Tuesday, December 16, 2014

Kids: most new methods to teach mathematics are dangerous pseudoscience

Czechia's most influential (?) news server iDNES.cz has been bombarding the readers with hype about new revolutionary ways to teach mathematics to schoolkids.
I placed this blog post at the top again after I discussed with the advocates at aktualne.cz.
This stuff is combined with constant calls to eliminate mathematics from the mandatory subjects in the high school final exams. Folks from a despicable Faculty of Humanities at the Charles University – a tumorous department that shouldn't exist at all – are constantly involved. You can imagine that I am terrified by all that and you haven't even heard any details.

Today, an 80-year-old chap called Milan Hejný is promoting his and his father's "new method" to teach mathematics (how "new" is a method whose founder was born in 1904?) in an interview titled The kid has mathematics inside itself, just listen to him or her, the father of a revolutionary method says. Quite an uncritical title, right? However, the content of the interview is trash.

By the way, this Mr Hejný of course opposes all mandatory mathematics exams, too. Whatever is needed to lick the buttocks of the least talented mathematically talented kids more intensely than ever before is embraced.

We learn that kids are not learning mathematics in the right way because the traditional methods "weaken their creativity" while they "lead to the order". You may imagine lots of absurd comments saying that children may find everything by themselves, the proofs of the Pythagorean theorem may be left to the college (if the students choose pedagogic departments), and the kids don't even have to be told how to add fractions with different denominators.

The punch line of this "hippie method" to teach mathematics – which makes all kids happier and vastly smarter, we're told – is the statement that the compassion with other people's families belongs to mathematics, too. Oh, really?

It seems very clear that this is a method not to teach almost any mathematics and get away with it. A method to peg the teaching process to the weakest students and completely ignore the smarter ones who are those who actually matter because mathematics has to be understood by a fraction of the society for which it becomes important, not by everyone.

Also, it is a method to leave the teaching of mathematics to people who don't understand mathematics well themselves – Mr Hejný tells us that he would always suck in mathematics and the feminist-style women promoting his stuff are probably even worse.

Concerning examples, I personally found it amazing that they would delay the Pythagorean theorem to the college. Every kid may learn \[

c^2= a^2 + b^2

\] for the triangle. It's beautiful. Some people and kids don't see the beauty but some of them do. It's the opinion of the people from the latter group that should matter. The Pythagorean theorem is essential in all of geometry and I think that by the age of 10 or so, all children should have been exposed to it.

I personally think that they should know at least some proof of the theorem. Because most kids are not Pythagorases, and let me be open, they are not Motls, either, they will simply not find their own proofs of everything. Leaving them without help means that they will be stuck where the cavemen were thousands of years before Pythagoras.

It's not just the statement. A proof of the Pythagorean theorem is a useful thing for one's mathematical thinking – and many proofs are beautiful, too. The theorem may be proven in many ways. A children-friendly proof is this picture:

Why is it a proof? The upper picture divides a fixed square – of area \((a+b)^2\), as you can see, but you don't need that – to one smaller square (it is a square because of the \(\ZZ_4\) symmetry of the picture under rotations by multiples of 90 degrees) whose area is clearly \(c^2\) where \(c\) is the hypotenuse of the triangle with legs \(a,b\) (you may see that this is what \(c\) is according to the picture), and four of these very relevant triangles of area \(ab/2\). You don't really need the bottom picture anymore because the decomposition of the square implies\[

(a+b)^2 &= c^2 + 4 \cdot \frac{ab}{2}\\
a^2+2ab+b^2 &= c^2 +2ab\\
a^2+b^2 &= c^2

\] Great. The proof is complete. I needed \((a+b)^2 = a^2+2ab+b^2\). If a kid finds this identity too hard, and by some time, she shouldn't, she can find the bottom picture of the proof useful, too. We just moved the four triangles of area \(ab/2\) to different places inside the bigger square.

The remaining area is clearly the same on both pictures (because it's the difference between two areas \(U-V\), and \(U,V\) are the same in both parts of the picture). In the upper picture, the remaining brown area is equal to \(c^2\), the area of a square, and in the bottom picture, the remaining brown area is \(a^2+b^2\), two squares. So \(c^2=a^2+b^2\) must clearly hold.

I can give you numerous straight and concise proofs like that. Some textbooks for children may fail to offer them any proof which is bad. But the solution is not to postpone the Pythagorean theorem to the college or something that Mr Hejný proposes.

He also mentions the addition of fractions with different denominators. Normally, kids are taught to compute sums like\[

\frac 25 + \frac 37 = \frac{2\cdot 7 + 3\cdot 5}{35} = \frac{29}{35}

\] but "children don't understand why it is", Mr Hejný says, so his method of teaching doesn't teach this general method at all and he boasts about it.

Now, if you're a smart enough kid, you must agree that this senile guy is just stupid. The fractions \(2/5\) and \(2\cdot 7 / (5\cdot 7)\) are demonstrably equal – we just divided each of the two fifths to seven equal pieces, but the total amount of pizza stayed the same. And similarly for \(3/7 = 15/35\). And \(14/35+15/35\) may be summed because we are adding pieces of pizza (\(1/35\) of a pizza) that are equally large, so the sum is just \(29/35\) – we sum the numerators and keep the denominator.

Fine. There may be some psychological barriers that prevent a child from getting these points. But he may get them later. Perhaps one year later. Perhaps ten years later. As an adult, if he is still using his brain, he will realize that the thing he was taught as a kid is actually quite logical and obvious. Hopefully, the kid learns similar things earlier than that. The kid may find out that it's possible to crack or understand or prove some of these things, and he will be able to do it himself in some other examples.

Perhaps, he won't rederive everytihng in mathematics. But there is no reason why he should. The point is that he will understand that such proofs probably exist and the formulae he was taught are probably right and rigorously provable. Mathematics works and he doesn't have to be afraid of it. But if the kid is never exposed to any identity like that, the chance that he will get these points is nearly zero. One must try! And of course that in the case of many children, one will fail to convey the message. But that's no excuse not to try.

One may discuss the question whether the kids should use systematic methods to mechanically calculate certain things, or more ad hoc, intuitive approach to each problem. You may imagine that as a kid, I would be a clear representative of the latter. I would find the solution to every school problem like that differently than the "standard method" and usually "differently than how I solved the previous one". Many of you must have used similar ad hoc methods. Imagine solving sets of linear equations through various methods. You couldn't even say which of the methods you used. And other children who were at most mediocre in mathematics couldn't understand your otherwise completely rocksolid arguments and methods to calculate. Not everyone is equal. You always knew that the problem was on the other kids' side.

But this "independence of the template" didn't occur because I or you couldn't learn the systematic method. Of course, I knew all the standard methods to the extent to which I cared. They were more or less obviously equivalent. One may use other methods, different methods in each case, because it just comes automatically to his mind and he's lazy to follow some mechanical procedure if he can live without it. Of course that I could imitate the mechanical methods perfectly, too, if someone wanted that.

However, if you have a kid that doesn't know how to solve a general problem, you simply shouldn't avoid teaching the general formula or algorithm to the kid. The only possible outcome of such an omission is that the kid won't have mastered this part of mathematics.

Think about the addition of the fractions as a major example. Concerning their knowledge and understanding, children may be roughly divided to five groups:
  1. The child discovers the method to add the fractions himself and finds a proof why it yields the right result.
  2. The child is told about the final result and is able to rediscover the proof why the formula is right.
  3. The child is told about the final result and a justification and understands both.
  4. The child is told about the final result and a justification but only understands the final result.
  5. The child is told neither and doesn't know how to add the fractions.
The groups were sorted from the "best mathematical talents". To a large extent, the membership in these groups is a matter of Nature. Some children are more gifted in these things, much like other (or the same) children may be gifted in other things.

But there is still some (hopefully upward) mobility. Some kids may find or understand the proofs after a delay. When they understand some proofs, they're more likely to try to find or at least understand proofs in future situations.

