A Conference On Pedagogy Of Mathematics

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

Most people believe that the method is named after Prof Milan Hejný who was of late celebrating his 80th birthday – as he reminded us several times – but it's really named after his virile individual bring upwards who has taught his boy Milan how to beloved mathematics. This gospel is beingness used past times 700 out of 4,100 basic schools inwards Czech Republic in addition to worshiped past times virtually all the mainstream media that write well-nigh these things.

The basic philosophy of the method is that the instructor shouldn't have got whatever authorisation inwards the class, he or she should do basically zip amongst the children allow lone to teach, in addition to the kids should play in addition to discovery all of import ideas past times themselves. At most, several standardized exercises from recreational mathematics are encouraged to live repeated. The instructor doesn't right mistakes when they're made, in addition to so on, in addition to so on.




Today's conference was masterminded past times Prof Vlastimil Dlab, a buddy who is also unopen to 80, who is a rather achieved if non renowned Czech Canadian mathematician, dissimilar Mr Hejný. His views are basically the same as mine – but that was also just about the instance of lots of other famous mathematicians I was honored to come across again.

There was Dr Jiří Rákosník, the manager of the Mathematical Institute where the lawsuit took house (we had some tensions during preparations but those evaporated today – for example, I should have got straight off believed his recommendation that PowerPoint shouldn't utilization dark text on crimson – the contrast really is also depression amongst most projectors); Dr Josef Kubát who is the chairman of the Czech Union of Mathematicians in addition to Physicists (of course of instruction I've known his bespeak years); Dr Jindřich Bečvář who was quoted as a transcend guru (teacher in addition to the writer of some of import pedagogic texts) past times 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 really a boss of a squad of contestants of a mathematical olympiad (which included myself) during a summertime camp, in addition to I am non certain whether it was xxx or 35 years ago. And some extremely smart ladies in addition to so on. If you lot don't see yourself here, it doesn't intend that I don't beloved you.

The morning time session was composed of 2 introductions, in addition to iii talks inwards favor of Hejný's method (by Hejný in addition to 2 teachers that promote the method); in addition to iii talks past times us, the critics – myself as a physics-like user of mathematics; an fantabulous woman individual mathematics instructor whose evaluations, goals, in addition to all comments just made perfect feel to me; in addition to Dr Jiří Pokorný from my Alma Mater whose verbalize was also bright in addition to who spoke mainly as a bring upwards of kids who needed to live tutored, specially when exposed to Hejný's method.




Now, Hejný's method may live somewhat "well-defined" but what isn't quite well-defined is the "non-Hejný's method". When you lot avoid Hejný's method, or when you lot utilization to a greater extent than traditional in addition to conventional methods in addition to advertizement hoc tricks to explicate something etc., you lot unavoidably exploit lots of memes, analogies, in addition to types of exercises that Hejný's recipe recommends, too.

So we, the critics, specially Dr Pokorný who made that point, agreed that some of Hejný's types of exercises could really good live helpfully used inwards all – perhaps otherwise "ordinary" – mathematics classrooms. Hejný's method mainly differs from the "rest of teaching of mathematics" past times having an overhyped trademark. But the "type of stuff" it's doing isn't also different from some other mathematics classroom types that don't have got whatever catchy names.

I sort of knew that in addition to that the give-and-take could mayhap degrade into some ho-hum technicalities – whether you lot should write the snakes (multiply past times three, write to the following box, in addition to so on) as trees or circles or whatever other particular similar that. But I am absolutely certain that fifty-fifty amongst Hejný's method, the give-and-take is well-nigh agency grander inquiry than that (even when 2 mathematics classes, Hejný's in addition to non-Hejný's, facial expression similar, in that place is yet some underlying vision in addition to philosophy that heavily affects the development inwards the large painting in addition to the long run), in addition to I tried difficult to opened upwards the original topics inwards a agency that focuses on the large picture.

