We had considered writing this post a week ago, then decided not to bother. Then, after Keith Devlin came up in conversation, we rereconsidered.

We’ve never much liked Devlin. He has written some things we like, including his excellent book on Fibonacci, but Devlin has always seemed loud and self-promoting, and not nearly as smart as he imagines himself to be. Devlin also happily assisted the American government to make their drones work better. One of those occasions when Professor Smarts might have thought a little harder.* Still, we have mostly just tried to ignore The Math Guy.**

About a week ago, at the start of Putin’s invasion of Ukraine, House Republican Adam Kinzinger did his best to start WWIII, by publicly advocating for a no-fly zone:

Unsurprisingly, there were plenty of people not so keen on WWIII who were eager to push back. Devlin responded to the push back:

Devlin’s idiotic tweet was hammered, of course, after which Devlin dug down rather than up:

Of course Devlin is not by a mile the most dangerous or the most idiotic person commenting on or contributing to this appalling situation. But, Devlin is our idiot. And Professor Smarts really should learn when to shut the hell up. Some guy called Joe Donnelly said it best:

*) In 2013, we asked Devlin a question at the end of a talk he gave on, among other things, working on drones, politely querying the ethics of such work. It is fair to characterise Devlin’s response as weaselly.

**) Disclosure. Devlin wrote a brief criticism of our first ever opinion piece on maths ed. Our piece was definitely amateurish and Devlin has an important argument, but he seems to us to have been pretty selective and snarky.

Spending all that time on Twitter can’t be any good for his brain.

Indeed. But Devlin suffered from Twitter Brain long before Twitter.

Do you guys know that Keith Devlin is also a major supporter of Slow Boaler? He straight up said that he’s big fan of Boaler. I can’t believe how a real mathematician who looks closely at her “work” can think that what she’s doing is anything more than a poor parody of math.

Was never a fan when he so snarkily responded to Stephen Wilson and Alice Crary Op-Ed (though that’s not without flaws). And was never a fan when he made like THREE separate posts on why multiplication is not a repeated addition, the first post, not even properly explaining what multiplication really is. Instead, it was filled with platitudes and self-aggrandizement (“we have to understand what multiplication is if we’re to survive in modern world” – paraphrased, obviously).

Thanks, Johnald. Yeah, I knew Devlin was a Boaler supporter. I can’t figure it out either, except that I think he so wants to show off his imagined superiority that he’ll grab onto any opportunity to demonstrate that he is A Better Kind of Mathematician.

And yes, his response to Crary and Wilson was snarky and obtuse.

And, yes, his repeated addition posts were silly and grandstanding, and wrong.

Actually, kind of curious, but in what ways is his claim “multiplication is not repeated addition” wrong? I mean, I don’t think it’s bad to start teaching multiplication as repeated addition (and for reals R, multiplication is kind of the unique extension to the repeated addition that still respect the usual rules), but I also think it’s fair to point out that multiplication for numbers that aren’t copies of integers (in a ring), it doesn’t make sense to think of them as repeated addition.

Anyways, just wanted to hear why you think it was “wrong.”

Thanks, Johnald. Of course Devlin is right. But he is also wrong, and more importantly wrong. And even to the extent he is right, what was his purpose except to prove that he is Professor Smart? Who did he help?

As you note, multiplication obviously begins as repeated addition. For a Year 2 kid, multiplication

isrepeated addition. (Just as, previously, addition was “repeated adding one”). As such, it is pompous, and wrong, for Devlin to make a contradicting universal declaration of what multiplication is or is not.Of course Devlin is also right, that, however presented, eventually multiplication becomes an operation in a field Q, and then a field R, and then a field C, and then a division ring Q. Oh wait. I went too far.

Which is the point. The fact that mathematics eventually provides nice and tidy, and sometimes necessary and deep, answers and generalisations, doesn’t mean these answers should always be showboated around as a universal truth, or that it is helpful to do so.

Now, to be fair to the Boaler-loving drone-perfecting Devlin, there is an important point behind what he is doing. For way too long, way too many students (and way too many teachers) will hang on to an impoverished view of multiplication, and it screws them up royally; even if it works, it works so clumsily, the kid hasn’t a clue what is happening. So, yeah, the point has to be made that there is a more sophisticated, cleaner and necessary view of multiplication. But these things should be done with care, in the right time and place, and without grandstanding.

