WEBVTT 00:00:00.920 --> 00:00:05.420 Good. Okay, so I am delighted to be chairing here today, not only because it looks likely 00:00:05.420 --> 00:00:10.940 to be quite a lively debate on whether technology is actually changing maths for the better, 00:00:11.800 --> 00:00:16.660 but also because both speakers have promised to be nice and take approximately 10 minutes 00:00:16.660 --> 00:00:21.340 to actually do their talks, but not be so nice in the debate, which is why they're actually 00:00:21.340 --> 00:00:28.280 speaking from different lecterns and sat either side of me. So I'll just introduce them briefly. 00:00:28.940 --> 00:00:32.800 We've got Conrad Wolfram, who is a physicist, mathematician and technologist. 00:00:33.680 --> 00:00:38.800 He's a strategic director and European co-founder of the Wolfram Group of Companies. 00:00:39.660 --> 00:00:42.960 Described as a place where computation meets knowledge and the math company, 00:00:43.480 --> 00:00:47.900 the Wolfram Group specialises at pushing boundaries in the intersection of computation, maths and knowledge, 00:00:48.300 --> 00:00:50.320 including making software such as Mathematica. 00:00:52.240 --> 00:00:57.480 Conrad is also the world's leading advocate for a fundamental shift of maths education to be computer-based, 00:00:57.480 --> 00:01:01.660 arguing that this is the only way to solve the global math education crisis. 00:01:03.260 --> 00:01:05.180 And to my left we've got Tony. 00:01:05.300 --> 00:01:11.200 So Tony Gardner has spent 45 years, in his own words, struggling to understand mathematics in all its guises. 00:01:12.040 --> 00:01:16.960 Alongside his own mathematics research he developed numerous unusual opportunities and resources for teachers 00:01:16.960 --> 00:01:18.880 and for interested pupils in schools. 00:01:19.120 --> 00:01:22.880 He spent 20 years working with teachers on curriculum development projects. 00:01:22.880 --> 00:01:37.963 He started and for nearly 10 years ran the national pyramid of competitions and summer schools that is now operated by the UKMT He was UK IMO team leader received the Paul Erdos Award was MA president and elected LMS 00:01:37.963 --> 00:01:47.643 education secretary. His 15 books are mostly still in print. So firstly, do you want to invite Conrad? 00:01:47.643 --> 00:02:00.883 I'm going to rattle through my arguments for computer-based maths pretty quickly, and there are fuller renditions of this on the web, so I hope it will be somewhat coherent. 00:02:00.883 --> 00:02:13.803 Let me start by talking about two subjects. One subject is more required, more aspired to, more used than ever it has been in human history. 00:02:13.803 --> 00:02:17.263 and there's another subject that's more despised 00:02:17.263 --> 00:02:20.003 causing more angst 00:02:20.003 --> 00:02:23.763 and more problems for governments and so forth 00:02:23.763 --> 00:02:25.223 than it ever has in human history 00:02:25.223 --> 00:02:27.943 and I'm talking about the two different subjects 00:02:27.943 --> 00:02:29.603 maths in the real world 00:02:29.603 --> 00:02:32.703 versus maths in education 00:02:32.703 --> 00:02:34.923 see in the real world you're doing a lot of modelling 00:02:34.923 --> 00:02:38.463 you're applying maths for 95% of the time 00:02:38.463 --> 00:02:39.963 to subjects that aren't maths itself 00:02:39.963 --> 00:02:48.203 and in education you are mostly cranking through calculations if you're lucky 00:02:48.203 --> 00:02:53.343 with the calculator so there is a fundamental difference between these two 00:02:53.343 --> 00:03:08.907 mathematics and my argument is that the fundamental difference is whether you use computers for actually doing the calculations let zoom out a bit Why are we learning maths And in particular why does everyone in the world 00:03:09.227 --> 00:03:11.747 is it deemed that everyone in the world should learn maths? 00:03:12.527 --> 00:03:14.567 Well, I think there are three good reasons. 00:03:15.547 --> 00:03:19.167 Firstly, technical jobs, which drive our economy and our livelihoods 00:03:19.167 --> 00:03:21.447 and our society so much in the last few years. 00:03:22.187 --> 00:03:23.587 Secondly, there's everyday living. 00:03:23.587 --> 00:03:30.127 just to survive in a modern economy and prosper you have to be much more mathematical than you 00:03:30.127 --> 00:03:36.027 did 20 30 40 years ago and third is what you might call logical thinking mind training whether for 00:03:36.027 --> 00:03:42.147 mathematics itself or for other areas a way of infusing a sense of this fantastic system of 00:03:42.147 --> 00:03:48.687 problem solving that has existed increasingly for the last few thousand years what is maths what do 00:03:48.687 --> 00:03:53.447 we mean when we say we are doing maths or learning maths? Well, I think there are four basic parts to 00:03:53.447 --> 00:03:58.527 this. You've got to ask the right question. What is it that you're trying to solve? You've got to 00:03:58.527 --> 00:04:03.847 turn that question, step two, into mathematics, mathematising as it's sometimes called. 00:04:05.027 --> 00:04:11.027 Step three is you've got to take that mathematised question and turn it into the answer in 00:04:11.027 --> 00:04:16.467 mathematised form, computation. And the fourth step is you've got to turn it back to answer the 00:04:16.467 --> 00:04:17.847 original question and verify. 00:04:17.847 --> 00:04:21.447 Crucially, you've got to make sure you got the right answer. 00:04:21.447 --> 00:04:22.227 It made sense. 00:04:24.926 --> 00:04:26.587 What we're doing right now in education 00:04:26.587 --> 00:04:39.010 is spending 80 of the time doing step three by hand Yet this is the one step that computers can do vastly better than any human after many many years of training 00:04:39.010 --> 00:04:41.750 And so you've got to ask why we're not using students far 00:04:41.750 --> 00:04:43.730 more for doing steps one, two, and four, which 00:04:43.730 --> 00:04:47.370 are the crucial steps that fail them in essentially all of the 00:04:47.370 --> 00:04:50.450 outside world, instead of spending all their time on step 00:04:50.450 --> 00:04:51.790 three. 00:04:51.790 --> 00:04:53.790 See, the crucial thing I'm saying is that maths is not 00:04:53.790 --> 00:04:54.730 the same as calculating. 00:04:54.730 --> 00:04:57.250 It's a much bigger, broader subject. 00:04:57.250 --> 00:04:58.730 And indeed, if it isn't that broad a subject, 00:04:58.790 --> 00:05:00.930 you've got to ask why everyone's supposed to be learning it. 00:05:02.290 --> 00:05:05.050 So I'm arguing calculating is not the end. 00:05:05.390 --> 00:05:06.910 It's the means to the end. 00:05:07.730 --> 00:05:10.110 It's the thing you'd like to get over. 00:05:10.270 --> 00:05:12.090 It's the thing that's now been liberated. 00:05:12.590 --> 00:05:14.150 For hundreds, thousands of years, 00:05:14.490 --> 00:05:18.030 the sticking point in applying mathematics was the calculating. 