Friday, December 19, 2014

Alternative teaching of mathematics: three problems

Mathematics is not just the mechanical elimination of a finite number of answers

Two days ago, I spent hours, literally, by discussions at about the Hejný alternative method to teach mathematics to the kids.

Off-topic: A group of 100+ engineers is actually building Hyperloop, Elon Musk's mach-one train that gets from San Francisco to L.A. in 30 minutes.

If I try to summarize some key points really concisely (some exchanges helped me to crystallize some of the points): mathematics is not just about the lessons that a human derives from the experience, but about the accumulated knowledge that dozens of generations of mathematicians have extracted from the experience and, even more correctly, from their pure thought. So mathematics can't be left to the rediscovery of each kid.

Now, there are differences between the kids and they will show up. Whether the kids at the top in a given subject – mathematics, in this case – master the subject well is more important than what the others do because those at the top are actually likely to use it. One may reduce the differences by forcing the kids who were not so good to spend much more time. But I actually think it is counterproductive. Kids – and adults – should better focus on things that make them happy and that they are good at. So I think that in a healthy situation, the less talented kids in mathematics will spend less, and not more, time with mathematics than their talented counterparts. Consequently, the gap will be even larger than it would be if everyone spent the same time with everything.

Some of the interesting exchanges focus on the child's thinking and early learning. Can the child understand abstract concepts? The word "abstract" may be interpreted either as "not concrete", describing general classes of things instead of particular elements of the classes (plus using variables to represent "not concrete" things); or as "imaginary" (in the sense of not being "real objects" in the "real world" but some invented ones, or mathematical idealizations of objects in the real world).

My view – which sounds almost self-evidently true – is that a child is ready to deal both with real objects and the abstract ones, in both interpretations of the word "abstract". In fact, kids of the kindergarten age spend the highest percentage of their thinking (if compared to other age groups) with "imaginary" things. Think about the childhood games, Baby Jesus who gives them gifts, and so on.

They are also able to deal with abstract categories, talk about "animals" and not just a particular animal species or a particular animal of a given species. In fact, the discrimination between "different animals" requires some learning, much like the discrimination between "an animal" and "a particular animal" in the logical sense.

Some 2013 UC Berkeley research (paper, fun talk on Bayesian babies' experiments) showed that toddlers are smarter than late kindergarten kids and able to look for totally abstract patterns in what they see or otherwise perceive. This ability is being suppressed later when the children are overwhelmed by too many concrete things in the real world.

For all these reasons, and because kids are so good at mastering complex mechanical algorithms (see the insane video above), playing various games, also games on tablets and computers, I am convinced that in principle, kids may learn any logically ordered material in the same way as adults who are starting with it. There is no need to "connect it with the real life". In fact, kids don't really distinguish "what is real life" and "what is not real life" too well; they need to get some experience to become good at it!

So all the claims that kids have to be bad when it comes to abstract variables, sets, generalizations of objects, formal operations, mathematical idealization of objects, and so on look self-evidently wrong to me. In principle, they are probably better in all these kinds of abstraction than the adults! They may have stronger CPU unit waiting to be used for something. And unlike adults, they don't care whether something is "good for practical life". They just play and do what's fun. That's also why half a billion of children in the world agree with me – an overaged child – that those who suggest that "theoretical physics isn't good for anything in their life" are old intellectually degenerated, non-creative witches who should be burned in the oven for the happy end to emerge.

But I want to mention a seemingly less controversial topic, namely three typical mathematical problems that the kids are learning in the Hejný method alternative schools. They aren't "totally different" from other problems used in other mathematical methods – and they cannot be. But they still reveal something about the philosophy of this approach to mathematics which I simply can't approve of.

(Dano sent us some results from a mathematical contest, the Mathematical Kangaroo, with some identification of the kids from the Hejný method classes among the (hundreds of) winners. There should be 10% of them among the winners because 10% of the classrooms use it. But the percentage was 8% for the 3rd graders and dropped to 3% for the 5th graders – which, if accurate, seems to look as a totally unacceptable 3-fold drop of the best mathematicians caused by this pedagogic method.)

Fine, the three problems.

This problem – one from the "spider web" category – doesn't seem to have any instructions, what the colors mean and what you should fill in the circles. It's like a question from the IQ tests. Correct me if I am wrong but I think that each of the three colorful arrows represents the addition of a particular (positive or negative) number. And you should figure out what are the increments assigned to each arrow; and what are the missing numbers that are being shifted by these arrows. OK?

