Concept vs Process

Mary and Jim have been married for 12 years, but they usually eat separate pizzas. Last week they bought two frozen pizzas from the supermarket, threw them into the oven and had them for dinner.  Mary ate half of hers and Jim ate 4 of his six slices.  How much pizza was left.

I have been programming computers since 1958, specializing in compilers and other forms of artificial intelligence. I can write a computer program that will set up the equation for this problem

x = (1 - 0.5) + (1 - 4/6  )

and give me the answer, that there were five slices left.  I could use the same techniques that I use with computers to program students.  So why do I insist on talking about the concepts involved instead of the process of solving the problem? Or, alternatively, why do other Math teachers spend so much time showing you how to work problems like this. Don’t they know this skill is not what employers are looking for?

If you read the Common Core Standard for Mathematics at http://www.corestandards.org, you will see that we need to focus on concept instead of process. Unfortunately it is easier to teach and assess process. It is easier to test whether you know how to “do a problem” than to test whether you “understand a problem”. We used to think that word problems, like the one above, tested your understanding of a concept. We now know that is not true. Students can be programmed (or trained, or taught, or whatever) to solve problems like this using the same techniques we use to program computers. Moreover, techniques that are the most efficient for doing problems, are often counterproductive for understanding concepts. For example, while it is very efficient to compare two fractions like 11/15 and 18/27 by cross multiplication to find that 297 is bigger than 285, you have lost track of the fact that you are comparing 297/405 = 0.733 with 285/405 = 0.703.

As long as we are talking about problematic issues, I have another problem. When a student asks me to show him how “to do a problem” I can’t get over the feeling that he wants me to entertain him. My answer is unequivocally a short NO! Just like you don’t learn to shoot baskets by having the coach demonstrate his style, you don’t learn to do mathematics by watching me show you how to do a problem. I want you to pick up the ball, show me how you hold it, bounce it in front of you a few times and then throw it toward the basket. My job is to help you improve on what you are doing. It is not my job to “show you” how to do it.

Now that I’ve got that off my chest. Let’s see if you can learn to do Math the Wright way.

J Nadas - January 2012