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What mathematical thinking looks like and why schools should teach it

Go to the Character of the Mathematical Thought list...
A week ago, Doug K. sent me an essay
Why We Should Reduce Skills Teaching in the Math Class
by Dr Keith Devlin, a British American set theorist and mathematics teacher.

Like many postmodern promoters of feel-good education, Devlin argues that we should reduce the teaching of all hard mathematics at school. After all, almost no one actually needs mathematics in his life so it's fine. This change will reduce the math anxieties and math phobia in the society, make the world a better place, and so on. At the same time, most people will understand what is mathematics, how and where it is used, they will have a positive attitude to it, and they will be ready to learn it as soon as they need some because math phobia won't be deterring them.

Please, give me a break.

Devlin identifies the teaching of actual skills, methods, and algorithms as the key villain and he promises us that in his world with almost no universal mathematics education, the following verbose proposition will hold true:
Tomorrow's citizens will appreciate the pervasive role played by one of the main formers of the culture in which they live their lives. As such, if and when they find that they do have explicit need of some mathematics, they will not start with the disadvantage of having to overcome math anxiety, as is so often the case today.
This paragraph reminds me of some statements by the EU authorities or the long clichés at an interdisciplinary conference attended by Feynman that could have been translated to English as "people read".

The paragraph says that people who have gone through no real mathematics education will understand how mathematics is important and why it is a key former of the culture? And they will be ready to learn and use mathematics as soon as they need it? Is he joking?

The reality is that many people have hard time to deal with mathematical problems because of 1) lack of talent, 2) lack of patience and motivation, 3) lack of background and training. Changes of the education system proposed by Devlin or others can't really change the first (biological) issue which is decided by Mother Nature, they can't mostly change the second parameter (personal character) either, but they will make the third parameter worse.

So people who haven't been tortured by mathematical education will lack some background which will make it harder, not easier, for them to deal with tasks that heavily depend on mathematics. This statement is particularly true for mathematics that builds insights on top of each other – like a pyramid or a skyscraper. This characteristic of mathematics as a subject is one of the things that make it really difficult for many people. You can't really memorize the insights in isolation – like in most other subjects – because they heavily depend on each other. Even more seriously, if you completely fail to get some important enough "stair" or "floor" in the pyramid of the mathematical knowledge, it may make you unusable for the rest of the mathematical education at all schools.

If a person hasn't trained himself to master the first three floors of the mathematical skyscraper – because he had no interest and he wasn't forced to do any of these things – he will not be able to quickly learn everything he needs to deal with the fifth floor of that skyscraper which is what he may very well need.

If I reiterate a previous statement with the policies in mind, it is a particularly good idea to reserve lots of time at school for the teaching of mathematics exactly because the mathematical insights depend on each other so heavily – and, consequently, one would have to spend a huge amount of time and energy by learning all the background that is necessary to master some task "at the fifth floor" that the life may bring.

In comparison with that, teaching of literature, zoology, geology, chemistry, and many other subjects is much less important because these subjects are closer to being composed of isolated insights and that's why when you need something specific, you need much less time to master it.

So I think that Devlin's statement that with no hard mathematics education, it will be easier for the people to master the mathematics that they suddenly need in their life is self-evidently laughable. But his first statement that the people who were not tortured by mathematics will have a better understanding of the character and importance of mathematics is equally wrong. It's just not possible. Let me write the sentence again
Tomorrow's citizens will appreciate the pervasive role played by one of the main formers of the culture in which they live their lives.
The problem is that for the person who has really done nothing like the application of a mathematical algorithm to think or calculate something, this sentence is nothing else than empty words. In the sentence that praises mathematics, you could equally well replace "mathematics" with "wakalixes" or even "feminism" – and because I am so incredibly generous, I haven't even mentioned that despite its length, Devlin's sentence actually doesn't contain the word "mathematics" at all.

If you have a person who has never done anything like adding two fractions, the sentence
Tomorrow's citizens will appreciate the pervasive role played by wakalixes.
sounds equivalent to Devlin's original quote. You just don't need the word "mathematics" at all. The sentence is just another empty and pathetic cliché that the students would be brainwashed to mindlessly repeat while feeling deeply moral and important – like all the bestial lies about the European Union that must exist to protect the world peace, about multiculturalism, about the ongoing catastrophic climate change, and kilotons of similarly toxic junk that have contaminated a very large fraction of the education systems in most of the Western countries and that have turned many kids into brainwashed, intellectually crippled parrots without any common sense or critical thinking.