So how do you solve it? There are numbers 3,6 that are known and a yellow arrow in between. Logically, the yellow arrow has to be the addition of 6-3, i.e. of 3. Write it at the bottom in the center above the yellow arrow. There is another, left-directed yellow arrow ending in the number 6. So its starting point has to be 3 as well.

Now, you may be stuck for a minute. But you will see that from the new number 3 in the upper right corner, you may get to 6 by three steps (down, left, up) and all these three arrows are red. So one red arrow has to be one-third of the difference 6-3, so it must be +1. You know the red arrow is +1, write it to the left bottom corner. This allows you to write 4 in the left upper corner, because it is the known 3 below that circle plus the red arrow's +1. And by a check, you may see that the green arrow is +2, e.g. from 5-3.

I hope that my solution is right LOL but I haven't verified it against the sources. You could solve it in a slightly different order.

It's just fine and the solution I presented above is routinely presented as a "logical" solution to the problem, I think, and this "logical" character is a source of pride for the champions of the method (and is contrasted with the "mindless memorization of formulae and algorithms"). What does it mean for it to be "logical"?

If you think about it, it really means that one randomly tries to find simple logical steps that allow you to write down a new number, just like e.g. in Sudoku. And even more typically, "logical" means that you try a finite (and usually very small) number of answers (numbers in a given circle) that could be filled in over there and you make some tests and find out which of them work and which of them don't.

To a large extent, it is a "combinatorial method" that directly and mechanically tests a finite (and small) number of solutions. And that's how the kids are trained to approach all similar problems.

But I am strongly convinced that this is a counterproductive way to interpret this "class" of problems. Why? Because the puzzle above is really nothing else than a problem involving a set of linear equations, and one needs to learn how to solve the full class of the problems that are equivalent to a set of linear equations.

The ad hoc guesswork I presented enough is simply no good for a general problem. One reason is that in a general enough problem, the unknown numbers simply won't be simple integers. In almost every application that is slightly realistic and not "engineered" to have simple solutions, there will be general enough rational, real, or complex numbers (or more complicated objects) as answers. Because the number of a priori possible answers is infinite, it won't be a good idea to try the possible solutions one by one.

The second difficulty is that in a general problem, one simply can't converge to the complete solution by the method "one straightforward step after another". In a generic enough situation, you will get something like a general set of 2 (or more) linear equations for 2 (or more) variables where no coefficients vanish.

So you simply have to learn how to solve these sets of the linear equations in general. You need to know how to use the substitution method, or how to construct the right linear combinations of the equations that allows you to calculate a variable and make a "step". If those steps are too "complex" for someone – well, then the fact is that the "elementary steps" taught in this method already fail to be enough for rather simple and generic problems, regardless of the hype said about the "logical" methods. The guesswork or ad hoc arguments could have worked in the special problem above but similarly solvable problems form a very small subset of the class of these problems. In the real world where numbers are often simple, it's a finite yet small fraction. Conceptually, they are a measure-zero subset of the problems, of course.

The kids are not learning mathematics in the sense of the methods to deal with classes of mathematically analogous problems. They are only learning how to solve crossword-like or at most Sudoku-like puzzles that were artificially engineered by someone to have "simple" solutions. I would even say that by being trained to solve elementary-to-solve engineered problems only, the kids are being trained to become gullible sheep who easily buy a cheap propaganda. The feeling that the sheep is "solving" or "verifying" something strengthens its feeling that the information has to be reliable – yet, the author of the "puzzle" or "propagandistic argument" clearly knows more things (or more subtle things) to control the sheep.

The second problem belongs to the class "snakes" of problems – one of the omnipresent ten or so classes of puzzles that are repeated all the time, for eight (?) years. The Czech words only say "Problem 2" at the top and "Solution:" (three times) at the bottom. What is this problem? It's a reverse problem of a sort where you either have to write down a missing number in an equation, or a whole operation – the operator with a missing number – so that an equation (given by two yellow disks and a blue one in between) holds.

So the first step tells you that "something" multiplied by 3 equals 12. "Something" is clearly four. Then there is a missing "operation" between 12 and 6. What is the "operation"? The operation may be either "subtraction of 6" or "division by 2" or "multiplication by 1/2". These three answers are tolerated.

Fine. What are we learning here? Is that mathematics? Is it really right to repeat the same "snake" puzzle 100 times or more?

Quite generally, the Hejný method teaches the kids that the "operations with numbers" are either addition or a subtraction of a small integer (or a simple fraction like 1/2), or the multiplication or division by a small (division: nonzero) integer (or a simple fraction like 1/2). I feel that it is a wrong lesson. There is nothing mathematically special about operations that involve integers as the other term or the denominator. Even more seriously, there is nothing mathematically special about operations that only involve one step.

