Computational Conversations EP#8 | Do either of us ever use the maths we learnt in school?

Hello, I'm Conrad Wolfram.

And I'm John McLune.

And this week's computational conversation is based on a question we often get. Have you, meaning you, John and Conrad, ever used the maths you've learned in school? Of course, many children ask, will I ever use this when learning maths? Very common question. And the timing of this particularly is because in Britain we have, or in England and Wales, we have a level results coming out, I think next week. And maths is, by the way, now in quotes, the most popular subject. So it seemed very topical to discuss whether these traditional forms of maths indeed have been useful to us. And particularly, actually, I noticed it's your birthday today, John, so happy birthday as well. Discussing maths on your birthday, that seems of the moment. So what do you think? Do you think, I mean, you did a maths degree as well as furthermore, and have you used that. I mean, and you work in a mathematical software company and for many years.

And I'm spending half of my time thinking about maths problems. So you would think I would be. If, if anyone's using the maths they learned, it would be me. You know, in. I gave this a little bit of thought knowing we were going to talk about this today, and it's not actually as straightforward an answer as I thought because I think there's a lot of things where some of the concepts that I took away from my school education and maths actually are really common, everyday things. And that's a separate question from whether the things I learned to do that I had to pass an exam to prove I could do are things that I. That I do. But I'm going to give you one that touches on both, I think, as a start point, which is Sohcartoa, you'll probably recognize the. I don't know if it's yourself in.

Every country, but yes, I don't know.

If this is just the way that I was taught to learn it, but it's the relationships of the angles and the triangles. The sine of an angle is the opposite over the hypotenuse. And the fact I still remember it as the acronyms shows that that was ingrained into me at the beginning. And that's actually one that's slightly the opposite. Because those rules, anytime I'm doing things that involve any kind of geometry or components of dimensions, those I come back to not because I compute anything or rearrange equations using them, but as a way of setting the problem up. And that's actually, I think, one where the Concept was the wrong way around for me because I was always taught it for angle chasing problems in triangles. And as an adult I now think about this as projection and components of things like forces in a particular direction. And it was never really taught to me as a projection issue. It was always taught to me as how to solve a triangle, which is perhaps.

Well, I think there is a thing.

I've learned, but it's recontextualized.

Yeah. So there's the actual, in a sense, content and then there's the thinking style. And then in a sense there's the process which are intermixed. And I must say for me, most of the detail, even at A level maths I don't really use. And in fact it's worse than that because some of the thinking wasn't quite right. Now, to be fair to it, at the time you and I did it, we were using algorithms that were then available. Although, you know, there were computers already, there were by, you know, decent calculators, we could have used many more algorithms than we did. But what we did is focus on the algorithms we could compute by hand, which was a pretty narrow set and led us to a particular style of thinking. So I mean, undoubtedly there's a pattern of thinking and you see this across people who are sort of quotes, well educated in STEM subjects, not mixture of maths, physics, particularly hard sciences. They have a particular style of thinking through problems, whether those problems are in business or in maths itself, which I think is certainly an extra dimension. It's certainly very useful. And interestingly, when we were at this stage of A levels, most people running large organizations were lawyers or philosophers or something by education. It was actually very rare to have a technical person running a large organization, certainly let alone a government, although that's pretty rare still. That has changed dramatically. And so I think that the pattern of thinking has a utility, but I.

Think at my slightly more at the coal face of maths. I think actually when I reflected a lot of the concepts that I picked up in school, like the basics of equation solving and simultaneous equation solving, which while as practical skills are utterly useless in my day job, you know, no problems are so simple, they're never univariate or just two linear equations. That the geometric thinking, the kind of the way that I see the pictures in my head that help me to understand these much richer problems that I actually have to deal with day to day, that all goes back to my school time, I think. And even that's followed through, I think the most profound course I did at university was the linear algebra course where just trying to think about things as, as abstract higher dimensional vectors and how everything can be thought or a lot of things can be thought about that way was very powerful conceptually, but not the tools. I know you quote this in your book, it's one of your favorite examples. When do you ever actually use a quadratic in work? And I can attest to that. I can't remember a time where I've had to solve a quadratic equation that was at all meaningful and I've certainly never done it by hand.

