Tony Gardiner’s Manifesto

A couple months ago, the comments on Post 1000 led to a discussion of manifestos on mathematic education. That inspired occasional commenter Solange Amato to link to a wonderful presentation by the irreplaceable Tony Gardiner. Gardiner’s talk was part of a conference, Thinking about Mathematics, held in Portugal in 2017; for a little about the conference, see here and here.

(UPDATE 23/11/25 Commenter JJ has emailed us, noting Anna Stokke’s podcast-interview with Portuguese mathematician, Nuno Crato. Crato was also Portugal’s Minister for Education from 2011 to 2015. It’s not exactly clear how this relates to Gardiner’s appearance in 2017, but it undoubtedly relates somehow.)

The video of the talk follows, and below that is my transcription (aided by auto-subs). The text is essentially verbatim, although I’ve tidied a sentence or two. I have not included Tony’s replies to a couple comments/questions at the end, although these are also of interest.

I’m not sure if this amounts to Tony’s manifesto, or if he needs one. Tony was forever manifestoing, pronouncing beautifully and wisely on mathematics education. (See here for some examples.) But, whatever it is, it is excellent.

Secretary of State, Director General, fellow delegates, thank you for the invitation. Though I have to apologise. I fear if you enjoyed the last session, maybe you won’t enjoy this one.

Mathematics is in some sense universal. Mathematics education, however, depends in many ways on local conditions. So please do not expect me to offer simple advice about what to do and what not to do, especially since it is impossible for a stranger to understand the swings in the Portuguese mathematics education pendulum since 2010-11. Open discussion and debate are essential, but it seems that perhaps many things have already been decided. So I can only offer some comments that might help you all work towards something better, maybe the next time you make a swing.

I have nothing to sell. I have no theory of education. I have no product that is guaranteed to improve mathematics learning and teaching in Portugal. And I am certainly not a prophet who will tell you what the 21st century has in store for us. I accept the facts as Cuno [?] drafted them. We live in a very turbulent, fast-changing world.

But I am also very old, and I remember almost identical things being said in the 50s and 60s by proponents of modern math, that everything had changed in mathematics, therefore, we must change things in schools. I remember the same things being said in the 70s and 80s about technology, calculators, computers. I remember in the 80s and 90s, the NCTM and other people’s focus on problem solving. In the UK, we’ve run our own experiments on many of the things which you are currently considering. Not one of these prophecies turned out to be accurate. Nobody managed to make their proposed changes work. So I may appear to you as a Neanderthal who looks to think more carefully about what it is that is permanent, that is eternal, that is reliable in mathematics education. So maybe nothing very exciting. I am sorry.

Conrad Wolfram has a much sexier message than I do. But he is crazy. He knows nothing about children except his own children – who are of course very clever – or about schools. Similarly, I would say Andreas Schleicher and PISA, you should consider very carefully and not too seriously.

Okay, on we go.

I think you face three serious dichotomies and you should be aware of them and be aware of the dangers of taking sides. Politicians and bureaucrats have, I think we almost sensed it this morning, a certain lack of sympathy with the difficulties that confront mathematics teaching. Somehow we are often viewed as the awkward children in the educational zoo. So it is very tempting for them to accept something which avoids the details of how one makes sense of multiplication and division, of fractions, of ratio, of algebra, where we fight and we bicker, because it’s important and it’s difficult. And it is tempting to reach for global competencies, to believe that problem solving, group work, critical thinking are somehow universal solutions, which are more interesting and more attractive than subtraction and decimals and trigonometry. But unfortunately in mathematics, I will argue there is no universal competencies that can avoid the nitty gritty, the difficulties of details in mathematics education.

Content in the end is the vehicle for trying to understand mathematics education and to deliver it more effectively. If there is a 21st century competency, I would argue that we have learned from the 20th century that there is one above all others. And this is mathematics. No one would have guessed in 1900 how much of the world would be taken over by hidden, invisible, but powerful mathematics. And we should remember also 2008 and the way mathematics can be trusted and abused, not only in banking, but in many other things.