Another way to get from 12 to 6 could be to exponentiate to the 0.721st power. Of course that I understand that these kids shouldn't know general exponentiation yet. However, a more realistic point is that the general interpretation of the problem is that there is a function \(f(x)\) that obeys \(f(12)=6\) and we are supposed to find out what is \(f(x)\).

If you interpret the problem in this way – and I haven't changed the problem and I didn't forget to tell you anything that the kids were told – it's obvious that the problem is silly. There are infinitely many functions \(f(x)\) that obey \(f(12)=6\). Even when we only deal with simple multiplication and simple addition or subtraction, it could have been \(f(x)=3x-30\) or anything else that works. A smart kid could could easily write ".3-30" to the circle, and add an extra comment at the bottom, "it is also a correct solution, idiot". I actually think that I would have written it if I were given that problem as a kid – and that better 10-year-old kids from the normal classrooms are probably better to construct this "prank", too.

This kind of a problem really "teaches" the kid that there is something mathematically special about functions that only involve one operation, either \(f(x)=x+b\) or \(f(x)=ax\). But it's sort of conceptually silly from a mathematical viewpoint. If you consider the class of functions that includes \(f(x)=x+b\) and \(f(x)=ax\) for all \(a,b\in\RR\), it is a very unnatural class – a union of a class of apples and a class of oranges – and the smallest mathematically natural (or closed under composition) extension of this union is to consider all linear functions \(f(x)=ax+b\). If you do so, there will clearly be infinitely many possible answers to the problem. The Hejný method teaches the kid to never consider the possibility of a subtler solution, a more complex answer, to never look beneath the surface. Even though they try to claim otherwise, this method tries to kill the kids' creativity and their desire to sometimes look outside the box, too.

You may see that this puzzle, I wouldn't even call it a mathematical problem, is pushing the kids into thinking that the functions and operators are naturally clumped into classes that are actually very unnatural from a mathematical viewpoint. Instead, this problem is natural not from a mathematical viewpoint but from the viewpoint of a special kind of puzzles that have their own rules of the game. Once again, all these puzzles always assume that things have been arranged so that the answers are nice and simple and the person who is solving the problem only has to check a finite number of possibilities.

But mathematics that deserves the name – whether it is studied as pure mathematics or it is some mathematics that emerges in the uncountable applications of mathematics – simply doesn't work in this way. The problems one needs to solve with mathematics in the real career of a mathematician, scientist, engineer, or even physician have probably not been fine-tuned by anybody to guarantee simple answers or step-by-step solutions.

So this is teaching completely wrong lessons to the kids and my guess is that the kids brainwashed by this stuff must exhibit a very clear disadvantage relatively to the kids who did learn proper mathematics – with symbols, equations, and methods to deal with equations – by the age of 12 or so. The schoolkids learning the Hejný method are not really learning mathematics that they may encounter. They are only learning how to solve a special class of puzzles that were designed as puzzles or the material for this method. The method only teaches itself, not mathematics, and the mathematical realm of this method is a zero-measure subset of the realm of mathematics.

This is also confirmed by the final problem. In each equation, you should only use two arrows – otherwise the number of solutions would be infinite even according to Mr Hejný.

The right-directed arrows are terms +1, the left-directed arrows are terms –1 (it's linked to the OK but overhyped teaching of the addition and subtraction using steps and translations), and you should write two copies of +1 or -1 into each equation for the equation to hold. If I am not wrong, there may always be several. Every +1 on the right hand side may be replaced by -1 on the left hand side and vice versa. But maybe you are only allowed to write one arrow in one box, I don't know. The description of the problem does seem to agree that there are several solutions and demands at least three.

Again, this is an extreme example of a fine-tuned, artificial problem. There is a very small number of potential answers – adding +2, +1, 0, -1, or -2 to the left hand side – and the kid is invited to go through the five options. Again, this is nothing like the general problem that the kid will need to solve a typical intrinsically mathematical problem of the same type.

It is much more similar to the kind of IQ problems from the Idiocracy movie and I believe that it may naturally be solved by kids in the kindergarten. But mathematics is something else than checking two or five possible answers.

By the way, I think that the people who become irrational when someone mentions that some equations have a large number of solutions – and I specifically mean the people who are upset about the claim that string theory has \(10^{500}\) solutions of a certain kind – are people who are imagining mathematics in the wrong way that the Hejný method tries to propagate. They always solve things essentially by trying the possible answers one by one. But mathematics offers vastly cleverer methods and of course that it's in principle possible to identify the right solution to a problem among \(10^{500}\) or infinitely many candidates.

