Rolfe Schmidt

Recently I’ve been thinking about how our university curricula can strip the excitement out of Math. After talking with my 4-year-old a little bit I realized that there are similar problems all the way down to the elementary levels.

My son loves playing with numbers and patterns. He seems to produce a steady stream of little discoveries. For example, one day he came up to me and said:

“Do you know some numbers have middles, and others don’t?”

I didn’t understand, so I asked him to explain.

“Well 3 has a middle, and 5. But 4 doesn’t have a middle. And 6 doesn’t have a middle, see?”

He wrote down the numerals 1,2,3,4,5 and circled the 3. Now I get it. So I asked him if 1 had a middle. He furrowed his brow for a little bit then laughed and said

“One has a silly middle!”

So he discovered even and odd numbers. He talked about this for days, and undertook very helpful projects like rearranging our encyclopedias so that the numbers “with middles” were on one side and the numbers “with no middle” were on the other. He was doing Math and having fun.

After seeing him run with ideas on his own a bit, the activist parent in me thought “imagine how much more fun he could have if I helped!” Well, I didn’t exactly tell myself that, but the spirit was there. So I bought a curriculum, Singapore Math, and we started having “Math Class”.

The Singapore Math books are great. They are nothing like the huge sheets of arithmetic problems I dreaded as a school kid. They introduce concepts like number, greater than, and less than by having kids explicitly construct maps between sets of objects. Addition and subtraction are introduced in a unified way as a relation between 3 numbers they call a “number bond”. They use ideas I never saw explicitly stated until I studied university level Math to make things easier for kids to learn.

So with an interested kid and a great curriculum, things must have really taken off, right? Well not quite. He enjoys some of the lessons and gets very frustrated by others. And whenever the exercises start to look like arithmetic drills, he turns off. For one lesson, he was asked to write ‘even’ or ‘odd’ next to the numbers 1 through 10. I had explained to him that a number was odd when it “had a middle”. But looking at the sheet full of blanks he just said he didn’t know. When I asked him one question at a time, he did fine but seemed bored and wanted to go do something else.

Part of the problem is that he is not too interested in reading or writing letters, and that is fine with me. But I still think that there is something about the “drill” – mastering the skill without thinking about why – that kills the joy. He can solve problems all day as long as they are his problems. When somebody else comes up with the questions, the spark isn’t there.

I don’t think this is particular to my son. How many of us revere the mystical powers of arithmetic the way early Indian mathematicians did? I’d guess not many who aren’t number theorists. We need arithmetic skills so they get hammered into us until we don’t even think about them. To motivate students we tell them it will help them buy groceries or design warranties or do some other mundane chore. Because we don’t think about the methods, we don’t appreciate them and we don’t really understand them.

But we need to do drills, right? I think that it is great that my son discovered “numbers with middles”, but I can’t think of anything I did to make that happen. And if I don’t know how to make him discover things, how can I make sure he develops the foundation of skills he needs? I can’t go through life thinking hard about every long division I perform, every square I complete, or every singular integral operator I bound. The problem seems even worse for me when I’m writing a computer program. Sometimes you just need to do things automatically to leave room in your mind to solve harder problems.

I’d like to think that the good teacher can find great ways to lead students into a life of discovery. But I’m not that teacher and I have kids who are learning right now. I can’t imagine trying to be this “good teacher” to a whole classroom of kids. At some point the student needs to learn that these drills will make life easier down the road. Even though they may not be fun right now, they will make Math more fun later. The younger a student is the less you can expect them to realize this, so it is easier to overdo the drills.

Whether the student is starting graduate school or elementary school, they need to master the tools without losing sight of the interesting questions. The teacher should encourage them to keep asking questions and remind them that the skills they are trying to master were discovered by people asking questions just like theirs.