So the question is, what do we need to know in order to prepare people a little better for the bumps that are ahead? And I utterly reject the big mistake of identifying what is needed in adult life with what the beginner needs when they start to prepare to grow up. You need to know what they might face, but that is not the same as what you give them when they are age 5, 6, 7, 14, 15. That’s a judgment one has to make.

In adult life, there are many unpleasant things, which Mr. Trump can tell you all about, but you don’t necessarily introduce them to your primary children. You bear them in mind. In childhood, you do many things which you do not need in adult life. I hope it doesn’t offend if I say breastfeeding. Later, you don’t need breastfeeding, but it seems to be a very appropriate way to begin. So what is mathematical breastfeeding? That is my question.

This dichotomy is complicated by other dichotomies. There is a version of progressive left versus reactionary right. Progressive left are often the liberal educators, often teachers and education researchers. The reactionary right you often see as the conservative mathematicians. That’s my picture, but this is not a helpful dichotomy. There certainly are two different approaches, but we do not choose between them and we do not take sides. Try to understand what each brings and how they have to work together.

And in mathematics education, those who work with children or with teachers see what is in front of them. That is students with their difficulties, their frustrations, their interests, their boredom. And this tends to focus one’s attention on what I call bottom-up. You are thinking about the needs of the students in front of you. And this is important in mathematics. You cannot just dump the mathematics on them. You have to think, where are they beginning? How can we make it acceptable, digestible to this young mind?

But there is a danger, especially in Western democracies, to overvalue the feelings of the student. Mathematics is important because of what it becomes when you have understood it more fully, what it can achieve, not only in adult life, but in science and in other ways. And so there is also a top-down version which is more to be found among mathematicians who understand the power of the mathematics itself. They tend to forget that this is not necessarily appropriate to feed direct to children.

So the concern for, if you like, the constructivist version of mathematics education, that children have to create an abstract mathematical universe in their heads. That requires one to be sensitive to the bottom-up details. But you also need to remember where you are trying to go. What is it that you want your 16, 17, 18-year-olds to be able to do, to think, to understand?

And so the bottom-up and the top-down versions have to come together. So do not take sides would be my first view. And if I have an urge, it would be for all groups to work together, even though it is sometimes uncomfortable. And to try to hold these two things in tension without discarding either the child’s view or the mathematician’s view, if I can put it that way.

And in particular, not to be misled by snake oil salesman. I don’t know how that translates into Portuguese, but when there is difficulty, there is always somebody prepared to sell you something which will solve your difficulty. The only question you should ask is, can I see how it actually works? So by all means learn from other systems, but they should be systems that have made their proposal work. Not systems that are dreaming that it might work, like British Columbia or whatever. There are other places going through the same process that you are going through. So look for places which have functioning systems, not ones which have promise but no practice.

TIMSS and PISA. All international comparisons provide very useful data. Internal data is often what you have from year to year, but it’s very hard to be sure that it does not degenerate. In England, we have had continually improving performance, but measured using tests which have got easier and easier. So the improved performance is just internal tests, so it’s very valuable to have the occasional international comparison.

You do have to be careful. PISA, there is something not quite right about PISA. It tends to move towards global competencies. It claims to test not mathematics, but mathematical literacy, which sounds very appealing to a politician or a bureaucrat. And if you read the definition, maybe you nod and say that looks very good. But the problems that they use don’t relate to the definition. And they are slightly awkward. They are good problems, some of them. You could use them in your classroom, and they would create some very interesting work. But they seem to me to be very bad problems for comparing your classroom with your colleague’s classroom. They generate activity of an interesting kind, perhaps, but not activity that you could easily compare. To compare one school with another school is more awkward. To compare one system with another system is even more awkward. Because the more interesting the question, the more dangerous the mark scheme. How is it marked? How is the student’s response graded? So there is something not quite right about PISA. But still, it gives information.