So I would conclude that the method – with its hatred against well-defined equations, identities, and algorithms – ends up teaching only how to solve contrived, artificial puzzles with a finite number of possible answers and admitting straightforward solutions. The kids are not really learning any new cleverer methods or patterns or theorems – the mechanical test of a finite number of possible solutions is apparently enough to get through the whole basic school. And if this is the whole philosophy of the pedagogic method and its "logical" character, and I think that it is, it simply doesn't deserve to be called "mathematics" at all.

And that's the memo.


  1. The kids can learn as much as the teacher can teach, most of the time. Young students can easily learn advanced algebra at age 7. They just need a capable teacher, and they will reach their goal. They don't know what their limits are initially.

  2. Great post! What do you think about the web site to help children learn mathematics?

  3. ""... focus on things that make them happy and that they are good at." This is the ideal but there must be a compromise between what the marketplace and pragmatism allow and what people really want to do. One problem with formal education is that teachers teach what is easy to teach and politically indicated instead of teaching what would be best for the students.

  4. Ok, let's rephrase it in such a different way: GR was never directly tested in its strong field limit, but its perturbative limits were checked out of any doubt. Even the existence of gravitational waves, despite of no direct detection, are settle after PSR B1913+16. Recently BICEP2 has shown quantum gravity perturbations on CMB, but those are still "only semiclassical" effects. Once observational cosmology is progressing quite fast recently, what would be (if one) genuine "string effects" one could (even indirectly) expect to able to observe ?


    Khan Academy creates objective competence by being an expert teacher. Soft approaches to learning do not seek to instill objective competence. They are heteronormatism problematizing homosocial othering (slathering pigs with lipstick). The eternal enemy of "good enough" is "better."

    Demand Equal Opportunity!
    Enforce Equal Opportunity!
    Inflict Equal Opportunity!

    Hey, Obama - who turned Sony inside out without even having a reliable source of electricity?

  6. Dear NumCracker, there are different strongly coupled regimes, depending on the coupling we mean.

    In weak gravity, classical GR may be linearized but it is a nonlinear theory and when the redshift becomes of order 100%, ideally at the black hole event horizon, it is a situation where classical GR is very strongly coupled.

    One may talk about the strength of the quantum effects, and they get important for the Planck-energy scattering of gravitons and similar situations. Exactly when quantized gravity gets strongly coupled in this sense, it breaks down and string/M-theory is the only framework that can yield any predictions at that point.

    I don't think one can see Planck-energy phenomena directly in any foreseeable-future experiment I can think of.

  7. Raphael Bousso also added a good comment this morning.

  8. kashyap vasavadaDec 19, 2014, 5:42:00 PM

    Thanks.Some time I will try to really understand Weinberg's proof, although I have lot of respect for Penrose also! What happens during contraction phase (if there is contraction) is still mystery to me. Although Frampton's work is very interesting,it did not resolve problems for me. Personally I would like contraction on Hindu religious ground! Ha, Ha, Ha!!!

  9. It is really interesting to get deeply into these ideas and contemplate the nuances.

    Sounds like you want to have a philosophical discussion...

    There really isn't any difference between philosophy, science, and mathematics. What they all have in common is that they try to describe things. All three can create consistent frameworks that may, or may not, correspond with nature. The problem arises when we want to know which of these many frameworks does correspond with nature.

    And just as there are bad philosophers, there are bad scientists and bad mathematicians. Sometimes I think we confuse the discipline with the disciples.

  10. Funny, as others too noted it's impossible to understand your (geocentric) ranting, actually.

  11. Pure politics incrementally infiltrating science via its very top journals. There are climate justice warriors in charge and anything they can do to install more gatekeepers affords them status. It's the science of psychology that needs to advance to better expose, understand and treat sociopathy. I'm actually long term delighted that the climate scam pulled so many people into intense adoption of trumped up and also irrelevant consensus, in public, because the public can understand it as a politicized fraud and be eventually much more savvy about activist scientists.

  12. I think understanding math is conceptual, and that is the beauty of it, making it quite fun.

  13. Advanced mathematics in France did spiral into abstraction.
    Usually one looks at particulars and becomes familiar with local models or effective theories, then uses this knowledge to generalize and to develop abstract systems that also work for the particular examples. Some mathematicians seem to bypass the particulars and jump immediately to the abstract. An example was Alexander Groethendieck, who was at home with abstractions, but uncomfortable with actual numbers. When someone asked him to show him how something worked with an actual prime number, Groethendieck said, "Fine, consider
    57....". This has since been called a "Groethendieck prime" :)
    Galois had a similar talent. You need both types of mathematicians.