So this is pretty sad, right?

I mean.

You'Ve worked for 30 years plus in a mathematical software company. You're actually doing, I mean you're both working there at strategic level, but you're also actually doing mathematics in quotes, if you can call it that, for all sorts of reasons. And yet you're basically not using most of what you would find or did find in your A level maths. And I mean we both have slightly more recent experience since we both have daughters who've done A level maths recently. So we kind of have been re exposed to it in a sense. And I would actually go slightly further, which is I think, I mean in a sense if the tools that we were studying were a means to an end of getting the concepts and what we actually use, I suppose that would be one thing. But I'm not even sure if they're doing that because I mean some of them are, but it's sort of almost by accident some of it.

That's what I'm saying. Yeah, I think there's a lot of that by accident because the concepts are useful. I mean another example is that, you know, I studied, you know, all the fundamental kind of mathematical functions, things like log power, exponential, factorial, sine, cosine, I.

Couldn'T remember formulas, I would call them the cut of fundamental things that were amenable to hand calculating that you were forcing.

Yeah, and I've never studied or used the apple folks one function or anything which I suppose is equally a fundamental, possibly more fundamental. But these kinds of functions do come up in real models. And while I can still remember these rules, the double angle formulas, do you remember this? Sine 2 theta is 2 sine theta cos theta, things like that. And I remember, I can still, if I thought about it, remember half a dozen of these things, utterly useless to me in practical work to know those formulas. The fact that I know, I have a intuition of how these functions behave, which grows faster, which are periodic, you know, that, that's some of that. But that seems like the intuition.

I mean, why not try a much wider cut of functions and even. Which were available then. I mean we've got obviously now there's vastly more computing power. There is a wider range of functions. There are new thing kids on the block like machine learning types of operations and things which didn't in practice exist then. But I mean there were plenty of functions then as well and plenty of them amenable to even calculators that we had at the time and actually wouldn't have much better way been to learn a wide range of those, play with a wide range to gain the intuition as to how a lot of them worked. We're not doing that now and there's very narrow intuition as to how these things work. And there's no intuition on things like machine learning because it's not covered. And it's very different to how traditional sort of deliberate algebraic manipulation works, et cetera, et cetera, et cetera.

I can give you straight because it's very topical. I was thinking about this deeply. Yesterday was one where I'm dealing with concepts I learned at school, but I got no intuition from school and that is linear regression. So I did once I spent a maths lesson, possibly a double math lesson, making a table of the sum of the squares of. I can't remember how you do this by hand. Finding a.

It's bloody boring.

But yeah, yeah, yeah, it was very boring. It perhaps ingrained some. No, it didn't even ingrain the principle of minimizing the errors. And we. Because we did it once and we're told see it works. I never learned anything about the influence of outliers and things like that. And yesterday I was grappling with a health modeling problem with one of our engineers and we had some horrible outliers and we still wanted to, you know, use the whole data set. It was giving us a line that was just useless. You could see intuitively a line of points, but it wouldn't fit them. And we were trying to switch to something that I was never taught about. I happened to read in a mathematica journal about 15 years ago was least median of squares as a robust way of finding fitting. But sadly we don't have that as a function built in. So we were trying to code that up from scratch and it turns out to be optimizationally a little bit tricky. But you know that that's sort of how to handle outliers and, and the way that it's good that you know the name, none of that.

You did cover medians. I mean non linear fitting. I bet you didn't cover at all in A level. I don't think I did.

Oh no, no, no. And even. Well, there were some things I think I can criticize for the perhaps the way I was taught because of course I learned median and mode. But they always just presented as here's three different ways of finding the average and here's how you work them out. And the idea of robust statistics versus non robust statistics was never a kind of a concept that we considered. And that's something that I've learned in my years at Wolfram is there's most of, is this fair to say the most of the actual maths I use I've Learned in the 30 years I've been working at Waltham, partly because I'm.

