Computational Conversations EP#3 | Maths education mistakes... why aren't people talking about them?

The linchpin for this, I think, is typically assessments. And who's responsible for assessments? Well, again, it's a mixed bag, but in the end, it's usually something governmental that has decided these should be the assessments based on advice from a whole range of people. Well, hi to everyone. I'm Conrad Wolfram.

And I'm John McLuhan. And this week's Computational Conversations podcast, we're going to be talking about maths education mistakes. And why aren't people talking about them? So the motivation for this was that I watched your recent talk at the WISE conference in Qatar, and you put up a new slide that I hadn't seen before, which listed a collection of maths education mistakes, and I thought there was a lot to unpack there, and that would make a worthwhile topic of conversation today. So there's a long list, but let's start with a more general question. Who do you think are making these mistakes? Is it teachers? Policymakers? Where does the blame lie for your list?

I think it's across the whole ecosystem of education, and that's part of the reason that they've persisted. So you can point fingers. In the end, the linchpin for this, I think, is typically assessments. And who's responsible for assessments? Well, again, it's a mixed bag, but in the end, it's usually something governmental that has decided these should be the assessments based on advice from a whole range of people. But I think one of the problems is that the system is totally very hard to make it change with time. So whenever there's an impetus on the outside, it doesn't necessarily change. And so that's why we've ended up with this subject, you know, the subject, the curricula literally not moving.

Well, that's why we're not moving. But don't you think these mistakes, as you claim they are, they come from some deeper belief in the background of what the world thinks mathematics is. They've become encoded in exams. But they come from somewhere, surely.

Maybe. I mean, I think that it depends on the country, the discipline, who you're talking to. I think there's a lot more confusion than there is outright hostility to change. I think the problem is many people can't see how to make the change. And this is not uncommon. One analogy I've often thought about using is the connection between, in a sense, starting a company. I think before in the 1970s, you want to start a startup somewhere. It was really tricky because you had to get funny. You had to put your house on if you had a House, mortgage it against. There were all sorts of things that stopped you starting it. And led by the US There was a move to change that because it was decided. Now most countries want to prove that they're somehow the place to do a startup in education. We're still back in that 1970s model. And there are people who are ideologically against, you know, the fact that you should use computers in education to compute for mathematics, for example, just like we will have people ideologically against students ever touching a computer AI, you know, in the future for subjects that are in the outside world end up being driven by AI. I think in the end, it's very hard to justify a chasm between, between the real world and educational world. The question is how we change the mechanics to get that aligned.

Okay, well, let's talk about that chasm then. So top of the list, you've said that one of the key mistakes here, I assume that the order matters here at the top of your list was that essence is confused with the mechanics of past moments. Can you explain what you mean by that?

Well, if you take mathematics, it's very, you know, it's dramatic, right. For hundreds of years, this idea of abstracting to this sort of logical mathematical form and process, which has allowed us to make decisions and come up with answers that we could never have done before. In a sense, the calculating step of that was the limiting factor. In most cases, really hard to do. Had to have humans do it. All the effort was to try and reduce the amount of calculation by being smarter about the abstraction.

Right.

And by, by the way, I had a, a teacher at school who was, by the way, a friend of Alan Turing's, an interesting character. And one of the things his, one of his sayings was maths is the art of avoiding calculation. So that was true for hundreds of years. That. So the mechanics, the moment there were, you had to know how to calculate by hand and, well, otherwise you couldn't, you couldn't use maths. Pretty much that was turned on its head over the last few decades by the best mechanization in human history of anything, pretty much. I mean, it's hard to come up with anything that's been gone beyond imagination any further than computers and calculating things. And we're now in a situation where that step is very cheap, incredibly well done, allows us to deploy all sorts of new things. So the mechanics have fundamentally and totally changed in the outside world, and that's what's driven us forward and allowed maths to be used everywhere in the outside world. That's the whole point. That's what everybody's worried about it in education.

So your teacher's essence is proven to be more true that we're trying even harder now to avoid actually doing calculations ourselves.

