**Most of the problems aren't the problem, the framing is**

I want some of you to spend some time by playing the games – or solving the mathematics problems – at

MATIKA.IN (English, play now)The website includes the interactive environments or "schemes" which are mostly equivalent to everything that pupils exposed to Hejný's method, the most prominent orthodox constructivist education method we know in Czechia these days, have to solve during their 8 years in the elementary school.

Helpfully enough, the website is available in English (and 7 other languages). It was coded by Andrej Probst, a young guy doing these things as a charity (who is mostly independent of the movement that spreads the method, as far as I can say), and is used by schools in Czechia, Slovakia, and Hungary.

In Czechia, there are some 4,100 elementary schools. 800 of them teach mathematics by Hejný's method. 200 of those have learned about the website and at least recommend it to the kids.

The website is some straightforward JavaScript but it looks fresh and helpful, partly because of illustrations by kovidesign. I surely believe that such websites should be used in conventional mathematics education – and in other subjects, too.

You will know basically everything about Hejný's method – and what it tries to teach the first graders (who get 7 years old sometimes during the school year)... and up to the eighth graders (14 years old sometimes during that last year) – if you spend an hour with this website, and if you learn about the broader philosophy of the method: The teacher always wants the kids to feel good, she never teaches any theory, there aren't any textbooks with theory, they don't learn any general rules or formulae, there are just textbooks with similar exercises, kids always work collectively and correct each other, they must learn everything by themselves.

The teacher is sidelined and ideally eliminated. This is in no way a Czech or Slovak (Hejný is mixed) invention. Americans may remember the math wars of the 1990s that were mostly about some radical constructivists' efforts to eliminate teachers or instructors from the math education.

It would be silly to denounce every problem on the page. After all, many of them are analogous to exercises that almost

*any*approach to mathematics education has to include. So in my eyes, most of the lethal defects in the method are in the philosophy (including the relative truth and the idea that kids should totally ignore everything that was invented by other people); in the excessive repetition of these puzzles we will discuss in detail; and especially in the things that are missing for the same reason (the time is already spent on the repetition of the standardized childish puzzles).

But sometimes, even what is included is just painful.

OK, return to the website. First graders do lots of stepping, sometimes on the horizontal floor, sometimes on staircases (see YouTube videos with stepping, some of them are kindergartens). The arrow to the right or left means to add or subtract one, respectively. So you're supposed to solve things like "10 leftarrow leftarrow leftarrow" and write "7" as the answer, OK? Without the jargon, the kids use some "Abelian group theory". They remember their state (location) before the steps, and after the steps (adding or subtracting one). That can help them to internalize the numbers and perhaps learn about zero and negative numbers, too.

I have nothing against this method by itself – as something that belongs "somewhere". I just think that the amount of time they (and much older kids) spend by marching and stepping is orders of magnitude higher than it should be. The kids are effectively trained to add larger numbers, like 8+9, by thinking about 17 steps, one by one, in their heads. The method is absolutely obsessed with the elimination or delaying any memorization – such as the fact that 8+9=17. You just can't get too far in mathematics if you're scared of similar "memorizable results" that dramatically speed up some problems or "trivialize" a class of tasks. In some sense, all the power of mathematics

*is*about this speedup and trivialization!

OK, first graders do see some conventional addition and subtraction of the small numbers, after all.

Then they deal with pyramids. There's a triangle of boxes, like in Pascal's triangle, and the top box or bottom box is assigned the number that is the sum of two boxes underneath or above this box. You fill in the missing integers. Again, there's nothing wrong about this kind of a puzzle if it appears in isolation. What's wrong is the obsession with this problem – which, in Hejný's classrooms, becomes basically 1/10 of all of mathematics of the elementary school.

Pyramids continue up to the sixth grade. The only progress is that one quickly gets from 3-level pyramids to 4-level pyramids; occasionally, you need to subtract because some other numbers in the boxes are missing, not the sums; and the sixth graders add not just integers but numbers like 7.9 – one digit after the decimal point. Is that the appropriate progress after four years of playing with this exercise? I don't think so.