The difference between the positive statement about "multiculturalism" and the positive statement about "mathematics" is that the statement about mathematics is actually true. Mathematics is really needed for a modern civilization to operate and at one level or another, it underlies all processes and events around us. But one can't actually know it, understand it, or even honestly believe this fact without actually having done at least some mathematics!

So, to pick a very specific example that emerged in several other debates about mathematics education in which I participated, finding the common denominator while adding or subtracting fractions is an annoying task for a majority of kids who are about 10 years old who are just being tortured by this stuff at school. It's one of the "focused symbols" of mathematics as the hated subject. It's hated because it's hard and indeed, some kids never quite jump over this hurdle. Some of them will learn it as adults and some of them won't learn it even as adults.

It's just how it is. Relatively to many other things taught to 10-year-old kids, the least common denominator is tough. It's tougher than to spell the name of William Shakespeare or analogous insights that are taught in other classes at the same time. It is an example of some relatively non-trivial mathematics.

But one of the main points of this blog post is that mathematics really deserves this prestigious name only once it starts to be at least somewhat non-trivial.

If mathematics were only solving trivial things, it wouldn't be mathematics. It is exceptionally important for the kids to learn an example of an insight that is non-trivial but may be settled, anyway. It is exceptionally important for them to learn that people are different and some of them easily master such things while others get drowned. So the kids simply have to be tortured like that. Their encounter with some non-trivial hurdles (where some kids unavoidably fail) isn't just an unfortunate side effect of the education that one should minimize; it should really be a main goal. The insight that there exists knowledge and methods that an average person doesn't know from the moment of his birth is critical, it is precious, and the citizens who fail to understand it aren't really civilized human beings. This insight has many corollaries. Kids learn that they sometimes need to work hard to achieve or find out something – and they also gain some respect to those who do work hard or those who have some special expertise, who may deserve a higher salary, or those who have contributed a lot to build our civilization.

At the end, the particular method to find the least common denominator isn't among the most important things. There are lots of similarly important things in mathematics – and other subjects. And it's even plausible that more than 50 percent of the people won't even need to calculate\[

\frac 12 + \frac 13 = \dots

\] in their whole life. ;-) But the general philosophical lessons – that difficult tasks can be solved, can be solved in the full generality, can be solved in finite time, but the time may be short or much longer, and despite the hard work, the question can be completely settled – are what is really important. And they just can't be "embraced" by the kid if the kid never goes through any particular example of the hard thinking.

Now, I would say that I don't really want the kids to calculate hundreds of expressions such as \(13/19-17/23\) – which almost no one will need, indeed. "Several" problems like that might be enough and then they can move on. But you know, this fast progress from one topic to another, conceptually very different topic (or method) is exactly what separates the kids who are simply not good at mathematics. If the kids spent 4 years with problems such as \(13/19-17/23\), almost all of them would learn it. But the smarter ones would suffer – and they would fail to advance.

So Devlin's proposal to replace mathematics classes with classes full of clichés about the importance of mathematics – which become virtually isomorphic to the classes where children parrot lies about the importance of climate change (I was absolutely terrified by this piece in The New York Times: I swear that in the communist Czechoslovakia of the 1980s, we have never been brainwashed to this unbelievable extent). People don't learn anything about the "true spirit of mathematics".

In fact, it's even worse when they are trained to say these clichés without actually knowing any mathematics. It will make them think that they understand mathematics even though they demonstrably don't. So lots of incompetent and uneducated people will feel and act very self-confidently about mathematics. I feel that this catastrophic consequence of Devlin's proposal isn't just an unfortunate side effect. It seems to be the very goal of it. It seems that Devlin wants the math phobic people with no interest in mathematics (and science) to take over and present itself as the core of the human civilization and the education system, too. And he wants the kids who like mathematics to be marginalized.

A country (or civilization) that allows such a counter-revolution will gradually become uncivilized again.

Let me be a bit more systematic. There are lots of methods that children should learn in order to get the background up to some fourth floor of the mathematical skyscraper – I haven't actually defined the floors exactly and I hope that you will understand what qualitative point I want to make, anyway. I've argued it's very important because mathematically loaded tasks and situations heavily depend on each other which is why it's a good idea to create as strong background for the children as possible.