Well yes indeed we can do it, but so now we can spend much more effort on these other steps. So this is an example where I think on the slide we have essence equals mechanics of the moment. This is the mistake. So. So that's wrong. The essence of mathematics in my view is a way a fantastic process for delivering answers and decisions that you'd never otherwise get and the precision to those. That's the essence of it. And the mechanics of it have changed over time and we have very new mechanics. We have machine learning which is effectively a new mechanics because of very high powered computers. What we shouldn't be doing in school is teaching some old mechanics which aren't that essence that they were just some vestige of the past. And by the way, my favorite, one more thing on this, I can go on about this particular one because it frustrates me quite a bit for ages, as I've often said about I enjoy photography and when I was at school you had to load a film into a camera. That's how you took photos. And now you don't. And if photography was a mainstream subject in schools, I think the first lesson would still be how do you load a film into a camera?

I'm wondering whether the essence of photography is changing even more with AI generated images, but perhaps that's going to taking us off topic.

Well, no, and, or video and still mixed together. So the essence evolves. Sometimes the essence evolves to beyond what you thought it was. And I think that that's true if you look at maths and AI actually.

So I think if we go back to your list, one of the least obvious ones, some of them are very self explanatory I think. But to me the least obvious is you've said that there's a confusion between driving and building. And I think that one needs some explanation. What do you mean by that?

Well, and if you take driving a car as a driver of a car these days, you're not typically the person who builds it or even maintains it. That wasn't true at the onset of cars. I mean most of the people involved in driving at that time were people who either were involved in building them, I mean whatever. And that's normal for very new technology. You don't separate the sort of the users from the developers. But when the thing gets consumerized they absolutely get separated. And that's partly what allows it to be a big driver in society that many, many people can partake because they can the drivers and not the builders. So take mathematics and computational thinking, computational literacy. This is a way in which we make decisions in society and we need a large fraction of society to be involved in that, because we are taking decisions based on how we think computationally. I mean, a lot of the decisions now, much more than in the past, are on that basis. So we've got to think very carefully about what do you need to know as a driver and what you need to know as the much smaller group, who are the makers, so to speak, the developers, and what is the interrelation between those in a particular case. And so a mistake is to say our mass subject of education is to try and show people how to invent the subject, how to build the car, how to machine a piston, when actually what you want for the math subject is driving. And when you drive a car, there are relationships between. I mean, the relationship is actually very interesting, right, because there's stuff about if you're just an average driver, so to speak, there are certain things, hopefully to avoid you having an accident, but then there are edge cases where it does interface quite a lot with the actual car. Like when you go around a corner too fast and you skid off the corner, and what you do, cars have increasing automation to handle that. But actually, and knowing a bit more about how it works helps you. And if you become like a Formula one driver, you get to know much more about the mechanics because you're much more on the edge of what can be done. And so that makes a bigger difference to what you're trying to achieve.

So let me push back slightly on that because I would argue that maths is not just driving a car where you just say the start and the end destination, and as we go to self drive cars, that'll be all you do. That it's perhaps more like Lego, I would argue, where you might not. You might argue there's no point learning how to build the blocks, but you're taking those blocks and you're still building a solution for your actual specific problem. And in a way, that's the power of maths is to give you all these building blocks. So aren't we also a consumer and a builder simultaneously when we're solving mathematical problems? And how does that fit with the misunderstanding as you pose it?

You can define as you want what the driver and the builder is. My point is what you're learning should be for whatever the current state of the art is in driving. So I mean a very good example with cars is manual gears, manual transmission. It was quite funny, by the way, because when I learned to drive many years ago, I thought, oh well, I quite enjoy driving transmission cars. But I'm not sure how long this is going to go on. Maybe they'll literally disappear within five years. Well, although they are fairly rapidly disappearing, they're still around, at least in Europe and that's many years later, decades later. So these things don't always disappear quite as fast as you want, but you always need to learn about, in a sense, the mechanics of the moment that you actually need. Now in the photography analogy, if you want to take photos 30 years ago, you have to know how to load a film into a camera. But don't confuse that with the essence. So there are things you need to know now about building, which you absolutely need to know. There are things that are the essence of understanding how to abstract. There are things about using computers at this moment that will change in 20 years time. Don't confuse all those. But you do need to learn today's mechanics, otherwise you can't operate. But don't learn yesterday's mechanics. That's the only thing I'm trying to.