As far as I can see, Mr Hejný wants to keep all the children in the fifth group. They don't know any systematic way to solve these problems and if they manage to solve one particular problem of this kind, it's some kind of an accident. Or a selective memorization of a few isolated answers they have encountered. It's just too bad.

The Hejný method is being used at about 350 Czech basic schools, about 10% (and 36% of schools use is at least partly, usually as a supplementary alternative method), and the powerful interests want to spread it. They cite some incredible successes which sound like direct contradictions because as far as I can see, they openly say that the outcome of the method is that the children don't know what I call mathematics. They may redefine mathematics so that the knowledge of the addition of fractions, sets of linear equations, or the Pythagorean theorem are not included. And many advocates of the method actually openly say that "logic is enough". Enough for what? Such a redefinition of mathematics is what I call a fraud, a cruel crime against the human civilization.

Of course, I am not the first one who despises similar new methods designed by self-anointed "teaching experts". Check Judging Books By Their Covers by Feynman who was a member of a textbook committee. They would add up the temperatures of stars just to have some "real-world things" behind the objects that are being added. Feynman explains why it is utterly idiotic because no sane scientist would "add the temperatures of stars".

Needless to say, this ("visual" and really populist) focus on drawing cute animals etc. when the goal is to add and subtract is also popular among the contemporary Czech crackpots who claim to "improve the ways how we teach mathematics". There is no science in it. Their methods are never really verified or independently rated. There is no reason to expect that these methods represent progress and they don't. It's not surprising that these methods are helping the results to deteriorate. They are methods designed by people who don't understand what they should teach themselves!

There are also all the fads about renaming concepts. E.g. "integers" were replaced by "counting numbers" in the books that Feynman would judge and he was annoyed by these details, too.

The Hejný method is also being supported by a Czech billionaire, I just learned. Very painful. The method's website www.h-mat.cz contains some information, for example the 12 key principles of the method.

1. Building of schemes

The proponents claim that children can think of many schemes to organize the world around them even if they were not taught. For example, they know the numbers, locations, and orientations of windows in their homes. They may also find out that they intuitively know concepts like "one-half" even if they have problems with fractions. And the method wants to build on these things.

Sorry to say but the memorization of windows in your apartment is not mathematics. It's pure memory and if one adds them up or does a similar exercise, it is at most kindergarten mathematics. Most people also learn the meaning of "half" and how to cut objects to halves but it's not mathematics, either. Mathematics only begins when these isolated claims and concepts are evaluated systematically.

Mathematics really says that \(5/7\) is as good a rational number as \(1/2\). If someone only learns to do certain things with "half" and not other rational numbers, it is a very clear sign that he or she is not thinking about the world mathematically at all. Mathematics is a collection of the logical relationships between concepts, objects, and patterns that may be applied in the most general sense. Being satisfied just with isolated notions about "half" or another number is not good maths.

At the end, people may know \(1/2\) more intimately than other rational numbers but if they don't understand that other rational numbers may be treated and dealt with analogously, they just haven't mastered basic mathematics. And the schemes? People may draw random schemes but what average children "automatically" do with schemes (of windows at home, for example) is not mathematics, either. Mathematics contains lots of relationships and laws that are not "completely obvious to everybody", that an average person "doesn't immediately rediscover", so if the average person isn't helped, he won't know them. And if he won't know the general laws and methods, he won't know mathematics.

2. Environments contribute to the accumulation of informal insights

That's the second principle. It isn't clear but they seem to say that the kids should spend all the time at home, in the family, in the bus, or in the staircase (building pyramids from cubes just like in the kindergarten) because they know these environments and they're not distracted by something else. What? Is that a joke or something?

Old environments have some advantages but new environments and new situations are often even more vital for the kid to learn something important. After all, the knowledge of mathematics is exactly something different from the memorization of a finite number of facts about the old situations and environments. Knowledge of mathematics is something that allows a human being to deal with new problems, environments, and situations. That's why mathematics is so far-reaching and its applications are so universal.

So this principle seems to be the exact opposite of the truth, too. It is very ironic – chutzpah – that these people want to claim that the "traditional" teaching of mathematics wants the kids to memorize things and not think creatively. But just the opposite is true. They explicitly say that they want the children to only memorize a finite number of facts about the objects and environments that they already know, like the number one-half, windows at home, and the staircase. Sorry, that's exactly what proves that mathematics hasn't been born in the mind of the kid yet. Mathematics is about the treatment of patterns that may appear everywhere so every attachment to a particular object or situation is counterproductive in the process of learning how to think mathematically.

There have been all the examples of children who were not able to solve a mathematical problem if animals were replaced by something else – even though the problem was mathematically isomorphic. Such anomalous problems largely follow from the fact that the children were not taught to think abstractly. They were not led to understand that mathematics applies everywhere. And constraining the children to a few well-known objects and environments is the "best" way to keep them limited in this sense.

3. Juxtaposition of themes

They say that the kids should pick their themes, all of them should be mixed, and mathematical regularities shouldn't be isolated.

Sorry, every single claim in this "principle" is just the opposite of the truth, too. Mathematics is about a sharp localization of some pattern, feature, or law. Once done right, the pattern or law is completely isolated from everything else, relevant information is completely separated from the irrelevant information, and the way how to do that in a particular situation is the whole arts of mathematics.

In the real world, idealized mathematical models of something fail to be fully adequate. Laws are inaccurate, objects and phenomena are mixing with others – but that's why these real-world situations are not fully captured by mathematics. The nature of mathematics doesn't change. By definition, it's sharp and its propositions are sharply true or false, and the reasons for something and the laws are isolated. If one builds a good enough mathematical model for a real-world situation, this isolation and sharpness largely holds for the real-world situation as well, but a kid who has never learned the idealized sharp mathematics can never see this fact. He will look at the whole world as if it were a complex pile of mushed potatoes governed by the laws of parapsychology, witches, superstitions, and consensus about them.

Different mathematical insights can't be replaced by each other so an effort to allow the kids to choose what they learn is just a different way to say that the kids are encouraged to ignore most of the material.

4. Support for the child's independent personality

It's a good principle, of course, but everything else that they have written shows that their claim that they support the independent thinking of the child is just a plain lie. Shockingly enough, they list quite some details of the complete destruction of the child's independent thinking.

This "principle" says that children – who are just officially learning mathematics – should nurture their right attitude to the problems of the society and the need for solidarity. Holy crap! Whether solidarity is right or wrong should be left to the child and even if it is not left to the child, such political brainwashing should surely get absolutely no time in the mathematics classes.

Most of the text in their description of this "principle" suggests that the mathematics classes have nothing to do with mathematics – they're the kind of postmodern discussions about everything and nothing where no one has to be right but the right group think with predetermined political opinions has to win at the end. Shocking trash that makes the education of mathematics during communism flawless and impartial in comparison.

5. Genuine motivation

They say that children don't have to get grades and they're always naturally motivated to learn, and all this stuff. It's just nonsense. Even children – like me – who were immensely motivated and in love with mathematics would find lots of things they found uninteresting at some point and were not motivated to spend too much quality time with that.

To some extent, it's OK. One doesn't have to learn everything and to the same extent. But it is very clear that the average children have much less motivation and they would effectively learn almost nothing, and the teaching of mathematics simply has to fight against this natural tendency not to learn anything.

That's why mathematics is a deeply unpopular subject among those kids who are not good at it. But this is inevitable and it's a sign that something is being done right. Children who suck at sports also hate sport classes but they're bullied and often afraid to tell you the truth – while the mathematics-ignorant kids are never bullied. They often determine the atmosphere in the classrooms. If everyone is OK with the mathematics classes, it proves that the teaching has been reduced to the common denominator.

The claim is that the Hejný method wants the kid to move in the environment that he or she knows which automatically encourages him or her to move further, and so on. It doesn't make the slightest sense to me. All people in [some countries where no one learns any mathematics] are moving in their well-known environments but they still do not know mathematics.

