I picked this story because I feel it shows the beginning of a lot of my strengths and weaknesses when it comes to my relationship with mathematics. Here goes.
My first experience with extended mathematical problem solving was in the fourth grade. My math teacher was Mrs. Adair. I was lucky enough to be put in her class with the fifth-graders for math time. On top of a little bit of nightly homework, we had P.O.W.s; I guess that means Problems of the Week, but P.O.W. is what stuck in my memory. I remember that they were always assigned on Tuesday and due on the following Tuesday, and each assignment had one problem, and five steps, which I can still vaguely remember, and here they are:
1. What is the problem? Write it in your own words.
The problem was: how many ways are there to get $1.21 using only half-dollars, quarters, dimes, nickels and pennies? There may have been some story I don’t remember, but it was definitely $1.21.
2. How are you going to solve the problem? Are you going to draw a picture, make a list?
I was a big fan of brute forcing problems, so I was going to make a list. But how? It seemed like there were probably at least 10 or 20 ways, so writing them down might be rough. I complained to my dad, since most P.O.W.s I could knock out in 30 minutes or an hour. He could have told me to suck it up, or to go away, but instead it turned into an Excel lesson. He showed me how to label the columns “Half-dollars,” “Quarters,” “Dimes,” “Nickels,” “Pennies,” and how to enter the numbers using the keypad at the right side of the keyboard. I started at it, starting with as many half-dollars as possible and working my way towards 121 pennies. He guessed there would probably be a few dozen.
3. Show your work. Be neat so that someone else can understand your work.
It turns out that this is not a good problem if you want your kid to give the one family computer back anytime soon. Several hundred rows later, each row representing a way of making one dollar and twenty-one cents, I proudly print the list.
4. What is the answer?
I take it as a good sign that I have NO IDEA what the answer was, other than that it was over 100 and less than 2000 (I think).
5. Did your plan work? What could you have done differently?
I thought my plan worked great! Good job me! (Now it seems mind-numbing and slow just imagining typing in all those numbers, but I was happy that I had an exhaustive method, a feat in itself.)
I would estimate that I spent around three or four hours on this problem (and my dad spent an hour with me). When the next Tuesday rolled around, I felt pretty pleased with myself, since there had been some talk about this being the worst P.O.W. ever. There were lots of different answers; no one had the same answer that I did, but most people had made a list and gotten numbers lower than mine; I had already checked mine over for double-counting, so I knew they had left something out and started feeling even more smug.
One girl, Sophie, had a number bigger than mine, but we ran out of time and all but the two of us were clearly tired of the problem. As recess began, she came up to me. I remember her making fun of my list, that she had a shorter way. She had taken all the ways to get a dollar, all the ways to get 21 cents, and multiplied them! This whole story sticks out in my mind because of this idea, even though it’s wrong. I told her that her way was wrong, that she must have double-counted, but I couldn’t explain why and refused to entertain the idea she might be right.
It turned out later that I was the closest, having left out only a few cases, but this story still comes to mind when I think about two things:
1. My own strengths and shortcomings as a mathematician. Sometimes I try to make up for in speed and brute force what I lack in elegance. I can come up with a plan, but it may not be the best plan and I can stick to it too long instead of hearing others out.
2. What does it mean for a mathematical idea to be a good idea? Does it have to be correct? I used to think that correctness was a necessary but not sufficient condition for a mathematical idea to be good, but now perhaps it’s neither?