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

So Hejný has principles that all the children should exclusively larn mathematical things using really specific situations in addition to environments – exclusively situations in addition to environments they already know in addition to are familiar amongst (in particular, they should exclusively count windows inwards their bird in addition to distinguish their gramps from their grandmother, Hejný proudly told us if I exaggerate the "exclusiveness" of these tasks just a trivial bit; if you lot desire to know, smartest kids "learn non-commutativity" past times seeing that mother's virile individual bring upwards isn't the same as father's woman bring upwards LOL). They should never generalize. They should solve problems past times guessing possible answers. Mathematics classes should learn the kids what to think well-nigh the guild – e.g. disrespect for government in addition to some social values. They should utilization non-standard notations. Children who know formulae are intellectual parasites. Teachers should exclusively allow insights that kids build themselves – no transferred cognition is permissible. The kids should memorize some beast codes for numbers – where a moo-cow equals 2 goats, 3 cows equals a pig, in addition to materials similar that. People should exclusively aid well-nigh the happiness of the kids inwards the classroom in addition to when the shape is over, everything is fine.

25 things similar that – taken from webs, talks, classroom videos, in addition to so on. I am mostly a theorist here, too. You tin imagine what I said. The exact negations of these – extracted in addition to concentrated – principles. Mathematics is all well-nigh generalizations. General methods to solve things must live taught when they exist. The mightiness of mathematics is the mightiness to travel inwards completely novel situations in addition to solve an unlimited number of tasks. The instructor has to right mistakes soon, otherwise they travel out of command in addition to the shape becomes meaningless chaos. Answers shouldn't live guessed. Notation in addition to terminology should improve live criterion because kids volition have got to larn it presently or later, anyway. Most things nosotros learned were flora past times someone else – just similar most products nosotros utilization were produced past times someone else. Kids loving formulae are kids loving mathematics in addition to should live utilization models for other kids. Mathematics classes have got no concern of becoming a novel politically right indoctrination course of instruction – fifty-fifty communists shielded the queen of sciences from Marxism. They have got no concern of becoming visual arts, either, fifty-fifty though some kids would clearly prefer some other arts class, in addition to so on in addition to so on.

So some of the philosophical assumptions were clearly outlined. My start speaking ally, the attractive plenty mathematics teacher, showed lots of mistakes in addition to omissions inwards Hejný's method textbooks, what the kids using this method don't know at i historic menstruum or another, how the teachers have got a difficult fourth dimension to adapt in addition to half of them leave, in addition to so on – lots of amazingly existent in addition to rather negative experiences from the existent life. She also confirmed my purely web-based inquiry (and theoretical derivations) indicating that kids coming from this method can't live expected to have got goodness results inwards conventional mathematical olympiads (starting amongst the Pythagoriad for the 5th graders). When several questions were asked, I warned high schoolhouse principals in addition to others that they should improve written report this number because they may human face upwards the start waves of this experiment – whole years of kids who may know fifty-fifty much less than those inwards recent years.

Dr Pokorný added lots of existent public information – in addition to explanations why, when he needs to tutor his kids, he is basically forced to ignore all the Hejný idiosyncrasies in addition to start from scratch. In the afternoon, a playful Slovak academy instructor of mathematics analyzed the principles, somewhat similar I did, amongst similar conclusions in addition to using perhaps harsher words than what I used – but he also controlled himself despite his previous announcements that he would have got been fifty-fifty tougher LOL. The mastermind of the conference Dr Dlab showed how teaching of mathematics is arts in addition to how arithmetic, quadratic, in addition to higher-order sequences (especially involving integers) may live helpfully used to opened upwards the children's eyes well-nigh many things. Dr Jindřich Bečvář gave a realistic painting of the condition quo inwards mathematics instruction which was rather frustrating, total of vicious circles, in addition to so on.

Lots of extra related topics well-nigh the instruction bubble, dropping requirements at schools, specially inwards other subjects (the requirements didn't driblet as apace inwards mathematics which is why mathematics is evidently fifty-fifty to a greater extent than unpopular than it was earlier 1989, for example, relatively speaking), comparisons amongst other countries etc. etc. were said during the discussion. It's a wide topic amongst lots of following topics. Kids' dependence on jail cellular telephone phones in addition to biological development ignited past times these fast changes were also discussed – I was i who tried to calm this particular worry.