6. Building on real experience

This item says that they should build on some completely random, unspecified experience and order the kids to perform some random silly tasks like to sew clothes for a cube in order to learn how many faces it has. Holy cow.

Children of our or other generations would do such things during arts classes and no one has ever claimed that it was mathematics. Because it is not. The children who sucked at mathematics didn't have problems with that and often liked it. But it was not mathematics. I think that they pretty explicitly say that in their schools, mathematics as we knew it almost completely evaporates and is replaced by another class of informal arts. It's surely popular with many people (kids and adults) but it is deeply harmful for the society and its future.

Kids have many different kinds of experience but mathematics only starts when the experience starts to be organized using strict, rigorously, universally applicable laws, formulae, patterns, relationships, and methods, and the mathematical ones are ultimately independent of all the idiosyncratic experience of individual babies. As long as they don't "get" the universal message that is independent of their personal specifics, they are not "getting" mathematics.

7. Happiness from mathematics helps

It surely does but happiness may arise for various reasons which are sometimes more legitimate, sometimes less legitimate. These folks explicitly say that what they care about is happiness experienced by the weakest kids in the classrooms. They are the true "consumers" of the method.

But sorry, the weaker the student is, the less likely it is for his or her happiness to be caused by genuine mathematical insights.

The true happiness occurs from the very same general realizations and patterns that the weakest students usually fail to get. And it is this genuine kind of mathematical happiness – as imagined by the students who actually like mathematics – that should be supported.

This principle includes some demagogic metaphors. Children who know formulae are labeled as "intellectual parasites". A Russian psychologist is mentioned as the authority to support the label. Holy cow. People with these opinions about formulae and those who know them want to conquer the education of mathematics at our basic schools! Flowers can't be made to grow faster by pulling them from the soil each morning, we are told. Well, I actually think that even this statement is invalid and gardeners would confirm that helping a plant to fight against Earth's gravity does help its growth. Higher CO2 levels surely help plants to grow and there are many other ways that may be looked for.

But their point is that the kid should never be asked to do more than what he wants to do completely voluntarily. This principle is completely wrong. There's lots of mathematics – and the amount keeps on increasing – and the kid must learn some portion of it and it's almost always more than what the average kid wants to learn voluntarily. Not to exert any extra pressure means to give up the efforts to teach mathematics to children.

8. His own insight is more valuable than an adopted one

Right. Except that almost no children are new Pythagorases so if the discovering of mathematics is left to themselves, they will know almost nothing.

They mention some intuition for integrals and areas and something like that and it's fine and important for a truly intimate understanding of mathematics. But these insights are completely random, don't follow from any broader pedagogic picture, and have nothing to do with the "children's own insights" because they're still something that the children have to be told.

And there's always lots of potential misunderstanding of mathematics by the very proponents of the method. So much of this informal mathematics that they want to teach is just wrong – either literally wrong or morally wrong in the sense of encouraging the children to think incorrectly about closely related situations.

9. Teacher: a guide and moderator of discussions

Wow. If you just leave a random group of children in the classroom for an hour, they won't learn any mathematics. If the teacher is competent, his or her knowledge of mathematics (and ways to explain it) is so much better than the knowledge of the average student that his "leadership" in the teaching process has to be by an order of magnitude faster than if the children were relying on themselves.

There may be incompetent teachers or students who are actually better than their teachers. That's another thing. The latter may play some role in teaching their classmates, perhaps. But one can't rely on such circumstances.

The average kid can't be considered the "boss" of the process, while the teacher would be just an assistant, because he or she is just incompetent to be a boss. If someone doesn't know how to approach a problem and guesses, it's not just about saying "Yes" and "No" to direct the thinking. Without knowing a right approach, one may be forced to make a thousand of guesses before one would be right – and the kid wouldn't know why it is right, anyway.

In this item, the proponent explicitly says that "the teacher is not explaining anything and he is not preparing the classes". Wow. Just to be sure, if someone memorizes an explanation, he may present it more quickly than what is comprehensible to a typical listener. So this is a problem one should be aware of. But the right approach is not to abandon preparations and explanations altogether. That's surely throwing the baby out with the bath water. Prepared, optimized explanations and proofs are still far more likely to be understood by a kid than unprepared non-explanations.

One of the promotional slogans is that kids taught by Hejný's method are ultimately led to a theorem and/or a proof when they were rediscovering a "piece of mathematics", but they do so in a way that is acceptable for them. That may sound nice but it's really a huge fundamental flaw in the method simply because almost all the variations of the theorems and proofs that children normally do are wrong. So tolerating this diversity is nothing else than tolerating wrong mathematics and sloppy thinking. Some diversity of people's thinking is desirable – and ultimately unavoidable – but it's harmful to artificially promote this diversity for the price of allowing wrong thinking.

10. How to work with mistakes and eliminate fear

Errors are natural and teachers should praise them because mistakes are activating thinking. Let's not point out errors but their reasons.

Those things may sound nice but in practice, they are unusable. If the teaching isn't sufficiently rigorous, wrong answers and methods are far more likely than correct ones and the exact causes cannot be localized. The exact cause is that the kid just doesn't have a clue how to approach the problem correctly! That's the cause. What do you want to do with that? How do you want to address the cause of the mistake?

In some sense, this principle encourages to treat wrong solutions and right solutions on equal footing. But that's simply not how one may have a chance to learn mathematics because the number of a priori possible wrong answers is vastly greater than the number of right answers. That's why the teaching must always be based primarily on positive insights, how things should be done.

Also, an important point is that while some mistakes are shared by many children, many others are specific and only comprehensible to one child or a few children. You simply shouldn't spend most of the class with a mistake of a particular child. After all, it is a mistake so there is no fully coherent justification why one should think in this way. A mistake of one child will probably look silly or incomprehensible to most other children (and the teacher). Only mistakes that are "sufficiently widespread" may deserve a special treatment.

And praising children for mistakes? What is the goal? It's clear that if a kid is praised for something, he or she is encouraged to do it. Why would one encourage mistakes? They would produce an even larger number of mistakes. At the end, the child is almost inevitably lost in the infinite forest of possible wrong answers. It's much better if he knows how to find the right one even if he or she doesn't quite know why the method works. He or she may learn it later.

11. Adequate challenges

Different children will have different success rate and that should be tolerated, I agree with that. I don't have any problems with the text of this principle but I think that it morally contradicts everything else written as a description of the method. Everything else suggests that there can't possibly be any time left for real mathematics as understood by the best kids in the classroom.

At the end, as long as the children are solving some real problems, however, they will see who is better and who is worse in mathematics. This unavoidably increases the self-confidence and happiness of some children and does the opposite to others. This outcome is right and a necessary sign that something is actually being learned. This outcome shouldn't be avoided just because it's bad news for some children.

12. Support for collaboration and discussion

It may be great if children cooperate, discuss, share their thinking and methods. But at the end, to solve mathematical problems correctly means to perform an intrinsically individual activity. One may hear solutions or explanations or arguments from others but he or she must always reprocess it for these things to sound logical to the listener.

One may also use partial results obtained by someone else and trust them. But what we do with them is again a mathematical operation done by an individual who uses some assumptions and produces some answers. What is important at the end is that the truth in mathematics has nothing to do with the people per se or their discussion or consensus – it is done objectively, impersonally, externally.

And I think that even if such recommendations of discussions etc. may sound innocent, their ultimate goal is to make the children think that mathematics is just another kind of vague politics where people are looking for agreement with many other people. That's exactly what mathematics definitely isn't which is why I tend to think that even this last point is extremely counterproductive.

As you can see, at least 10 or these 12 principles seem almost completely unacceptable to me. Similar programs to "teach mathematics in new ways" are postmodern sleight-of-hands designed by people who don't really understand mathematics, its particular content, even the meaning of the word and reasons why it's important, and its relationships with the real world themselves and who are doing nothing legitimate to validate their hypotheses about the success of one method or another.