Mostly one I want to come on to. What have we learned from actually using maths in our day job, so to speak, and also using mathematics tools? I mean Mathematica in particular has that taught us anything? I mean, I think. Let me answer that for Mathematica because although I was fed it, you know, by you know, from the siblings, so to speak, at extremely early stage, I think one thing it did put in context for me was the breadth of mathematics that existed out there. And even this was back in 1988. And secondly, there was a very interesting set of structures, things that I hadn't understood were connected to each other and how they might be connected conceptually and sort of ways to think of things and think of how mathematics is and the process of mathematics, none of which I had any inkling of in practice from my maths and further maths A levels and not really even from sort of. I mean I did natural sciences for a couple of years at Cambridge and then maths. And although it was more coverage, it was segmented into, you know, particular things you were studying at that given time. So that bigger picture really didn't present itself and has been very useful. I think that sort of piecing together of, you know, there's a way to think about data structures which is actually very general and quite helpful, basis sets, how you're setting things up, etc. Etc, etc. So I think I learned quite a lot just from looking at a very general system like Mathematica. And it's very nice structuring. I think it's somewhat unique to Mathematica. Probably that one would get that.

Yeah, yeah, for me, I think there's been some of that. I spent a Lot of time. Partly because of some of the things I do for the company. Like in a couple of hours I'm giving the what's new in version 14.1 talk. I'm constantly having to learn new bits of maths as some of our very smart engineers implement things I've never heard of. And there's a motivation to learn this kind of breadth. Having a common representation I think has helped me to hold maths in my head in a way that didn't exist at school where when you talked about algebra problems they would seemed like fundamentally different beasts than when you're working with matrices. Actually matrices is another one that I use a lot but, but none of the stuff that I actually was taught how to do by hand.

And I want to see you inverting a three by three matrix by hand.

I can do it. I haven't done it. I know how I'm one supposed to do it, but I've never actually tried it and it, it will become horrible at 4.

I had to invert 3 by 3 matrices in my time.

Well, my daughter does. That's apparently part of the A level. I'm surprised to be able to do.

That at the time.

And I tried to show her the row reduction technique which scales much better as things get bigger than the way she's taught, which is a recursive technique. And she didn't want to know because she couldn't use it in the exam. But the other thing that I think I've got out of learning stuff through Mathematica is the cross silo cross domain computations. And if you look at the more recreational maths blogs that I've written over years, a lot of them actually are trying to join disparate bits of maths. So I'm doing things like, well, what happens if you use a Newton rational Jackson type recursive convergence on an image? Or what happens if you think about word problems as a graph and trying to put one kind of data into another kind of structure and kind of asking questions about that. I'm very into recursive things as well. I think that's part of it, which isn't something that I particularly learned at school.

Why is that? Okay, well, so let's just pick on that for a second. What's caused that? Just because you happen to like them as a theory of life or because you've found them quite successful or because they're sort of kind of neat or what.

I think it's, I think that it's just I like the concepts of Things that go like, is the snake eating its own tail? Kind of concept. What happens if you feed something itself? And that kind of concept in maths, you know, I find all kinds of interesting things pop out of what happens if you put things around in a circle. I mean the only one I can think of that I learned at all like that, and actually this was quite a. Again one which while I learned to do it by hand, which I never do now was a useful lesson, was Newton Raphson. And this is route finding in, you know, you take a trajectory on the curve to make your next guess and then you repeat the process. So it's the only iterative recursive process that I think I learned at school. And actually optimization of functions is something I do a lot of. And worrying about yesterday's problem with trying to optimize least median of squares is an optimization problem. And you know, I never learned about local minima versus global and how to do it in multiple dimensions. I've learned all of that since. But that concept of iteratively hunting down a route, that's been a useful concept.

So a couple of things. So in computer based maths we formulated a set of objectives or outcomes that we want from people learning this and one of the things we did deliberately, and this conversation makes me particularly happy we did this, is that we separated the idea of the concept of different bits of mathematics from the tool set. There are actually many other. I think there's 11 dimensions of outcomes we have of which concepts are ones and tools are another. Obviously the concept sort of a. The idea is, and we'll flesh, we can flash this up, but the idea is that we've got, you know, the I, you can go down what you might want to know about the concept and sort of, you know, do an outer product of that with the actual concepts that you might think are important, which by the way is never done in the write up of curricula. But they kind of list them linearly down. You must know this about equations, you must know this. But they don't even do that. They say, not about the idea of an equation. They say you must know how to do the following with the quadratic, which.

