As I’ve said before, I am addicted to Quora.
It’s not because of the intelligent questions but just the opposite. I’m a fan of what are said to be nonexistent — namely good stupid questions. I find there’s nothing better than a good stupid question to help wake me up out of a stupor.
Here’s a nice recent example: “How can we teach complex mathematical problems effectively to students (including ourselves) without using words or symbols, but just using images or actions related to the problem itself?”
Now, actions can be pretty much anything, including uses of the forbidden symbolism or words. Talking is an activity, but let’s respect the spirit of the question and rule language out altogether, even sign language or proto-sign language (hand-waving, pointing, nodding of the head, etc.).
I have to admit that, as someone who has tried to teach mathematics to undergraduate business admin majors, there was really no natural use for language anyway, apart from grunting and belching. It was just needed to get their attention, and when they turned to look at you, I would draw some squiggles on the board — pointlessly numbered 1 to 5 — and then indicate by hisses, snarls, snorts and foot stamping that the squiggles were the assignment due next Friday.
One of the things that always fascinated me about these exchanges was that even the most promising of my students took it for granted that the squiggles were legitimate problems in mathematics. They would make serious efforts to solve them, usually using guesses from their repertoire of numbers they had seen in the past, like 2, 5, 8.5, pi and the square root of two. Some really clever guesses consisted of permutations of the squiggles themselves.
Grading was always easy. I would grade on the curve, meaning I would construct a Gaussian distribution of grades and then randomly assign them to the students’ papers. The most popular method of assigning random grades used by college math teachers is, by the way, not coin tossing, as you might think, but tossing all the papers at once down a stairwell and arranging them according to distance traveled. Grading is the fairest when it is maximally meaningless. That’s why Gaussian distributions have always been preferred, because they are typical distributions of meaningless accumulated errors.
Let’s face it. Language only works when you are talking to someone who already knows what your words mean. For example, say you want to tell a two-year-old, “Don’t pull on the cat’s tail.” I don’t know about you all but, when I was two, I lived in a place that was devoid of cats — didn’t know what they were, didn’t know they had tails. Therefore, the sentence would have been wasted on me.
Talking about complex mathematics to business admin majors or likewise is a similar situation. It’s not that they don’t know anything. Hardly; it could be that they know too much. They just don’t happen to know what those particular words mean.
And then there are symbols. They aren’t even meant to be like words. Half the time you can’t even pronounce them. What does a < b mean? Is the letter at the pointy end bigger? No? Whoa, so random.
I love how the Quora question suggests we could learn mathematics more effectively if we ourselves avoided the use of our own language and symbols. This clearly implies we don’t know our own language and symbols and can only be misled by them. I think that’s spot on. I know never to talk to myself; I’ll only confuse me. I should stick to images or actions related to the problem at hand, like scratching, sneezing, building a birdhouse or starving to death.
Images should, of course, be carefully policed. No thought balloons, captions and certainly no equations. Stars exploding, kittens and bunnies being adorable, sunrises, sunsets, forest scenes and mountain meadows should all be fine, I guess. Just don’t wrap them in words.
I would go so far as to say that this approach to understanding complexity ought to be applied to more than mathematics. Does politics confuse you? Avoid your words. Rocket science? History? Keeping the “Star Trek” universe straight in your head? Keeping words and symbols out of any of it is good, sound advice.
I hope I have rescued a mind or two today.
Read more of the Oct. 4-10, 2023 issue.