Indeed, the "progress" in children's mathematical abilities during the last 20 or 30 years has been demonstrably negative even though it's pretty much the same kind of people who are self-described experts in teaching (and not only self-described; there are whole schools that produce such "experts" and give them degrees). This is not science. They are not real experts and it is shameful to call them experts. Instead, they are co-responsible for the continuing degeneration of the Czech nation and perhaps the human society. Almost no really positive changes have been made. If one restored the textbooks and methods used in the 1950s, it would probably dramatically improve the children's knowledge of mathematics.

A TEDx talk by Prof Hejný, in Slovak language (and Slovak capital), with English subtitles. When I quickly checked the video, it's about claims like "a girl has an A from everything except a B from maths". It must be because she is "stubborn", we learn! Can't it be that she is just worse in mathematics? Or is there a law of Nature that the grade from mathematics can't be the worst one? One may be good at mathematics and bad at mathematics if she is stubborn or if she is non-stubborn. These things are independent. One shouldn't get an A from mathematics for being stubborn.

Also, he repeatedly says that children are taught to imitate and reproduce, to parrot. But the actual reason why a good schoolkid says the same answer to a question is because it's the right one and mathematics is rather decisive about the question what is right and what is wrong. When people are doing mathematics right, they just end up with the exact same answers to most well-defined questions. And the task of teaching mathematics is to make the child able to get the same answers that will sound like he or she is parroting if the same problem is discussed by many but he or she will be able to do it in millions of other situations he has never heard of, too.

There may exist several or many ways to get to the same final result. Two ways to solve a problem may differ by local details or conceptual differences. There are multiplicities. But it's still true that "what is the right solution" and "what is the wrong solution" are sharply separated sets and most randomly proposed modifications of a solution will end up being wrong. In practice, the derivations done by many people may look almost exactly identical simply because there's often not too much freedom. There is nothing wrong about it. It's a symptom of mathematics' being well-defined and objective.

A huge portion of the talk (6:30-10:00 or so) seems to be about how to add 2+3. You shouldn't add 2+3 apples. You must add 2+3 steps and the other kids have to slap their hands. It reminds me of some 1995 masses at a charismatic church/sect of an ex-GF. ;-) 2+3 steps, slap slap slap, wrong, once again. An introduction to dancing classes but not mathematics. (I have nothing against slapping and dancing and screaming in the class. It may help children feel alive and breathe. But one should still recognize that it is not the main goal and most of this screaming and slapping is likely to be useless for the learning process and much of it may be very distracting, too.) OK, one girl made 2 steps and then 3 steps, and the second girl is supposed to catch up with the first one, so the kids have to decide how many steps she has to do. Someone says 4 so of course, the wrong answer is preferred and she makes 4 steps. Some of the kids think that she is not far enough, thus disfavoring the answer 2+3=4. But it looks pretty close, especially because the steps don't have to be equally long.

So that was an example how to teach that 2+3=4, or 5, the result isn't really important and the kid has the right to believe whatever she wants and preserve her personality. A. ;-) No, sorry, more seriously, it's just another great example why the repetition of the right answers like 2+3=5 is important and it's totally wrong to be against it.

He also claims that children don't know why they say that a cube has 8 vertices. I don't believe that. How many children aren't able to count vertices of a cube after they are told it has 8? To verify it? If they can't verify it, they can't. It's still useful that they're exposed to some facts, like "a cube has 8 vertices", earlier than they would be without any "leadership".

I agree with him that the steps are better than apples for explaining negative numbers. But the steps are just the real axis and I am confident that it is in principle how negative numbers are explained, anyway. It can't be done otherwise. Positive and negative numbers form an additive group – isomorphic to the group of translations. So of course that every visualization of a number with a sign is some translation, some change that may be done in two different directions. Steps back and forth or profit/loss are two real-world examples.

It is very clear that when he recommends to constantly praise the children with all kinds of partly or completely wrong, sloppy, idiosyncratic answers and solutions to the problems, there are some children who won't be praised. Those who happen to approach the problems in a mature, correct, rigorous way – the way identical or similar to the way how competent mathematicians would approach the problem. Those children will be dismissed as parrots. But they could have rediscovered the standard method independently as well and they're most likely to be the best ones because the standardized formulae-based approaches are the result of centuries of careful refinements.

There is ultimately some answers and solutions and even some methodology – or a few methodologies – that are vastly better, more reliable, more universal etc. than what the random alternatives that random classmates will propose. Hejný's whole method seems to be about the equality of the superior and inferior methods and sometimes even answers (and sometimes even about the superiority of the less talented kids' approaches), and that's just wrong, wrong, wrong because the whole goal and power of mathematics is about the very sharp discrimination of answers and propositions that are incorrect and those that are correct.

While I have listened the TEDx talk above in its entirety, I could only withstand a few sentences at random places of this 80-minute long 2014 talk (in Czech). When I heard that saying that the area of the right-angles triangle is \(ab/2\) is not knowledge or an insight but just a "prosthesis of knowledge" (which he opposes), I couldn't go on. This is just so incredibly stupid. The whole point of mathematics is to produce similar "prostheses" of everything, to see abstract and universal patterns and regularities in almost everything, and those can and should be described by symbols. The symbols represent the patterns and features that millions of situations have in common. One may use different symbols (which is usually just creating havoc) but until a child develops a way to think about these different situations in a unified way, being aware of their being isomorphic, he or she is simply not thinking mathematically and he won't be able to apply mathematics in any new situation, either. It may take time for a child to realize the actual connection between properties of real objects and mathematical variables (expressed by letters etc.) but the fact that this realization is nontrivial doesn't mean that it could or should be avoided in the didactic process. It is absolutely essential.

The area of that triangle is \[

{\rm Area} = \frac{ \text{firstleg} \cdot \text{secondleg} } { 2 }

\] and we just replace the words by symbols, \(A=ab/2\), to save time. That formula may be proven by cutting a rectangle to two equal parts along the diagonal. Is that hard for a kid? It is hard to know why. It may be hard, anyway. But it is important. The general formula expresses the understanding of the general problem. It is a template to deal with lots of isomorphic problems – as a template, it is indeed a "prosthesis" but "prostheses" are not bad, they're the very point of mathematics. The "prosthesis" is a good word because it correctly conveys the point that we are distilling the "relevant idealized mathematical essence" of the problem while omitting all the dirty real-world "flesh". That's indeed what mathematics does. If you demonize these "prostheses", you are missing the whole raison d'être of the mathematical reasoning. Mathematical reasoning is not just some fuzzy collection of random experiences. It is a rigid skeleton reaching almost everywhere, and the more rigid and organized it is, the more it deserves to be called mathematics.

Another discussion in the talk above was about a special problem involving one-half of one-half of a pie, or something like that. I would say: a special problem that may be solved by some special arguments that are not usable for other fractions. And he wants to promote such special ad hoc arguments. But that's just wrong. People may find the right solution of similar problems but as I have already said, there is an actual huge class of problems that are equally easy from a mathematical viewpoint, and the kids simply have to learn the general formulae or algorithms because a quarter of a pie is not the only fraction they will need in their life, especially not if their life will depend on mathematics.


  1. They are the kind of people who are "proud of not know mathematics", I hate them from I was a child, among other things, because the reverse doesn't exists, there is no people "proud of not know History" (or Literature, Linguistic, Art, etc, for the matter). I understand that not every one has the capacity for abstraction to be exposed to Mathematics (or Physics) in the childhood, but this fact can not become an excuse to prune educational content (never has been). Also you can always find a way to make math attractive to kids, especially in the age of information and digital content ... for example, in the case of the theorem of Pythagoras, check out this video, it is impossible not to understand:


  2. Exactly, the pride about ignorance is only directed at mathematics (as a pure science and as a tool to do other things). Mathematics is being selectively bashed like no one else.

  3. Of course it isn't, but you can bet that people not inclined to rational thinking (i.e. abstraction, logic, etc...), will understand what the theorem says. There will be time to teach the subtleties.

