Possibilities and fears.

An example

It is possible to introduce problems with minimum- or maximum-calculations long before the concept of derivatives is introduced. Problems including iterations and open-ended problems can be introduced at a quite basic level due to the calculation capacity of the calculator. Problems can become quite complicated and still solvable with just the four rules of arithmetic. Look at the example.

When Linda was born, her grandmother decided to give her 100 each Christmas. On the New Year following this the money was put in Linda's bank account giving 6.75% interest per year. She received this Christmas gift for the last time when she was 18. The New Year after she turned 19 she was allowed to use the money. How much money did she then have in her account?

One solution to this problem is hinted at in the figure:

If we start out with the value 0 in x, which was Linda's capital when she was newborn, we can enter the formula and obtain the result by just pressing EXE 19 times (depending on what calculator we use).

The traditional way to solve a problem like this is to use a geometric sum. You will therefore never see an example like this in a chapter of a mathematics textbook where the factor of change is introduced. You will see it in the chapter dealing with geometric sums, and then as a very advanced example. I know that less than 50% of the students would solve this problem this way in a test just when this method has been studied, and I bet that very close to 100% would not be able to solve it one year later.

Use of iteration when solving this problem (as above) instead of a geometric sum makes it much easier and, from a students point of view, much more logical to solve. The problem as such lies quite near to the reality that every person needs to master in the modern society, so I cannot see why its solution must be hidden in so much mathematical abstraction.

I gave this problem (in the context of iterations) to teachers I had on further training. When I said: "You have never heard of geometric sums" the teachers were unable to solve the problem. (Well, they might have eventually, but we didn't have too much time.) What I find serious about this is that even teachers in mathematics have the opinion that there is a one to one relationship between problems and methods / formulae.

Broman - 5 OCT 1996



There is a PDF file available for this paper.


Gómez, P. & Waits, B. (Eds.) (1996). Roles of calculators in the classroom.

Mail comments to Pedro Gómez: pgomez@uniandes.edu.co