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A conference on teaching of mathematics

I just returned from a day-long conference called "New Teaching Methods in Mathematics?" that was primarily dedicated to an explosive phenomenon in the Czech basic schools, the so-called Hejný's method to teach mathematics (TRF texts on Hejný).

Most people believe that the method is named after Prof Milan Hejný who was recently celebrating his 80th birthday – as he reminded us several times – but it's actually named after his father who has taught his son Milan how to love mathematics. This gospel is being used by 700 out of 4,100 basic schools in Czechia and worshiped by virtually all the mainstream media that write about these things.

The basic philosophy of the method is that the teacher shouldn't have any authority in the class, he or she should do basically nothing with the children let alone to teach, and the kids should play and discover all important ideas by themselves. At most, several standardized exercises from recreational mathematics are encouraged to be repeated. The teacher doesn't correct mistakes when they're made, and so on, and so on.




Today's conference was masterminded by Prof Vlastimil Dlab, a buddy who is also close to 80, who is a rather achieved if not renowned Czech Canadian mathematician, unlike Mr Hejný. His views are basically the same as mine – but that was also approximately the case of lots of other famous mathematicians I was honored to meet again.

There was Dr Jiří Rákosník, the director of the Mathematical Institute where the event took place (we had some tensions during preparations but those evaporated today – for example, I should have immediately believed his recommendation that PowerPoint shouldn't use black text on red – the contrast really is too low with most projectors); Dr Josef Kubát who is the chairman of the Czech Union of Mathematicians and Physicists (of course I've known his name for years); Dr Jindřich Bečvář who was quoted as a top guru (teacher and the author of some important pedagogic texts) by all my undergraduate classmates who did the teaching (and/or mathematics?) branch of my Alma Mater already 25 years ago; Dr Josef Polák from Pilsen who was actually a boss of a team of contestants of a mathematical olympiad (which included myself) during a summer camp, and I am not sure whether it was 30 or 35 years ago. And some extremely smart ladies and so on. If you don't see yourself here, it doesn't mean that I don't love you.

The morning session was composed of two introductions, and three talks in favor of Hejný's method (by Hejný and two teachers that promote the method); and three talks by us, the critics – myself as a physics-like user of mathematics; an excellent female mathematics teacher whose evaluations, goals, and all comments just made perfect sense to me; and Dr Jiří Pokorný from my Alma Mater whose talk was also brilliant and who spoke mainly as a parent of kids who needed to be tutored, especially when exposed to Hejný's method.




Now, Hejný's method may be somewhat "well-defined" but what isn't quite well-defined is the "non-Hejný's method". When you avoid Hejný's method, or when you use more traditional and conventional methods and ad hoc tricks to explain something etc., you unavoidably exploit lots of memes, analogies, and types of exercises that Hejný's recipe recommends, too.

So we, the critics, especially Dr Pokorný who made that point, agreed that some of Hejný's types of exercises could very well be helpfully used in all – perhaps otherwise "ordinary" – mathematics classrooms. Hejný's method mainly differs from the "rest of teaching of mathematics" by having an overhyped trademark. But the "type of stuff" it's doing isn't too different from some other mathematics classroom types that don't have any catchy names.

I sort of knew that and that the discussion could possibly degrade into some boring technicalities – whether you should write the snakes (multiply by three, write to the next box, and so on) as trees or circles or any other detail like that. But I am absolutely sure that even with Hejný's method, the discussion is about way grander question than that (even when two mathematics classes, Hejný's and non-Hejný's, look similar, there is still some underlying vision and philosophy that heavily affects the evolution in the big picture and the long run), and I tried hard to open the main topics in a way that focuses on the big picture.

My 25-minute-long talk was called "25 blunders of Hejný's project". 12 of the 25 blunders were the 12 main defining principles as listed at h-mat.cz. I had to summarize the points in some way so of course that some Hejný fans called it demagogic. But in the subsequent talks, they could have noticed that lots of my reviewed principles were exact quotations and the rest was morally accurate, too.

So Hejný has principles that all the children should only learn mathematical things using very specific situations and environments – only situations and environments they already know and are familiar with (in particular, they should only count windows in their apartment and distinguish their grandfather from their grandmother, Hejný proudly told us if I exaggerate the "exclusiveness" of these tasks just a little bit; if you want to know, smartest kids "learn non-commutativity" by seeing that mother's father isn't the same as father's mother LOL). They should never generalize. They should solve problems by guessing possible answers. Mathematics classes should teach the kids what to think about the society – e.g. disrespect for authorities and some social values. They should use non-standard notations. Children who know formulae are intellectual parasites. Teachers should only allow insights that kids make themselves – no transferred knowledge is permissible. The kids should memorize some animal codes for numbers – where a cow equals two goats, 3 cows equals a pig, and stuff like that. People should only care about the happiness of the kids in the classroom and when the class is over, everything is fine.

