Two, four, six, eight. What comes next? If you said, Who do we appreciate?, you may have spent more time at school football and basketball games (rah, rah!) than in math class. What comes next is, of course, ten. Any dummy knows that.
I’ve forgotten exactly when we hit sequences like this in school (late grade school? early junior high?), but I remember my frustration pretty precisely. Evidently not as smart as ‘any dummy’, I was lousy at seeing the pattern the answer guide provided and on which the teacher insisted.
Couldn’t the next number be fourteen? Take a number, add two, then add the previous two numbers and repeat? Nope.
Couldn’t the next numbers be six, four, two, four, six, eight? Up, down, up, down? Nope.
It drove me crazy. Not that I was always wrong, but my hit rate was abysmally low. Able to memorize operational rules, to understand basic concepts, and to apply some intuition to solving problems, I did pretty well at other math class tasks, but in this area I never broke the code.
So imagine my feelings in a calculus class many years later, when we proved—proved—that every such sequence has an infinite number of correct answers to the question, ‘What comes next?’. Infinite. Remembering my juvenile frustration, the mature me was, simultaneously, overjoyed and outraged. Overjoyed, when I realized that my non-standard solutions had, indeed, been right. (Was it too late to get credit for them?) Outraged, when I wondered what the heck the curriculum developers had been thinking by insisting on One Right Answer.
Would it have been so hard to change the question from ‘What comes next?’ to ‘What could come next?’ To challenge us to see more than one pattern, one possible sequence? To impress upon us that a correct answer means nothing, if you can’t explain it? To expose us to the idea that simpler explanations are somehow better than complex ones, to introduce the concept of ‘natural continuation’ and the beauty of elegant solutions?
Maybe grade school and junior high are too early to think such deep thoughts. Maybe we only think these thoughts are deep because we don’t think them early enough.
In every discipline, we move from ‘no control’ to ‘conscious control’ to ‘unconscious mastery’. As preschoolers, we learn to count, painfully, carefully, meticulously. In school, we learn to add (painfully, carefully, meticulously), by relying on our now unconscious mastery of counting. Eventually, adding, subtracting and (for some people) more complex arithmetical operations also become second nature through extended exposure and regular use.
As post-schoolers, we learn a second language by memorizing vocabulary so that we can read signs and point to what we want on a menu. Then comes the ‘painful conversation’ stage, where we can put together sentences with some forethought and deliberation. Eventually, we can converse without having to translate everything we hear and say.
If basic arithmetical operations and other languages can become second nature through extensive use, why can’t critical thinking skills become second nature to us in the same way? Why can’t we think deep thoughts about what could come next, rather than learning to look just for the One Right Answer?
Richard Feynman, one of the 20th century’s great physicists, told a story about himself as a preschooler. Playing with his red wagon, he jerked its handle to get it moving, and the ball in the wagon rolled to the back. Why did the ball roll backwards? he asked his father. It’s called ‘inertia’, his father replied. No one knows why it happens.
A special case, to be sure. We can’t all be like Richard or his father. But all of us—children and adults—can learn to think better. And the earlier we start, the better.