So I think that we, the critics, have got made a persuasive instance against Hejný's method – in addition to fifty-fifty at the storey of authorities, people (e.g. Czech TV in addition to other media that were present) could have got seen that the elite of the mathematical in addition to mathematics-teaching institutes in addition to associations inwards Czech Republic basically think the same thing in addition to they experience that Hejný's method is going mostly inwards the incorrect direction.

What was every bit remarkable was the almost consummate inability of the fans of Hejný's method to explicate what the method is or why it's supposed to live good. Hejný said just a yoke of full general clichés well-nigh the demand to beloved children etc. – I've talked to him for the start fourth dimension but I've exclusively heard the things that I've heard from those videos, anyway. He hasn't really answered whatever of the 10 questions that were asked to him past times several folks (I asked one, what he would similar to hand off to a hypothetical ingenious 8-year-old hereafter man child Euler from the Czech basin, who has mastered the adult mathematics at his historic menstruum in addition to wants to travel farther but who happens to sit down inwards Hejný's classroom; it remained 100% unanswered). At to the lowest degree Hejný agreed to sit down inwards the panel during the press conference, along amongst 4 people who may live described as opponents. None of his allies agreed to debate.

The teachers – fans of Hejný's method – were (even) much to a greater extent than incapable of contributing anything. The 2 other pro-Hejný verbalize showed us videos of kids from the classroom. How do kids solve some problem? So for example, you lot have got 2 numbers \(x,y\) in addition to \(x+y=17\) in addition to \(x-y=3\) (they don't utilization this linguistic communication but I promise you lot handgrip it's the best one). The results are 10 in addition to 7. So for 20 minutes, the kids chaotically endeavour to notice the right solution past times some guesswork. Well, it was extremely inefficient when it came to time. And when someone flora something important, it wasn't elevated. When in that place were mistakes, they were never corrected. Influenza A virus subtype H5N1 kid is to a greater extent than probable to larn something incorrect than something right because the number of incorrect things said during such episodes of chaos are much higher. The teachers didn't say much. They just made it implicitly clear that they back upwards the chaos.

There was some other work – a cyclic serpent of a sort which may live solved past times an equation. So some smart plenty kid flora the primal in addition to solved it using \(x\) in addition to equations. Others who answered at all just copied the outcome from the smart kid. (Hejný's method also prefers teamwork at all times in addition to I wasn't the exclusively i who said that it unavoidably turns most of the schoolkids into freeloaders spell the under-appreciated drivers of the squad gradually lose motivation, too.) The instructor was surprised where this bully thought has come upwards from. Well, allow me tell you lot where, some people including me pointed out. The kid has learned the mathematics properly from other sources at abode – either inwards solitude, amongst some improve books, or amongst some parents or someone else. There's really basically no equation amongst \(x\) inwards those textbooks so this is what had to happen. Those tutors are the existent teachers in addition to the exclusively goodness thing that the Hejný's method instructor did was to allow the smart kids (tutored past times someone else) to learn others inwards this otherwise failed classroom. I think that the parents of the "smart kids" are those who should live really getting the salary instead of the "mathematics teacher".

You could see that those teachers didn't give a damn whether the kids endeavour to estimate the solution or whether they original whatever full general plenty method – which would live equivalent to the solving of equations. These strategies to approach the problems clearly differ past times several years of sophistication in addition to maturity but the teachers just don't give a damn well-nigh the difference. Guessing integers – in addition to perhaps copying from a classmate – is just as goodness as mastering methods to solve linear equations in addition to equivalent problems.

But what was fifty-fifty to a greater extent than spectacular was the whining past times the social jurist warriors – in addition to I think it's totally accurate to depict them primarily past times this phrase – who were attacking the Hejný's method's skeptics in addition to the organizers. During the periods for questions in addition to the full general debates, in that place was a dozen of genuinely intend rants over there. About 70% of them were addressed to me. They were normally really long in addition to extremely personal. Those people have got apparently decided to travel in that place in addition to assault me because they had already known me from my Czech weblog that has acquire a beacon of the Hejný's method's skeptics. So the organizers were called heretics for having dared to invite me in addition to materials similar that. What I said were claimed to live distortions, in addition to so on. Everyone who has studied it or at to the lowest degree followed the conference could have got checked that specially given the 25-minute fourth dimension limit, my summaries were extremely honest, the evaluations were right of the money, in addition to all my critics were just completely total of šit.

