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:

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:

This is an immensely useful result. One type of numerical example (taken from a popular precalculus text) is this:

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:

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.

Here are some

Suggestions for your effective use of examples

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