  14. Funny, as others too noted it is your (geocentric) ranting which is impossible to understand.

  15. I agree that most kids should just learn basic arithmetic and let it go at that.

  16. Do you think they should just learn the present tense and not read literature?

  17. Rubix cubes are an interesting subject to teach. To work out how to solve a cube front scratch you need to understand a bit about groups, and cognates. ie. Change all bar one on a face, rotate the face, reverse the process, and you've flipped or rotated.

    Explainable in a non mathematical way. So are groups with some paper. Turns out to be a rich area far better than a IQ style puzzle.

    You can treat it as art, as all sorts of things.

    Completely abstract, but a fun part of maths.

  18. My opinion is that kids need to "feel" mathematics not learn it. Excuse me while I go and take a bong hit.

  19. Thank you, Lubos. Excellent argument and explanation.

  20. Agree with whom, Luke? I haven't met anyone in these debates who would be willing to exterminate mathematics from schools to *this* extent.

    My opinion is the opposite one. The basic school kids should be exposed to the maximum amount of mathematical concepts that have ever made it to the basic schools. Different kids will get different percentage of the material -perhaps the average could be close to 50%- and this should be considered OK, a sign that they are learning something.

  21. It's a rich puzzle, indeed, more mathematical than e.g. chess, I would say, because of the links to combinatorics, group theory (including reversability) etc.

    But it's still a particular and very special class of puzzles in recreational mathematics. Quite generally, like I wrote in the case of the Hejný method, it seems counterproductive to me if children are led to visualize the word "mathematics" as "recreational mathematics". I know too many adults who are doing that, too, and they of course don't have any clue about the real power of mathematics and its connection with anything in the real world.

    Recreational mathematics trains one's mind - like the real one can - but it's mostly no good for the real "business" or "work". That's why it's called "recreational". In my opinion, the purpose of mathematics at schools is not to train kids' brains in "any way" - the pure CPU power of their brain is actually maximized when they're toddlers and can't be changed much - it is to make them able to direct their brain CPU power in directions which are important from a mathematical viewpoint.

    Recreational mathematics directs the CPU power in a direction that is not equally important (except for chess masters etc. who can make living out of these things). Recreational mathematics are often contrived special problems invented to look like fun problems with fun solutions - for their entertainment value - and not for the method or the answer's importance for understanding something that actually naturally emerges in life or science.

  22. Mooloo, I am convinced that kneemo is right and you are just wrong.

    First, what one learns as a kid is remembered much more safely than what one learns later. When one learns a language as a kid, he really knows it. The case of algebra is virtually and obviously nearly isomorphic!

    Your suggestions that algebra - something that really underlies the whole Universe - is meaningless is just dumb and I won't honor it with an extra response.

    When you say that the Chinese are only "rote learning", you are completely distorting the actual differences. What the Chinese are actually better at is exactly the nontrivial CPU-based pure mathematical intelligence, the g-factor - the equivalent of 10 extra IQ points in these important matters. They may also be "rote learning" but it's a consequence, and not the cause, of their mathematical skills.

    The Chinese may be worse in self-confidence or the ability to verbally impress others, or things like that, but your idea that they are inferior in some mathematical skills is just a silly fairy-tale that mathematically inferior nations invent for themselves to look better.

  23. Sorry off topic.

    Lubos have you seen Polchinski's comment for you in PW's blog? What are these trackbacks? Are they important?

  24. Dear Giotis, no, I haven't, I am avoiding that blog and especially things like the owner's interactions with people whom I consider good physicists because I have always felt that from their viewpoint, the disagreement is just one game and totally different things like those people's comfortable chairs or political ideology are more important for these people than the truth about string theory.

    BTW Polchinski had a review of dualities some two days ago, I linked to it in the updated version of this blog post above.

    Trackbacks are (semi)automatically added hyperlinks from the arXiv papers' and similar pages that link to external websites that have linked to the paper. ;-)

    I used to care about them for some year in the past, and manually added them using a script, but I think it's a waste of time.

    Of course, PW is a dirty parasite so he uses every opportunity to promote his dishonest rants - links from the arXiv are great. I am the opposite kind of a guy, a good guy who is helping the people to be directed to the arXiv and science in general.

    The trackback traffic has never been a detectable fraction of my traffic. You should understand that the overall arXiv traffic is just 10+ times larger than the traffic of this blog, which means that a single arXiv paper's traffic is 10+ times smaller than the traffic on this blog, and only O(1%) of the visitors of a paper actually click at a trackback.

