Computational Conversations EP#5 | What's the point of the 12 times table?

Welcome to another computational conversation. My name is Conrad Wolfram.

And I'm John McLuhan.

And today we're going to talk about the 12 times table and actually times tables in general. And this was kicked off because many years ago, I'm thinking more than 10, we had a rather hapless education secretary or minister, I think, in Britain. Maybe we've had one since too. But at the time we did, who was bitterly complaining that people were only learning upstairs 10 times tables, I think, and not 11 and 12 times tables. And at the time, you, John, I think, wrote a blog post about whether this actually was right, and I wondered whether that stood the test of time and whether we could discuss a bit more about whether we do need to learn our 1112 times tables and how you analyze this.

Yeah, so when that happened, I think he was arguing that all primary school pupils should learn to their 12 times table before they moved up to the next year. And for him it was a simple want to raise standards and improve maths and that kind of motivation. But to me, it came as a bit of a surprise because I always understood that the reason why we learned 12 times table was imperial units in money. We used to have 12 pence a shilling and 12 shillings in a pound. And of course, you and I were born the year that that was phased out in UK money system. So even when I was learning the stuff, when I was six, seven, whatever it was, even then it was already anachronistic. So this is way later than that. The idea seems to be coming back. And so my argument was basically, why do we have maths at all? We have maths so we don't have to know the answers to things. We can work them out. Now, of course, there's some value in times tables, but the question becomes, where do you draw the line? If it's worth learning your 8 times table, is it worth learning your 9, is it worth learning your 12, is it worth learning your 17 times table? So I don't think there's any argument that there's no end to memorizing mathematics, because otherwise maths is pointless.

When you say learn, there are lots of different forms of learn, right? So you say, well, we have mathematics, so we don't have to learn. But I mean, mathematics, you are learning mathematics. There are facts you need to know about mathematics in order to apply it. I think what you're talking about here is rote learning that six sevens are 42, simple factual recall.

So there's no process to your timestable. You know, nine nines are 81. I haven't had to go through any process to do that. I'm just recalling the fact. And of course, maths is most useful where there is no answer, it's never been worked out before. It's a great mechanism for generating new answers and new facts that you can't actually know in advance. So timetables is sort of a special case where we think it's worth just knowing the answer rather than having a mechanism for working it out.

So I looked at your blog post, which I haven't done for many years. It's a sophisticated analysis, you know, involving various assumptions about, you know, errors when you estimate things. And I think the first thing that struck me is how hard it is to figure out what you need to know. So you'd think this was a pretty simple question, right? It's kind of like, should you have this in the curriculum? Should you not have it in the curriculum? And as we just talked about, what does it mean? I mean, does it mean knowing the idea of multiplying, which you clearly do need, it's an important function in maths, or does it mean factual? I mean, as in memory recall. And if so, up to was.

Yeah. And in the end I think I started with a premise of what's the point? Why is anything about this at all useful? And that's a fairly simple argument. I think that there's three reasons one might want to know this. One is that these things come up so often that you have to know them. That's a perfectly valid reason. You don't want to always have to reach for pencil and paper and do some long winded calculation or a calculator.

The second, I've forgotten about those, but yeah, carry on.

Well, the second and really the second and third reasons that I proposed were that they are a component in some bigger infrastructure. And if you look at where else we go in multiplication, most people learn, there's lots of different algorithms, but most people learn at some point multiplication in columns where you're processing one digit at a time. And there there's a really simple answer. You can't do that algorithm without your 10 times table up to your 10 times table, because every digit, presumably nine, well, 10 is kind of a special case because is it managing number position or is it 10 times? And then they amount to the same thing, but it's only one to nine. Plus you need to understand the concept of number position. So there's a really clear answer for not stopping earlier. You can't do the more general algorithm without basically knowing your 10x tables and below. But then it doesn't help in that algorithm any further. So I look for what other places where it might be useful. And I centered a lot of my arguments on a approximation that I can't work out. 1242 times 1191. But I knew that's roughly 12 times 12 and some decimal places to count to know whether it's thousands or tens of thousands or hundreds of thousands. So being able to turn it into a similar, not quite exactly the same, but close enough problem that is a times table thing actually is a valuable thing. And that's where I did all the technical analysis of how much return do you get for the more effort you take to learn, you know, how. How much more accurate are you knowing your 17 times tables and below compared to knowing your 20 times tables and below? And, and it turns out it's a losing game from, from the beginning.

