Understanding Mathematics by Peter Alfeld, Department of Mathematics, University of Utah

Do you often wish your teacher would do more examples?

Examples certainly have their place, and they are an essential part of teaching mathematics, and of understanding mathematics. On the other hand, teachers often are reluctant to spend too much time doing examples, for several good reasons:

• First of all, you can do as many examples as you like by yourself.
• Second, examples take away class time that could be spent on understanding the underlying concepts.
• Third, examples may encourage a recipe type approach to learning. ("Just show me an example and I'll know how to do it.")

Moreover, there's often a discrepancy between what a teacher thinks of as a good example, and what a student expects as an example. Students often like to see numerical examples (presumably because it shows them how to answer exam questions) whereas teachers like to put emphasis on illustrating concepts and applications.

Here's an example for the difference between examples. Take the law of sines:

• In a triangle, the ratio of the sine of an angle and the length of the opposite side is the same for all three angles.

Given a triangle with C=10.3 degrees, B=28.7 degrees, and b=27.4 feet, as shown in the figure, find the remaining angle and sides.

This is a typical textbook example, made up, boring, and trivial (that particular textbook has lots and lots of examples like that in the exercise section). It's something any student should be able to do at home to any extent desired. Here is what I would consider a better example:

You drive along a straight road. At some point you see a mountain that makes an angle A with the road. You drive on for c miles. Now the mountain makes an angle B with the road. How far are you now from the mountain, and how far is the mountain from the road?

The particular textbook I'm referring to has a few exercises like that, but with specific values for the two angles and the distance traveled.

I believe the second example is better than the first for the following reasons:

• It is interesting. It's something you actually might do some day. The road may be the shore of a river and the mountain a rock on the other side, and you may want to compute the width of the river, but it's the same problem.
• It's not cluttered with unneeded notation. One could even omit naming the angles and the distance covered, and have the student make up the notation. (That would be a better exercise, but harder to grade.)
• It has general values of the parameters, rather than specific numbers. That means if you make a mistake you can more easily recover. If you compute the distance to the mountain as c squared you know something is wrong since the unit of the distance should be miles rather than miles squared. Also, by looking at an expression you can tell where it came from, whereas a number like 3.95625 carries no information about its history.
• It requires combination of various things to get the complete answer. In this case you need the law of sines to tell the distance to the mountain, and then you need to know that the shortest line from the mountain to the road meets the road at a right angle, whereupon you can engage in a complicated piece of algebra based on the Pythagorean theorem (I did that once in class) or just apply the law of sines again - which is very gratifying when you recognize the possibility (which I did after class).

An exercise based on the law of sines, and going beyond just doing an example, is:

Make a table that lists all combinations of three angles and sides that allow all other angles and sides to be determined by the law of sines. Comment on your table!

In the textbook I keep referring to they go through all possible cases in the body of the book! Students may get the impression they need to memorize those cases. Of course not! You can always figure out if you have enough information to determine the rest of the triangle's geometry by the law of sines. If you do do the above exercise you'll be sure to understand the issues.

Suggestions for your effective use of examples

• If you are lost in class, ask the teacher to do an example. The teacher might actually do one, refer you to examples in the book, suggest some for you to do yourself, or offer to meet with you privately after class. The worst that can happen is that he tells you there's no time (which would not be a constructive response).
• At home do examples yourself. Most likely you can find many in the textbook. Put emphasis on getting the most mileage out of each example, rather than doing a lot of them. In particular, for every numerical calculation get independent evidence that you have the right answer. In other words, check your arithmetic and your algebra! (In the context of the geometry of a triangle, draw a picture using a ruler and compass, or perhaps just a rough sketch, and see if your numerical results are consistent with your drawing.)
• Use your judgment to decide which and and how many of the textbook examples you wish to do. If you are sure you can do them don't waste your time. If you are having trouble with one, do another one like it. If your answers don't check out, figure out why.
• Make up your own examples. Generalize numerical examples.
• Always ask yourself what each example taught you.

Fine print, your comments, more links, Peter Alfeld, PA1UM

[16-Aug-1996]