    It's not a big issue but I do think that the arXiv administrators have made arrangements that *automatically* add the trackbacks to tons of shitty weblogs including PW's NEW. I was never among the automatic ones and it is even possible that I wouldn't be allowed to "post" the trackback using the online tools I used to use. I haven't tried it for 5+ years and I don't care. All the behind-the-scenes arrangements in similar institutions seem completely immoral and corrupt and I better don't want to learn details because I was *always* shockingly disappointed when I did so.

    So if there is some evidence that Ginsparg or Polchinski or perhaps Witten are actually friends with the Lithuanian Nazi spoiled brat and indirectly help him to promote his anti-science delusions, well, I can imagine it's true but I just don't want to actively torture myself by being reminded it's true. The world is a shitty place.

  25. Only one short remark. As physicists J. Ellis and G. Ellis share the same family name, maybe it would be useful (for non-physicists, at least) to write here G.Ellis. (Once upon a time I had the privilege to meet J.Ellis.)

  26. I always wondered, why Witten and others not just make the Übertroll and other subjects getting blown out of social existance....
    Making even friends with him would be very self-distructive indeed.

    I dont understand the intentions of the ArXiv admins, tollerating links to rants of laymen concerning the topic of the paper at hand who do nothing but spit on the science the ArXiv is meant to support. This is very inconsistant to say the least ...

    Our developper contacted them concerning some technical issues, as we are trying to get automated trackbacks for PhysicsOverflow too (as MO has and TP.SE had). But all he got was an automated message that his questions will be processed already weeks ago. Seems the ArXiv (admins) care more about troll-blogs ... :-/

  27. Thanks for that reminder for everyone! ;-)

    John Ellis is the 2nd most cited HEP physicist in the world - at least a few years ago, it was so - and has done lots of work on SUSY etc.

    Concerning popular physics, John Ellis is actually the person who has introduced the term "theory of everything" to the common popular physics discourse!

  28. Dear Dilaton, I used to have a mixed feeling about this Witten's and others' silence, but it morphed to a rather clearly negative one.

    One could have thought that it was just about those big shots' being decoupled from the public discourse, focusing on technical work, and their modesty.

    But it seems much more accurate to say that it's their focus on their own personal careers, image, the opinion that they don't give a damn what happens after their era, and often open alliance with people who are hurting science (and making living out of that) because of various collective interests and/or ideological overlaps.

    I think that these folks should be ashamed because they have really enabled these things. I have never received any detectable support from them, except for a local one from my Harvard colleagues, and they allowed this hostile activity grow to dimensions that almost beat string theory research itself today.


  29. Its like learning scales before trying to play brahms...or michael jordan learning the discreet skills necessary to play basketball, before being able to put it all together in a game, against opposition.
    Even picasso learned to create representational art before he rejected it.
    Leave it to people who dont know it, and dont have to use it in order to make a living, to figure out a new way to teach math.

  30. Yes, the Übertrolls have spawned tons of subtrolls who are not much less agressive and damaging, and they have penetrated everywhere ...

    So I fear that at present, the only way to blast this witch-hunt movement against HEP would have to be nuclear ...

    Anti-HEP trolls are like the replicants in stargate:

    Dangerous mechanical vermin, that reproduces and spreads everywhere quickly, if you dont manage to savely destroy the first one upon sight ...

    Unfortunatelly, Witten and others missed (or intentionally neglected?) the change to do the right thing as still was time for it ...

  31. No. I kinda like the idea of this 'new math'. When these 'dumb as dogshit' students do poorly in tests and need to do remedial math for entrance exams to college etc. I will make $65 an hour tutoring them.

  32. Luboš is correct but perhaps too kind. Math notation is a language and like all skills children learn more quickly. Children forget what they learn only if they do not use it daily. My first language was German but at age 3 my parents stopped using it at home so that I would learn English for school. I had to relearn German years later and never was fluent again.
    By the time kids are in high school, many of them have been taught little math but have been told it is hard by people who were similarly told.

  33. Short term ok. Long term, I find it de-motivating. I have given it up these days because I like a fire in my belly, more.

  34. yes you are so "right on" as my generation said. My then brother in law was as illiterate in math as most.
    We both took took at least an hour the first time we tried to do the Cube. A few days later I still hadn't figured It out. He was able to solve it like the boy above. But when I asked him how, he just took another hit from the bong ( Like you said, Bill:-) ) and said " you need to be one with the cube".
    When he left to use the bathroom I "altered" the cube by peeling off the colored square tiles and "solved" it by sticking them on other faces. He predictably looked confused after returning to the room It appearing that I had mastered it :-)

  35. That's quite some income. Here I would be getting $10 per hour at most. ;-)

  36. Good for you to have learned German, English, and algebra this early!

    The life would have been smoother if I (and similarly others) learned English as kids, too. Speaking as a native speaker, without the accent and with the right propositions after "explain" and without invalid usages of "would", has clear advantages.