00:05:18.190 --> 00:05:19.370 The only way you could do it was by hand, 00:05:19.410 --> 00:05:21.170 and that limited you into what you could do. 00:05:21.850 --> 00:05:22.550 Now it isn't. 00:05:22.550 --> 00:05:28.810 and one thing we've got to be very clear in distinguishing is the mechanics of the moment 00:05:28.810 --> 00:05:34.710 it's been a long moment hundreds of years versus the actual underlying essence of the subject of 00:05:34.710 --> 00:05:40.410 mathematics and I believe those have got quite severely confused and they've got confused for 00:05:40.410 --> 00:05:44.710 very understandable reason we've got the most tremendous mechanization of mathematics 00:05:44.710 --> 00:05:51.610 of any subject that has happened I think in any reasonable part of history the computer has 00:05:51.610 --> 00:05:53.950 completely mechanized this calculating step. 00:05:55.570 --> 00:05:57.630 Now, one of the things we've got to be clear about in this 00:05:57.630 --> 00:06:00.270 discussion is what are the outcomes we really want from 00:06:00.270 --> 00:06:00.890 mathematics. 00:06:00.890 --> 00:06:02.290 We've been trying to construct some of these. 00:06:02.290 --> 00:06:04.010 This is a n dimensional 00:06:04.129 --> 00:06:05.589 I can't remember how many dimensions it is. 00:06:06.849 --> 00:06:07.849 Construct of outcomes. 00:06:07.949 --> 00:06:09.529 And I'm not going to go into the detail of this. 00:06:10.069 --> 00:06:15.449 But there are many, and they are way beyond knowing how to use what I would call the tools of mathematics. 00:06:15.969 --> 00:06:19.709 And they extend to concepts of mathematics and things that's come up a lot today. 00:06:19.889 --> 00:06:21.529 Confidence to tackle new problems. 00:06:22.209 --> 00:06:22.649 Interpreting. 00:06:23.949 --> 00:06:24.929 Instinctive feel of maths. 00:06:25.029 --> 00:06:25.849 Critiquing and verifying. 00:06:25.849 --> 00:06:29.069 These are all crucial outcomes from a general purpose mathematics. 00:06:31.029 --> 00:06:33.249 Now, I said at the beginning, we want... 00:06:33.249 --> 00:06:36.669 I'm talking here about mathematics for everyone. 00:06:37.369 --> 00:06:44.549 And people often say to me, but surely if people want to learn the mechanics inside here, that's an important thing. 00:06:44.609 --> 00:06:46.609 And yes, it is important. So is ancient Greek. 00:06:47.269 --> 00:06:53.489 The question is, would you spend 21,000 lifetimes a year making people learn ancient Greek around the world? 00:06:53.509 --> 00:06:55.449 And I would say no, that's not justifiable. 00:06:56.329 --> 00:07:01.569 I think the history of hand calculating, as I would like to characterize much of our maths curriculum, is in that category. 00:07:02.189 --> 00:07:05.429 If people are interested in it, it's fantastic that they should pursue it. 00:07:05.429 --> 00:07:10.069 I'm up for one minute suggesting that people who get engaged and interested shouldn't pursue it. 00:07:10.069 --> 00:07:17.349 But I don't think that is the subject that is the general purpose, important subject, either at a high level or at a low level that people need to do. 00:07:18.409 --> 00:07:22.729 Please be clear that I'm talking here about computer-based, not just computer-assisted. 00:07:23.529 --> 00:07:24.369 So what do I mean? 00:07:24.369 --> 00:07:29.889 computer-based means the subject of mathematics itself in the outside world 00:07:29.889 --> 00:07:42.132 fundamentally changed because computers do the calculating what a lot of people talk about computers in education and maths is they talk about replacing the teacher with the computer or somehow supplementing them 00:07:42.572 --> 00:07:46.032 I'm not in any way against doing that where it makes sense. There are many things 00:07:46.032 --> 00:07:49.892 technology can help with in the process of learning. I'm talking about 00:07:49.892 --> 00:07:54.012 the actual subject. So this is different from the discussion about history. I don't think history 00:07:54.012 --> 00:07:58.012 has fundamentally changed with computers. Certain aspects how you do it have evolved 00:07:58.012 --> 00:07:59.212 but I don't think it's fundamentally changed. 00:07:59.372 --> 00:07:59.972 Maths has. 00:08:00.132 --> 00:08:01.172 The subject is different. 00:08:02.632 --> 00:08:04.252 And the key point here is, 00:08:04.472 --> 00:08:06.552 you know, a better deployment of the wrong subject, 00:08:06.632 --> 00:08:07.672 however well you teach it, 00:08:07.732 --> 00:08:09.192 it's not going to make it the right subject. 00:08:10.052 --> 00:08:12.572 And I think that's one of the positions we're currently in. 00:08:13.672 --> 00:08:15.072 Another thing I think is a mistake 00:08:15.072 --> 00:08:16.472 is we currently have what I call 00:08:16.472 --> 00:08:18.472 mechanics-centred mathematics. 00:08:19.252 --> 00:08:20.552 So you look down a curriculum, 00:08:20.672 --> 00:08:22.132 it talks about inverting matrices 00:08:22.132 --> 00:08:23.732 or simplifying thirds, 00:08:23.812 --> 00:08:25.032 one of my particular dislikes. 00:08:25.872 --> 00:08:27.612 But the point is, 00:08:27.612 --> 00:08:29.192 That's about the internal mechanics. 00:08:30.152 --> 00:08:33.012 That's about machining the piston if you're learning to drive a car. 00:08:33.992 --> 00:08:37.432 What we should be doing is problems, problem-centric mathematics 00:08:37.432 --> 00:08:40.432 that tie back to the conceptual understanding people need. 00:08:40.572 --> 00:08:41.632 You start with the problem, 00:08:42.152 --> 00:08:44.632 the thing that people might actually be interested in solving. 00:08:45.272 --> 00:08:47.932 Whether that's how many levels am I separated by on Facebook, 00:08:48.352 --> 00:08:51.292 what's a beautiful shape, perfectly rational problem. 00:08:52.112 --> 00:08:55.852 How much do I need to compress a photo or video or music before you notice? 00:08:55.852 --> 00:09:00.752 You know, these are things that actually people have to do every day. 00:09:00.752 --> 00:09:17.495 And of course with a computer you can actually play with these things You can see what happens when you compress an image more or less and how it affects it you know without necessarily doing all the calculations yourself you can go much further it a much richer thing more practical is more conceptual in the modern world the 00:09:17.495 --> 00:09:20.775 mathematics we need in the outside world is highly conceptual it's got much 00:09:20.775 --> 00:09:27.395 tougher than it was 40 50 years ago so if we mimic that we are also gaining the 00:09:27.395 --> 00:09:30.855 rigor in quotes although I dislike that word the conceptualism the 00:09:30.855 --> 00:09:35.755 intellectualism that we need. It's not a choice that both are possible at the same time. And 00:09:35.755 --> 00:09:42.095 that's a rare possibility, I think, in any subject. Please be clear about this. If you 00:09:42.095 --> 00:09:48.295 remove the computer from mathematics, you remove most of the context. What do I mean? 00:09:48.955 --> 00:09:53.955 The reason that mathematics is so predominant in the world today is because you can apply it to 00:09:53.955 --> 00:09:58.095 many subjects that you couldn't apply it to 50 years ago, like biology, like social sciences, 00:09:58.095 --> 00:09:59.555 stuff with hair all over it. 00:10:01.455 --> 00:10:04.015 If you remove the computer, you can't do that. 00:10:04.135 --> 00:10:06.615 You can't do data analysis without a computer. 