I can tell you the example I always think of with this one, which is derivatives that you probably did this, which is derivatives from first principles where you draw the tangent and you let the length of the tangent limit down to 0 and out pops 2x from the x squared curve that you started with. And that blew my mind. And, and of course the concept of derivatives. I use, I don't know if every day, but it's just a core part of the language of doing any mathematics. But looking back the next six months, I learned how to use chain rules and product rules and all of these different ways and the special rules for what happened when you set the derivative of octan and things like that. Total disinterest to me. I get the idea that there are some rules that generate the derivative from an expression. I'm happy to trust Mathematica or some lookup on the web or whatever your technique is to get that. I don't.

And also I'd even do it backwards for the derivation. I mean, I think derivation is rather interesting in that case. But I wouldn't start with that. I'd start with actually using things for most.

I don't think it's interesting. Why is it interesting? How to work out the derivative of sine of 3x squared.

No, no, no. I mean your point about the basic thing with the, you know, the X squared goes to 2x.

Oh, yeah, yeah.

But I wouldn't start with that. And that is how it was started. Let's tell you about derivatives. This is the derivation. Why start with that? I mean, I think that particular one is in fact, kind of interesting. And I think you want to poke. I mean, one of the things I think is interesting is to take most people on a journey of use the mathematics because it's useful for your life, for making decisions, for, you know, for conducting your life. It's a way of thinking. Which is why I quite like the term computational thinking. Now. Every so often you want to poke into the insides and. Because firstly. Well, firstly, I think it may trigger a bunch of people to say, oh, actually, I'm rather interested in the insides. Let me go and become a maths researcher, because I kind of like that. I'm all for that. For others, it will be like, okay, well, I looked at it, it was good to know that it existed. And that was that, basically. And that's fine, but it's completely reversed and that's a mistake. I think at the moment that's percolating. And it's particularly mathematics because the point of mathematics. So I want to come on to A levels, back to A levels for a moment. I mean, and maybe we can do a future podcast. But there's this whole thing that they're the most popular A level in quotes. And I mean, I think one has to be very careful about the word popular. I mean, popular is a term that suggests that people of their own well informed volition choose the thing because they think it that you know, because it gives them sort of enjoyment. But maybe you don't have the same.

Definition but to me it does make it sound like they like it if it's popular.

Right? To me that's not quite right. I think they think that it's important to them because they'll get a job and that everybody's been telling them if they don't learn their math properly, you know, they'll have. They'll struggle much more. So they're kind of. A lot of people are doing it because they think it's important and obviously they want to do well. So I think we need to be very careful about term. I think we also need to be very careful about the cause and correlation of. You know obviously everybody who's vaguely technically orientated at this point is forced to do well, forced told that if they don't do maths they're really foolish and they won't be able to do degrees they want to do and so forth or get jobs. So obviously they do that. So then you say ah, well the people who have got maths A levels and maths degrees they earn more and they do all these things. Now whether that's correlated to the fact they were already interested or whether that's caused. I question far more that a lot of that is actually being caused by the mathematics in many cases the mathematics learning.

And I think one thing that, you know, the choices that are offered at A level that is the maths course lumps together all the things that what are maths? And actually I'm just going to wind back to your previous example because it just made me realise actually perhaps the prime example of something I've never used. I don't totally agree with you that that's a bad point to enter derivatives but actually that's not the point. I entered derivatives because in order to do that limit I think I spent a month beforehand learning the formality of limits and that is absolutely something I have never used. The idea of sticking a big number or in small number in it roughly becoming something. Yes, in a hand wavy way. But the formal calculations.

Why is the formal.

