Thinking Mathematically and Fermi Problems

Getting students to "think mathematically" and apply what they have learned outside of the confines of their textbooks can be a challenge.  The problems with the way we usually teach math have been well stated in a long article here and a short video here.  As the video explains, math problem solving becomes something like using a cookbook; a) what kind of problem is this? b) what is the formula or process? c) crank out an answer d) check answer in the back of the book.  While a certain amount of this needs to be done in order to build skill, the student needs to go beyond this kind of math practice.

I have been thinking about this issue for a while, but recently I asked the Headmaster and Math Chair at Coram Deo Academy for some suggestions. He suggested using Fermi Problems.  I took his advice and began to give my 10 year old some of these problems.  Their effectiveness even exceeded my high expectations.  In fact, I told my son that I was planning on giving him one of these problems every week or two and after doing the first one, he said he wanted to do one a day.  Here is one of the problems I gave him:

How much unpopped popcorn would you need to fill your sister's closet with popped corn?

I offered no suggestions.  His task was to figure it out and to write down each step he used to solve the problem.  The process is more important than the exact answer.  In fact, the answer is meant to be a rough but logical estimate. Here are some of the things he did:

1. He found the ratio of unpopped to popped corn - he knew how much corn we put in the popper and how large a bowl of popped corn resulted.
2. He then used water in an amount that took up the same volume in the popcorn bowl and poured it into a rectangular container (for easier calculations).
3. He measured and calculated the volume of the rectangular container.
4. He measured and calculated the volume of the closet.
5. He divided the result of step 4 by step 3.
6. Then he converted the result back into cups of unpopped popcorn.

He had to be creative, use several different skills, and engage his mind to solve this problem.  I doubt he will ever forget the concepts he used and he also got real joy out of solving this problem.  Now I have to get busy and come up with more.

Enrico Fermi