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
First of all, you can do as many examples as you like by
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:
This is an immensely useful result. One type of numerical
example (taken from a popular precalculus text) is this:
In a triangle, the ratio of the sine of an angle and
the length of the opposite side is the same for all
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
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
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
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
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
Always ask yourself what each example taught you.
Fine print, your comments, more links, Peter Alfeld,