But later in the essay, I argued that the children's getting the "general spirit of the mathematical thinking" is even more important than the skeleton of the skyscraper that can be built upon. The general spirit isn't important just in mathematics, economics, engineering, or natural science: It's really important for the citizens' independent and critical thinking, for their ability not to be fooled, to pick a good representative, to pick a representative who is really good for them, and so on.

An important point is that a kid who just "generically plays" and "faces and solves some generic practical hurdles in its life using common sense" isn't thinking mathematically. The kid may have more or less talent, more or less interest in mathematics but the important thing is that the practical life of an ordinary person and his or her common sense (with some numbers appearing on phones, houses, and microwaves) is simply not the same thing as mathematics or the mathematical thinking.

It would be a huge tragedy if the schools became fake institutions that try to claim something else and that give you a "stamp" or a "diploma" certifying that your common sense is enough for you to be educated in mathematics at the high school level or something like that.

What are the characteristics of mathematics and the mathematical thinking that make it differ from the ordinary person's thinking or common sense? Let's list a few of the key ones. They're somewhat overlapping but different enough to deserve thirteen or so boxes (while doing mathematics, one should also understand that 13 is unlikely to be unlucky, after all).
  1. Mathematical propositions, questions, and answers have to be formulated very precisely and a mathematically thinking person realizes that details often matter a lot.
  2. Mathematical propositions are either true or false and these two truth values are sharply separated.
  3. Disciplined thought. At least in mathematics classes, one must avoid distractions, vagueness, wishful thinking, laziness, and think (perform sequences of controllable steps) about problems for their own sake, whether or not they are practically relevant. Thanks to Edward Measure.
  4. The truth in mathematics doesn't depend on the society, the social consensus, feelings, political power, or the brute force of muscles. Mathematics is apolitical, universal, timeless, transnational, race-free, asexual, gender neutral, and resistant to social fads. Thanks to Eclectikus for the refinement.
  5. The truth value of the propositions may be completely settled so that in principle, you may erase all your remaining doubts and worries once and for all.
  6. The truth may be demonstrated by a mathematical proof, a possibly elaborate and quantitative counterpart of what attorneys may offer in a court, and pupils should get familiar with this methodology. Thanks to John Archer. (The previous point is mostly about short calculations, this one is about long structured proofs.)
  7. Patience. Even simple questions sometimes require very difficult – and often not straightforward or unexpected – procedures to be answered, solved, proved, or calculated.
  8. On the contrary, clever mathematics is also often able to dramatically simplify a naively difficult problem, beat the brute force, and save lots of energy otherwise consumed by brains and muscles. That's a part of the reason why mathematics allowed to create a more effective economy. Thanks to Tom Vonk.
  9. Even seemingly non-mathematical problems and statements (e.g. those involving geometry or hope [probability]) may be converted to numbers, variables, and other mathematical objects.
  10. Mathematicians try to generalize a problem and solve it in generality. They are not too interested in ad hoc solutions to very special situations and cases even though those are often enough for an ordinary person facing practical tasks.
  11. Mathematicians always try to generalize the concepts and structures themselves. The mathematical thinking allows one to discover patterns and relationships between things and ideas that used to look independent and unrelated.
  12. When one is asked to find a solution, mathematicians naturally try to find all of them and prove that no others exist.
  13. Mathematicians find it standard to verify that their solutions satisfy the conditions formulated in the problem they were solving.
If you find some important omission in this list, I will be grateful if you point it out and I will be allowed to add it.

So if Devlin were recommending to cancel mathematics classes entirely – and instead, replace the sentence "feminism is great" in another subject with "feminism and mathematics is great" – it would mean a collapse of the civilization as we know it. So I suppose he didn't really mean that. I suppose that he wanted to omit some methods and teach the kids to appreciate at least the list of the features of the mathematical thinking that I wrote above – or something equivalent.

But again, it's not really possible to know, understand, or even believe the validity, usefulness, and sometimes critical importance of the points above without actually doing at least some hard mathematics, without getting one's hands dirty – and without antagonizing some kids who really dislike it, usually because they are simply not naturally good at it.