25 things like that – taken from webs, talks, classroom videos, and so on. I am mostly a theorist here, too. You can imagine what I said. The exact negations of these – extracted and concentrated – principles. Mathematics is all about generalizations. General methods to solve things must be taught when they exist. The power of mathematics is the ability to survive in completely new situations and solve an unlimited number of tasks. The teacher has to correct mistakes soon, otherwise they get out of control and the class becomes meaningless chaos. Answers shouldn't be guessed. Notation and terminology should better be standard because kids will have to learn it soon or later, anyway. Most things we learned were found by someone else – just like most products we use were produced by someone else. Kids loving formulae are kids loving mathematics and should be role models for other kids. Mathematics classes have no business of becoming a new politically correct indoctrination course – even communists shielded the queen of sciences from Marxism. They have no business of becoming visual arts, either, even though some kids would clearly prefer another arts class, and so on and so on.

So some of the philosophical assumptions were clearly outlined. My first speaking ally, the attractive enough mathematics teacher, showed lots of mistakes and omissions in Hejný's method textbooks, what the kids using this method don't know at one age or another, how the teachers have a hard time to adapt and one-half of them leave, and so on – lots of amazingly real and rather negative experiences from the real life. She also confirmed my purely web-based research (and theoretical derivations) indicating that kids coming from this method can't be expected to have good results in conventional mathematical olympiads (starting with the Pythagoriad for the fifth graders). When several questions were asked, I warned high school principals and others that they should better study this issue because they may face the first waves of this experiment – whole years of kids who may know even much less than those in recent years.

Dr Pokorný added lots of real world data – and explanations why, when he needs to tutor his kids, he is basically forced to ignore all the Hejný idiosyncrasies and start from scratch. In the afternoon, a playful Slovak university teacher of mathematics analyzed the principles, somewhat like I did, with similar conclusions and using perhaps harsher words than what I used – but he also controlled himself despite his previous announcements that he would have been even tougher LOL. The mastermind of the conference Dr Dlab showed how teaching of mathematics is arts and how arithmetic, quadratic, and higher-order sequences (especially involving integers) may be helpfully used to open the children's eyes about many things. Dr Jindřich Bečvář gave a realistic picture of the status quo in mathematics education which was rather frustrating, full of vicious circles, and so on.

Lots of extra related topics about the education bubble, dropping requirements at schools, especially in other subjects (the requirements didn't drop as quickly in mathematics which is why mathematics is obviously even more unpopular than it was before 1989, for example, relatively speaking), comparisons with other countries etc. etc. were said during the discussion. It's a broad topic with lots of adjacent topics. Kids' dependence on cell phones and biological evolution ignited by these fast changes were also discussed – I was one who tried to calm this particular worry.

So I think that we, the critics, have made a persuasive case against Hejný's method – and even at the level of authorities, people (e.g. Czech TV and other media that were present) could have seen that the elite of the mathematical and mathematics-teaching institutes and associations in Czechia basically think the same thing and they feel that Hejný's method is going mostly in the wrong direction.

What was equally remarkable was the almost complete inability of the fans of Hejný's method to explain what the method is or why it's supposed to be good. Hejný said just a couple of general clichés about the need to love children etc. – I've talked to him for the first time but I've only heard the things that I've heard from those videos, anyway. He hasn't really answered any of the 10 questions that were asked to him by several folks (I asked one, what he would like to happen to a hypothetical ingenious 8-year-old future boy Euler from the Czech basin, who has mastered the adult mathematics at his age and wants to go further but who happens to sit in Hejný's classroom; it remained 100% unanswered). At least Hejný agreed to sit in the panel during the press conference, along with 4 people who may be described as opponents. None of his allies agreed to debate.

The teachers – fans of Hejný's method – were (even) much more incapable of contributing anything. The two other pro-Hejný talk showed us videos of kids from the classroom. How do kids solve some problem? So for example, you have two numbers \(x,y\) and \(x+y=17\) and \(x-y=3\) (they don't use this language but I hope you agree it's the best one). The results are 10 and 7. So for 20 minutes, the kids chaotically try to find the right solution by some guesswork. Well, it was extremely inefficient when it came to time. And when someone found something important, it wasn't elevated. When there were mistakes, they were never corrected. A kid is more likely to learn something wrong than something right because the number of wrong things said during such episodes of chaos are much higher. The teachers didn't say much. They just made it implicitly clear that they support the chaos.