    But that wouldn't be me! ;-)

    Aside from algebra, there are also programming languages. So not shockingly, I would learn BASIC before everything else - that was my first foreign language and my idea about English, too. ;-) RUN, LIST, INPUT, GOTO, GOSUB, RETURN. PRINT, POKE, PEEK - funny when those words were not parts of any conversation in full sentences. :) Assemblers 8080, 6510, Pascal, and not so many clear languages afterwards LOL.

    If the question were whether kids should learn 2 human languages or 1 human language and algebra (notation etc.), I would surely say the latter!

  37. Rehbock, your solution to the Rubik cube was an example of thinking outside the box! ;-) I did it with one cube as well once but I still learned to solve it quickly enough so that the kosher method would be faster than peeling.

    More seriously, your brother-in-law and similar folks probably have to have a lot of talent for mathematics. But their spare intellectual abilities are available for things like the Rubik cube.

    The potential to learn things - either Rubik cube or creative invention of new high-energy physics theories or whatever - is something else than the knowledge of mathematics itself, and the latter is simply also need and mostly needs to be taught, not hoped to be automatically given from the heaven.

    Some curious people who are highly into maths will be self-didacts. But that's learning, too.

  38. Right. Not to mention lots of writers of fiction who first read others' books before they write their own, if I avoid the word plagiarism. ;-)

    Your last sentence describes the real state of affairs. Math teaching (here...) has been largely left to the people who don't really know how maths is used at the level "above the schools" for which they design the education.

    So it's like if you get piano lessons from someone who has never heard or played any decent composition. Just the elementary etudes and finger drill. With this background lacking the "transcendent" knowledge, the pupil is likely to learn some wrong lesson that are counterproductive for the real playing, right?

  39. from woody allen (who proves that everyone can be right, at least once in a lifetime)
    "Those who can do, do.
    Those who can't do, teach. Those who can't teach, teach gym."

  40. Isn't it funny how "teaching to the test" is almost universally derided, until that is, someone really needs to pass a test? Then people pay a ransom to be taught to the test.

  41. I didn't know the last sentence and it is good! ;-)

  42. he also said that 85% of life is showing up on time...I would add to my classes "the other 15% is having sex with your girlfriend's daughter, I guess"
    this was back in the day.

  43. I have subscribed to it for my 14 yrs old daughter. It is good as an exercise, but does not teach directly. It is good in the sense you don't have to check their work in detail, yet you can use it as a reward system. You can ask them to finish certain numbers of question with some certain score so you may buy them what they want.

  44. My general theory of learning goes like this. Anybody can do anything, almost. My analysis is that people early on have a history like an analytical function, few start points can affect the whole future. It is not strictly about intelligence, it is about some random events that you sort of "cling" to some of them just out of curiosity and environmental(social) pressures. This happens even when we are old,like maybe you are not much into country music, but one day you run into one of them in a friends house. Next, you see yourself becoming a fanatic.

    The same process happens to us as we grow up. Typically we have multiple interests but with time certain interests percolate to a bigger size than others, as a function of attention to them. More attention/satisfaction then leads to more attention, that is how interests grow. Then you might hit a critical point where all other interests become harder to maintain, and your specialization takes control.

    I always thought of myself as this dumb geek only good in science. However, during the start of the Masters program, I got burnt up with science doing so much. So I started trying poetry, painting, music, anything but science/math. I was so surprised to find out that I can be very good at them and even love them. Same thing happened with business, I always thought it was something that no good moral(and stupid) people do, then I became an expert in it. That was mostly because of a severe social pressure, but then I started to really like it. That is why I truly believe that anybody can do anything, that is what I tell my children.

    Now, part of these influences(acting as random events) can be the parent, teachers, friends or some extended family members. The earlier the influence the stronger its hold.

  45. Even if some genes were needed, it's rather likely that *your* children can do most things that you could and can! ;-)

  46. I have five brother. One of them I consider as the dumbest human in history, the other is a total failure, another weird religious guy, forth he is owner of a very big business in kuwait, the fifth is the minister of housing( though he barely has a HS degree, that is how our government is run!!!)

    Sure, genes play a role, but I can see environment has a heavy grep on our lives.