TIMSS gives much simpler problems, much more recognisable as school math. And so more limited, it doesn’t really reach into the 21st century competencies that we have heard about so much. But what it tests gives, I think, more robust information. And if we look at TIMSS, unfortunately Portugal took part in ’95 and then again in 2011 and not in between. So two data points is always a little awkward. But the fact that they give very different numbers can only lead one to say that something good or quite good had been happening.

But to interpret the second data point is very dangerous. In 2003, England saw a massive jump. Not as large as yours. Yours is 100 points in TIMSS, I think. Is that right? From ’95, a hundred in the average school. In 2003 England had, from 1995, I think, a 48 point jump. So it’s similar in some ways, not as dramatic, but to interpret this was very delicate.

Fortunately we then have 2007, 2011, 2015, which allow you to see whether this is a random variation or whether it’s a steady improvement. So be patient about interpreting such numbers. But on the face of it you have much to think, be pleased with and to work on, and to build on. Don’t throw away. Try to improve without throwing away.

So, I claim the modern world is very different from the world I grew up in, but some things seem to be very similar. The way my children bring up their children, the way they play with them, the way they read stories to them, the way they physically interact with them, these look very familiar. Even though the world they are preparing for is rather different.

So certainly organisms in the early stages of life, you should hesitate before you throw away the old methods completely. Better to improve them slightly. I think I would claim, as I’ve said before, the only 21st century shortcut that I am aware of really is: the importance of language is the same as it ever was, but the importance of science and in particular of mathematics should now be very clear. And so we then face the challenge of how to teach these disciplines more effectively. Not dramatically differently, not throwing things out. Be a little patient.

The only systems that I know – and individual schools – that have made a successful distinction in the way they educate their students have essentially taken the old targets, the old goals, removed a small number of things but achieved the old goals better, by adding to the original structure. I know of no system which has rejected the original structure and replaced it by interdisciplinary work, by projects and so on, which has made this succeed. There are plenty of examples where they have tried it and failed, from which you might learn.

So a little bit about my background. I am a research mathematician. I’m not a teacher. I once said I would not last a week in an ordinary school and promptly two teachers in the audience arranged for me to come and work in their school for one week. Within the first day, the head teacher had two complaints from students, which they handled very comfortably. So I’m not a teacher. My wife is a teacher and one of my children is a teacher so I have lived in the world of mathematics teaching very closely for 50 years and I have interfered in schools for 50 years. I started working in the university in Tanzania in 1968 where I was required to work, introduce African teachers to modern math. And I realised very quickly this was a difficult task, and some of the problems. And since then I have got involved on different levels, but I’m not a teacher. So I’m an outsider who observes.

My original goal. I did a PhD in Germany and had contact with a lot of Soviet mathematics education materials. And I went back to England in 72-73, and wanted to try to make these work in England. So my goal all the way through was to learn from other countries as best I could. When I go for a research visit for three months, six months to a country, I get into schools to try and feel what’s going on. I don’t understand the system particularly but I get a bit of a feeling. So I wanted to improve things and I started by essentially throwing everything out in my own head, and my wife and I tried to draft what God would do. But you learn very quickly that this is hubris, this is pride. And increasingly I realised I had to learn from teachers.

So perhaps my most valuable experience was to work on the School Maths Project with a large team of teachers, working in their own time, where I was the only mathematician. And we’d meet at weekends and write materials. We produced some wonderful materials. But they turned out to be a disaster. They were used in 60% of UK schools, but we made a mistake. We forgot the central role of the teacher.

At that time, primary schools had no curriculum. We had no national curriculum until 1988, and so they would arrive in grade 6, 7 with very different backgrounds, and we wrote materials which allowed them to slot in where they fitted. It seemed like a very good idea. So the textbook was just lots of little booklets and the students could do those which they needed to do. And it destroyed the mathematics classroom, because the teacher was left administering thousands of different booklets with students all doing different things, and nobody knew what they were learning. So one learns from this: you can have a great idea but it can go wrong, because you forget something.