There was another problem – a cyclic snake of a sort which may be solved by an equation. So some smart enough kid found the key and solved it using \(x\) and equations. Others who answered at all just copied the result from the smart kid. (Hejný's method also prefers teamwork at all times and I wasn't the only one who said that it unavoidably turns most of the schoolkids into freeloaders while the under-appreciated drivers of the team gradually lose motivation, too.) The teacher was surprised where this great idea has come from. Well, let me tell you where, some people including me pointed out. The kid has learned the mathematics properly from other sources at home – either in solitude, with some better books, or with some parents or someone else. There's really basically no equation with \(x\) in those textbooks so this is what had to happen. Those tutors are the real teachers and the only good thing that the Hejný's method teacher did was to allow the smart kids (tutored by someone else) to teach others in this otherwise failed classroom. I think that the parents of the "smart kids" are those who should be actually getting the salary instead of the "mathematics teacher".

You could see that those teachers didn't give a damn whether the kids try to guess the solution or whether they master any general enough method – which would be equivalent to the solving of equations. These strategies to approach the problems clearly differ by several years of sophistication and maturity but the teachers just don't give a damn about the difference. Guessing integers – and perhaps copying from a classmate – is just as good as mastering methods to solve linear equations and equivalent problems.

But what was even more spectacular was the whining by the social justice warriors – and I think it's totally accurate to describe them primarily by this phrase – who were attacking the Hejný's method's skeptics and the organizers. During the periods for questions and the general debates, there was a dozen of truly hateful rants over there. About 70% of them were addressed to me. They were usually very long and extremely personal. Those people have apparently decided to go there and attack me because they had already known me from my Czech blog that has become a beacon of the Hejný's method's skeptics. So the organizers were called heretics for having dared to invite me and stuff like that. What I said were claimed to be distortions, and so on. Everyone who has studied it or at least followed the conference could have checked that especially given the 25-minute time limit, my summaries were extremely honest, the evaluations were right of the money, and all my critics were just completely full of šit.

I haven't used any word like šit over there. I was clearly told to be polite, I expected myself to be polite as well, and numerous people told me that I succeeded. But I did defend myself in some way, saying that regardless of myself, it's right for people like me, with the similar background and experience, to co-decide about such issues (how mathematics should be taught). It's plausible that the first, male critic of mine was one of the insufferable commenters on my Czech blog. I don't remember these people's names. So apologies if he expects my personal hatred to be reciprocal – I am just unable to do it for him. If he were a commenter from my Czech blog, I consider him just another anonymous, obnoxious, and irrelevant troll and today's interactions were just another confirmation of the rationality of this classification.

About three ladies who made "seemingly general" critical comments about the conference betrayed the fact that their anger was directed against me as well. But they had to suffer because almost all the speakers were really saying stuff that was more or less equivalent to mine, just formulated in a somewhat ambiguous way that doesn't make them so angry, but I suspect that their peabrains were marginally able to figure out that the whole room was basically against them. ;-)

But many of the criticisms were targeting the organizers of the conference such as Dr Rákosník, even for things unrelated to me. The conference was called "New methods to teach mathematics?". So they expected some general methods to teach to be outlined there, they told us, and they were totally disappointed. Holy crap. First of all, one problem is that these ladies – and unfortunately, way too many similar ladies as well as men – only read the titles. They're extremely superficial. If they could read something from the program that was available, they would have understood it was clearly mainly a debate comparing claimed positives and negatives of this particular method. Moreover, the question mark in the title itself makes it clear that the event wasn't what they said to have expected.

But what these ladies have said was much lousier than that.

The female teachers who were giving the "positive talks about Hejný's method" were absolutely shocked by the "revelation" that there could have been questions or a debate. No one has told them that someone would disagree with them over there. I kid you not. Well, first of all, this proves that they're illiterate because things like that were clearly enough written on the program. Second of all, it shows that they're absolutely incapable of doing any honest intellectual work. They just make meaningless talks playing some uninteresting videos with children's generic, partly dumb and partly semi-smart, reactions to a mathematical problem, and it's just considered enough. They don't have to think, they don't have to know anything about mathematics, they don't need to do anything. Hejný is considered "cool" enough by their fellow social justice warriors that this doing nothing is basically enough. They may be giving talks as if they were experts on teaching mathematics although they are clearly unable to answer a single question about the teaching of mathematics, even through their own method, and they couldn't even consider to be in a panel that receives questions.

