Friday, March 25, 2011

Michael Green: math classes are boring

Can and should maths and physics be made much more appealing for schoolkids?

According to the Telegraph, Lucasian professor of mathematics in Cambridge, string theorist Michael Green, has agreed with Barbie (who was silenced by obnoxious feminists in 1992):
Children bored by 'tedious' maths lessons
Well, I am somewhat less certain about the problems he has identified and about the proposed solutions.

Green's idea - shared by many others - is that the math education at schools degenerates into "drudgery" and "boredom" which is why pretty much everyone, especially the girls, want to bid farewall to the exact sciences. Permanently.

Do I agree? Well, I mostly don't. First of all, I surely don't think that the kids should be led to listen to hip-hop instead of working equations because mathematics is simply not hip-hop. (But maybe, they should do both things simultaneously - who knows.) Second of all, I disagree with the notion that the typical groups - such as girls - are more likely to remove maths from their lives because it's "drudgery".

The right kids to learn maths look like Jacob Barnett, a cool 12-year-old student of the Indiana University - Purdue University Indianopolis. Check his talk on back-of-the-house calculations of integrals. Via David Hain

In my life, I've had hundreds of classmates and almost one half of them were girls. Add dozens of people I've been tutoring. Many of them were highly educated - they have learned lots of things. If they had problems with maths, it was exactly because maths could not be turned into "drudgery".

Instead, maths has always required a pupil to independently think, to understand what objects meant in various situations and which of the numerous methods to deal with them is appropriate given some additional information. This situation is very different from many other subjects in which the ultimate "drudgery", namely a mindless memorization of sentences (or pictures or motion), is a nearly universal key to knowledge.

Are kids tortured with unnecessarily boring and/or uselessly abstract things?

It will surely sound as a heresy to many readers but my answer is mostly No. Let me discuss a couple of topics. Children could be tortured by Bourbaki-style nitpicking and uselessly rigorous maths - and by meaningless ideology that a pedagogical Big Cat randomly invented - except I don't think that they are.

In particular, children are taught basics of set theory - sets, intersections, unions, being elements, and examples with sets of numbers. In spite of the criticism that's been raised about those "modern" math disciplines, I actually think that those matters belong to maths and must be taught. Children also have to be taught addition, subtraction, multiplication, division of positive integers, all integers, rational numbers, decimal points, real numbers, distributive law and many other aspects of algebra with addition and multiplication, linear functions, their sets, quadratic functions, quadratic equations, rational functions, geometry, analytical geometry, properties of triangles, squares, lines, solids, polyhedra, spheres, trigonometric functions, and so on, and so on.

I don't claim that the proportion of the material is perfectly chosen but I believe that from a broad perspective, the children are pretty much being taught what they really need for any applications or even formal studies of maths. There could be more diversity and experimentation when it comes to the selection of topics but I believe that the schools that would ultimately teach the kids some valuable things wouldn't differ that much.

Mechanical arithmetics

One type of "drudgery" clearly deserves a special treatment - it's the calculi, large sets of tasks to compute sums or differences or products or ratios of many numbers or complicated numbers and similar things. Not sure about English but we call this discipline "merchant's calculus" in Czech.

Children are sometimes being tortured with this stuff - and I essentially disliked it, too (even though there were situations in which I could be the best guy in this stuff as well - a status I surely tried to avoid later). However, it's exactly the part of maths that a large portion of the schoolkids may succeed to learn. In fact, the non-mathematical types may sometimes get even better than the mathematics-loving kids.

Now, is the "merchant's calculus" the same thing as maths? This is a prevailing opinion among a large fraction of the society - I would even say a majority of the society - or, equivalently, it's the opinion among the people who still haven't a slightest clue what maths or exact sciences are about, what they're capable to do, and what is actually essential to be good at them.

But I am convinced that the kids and folks who are actually forced to study some maths and who may have a problem with it are well aware of the fact that mathematics is not primarily about "merchant's calculus".

Back to Green's theses

He is quoted as saying:
[Green] said teachers failed to present the “glamorous” side of the subjects as classes often descended into “drudgery” and “boredom”.
Now, the presentation of the "glamorous" side of maths has two levels: it's about the personal excitement of the teacher that may be infectious but it is often not; and about the actual ability of the teacher to convince the students that the side of maths or the whole mathematics is actually "glamorous". The second question depends on the belief in the glamor - if one doesn't see it, he doesn't necessarily believes that it exists.

There are obviously negative aspects of the situation - for example, when the teacher either hates or misunderstands maths herself or himself, it's obviously bad and it is likely to produce bad results, too. However, if the teacher sufficiently knows and likes his or her subject, there is still the remaining question whether the curriculum is pushing him or her in a wrong direction.