00:10:06.695 --> 00:10:07.995 It doesn't make a lot of sense. 00:10:08.635 --> 00:10:11.375 It's no good trying to learn how to do it with 10 data points 00:10:11.375 --> 00:10:13.455 because that's not what anybody does, 00:10:13.495 --> 00:10:14.775 and you don't have any hair over it, 00:10:14.775 --> 00:10:17.235 and all the problems you actually get in the real world don't crop up. 00:10:18.495 --> 00:10:22.155 So understand that if you remove the computer, you remove the context, 00:10:22.255 --> 00:10:25.235 and the context is where a lot of the intellectual and conceptual interest, 00:10:26.075 --> 00:10:27.795 I think, in the subject for many people is, 00:10:27.795 --> 00:10:29.735 except for people who really, really want to go on 00:10:29.735 --> 00:10:31.095 and become pure mathematicians. 00:10:31.095 --> 00:10:43.858 And even then I think there a lot of interest there Another crucial part of the context is programming People ask me sometimes why is there less algebra in the outside world Well because a lot of it turned into programs 00:10:44.858 --> 00:10:48.678 And I compare programming not only as a way in which you instruct a computer, 00:10:48.758 --> 00:10:50.598 but as a way in which you write down mathematics. 00:10:51.798 --> 00:10:55.518 To me, programming is to maths a little bit like composition is to English. 00:10:56.218 --> 00:11:01.278 It's a fundamental building block of today's machinery for doing mathematics. 00:11:02.258 --> 00:11:07.298 So I think I'm making a fairly general point here about mathematics to sum up, 00:11:07.298 --> 00:11:12.118 which actually is a point about education in general, but it happens to be extremely, 00:11:13.038 --> 00:11:18.458 in a sense, mathematics is its biggest case, certainly at the moment. 00:11:19.278 --> 00:11:23.298 You've got a choice when new mechanization or automation comes into a subject 00:11:23.298 --> 00:11:26.958 in terms of how you educate people for that subject. 00:11:27.618 --> 00:11:28.678 And the choice is this. 00:11:28.678 --> 00:11:32.258 do you decide that you want them to replicate the machinery 00:11:32.258 --> 00:11:36.278 and learn how to do what the machinery now does 00:11:36.278 --> 00:11:38.678 or do you want them to stand on that new machinery 00:11:38.678 --> 00:11:40.398 and go much, much further? 00:11:43.278 --> 00:11:45.458 History shows us that the second 00:11:45.458 --> 00:11:47.358 is the only possible long-term outcome. 00:11:48.198 --> 00:11:50.998 You cannot get everyone doing what all the machinery does 00:11:50.998 --> 00:11:52.258 nor do you particularly want to. 00:11:53.538 --> 00:11:56.798 There is a difference between learning the mechanics inside 00:11:56.798 --> 00:12:00.458 and learning how you operate today's mechanics to get to the essence of the 00:12:00.458 --> 00:12:04.778 subject and that's what I'm arguing for and now what's happened in the last 30 00:12:04.778 --> 00:12:08.138 40 years to mathematics is it got automated it got all 00:12:08.257 --> 00:12:11.897 automated more than any other subject in recent times by a computer. 00:12:12.317 --> 00:12:15.857 A fantastic, so to speak, tool of automation for mathematics. 00:12:18.157 --> 00:12:20.037 Okay, I'm just at the end here. 00:12:20.597 --> 00:12:23.677 So my argument is computer-based maths is inevitable in the world. 00:12:25.057 --> 00:12:28.157 And the question is when and how and who first? 00:12:28.237 --> 00:12:34.177 Well, Estonia, we're starting off with Estonia to produce an early start at this. 00:12:34.177 --> 00:12:38.317 and either mathematics will turn into ancient Greek 00:12:38.317 --> 00:12:41.997 or it will turn into the mainstream subject it should, 00:12:42.457 --> 00:12:44.997 applying itself to the real world and to what everybody needs. 00:12:45.897 --> 00:12:46.177 Thanks. 00:12:46.277 --> 00:12:48.657 And so hopefully, I'm hoping we can all unscramble 00:12:48.657 --> 00:12:52.117 what mathematics is really about and stand on its power. 00:12:52.357 --> 00:12:52.737 Thank you. 00:12:52.737 --> 00:12:53.397 Thank you. 00:12:53.397 --> 00:12:54.297 Thank you. 00:13:04.177 --> 00:13:07.997 I don't recall any promise to be nice. 00:13:10.557 --> 00:13:17.217 I have to say it's a long time since I've spent an afternoon where I have heard so much confusion. 00:13:19.777 --> 00:13:24.757 I prefer to respond spontaneously but to try to support discussion. 00:13:25.197 --> 00:13:26.937 This was all prepared in advance. 00:13:27.137 --> 00:13:33.457 Fortunately, Conrad has preached the sermon many times so one has some access to what he might want to say. 00:13:34.177 --> 00:13:46.900 and I will try to respect the ten minutes Mathematicians appear not to understand the number five or the number ten today So I will focus essentially on one thing 00:13:47.980 --> 00:13:52.340 But it's not the whole story, but it's more relevant than most. 00:13:53.700 --> 00:13:57.600 The title of this debate asks whether technology has changed mathematics for the better. 00:13:57.600 --> 00:14:02.060 there is in fact no disagreement 00:14:02.060 --> 00:14:05.420 that technology has changed the focus of mathematics 00:14:05.420 --> 00:14:07.540 and the extent of its applicability 00:14:07.540 --> 00:14:09.440 often in dramatic ways 00:14:09.440 --> 00:14:14.500 Conrad and his colleagues are at the cutting edge 00:14:14.500 --> 00:14:15.840 of this fast developing world 00:14:15.840 --> 00:14:17.260 and do truly impressive work 00:14:17.260 --> 00:14:19.480 but that's not in education 00:14:19.480 --> 00:14:23.140 this does not mean therefore 00:14:23.140 --> 00:14:24.480 that they should be taken seriously 00:14:24.480 --> 00:14:26.960 when they talk arant nonsense about the question 00:14:26.960 --> 00:14:29.440 how should mathematics be taught to beginners? 00:14:31.560 --> 00:14:34.320 Nevertheless, if you've even started to read your homework sheet, 00:14:34.960 --> 00:14:36.480 you'll see that there is complete agreement 00:14:36.480 --> 00:14:38.360 that something in England needs serious attention. 00:14:39.860 --> 00:14:43.360 It's just that the missing ingredient is not computer-based mathematics. 00:14:46.240 --> 00:14:47.320 What is mathematics? 00:14:49.540 --> 00:14:51.760 Mathematics is a very precise way of thinking 00:14:51.760 --> 00:14:53.320 about certain mental constructions. 00:14:54.180 --> 00:14:56.540 It's a mental universe with its own rules. 