As another example, think of fractions. Kids find them hard because they are hard. Sure, they can take pieces of goddam pie until they’re stuffed to the gills, but grasping the idea of fractions as numbers, and manipulating them with skill and reliability, and a modicum of understanding? That is not easy. Part of the reason it is not easy is because it is difficult to even say what fractions, as numbers in and of themselves, are. Sure, you think get 1/3, and you think you get 2/6, but then it turns you may not actually get that 2/6 and 1/3 are the exact same thing.

So, what do you do, what can Marty the Mathematician do, about the teaching and learning of fractions, apart from advocating the giving of a few neon-light examples and setting *tons* of carefully selected exercises?

One thing I can do is wage war on pies. Another thing I can do is wage war on the poisonous expression “equivalent fractions”. And a third thing I can do is to point out that a fraction is really an equivalence class of ordered pairs. Two of these could be helpful; the third is self-evidently not.

Hi Marty,

“…even to the extent he is right, what was his purpose except to prove that he is Professor Smart? Who did he help?”

Lmfao. Classic dunk on Devlin. If you rearrange the letters of his name a little, you get DEVIL-n… Maybe he IS the DEVIL… n.

I basically agree with the premise. So I was thinking about the Newton’s laws versus Special and General Relativity. Sure, are Newton’s laws *wrong* in the sense that it fails for high speed objects (or high mass, whatever)? Absolutely. Is this still a useful (and philosophically valid) looking at our everyday (or not so everyday) world? Absolutely. I think of saying that multiplication is not repeated addition as akin to saying Newton’s laws are wrong – if anything, General Relativity *generalizes* Newton’s laws the same way the multiplication on reals (or complexes, or quaternion division algebra, or abstract rings, whatever) generalizes the repeated addition.

And also, I want to add that in a Peano system (which is unique up to unique isomorphism), you can certainly define multiplication as repeated addition (and addition as repeated successor). Though professor Stanford-Smart will (and did) say that that’s still not repeated because you’re using recursive operation (jesus, with the pedanticity). Anyways, that system (together with the addition and multiplication defined), is a thing that just is – it just sits there. And it also happens that that system embeds into R uniquely, which, arguably is a different kind of system (unique complete ordered field). So *even from THAT point of view*, Devlin is still wrong to say that “multiplication is not repeated addition.” If anything, it should be “multiplication *in R* is not repeated addition,” whereas, multiplication *in N* IS repeated addition.

And of course, you’re 100% spot on with it totally not being helpful for 2nd graders. I just wanted to point out the *mathematical* problems I see as well, simply because his whole excuse of eschewing this easier view was that “multiplication is repeated addition is mathematically incorrect” and I am saying that “multiplication in N *really is* repeated addition, case closed.”

Thanks, Johnald. I guess the the only thing to emphasise, in very partial defense of Devlin, is that multiplication as its own operation in N, then Q et al, is pretty fundamental. So, the step from multiplication as repeated addition to its own thing should be undertaken, or at least begun, pretty early on. But, God, his showboating gets up my nose.

Someone named their kid Slow?! My words fail to represent the depth of my sorrow for a kid growing up with that name.

Are you serious? “Slow Boaler” is a cricket joke name I came up for Jo Boaler.

He thinks Multiplication is not repeated addition?! What else could it be?! I started the math teachers group The Defenders of the Order of Operations. If you’ve been on facebook math posts, you’ve likely seen us trying to convince people to do math correctly. I really wish people would be more willing to believe teachers, or at least be more malleable.

Thanks, Aaron. But in defence of Devlin (and don’t expect it to happen again), how do you make sense of β2 x β3 as repeated addition?

I just recently discovered this place, and read about this DEVILβ¦n, so I sought him out on π. I am SO glad I never attended any course with him teaching. The depth of my sorrow for any child he teaches Arithmetic to can not be expressed accurately in this clunky language. Shall I make something up? Perhaps the newly made up word flargle may suffice. I feel very flargle for those kids.

Aaron, you’re welcome to your Twitter fights, and have fun, but not much is going to be gained by trying to broaden the battleground to include this blog. I’ll choose the battles for this blog.

Of course, it’s your blog! Thank you for informing me that you are uninterested in more examples of his Twitter/x shenanigans. You are, of course, free to remove my comment(s) and I will just move along. Have a great day! π

Ooooh. Well I learned yet another thing. Thanks. :). I have No idea who Jo Boaler is. I’m new here. Expect more evidence of my ignorance, you have been warned. π