Well, I think it's usually not. Why? Some results in maths may look "glamorous" but one may only appreciate the glamor after some hard work - or after some "revelation" that often takes place in the solitude. The real challenge is to make the student able and willing to go through some sufficient "drudgery" that is actually necessary to get convinced that an ultimately simple result is correct.

Some results may be guessed - or even proved - much more efficiently than through the most standard, mechanical methods to calculate things in the same class. The more creative and talented student you have, the more likely it is that he or she can apply more elegant methods. Of course that I think that in general, students shouldn't be punished for using creative and diverse yet correct ways to think about a problem.

On the other hand, if the unusual shortcuts can't solve a generic element of a class of problems, while a standardized method can, the students simply have to learn the general method, too. Just think about a random but particular class of problems: sets of linear equations. I don't know how one could avoid the teaching of one of the standardized methods to solve them.

Do children have to stay narrow-minded?

Green also says:
You can't imagine [maths'] beautiful elegance and way of describing the world at that stage. When [pupils] go to school and choose maths, they don't know enough about the subject and the way it developed. Some of them don't want to know.
Is that right? Well, at every stage, one can imagine the elegance of a certain portion of thinking or maths - and the level should be getting increasingly sophisticated with time. I thought that schools should be - and at least partially are - contributing to this expansion of the students' horizons. The history of maths is a largely independent issue but I think that the same thing holds for the history: children should be taught increasingly detailed things about the way how mathematical concepts developed, too.

Did I understand well that Green wanted to strip the schools of this role? That's how the paragraph above looks like.
I never understand why anyone wants to do maths, having been exposed to it before the age of 10; the drudgery they are exposed to. It’s difficult.
Maybe they like it. And maybe even if some of the material may be classified "drudgery", some of the kids still like it more than their classmates - and they're better at it than their classmates. That's how the world works. People are not equal, they will never be equal, and it is completely correct that people do XY if they like XY more than other people do. I mostly don't understand why people volunteer to torture themselves in fitness clubs for hours - but many other people actually do understand it. ;-)

Green mentions that his daughter isn't too interested in maths and he apparently feels uneasy about it. Well, he shouldn't: that's how most girls feel and just because her father is a Lucasian professor doesn't mean that she has to be totally different when it comes to this point.
I see it with maths, and there’s a real problem with physics, to convince [pupils] that science isn’t geeky, especially girls.
Except that mathematical sciences are geeky - and being geeky is sexy, too. I think that Green underestimates the girls' ability to figure out what maths - and math-dominated approach to life, if I can use these big words - really mean. They find out the approximate answer and they decide they don't like the broad content and image of maths.

If someone primarily cares about the "cool" behavior of people, be sure that musicians or actors or similar professions will always beat the scientists according to his or her criteria. I think it's completely logical and inevitable - and the person is judging the world rationally. Scientists are undoubtedly more geeky than non-scientists.

A scientist is someone who is much more likely to differ from a random chap on the street when it comes to his priorities, ways of thinking, moral values, or even behavior. This is just a fact - a fact that has both positive and negative consequences - and the point that the societies should actually try to make is that this difference is very positive because of many reasons. It is important for the progress of our society and its identity - even though mathematics and science is not actively developed by everyone.

The idea that all people in the society will be mathematicians is unrealistic and egalitarianism in math education is downright harmful.

Where did the result or formula come from?

Let me return to one of the topics from the beginning of this text. Many people who are not exactly into maths hate various methods taught in maths because they find them "arbitrarily". If you don't know how a method or formula was derived, there are many apparently conceivable ways to "mutate it". And you may ask: why aren't we taught another mutation?

For example, the largest exceptional simple Lie group is 248-dimensional. Why it's not 1917-dimensional, for example? Some people may view those facts as "unfair". After all, the number 248 got an advantage while 1917 was discriminated against.

Well, of course that if you know some Lie group theory, you will be able to prove that 248 is right while 1917 is wrong. But even in much simpler examples than the dimension of E_8, the required proof is often complicated enough so that it's been decided that the children won't be trained to understand or derive the proof: they're just told what the right number, formula, or method is.

Again, I think that in many cases, it is a legitimate attitude. Most of the children will end up being "at most users" of the formulae, e.g. the formula to solve a quadratic equation, so they don't need to know elegant ways to derive it. The explanation is the same as the logic why people are not taught the architecture of microprocessors or their machine code in their programming classes: those questions are largely separated and they won't need the answers to them.