But, but, but I think what one thing that's also changed is, as you correctly say, I mean, as of today, I do people ask me, you know, do you think times tables are worth learning at all? Forget how far up. I think my answer is yes, because I use them often, but I mainly use them for estimating.

Yeah.

Not for precise calculation. I mean, I might use it for precise calculation, but it's sort of, you know, if I happen to want to, you know, precisely know six sevens or whatever I said before, it's probably the only one I know, 42, then it seems useful, more useful to know it's roughly 40. It would seem to me in most things you're doing, because often when you're discussing these, you use multiplication as a standard thing. Although you might ask why you don't learn by rote standard editions, for example, and standard divisions. I don't really know.

Well, I mean, some of that does get taught where actually this is something that's changed since our day. I was never taught this way, but certainly my children were taught these kind of number decompositions that knowing that seven is three plus four as a way of being able to break problems down more quickly into things like timestables. But again, it's like, how much of that do you know? It's actually really useful to know all the numbers from 1 to 10 and how they break down and how you could, you know, turn them into, you know, eight plus one for nine or nine. But it's not really useful to immediately.

Why is that useful?

What? When you have things like, you know. Yeah, but simple things like, you know, if you were doing like 12 times 13, which isn't a table. I know, I know immediately that's 12 times 10 plus 3 times 12. You know, that kind of being able to find a useful number ref point where you do know the answer and know what's left over is kind of a useful tactic. But again, for what I mean, so.

One of the things that's changed, obviously, since we were at school is, I mean, you really do have a calculating device with you essentially at all times. I mean, I think it's an interesting question, actually. In the last decade, let's say, when have I not had a machine with me capable of doing calculations when I possibly needed it? I mean, I don't know, but it seems extremely rare to me. I mean, you even calculate on your watch. Well, actually, there were calculated watches, I remember when we were out in school. Right. But they're now integrated with other things.

I was very jealous of those.

Yeah, they weren't that, of course, they.

Were incredibly difficult to use because you needed to poke the buttons with a pen because they were so tiny.

Well, I know I didn't actually have one of those, but I did have the latest Casio calculator, I remember, with tape backup for its programs. But back to the point, which is that if you've got a piece of paper in front of you now, at least in a reasonably developed country, et cetera, et cetera, more or less, you've got a computer in front of you of some description, it might be a phone. So it seems to me the game shifted a bit because even when we were at school, there was some argument for knowing fast recall of precise answers like this. It seems to me now, if it's going to take you more than a little while, I mean, you know, I don't know what a little while. There's a few seconds. It's not clear it's that useful. And mostly it's useful for this estimating Now. Now, a bunch of people would claim, I think, that, you know, there's been a lot of study of the neurology or the neuroscience of this, and it's really important that you learn this stuff for your brain development.

Yeah, but that argument goes for learning anything. You could learn the scores of your Pokemon characters in battle, and that would still be practicing your brain memorizing things. We mostly try and use education to teach things that are also useful.

Well, I mean, maybe. I mean, I often quote to people who want to retain the current state, you know, the status quo, without thought, so to speak. I love asking them Whether they could recite the pluperfect subjunctive of amo in Latin to me, which at some point I guess I could do, but certainly can't anytime recently. And the point about that is, in the 1950s, certainly here in the UK, if you're at a grammar school or a private school and you wanted to go to a reasonable university, I think that would have been one of your requirements. Now, as you say, there's some correlation, one would expect, between the ability to be able to learn precisely something, whatever it is, and deploy it in some way, and generally being able to learn things. But it seems a bit, when we've got so much in the real world to learn that's very complicated and getting more complicated, it does seem rather spurious to insist on something that seems parochial and basically without practical use.