The antagonization of some kids – perhaps a majority of kids – is an unavoidable sign that something important is actually happening. Dozens of percent of kids will always unavoidably start to hate mathematics once they are first asked to find the least common denominator – and maybe earlier than that. But it's just totally wrong to create the illusion that these are the kids who should determine what is taught – and, indeed, whether anything is taught at all.

Instead, it's the kids who are capable of learning something, whose mathematical skills – and, inevitably, their understanding of the "mathematical methodology and philosophy" as well – improve after the classes. The key group of schoolkids in ordinary schools isn't the future theoretical physicists and set theorists – who will probably find their path towards mathematics without much help from the schools. They represent less than 1/1,000 of the human society. They will tend to rediscover everything themselves and lovably play with all the mathematical things.

The key target group of mathematics classes at basic and high schools are those 10%-30% – I don't know exactly and there's no "only correct number", of course, so this is just an effective or qualitative estimate – of the schoolkids who actually have the ability to learn something, find the least common denominator a few times in their life or do something similarly hard, learn the general lesson or spirit of mathematics, and who can mostly learn how to apply particular methods and strategies of thinking even though they haven't rediscovered them independently and they often fail to deeply understand why they actually work.

This 10%-30% of the people will become engineers, natural scientists, economists, programmers, or they will choose another occupation from a list in which the mathematical methods and thinking is simply needed. Those are those who matter and the future prosperity of a nation or the mankind will non-trivially depend on the quality of these people's background (and on other things, of course). To emphasize the fact that a big group – and perhaps a majority – doesn't develop any love to mathematics and fails to master many methods is a pure distraction. These people are simply not important.

The misunderstanding of the group that is the actual audience of mathematics classes – and why this group and its numeracy is important – may be the main reason why Devlin wrote his completely wrong essay. Why do I think so? Well, it's because I see references to the "majority" everywhere in the essay. Just the word "majority" appears five times (and vaguely equivalent statements appear about 100 times in the essay):
  1. "a general antipathy toward mathematics in the majority"
  2. "and we wring our hands endlessly when, for the majority of our students, we fail"
  3. "we simply turn (what I think is) the majority of people off mathematics altogether and produce significant math anxiety in far too large a minority"
  4. "By changing our present education system radically so that, for the vast majority of students, the primary goal in the mathematics class is to create an awareness"
  5. "apart from a tiny majority"
I've included the fifth one as well although I am doubtful that there are too many "tiny majorities".

But misprints notwithstanding, the point is that the school just isn't democracy so the majority's attitude towards mathematics isn't what necessarily matters. The schools exist to educate, cultivate, train, or prepare a generation of students as a whole for their life and for their thinking. The generation needs and will need mathematics – although it will only need it through a minority – and that's why all the kids must get the opportunity to see what the mathematical thinking encompasses, why they need to be tortured by some particular albeit often boring "hard mathematical work", and why the schools must struggle that at least some of them will take these things seriously, learn some of the methods, and – especially – learn something about the general spirit of the mathematical thinking.

P.S.: Devlin has obviously written lots of texts whose message is supposed to be similar. Two months ago, in Edge and The Huffington Post, he argued that all methods he has ever learned in mathematics courses became obsolete. This is just plain bullšit. I can swear that I've needed an overwhelming majority of the things that were taught to me at all the mathematics classes at all levels I had to undergo.

But even when some things get obsolete, what's the problem? For example, I probably belonged to the first generation of kids that didn't really learn to work with the slide rule or slipstick (in Czech, we call it the logarithmic ruler). Some 40 years ago, slipsticks were replaced with lipsticks to do science.

Would I protest if I had been taught to work with the slide rule? I think it's great. My grandfather probably taught me to work with that when I was 8 or so. It was very interesting. It was replaced with other things – perhaps calculators, although the equivalence isn't perfect – but it doesn't hurt to learn the old technology. You know, if you play with the slide rule, you will understand why \(x^y\cdot x^z = x^{y+z}\) where the base may also be \(x=e\) – and you are unlikely to learn this identity if you play with a calculator. Moreover, many of the newer replacements may only be easily understood by those who understand their predecessors.

So I am baffled by this very logic. The fact that there is progress doesn't mean that we shouldn't teach mathematical methods. Instead, the actually important and true causal relationship between these two phenomena is very different. There is progress partly because people who have learned mathematical skills are helping the progress to materialize.

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