However, what I find important is to force every single child to go through enough "drudgery of proofs and derivations" to understand the much broader philosophical point, namely that there is usually a damn good and unavoidable reason why the answer is something rather than something else. If the kids think that all of mathematics is about some unsubstantiated harassment of the number 1917 by the evil and oppressive number 248 that was randomly chosen for no good reason, they can't possibly like maths or understand why maths is legitimate in the first place.

My current point is much more general: if there is some hard work that a student can do and that will push his opinions or expectations about a much broader class of questions in the right direction, he or she should be forced to actually do the hard work!

A few paragraphs above, I mentioned a general but widespread myth that the details of the formulae and numbers that appear in mathematics are arbitrary and unfair. This is obviously a devastating myth that prevents its holder to develop any positive relationship to maths. Still, this myth is fully compatible with - and, to some extent, directly produced by - some recent popular degenerated ideologies usually classified as political correctness. And it should be fought against because it's wrong and very influential at the same moment.

There are many comparable but more subtle myths that should be fought against, too. For example, at a slightly higher level of knowledge of maths, there is a widespread perception that "there must exist a straightforward method to find the answer to any question." Well, if there's one, no one has surely found the general recipe how to find the general method for a given problem.

Most likely, such a method doesn't exist. But many people conclude that if there's no mechanical way to derive an answer to a question, such an answer can't be quite right or quite reliable. But of course, it can. In many cases, one may "guess" the right result or formula - and fully prove that it is right. Again, this general lesson should be conveyed to the students.

It's usually being assumed that the kids can never learn "general" lessons - they must always focus on some particular systems and do limited operations with them. It's surely true to some extent; on the other hand, there comes a point at which the child simply has to be able to deduce general lessons and it's important to guarantee that they won't be completely wrong lessons. It shouldn't happen that the kids can't see the forest for the trees.

More abstract notions such as the forests must ultimately be offered to the kids, too. But yes, this "abstract development" should only appear at points when they can already imagine lots of individual examples behind them.

Drudgery in biology, evolution

While I found the composition of the material in maths - and, to a lesser extent, physics - in the basic school and high school curricula acceptable, I may be more critical about the way we teach chemistry, biology, and other subjects.

For example, there's this contentious issue of evolution. When we were 10-13, we learned lots of details about various groups of animals and plants and their anatomy and physiology. I think that there have been too many "trees" in this discussion and the "forest" could have remained invisible. The amount of worthless memorization was equally high in chemistry.

In this context, I actually believe that the evolution-creationism debate is more exciting for the kids - and what they would learn out of this topic is arguably more important for the evolution of an intellectual personality.

In a free society, the children who are destined to prefer creationism will ultimately find and prefer creationism, anyway. That's why I find it silly and counterproductive to try to censor it at school. Of course that I think that the punch line of such classes should be that evolution is right and creationism is wrong; but I see absolutely no reason why the students shouldn't be exposed to the ideas and would-be arguments for creationism because they will eventually be exposed to it, anyway.

A much more general point, one that applies to all subjects, is that the students should have a reasonable idea about the huge proportion of the theories that have been believed to be true but turned out to be false, and fundamentally false. People should be led to understand that many things written in the newspapers etc. may still be wrong; on the other hand, they should also be led to understand that in many contexts, very robust arguments exist that may de facto eliminate all realistic doubts.


So in my opinion, the teaching should be viewed as a collection of important examples leading the student to learn how to independently and properly think and to get rid of the most hurtful, universal myths about science and logical thinking. Contentious - and therefore appealing - topics shouldn't be avoided. But hard work and mechanistic methods whose importance is sufficiently universal can't be avoided, either.

Of course, separate discussions should be dedicated to "training" for particular tasks in the future and "general education". For obvious reasons, the priorities are different in the two classes.

Students should be encouraged to appreciate the importance of maths and science - despite, or perhaps even because, their being different from the attitudes of a Joe Six-Pack. However, I am afraid that even if the education system were made nearly perfect, some people would end up hating maths - and it wouldn't be too different people from those who hate it today.

And that's the memo.

Conrad Wolfram and computers

Conrad Wolfram recently addressed the math teaching questions in a TED talk. He believes that one has to stop teaching calculating - people need to learn maths. An OK slogan. And computers are the messiahs here, he think. Well, maybe. Or not.

Well, I am afraid that there's a lot of computers around us but computers are not yet maths. Half a billion people spend lots of time with Facebook but this activity is not maths. Computers may replace us in some truly mechanical computational activities - I agree with him. However, I think that one can only do "formulating of the problems" etc. well if he actually knows how to calculate things themselves.