Well, if you're going to learn two things, why not learn the one that's more useful, whether that's culturally useful or technically useful. But Latinx example is a good one, because I still remember amo, amass, amat, amama, sematissamant. But I can't use it, not least because I've got no Latin grammar vocabulary left. But also I can no longer remember what those word conjugations apply to. So it's useless fact information.

You have a Volvo car, which I hope you know, comes from Volvo in Latin to roll. Right. And I believe.

But it was not useful in my purchasing decision.

Okay, well, that's fine. But I. I Volvo Volvari, volvi, volutum. If I remember the. The past participle writers to do with revolutions and things. But I remember anyway.

But I'm going to go back to why we're on topic, which is I want to draw you up on the fact that I think for very small numbers we are routinely making use of those timetables. They're just so ingrained. You don't notice that you bring cakes out for your four guests. And if you're going to do two cakes each, you don't want to turn up with seven and have one person, one shot, you know, immediately it's eight. You're not doing multiplication in your head, you just know four guests, two takes eight cakes are needed. You're just not thinking about it. But as soon as they get beyond those kind of practical small numbers, which again tends to be kind of under 10 or under 20 as the outputs, you're right, we mostly go to approximation. I don't do the same calculation for the tip when I go to a restaurant with a Bunch of people. I look at the bill and go, well, we need to raise so many pounds tip roughly and that's going to be roughly three pounds each, Whatever, everyone chip in. And the fact that it might be slightly under or over the 15% target doesn't really matter. It's got to be roughly. Right.

And also it's. Well, with all these things one wants to ask upsides and downsides, what's the worst that can happen if you don't know your 11 times table in the modern world? I mean, it's kind of like approximation.

Techniques, which is what we actually do.

Yes. And as you say from your argument, I think up to 10 times table really is quite useful for all sorts of reasons, including being able to manually multiply bigger numbers, which one doesn't do very often at all. But I guess it's useful to be able to do it in print. Sensible still, presumably. But it doesn't seem that useful above that. And it doesn't seem like it would ingrain any greater thinking. I mean, even if you could argue there's thinking ingrained somehow from up to your 10 times tables, it doesn't seem like 11 and 12 is going to ingrain much more than that. In a sense, the upside seems quite small and the downside seems almost non existent in the modern world. Now, as you say, when we were dealing with money, which was carefully, and other things, weights and things which were typically sort of somehow based around 12, I mean divisible by 3 and 4 because 12 is so great, because you can divide it 3, 4, 6, etc. So it's very easy to do hand calculations, which I think is why a lot of these things were set up around 12s in one form or another. Again, that made more sense because it was just a practical thing which has now been lost.

But.

And what's funny is the way that this was trotted out at the time by the minister as a great rendition of, in a sense, rigorous maths.

Yeah.

And you know, to me it's just sort of mindless misunderstanding maths. And it's kind of like he failed to do the mathematical analysis that you did in your blog to ascertain why you actually wanted to do this.

Now it was always better argument, which is a misunderstanding of optimization, I think.

Yes.

I mean it's interesting when you at the time, a whole bunch of people of course engaged in the conversation and some more constructively than others, should we say. But there were a few people who tried to make an argument that there was value in the 12 timetable for its own intellectual merits. And those tended to break down, I think, in most cases, in terms of misunderstandings. There were those who said, oh, yes, well, you've got to learn things like it's 10 times something, which, you know, plus two times something, which is my point about. You're not recalling it now. You're using an algorithm which is defeated by learning it. There were people who pointed out how important 12 was in the real world, like, you know that. But mostly they were about divisors, not about multiples.

That.

The fact that you can divide 12 neatly is really interesting about it. But if you use the multiples argument, then why aren't we still using. Was it the Abyssinians who use sexagemal base 60 counting, which has even more divisors? But we don't do 60 times table, which in fact is more useful because 60s fit into a circle 360 degrees better than 12s do. It is a divisor, but it's not a useful divisor. So they tend to be quite spurious. And then there were the ones who said there were patterns, which is an argument for 11, that it's kind of neat that it's 11, 22, 33, that you get a pattern. And as an early discovery that patterns sometimes emerge from processes, maybe that's interesting. I don't think it gives any great insight. But that quickly breaks down with 12, which doesn't do anything very interesting very quickly.