Triangles are virtually the same thing as pyramids except that more numbers are missing. That would make it too hard or ambiguous so you're often given the list of numbers that are missing – in the wrong order. Or you're told some extra condition that the missing numbers satisfy. So all these problems end up being "solved" by brute force, simply by trying all possible permutations or small integers that can be filled somewhere. That's how completely uninformed people may solve Sudoku – after all, it's almost the same thing. But there's no mathematics in it because mathematics only starts once you find some cleverness to deal with the simple objects around you.

Triangles are a bit less straightforward than pyramids so they continue up to the sixth grade. Six years of repeating the same kind of problems that you're supposed to solve almost entirely by brute force, without learning any methods, formulae, tricks, anything. You know, mathematics has a lot of levels of abstraction – everyone who is close to mathematics must totally know what I am talking about. The kids in Hejný's classroom always stay on the ground floor mentally; they only get more experienced with the ground floor.

Snakes and spider webs (they look like similar problems). Boxes are connected with arrows. Each oriented arrow means "plus 3 equals" or "times 4 equals" and you have to fill in the number to the boxes or to the arrows. Thank God, that environment only continues through the third grade. Spider webs are a bit more complex and "two-dimensional", snakes are mostly one-dimensional. In spider webs, only addition is trained, it seems, and it continues through the fifth grade. In the spider web above, only three colored "additive arrows" are used.

*Fill in the numbers so that the sum of 3 neighbors is always 8. Obviously, the rule only works for 3x1 or 1x3 neighbors, not for L-shape neighbors around a corner.*

Neighbors. Another puzzle similar to pyramids or triangles but the numbers are in a rectangle and kids memorize some particular conditions on neighbors that has to be true in all the problems of this kind. You may see that the exercises become increasingly arbitrary and kids are memorizing an increasing number of nonsensical rules that have nothing to do with mathematics or its application anywhere in the world. Neighbors continue to the third grade.

Fairground (well, I think that the correct translation is an exhibition hall but who cares). You go along a path in a 3 by 3 square and the digits 1,2,...9 successively appear on your path. 2 or 3 digits are there, you need to complete the remaining 7 or 6. A straightforward puzzle but it "enriches" the kid in a direction that has little to do with what I call mathematics. Like in most similar problems, one just tries the possibilities by brute force (it's usually trivial).

Buses. There are 7 people in the bus before one stop. Then 3 people get in or get out. How many people are in the bus afterwards? Through several stations, you need to keep track of the number of people in the bus. The first graders have about 10 people in the bus. The exercise appears in the sixth grade as well – and the progress? The sixth graders have 18 people in the bus or so. Wow, what an impressive progress in just 5 years.

Word problems. John has 10 dumplings, Anne has more by 2, how much is 2+2? The answer is 22.

Now, to wrap the first grade, there are lots of "games" that are variations of Don't get angry, buddy, a German children's game popular in Czechia where you roll a dice and advance your piece along a path. Sometimes you may have to solve some problems at some positions. Shockingly, these games continue up to the eighth grade. When I was a kid, no one pretended that "we were learning mathematics" when playing games like Don't get angry, buddy. ;-) But this is an example of the general goal: to sell ordinary games that kids play outside school as "mathematics".

OK, so lots of these problems have appeared in the first grade! So the first grade could have been OK. Kids aren't learning any "real mathematics" in the first grade, anyway? Indeed, I think that the method becomes increasingly indefensible if you look at the older kids. You just see that there's no progress.

OK, you get to the second graders. What is added?

Dog handlers. One counts how many dogs an owner has and how many eyes the dogs have. I could solve 99% of the problems immediately but I just couldn't figure out what I should add into a 4 by 4 table with 20 in the lower right corner. The problems looks totally underdefined to me. I suspect that they're adding eyes and dogs – probably because they didn't have enough apples and oranges. ;-)

There's some "multiplication square", it's like the snakes or spider webs with the operations' being multiplication and some special rules. Again, this type of puzzle – with the exact same geometry of the square and basic rules – is taught between 2nd grade and 5th grade.

Biland. The kids are basically learning to convert small enough integers to the binary form. It continues to the third grade. It's an exceptional case in which one could argue that too young kids are exposed to some material. But maybe it's OK. Binary code may be important for some portions of computer science but it's a classic "dead end" that leads nowhere.

Parquettes. Cover some rectangle with several prescribed tetris-like or smaller pieces. Is that really mathematics?