Get across when I talk about this. I find the one that is the most sort of firmly held belief out of this list, I think that a lot of people would cling to is the one you have second on your list, which is that you've got to get the basic straight first, by which they mean got to learn maths on pencil and paper and then you can be allowed to use the computer. And I hear that over and over again that computers are great. Yes, we should absolutely use them for doing real work. But you've got to get the basics straight first. How do you respond to that?

Well, basics of what? And go back to these analogies. We use basics of driving a car, learning how to machine the piston. No, they're just not. And by the way, just to force that analogy even further, if you want to become an automotive engineer right now, learning how to machine a piston is probably not where you want to go. That's not your number one thing to learn. It's unlikely to be.

So what are the basics then? If the hand calculating isn't it, what are we replacing as the basics?

The basics are, in my view we have this four step process that I summarize maths as being, which is define abstract, compute and interpret. And that is Very much in the basics. That's like, can I apply this four step process? Do I understand, do I understand where it goes wrong? Can I use it efficiently? Can I use it with computers in an appropriate way, with AIs? That's the basics, in my view, that we need not. Do I know how to do formal long division? It is funny how people mess these things up sometimes, in my view. I was having a conversation with somebody yesterday and it was like one of the things we're trying to assess is, you know, how quickly people can learn, sort of, I don't know what the, you know, something like 7, 8 plus, you know, 13, 14 or something like that. How, how do you add those up? And you know, can you see that? That's roughly two. Quickly, that seems useful, the idea that there's a fraction, you have a rough idea of the fraction, the actual formal addition of those that everybody has to learn around the world. I've never done that. I mean, when have I ever actually used that, actually added precisely two fractions of that sort? I mean, I've turned them into decimals, I've approximated them, I've put in my computer, I've done all sorts of things to get an answer, but I've never formally done that other than in education.

So I think one of the things people often say is that doing it by hand somehow teaches you how it works and gives you a greater understanding. And I made a rhetorical mistake at a conference once of maths teachers where I said, who here actually understands why long division works? And a sea of hands went up and I looked very silly. And then in the coffee break afterwards, I asked four or five of these maths teachers explain to me why does it work? And they started to tell me how you do it. And I said, no, no, I said, why does it work? No, you know, what is the understanding that it's based on? And no one could explain to me why long division works. They could only tell me how to do it. And indeed, I think that's kind of.

It'S like, it's like part of this.

Confusion that they think they know.

Remember a nice bit in Surely you're joking, Mr. Feynman, the autobiography of Richard Feynman, the famous physicist who's funny enough, my brother's sort of PhD supervisor in some sense at Caltech. But there was a nice bit where he said that his dad had told him all these friends knew the name of various birds, but they didn't know how they functioned. And there's a difference between Knowing the name of something and knowing what it is. And this is the same kind of dichotomy. There's a difference between knowing how to turn the handle and understanding in some more depth how it relates to other things. And then there's a priority. I mean, it's like we don't have infinite time, we don't have infinite understanding, and particularly not for many people. And we also, most importantly, don't have infinite interest. It's like if the thing is bloody boring to start with, many people will switch off. Or if they find it bloody boring, you might find it very exciting. Right. So that's an important criterion for how we deliver.

Anyway, so as a comment on the list, I just wanted to add that for me, the one that speaks to me most is that you describe your list trustworthiness, being confused with by hand. And as a practitioner, I spend a lot of time trying to figure out is the answer right. And it's a really challenging problem that one has to be quite creative sometimes to figure out, do I believe this calculation that I've produced?

And where do you get that wrong? So if you take the four steps you're talking about, where do you tend to get that wrong most? In the. Define the abstract, the calculus, interpretation.

Yes, I think that probably I make the mistakes in the abstraction where I have some conceptualization that's flawed or just plain wrong. And more often it's in the mechanics. It's actually in writing the code and executing it, or it's in the raw.