I mean, the comments I'm just looking at here from 11 years ago are quite fascinating. I mean, it says, for example, up until the 1950s, many Scottish schools taught up to 14 by 14 since stones and hundredweight were in common use, et cetera, et cetera. But as you say, a lot of arguments here about the multiples, about number of degrees in a circle and so forth. The thing that's, I suppose, most interesting to me, thinking about what you need to learn in mathematics, is just how difficult this is to figure out. I mean, this is the times tables, right? It's kind of like, you know, yeah.

It'S basic as you can get.

It's basic as you can get. And in the end, you might say, well, why are we bothering? You know, if you teach people 11, 12 times as well as 10, it's not a big deal. Although, again, this was the reason we're bringing this up is because it was highlighted as a rigorous thing that's required, which is, in my view, absolutely absurd. But if you think about this, onto lots of other areas of even basic maths, you realize why it is we're stuck with things because they were always done in the past.

I mean, even at a basic level, it's a time budget question that takes time to drill this into children and that time could be doing something more interesting, more useful. I mean, there's nothing, there's nothing intrinsically wrong with it. Apart from that. There was somebody, I think.

Well, no, except you lose a lot of them. A lot of them think it's pointless.

Well, yes, yeah, some have turned off it. That's the negative.

You know, they're not doing things that they might find more interesting and you know. Yes, it's taking bandwidth away from other things and their time is precious. So basically it is important and if you're going to force everyone to do it, which is what we're doing, it's.

Forcing is the problem. There was somebody who commented that his daughter knew up to the 20 times table and really loved it. And that's great, you know, I'm a nerd. I learned PI to 10 or 12 decimal places. It's not useful for anything.

Somebody at my place knew. I remember as a friend of mine, you knew it to 60 places seemed utterly pointless, but.

Well, it is pointless. But if it's fun, who cares? It's not harmful. But when you make, you know, if we made everyone learn 60 decimal places of PI, that would be a complete misuse of teaching time.

So here's another bugbear for me which is adding fractions precisely. So when was the last time you added 5, 6 and 78 precisely? Now, much more important to me is to know that five, six plus seven eighths is sort of a little bit under two. That seems a much more useful thing to know. But I can't actually ever remember a time when I've ever added fractions precisely. I mean, possibly a half and a quarter. I suppose to know that's three quarters, but that's the extent I would change them into decimals if I had any problem with them.

Well, I mean, here's my theory on this. I haven't studied the background to fractions, of course, go back a long way, but the reason I think they were important in our educational system was that they were a quick way to be accurate. Because doing decimal calculation on pencil and paper is much more expensive to achieve less accuracy that if you can work out, and I'm not going to embarrass myself by trying to say what five, six plus seven eighths is in, you get an accurate answer in relatively small effort. But that balance is Completely flipped now. Now it's a lot easier to do decimal arithmetic because of computers. So, you know, you can do the calculation to 20 decimal places in a more systematic and easier way than trying to do, you know, the fractions, which a lot of computing systems don't.

But this is the whole of math. Just so a clip, by the way. It's 41 over 20. It's 4124. So just Mathematica.

I wasn't going to get that live. No.

As my teacher at school, who was a friend of Alan Turing's, used to say, maths is the art of avoiding calculation. And at that point, it certainly was hundreds of years of effort so that you could take a problem, turn it into the abstract math, get an answer by calculating the answer, and interpret the result back to a useful decision. And the problem was, of course, that most things were not amenable to hand calculating. So people spent a lot of effort trying to work out or abstract it in a way that allowed them to calculate it. And that's exactly what you're describing with fractions. That era has passed and we haven't reacted to it. But it's funny how the thinking persists that we need this even though we have the machinery ready to do it.