*The symbols "ice cream, cat, peace, camel, and bird" actually denote "mouse, cat, goose, dog, goat".*

The most shocking addition: daddy Forrest. It should really be granddaddy but OK. Kids have to memorize that 1,2,3,4,5, [??], 10, 20 should be written as mouse, cat, goose, dog, goat... [sheep, ram], cow, horse. See this video by a bunch of fifth-graders explaining the values. The kids have to memorize special icons for every animal – a new system to write digits and beyond. On matika.in, this continues up to the fourth graders but classes have been seen where fifth graders still do it – after all, the video I just linked to was created by fifth graders.

(I think it's no coincidence that the icons look similar to the letters in Glagolitic script, the first alphabet used on the Czech territory since 863 when it was brought here by St Cyril and Methodius who constructed it artificially even though they could have simply added some accents above Greek letters to enrich the Greek alphabet and make it usable here. OK, that's basically what Russians did when they invented the Cyrillic script – and we did the same with the Latin alphabet. The Glagolitic letters were rather stupid, excessively symmetric symbols. Should all the schoolkids in Czech or mathematics classes learn the Glagolitic script again? Would it bring something to them? Maybe if a few kids learn such things, it's fine, but as a mandatory stuff for everybody?)

Every kid who is at least above the average in mathematics must know that this is just plain retarded and has nothing to do with mathematics. But in Hejný's method, tons of problems are being solved where you have to use the animals and their icons instead of regular numbers. "Mouse plus goat equals cat plus goose": erase one animal so that it's true. You have to erase the mouse. But what do you exactly learn by this game based on the assignment of bogus values to animal species? Clearly, kids who love to memorize nonsense may love it but smarter kids suffer. They may have literally physical obstacles that prevent them from memorizing similar bogus stuff. I was surely like that and I've heard about kids who suffer in such contexts, too. The smart kids are clearly the "villains" who must be punished.

Games, board games everywhere. Don't get angry, buddy.

Third grade.

Indian multiplication is added. You use an ancient, alternative method to calculate 85*23. Should that alternative really replace the normal way how we calculated such things? The Indian multiplication isn't just some curiosity that appears in one or two classes. It's an environment studied repeatedly between 3rd and 6th grade as if it were a foundation of mathematics! Just to be sure, they also learn the normal method to multiply larger numbers. There's some quasi-conventional division with remainder as well. As far as I can say, the ancient Indian method only differs from the normal modern one by "details" and the teaching of two methods may only confuse the kids.

Cards. You use some cards with digits to create a number that is closest to another one and stuff like that. You can do various things with digits and cards with digits but is that really the young mathematician's way of dealing with digits? It reminds me of the cave men who found the Škoda car. ;-)

Cycle route. Pick segments in a map so that the route doesn't intersect itself. An OK puzzle, the relationships with mathematics is limited. Hours ago, you could find YouTube videos from such classrooms with a cycle route. The teacher was doing literally nothing and the kids didn't seem to know what they were doing. Just an hour of chaos.

Fourth grade.

ABCD – algebrograms. A,B,C stand for digits, you're given some constraints, find the value of A,B,C. As in other cases, a normal problem when used in isolation. Here it is trained in the fourth and fifth grade. It's not too much time but it's still a bigger exposure to this kind of puzzle than what I find appropriate. (We encountered it in the fifth grade, too.) In principle, exercises like that lead to a set of equations with many variables. In practice, the kids don't solve them as equations but as a combinatoric problem that needs brute force to test a finite number of possible answers.

Area. They're calculating areas except that in all cases, it seems just counting the integer value of some boxes in a grid. If some geometry doesn't fit into a grid (2D grid or a 3D grid of cubes), the kids probably learn nothing about it at all.

I have already covered over 1/2 of the environments or templates of the problems. It goes on and on. Mostly the same environments that were explained above are repeated throughout the elementary school, some of them sometimes end, others appear. But where do the kids get?

You see some slightly advanced stuff – primes and factorization – in the seventh grade. But the seventh grade and eighth grade also teach kids to do "digit sums". Now, digit sums are a very immature activity to do with a number. It's exactly like the cave men who found the car. Unless you want to learn whether something is divisible by 9, you don't do digit sums because it's really a very unnatural operation from a mathematical viewpoint. (They don't learn the rules about the divisibility by 9, I think. Remember they don't learn any rules that they couldn't discover by themselves.)