I haven't used whatever tidings similar šit over there. I was clearly told to live polite, I expected myself to live polite as well, in addition to numerous people told me that I succeeded. But I did defend myself inwards some way, maxim that regardless of myself, it's right for people similar me, amongst the similar background in addition to experience, to co-decide well-nigh such issues (how mathematics should live taught). It's plausible that the first, virile individual critic of mine was i of the insufferable commenters on my Czech blog. I don't call back these people's names. So apologies if he expects my personal hatred to live reciprocal – I am just unable to do it for him. If he were a commenter from my Czech blog, I consider him just some other anonymous, obnoxious, in addition to irrelevant troll in addition to today's interactions were just some other confirmation of the rationality of this classification.

About iii ladies who made "seemingly general" critical comments well-nigh the conference betrayed the fact that their anger was directed against me as well. But they had to endure because almost all the speakers were really maxim materials that was to a greater extent than or less equivalent to mine, just formulated inwards a somewhat ambiguous agency that doesn't build 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, fifty-fifty for things unrelated to me. The conference was called "New methods to learn mathematics?". So they expected some full general methods to learn to live outlined there, they told us, in addition to they were totally disappointed. Holy crap. First of all, i work is that these ladies – in addition to unfortunately, agency also many similar ladies as good as men – exclusively read the titles. They're extremely superficial. If they could read something from the computer program that was available, they would have got understood it was clearly mainly a ground comparing claimed positives in addition to negatives of this particular method. Moreover, the inquiry grade inwards the championship itself makes it clear that the lawsuit wasn't what they said to have got expected.

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

The woman individual teachers who were giving the "positive talks well-nigh Hejný's method" were absolutely shocked past times the "revelation" that in that place could have got been questions or a debate. No i has told them that someone would disagree amongst them over there. I kid you lot not. Well, start of all, this proves that they're illiterate because things similar that were clearly plenty written on the program. Second of all, it shows that they're absolutely incapable of doing whatever honest intellectual work. They just build meaningless talks playing some uninteresting videos amongst children's generic, partly dumb in addition to partly semi-smart, reactions to a mathematical problem, in addition to it's just considered enough. They don't have got to think, they don't have got to know anything well-nigh mathematics, they don't demand to do anything. Hejný is considered "cool" plenty past times their swain social jurist warriors that this doing zip is basically enough. They may live giving talks as if they were experts on teaching mathematics although they are clearly unable to respond a unmarried inquiry well-nigh the teaching of mathematics, fifty-fifty through their ain method, in addition to they couldn't fifty-fifty consider to live inwards a panel that receives questions.

So Dr Rákosník has also concluded past times maxim that he wanted to larn what the method really was in addition to what is its destination or justification but he couldn't because their presence in addition to talks were basically useless – he used somewhat to a greater extent than polite words but he responded inwards anger, too.

What is the powerful community that has promoted this nobody (former kid who needed to live tutored past times his dad in addition to that's how the kid became famous almost lxxx years later) from an inferior acre – teaching methods inwards mathematics – to a novel Son of God? It's evidently lots of the SJW types inwards the media, mostly woman individual ones, who have got just hated mathematics throughout their life, specially because they have got e'er sucked at it. (I haven't said explicit sequences of thoughts similar that today which could have got been also personal in addition to provoking. And fifty-fifty here, I am talking well-nigh principle, I evidently don't have got whatever personal human relationship to anyone whom I describe.) They have got hijacked much of the instruction arrangement in addition to contaminated it past times pro-European-Union, genderist, environmentalist, in addition to similar types of shameless indoctrination ("cross sectional themes"), in addition to at nowadays they just desire to destroy fifty-fifty the classes of mathematics itself in addition to convert them to some other lesson inwards indoctrination where no difficult cognition is expected.