Just to let me finish the point I was trying to tie it back is that I think there is the branch of maths that is heading for what colloquially we would call pure maths, which is the research math for its own sake where things like limits are very important. Because if you don't get that stuff formally and rigorously right, you get the wrong results versus the more sort of applied practical maths, which is in the end, what I do every day is solving real world problems where the formality of a formal limit is no more useful than saying, well, if I put in a number that's very small, then it's approximately this. Where just knowing do things go up or down as you head towards infinity or zero is important, but proving that they stabilize at zero.

I found that more interesting. So one question is why the formality of that proof, or exactly proof, but the formality of that was less interesting to you and the formality derivative thing was more interesting and how one distinguishes what to do. I mean, I think we can both agree those shouldn't be the primary focus because the primary focus is applied on a vastly larger number of things. One of the problems is there's a limited time, and so you want to cover a vastly larger range of things than you have time to be able to go and go into their derivations. Now, on the other hand, I do think it's good to see the occasional derivation, just like I think it's good if you drive a car to have a look under the bonnet or the hood if you're in America of the car, because it's useful to know roughly what's there and something about how it works. And for some people it'll also be very interesting and they want to become car designers. And so I feel a bit the same here. But then there's a question, what would you pick? And to some extent, the eye of the beholder. One thing looks neater than the other, the other one seems a more amazing result.

And maybe it's just because I've never been very interested in proofs and I think a lot of the maths education is on a journey to being able to prove things, which is, I think, a very pure maths way of thinking about what maths is and what it's for. To me, there are levels of how convinced I am and a proof might be at the top of that.

A key question in applied things is whether something is actually true, which is not the same thing as whether you've proved it.

Well, is it true or is it just. Is it practical? I mean. Oh, you mean like it might be true but unproven?

Yeah, yeah, no, but, well.

But to.

Me there are levels below a philosophical discussion about proof. I mean, my view of it is that the things that are there are often bugs in proofs, which means that what you thought was proven was actually untrue. And then There are other things where it's very hard, if not impossible, to come up with a formal proof in a traditional mathematical way. But there are many other ways of ascertaining that it really is very highly likely to be true.

Well, a while ago I saw a Reddit thread where somebody was trying to get examples of things that conjectures that were thought to be true and had stood up to qualitative testing at some serious level, but then later been proven not to be true and no one could think of any examples. I'm sure there are, but the fact is that if you take things like the collapsed conjecture and calculate it for the first 10 billion numbers and it's still holding true, I'm getting kind of convinced it's probably true, at least within all practical space. The fact that it's never been proven is an intellectual curiosity that it would be nice to see. But it's not going to change my world to know that this thing that I think what you're true is true.

And we should probably round out in a moment. But a level maths, one of my problems with it, and this is we know the English situation better than many other countries, so that's why we're discussing. But I think actually it's very similar in essence in many other places. We've got a vast cohort doing it. The reason they're given for the vast cohort doing it is practical use in jobs and that it's crucial for the future jobs. Yet I think it's worse frankly when I've looked at them than it was when we did it, because I think when we did it was a somewhat more specialized thing, particularly further maths, which is the add on when you do if you're trying to do hard science or maths at university, at sort of a high level university. And that really was for people who were very interested in the mathematics itself, or at least something like physics. I think what's happened is this has become far more of a sausage machine, to the point where it's more procedural. I mean, what I saw Sophia doing in high level maths was very procedural. It's also further away from what you're actually doing practically because computers got more powerful and we've got many new ways to do things which are a different pattern of thinking. So it's actually even the wrong. I mean, when we talk about did it get us thinking in the right way, did it get us logically reasoning? Yeah, but with kind of the wrong tools of the trade, with kind of too narrow a range of possible constructs. So rather a narrow form of mathematical or computational thinking compared to what we ought to be doing. And it's actually very hard later to, if you're very used to a particular narrow pattern of thinking, I think it's actually very hard to expand that later. I think it's actually easier to, if you have a much wider pattern of thinking to be able to say, well actually we can only use this in this case because the machinery supports that or whatever it is. So on all of the other, this.

Is back to your point by the way, getting intuition and experience as well is you've got to apply these things to know where they fail.