  4. Right, while it's not a mathematical proof, it's still something that has a big power to convince the people that it's right, and it's really enough in many cases.

    Most people in their lives will be enough with some circumstantial or experimental "proofs" of many things. These possible "experience-tainted" ways to teach mathematics may be underrated.

    However, I still think it's important that kids know by the end of the basic school what the Pythagorean theorem says. Comments that such formulae or classes exposing them are inevitably boring are just proofs of someone's mathphobia.

  5. Sometimes I wonder, if this "New Age" nonsense is a phenomena rather limited to the "Western" civilizations such as US/Europe ...

    I often get slightly the impression that in other parts of the world, knowledge of deep things and good education are more appreciated (by those who can get it), knowledgeable and bright people are more respected, etc (?)

  6. I fear the same, I get that impression too, specifically in the case of Indian (well, South Asian people in general) schools and colleges, I tend to think that they are inmunes to New Age-ism, maybe because they circunscribe it to religion and spiritualism.

  7. Throughout the US they use these new mathematics teaching methods, while our students test scores keep dropping compared to Asian countries and other places where math is taught correctly. I once read a report from a math professor at a college in our state of Washington. He said that the state started using a new math teaching method in 1998. Each year thereafter a greater percentage of students needed to take a remedial math courses when entering college because of their poor math abilities.
    By 2010 very few students benefited from the remedial math courses, because their ability to learn math had been permanently damaged by the crazy teaching methods that had been used in their prior schooling. Essentially, the only students who did well were those who had received private tutoring.
    I think that these types of problems are caused by our liberal elite's desire to proclaim everyone as being equal. This obviously is not the case in sports-or in anything else. Surely, not every student should learn math beyond simple arithmetic. There are many algebra students who never really learn to add and subtract signed numbers, But we should not "dumb down" all students to the lowest level which seems to be the desire of our liberal elite in their efforts to promote total and complete "equality".

  8. It started in 1917 with the praise of an upside down urinal and deification of Duchamp who signed it to rebder it into fine art. Steven J. Gould's wife carried on the tradition by fractal analyzing Duchamp's spaghetti strings thrown onto a canvas. Likewise the mirror neuron Nobel prize winner went on to claim abstract art was real by comparing it to truly boring control images instead of using a proper controls of randomized versions of the same "art" styles. This all prevents both little kids and adults alike from declaring that the emperor has no clothes.

  9. Bucky Fuller condemned the Cartesian convention of turning everything into squares, so I hereby present a triangular version of the theorem:

  10. lubos:
    I made my living for while teaching math that I learned as a child, in elementary school to college graduates so they could get passable scores of their GREs and GMATs...
    I also taught statistics to undergraduates (primarily analysis of variance) for the better part of a decade
    I believe that in order for kids to learn math:
    1) most must be taught by someone who knows math...in the US, this is simply not the case...most elementary and secondary school teachers' grasp of mathematics is abysmal...simply abysmal.
    2) the curriculum must require mastery of the subject. If it can be avoided, most students will do so. Here many districts do not require 4 years of mathematics in high school, the result being that not only are colleges now in the business of teaching "college algebra" (whats that?) but also offer "pre algebra" I assume thats the stuff I learned when I was ten years old or so.
    Our culture (here in the US, at least) celebrates ignorance...
    who needs to learn about greek men? After all, there dead, and white too.
    People here believe that Columbus proved the earth was "round" forgetting that the greeks knew this, as well as its diameter thousands of years ago.
    people here are ignorant, and proud of it too.
    so, I dont think mathematics is the only field of culture into intellectual endeavor that is under attack.
    Historical knowledge-the ability to put our our culture into the context of human history is derided students here dont know who "we" fought the revolution against...they dont know where the pacific ocean is...cant locate europe...
    dont know who fought in our civil war...our ignorance is astounding, as is our pride of it.
    sad, really sad.

  11. Prof. Hejny never said in the interview, that he would postpone learning Pythagorean theorem until college. He just mentioned, that he showed to his math pedagogy students, how to re-derive it for themselves on elementary school level. Actually in the very next answer he explicitly said, that the only thing he wasn't able to let elementary school students to re-derive for themselves, was irrationality of \pi.

    I'm not going to discuss Hejny method with you. From what you wrote, you obviously don't understand the method, but has the audacity to claim, it's bad, so you're basically a crackpot, regarding Hejny method. And you, as physicist, should know, it makes no sense, to discuss physics (or in this case Hejny method) with crackpots. However, I will leave short comment to think about: I organize math competition for elementary school students. The students, that use Hejny method are, as a rule, among the best competitors, often beating the best students from best math school in the country. Similar results are for Klokan competition, for whole classes. So whatever logic makes you feel, Hejny method doesn't teach you any mathematics, it's flawed.

  12. I don't think this is pseudoscience. I think it is better described as insanity.

  13. Let me first thank you for this blogpost. As a parent I will surely do my best to teach math to my child and to prevent it from being demaged by perverts of Mr. Hejný type (including equaly wrong new handwriting see http://cs.wikipedia.org/wiki/Comenia_Script).

    However, since you have already done the very important work of pointing out the mystakes and errors, I would like to offer something which I consider a good example and still not a mainstream one - this is an excelent book by Paul Lockhart on elementary mathematics:

    It goes as far as explaining calculus and I believe the book can be recommended for interested basic school children.

    Interestingly, Paul Lockhart wrote a lament on mathematics teachhing too :D
    see e.g.


  14. Call it what it is, Lubos: the dumbing down of math. And do realize that this dumbing down is being done deliberately in order to create wage depression among American/Western STEM workers. That way when American/Western tech companies hire foreign workers, mostly from India, to do STEM work at a fraction of the cost, they can simply argue that you really don't need to be all that smart in math to do STEM work.

    Why do you think top tech CEOs like Mark Zuckerberg support Obama's version of immigration reform, and in Zuckerberg's case, to the tune of $50 million (see link below)? He'll get a lot of low-wage, yet highly-skilled STEM workers out of the deal.

    Now if these tech CEOs were really among our so-called best and brightest, they would know that most of what they do in the corporate suite can be automated, freeing them up to do the more intellectually challenging work that STEM workers do. But they’ll see to it that this never, ever happens because the truth is that they really aren’t among our best and brightest. They really aren’t smart enough to do STEM work and the money is just too darn good for them to ever admit that their job in the corporate suite can be replaced by a few finely-tuned robots, developed by their STEM workers of course.


  15. Dear Dano, could you please show me a specific URL describing the math contest, the problems, and the contestants who used this method?

    I think that the irrationality-of-pi proofs are clearly college-level stuff, to say the least. This is way beyond the difficulty of other things he is questioning, like the addition of the fractions.

    Thank you very much

  16. Nice post. And I am curious what are your thoughts on talented kids being required to do a lot of math "busy work", e.g. pages of problems on adding fractions. Does it instill good discipline and concentration skills? Or can it ruin a child's potential passion for mathematics?

  17. In other words, maybe the Asian nations are better than the Western nations in figuring out that the "New Age" actually means "Very Old Age", when it comes to most aspects of it. ;-)

  18. The silver lining is that the "hot trends" in Czechia are almost the same as those American ones you describe, so we're happily a part of the Western civilization with all this great influence of the "liberal elites", or the Prague Lumpencafé, as they're sometimes called here e.g. by the current president. ;-)

  19. It's the same here. The incompetency of the basic school teachers is obvious. If they were competent, most of the problems would be non-problems. Like the problem that some textbooks don't really allow children to understand why the Pythagorean theorem is true.

    Now, a normal reaction by many kids in this situation is to ask: Wow, why should such a strange identity (Pythagorean theorem) be true? And the teacher may know some quick answer that would satisfy them, or prepare it for the next class where she spends a minute by answering the previous question. It's not a problem. And of course that not everyone will understand any proof. The proof is arguably harder than the claim itself.

