We had been aware of Eugenia Cheng, of course, as the happy new face of mathematics popularisation. We hadn’t paid her much attention, however, until seeing her in a very bad maths documentary; the badness wasn’t Cheng’s fault but she fit right in. Then, in synch with the appearance of her weirdly pushy new book, Cheng wrote a very silly article for the Guardian. Now, with her most recent WSJ column (Murdoch, paywalled), Cheng has tripled down on her silliness.
Cheng’s article is titled,
Solving Our Problem With Math
Our problem is then made clear by the subtitle:
A rigid focus on the ‘right answer’ is needlessly turning students off from a field that’s all about asking interesting questions
Even ignoring the scare quotes, that is an astonishingly stupid sentence. The scare quotes are priceless.
Cheng’s column begins well enough, noting the clichéd complaint of “when are we ever going to use this?”, and the boring and foolish non-solution of teaching kids to do their taxes and whatnot. But then things begin to go off the rails:
There was another meme that went around during the COVID-19 pandemic, depicting a math teacher teaching a class about exponentials, and some bored students saying “When are we ever going to need this in life?” Unfortunately, when the pandemic began it would have been helpful if more people had understood exponentials.
As if it takes more than about two minutes for anyone to understand the implications of exponential growth.
Instead, when scientists tried to point out, using exponentials, that it looked like things were going to get bad, far too many people thought the scientists were scaremongering or making things up.
Um, maybe it was also because the scientists were making things up?
This isn’t good, but then Eugenia gets into the nonsense proper, with the “rigid” focus on right and wrong answers:
… this image of a rigid world with clear answers is a very limited view of what math is. Thanks to this image, too many people are being put off math unnecessarily; they’ve only been shown some very narrow, unimaginative version of it, a version that doesn’t allow for any personal input and curiosity of their own.
What the personal input into 2 + 3 or 6 x 8 might be, we’re not quite sure. But Cheng has ideas:
I would like to show math in a different emotional light. I want to encourage and validate the asking of questions, the ones children may want to ask but which math class doesn’t seem to answer, the ones that made people say you should just buckle down and do your homework. The ones that make some people feel they are not a “math person,” because the people who did well in tests didn’t seem to be asking those questions. Questions like: Why does 1+1=2? Why can’t we divide by 0? Where does math come from? How do we know it is right? These questions are not usually on the curriculum.
First of all, these questions are not on the curriculum, or are just on the fringes, because they do not easily lead anywhere. Secondly, of course kids can ask such questions; it is only in Cheng’s Dickensian fantasy that such questions cannot be and are not asked. Thirdly, in order to ask and ponder such questions, and meatier mathematical problems, in a meaningful manner it really helps to know some maths. Fourthly, what is Cheng actually suggesting: twelve years of just gazing at the stars? (29/08/23 Fifthly, Ed Barbeau, below, has made the obvious but critical point that questioning mathematics in the way Cheng is advocating will confuse the hell out of the majority of students, for no significant gain.)
Maybe so. Cheng continues:
One problem for many children is the prominence of times tables.
Of course. And, for the millionth time, the tables are not a “problem”. They are simply not that hard, and they are, of course, necessary. The problem is a pandering to children’s natural inclination to inattention.
It can seem like memorizing times tables is a key part of being a good mathematician. This is not the case at all.
That part is too weird to even critique.
I have personally never memorized my times tables.
Eugenia Cheng, mathematician and concert pianist, who must have memorised dozens of musical scales to perfection, never memorised her multiplication tables. Yeah, right.
Cheng follows up with a similar story about her PhD supervisor, struggling with his tables as a kid. Maybe, although it feels like the’re gilding the lily. But even if true, so what? So Cheng and her supervisor made their way to be mathematicians without memorising the tables? Mere mortals cannot.
Having taken the obligatory bash at the times tables, Cheng goes back to asking her questions.
Take “1+1=2,” for example. This might seem like an obvious mathematical truth,
Well, more properly a definition, but go on.
but there are contexts in which different outcomes occur. If you mix one color of paint with one other color of paint you get one new color, not two.
Paint. You’re wasting our time with buckets of paint.
If you turn a piece of paper over once and then turn it over one more time you get back to where you started. These are valid outcomes that are also studied by mathematicians; questioning the equation 1+1=2 leads us to deeper mathematics.
This is at least mathematics rather than paint. And sure, teaching kids some modular arithmetic is a great thing. It’s also been pretty standard for about a thousand years. And, if you want kids to gain some proper sense of modular arithmetic, perhaps they might want to be in possession of a few facts and skills on regular arithmetic? Just a thought.
After a brief digression into the different emphases of college mathematics (as if this were somehow a surprise, and as if it didn’t depend upon the foundations laid in school), Cheng combines her bashing of the tables with her love of questions:
Instead of asking “What is 6×8?” we could ask them “Show that 6×8 = 48.” I can intone “six eights make 48” without engaging any part of my conscious brain. But what matters is that if someone questioned it, I could provide several different explanations to back up my answer.
God, this dumb. Yes, of course kids should, must, learn about factorising and rearranging factors and so forth. But they also must learn that 6 x 8 = 48.
This is really the point of logical rigor and abstract math. It enables us to package up ideas and treat them as building blocks so that we can understand increasingly complicated concepts.
School kids aren’t engaging in anything remotely like logical rigour and abstract mathematics, at least for many years. If and when they do, they will only be able to do so successfully if they have the proper grounding in the mechanics, how mathematics works. If you’re not really clear on the “how”, you’ve got Buckley’s hope of mastering the “why”.
And finally, mercifully, Cheng ends:
Then, by continuing to question and explore, we can create things that are further and further from those basic building blocks—like starting with the idea of exponential growth and ending up with an understanding of how viruses spread.
Yes, once more with the two-minute exponential growth thing.
Cheng’s ridiculous column brings to mind a blog post by Scott Alexander at Slate Star Codex, on not liking mathematics. We’ve already written on Alexander’s post, but here is a paragraph that we didn’t quote:
Certainly I love the sort of math that doesn’t involve doing actual mathematics. I love reading about Moebius strips and Klein bottles, discussing the implications of Cantor’s discoveries about infinity, even playing around with fractals and tessellations and other forms of mathematical art. It’s just that when you put actual equations in front of me, with numbers and symbols and variables, my brain melts.
Of course Möbius strips and the Cheng stuff is great to discuss. But to discuss the majority of this stuff beyond a party trick requires some mathematical sense. Which is not achieved by party tricks.
Wouldn’t it be nice if we taught students so that, when confronted with actual equations and numbers and symbols, their brains didn’t melt.