Well, quite. But I think even for the people who are die hard maths people who really want to study the subject itself, it's sort of got proceduralized and rather watered down. So some of the hard conceptual things that the real maths people would find interesting, which by the way is not the reason given for why everybody needs to be studying this and why it's quote so popular, are missing or they're certainly missing in conceptual essence. I mean, by that I mean, you know, they've kind of got proceduralized. They might still be there in terms of they're listed on the curriculum. So this is a big worry for me because I think that I'm afraid, I mean look, very good luck to everyone who's trying to do maths A level and I hope you get your A star or whatever it is that you wanted and particularly that maths, if it isn't your main focus, isn't something that holds you back because that is a big worry I have. There are a lot of very able computational thinkers out there or scientific thinkers or economists or whatever, lawyers who've been told they have to have A level maths because it's kind of a crest that they have to carry. But actually in reality the A level maths content and the thinking patterns and things aren't really what they need to be able to do. And therefore if they fail at that, it shouldn't be necessarily a failure for what they want to do and they're being held back by the requirement of this particular crest. So I'm quite worried about those people and I sort of say if you haven't done as well as you might hope, don't think that's the end of everything. I think things are moving a little bit in this front and I've done this often in the past, but I worry about the comparison with Latin in England. You know, everybody thought Latin was a really important system of thinking. And again, there are things in learning Latin deeply that are useful for how patterns of thinking. But whether you memorize the subjunctive pluperfect of amo, which is a verb in Latin, and can recite it and utilise it in an examination, it seems pretty far removed from, you know, knowing how to do biosciences at Oxford.

Well, let me. As we wrap then, let me give you one more example, just because it was timely and amused me that one thing that I have never used on actual data I've done. I've done it for. Used it for marketing diagrams for concepts, but Venn diagrams for data. And as you know, your daughter Sofia is in the office today doing some work on some biosciences data that she has. And she came to me and said, how do I do a Venn diagram for four categories where there's obviously two to the four, 16 different overlaps? How can I make that understandable? And we grappled with that on the whiteboard for a little while and decided probably Venn diagrams weren't useful. So even kind of 40 years after I've been taught something, sometimes they still pop back up and need rethinking about. So maybe maths education will all still prove to be useful in the long run.

Maybe. But in fact, I did briefly talk to Sophia about this and I mean, she's a, you know, just finished her second year at Cambridge and she's done maths part further maths maths in her first year, maths for biologists in her first year at Cambridge. Has any of this been relevant to the project she's doing now? No, frankly not. I mean, what she's trying to do now is take some data set that she's been handed from some, you know, some medical data based on looking at death rates after particular disease and treatments, and she's got a bunch of this data and she's trying to find interesting things out of the data. I mean, that's fundamentally what she's doing, Right? And, you know, she's obviously trying to do it with Computer Mathematica, I'm pleased to say. And, you know, the question is, is she set up to do that? And the answer is no. I mean, there's almost nothing in her training mathematically that are. So do it. I think there's quite a lot in the training of, you know, understanding just from life that certain things don't mean certain other things, being a bit skeptical, which she's quite good at, particularly when I tell her anything but this thing about applying them in mathematics is, you know, and what she learned is almost totally absent. I mean she never standed in her education a data set and told go figure out what you can from this data set ever.

I think this is again the split between pure and applied maths that there's two the, the kind of the maths education is owned by the purist that my, my daughter as you know, is also at Cambridge and is doing computer science. She didn't, couldn't be persuaded down the maths route and they had to sit some maths courses but she did number theory. So she learned how to do proofs about divisibility and of co factors of numbers and, and kind of abstract problems that she could then prove using some of these number theory arguments. I could write down a dozen better bits of maths that I know that would be useful for computer sciences. And that seemed almost the bottom of the pile to do number theory maths. But I guess that's whoever in the maths department said we'll teach your computer scientists some maths. That's the thing they're interested in. And it's not focused on application, at least not for her subject, same as Sophia's.

