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.
Excellent article! From my position in life, I would add a few words to your last paragraph. “But all of us – children and adults (even VERY old adults) – can learn to think better”. And if we can achieve this, the happier our latter years will be. It is really sad to see the grief people carry around because “the one right view” they learned maybe 80 years ago, is not operative in their current world. The fact that there could be MORE than one right view, often has never been considered! What about a remedial course in basic thinking?
Marjorie – Maybe if we can stand to stay open to new ideas, the world offers us that remedial course every day! Old and young, alike.
We can learn critical thinking patterns, yet as we evolve we find out they are limiting our thought. We can just let the brain do its thing. Ask the Blinkmeister, Malcolm Gladwell.
Gary – I haven’t explored Gladwell’s stuff. Maybe it’s time…
Reminds me of high school math teachers. I could classify them into two types. Those who insisted that a solution to a math problem had to follow a certain set of prescribed steps and those that allowed one to have an alternative, equally effective approach. Fortunately, I did encounter a few in the latter category.
And to Marjorie – there is nature and nurture, and unfortunately too many are born with “the one right view mentality”. How do you fight what is in the genes? I wonder if in the next 5000 years humans will evolve so that MORE than one right view is tolerated (with of course my view still being the preferred!).
Dave – Good teaching is really hard, that’s for sure. So easy to just dump the information or methods and leave it at that. Much harder to encourage thinking. I don’t know whether we are ‘hard wired’ to think there’s one right way – but any nurture in this respect sure seems to fall on fertile ground.
I couldn’t have said it better — but I will probably try to, one of these days, anyway…
Jim – That’s OK, maybe that brings us close to even in this regard.
Isabel, I love the fact that this post is about improving the quality of thinking. But I worry about some of our young people (now I’m starting to sound like my mother) who are at the mercy of some really goofy ideas in teaching methodology. I spoke to a teacher last week who was extremely frustrated because her principal has forbidden the use of text books as teaching aids. They are taught by power point slides! The teacher was so desperate to share an idea that couldn’t be reduced to bullets on a slide that she tore 2 pages out of a text book, scanned it into the power point presentation and presented it to her students that way. Maybe I’m old fashioned but I can’t understand how we will teach children to think if we don’t give them an opportunity to read.
Susan – Yes, it seems to me, too, that reading extended content is an important part of learning to work with ideas: following complex arguments, remembering points made maybe pages earlier, making connections. So a lack of textbooks (or any books? yikes!) seems like a Bad Idea. Maybe it goes deeper than that, though, in our culture. Neil Postman has argued compellingly that the advent of TV has destroyed our attention spans. A few generations ago, people used to listen to day-long political speechifying. Hard to see anyone sitting still for that today! Whether today’s young are any worse at thinking than we were, I don’t know. I always suspect this perspective when it comes from a middle-aged or older person – it just plays too true to the ‘age & stage’ bias. I’m not even sure how we’d go about measuring such a thing, so it’s just as well that others are charged with managing the school system. At an individual level, wherever we are, we can get better.