So Dr Rákosník has also concluded by saying that he wanted to learn what the method actually was and what is its goal or justification but he couldn't because their presence and talks were basically useless – he used somewhat more polite words but he responded in anger, too.

What is the powerful community that has promoted this nobody (former kid who needed to be tutored by his dad and that's how the kid became famous almost 80 years later) from an inferior field – teaching methods in mathematics – to a new Son of God? It's obviously lots of the SJW types in the media, mostly female ones, who have just hated mathematics throughout their life, especially because they have always sucked at it. (I haven't said explicit sequences of thoughts like that today which could have been too personal and provoking. And even here, I am talking about principle, I obviously don't have any personal relationship to anyone whom I describe.) They have hijacked much of the education system and contaminated it by pro-European-Union, genderist, environmentalist, and similar types of shameless indoctrination ("cross sectional themes"), and now they simply want to destroy even the classes of mathematics itself and convert them to another lesson in indoctrination where no hard knowledge is expected.

But it's not really just outsiders. Today, we were also exposed to several "insiders", teachers from basic schools and maybe a high school who hated the conference and who love Hejný exactly for the reason that is explained by the most frequently repeated Hejný's slogan from the conference – his slogan that "mathematics teachers don't even have to know mathematics". Why do they love Hejný and his slogans? Because they are these teachers – mathematics teachers who know nothing about mathematics. They know nothing about mathematical logic, rational thinking, proofs, or fair argumentation, for that matter. Or about reading anything below the title. They know how to spend some hours with kids in the classroom and exploiting the protection by Hejný's indefensible aura, they simply earn the average salary for doing nothing useful at all. Can you be shocked that they just instinctively defend (without any arguments) everything that is Hejný's and attacking everything that sounds as anti-Hejný?

Well, it has become easy to flood the schools with lots of such teachers who shouldn't really be there. Hejný's and other types of non-teaching are both tolerated and much easier than some real hard pedagogic work – so it's guaranteed that their percentage is increasing and will be increasing.

At some points, these people were trying to switch the debate to their usual regime whose basic assumption is simply hatred towards mathematics and the promotion of the hatred towards mathematics. "What will you do with the large number of kids who hate mathematics at school?" Well, I think that most of this hatred is deliberately social engineered. So what needs to be done is perhaps primarily to fight against the statements that mathematics is hated or should be hated. Mathematics is hated by some people and kids but it's also loved by another group, arguably much more important, group of people and kids, and those are those who should have the upper hand if the status of mathematics is supposed to improve. For decades, we have followed the recommendations of similar SJW-like people and the results were clearly getting worse. It's reasonable to expect that the status of mathematics will be getting even worse if these people continue to influence these matters.

It may be fun to spend a polite day with these people, pretending that we are together looking for some nice and clever ways how to improve the children's relationship to mathematics and their mathematical skills. But the truth is that we're not really a team that is looking for something that will elevate the status and knowledge of mathematics among the kids. Instead, what's going on is that there is a clear battle between those who want to preserve and re-strengthen mathematics at schools; and those who are actively doing everything they can for the present deterioration to continue and even accelerate. This is a war of its own kind and the evil people have to be defeated, perhaps by some aggressive enough new minister of education (perhaps by Václav Klaus Jr), otherwise there can't be any real progress.

And that's the memo.

P.S.: Aside from Ms Mazáčová, the totally smart and sensible teacher who talked after me, there were actually several additional intelligent teachers and even a very young, fresh male teacher who is still really enthusiastic about the teaching process – and who is also in touch with YouTube, as Prof Dlab and myself defined ourselves. ;-) I wish he will preserve his enthusiasm for quite some time and I recommended him to learn rapping and start a successful YouTube channel to enhance the people whom he infects with the good attitude to mathematics from dozens in his classroom (that he probably rightfully bragged about) to millions on YouTube – otherwise his nice contribution has no chance to beat the elephant in the room with millions or billions of views, YouTube videos that generally encourage the kids to hate mathematics.



Off-topic: Elon Musk is worshiped for having sent his Tesla car to a trash bin in the Universe so that it doesn't contaminate Earth. But that's nothing spectacular. Managers of French Škoda, the Volkswagen-owned Czech car brand, launched a Škoda Superb towards Mars. The driver was filmed while greeting the inhabitants of Mars, too.

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