Fractions you kind of need to separate into. There is the abstract kind of intellectual structures of fractions, which maybe there is still value in studying versus the practical tool for getting answers. Where I think in the same way that slide rules and Napier's bones have all been swept away by better technology, I think fractions are now at that point where they're not really the practical way of doing things, apart from in simple cases like half of a half, I kind of feel like those kind of regular numbers, just like timetables, the things you hit all the time. Often when you take a recipe that's got half of something and then you want to do half a recipe. Yeah, those kind of very frequent small numbers. But the, the five, six and seven eights, that's not a regular kind of combination. Why learn the algorithms for that anymore?

Yeah, I don't know. I mean, that particular one with adding fractions, a lot of children get very stuck on. I mean, it's just a technique of, you know, common denominated things that they just don't find it very easy to master. And the problem with this is that, you know, if they don't happen to master easily, they get upset and they lose, you know, they, they, they lose confidence and all sorts of other things. And, and that would matter. In a sense, it would be worth pushing through that in a case where it really matters today. But since I don't think it does matter today, I, I think we, we need to be very careful about doing that and, and I worry about it. But what really troubles me is how we can get to a point where people do the sort of analysis you did on whether you need 11, 12 times tables on the whole curriculum. I mean, we've been doing some of that. I think one of the things we've been doing in our outcomes list. Right. Is to try and separate out, abstract out the levels of knowing. So our point about the fractions, or indeed the times tables is there's knowing it exists. I mean, knowing that multiplication exists, knowing that fractions exist, knowing that addition of fractions exists, then there's sort of knowing when you'd use it, and then there's knowing different ways of working it out, either precisely or approximately, and knowing when you can get caught out by it. Like, for example, you know, the fact with fractions that when the bottom number gets bigger, it doesn't mean that the number is bigger.

Yeah. For example, I think fractions are a good example as well, where that desire to deliver the practical calculation and make you good at that is actually squashing out the intellectual benefit of it. Because to me, almost the central discovery that one needs to understand with adding fractions is that you can only add things that are the same. They've got to be the same kind of thing. And I think units is a better place to understand that. That adding one inch to one meter is not two somethings. Well, it might be two somethings, but it's not two meters or two inches that you've got to understand. You've got to make them the same before you can add them. But so much of the desire to teach the process means that we end up doing things that ignore the intellectual reason. I'm sure you were taught. I was only taught dividing fractions. You turn the second one upside down and you multiply. And as a process, that's really easy to remember and really doable. But why are you turning upside down? Well, yes, there's a good intellectual reason for that, but that's not what's taught. The desire to make sure you can do the process overwhelms the fact that the inverted fraction is kind of the. Well, if I go back to my kind of university undergraduate stuff, I guess it's the kind of inverse element of a, of a, of a ring or a group. I forget now that's also part two.

How it's written down that there are these, you know, different ways of writing down divisions, fractions, whatever. And there's a particular trick of the notation as well.

Yes.

And by the way, I did the same thing with equations, right. So I knew for a long time that, you know, x plus 2 equals 4. If I move the 2 to the right hand side of the equation, it changes its sign. I didn't know what the hell I was doing. Worse, I got quite confused and messed up a bunch of them because when they were multiplied or something, I didn't really know what I was doing. I didn't understand. I was trying to do the same to both sides of the equation, but nobody really explained to me why you couldn't square both sides of the equation. That wasn't clear. So I think there's a big issue also about the concepts of what's being learned versus the actual mechanics of doing it. So the concept, the idea of equations.

Or the idea of mechanics of learning, which also can obfuscate both of those.

Yes, well, indeed. And of course AI tutoring things may adjust that. I mean, I'm hoping that one thing AI tutoring will do, whether or not one keeps some of these things in that we think are probably a waste of time. One would hope at least AI tutoring could optimize the learning for people who get stuck on them. Because I think sometimes the diagnosis of why people are stuck is quite unclear. I think my daughter, if I remember correctly, Sophia, was. She was very good at geometry, for example, and could help everybody else in geometry. But she, for some reason, I think, had missed the lesson. I think it was adding fractions, but maybe I'm wrong about that. I can't remember really. And so she just. I don't know, she just missed the thing about that. And she got into trouble because they didn't diagnose properly that it was just that she'd missed that lesson. It wasn't that she had some deep understanding that was missing. And I think that keeps happening with children all over the place. That's bad in them not continuing with computational learning that they could continue with.