    Some wrongness in the existing "boring classes" may be that the teachers have memorized something that they don't understand themselves. But in that case, one can't fix it by being more playful. It's a hiring/personal problem. If someone doesn't have the knowledge or experience or intelligence, she won't be able to bring the heureka moment to the kids in a boring way, just like in a playful smiling way.

    Otherwise, I could dismiss the humanities, but when it comes to the old insights about Nature, the ancient Greeks and Romans were just great and if we were copying their arguments for many of these things that haven't been disproved, it would be superior. Like your round Earth examples - which go back to the ancient times.

  20. Cynthia, a very interesting theory. Why is it being done? Who benefits if STEM jobs would be selectively outsourced? I am not sure it's actually happening to the U.S. already now because the STEM education is still the kind of activity that should be superior in the West - that's really a primary reason why West got so ahead of others, isn't it?

  21. "Allah-sponsored engine" LOL.

    Yep, there should be a "minimum's idiocy state" (and "maximum's intelligence state") that ultimately make similar all human in the IQ's extremes, or, let's say, that idiots (and geniuses) are qualitatively similars everywhere ;)

  22. Dear Brian, there are diminishing returns and it's obvious that torturing anyone with some repeated mathematical operations that he or she has in principle mastered is counterproductive. If a child needs to get faster in that, he will become faster by training, but there's no point in trying to do it preemptively. Doing things like that well should be a side effect of someone's work on something that is motivated by something else, not a goal.

    I think that there are certain mechanical operations that everyone may master and this torturing by mindless mechanical exercises is actually being done in order to punish the more talented kids, too. I don't want to enumerate my relatives etc. ;-) but it is the general idea of mathematically illiterate people about mathematics. They think that a good mathematician is someone who learns to do some 2nd-grade mathematical operations quickly or often, as an animal in the circus.

    But of course, mathematics actually becomes interesting - and mathematics-like - when it gets to completely different, more abstract and conceptually complex things that are not understandable to the 2nd graders at all.

    However, my experience - not just my personal one but especially the personal one - suggests that passion for mathematics can't be "killed" by torturing a child with some repeated mechanical drills. The child may still recognize what he likes and what he finds annoying. So if some mechanical drills are boring and he likes something else in mathematics, he just ignores the mechanical drills just like ignores a completely different subject at school.

    My point is that while there is a big correlation between a child's attitude to different topics taught in mathematics, the correlation is not perfect. Moreover, the correlation is a consequence of the similarity between the pieces of maths and their dependence on the same traits of the child. However, this correlation cannot act as a cause that damages the image of one topic by another topic.

    Take a different example. Sports. Someone may hate jumping or runing but he likes soccer and swimming, or vice versa. I think that he still distinguishes the different things. You won't make a kid hate soccer just because he is forced to swim, will you?

  23. For me real problem with any novel method is how does it work in real average situations. The method may allow gifted students better realize themselves but fail spectacularly with average ones. The method may flourish with good teachers in test classes but be harmful when employed with average teachers.

  24. On the one hand they are done to ensure that kid understand the stuff corectly but IMHO the discipline and concentration you mentioned are actually important aspects that are often overlooked. I knew some people who not only couldn't force themselves to do any nontrivial calculation but even if they did they made mistakes at every step. Even though they were very clever guys they still failed because of it even using Mathematica. After all it's not fun all the time sometimes you need to do unpleasant work

  25. strictly speaking...Dec 14, 2014, 8:27:00 PM

    Using squares to demonstrate the Pythagorean theorem is useful because it explicitly illustrates algebraic squaring.

    If you just want to show off the theorem up to a constant of proportionality, my preferred way to do it would be to just draw a height to the hypotenuse from the opposite vertex, and observe that it divides the triangle into two triangles that are similar to the first one.

  26. Derek Muller did a PhD in physics education, and suggests:

    "Luckily the fundamental role of a teacher is not to deliver information, it is to guide the social process of learning. The job of a teacher is to inspire, to challenge, to excite their students to want to learn":


    Lubos, have you ever thought of using http://www.patreon.com/creators/ to get regular funding for your blog from the public?

  27. Similar to Patreon:


    Though they have a strong limitation: "[you] can not collect any more than $5 million total dollars annually" ;-)

  28. For your last question, I think there are at least three reasons.

    1-) Even in computer science usually the most rich ones are not the best programmers but the ones who can successfully turn it to business. This is a different kind of skill than STEM skills. Of course there are people with both skills, I think founders of Google and Oracle are good examples.

    2-) This is just from my personal experience, but most of the STEM people are shy ? I am using this world in general sense, in the sense that they can't influence people easily and communicate with them easily. Managers are much better in this regard.

    3-) It is hard to measure a STEM guy's productivity. A CEO can say that I managed this companies and got this profit. But it is much harder to measure a STEM guys productivity. First, what they produce is usually mixed with what other people produce: even there is a successful product it is generally product of many people so it is hard to measure each person's contribution. It is obvious that CEO's don't have this problem. You should be able distinguish yourself to get your worth. Second reason is that it is hard to understand quality of a STEM work. Now I am sure there are people who can solve this two problems, but in market most of the managers can't so this lowers STEM people's salary.

  29. This is happening because more and more money and power is shifting to management and away from labor, Lubos. That's all because management has come to be viewed as an extension of capital. With the corporatization of healthcare, you are already starting to see this happen within large hospital systems. Hospital administrators are now being paid more than the top hospital surgeons. Some will argue that this wouldn't be so had surgeons not become hospital employees. But this argument won't go far because the same thing can be said about hospital administrators: they too are hospital employees. It's rather arbitrary and thus illogical as to why hospital management is viewed as an extension of capital, while hospital surgeons are viewed as a labor cost. If anything, it should be the other way around. Unlike hospital management, hospital surgeons bring in money to the hospital, making them a natural extension of capital. Before you accuse me of being a Marxist, please hear me out.

    Let me start by stating the obvious: people in management draw a paycheck from their employer just like those in the labor force do. In other words, those in management are working stiffs just like people in the labor force are. The only exception to this are the people in management who own the company outright or have controlling shares of the company, but the vast majority don't.

    The common thread here seems to be the corporatization of the workplace. We can't do much about undoing corporations, they are here to stay, but we can change corporate culture and how we view everyone who works for corporations, from the CEO on down. If we don't, one of two things is going to happen, both of which are detrimental: 1) labor is going to rebel and management will respond by installing a militarized police force within the workplace or 2) management will become so clueless as to how their products are made or how their services are delivered that they drive their company into the ground. I prefer the second outcome. That way labor can step in and take over management. Then they can put management on a level playing field with labor, saving the company a lot of money. Or, they can simply turn management into a robot, saving the company even more money.

  30. It is simple. Imagine a Medieval property.

    Whenever peasants complain the aristocrat overlord removes heads from some of them and/or rapes their daughters.

    Unless he pisses off enough peasants by killing too many and/or raping too many, what are peasants to do?

    They will keep working in the fields and other aristocrats may congratulate their guy on good management skills.

  31. I see your point 2 as a very polite statement that the power is taken by those who are more ruthless.

  32. My daughters' schools didn't do anything this crazy, and they even did a few things right, but I still find much to criticize in how they were/are taught math. The schools in our school district do test the kids and place the more advanced ones into math classes one, two, or - in exceptional cases - even more than two years ahead of the classes for their age group. That is the good part. The not so good part is that even if you are placed ahead of your age group in a math class that is not well taught, you are still exposed to inferior math education, just sooner rather than later. The big problem is that math teaching here is too computation-oriented, so they are not exposed to enough proofs, which not only results in inferior mathematical proof skills, but also completely hides the beauty of mathematics, so even the children with an above-average ability in math it may end up disliking it.

  33. Do any of these education wizards have any children of their own? If so, I pity their kids, but their crazy theories suggest that they've never actually been responsible for making sure a child learns. There's _lots_ of repetition as you try to keep the focus on the important bits. When the kid is eager to learn it's wonderful, but one of the big problems of being incompletely educated is that you don't know which parts need to be mastered first.