  47. I hope that most of the brothers can't read, or at least read English, otherwise you're at risk! ;-)

  48. Lubos, you wrote:
    “The imprimatur of science should be awarded only to a theory that is
    testable. Only then can we defend science from attack”
    Perhaps the next experiments will slightly change our view on the world.
    “Experiments to determine the mass related Lightspeed extinction volume around the Earth and
    around spinning objects in the Lab.”
    According to Einstein’s relativity theory, is the speed of light for every observer the same in all
    reference frames.
    However, there seem to be incidental differences in the lightspeed if we observe the outliers of GPS
    satellite to CHAMP satellite distance measurements of 180m.
    At the same time in the literature I found tiny structural but characteristic unexplained
    irregularities in Planetary radar-pulse reflection measurements, made by I.I. Shapiro in 1964, between
    the Earth and Venus.
    Both observations support the idea of the existence of ellipsoidal lightspeed extinction
    (or vacuum adaptation) volumes around massive objects like the earth. Such a volume I will call LASOF
    or Local Asymmetric Oscillating Vacuum Frame.
    Other historic lightspeed experiments support the idea that all objects with mass are equipped with some
    extinction volume.
    As a consequence, I propose new triangular trajectory lightspeed comparison experiments between the
    earth and dual satellites or dual balloons and even in the laboratory to support these lightspeed extinction and adaptation ideas.

    ( see my my vixra article)

  49. A friend in Australia hooks up tutors with students. He's been organising this for years and suggested I do it when I go back to Australia next year. The cost of living is high in Australia. A casual in a shop etc. gets $22 an hour. I do some casual english teaching at the moment , in China. I charge 200 RMB an hour and do upto 6 hours a week. I'll see how things go in Australia.

  50. AFAIK John Ellis is now #3 of the INSPIRE database if Simon White counts as a member of the HEP community. Now, if you exclude all the big experimental collaborations from the counting, your top ten should look like: Witten, Ellis, Weinberg, Nanopoulos, Vafa, Wilczek, Seiberg, Brodsky, Maldacena, Georgi, ...

    With the same exclusion criteria, Georges Ellis is a bit lost at see, somewhere behind the 200th most cited HEP physicist ever...

  51. I completely agree with the main thrust of a posting by Lubos Motl. Hell has officially frozen over. Fundamentally speaking one would have to admit that M-Theory, String theory etc are at the very least plausible. As a fellow on the site where I blog put it they reproduce GR and the SM of particle physics. By proxy they gain a degree of verification from the SM and GR.

    The only thing I would add is that a theories unique predictions need to be confirmed in addition to its ability to postdict known physics. The efforts of Mersini-Houghton and Pfeiffer to apply String theory to black hole formation, once refined could supply a observationally verifiable prediction not made by any other theory (I think black holes do exist though their model must have numerical stability problems).

  52. "It's just a technical fact that the experimental verification is hard and it is probably impossible to build experiments that could "settle" the question about the validity of the theories in any foreseeable future."
    Although you rephrase this in a number of ways, this seems to be your central point. Since you don't seem to understand the problem with this, its because if something is a nonfalsifiable theory, then it is by definition unscientific. There is nothing that can ever demonstrate it being right or wrong, which makes for a useless theory.

    This is the entire basis of science. It is both anti-science, and extremely worry some when people are trying to change the criteria to how pretty they feel a theory is as evidence for it being true.

    "But if a theory is not killed by falsification yet, it is alive."
    It would help to read one of the citations in the original article, "Not Even Wrong".

    If we are to convert to this " postmodern science" way of thinking, we may as well just say "God is the answer to everything", and cancel science altogether. Its beautiful, elegent, explains everything, and will never be falsified.

  53. If a theory can't be falsified for 1 year, 5 years, 50 years, or 1,500 years, it does *not* mean that there is anything wrong with the theory.

    Indeed, the whole point of science is that theories remain viable up to the point when they are falsified, and then they are abandoned.

    I am an enthusiastic supporter of the principle that things proven wrong and useless must be thrown to the trash bin which is why I instantly banned you after I wrote your "comment".

  54. This is a really good post. Must admit that you are amongst the best bloggers I have read.

  55. "They completely misrepresent that status of string theory. String theory is not 'supposedly' the only game in town. It is the only game in town. And it doesn't contain just a 'grain' of truth. It contains the whole truth. Due to its basic mathematical properties, it cannot be just another approximation that has to be deformed to obtain the right theory. It must be the right theory exactly." Wow - amazed that the author of this blog was able to falsify spin foam and other approaches to QG - he must know something that the rest of the physics community does not?