A shocking combination is that eighth graders are supposed to play "Don't get angry, buddy", and it's decorated by tasks to compute the digit sums. This is really retarded.

Eighth graders get the task to solve a set of linear equations with two unknowns. I am not sure how they can suddenly do it, without any theory and with the childish problems they did before. There's also a task to solve a quadratic equation. But I think that they just never learn anything such as the general formula – or what the discriminant is. So they just guess the possible solutions and the problems are constructed so that the roots are 7 and 8 so there's a chance for them to guess correctly, after all. If the roots were more complicated, they would have no chance.

And we're talking about eighth graders.

I can't believe this. When I was a third grader, I heard about the general solution to the quadratic equation, it intrigued me, and I derived the general formula, and was somewhat happy about it. But most kids never derive such things. They still need a huge amount of things that someone else was able to derive and discover; and that they can't derive or discover but they may learn how to use it. It's just completely sick to demand – and Hejný's method demands it – that kids may only be taught what they discover themselves. It's exactly as stupid as to demand that people can only drive cars that they have made themselves. The greatness of important inventions and discoveries is

*exactly*in their ability to be used by lots of people who weren't able to invent or discover them.

The eighth graders' magic additions, magic equations, and magic square don't intellectually surpass the problems that already appeared in the first grade. It's the same kind of standardized schemes, environments, there are no new ideas, no new levels of abstractions, no building on the previous insights. So the pupils learn basically nothing deep during these subsequent 7 years. You know, if it were mathematics, they should learn lots of formulae, rules to logically think, methods to prove and disprove things, numerous particular proofs, algorithms to draw and calculate areas and convert completely new problems and situations to variables and equations. They just learn nothing of the sort.

They just spend their 8 years with the same kind of puzzles that the first graders or kids in the kindergarten may immediately understand. They are just substituting somewhat larger numbers, somewhat larger animals to the collection of granddaddy Forrest's animals, add fractions of the form x/2 and x/3 to the integers that were present from the beginning, and that's it. It's bad not only because they learn nothing advanced; they don't even learn that something that is more advanced exists. So they end up thinking that the greatest mathematicians in the world are those who can compute not with 20 but 100 or more animals of granddaddy Forrest. ;-) Such kids just end up being intellectual cripples.

The progress is simply not equivalent to the progress in a conventional classroom. But if I had to say what the alumni of Hejný's elementary school are roughly equivalent to, well, I would say that the old eighth graders from Hejný's classroom are roughly equivalent to the conventional fifth graders.

By constantly repeating these childish problems and puzzles, it's surely the case that numerous slower kids "get it" after 8 years, even though in the conventional classroom, a smaller fraction would "get it" after 5 years. But lots of kids who had As (and sometimes Bs) in the conventional classroom are really learning much more advanced, mature stuff in mathematics. And kids with the same or similar skills at age of 14-15 would be basically non-existent if Hejný's method became dominant.

This would be a catastrophe for the nation because lots of occupations really

*depend*on the former kids who got As or at least Bs in mathematics in conventional classrooms – and the knowledge that they could have accumulated at the higher rate that was expected. Kids who got somewhere – as kids or adults – had to use lots of skills and methods that they were able to reproduce but they just didn't exactly understand, especially not "instinctively", how they were derived or why they work.

If you slowed down this whole progress by a factor of two, and if the high schools and colleges adapted to the slowdown in the elementary schools – and obviously, they will have to adapt and slow down as well if Hejný's method becomes really widespread at basic schools – it would mean that the college alumni in 2025 will be literally equivalent to high school alumni in 2010. So all the jobs that had good reasons to demand university diplomas in 2010 will be impossible to meaningfully fill. (Maybe, full professors in the new Hejný system will be equivalent to the contemporary bachelors. And there will also be a Hejný-Nobel prize which which will be on par with the contemporary masters of science.)

Sure, it's convenient for kids, especially those who had trouble with normal mathematics; for their parents who are rich enough to fund the kids even when they grow older and the immediate happiness of the children is more important than other things; and for the teachers who don't really have to do almost anything in Hejný's classroom. But it's not a good way to spend the taxpayer money and it's a threat for the future of the Czech economy and the intellectual credentials of the whole nation.