So we should probably round out here. I mean the sorry news is I think the answer to the question we posed, you know, how much use has the maths we learned actually been. It's a bit marginal. Yeah. And it oughtn't to be. I mean the really frustrating thing is it oughtn't to be any. I mean if you take Latin, I think even as I've often said about it, in 1950s it was even in England people didn't go around speaking Latin to each other in the 1950s. So it was, it was clearly a proxy and you could argue it was good or bad proxy for learning philosophy or learning thinking. Maths shouldn't need to be a proxy. It's there practically in the outside world everywhere. So neither should it need to be a proxy and nor should it be failing to do the thinking job the logical, the way of thinking with the tool sets we've got and without them. And I think it's really not quite hitting any of those in fact by a long margin.

Well, there's no need for a proxy when maths really does have all kinds of really useful and interesting and challenging applications in the real world.

Why haven't we in the sense that we are dead center in the, the maths world. I mean we couldn't be connected to it in More ways than we are. Literally. I mean, just to make that clearer, I mean, you know, not only we build mathematical software and do mathematical computational consulting in all sorts of fields, but we hire many people who are mathematical. We try to build out new theories of math. I mean, literally, of course, we have millions of customers around. We're probably more connected to math than any other organization on the planet. And yet two principal people in the organization who have maths degrees and maths A levels and things aren't really using most of that stuff directly. And that seems a real shame. So let's hope everybody can listen to this. If you are connected to a government, a foundation, anybody who can make change top down, or indeed an organization that needs people to learn more on the job, we'd love to hear from you actually, because both from the point of view we might be able to help you, but secondly from the point of view of pushing this objective of changing what we do to make everyone's life better, hopefully make better decisions happen in the world. Well, look forward to seeing the next computational conversation. Thanks very much. Next time, John and I were talking about how whether maths that we learned in A level, the A level results coming up next week. As far as I know whether maths we learned in A level or in our maths degree was any use whatsoever. We wondered whether we could just ask you, since you're in the office today, whether we could ask you for a moment your observation about this just for a few couple of minutes maybe, given that we also mentioned you and Hannah in this context. So maybe you should, since we've only got the setup for two, you could take over here for a moment. Probably the wrong height.

The younger Wolfram version.

Yes.

Okay, ask away.

Okay, so you've done some service courses at university in Mass. And you've also obviously you did maths A level, right?

Yeah.

Did you do further maths as well?

I did half further maths.

Okay, so you've got a pretty full maths education within your main degree. Which of those things are useful? What are you actually using and which are the things where. Well, let's start. Which are the things that are most useful?

Well, I think there's a difference, first of all between within my degree and within my career skills I might use in my life because the. I also had to take maths for my degree and a lot of that maths is based on a school curriculum type maths, which. So some of my A level was useful for that course. Have I used it since?

No.

Even so. Even the course that is designed by the university for my course I haven't actually used since I had to sit in those lectures.

So we haven't used it in your course? No, I haven't used it as part of your course, but it doesn't have any connection.

Part of my course which is supposed to, you know, it's only for people who are then going further on in the course. And I haven't used it at all. Which is mainly.

What was the, what was the topic they had you do as the maths for?

Yeah, so I mainly study biological sciences. So my maths course was called Maths for Biologists which seems like a good idea. You know, you have a lot of people coming in with it.

Sounds very sensitive.

Biology. And nowadays, you know, you have to as part of that be good at kind of data analysis and actually getting things out of your, out of your data. So most of it was data analysis, but all sort of old fashioned stuff. So a lot of like binomial distribution, a lot of. Kind of ANOVA tests which are used to see if something is significant or not depending on different variables that you may or may not include. But knowing how to do that all by hand and things like population dynamics, you know, drawing graphs by hand to see how a population might change if you add a predator in or whatever.

But you never modeled the differential equations which are classic. Predators of prey is a classic differential equation example to kind of.

Oh yeah, I'm sure we did, you know. But since then in my second year I have. Well, and also I'm doing a research project at the moment. I haven't used any of those things when I've actually been trying to analyze data, which is quite worrying. I sort of, you know, I know that I want to get out of my data. Okay, has this, does age have a significant impact or you know, sex like whether you're male or female? It doesn't. I haven't used any of the actual maths I might have used.

Well, it does feel like ANAVA ought to be useful there. Although learning it by hand seems crazy because it's spinning even on the computer to do aniva. Well, let alone doing it by hand.