00:14:56.960 --> 00:15:05.240 The ideas, methods and results of mathematics unfortunately do not seem to be hardwired 00:15:05.240 --> 00:15:17.524 into the brain but they emerged in human culture because they commensurate with the brain potential architecture which is plastic in young children 00:15:17.524 --> 00:15:19.724 and can be moulded. 00:15:21.204 --> 00:15:24.044 But because the brain's architecture can be shaped, 00:15:25.444 --> 00:15:26.744 mathematics can be learned. 00:15:28.544 --> 00:15:29.944 And once it's been learned, 00:15:30.004 --> 00:15:33.324 it can be used to solve an impressive array of problems. 00:15:33.324 --> 00:15:42.364 Among the ideas, methods and results of mathematics, some provide a natural way in and appear to 00:15:42.364 --> 00:15:47.724 hold the key to pupils constructing and mastering this complex mental universe. 00:15:50.724 --> 00:15:58.624 The known ways of improving maths education focus on mastery of these key ideas that allow 00:15:58.624 --> 00:16:01.504 pupils to enter into this mathematical universe. 00:16:03.784 --> 00:16:09.484 So for the foreseeable future, the adult world is likely to be dominated by computing. 00:16:10.404 --> 00:16:16.304 The question we therefore need to consider is how relevant is this to how we teach mathematics 00:16:16.304 --> 00:16:16.964 to children? 00:16:19.844 --> 00:16:26.284 Conrad's popular pitches, and we saw it earlier today, constantly repeat that math is not 00:16:26.284 --> 00:16:27.584 equal to calculating. 00:16:28.624 --> 00:16:36.224 that math is much greater than calculating, much more than. Who could disagree? I've devoted 00:16:36.224 --> 00:16:50.607 my professional life to giving substance to the variant math is calculation plus dot dot dot what goes in the space This is much harder than it may appear but the venture is not helped by 00:16:50.607 --> 00:16:57.107 sloganeering, such as math has been liberated from calculation, but that liberation didn't get into 00:16:57.107 --> 00:17:07.947 education. The quotes are mostly quotes. For the adult who's using mathematics, it may 00:17:07.947 --> 00:17:14.107 sometimes be true that calculating is the limiting part which computers can now do for 00:17:14.107 --> 00:17:19.927 the adult. But the learner is in a very different position, for she is faced with the task of 00:17:19.927 --> 00:17:27.067 constructing his or her own mental universe of mathematics. So one's faced with a completely 00:17:27.067 --> 00:17:32.907 different problem. Each such learner is like somebody struggling to play the violin or piano, 00:17:33.467 --> 00:17:37.947 where contrary to common belief the goal is not really to produce a perfect performance 00:17:38.507 --> 00:17:45.947 for which you could buy a CD or something, but to train the brain to coordinate and interpret the 00:17:45.947 --> 00:17:51.867 dots on the page and the markings, the bowings, the fingerings in the score with decisions in the 00:17:51.867 --> 00:17:57.687 brain which sends instructions to the fingers whose effects are monitored by 00:17:57.687 --> 00:18:05.847 the ear an astonishing process with the aim that gradually this astonishing 00:18:05.847 --> 00:18:10.227 complex of skills can become robustly available for subsequent similar 00:18:10.227 --> 00:18:12.267 challenges. This is a 00:18:12.386 --> 00:18:16.726 Completely different problem compared with the adult and the way they use mathematics. 00:18:18.346 --> 00:18:23.186 This musical calculation is not some kind of redundant optional drudgery. 00:18:24.106 --> 00:18:28.386 It's the way the relevant skills get wired into the brain. 00:18:30.626 --> 00:18:40.466 Success in this astonishing process depends on extensive, carefully structured and progressively more challenging repetition. 00:18:40.466 --> 00:18:45.126 and as Tim said this morning, not barren repetition, 00:18:45.786 --> 00:18:51.706 but progressive, imaginative, satisfying, challenging repetition. 00:18:52.846 --> 00:18:55.806 Without this repeated exercise of basic musical calculation, 00:18:55.806 --> 00:18:57.946 there will be no music worth listening to 00:18:57.946 --> 00:19:00.346 and there can be no worthwhile music making. 00:19:01.206 --> 00:19:03.026 The same is true in mathematics. 00:19:04.166 --> 00:19:07.766 To suggest that such activity is somehow limiting or demeaning 00:19:07.766 --> 00:19:10.226 or irrelevant or redundant 00:19:10.226 --> 00:19:12.806 is to demonstrate ignorance of human development 00:19:12.806 --> 00:19:14.186 and of human culture. 00:19:15.186 --> 00:19:17.066 So the real choice would seem to be, 00:19:17.506 --> 00:19:19.706 is it enough to reach for the nearest sexy headline 00:19:19.706 --> 00:19:22.166 or technological silver bullet in education? 00:19:22.866 --> 00:19:25.526 Or should we respond rationally to current failures 00:19:25.526 --> 00:19:28.626 that are amply documented and are agreed on all sides 00:19:28.626 --> 00:19:32.066 and to engage in a serious analysis of what's missing 00:19:32.066 --> 00:19:35.306 and what is needed on the level of the learner? 00:19:35.306 --> 00:19:38.386 if we choose the latter 00:19:38.386 --> 00:19:53.589 we discover where we agree that mathematics is indeed much more than calculating but that calculation is a crucial part maybe the crucial part of mathematics education 00:19:53.589 --> 00:19:59.389 because mathematical ideas get wired into the brain through engaging with and mastering 00:19:59.389 --> 00:20:06.029 carefully designed sequences of calculation, something that requires, as Tim stressed this 00:20:06.029 --> 00:20:11.169 morning more time and more imagination than we think. If you look at what happens in Singapore, 00:20:11.349 --> 00:20:20.369 China and Flemish, Belgium and Russia, they spend time on these things not to train people like sort 00:20:20.369 --> 00:20:26.169 of rats, but to develop the imagination and the creativity. It's very, very different. 00:20:27.449 --> 00:20:32.409 The astonishing outcome of our own recent neglect is visible to us all on our homework sheet. 00:20:33.069 --> 00:20:34.929 These problems reveal the horrible truth 00:20:34.929 --> 00:20:37.109 that we have deprived our children 00:20:37.109 --> 00:20:38.869 not just of some missing ingredient, 00:20:39.029 --> 00:20:40.609 but of the much more basic prerequisite 00:20:40.609 --> 00:20:42.369 of engaging with the necessary, 00:20:42.589 --> 00:20:44.429 carefully designed sequences of calculation. 