    IIRC Plato thought you didn't teach a child, you just guided him to remembered what he already knew. So goofy theories are nothing new, but I don't think the Greeks tried to turn that one into a national program.

  34. Seems like a great program for a big brother society ruled by aristocrats. Keep the sheeple happy and turn as many children as possible into sheeples. Seems to me somebody must have a program : the elite's children will be going to special schools of course.

    Usually I do not like conspiracy theories , "What can be explained by human idiocy needs no conspiracy". But your mention of support by a billionaire makes me suspicious. All those "club of this and the other".

    And some personal experience:

    Up to the age of 14 I had a very bad math teacher ( a lady, widow of an officer killed in WWII) in the gymnasium. The result was that I thought, as they do, that mathematics was a parrot business of hit and miss answers and my grades were around C.( At that time I was reading a book a day from the school library). Fortunately the third year of gymnasium we got a great teacher. I really had an experience as Paul on the way to Emaus: the scales fell from my eyes and I was enamored with maths ever since and grades went accordingly.

    It is a pity, all these mathematically inept people making mathematical curricula. My daughter spent third grade in a public school in Switzerland where they were into "new math" sets etc with examples with animals etc. When we came back to Greece I had to get her a tutor to reach the level of her class at home, long divisions etc.

  35. Dear Mikael, yes, I think that in principle, everyone having access to the full initial state may probabilistically predict the observations or perceptions done by any observer with any fate.

    It's just important that these observations are different in general and they can't be unified to histories involving all of the observers simultaneously. It's the usual complementarity.

    If an observer (already) inside the black hole does another observation, you may say that he collapses some observable - effectively replacing the wave function on the black hole microstates by something else. But it's still important that these observations and collapses - split in consistent histories happening inside, if you wish - are not juxtaposed with the splits done outside. In this way, you wouldn't get a set of consistent histories.

  36. Dear John, I agree with those things.

    Cynthia plans a typically working class' warfare against the managers etc. It's silly. Of course that they get better paid than technical STEM people because they are ultimately doing almost the same thing as full-fledged owners of companies, and many of them could be ones, so of course it's similar. Even their incomes are. They're responsible for the money of the company, at the end, so of course they must also get a significant compensation for that job.

    And STEM folks are shy and the work is hard to quantify in the money.

    I just think that avoiding mathematics doesn't make one a CEO, does it? At the end, I am convinced that even today, the average income of those with math/technical training is higher than the average salary of all those without it.

    When I was asking about the selective STEM outsourcing, I wanted to compare STEM workers with non-STEM workers who were getting similar salaries before that.

  37. Dear Anna, long division is fun. Maybe we need less of it because everyone has calculators, and so on. But we clearly need to go in that direction.

    The real difficulty for many children is not torturing them with any particular fixed thing, like long division, but the very process of going further and further in complexity and abstraction.

    Sets, intersections, unions, they are great, and so is mathematical logic. But they are skeletons without beef and at the end, they are extremely simple skeletons. It's true to say that logic - and perhaps set theory - is some kind of a basis or the basis of mathematics. But just the basis is just too little.

    Moreover, it's questionable whether a child can really get the point of all these things. Adults may understand why set theory "contains all possible mathematical axiomatic systems", or something like that. But that's a rather abstract insight. I think that when I was exposed to this playing with sets etc. at age of 7 or something like that, I thought that it wasn't mathematics.

    It's similar like the abstract notions in physics about energy conservation etc. It only starts to make sense after the child accumulates some material so that the general usefulness of the concepts like conserved quantities may be appreciated. Incidentally, here I am saying similar things as Hejný. He also wants the generalizations to be found only when it's needed.

    Except that a vast majority of the children can never induce the generalization correctly if they are not carefully led. Even rather smart kids - and even college students - may need to be told what is the right generalization of something, otherwise chances are too high that they make something incorrectly. So I don't believe that any of these things may be done without leadership of the teacher, her will to interrupt the student etc.

    The defect of the set-theory-based thinking is that the creators brutally underestimate the number of levels of abstractions that can be built in mathematics, even within set theory.

    Try to understand path integrals, just for fun, in set theory. At the end, path integrals are as "computational mathematics" as you can get. But to even describe the objects that are computed, one needs to compose sets to create integers; rational numbers and then real numbers; functions of real numbers; functionals of real-valued functions, and summing of infinitely many values of these functionals over all possible functions. And in real mathematics, one may make every space in the middle curved, nonlinear, add indices with multiplicity of objects (several variables, several functions), and so on.

    And while all these objects look just "huge", there are completely different ways to look at the whole object from a different angle where it seems very "elementary". Take the relationships between number theories and zeta functions in the complex plane, millions of things like that. Clearly, an overwhelming majority of the time one has to spend is about something else than just learning sets and their intersections and unions. The real mathematics doesn't really start there yet.

    I would have very mixed opinions about all technical questions like whether geometry should be taught synthetically or analytically, and at what age. Vectors or how to construct something with compasses, and so on, and so on. These are interesting questions and some diversity will always be there. But I think that this Hejný stuff is a way to avoid all sufficiently advanced, abstract, and rigorous mathematics, and if this were done in a whole nation, it would be pretty tragic.

  38. Hejný has a son, and maybe other children, and the son was exposed to this teaching much like Hejný himself was taught by his father.

  39. "a lot of people"... indeed, which means not everyone in social science hate math.

  40. Just a comment apropos "10. How to work with mistakes and eliminate fear"

    [N.B., I'm not responding in opposition to what was written under this subheading.]

    Berating of children for having failed to think logically in a way that produces a correct or workable result (or for logical - not just mathematical - mistakes) will do nothing but harm. Such behavior by 'evil' parents/teachers will tend to condition-in a fear of (and a reflexive reluctance to/avoidance of) asking for help with such intellectual hurdles (or barriers to intellectual growth), not least mathematical ones.

    Speaking entirely personally - though still largely on topic:

    I have since long perversely praised myself (being a narcissist since birth) NOT for having fertilized and grown my mathematical faculties but FOR having managed to arrive at a (nevertheless) relatively rational, science-aligned, effectively philosophy terminating, enjoyable (partly teasing in tone), fairly philanthropically and (potentially) politically oriented outlook on mainly (but not only) ourselves.

    I arrived at this -- what I flexibly refer to by a diversely derivable (except not from anything unflattering) string (of capitals) -- outlook less by a tactically tweaked and employed "Principle of Tolerance" than by perceptively allowing Mother Nature's most fundamental 'method' to be played out; Namely, her method of "error plagued trials".

  41. I'm not proposing class warfare in the workplace, Lubos. What I am proposing is that labor and management be placed on a level playing field. That way both sides of the workplace divide can work together as a team to improve product development or consumer services, ultimately making more money for the company. Now if management doesn't agree with this, that means they really don't care about making money for the company. They only care about making money for themselves. If that's the case, let the STEM workers use their skills to replace management with a robot.

    And it's simply not the care that the technical skills of management are equivalent to that of labor. Believe me, my manager lacks the technical skills to do my job. She might have some of the technical skills that I have, but she doesn't have enough of these skills to function as a competent employee. That's true for most of the managers where I work.

    Regarding "management being responsible for the money of the company," often time when management mismanages the company's money, they don't get fired. Sometimes they even get a raise or a promotion. This wouldn't happen if we still had a true capitalist economy. What we now have is an economy that's based on crony capitalism. The other day I heard Carly Fiorina describe what in happening in our healthcare system as crony capitalism. She's right, crony capitalism is undoubtedly rampant in healthcare, but it's rather hypocritical of a crony capitalist such as herself to accuse others of engaging crony capitalism. Recall that it was Carly Fiorina who nearly ran Hewlett-Packard into the ground and walked away from the disaster richer than ever.

    Regarding outsourcing, companies do this for one of two reasons. They either do it to reduce labor costs, or to avoid the headaches of managing employees. But having another company take over your duties of managing employees only adds to your management cost. This is something that has always bothered me about American businesses -- they often fail to see management as a cost center. At least this is true in the hospital industry.