So increasingly it became clear to me that, if you like, the top-down view, the mathematicians view, I will call it, the idea of where we are trying to get, and the bottom-up view, which is sensitive to the interests and the needs of students and schools, have to come together. They will be in conflict to some extent but they need to be both in the room in order to not to lose sight of where you are trying to get.

Also in England, I don’t know about Portugal, but in England we tried to individualise instruction so that in each class the teacher was supposed to accept some were very good students and some were not so good, and to offer somehow a different program for these different groups, and this also was a disaster. If possible, my conclusion is that one has to learn from the Far East and from large parts of Europe, to keep the class together for as long as possible, to teach them as a class, as a social unit. That is part of the joy of the mathematics the interaction, everybody on the same page, even if some read it more quickly than others.

And, I think you are very lucky to have in some sense the natural distinction between grades 1 to 9 and grades 10 to 12. So they present rather different problems and I hope that you will be able to try to design grades 1 to 9 so that students stay together as a group for as long as possible. Whether this creates problems in grade 8, 9, I don’t know. That’s for you to decide and how you address this. But a common curriculum for that kind of period has great advantages, for the teacher and for the students. In grades 10 to 12, that’s a different story, but I shan’t address that here. So the lesson is everyone can and must grasp the basic abstractions.

So we have touched upon TIMSS in Portugal, but we have not said the really crucial thing which you may like to reflect on. I do not know whether it applies in Portugal. But in England we had this dramatic improvement, apparently, in TIMSS average scores in grade 4. And everybody thought this was rather good, and it continued. 2003-2007 also increased slightly, 2007-2011 also increased slightly, 2011-2015 also increased slightly. So this is not something you can dismiss. Something had improved in primary school. I shan’t bother you with what it was, but it was undeniable.

But the important thing is that in grade 8, performance changed not at all, and was weak. I think, in PISA, your performance increases but not so much, is that about right? It’s slightly up but not very much up. And I think the message I would ask you to think about is: what is the point of an improvement in grade 4 if it doesn’t lead to a similar improvement in grade 8, and in grade 12?

This is top-down. The goal is where you’re trying to get. I am not very interested in people who achieve enormous things in grade 2 and grade 3. Well, I love them dearly, but all parents want their children to do brilliant things in grade 2 and grade 3, but it’s not what matters. What matters is what you get to at the end. In a marathon it doesn’t matter who’s ahead after 100 meters. That’s the wrong model. So what is the point of a dramatic improvement in grade 4? You want it to feed through to grade 8 and to grade 12.

Now I can only tell you what I think went wrong in England. There were great pressures on primary teachers to improve certain calculational skills and mental arithmetic in grades 1 to 4. But because there were pressures on them to get results for students to answer grade 4 type questions, they encouraged grade 3 type thinking. Because the easiest way to get an answer in grade 4, for many students, is to use what I call backward looking methods. Finger counting is the obvious bad example. For a long time in primary school, you find students still using finger counting to get answers. And they get answers, but they are handicapped. They will not be able to progress. Somehow if you want to calculate you have to remove finger counting and work with place value in the head. So what we forgot was to create incentives to get teachers to push students onto creating structures in the head, which would support development towards grade 8 and beyond.

Yep? So that’s a confession of our own failure and we’re still paying for it. So I think a balanced curriculum has to focus on achieving serious mathematical goals at age 16 to 18 for large numbers. It’s not just the specialists who need serious mathematics in grades 11 and 12. Though, I confess, I haven’t had very clear answers but it seems to me you have an enormous number of people trying to do Mathematics A who might be happier doing something different. How you get to that point, I don’t know. So don’t throw away Mathematics A, but see if you can find a structure which allows a little bit more flexibility, rather than maybe requiring too many people to. I don’t know, it’s a local condition for you to consider.