00:20:45.709 --> 00:20:48.849 In the absence of this essential groundwork or staple diet, 00:20:48.849 --> 00:20:50.869 all attempts that we should focus on 00:20:50.869 --> 00:20:54.729 the missing ingredient in math is calculation plus 00:20:54.729 --> 00:20:57.849 are no better than Marie Antoinette's infamous declaration, 00:20:59.049 --> 00:20:59.969 let them eat cake. 00:20:59.969 --> 00:21:02.069 we're offering them cake 00:21:02.069 --> 00:21:04.089 because we think cake is a good thing 00:21:04.089 --> 00:21:04.829 it is indeed 00:21:04.829 --> 00:21:07.729 but in the absence of a basic diet 00:21:07.729 --> 00:21:09.189 which you see on your homework sheet 00:21:09.189 --> 00:21:20.492 is the position we got to cake is not the answer I like cake But it not the answer That why my own efforts over time have shifted 00:21:20.492 --> 00:21:23.652 from the supplementary enrichment activities, 00:21:23.792 --> 00:21:25.652 such as a book in Understanding Infinity 00:21:25.652 --> 00:21:27.792 or Mathematical Puzzling or Discovering Mathematics, 00:21:28.032 --> 00:21:29.332 to the much more mundane, 00:21:29.892 --> 00:21:32.012 but much more basic and essential series, 00:21:32.292 --> 00:21:34.212 Extension Mathematics, that some of you may know. 00:21:35.612 --> 00:21:37.972 Today's date risks confusing three separate questions. 00:21:37.972 --> 00:21:48.352 Have computers influenced mathematics and the way it's used in the real world, where there is probably complete agreement, though Conrad would understand the profundity of the change more than many of us? 00:21:49.652 --> 00:22:02.772 More relevantly, can individuals or small groups learn elementary mathematics more richly than in the past by exploiting, in either limited or extended way, the power of computers? 00:22:03.772 --> 00:22:08.132 The evidence suggests that the answer is a definite yes, or at least sometimes. 00:22:08.992 --> 00:22:12.752 For it depends on having access to a privileged level of support 00:22:12.752 --> 00:22:17.432 for the increased power that is available to lead to improved outcomes. 00:22:18.292 --> 00:22:21.712 And this then has resource implications both of money and of competent manpower. 00:22:22.292 --> 00:22:25.912 Which brings us to the question that we really should have been asked. 00:22:27.192 --> 00:22:27.632 I see. 00:22:30.192 --> 00:22:31.252 Which is at the bottom. 00:22:32.772 --> 00:22:39.452 Can we honestly recommend such an approach on a large scale, either now or in due course after suitable preparation? 00:22:40.572 --> 00:22:52.875 The answer should be clear We cannot recommend it now Our position is far too primitive and insecure And until we begin to take seriously the question of what I would call PPD 00:22:53.556 --> 00:22:55.296 permanent professional development, 00:22:55.296 --> 00:22:59.695 and really challenge what our serving teachers know, 00:22:59.776 --> 00:23:00.916 at secondary level in particular, 00:23:01.516 --> 00:23:05.736 then any such moves in the foreseeable future will lead to inevitable degeneration. 00:23:06.195 --> 00:23:06.516 Thank you. 00:23:10.516 --> 00:23:16.296 So they've roughly kept their promise 00:23:16.296 --> 00:23:19.076 and we have about 10 minutes for questions now. 00:23:23.636 --> 00:23:25.916 I'll try and add in some responses as well, 00:23:26.155 --> 00:23:29.176 either first or as the questions come up. 00:23:29.176 --> 00:23:31.195 I'm sure these two would quite happily sit here 00:23:31.195 --> 00:23:33.396 and argue for the rest of, well, this evening. 00:23:33.776 --> 00:23:35.496 I was wondering if anybody had any questions 00:23:35.496 --> 00:23:36.655 from the audience at all. 00:23:37.716 --> 00:23:38.396 Come on, Mick. 00:23:38.396 --> 00:23:47.296 Well, I've really got more of a comment. 00:23:49.916 --> 00:23:58.136 I must say that I'm firmly on Tony Gardner's side in this debate here. 00:23:59.076 --> 00:24:00.316 Neander calls together. 00:24:00.316 --> 00:24:16.416 I was a pure mathematician and turned into now a mathematical software engineer is what I call myself. 00:24:16.514 --> 00:24:30.514 I, at the moment, have been teaching for five years the first year numerical mathematics course at Cardiff University School of Maths, 00:24:30.514 --> 00:24:45.514 and have become more and more dismayed by the lack of what I regard as basic skills among my class. 00:24:45.514 --> 00:24:47.494 Can I ask, what are those basic skills? 00:24:47.954 --> 00:24:49.614 Well, I'll come to that in a moment. 00:24:49.614 --> 00:24:57.454 And I'll just say that I love computers. 00:24:58.154 --> 00:25:00.974 I use them and have used them a lot. 00:25:02.754 --> 00:25:08.394 Two days ago, I was internal examiner for a PhD viva, 00:25:08.394 --> 00:25:18.974 of which the project was to model flow of viscoelastic fluids, 00:25:19.694 --> 00:25:29.234 including such things as the candidate had used the university's high-performance parallel processing machine 00:25:29.234 --> 00:25:34.054 to determine such things as under what conditions, 00:25:34.054 --> 00:25:42.154 if you drop a ball bearing into a beaker full of a viscoelastic fluid, 00:25:42.794 --> 00:25:56.118 does it actually bounce before it settles down and starts sinking at constant velocity And that is no mean feat computationally using I may say my favourite language Fortran 00:25:57.398 --> 00:26:05.298 And on the other hand, I ask my students, 00:26:05.298 --> 00:26:12.818 who I teach things like solving equations f of x equals 0 numerically. 00:26:15.318 --> 00:26:18.658 A total class of about 112. 00:26:19.518 --> 00:26:21.738 I have 75 in the class. 00:26:22.458 --> 00:26:23.738 And I ask them to say, 00:26:24.658 --> 00:26:32.038 why does y equals x cross y equals e to the minus x 00:26:32.038 --> 00:26:34.918 somewhere on the positive axis? 00:26:35.298 --> 00:26:39.378 and I'm sure all of you have a mental picture, 00:26:39.938 --> 00:26:42.438 a line going up and a line coming down, 00:26:42.998 --> 00:26:47.718 and you know why that has an answer. 00:26:48.758 --> 00:26:51.738 And when I was at school 55 years ago, 00:26:52.598 --> 00:26:53.758 in the lower sixth, 00:26:54.918 --> 00:26:57.398 I think everybody in my class 00:26:57.398 --> 00:26:59.818 would have immediately had the skills 00:26:59.818 --> 00:27:04.138 to draw the one curve and the other curve 00:27:04.138 --> 00:27:24.161 and say and even say well after a bit of thought well y equals x starts off below e to the minus x when x is 0 and sufficiently further on I think I got the rough idea Can I respond to a few of these things Because I kind of itching to do that We still have time 00:27:24.381 --> 00:27:33.841 So my point is that any amount of the sort of computer-based stuff that you were talking about 00:27:33.841 --> 00:27:41.061 will not give those skills which Tony Gardner has identified as the kind of skill 00:27:41.061 --> 00:27:44.001 which has to be developed over time. 00:27:44.121 --> 00:27:44.761 I've got it. 00:27:44.861 --> 00:27:45.081 Okay. 00:27:45.341 --> 00:27:45.561 Right. 00:27:45.701 --> 00:27:46.541 I've got the question. 00:27:46.641 --> 00:27:48.661 Let me try and address a whole bunch of things. 00:27:48.761 --> 00:27:50.561 All this may come out in a semi-random order. 00:27:51.641 --> 00:27:54.441 Look, there's experience, which I think is what you're talking about, 00:27:54.481 --> 00:27:57.101 of different situations and being able to handle those. 