After reading Lockhart’s Lament (see yesterday’s post) you may wonder what my experience with math in school was and how I got to be a mathematician. Here are a few, rather disjointed vignetes :
Once (upon a time) in 1st grade my teacher was teaching us about multiplication. She said that she had a little secret that she knew: when you multiply a number by one you get the same number that you started out with! So there you go, we should have no trouble multiplying by one now. I thought that it was great that she had discovered this fact and I wanted to get in on the act also. I had noticed a pattern myself: when you multiply by two you get double the number that you started out with! Now there was a fact that my teacher hadn’t yet discovered! So I went to her and told her this little secret that I had learned. I thought that surely she would gather the class and tell them that: hey look, here is another little trick! She never did. I wasn’t crushed, but I was puzzled. After all, at the time at least, my secret seemed just as good as hers. I mention this for two reasons:
1) Because of the incident I know that I had a natural, intuitive understanding of what “doubling” meant before learning about multiplying by 2, and, at the time that I was first learning multiplication, these two things seemed like distinct and separate operations. (If I didn’t remember this incident, I wouldn’t remember how I thought about doubling and multiplying by 2 at that age).
2) Also at that age I thought of math as a process of discovery that everyone was allowd to participate in (which it is!). The system hadn’t yet drummed that idea out of me. I guess that the system never really did drum it out of me completely.
My second memory comes from first grade also, I’m pretty sure, but it could have been second grade. We were learning how to add multi-digit numbers and we were supposed to be working on a sheet where we calculated dozens of such sums. I thought it was cool that the technique we were using would let you add two numbers no matter how big they were! So I flipped my sheet over, wrote down two really big numbers, and started adding them. My teacher caught me before I had gotten very far and made me turn my sheet back over and work on what I was supposed to be doing.
1) Why did she do that? Wouldn’t I have gotten just as much practice from working out my own sum? Yes! And I would have been having more fun. Again, I was participating in discovery for myself. But, of course, it would have been harder for her to check my work for accuracy…
2) I was having some fun with the playground in my mind! These big numbers had nothing to do with any real-life application. Kids just like big numbers! In fact kids love big numbers. My niece really likes the number 1,082. It was a kid who made up the words googol and googolplex. Why don’t we let kids play with numbers in a math class?
The next scene is from fourth grade. My teacher called two students to the front of the room. (I think that she had arranged this with them ahead of time). On the board she had a problem for each of them. One was to calculate $\latex 7+7+7$. The other was to calculate . They set about calculating. Well, I should say that the first student set about calculating . The second student simply wrote down . When they had both finished, our teacher said “See, multiplication is just a fast way of adding.” I was appalled. I thought about what would have happened had it been me up there. I didn’t have memorized, so I would have had to calculate . It’s the exact same problem! There is no easier way (that I was aware of at the time) to calculate other than calculating . Of course, if you already have the answer memorized, then of course you are going to be able to do it quickly. You just write it down! I don’t know if it was this demonstration, or something else (or just the fact that I am bad at memorizing facts) that turned me off to memorizing my times tables. In any case, I remember staying after school (or maybe it was staying in from recess) in order to learn my times tables with the other slow kids.
My comments here are:
1) Looking back I realize that the teacher could have had a much better demonstration if she had given something like to the first kid and to the second, or, even better, for the first and for the second, then we would have seen the power of this cool little algorithm for multiplying. You know, the one that you learned–just do it a digit at a time.
2) I don’t think (and I could be wrong on this) that multiplying is just a quick way to add. It certainly can be a quick way to add, but I think that going from adding to multiplying is a pretty big leap. Multiplying is a whole other animal. It is related to adding, and it interacts with adding in interesting ways, but I think that it is more than just fast adding. (Like I said I could be wrong. This is a philosophical issue, as much as a mathematical one.)
That’s enough for now, but look for more such vignettes in later posts.