Data I'm going to push, I think. Yeah, I'm sure you do. You know, there are very visible mistakes there and so forth, but I think there's catastrophic mistakes some of the time with defining, I mean, I think, and interpreting, but not necessarily yours, but in general, but all of us. I mean, the problem is that I suppose one of the things I'm most worried about is if you take early uses of computation, predicting eclipses, for example, which, by the way, I know my brother's just written a book for the upcoming eclipse that's going across the U.S. i don't know when.

A few months time, maybe March, April, I forget.

Yeah, full solar eclipse. That was a very early success of computation. But there's a lot of complexity to it, actually, but there's a certain cleanliness to it. By the time you're applying these techniques to biology, people, societies, lockdowns, pandemics, it's incredibly messy, incredibly messy. And then you get all these problems about what is it exactly that you're trying to figure out. And if it's the eclipse, you might want to say, well, I want to figure out when I'm going to see. When I'm going to see the sun blacked out a particular place and where is that going to be? There's quite specific. If you're asking about, well, the pandemic's a good example. What's the best outcome and what are the tools at our disposal and can we model what the effect will be, each of those, and how it will affect what we actually want? So the definition of that is a big problem. I mean, the outcome could be we want to prevent as many deaths and bad disease as possible at that point, excluding all other factors. Or it could be, well, by doing that, there are going to be other factors, like, you know, economic and other medical factors. And so that's a problem. And so do we take those into account? How much do we take those into account? And are we honest about how we're taking those into account? So I think these definitional things are, you know, are very key to a lot of the failures to use computation and they are almost totally absent in education right now. And in a sense that's a scary misunderstanding of trustworthiness that has crept in so.

Well, I know that watching my daughter prepare for UK A level exams in maths, that she's applying one of the two tests I use that worked in education, which is if your answer comes out something simple like square root of X is probably right and if it becomes a large piece of algebra, it's probably wrong, but I know that doesn't work for me in my day to day job.

Yes, well, indeed.

We should wrap in a moment, but let me just end with one final question then, which is, we've got a list here. They're all really interesting. Indeed. But which of these do you think is the most damaging out of the list? If you could only fix one of them.

Yeah.

Which one would you wave your magic wand at and get everyone to suddenly wake up to and see the truth about?

I think if you talk about math specifically, I think the idea that basics equals by hand, I think is pretty catastrophic.

The one that seems to be most firmly held by everyone.

Yeah, I'm not sure it is. I think this.

You think everything else stems from that. Is that actually just the root problem?

Well, I mean, the reason I say strictly mathematics is because the machinery has changed so much. The machinery was by hand for calculating and now it's by computer and that dichotomy is extraordinary. So that's not true necessarily of all the other cases. But I do think that we. So that's why I specifically say. I mean I think one of the key points here is that on the slide is avoid for AI on every other. On every subject. So the problem is we've had 20 or 30 years run up where maths has been diverging in education in the real world because of new machinery and we haven't taken the steps to fix that really micro steps at best. We've just got major new AI that's happened. We're entering what I would call the AI age, new industrial revolution. It's going to affect all other subjects in some way but we don't quite know how yet. We don't know what the real world's going to look like but we've got this intransigent education system but it's got to learn from maths how to adjust. We've got to fix the maths but we've also got to go and adjust how it gets fixed so that we can alter that in the future. So in a sense, I suppose another one that's very important in this list is what I, you know, innovation led evidence rather than evidence led innovation. You can't, if you want to make a major change you can't have the evidence first because that's just not how it works.

Right.

You've got to only can you make micro changes with evidence sort of baked in front of you. You've got to guess then check then see if you've got the evidence. It worked that way round and I think that'll be a critical thing as we go forwards.

Okay, great. Well we should wrap there and perhaps. So hopefully I've convinced.

But at least, at least it was a good slide. I enjoyed making the slide actually and I hadn't put it together quite like this before so anyway, it's good to talk about these things.

I do think it captures a lot of what we've been talking about for the last few years.

Well that's good. Great. Well see you next time.

Bye bye everyone.