In the UK, the most striking positive example we have, I’m afraid, was imposed politically by the previous government and continued by the present one, which looks to the Far East to see how mathematics is taught in Singapore, in Shanghai and elsewhere. And on the whole there is it is very difficult to copy. In Shanghai, primary teachers teach two lessons a day. They work the rest of the time marking and remedial work, so that this is their way they can make sure all the students are held together. So those who did not succeed in the morning get some support in the afternoon, and the next day everybody is back at the start line more or less.

Primary lessons that I have seen in England from Shanghai maths teachers have been quite impressive. They focus on one idea, they do not lecture, they involve, they’re very focused on what they are trying to achieve, and on the whole I think we can learn a lot from them.

The secondary lessons, grades six grade seven, that I have seen all fail in England. Again, they focus on one idea, they approach it in imaginative ways, including variations that require a deeper thinking, but they assume that students have achieved a level of numerical fluency in their mental universe of number, which is not there. So that is our fault, but it means it’s rather difficult to copy simply into secondary school.

More interesting is the Singapore maths textbooks project, where we have for 20, 30 years moved away from using textbooks. Maths teachers now view maths textbooks as embarrassing, a sign of unprofessional behaviour to use a textbook. They have difficulty seeing that a textbook actually liberates the teacher to focus on teaching the mathematics. So it provides a map, a skeleton, which you can use, it’s been tried and tested so you don’t have to go and make up your own exercises. And we’ve got into a hole with that, we use textbooks very very little here. So to introduce Singapore based textbooks rewritten for the UK curriculum has been a huge support to teachers, those who use them. The teachers have begun to understand what they have always been expected to teach. They can see a progression which previously they have never seen. They have only seen curriculum items in a list, and now for the first time they see a whole, an organic discipline. And many of them are finding it very emotional, very moving, very thrilling, very refreshing. And it’s wonderful to see them using these materials with their students. Holding the classes together at least until grade 4, 5 where there begins to be a little problem. Okay? So that’s something somebody might want to learn from.

The approaches are open. They are not prescriptive. They always start with a discussion. They adopt a kind of Confucian version of Jerome Bruner’s Concrete-Pictorial-Abstract. So each topic will start with something concrete, which may be a picture, it may be an activity. This is interpreted, well, discussed. And many ideas and answers are proposed, but there is always a target idea. Which is not always made clear to the English teachers, but the textbook authors know it. From these options that come from the discussion, one is going to recur more often than the others. And the aim is to try to get this to take root. And this is done partly through pictorial representations, which use a small number of models to consistently capture different aspects of elementary mathematics. So the bar model, the strip model, is perhaps the best known, but there are a few other partition models that are used. So the concrete is infinitely varied, the pictorial is much more restricted, with three or four main models that are used as an intermediate stepping stone. The abstract there is then essentially the standard mathematical representation.

And the idea, which we have forgotten, is that every student has to get to the abstract. We have left far too many students, this is finger counting if you like, far too many students struggling with concrete and pictorial, because they can get answers from these. But what they cannot do is progress to the next level. So that model is used fairly consistently and in particular the emphasis is on variations. Now this has become very fashionable but the message in mathematics, what has been weak in English school mathematics teaching, is the temptation to treat every topic on the simplest level. Only. And then you discover later on that what you need for progression, is not the simplest level but various more elusive variations on that simple theme. And so these have tried to focus on variations.

I think much of that you can read on the website afterwards, perhaps. But I will repeat the comment: I think there may be 21st century skills but none of them can be taught separately from the mathematics. So I encourage you to think how to teach new competencies or skills that you think matter through the mathematics, rather than separately from the mathematics. What cannot work, and I suppose I have to say this, is either to simply stuff more content into the same suitcase, as in an airport or going on holiday, or to treat the mathematics curriculum as a piece of string and to cut 25% of the end. This you cannot do. You can reduce and you can increase, but the curriculum is an organic structure. You have to start again and make sure that the things you reduce don’t have knock-on effects elsewhere.

Jolly good. Thank you very much indeed.