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.

This is hilarious. Thanks for posting, Marty. The article by Eugenia is a subconscious, unfiltered dump of thoughts. Incredibly, simply incredible. It is like she got up in the morning and all of a sudden realised that all her life she didn’t know how deep and intriguing timetables might be. I am sorry for Eugenia; she probably had a bad math teacher in school who didn’t explain to her that understanding and practising timetables inevitably leads to memorising them. Somehow, abstract thinking also doesn’t visit Eugenia too often, as she doesn’t seem to associate the experience of learning music with the experience of learning math. I can’t meet Eugenia, but I would love to ask her a question: If in music, putting your fingers on the piano without producing sound enough for a concert pianist? One shouldn’t care about the ‘right music and notes’. One should care about striking the right pose and asking the right questions about music, and that should be enough. I am not sure if it is even possible to criticise this so-called writing. Incredible, simply incredible. To write such stupid things, one must be a very educated person.

Eugenia Cheng is sowing so much confusion that it is guaranteed that the kids will not know whether they are coming or going. I suspect that most kids will take for granted that 1 + 1 = 2 once they know how to count and examine concrete instances of this. Comparing with turning over a page twice is to introduce a complication by linking two separate things that makes no sense for the children.

As for “right answers”, hers is a hobby-horse that has been flogged too many times by those who object to the authoritatian character of mathematics. One of the good things about mathematics is its certainty (which I think a lot of kids appreciate). However, what is true is that you can get an answer from different perspectives, so that it is possible to ask children whether they can see the result another way. Also, this ideas encourages the teacher to take a wrong but promising answer from a pupil and make it into something respectable.

As for the times table and the like, my experience with children is that many of them they like to know stuff and take pride in what they know. At the risk of being a dinosaur, the old fashioned techniques of rote, memorization and chanting can be used judiciously to cement things. Mathematical memorization is not like memorizing the name and capitals of the states of Australia; the things that you are asked to memorize lie in a structure that can be used to buttress and check the memory. It is a little like a space ship exploring the distant parts of the solar system by using the gravity of the larger planets to slingshot it on its way. Every multiplication fact lies in a nest of mutually validating relationships.

” Also, this ideas encourages the teacher to take a wrong but promising answer from a pupil and make it into something respectable.”.

Wrong, but promising? What would be an example of that 2+2 = 5?

I can imagine such answers (more accurately, solutions), but Cheng is overplaying it. She is pretending that very standard teaching is some new, magical insight.

I can see how the ‘wrong solution’ might and actually must be discussed in class. Those can always provide an excellent passage to learning. That is a standard process of understanding. However, solutions and answers are different things. We can discuss why Glenn Gould played with straight fingers and used his low chair and all the problems he developed due to it.

I am trying to envisage a situation in which a pupil might suggest that 2 + 2 = 5. Has any teacher among you come across this mistake in practice, or is this just a bogeyman of the educationists? However, if it happens, I would hope that the teacher might try to ascertain what is on the pupil’s mind, because there may be a misperception at stake that needs to be revealed. Basically, does the pupil really understand what addition connotes, or is he just working from memory.

This reminds me of a thing we used to do as kids to show that 6 + 5 = 11. You display the fingers of both hands, and one one hand you start counting backwards along the fingers from 10: 10, 9, 8, 7, 6. And then you put up the other hand and say: “and plus 5 is 11!”. This was a schoolyard thing; I have no idea where it initiated and whether it was ever used in a classroom. But it is an interesting thing to unpack.

Another example is a riddle from my schooldays which interestingly made a recent reappearance (the price quoted will give you an idea of its antiquity): Three men go into a hotel and book a room for the three of them. The desk clerk charges them $10 apiece. Somewhat later, the clerk realizes that he overcharged them, that the cost for the room should have been only $25. So he gets the bellhop to return the extra $5 to the men. However, the bellhop figures, “What the heck! Those guys won’t know the difference. I will give them each a dollar back and keep the other two for myself.” Which is what he does. So the three guys each pay $9; that’s $27; the bellhop gets $2. That’s $29 altogether. What happened to the other dollar?