But please, be more than free to take the problems from Hejný's method and spread them in your country, too. While I think that the blanket application of the method would be rather devastating for every civilized nation, I also think that there are always pieces of the methods that you may pick or helpfully use in your education.

**Bonus: the levels of abstraction in mathematics**

I have made it clear that the method ignores the characteristic of mathematics (and physics) that one needs to build on the previous insights, and get familiar with increasingly abstract levels of abstraction. That makes mathematics (but also physics) different from common sense. It makes these subjects different from a set of isolated insights – which is what many other subjects are all about (memorization is always enough; and skipping some classes usually doesn't hurt later).

What do I mean by these levels of sophistication?

One starts to learn small integers. And addition. At some point, he learns numbers greater than 9. You can write them using several digits. You may suddenly do things with higher numbers or several digits at the same moment. Of course, they sort of get there in that alternative classroom. They also understand the generalization to negative numbers etc. although I feel that it's really hard to find negative numbers in Hejný's books of exercises.

But the real mathematics doesn't end there. One learns that aside from integers, there are fractions. Fractions already become almost outlawed in Hejný's classroom. And if fractions appear, they have small integers in the denominator. The general methods to add fractions and do other things are being delayed and suppressed. We could call fractions the second floor of the mathematical skyscraper.

Rational numbers may be extended to real numbers. The Hejný kids learn something about real numbers but they don't practice the decimal system. I suppose it's being assumed that they use calculators for everything. But calculators can't teach them about various relationship between rational and real numbers, about solutions to simple problems that shouldn't require any calculators, and so on. I think that calculators should justify the reduction of time that kids spend by doing numerical calculation because it can indeed be done by machines; but I still think that they should learn the same "theory".

Adults' mathematics extends the fields – to complex numbers, quaternions, Octonions, but also other algebraic structures, rings, semigroups, groups, vector spaces, algebras, Lie algebras, manifolds with atlases, fiber bundles, sheaves, derived categories, be my guest. Obviously, Hejný's method is meant to be for elementary schools so there's no room for those advanced structures. But the kids don't learn "any" structure whose nature isn't obvious to the smart kids in the kindergarten. And that's a problem because the kids aren't just ignorant about particular advanced things, they're ignorant about the existence of advanced things.

Now, extending the "set of numbers" is far from the only direction how new floors are being built in mathematics. In some sense, they're still some of the most trivial ways to extend and generalize in mathematics.

Elementary schools should teach variables, the power of \(x\). It's really a taboo. You can find \(x\) and \(y\) in the seventh grade but none of these things are really trained so the kids can't get what it's good for. One should learn how to manipulate with equations. How to understand that two lines are exactly equivalent to each other. Sometimes, the manipulation only works in one direction and one needs to check the candidate results. Some of them will work. This kind of logical thinking isn't trained.

Now, there are general formulae where you may substitute any value. Formulae for solutions of equations or sets of equations, formulae for circumferences, areas, volumes. All these things are suppressed for ideological reasons. Formulae are evil, Hejný and his disciples repeat. In fact, they even say that a kid who knows some formulae is an "intellectual parasite", I kid you not. But in reality, they're absolutely essential in mathematics in the normal meaning of the word.

Aside from exercises designed for schools, mathematics may actually be used in lots of situations in the real world – to some extent, in all of them. This requires the kid to be able to extract the mathematical description of the relevant real-world questions, learn how to overlook the distracting features of a specific situation that have nothing to do with the calculation, and how to use the general mathematical ways to solve the essence of the problem. Again, this very philosophy is considered a taboo. The kids are never supposed to solve "problems of any new kinds". The repetition of the standardized exercises or "environments" is where the kids should live forever. In some sense, these environments are new examples of the "safe spaces". The kids are being protected against the real world!

In mathematics, there are lots of general formulae and rules to solve classes of problems and Hejný's kids aren't learning them at all. So they can't even start with the numerous levels of abstractions and broader lessons that arise further in that direction. What do I mean?

