one argument, and the second a simple arithmetic calculation combining multiplication and addition. So why do students find this to be such a brain teaser?

They lack the analytical thought process to tackle the problem. Students don’t see a problem with bits of meaningful information that can be extracted. They see the problem, but they don’t see into the problem. I’ve walked students through simple data distribution problems over and over, but I always have students who blank out when they get one on an exam or a project.

The scores on an exam are normally distributed with a mean of 75 and a standard deviation of 10. If the professor wants 10% of the class to receive As, then what is the minimum score a student can get and receive an A on the exam?

It’s exactly the same kind of problem. We have the mean and stdev. The P(right area) is 0.10, and the P(left area) is 0.9. We use NORMSINV to convert the P(left area) into a z-score, then a simple calculation to turn that into the answer (which is 87.8, by the way).

Students find problem solving difficult, not because they’re stupid (they aren’t), and at least in many cases not because the problem is overly complex or difficult, but because they haven’t been taught how to think analytically. Admittedly, schools weren’t great at teaching analytical thought when I was a kid — in fact, they didn’t teach it at all, but expected you to pick it up. I don’t have a rosy view of my own education. I remember the trouble a lot of kids had with “story problems.” But it’s undeniable that otherwise sharp students have more trouble with basic analytical thought than people my age did. We’ve been watching it degrade over the years.

Analogy, pattern recognition, and categorization are analytical thought. It isn’t obvious to many students that the second problem is of the same type as the first until you point it out. This again goes back to analytical thought. They don’t recognize the pattern, see the similarities, make the analogy, or categorize the problems effectively because they lack the ability to think analytically.

I suspect that going back to teaching geometry and doing proofs in math classes from elementary school on would help. But I’m not sure it would be a panacea. Thoughts?

(Speaking of education, by the way, if you’ve seen the new cheating report, read this.)

rightwingprof :: Dec.02.2008 :: Math, Education :: 2 Comments »

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