00:27:57.181 --> 00:28:00.841 Some of the experience needs to be done because you actually do the thing by hand. 00:28:01.601 --> 00:28:05.121 And some of it is because you use a computer to gain much more experience 00:28:05.121 --> 00:28:06.821 than you could do by doing it by hand 00:28:06.821 --> 00:28:09.361 because you do many more cases and they're harder and so forth. 00:28:10.121 --> 00:28:11.121 There is a dividing line. 00:28:11.201 --> 00:28:14.761 I'm not suggesting people should learn absolutely no calculation by hand, 00:28:14.981 --> 00:28:17.581 but I'm suggesting you've got to be incredibly clear 00:28:17.581 --> 00:28:19.661 that the fault should be the computer, 00:28:19.861 --> 00:28:22.901 unless there's a really good practical reason to do it by hand, 00:28:22.921 --> 00:28:27.281 or unless you can really exactly point to the conceptual framework 00:28:27.281 --> 00:28:30.601 that you will need going forward to have done that by hand. 00:28:30.641 --> 00:28:32.801 And I think if you think about the curriculum from that point of view, 00:28:33.361 --> 00:28:35.061 you build a very, very different curriculum 00:28:35.061 --> 00:28:37.361 that goes much further, much faster, 00:28:37.361 --> 00:28:43.421 and get deals with concepts much faster let me deal with another few talking of 00:28:43.421 --> 00:28:56.024 analogies yeah should we take some more questions first well okay I wanted to just pick up on the analogy thing which perhaps might help Let take other questions first I wondering if we coming back to Bennett original proposal 00:28:56.024 --> 00:28:57.284 about the different ages, 00:28:57.484 --> 00:28:58.884 and we're talking different age groups, 00:28:59.004 --> 00:29:00.004 perhaps different topics. 00:29:00.784 --> 00:29:02.944 Not age groups, but dividing lines, I think. 00:29:03.084 --> 00:29:03.484 Dividing lines. 00:29:04.304 --> 00:29:05.124 Okay, any more questions? 00:29:07.424 --> 00:29:09.184 I think there's a very interesting question. 00:29:09.504 --> 00:29:10.944 National Numeracy, Mike Ellicott, 00:29:11.384 --> 00:29:13.904 between where Conrad's computer-based maths 00:29:13.904 --> 00:29:18.364 kind of kicks in and what are the essentials that we need to learn in advance of that. 00:29:18.524 --> 00:29:21.984 And the point I'd like to make kind of in response a little bit to the previous question, 00:29:22.484 --> 00:29:26.644 you know, only 22% of the adult population in the UK are working at level two, 00:29:26.904 --> 00:29:28.824 which is the equivalent of A to C at GCSE. 00:29:29.124 --> 00:29:33.404 So we have a vast sway, the mass of the population in this country 00:29:33.404 --> 00:29:37.024 who have been switched off from maths at a very early stage 00:29:37.024 --> 00:29:38.824 or are working at a very low level. 00:29:39.084 --> 00:29:42.564 We need all of those people's skills to improve. 00:29:42.564 --> 00:29:48.304 And I think mathematicians and the maths community needs to address that whole area too. 00:29:48.664 --> 00:29:53.544 And I think Conrad's construct of the four questions, or four elements of mathematics, 00:29:54.004 --> 00:29:56.624 is what mathematicians at the top end are doing. 00:29:56.824 --> 00:29:59.364 But it's absolutely what people are doing in their everyday lives. 00:29:59.444 --> 00:30:02.084 And we need to try and support and strengthen that. 00:30:03.524 --> 00:30:04.204 Thank you. 00:30:04.804 --> 00:30:05.484 Down here. 00:30:09.444 --> 00:30:10.124 Thank you. 00:30:10.124 --> 00:30:20.524 I don't think that there is a chasm here in reality, but I do think that what Conrad is saying, 00:30:20.643 --> 00:30:24.543 and what Tony is saying, is that it should be non-negotiable 00:30:24.543 --> 00:30:31.183 that we start teaching rather better for a deep and robust conceptual understanding. 00:30:31.903 --> 00:30:36.243 I think part of the difference lies in the fact that I would argue 00:30:36.243 --> 00:30:44.163 that the evidence we have leads us to think that actually we need some fluency 00:30:44.163 --> 00:30:50.343 and some robust skills in order to develop those concepts in depth. 00:30:50.643 --> 00:30:57.583 and that we ought to be concentrating on how we can do that rather better in our classrooms. 00:30:58.103 --> 00:31:00.483 And that's about teacher development, actually. 00:31:00.843 --> 00:31:03.043 Let me respond partly to that. 00:31:03.163 --> 00:31:04.183 I don't disagree. 00:31:04.443 --> 00:31:07.223 I think the question is what's robust and what's basic. 00:31:07.943 --> 00:31:10.923 And I think that's fuzzy for many people. 00:31:11.483 --> 00:31:15.903 And I think that I would have a different basic, I think, to what Tony would have and to what various other people have. 00:31:16.163 --> 00:31:18.583 Let me come back on the analogy, because I think it might help with this. 00:31:18.863 --> 00:31:21.823 I'm not sure I completely concur with your music analogy 00:31:21.823 --> 00:31:23.643 in terms of its representation, 00:31:23.883 --> 00:31:25.123 but let me do one I use, 00:31:25.363 --> 00:31:27.123 and we'll see if we can figure out the music one too. 00:31:27.983 --> 00:31:28.383 Photography. 00:31:29.263 --> 00:31:31.083 Is the essence of today's photography 00:31:31.083 --> 00:31:33.903 learning how to coat a plate with chemicals 00:31:33.903 --> 00:31:35.643 or indeed load a film into a camera? 00:31:36.583 --> 00:31:37.603 I don't think so. 00:31:38.243 --> 00:31:40.383 I think those were mechanics of that moment. 00:31:41.083 --> 00:31:42.423 They were absolutely essential. 00:31:42.943 --> 00:31:43.843 If you wanted to be a photographer, 00:31:43.903 --> 00:31:45.743 you had to develop films and know how to do that. 00:31:46.323 --> 00:32:03.426 Life has moved on The essence of photography is a creative subject to do with representing the world I guess one would say in a particular way You can define it in various ways But that the kind of outcome and the mechanics have changed And I think the idea that you need to start by learning 00:32:03.426 --> 00:32:11.586 how to load a film in order that you can appreciate the photography is the wrong basic idea. Now, 00:32:11.766 --> 00:32:16.586 you can argue about maths. You can argue that we have no evidence that if you don't learn the basic 00:32:16.586 --> 00:32:22.586 construct, what you're calling the basic constructs, that you can then go on to do more complicated, 00:32:22.866 --> 00:32:27.426 what we currently consider conceptually more complicated things. The fact is nobody's tried 00:32:27.426 --> 00:32:31.766 it before. And the reason they haven't tried it before is because they've assumed that's what you 00:32:31.766 --> 00:32:40.046 have to do up front before you try and take, think about sort of higher level things like risk or 00:32:40.046 --> 00:32:47.106 significance, concepts, or what an equation is or why you'd use it. What an equation is or why 00:32:47.106 --> 00:32:51.686 you'd use it is fundamentally a different thing from solving a quadratic equation by hand using 00:32:51.686 --> 00:32:57.886 a procedure that you've learned by rote. They're different. Now, I'm not suggesting that people 00:32:57.886 --> 00:33:03.086 shouldn't learn any of the calculating end of that, but I am suggesting we should absolutely 00:33:03.086 --> 00:33:09.806 focus much earlier on many more of these conceptual pieces of framework that I think people don't 00:33:09.806 --> 00:33:14.426 They know some of them how to put the numbers in a formula to solve a quadratic equation. 