Well, I get more concerned when the actual techniques of teaching obscure the inner truth that they're claiming that you'll somehow learn from this. Again from the blog, some people argue that it's really useful to be able to see the patterns. And I recalled that my daughter had been taught the seven times table, which of course is quite a tricky one, maybe because the patterns aren't so obvious. By a poetry mnemonic. I can't remember. It was something about cycling Susie. I can't remember what the poem was. Cycling Susie goes to heaven. One times seven is seven. And it was like all rhymes to help you remember it. So as a teaching exercise to be able to recreate that table for a.

Test, nothing to do with the maths.

Nothing to do with the numbers, and in fact it's distracting you from the numbers by saying what you need to do is be able to recite this poem and then you'll get a tick that says you've reached some successful criteria. And this is what confidence in maths.

All of that would be okay if it's really necessary and maybe it is with seven times table. But as you say, that falls off very rapidly. So one has to be very particular as to what it is that's actually needed. Now I do think now is an inflection point in all of this, not just for maths, but for everything in the sense that we have, you know, modern AI's coming. I call them modern now. They won't be modern in 10 years, but the LLMs statistical AIs on top of our symbolic Wolfram type AI type way of functioning. And I think that opens up the whole question to a much broader, deeper issue as to what is it as humans we really, really need to learn? At this moment it's not clear. And I know in computer based maths outcomes, things we've been trying to, as I was saying, abstract this out, which is not done in curricula usually to try and work out what it's kind of, you know. And back to some of the things we've been talking about. In a sense, these are a very simple version of what we have coming up. You know, now we have calculators or computers or whatever we have for doing multiplication. And obviously this is a question of already decades ago, what is it we need to learn about multiplication and times tables? Essentially what your blog was about. Now we've got the same question about differential equations and we've got the same question, of course about machine learning, which is interesting because it in practical terms did not exist before very recent computers which had the power to execute it in a useful way. So of course machine learning is a case where you can't do it except with the computing machinery. So you've got to. There's no way you can learn what it's doing by hand.

Well, not in a practical way. Not in a real.

No. Well, not in any useful way.

In a way, a Bit like how you can understand how times tables work, but that doesn't actually really help you in a practical way when you're multiplying thousands of numbers together to build a bridge.

No, but at least does help you for, you know, seven times seven. Right. Which, as you say, is a building block that's useful for certain kinds of estimation, things like that. Not something to be put on a pedestal, but of some utility.

But it comes back to where I opened that blog, which is to ask the question, what is the point of this? And does it still fulfil that point? Because there's this idea that maths is timeless. And of course, the truths of maths that have been discovered or invented are timeless, but the usefulness of different bits of maths to know how to do by hand is not timeless. And that changes just in the same way that slide rules have got swept aside. Maybe machine learning will only have its time for a few decades before it gets replaced by something more useful and more practical.

Well, I think this is this thing about the essence of what we're trying to do versus the mechanics of the moment. Although in case of maths, of course, it's been a moment that lasted hundreds of years in terms of hand calculating being the only way to do it, and that's now been overturned. So that's good. That probably opens us up to a future, a future discussion about what we can learn from maths for the future change of all subjects in the AI age. And, you know, whether there's anything we can, I mean, one thing to learn is that the educational system has not got itself reformed around the changes that machinery brought maths. So maybe that could be learned before we get to a point where all other subjects are being transformed at a rate of knots for the real world, as it then will be for what you do, you know, with those other subjects which literally happened decades ago already for maths and is continuing to happen, but we still haven't learned the tale of that. So maybe that's a future discussion we can have.

Love to have that.

Good. Great. Well, until next time, join us for the next installment on computational conversations.