But it's non really just outsiders. Today, nosotros were also exposed to several "insiders", teachers from basic schools in addition to maybe a high schoolhouse who hated the conference in addition to who beloved Hejný precisely for the ground that is explained past times the most often repeated Hejný's slogan from the conference – his slogan that "mathematics teachers don't fifty-fifty have got to know mathematics". Why do they beloved Hejný in addition to his slogans? Because they are these teachers – mathematics teachers who know zip well-nigh mathematics. They know zip well-nigh mathematical logic, rational thinking, proofs, or fair argumentation, for that matter. Or well-nigh reading anything below the title. They know how to pass some hours amongst kids inwards the classroom in addition to exploiting the protection past times Hejný's indefensible aura, they just earn the average salary for doing zip useful at all. Can you lot live shocked that they just instinctively defend (without whatever arguments) everything that is Hejný's in addition to attacking everything that sounds as anti-Hejný?

Well, it has acquire tardily to alluvion the schools amongst lots of such teachers who shouldn't really live there. Hejný's in addition to other types of non-teaching are both tolerated in addition to much easier than some existent difficult pedagogic function – so it's guaranteed that their pct is increasing in addition to volition live increasing.

At some points, these people were trying to switch the ground to their commons regime whose basic supposition is just hatred towards mathematics in addition to the advertisement of the hatred towards mathematics. "What volition you lot do amongst the large number of kids who loathe mathematics at school?" Well, I think that most of this hatred is deliberately social engineered. So what needs to live done is perhaps primarily to struggle against the statements that mathematics is hated or should live hated. Mathematics is hated past times some people in addition to kids but it's also loved past times some other group, arguably much to a greater extent than important, grouping of people in addition to kids, in addition to those are those who should have got the upper manus if the condition of mathematics is supposed to improve. For decades, nosotros have got followed the recommendations of similar SJW-like people in addition to the results were clearly getting worse. It's reasonable to facial expression that the condition of mathematics volition live getting fifty-fifty worse if these people hold to influence these matters.

It may live fun to pass a polite twenty-four hr menstruum amongst these people, pretending that nosotros are together looking for some overnice in addition to clever ways how to improve the children's human relationship to mathematics in addition to their mathematical skills. But the truth is that we're non really a squad that is looking for something that volition elevate the condition in addition to cognition of mathematics amidst the kids. Instead, what's going on is that in that place is a clear battle betwixt those who desire to save in addition to re-strengthen mathematics at schools; in addition to those who are actively doing everything they tin for the introduce deterioration to hold in addition to fifty-fifty accelerate. This is a state of war of its ain form in addition to the evil people have got to live defeated, perhaps past times some aggressive plenty novel government minister of instruction (perhaps past times Václav Klaus Jr), otherwise in that place can't live whatever existent progress.

And that's the memo.

P.S.: Aside from Ms Mazáčová, the totally smart in addition to sensible instructor who talked after me, in that place were really several additional intelligent teachers in addition to fifty-fifty a really young, fresh virile individual instructor who is yet really enthusiastic well-nigh the teaching procedure – in addition to who is also inwards touching amongst YouTube, as Prof Dlab in addition to myself defined ourselves. ;-) I want he volition save his enthusiasm for quite some fourth dimension in addition to I recommended him to larn rapping in addition to start a successful YouTube channel to lift the people whom he infects amongst the goodness mental attitude to mathematics from dozens inwards his classroom (that he in all probability rightfully bragged about) to millions on YouTube – otherwise his overnice contribution has no lead chances to rhythm out the elephant inwards the room amongst millions or billions of views, YouTube videos that to a greater extent than often than non encourage the kids to loathe mathematics.



Off-topic: Elon Musk is worshiped for having sent his Tesla car to a trash bin inwards the Universe so that it doesn't contaminate Earth. But that's zip spectacular. Managers of French Škoda, the Volkswagen-owned Czech car brand, launched a Škoda Superb towards Mars. The driver was filmed spell greeting the inhabitants of Mars, too.