  42. Dear Cynthia, employees and managers in pretty much every alive company work as a team but that doesn't mean that they're getting similar salaries.

    It's not true even in soccer *team*, check the diversity of the salaries in FC Barcelona now:


    I assure you that if managers could be replaced by robots, without a loss, the change would have already taken place.

    Similar suggestions of yours are the standard Marxist utopia crackpottery. The managers or capitalists *do* play a more important role for the economy than other occupations.

    Before 1989, we already had *exactly* the system that you propose in which labor is also deciding about management etc. because it has to be easy. Thank you for your offer but I will never allow something like that to be repeated.

  43. Please call them "neoliberal elites," a true liberal wouldn't stand for such nonsense!

  44. But tech companies don't respect people that are good at math, Shannon, otherwise they wouldn't seek out the lowest paid people on the planet to do STEM work for them. They'll pay the slicksters in marketing and advertising more than they'll pay their STEM workers. This is totally backwards because unlike the company slicksters, STEM workers add sustainable value to the company. Tricking consumers into buying something is simply not sustainable.

    And I don't know about you, but companies that engage in this and other forms of in-your-face deceptions really, really bother me. It's like having a gnat buzzing in your face and no matter how many times you bat at the darn thing, it just won't go away. But eventually consumers will get sick and tired of buying products and services that have more sizzle than steak and will avoid all forms of marketing and advertising being thrown their way, causing this all hat, no cattle business model to collapse into a black hole the size of Texas. Hopefully I'll live to see that day.

  45. may be but who decides who pays... so they are right even they are wrong...:)

  46. "... slicksters in marketing and advertising ..."

    Even though I only like doing technical work myself, I recognize that marketing is needed to sell the product. Without it, all the technical talent the company may have is of no use.

  47. http://dailycaller.com/2013/11/26/prof-corrects-minority-students-capitalization-is-accused-of-racism/
    "all knowledge is subjective and based on one's position in society"

    All societies must set their standards of knowledge and abilities upon the diversity of their most failed contained race. Anybody who disagrees is thus proven unqualified to comment. There is no reason to build with right angles and plumb walls other than historic White Protestant European racist paternalism. Free your mind!

  48. A lot of companies hire the services of statisticians and mathematicians for their research and development businesses. These statisticians can work from home and are very well paid by the way.
    Same as you though, I don't like marketing and advertising, or even selling jobs, they are not noble occupations. It is all about how one can screw someone else. Unfortunately it is the world we live in, especially in the US where people are considered as consumers only.

  49. But if you've got a great product or service, you don't need to market it to get people to buy it. If it is great in and of itself, it will sell itself without you having to lift a finger to market it. Now if your product or service is relatively unknown you might first need to market it a bit, but as soon as the word gets out about how great it is, then you can cut your marketing expenses back to a minimum.

    Frankly, I'm very leery of buying anything that has been marketed to the max. It usually means it's lousy or not worth buying. For instance, you know that ObamaCare is a lousy product because so much money had to be spent to market it. In fact, the product is so terribly lousy that it practically had to be given away, on the taxpayers' dime of course.

    Presumably there are some companies that have a product or service that is so amazingly good that they really don't need to market it at all to boost its sales, but they'll market it anyway because there a tax advantage or a PR purpose for doing so.

  50. "But if you've got a great product or service, you don't need to market it to get people to buy it."

    The signal to noise ratio is rather low in modern society, and competition is rather stiff. A company absolutely needs people who are good at selling. I didn't realize this was controversial. :)

  51. I fully agree that this "observer complementarity" may be an important aspect. But could it not still be, that some powerful non perturbative method exists, which allows one to calculate the wavefunction for all observers at once? One obstacle I see is that the parameter t in psi(t) does not have an observer independent meaning. Therefore I guess some new formalism would be needed.

  52. I have different perspectives on the subject. After graduating from HS in my hometown in the ME I got a government scholarship to study engineering in the US. I was so excited to meet these great people, after all I remember the Apollo moon landing in 1969. I was in a shock in college, I would ace all the math exams and my american counterparts were so primitive, they couldn't do even basic algebra properly, what a disappointment.

    But later as I spent more time I understood the mechanism of the system. Since the population is huge and there are so many colleges and universities, the system is only interested in the cream self motivated people. That is because the US has so many companies(and universities), but the highly technical jobs are few. However these companies need a very large supporting staff, a lot of engineers of the "mechanical" jobs not innovative ones. These companies can always outsource the innovative and highly technical jobs with the best in the world. Nothing else makes sense.

    Trying to instill the love of science and mathematics in particular in my children was a loosing battle. A battle against society and schools both public and private( as in american schools with disastrous outcome). The teachers themselves were from a failed system. My daughters are in the medical field, my son is an engineer(because of a lot of effort on my part).

    A lot has been said about India, However, on a trip to Bombay with my Indian friends, we went to a college to meet a friend of theirs. I saw this gigantic crowd in the college yard chating and I was there for three hours. the bell must of rang four times and nothing seem to ever happen. Then I asked my friends what is going on, why aren't they heading to the class. He replied, what the hell for, they will graduate anyway!

  53. Missing a zero in the title.

    On the 10^500 IIB flux vacua; is the instability of the KKLT construction not certain enough to say that these vacua don't really exist?

    It would be somehow aesthetically pleasing if they reduced to the same amount as M-theory on G2 and "decorations" contained equal information to "geometry" :)

  54. The title is right this time. It's 50 million this time.

    Once, in the body of the text, I wrote 500 million incorrectly, I guess. There is also a number 470 million for the number of building blocks in a construction of Calabi-Yaus.

    Well, I won't be fixing that, the number of readers who care is too low.Everyone who is not sure about 50 or 500 million in the blog post should check the paper. ;-)

  55. I always like and carefully read with excited interest (often more than once) such well explained posts about the cutting edge state of the art of all arts, they have been rather sparse say the last few months.

    But conversely to what I am doing now ;-), I usually rather shut up my recently too big mouth if I dont think I have something substantial to add ...

    Maybe other TRF readers do silently highly enjoy such posts too; thinking that it should be left to competent enough friends and colleague of Lumo s to discuss below them...

  56. Going from 10^500 to half a billion is a huge improvement.

  57. kashyap vasavadaDec 17, 2014, 2:02:00 AM

    Most likely, I do not understand the real reason for argument. But once it goes beyond 1 (unique), does it really make any difference whether it is 10^500 or few million? I would be interested in someone's comment on this.

  58. Physics was never "at" 10^{500}. People who believe this was the likely "shortlist" of candidates did so because they think that a larger number is better. But a rational justification of such an attitude doesn't exist.

  59. Does it make difference for what?

    There are surely differences between the numbers, right?

    If we could calculate the spectrum of the vacua enough to check whether it's right, we would be almost certainly guaranteed to find the right one - and decide whether there is a right one - if the number were a few million.

    If we had to deal with a googol or more "equal" candidates, that outcome wouldn't be guaranteed. Some people say that the localization of the right one would be impossible, but I of course disagree.

  60. Would it be possible, for instance, to Monte Carlo sample all those possible compactifications ? For instance, I guess string gas cosmology is a nice try based on general thermostatistics arguments ... but, how to systematically and exhaustively could one find our real physical vacuum in such a huge set of superselection sectors ?

  61. This is probably a weak point, since I don't know the details behind extracting matter content from any compactification, or numerical K-theory and similar things, but anyway:

    A few million seems fairly testable; computers test a few million of something or other every single day. It's pretty easy to generate a few million primes in minutes on an average laptop. If there's some way of teasing a simple experimental phenomenon out of these geometries, then we might be able to actually brute force a serious amount of the data here, if it should come to that. 500 million is a lot to go through one by one, but it stands some sort of chance of being tractable. 10^500 could never be seriously explored in earnest without using generic theorems to slash away 99.999999...% of the possibilities. Which is of course a totally legitimate and very useful thing to do, but is also extremely difficult.