So it's sort of like it gave me good overview of the principles that. Okay, I want to know if something, if you take out this variable, does it actually make a difference? But it didn't really help me in actually here's my data. How do I actually do that? Which is.

So there was no sort of practicality of actually working with a real Data set and cleaning it and processing it.

Not really, I mean a little bit with R, but they give you sort of pre cleaned data.

Yeah.

So that's actually very different from what you would get from an actual experiment.

Yeah. So it feels like if you were learning car mechanics, never being taught what to do when something's rusted.

Exactly. Yeah. Which I think biologists, you know, we're most biologists, are quite scared and insecure when it comes to any sort of data analysis and maths. There are most people who sort of thought they were good at science early on and then decided that they didn't like maths and so decided to do biology and now they're being forced to do maths again and sort of panic. But actually if it were presented in a way that wasn't, you know, it doesn't have to be that scary. And actually the hand calculating makes it seem so much worse than it needs to be.

Yeah. So we're sort of drifting off the topic of the, of the topic. And actually this is the question of, you know, I have this belief that there are people out there who are good at maths conceptually but can't do the, the, the symbol shuffling and, and process without dropping minus signs. And they lose confidence because they think they're bad at maths. But you've got people who can be very processy and understand nothing who think they're good at maths and you've got a kind of selection bias there of the people doing biology are probably exactly the ones who fell out with the things that they're now being forced to do when in fact the concepts of handling their biological data is probably interesting to them and not scary.

Yeah, exactly. And I think it's really sad because there's so much, there's so much potential and in the last, you know, 10 years that really is a barrier that doesn't exist if you have some kind of software, you know, that really, that sort of translating. It has been solved by people who know how to do it, but not applied in education at all.

Yeah.

So you can't really access it unless you. You know, I've been fortunate that I, I live with someone telling me about it every day. So I know that it exists, but most people don't. Right. And that's not their fault. I mean that's what education should be about is saying, hey, we've got these great tools that you can use.

And there's, I think to me, for once you've described it. I think maybe the, as I'm Kind of chewing it over in my head that maybe things like the binomial distribution are actually the worst bits there. Because from a maths point of view, these kind of ideal perfect distributions that have been well studied are interesting as mathematical objects, but in the real world, in the biological world, everything is real data. Everything's not quite anything. Most of it's vaguely normal or multimodal normal, but it's all. In the end, it's working with distributions of data. Seems like the interesting thing. Not idealized, named.

Well, the point, you know, the point is biology is about looking at what's not normal, why it's not normal, and if that could be useful to us, if it's important, which is, you know, maths is all about sort of like finding the perfect, the unifying things, where it works. Biology, in my opinion, is more about why, why does it not work, or why have these things developed, you know, two ways to achieve the same thing but done it differently. You know, that, that it's about finding the kind of weird bits in the pattern rather than.

Yeah, that's a good observation. And you know, I, I feel like also part of the reason why these things, these pure things get into the biospace is in the old days it was to deal with complexity that you couldn't deal with the data, there was no way of processing it. So the only thing you could do is say, well, it's roughly this normal distribution. And now I've got something I can write down in two numbers that I can work with. But as you say, everything that isn't normal about it is what's interesting. So if you don't work with the real data, if you make that simplifying step of saying, let's approximate it with the normal distribution, you're absolutely throwing away the interesting bit of it in order to keep the dull part.

You know, we've. In biology, right, most of that kind of normal stuff has been worked out we really don't care about. It's like useless to look at the obvious trend because most people have done that. What you wanna. It's more about looking at the kind of. The detail and the small bits that have changed in the trend. That is where like the majority of biology, biological innovation is happening.

Great. Well, I think that has fitted very much with what your father and I have been talking about for the last 20 minutes.

I hope that I summarized all right.

Yeah, I think that's really kind of interesting as well about this, you know, this not normalness, which is sort of contrary to the whole maths education world of there is a right answer and everything is perfect and it's the mess is actually where the interesting is. That's really useful insight, I think.

Yeah.

So thanks for that, Sophia. I hope you don't mind the fact that I talked about our Venn diagram discussion.

Yeah.