Let me give an example from an experience in a classroom. There were models of the regular 3-dimensional solids suspended by strings from the ceiling and the question arose as to the number of edges the dodecahedron had. The children knew that there were twelve pentagonal faces. One student said that it was 60, since there were twelve faces each with five edges, and 60 = 12 x 5. There was a tetrahedron sitting on a table, so we looked at this and saw that the same argument would justify its having 12 edges. However, it visibly had 6. So somewhere there is a fish in the reasoning. This was followed by quite a lively discussion among the pupils.

So I think that this illustrates the moral that if a student gives a wrong answer, often you can often do better than to simply growl at him.

Thanks, Ed. The confusion thing is obviously a really important point, which I overlooked. I’ve added an edit to the post.

Marty has got it in one line: “…pretending that very standard teaching is some new, magical insight.”

Hey, unrelated to this post and maybe you’ve covered this already, but what do you think of the constant colorification, fluffing up with text that has no value, inclusion of web-based “learning tools” and the like in math (and other subjects, most notably physics) textbooks? Compare a modern textbook aimed at school students versus something published maybe sixty years ago, maybe in the soviet union. The difference in quality is stark.

Hi anon (if you comment on other posts, please try to choose a less others-will-also-choose-it name),

Of course I agree with you entirely. Textbooks now are much worse, in all the ways you suggest. And no, I don’t believe I’ve written specifically on the comparison. Not obvious how to do it, but well worth pondering. Thanks for the suggestion.

>please try to choose a less others-will-also-choose-it name

Alright 🙂

Very funny! Reminds me of the guys from Hootie & the Blowfish being first interviewed in Australia: “Were you ever worried that the name might have been taken?”

Recently in classes for years 7 and 8 students, I have been using Isaac Todhunter’s “Algebra for beginners” (1870). No calculators of course; after all, students in 1870 did not use them. I give the students problems from one section; usually about 15 problems – plenty to keep them busy for a lesson; they are drill problems that get increasingly more difficult. The students seem to enjoy them; they throw themselves into the problems – and into the ensuing debates when students get different answers. The students seem to be more confident in arithmetic as we progress. Here is a typical problem from p. 11:

If , , , , , find the numerical value of the following expression.

.

I place great emphasis on setting out.

Thanks, Terry. Can you characterise why they work better than current texts? Also, how did you come to choose Todhunter? Presumably other school texts of the era would have been similar?

Marty, I know you hate Youtube clips but I think the following is relevant. (It’s 10 minutes, can be very frustrating but bear with it. I think the punchline is worth it):

I know it, it’s a great clip, and it’s still a shit lazy comment.

~or

We can view this video as illustrating an issue we have with the teaching of Maths in the USA.

Conceptually, both the teacher and the punkazz kid had a point.

It was the + and = sign that got in the way of the two communicating.

When I was in first grade the teacher wrote this on the board:

11

11

11

Then she said “add them”.

I wrote them down as she had written them, and came up with 6.

Instead of telling me I was wrong, she asked me how/why I came up with 6.

Using my first and middle fingers I counted them “2, 4, 6”

In response, she wrote next to my writing on my paper:

11

11

+11 (with a line under +11)

To that, I answered “33”

Then we had a talk about the usage of symbols, and how they clarified things.

She knew how to get into my head, and she’s the one that taught me how to work the arithmetic tables: how addition & subtraction, multiplication & division work together.

My parents didn’t let me have a calculator, and to this day using on gives me hives, maybe even boils.

– I do most of it in my head, then write it down for review.

I know the video was comedy, and overall I agree with the sentiment.

However, I’ve had more than one pursed lipped instructor tell me I’m WRONG when my logic was solid.

And, when I’ve explained the validity of my argument (eg: let them “hang themselves with their own words”), I’ve been kick out for the day.

– people say Mathematicians have a strange sense of humor, or, at least that’s what they say to me……

Maths is logic, and you can’t be logical if your foundation isn’t strong.

Eugenia has far more schooling than I do, and I think she’s great, and….I know that learning the tables & formulas until they are ingrained is vital.

Yes, it’s “Old School” but there’s a reason that Old School has been standing long enough to become an old school.

Thanks, Andrew, but the bratty kid didn’t have a point.