00:33:15.306 --> 00:33:17.466 But most of them don't know when they'd use an equation. 00:33:19.746 --> 00:33:28.769 Yeah just brief The most frustrating thing in such a discussion is being misrepresented or misunderstood even by one supporters 00:33:30.929 --> 00:33:34.129 I did not mention photography and photographic plates. 00:33:34.369 --> 00:33:35.849 I mentioned music for a good reason. 00:33:37.049 --> 00:33:39.629 So I don't have a view on photography. 00:33:41.169 --> 00:33:42.369 I have just checked. 00:33:42.509 --> 00:33:45.009 I did not use the word basic at any point. 00:33:45.009 --> 00:33:48.829 I did not use the word skills at any point 00:33:48.829 --> 00:33:52.029 I did not use the word fluency at any point 00:33:52.029 --> 00:33:55.809 I think somebody's been reading too much of the curriculum drafts 00:33:55.809 --> 00:33:59.549 I used other words 00:33:59.549 --> 00:34:03.669 so let's be a little careful before we refute things 00:34:03.669 --> 00:34:06.809 or even embrace things which weren't said 00:34:09.389 --> 00:34:10.929 on the music analogy 00:34:10.929 --> 00:34:12.009 any more questions? 00:34:15.009 --> 00:34:26.769 Ursula Bowden, I'm a maths teacher from a school in Oxford. 00:34:28.129 --> 00:34:29.609 Tony, I enjoyed your talk. 00:34:29.949 --> 00:34:32.449 I think it got the nearest that anybody has today 00:34:32.449 --> 00:34:34.809 to actually addressing what I thought I'd come to listen to, 00:34:35.309 --> 00:34:37.289 which was what's wrong with the maths we teach, 00:34:37.669 --> 00:34:39.209 not whether it should be on a computer, 00:34:39.369 --> 00:34:41.589 but actually what's wrong with the curriculum that we're teaching 00:34:41.589 --> 00:34:43.529 and the ways that we're doing it. 00:34:43.529 --> 00:34:44.969 Maybe I'm misrepresenting you. 00:34:44.969 --> 00:34:46.189 If I am, I'm sorry. 00:34:47.429 --> 00:35:00.953 You probably tell me off later I used computers when I did my degree in maths I did a course on statistics Maybe it isn maths but it a good illustration I got taught to use a computer I learnt to press the buttons and get a chi result 00:35:01.453 --> 00:35:03.553 I had no idea what I was doing. 00:35:03.913 --> 00:35:06.293 It was only when I learnt to do it by hand 00:35:06.293 --> 00:35:10.253 that I then went back to the computer to find out what to do. 00:35:10.693 --> 00:35:12.333 So I think there's room for both, 00:35:12.553 --> 00:35:16.793 but I think we have to be careful that we do need to learn the basic constructs. 00:35:16.833 --> 00:35:18.233 We do need to understand the basics. 00:35:18.233 --> 00:35:19.773 No, but that's my point about basics. 00:35:20.333 --> 00:35:24.073 Absolutely, just button pushing when you don't know what you're doing it for 00:35:24.073 --> 00:35:28.453 or how you'd use it or where it goes wrong is no good. 00:35:29.373 --> 00:35:32.693 But what I'd argue with the basics are understanding that, 00:35:33.373 --> 00:35:36.153 and that is not necessarily the same as how the computer does it 00:35:36.153 --> 00:35:37.513 or how you would do it by hand. 00:35:38.093 --> 00:35:41.773 The fact is people don't do chi-squared distributions by hand in the outside world. 00:35:41.913 --> 00:35:44.693 They do them with computers because they're doing them with lots of data. 00:35:45.413 --> 00:35:49.593 And the way that computers do them is completely different to the way people do them by hand. 00:35:50.073 --> 00:35:55.573 So what you absolutely need people to understand, I completely agree with this, is how you deal with it. 00:35:55.613 --> 00:35:59.013 You can't assume the computer's going to get it right, because it depends what you've done with it. 00:35:59.053 --> 00:36:00.993 You need to understand how to handle that machinery. 00:36:01.473 --> 00:36:03.093 And that is absolutely not happening. 00:36:03.093 --> 00:36:08.953 And I would argue some of the financial crisis issues were because people were forced to do, 00:36:09.093 --> 00:36:11.633 they were used to doing the process they learned by hand. 00:36:11.633 --> 00:36:20.273 They weren't used to dealing with much more complex things through a computer and then verifying the results and knowing how to handle machinery where they didn't know what was going on the inside. 00:36:20.973 --> 00:36:24.473 In the end, machinery always means you don't know what's going on the inside. 00:36:25.091 --> 00:36:28.171 And you've got to decide, if you really want to learn everything that's going on the inside, 00:36:28.231 --> 00:36:31.051 you're not going to get very far as the machinery gets more automation. 00:36:31.631 --> 00:36:36.451 And that's true of, you know, whether you drive a car or an airplane or many other walks of life, 00:36:36.491 --> 00:36:40.171 where machinery and automation have taken us much further. 00:36:40.451 --> 00:36:42.891 But you lose something with that. That's for sure. 00:36:43.051 --> 00:36:47.811 You lose the understanding of what happens on the inside, but you can gain a lot more as long as you do it right. 00:36:47.971 --> 00:36:51.791 I mean, as long as you don't ignore the fact that you can't just assume it's a great black box 00:36:51.791 --> 00:36:53.671 and it'll always get things right 00:36:53.671 --> 00:36:54.891 and you don't have to look at the answer. 00:36:55.531 --> 00:36:57.511 Okay, and before we go on to solving the world's recession, 00:36:57.951 --> 00:37:00.071 there was another question up there, wasn't there, I think? 00:37:01.571 --> 00:37:02.571 Maybe less of a question, 00:37:02.671 --> 00:37:04.491 but I'm just going to throw something back to the two of you. 00:37:04.891 --> 00:37:05.911 My name is Vineka Totia. 00:37:06.051 --> 00:37:07.071 I work at the Nuffield Foundation. 00:37:07.831 --> 00:37:09.291 So given your analogy of machinery, 00:37:09.951 --> 00:37:11.431 the question I would pose to Conrad is, 00:37:12.351 --> 00:37:14.011 you can't give high-power machinery, 00:37:14.151 --> 00:37:16.851 whether it's a car to take an analogy of locomotion 00:37:16.851 --> 00:37:19.051 or the camera to a learning child. 