Well, some problems need a lot of space to be formulated but the solution is simple. When I was 6, my grandfather asked me how much is 34+78-56-29-34+56+29-78. I don't remember the numbers, they may have been smaller. I got the correct result, zero, and he told me: But look, I can get zero easily, by rearranging the terms. You can find pairs that cancel. My general point is that the solution may often be shorter than the formulation of the problem. Some things are trivial to solve if you're clever. And when you're learning mathematics, you should have the goal to be more clever in this sense.

On the other hand, some problems, like Fermat's Last Theorem, may be easily formulated but it's incredibly hard to solve them or find a proof. Even if you don't know any really mature examples of that, you should be persuaded about the general fact that solutions may be extremely complicated and it's sometimes OK to spend hours or centuries on a problem written on one line. It's important because people realize that mathematicians etc. cannot be paid just for the number of characters used to formulate a problem. It's needed for the people to have at least an adequate respect to some folks doing intellectual things – whose content isn't understandable to the public. There may be some valuable essence in that work even if it seems that they're just solving something very simple.

Let me return closer to the beginning. I talked about the extension of integers to rational numbers and real numbers. But there's also the increasing sequence of operations. You do addition, subtraction, multiplication, division. But already at the basic school, kids should learn something about exponentiation. Clearly, Hejný's classroom teaches nothing like the identities involving powers let alone logarithms.

But then you have functions that are "not rational", starting with logarithms and exponentiation. You need to learn how they work for non-integer exponents, how they're inverses of each other, how they're used to calculate the interest in banks. And then there may be special functions. And operations one may do with functions, starting with derivatives and integrals. And differential equations etc. Again, most of these things don't belong to the elementary schools. But the kids don't even learn any kids' version of the insight that increasingly non-obvious operations with numbers, generalized numbers, and operations with operations (functions) may be made and there may be good reasons for that. The kids in that alternative method don't really understand anything about the value of any of the stuff they learn. All they learn is some recreational mathematics that is only justified because the teacher wants the puzzles to be solved.

One other branch of the skyscraper is the probability theory and statistics. So the kids do some things in the direction. They roll dice for an hour and calculate how many times they got 6. But they don't learn any laws. It's forbidden for the teacher to tell them the laws. So they don't really learn anything like general combinatoric numbers, rules for probabilities of independent things that multiply, let alone standard deviations, linear regression etc. An average person cannot rediscover most of these things which is why no one is allowed to learn such things in Hejný's classroom.

The kids who have to rely on the direction they're led to end up believing that it makes sense to number granddaddy Forrest's animals, and it doesn't make sense to do what advanced mathematicians are doing. It would surely be silly to consider general functions and do operations with the functions – like integration. The school just teaches them wrong ideas about what is meaningful, useful, and natural to do; and what isn't.

The method's focus on the brute force is entirely pathological for the kids' opinion about "brain activities" in general because the kids measures everything by the amount of brute force. It's being assumed that every problem is "solved" by trying all possibilities for answers. When the number of possible answers is infinite, e.g. when it is a generic real number, the kids are screwed. But methods to get the answers right away – or at least faster, by many, 100, or a googol of orders of magnitude – often exist. They should really be the content of mathematics education but they're treated as taboos by Hejný's method, too.

What actually makes me rather emotional is that I know lots of people who really believe that rational reasoning doesn't work, physics and science don't work, and mathematics is at most useful for the solution of some special puzzles in recreational mathematics. They misunderstand everything about mathematics and science and why the whole Universe basically works on the basis of rules that may be mathematically formulated and mathematically analyzed. And what makes me angry is that I feel that Hejný's method is designed to "confirm" these utterly idiotic misconceptions about the power and role of mathematics, mathematically formulated laws, and mathematical reasoning in the whole world! While the puzzles may be fine in isolation, the message conveyed in between the lines is a set of misconceptions that a person who doesn't understand mathematics at all may believe. Hejný's method is a political program to legitimize the stupid people's opinions about mathematics.

The kids end up being innumerate at all levels (starting from the multiplication tables and ending with all the levels of abstraction sketched above) and if they will ever represent a substantial fraction of the productive generation in a nation, the nation will cease to be civilized. Maybe such a nation will still provide the world with sufficiently many workers who may press some buttons in a factory. But it can't expect to be among the nations with the highest GDP per capita.

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