He also defends the specialization and fragmentation of knowledge - for an obvious example, people can learn to drive cars without learning the engineering. I agree with that. This specialization and separation of disciplines that used to be inseparable is an inevitable by-product of the expanding human knowledge. On the other hand, I do think that maths is exactly what attempts to overcome this fragmentation! Maths may really be defined as whatever all those branches of science and engineering have in common. It's not a coincidence that "polymath", the word for someone who knows all/many disciplines, is related to "maths" (the root "math" is "learning"). So I don't think that kids are exactly learning maths if they're led to think that it's just a very special tool to solve very isolated subproblems. And I am not sure whether Conrad Wolfram and I mean the same thing by "maths".

CW also fights against the notion that "computers dumb maths down." It's a closely related question to the previous one. While I agree that manual computations and derivations may be equally dull as "pressing of the buttons", I also think that it's important to know what the computers are actually internally doing. I don't claim that kids should spend lots of time e.g. with mechanical calculations; but I do think that they need to be familiar with it.

And I think it's actually exciting to know how computers and programs and operating systems internally work - and we still need many people who understand such issues. In some sense, a big portion of the people who will need maths at all will also need to look "inside". For this reason, the attempt to "cut" maths and reduce it to some "management of computers" is a part of the problem. That's why the interest in maths is decreasing.

As you can see, it becomes strange why your humble correspondent wrote that uncritical comment on that website. Well, frankly speaking, I hadn't listened to the talk carefully when I wrote the comment.

CW even thinks it's "nuts" to use computers to teach kids to do certain mechanical procedures. I surely don't think it's nuts. Even if this "class" were meant to explain what's happening inside a computer program, it's damn important. If I take CW's philosophy to the limit, kids should be taught maths which is equivalent to "if you need to know something, just press some computer buttons." But this teaching is not maths. It's vacuous instead.

The real core of the problem is that every sufficiently understood mode of thinking or procedure may be made automatical and solved by computers. But that doesn't mean that the people have become unnecessary. They have surely not.

CW also says that it's useless to teach "clean maths" because the real-world problems are "messy". I completely disagree with this point. The real-world problems are only hopelessly "messy" as long as the people haven't managed to deconstruct them into clean maths. In fact, (applied) maths skills partially are all about the ability to see the clean mathematical patterns behind the seemingly messy real-world situations. The simple problems such as sets of linear equations are ultimately core components of the answers, even if the questions deal with the "messy" real world.

Conrad Wolfram says that computers haven't dumbed down the education; instead, we have dumbed down the problems. I disagree. It's just utterly logical and correct to teach, for example, sets of linear equations before sets of nonlinear equations. The latter usually can't be solved analytically. But sets of nonlinear equations are just a "minor" generalization of the sets of linear equations and one learns most of the things that "matter" already when he learns the linear case. Moreover, in the real-world practical applications, linearizations and other approximate schemes are damn important and often sufficient to deal with the reality. So it's always more important to learn the "dumbed down" - I would call them "fundamental" - problems.

This point of mine because even more obvious when CW talks about the need for the kids to "feel the maths". There is a potentially infinite set of possible problems and if one plays with them one-by-one, he will only "feel" one of them but he will completely lose the "universalistic" part which is what all maths is about. It's just important to "feel" that there's ultimately a set of linear equations - at least approximately - behind many superficially very different problems. This outcome can't be achieved without teaching sets of linear equations. If people play with the 3D graphs of a function, with a few parameters, they will only learn one function. Still, some functions are more important than others.


  1. Although I loved math, and ended up majoring in it at Yale, there are a couple aspects of the standard U.S. curriculum that strike me as outdated.

    First, we spent a long time on the theorem/proof techniques of Euclid. This is pleasant stuff for those who are born to love math, but it isn't really very important. I think students should be taught some of the essential geometrical "tricks" like triangle similarity, and see how they're applied to get some cool results, rather than building the subject up incrementally from Euclid's axioms.

    Also, I don't really see the utility of much of what one does in grade-school algebra. It is just tedious drills with polynomials, and no meaningful uses of it are really provided. I would rather see kids spend that time gaining a better knowledge of trigonometry, which is so useful in so many areas, and perhaps financial math and probability, during the course of which some aspects of algebra would naturally be introduced.

  2. Dear Prof. Michael Green,
    I agree with you in most of the points in your above story. So I suggest you to see some of my comments on the Physics World, UK, listed below.

    1) My comment of 08 February 2010,
    2) My comment, 29 June 2010, Newton's thinking at 60+,
    3) My comment of 13 September 2010,
    4) My comment of 03 March 2011, Pseudo-Foucault pendulum,

  3. People who state that they like "science" but don't care for "math" wind up susceptible to believing nonsense like "global warming."

    Ideas like "global warming" have an appeal to people who have no concept of magnitude or the application of physical law to the world.