00:37:19.631 --> 00:37:20.751 So my question to you is, 00:37:20.751 --> 00:37:22.451 what sort of research program do you want to put in place 00:37:22.451 --> 00:37:25.811 so that before they can be given the car or the camera, 00:37:26.331 --> 00:37:28.191 they can develop the skills that you would like them to develop. 00:37:28.731 --> 00:37:31.111 As to Tony, I have a similar response. 00:37:31.211 --> 00:37:33.051 I really like the analogy between maths and music 00:37:33.051 --> 00:37:34.791 and marks on the paper and converting it. 00:37:35.291 --> 00:37:37.971 The difference I found in sort of my brief experience with teaching is 00:37:37.971 --> 00:37:41.991 the music engages the child to want to continue with the harder bits. 00:37:42.451 --> 00:37:44.011 And what mathematicians will need to think about is 00:37:44.011 --> 00:37:45.471 how can we keep them engaged? 00:37:45.471 --> 00:37:47.891 You've not had five children trying to play instruments. 00:37:50.751 --> 00:38:03.075 On the power of the machine I mean I think Conrad takes a provocative position publicly 00:38:03.295 --> 00:38:06.095 In fact, he then sort of backs off, which may be a good tactic. 00:38:07.035 --> 00:38:13.815 But, for example, the Estonians set up, the Estonians have an excellent mathematics education 00:38:13.815 --> 00:38:18.535 structure, and they're not replacing it with computer-based mathematics, as I understand 00:38:18.535 --> 00:38:18.895 it. 00:38:18.895 --> 00:38:23.575 They're developing a statistics curriculum, which is computer-based, 00:38:23.575 --> 00:38:25.495 which makes perfect sense. 00:38:26.235 --> 00:38:29.595 But I suspect they're not beginning it in year one. 00:38:30.235 --> 00:38:32.335 I suspect they're beginning it at a high school level, 00:38:32.415 --> 00:38:35.195 I don't know, junior high school level or upper primary, 00:38:35.915 --> 00:38:39.255 when they've established what the rest of us realise 00:38:39.255 --> 00:38:40.915 perhaps needs to be done first. 00:38:41.575 --> 00:38:45.055 And I entirely agree that statistics done by hand, 00:38:45.055 --> 00:38:49.395 It's a bit like D1, which is the biggest abomination in the universe. 00:38:49.675 --> 00:38:55.075 Finding shortest paths in a graph with six vertices is just nonsense. 00:38:55.895 --> 00:38:58.875 So there are situations where you certainly need, 00:38:59.395 --> 00:39:01.275 and again, I think Conrad's been misrepresented, 00:39:01.695 --> 00:39:04.195 computer-based does not mean using a computer. 00:39:05.095 --> 00:39:10.415 Computer-based means actually allowing the computer to reinterpret what needs doing 00:39:10.415 --> 00:39:14.015 and what the human's function is. 00:39:14.015 --> 00:39:21.515 And it is a matter of timing, what needs to be done first, and it isn't you should do it from age 13 onwards or something, 00:39:21.515 --> 00:39:32.618 because what I was trying to say applies at age 5 at age 7 at age 11 at age 13 at age 16 at age 45 with different things 00:39:33.238 --> 00:39:43.058 I think a mature adult coming to a new subject might be better off scribbling on paper for a bit before they engage the full, 00:39:43.138 --> 00:39:47.618 but they get more fluent, more willing to use the software as part of their fiddling. 00:39:47.618 --> 00:39:49.838 and that's a slow process. 00:39:50.098 --> 00:39:52.898 But I think the kid beginning is completely different 00:39:52.898 --> 00:39:54.338 from the adult's position. 00:39:54.518 --> 00:39:56.118 That's all, and it seems to get forgotten. 00:39:56.518 --> 00:39:59.578 So I think what we've got is the question is not necessarily 00:39:59.578 --> 00:40:02.818 whether to use technology, but how to use it 00:40:02.818 --> 00:40:05.378 to support effective learning in the classroom. 00:40:06.358 --> 00:40:07.258 Tony, any comments? 00:40:07.258 --> 00:40:08.218 Which bits of the subject? 00:40:08.758 --> 00:40:11.298 I mean, partly it's the question about proxies as well. 00:40:11.298 --> 00:40:11.998 Just to sum up. 00:40:12.318 --> 00:40:13.958 Okay, two summaries. 00:40:14.578 --> 00:40:15.018 Proxies. 00:40:15.018 --> 00:40:20.818 I mean it's not that I think that the subject in education should be absolutely identical to the 00:40:20.818 --> 00:40:23.918 real world subject because there are things you need to do in education that are different but I 00:40:23.918 --> 00:40:28.678 think you should start from the real world subject and not try and teach a complete proxy to that as 00:40:28.678 --> 00:40:33.638 the default assumption I think that's a mistake because the real world subject is highly conceptual 00:40:33.638 --> 00:40:39.998 highly practical and used it's not you know in that sense it's different from Latin um second 00:40:39.998 --> 00:40:43.458 thing is back to the music analogy I mean one of the things about music is I mean you can argue 00:40:43.458 --> 00:40:45.678 about what the outcomes are from somebody learning music. 00:40:45.678 --> 00:40:48.318 Most of the outcome is not they become a concert pianist 00:40:48.318 --> 00:40:49.618 or whatever. 00:40:49.618 --> 00:40:52.198 It's enrichment of their life because they enjoy playing 00:40:52.198 --> 00:40:52.698 music. 00:40:52.698 --> 00:41:06.321 And they do pretty early start playing actual pieces They start using music which is what they going to go on doing The pieces are simpler than they be and they might do some scales to help them play those pieces and later they do some more theory But that the order 00:41:06.501 --> 00:41:10.341 They're doing things where they're practically doing the thing they're ending up going to 00:41:10.341 --> 00:41:16.241 do in the end to start with, at least as close as they can. And I think that's an important 00:41:16.241 --> 00:41:17.581 lesson for us in maths too. 00:41:19.821 --> 00:41:21.041 Thank you. Tony? 00:41:21.041 --> 00:41:23.681 One closing comment. 00:41:24.041 --> 00:41:27.701 If any of you or your colleagues 00:41:27.701 --> 00:41:32.941 try to make sense of the question that this debate has raised, 00:41:33.041 --> 00:41:34.641 namely how to make best use of it, 00:41:34.921 --> 00:41:37.121 how to improve what we do, 00:41:37.661 --> 00:41:42.681 please let us not commit the traditional English sin 00:41:42.681 --> 00:41:47.041 like MEP, does this ring any bells, 00:41:47.041 --> 00:41:50.841 microelectronics project, can computer-aware number, 00:41:51.041 --> 00:41:57.661 neither of these were ever evaluated in a public way long term. 00:41:58.021 --> 00:42:04.601 Please experiment, set something up, and test it to see what the long term effects are. 00:42:05.421 --> 00:42:07.921 That's what the community needs and that's what we've not had. 00:42:08.301 --> 00:42:09.561 We have gimmicks. 00:42:10.601 --> 00:42:15.841 Short term, short term funding, and we see what happens. 00:42:16.541 --> 00:42:19.041 Those of us who are old enough have seen what happens. 00:42:19.041 --> 00:42:20.101 Thank you very much. 00:42:20.101 --> 00:42:22.501 Thank you. I'll pass back to Bennett on that one. 00:42:22.501 --> 00:42:24.501 APPLAUSE