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Can kids learn to think mathematically from granddaddy's animals?

On Saturday night, we had a reunion – the end of elementary school after 30 years. Lots of beer, memories, personal stuff. I always discuss some serious topics. So one classmate (DS) holds impressive 3 bitcoins and is a full-blown hodler ;-) while your humble correspondent and another classmate (JK) were arguing why the Bitcoin pricing was a bubble and what it meant.



Granddaddy Forrest

I asked lots of people about Hejný's method to teach mathematics. (Teachers must be silent in the method, kids must invent everything by themselves, they solve some 10+ types of problems in recreational mathematics for 8 years, without any conceptual progress, and at the end, they tell you how much they love and understand mathematics because of this method.) By the end of the exchanges, 10 people were familiar with the topic, 8 of them were familiar to start with. Only 2 were sort of positive about that "constructivist" method in education – and one of them (VK) arguably changed his mind to a large extent. The rest was highly critical, just like I am.

In March, I discussed particular problems, as seen on the matika.in website. All of them are recreational mathematics of some kind and they are supposed to be solved by guesswork – by the trial and error. That brute force strategy is a typical non-mathematical approach to the problems – mathematics is all about searching for patterns and clever things to solve otherwise hard or unsolvable problems.




The champions and opponents of the method disagree about all those problems as well although some of them could be used in a wise classroom, too. But nothing polarizes the two camps as clearly as the Daddy Forrest. Search for that phrase on the matika.in website and try to solve some of the problems. Daddy should really be "granddaddy" (děda), some old guy from the family who lives in the countryside, who owns animals, and whose name is derived from the "forest" (les-Lesoň).




This whole "environment" of Daddy Forrest's animals is using animal codes for animals that represent small integers, up to 20. You may search for the pictures on Google Images. The numbers 1,2,3,4,5 are replaced with a mouse, cat, goose, dog, goat. 10 is a cow and 20 is a horse. There are some other animals, too. The textbooks contain tons of colorful pictures of these animals. In the classroom, they use some stickers with the pictures of the animals that may be attached to a board. On top of that, children have to memorize how to write and read some icons or quasi-letters that represent each animal.

The problems are of the type: place two cats and a goose on one side and five mice and a dog on the other. Which side is stronger? Or: remove one animal from such an "animal equation" so that the equation holds (they don't use that language).



Now, opponents of the method such as myself usually say that it's nonsense and there's nothing about mathematics that the children learn from this activity. It's arguably the single most obscene example of the nonsense that is being pumped into the children and that is being marketed as mathematics. On the other hand, the people who defend the method – or people who have the natural tendency to defend it – often praise it as a great idea that teaches kids to think mathematically.

Who is right? Of course the opponents are right. But what do the others say? A classmate VK turned out to be a fan of the method – we have been sitting next to each other for some 8 years when we were kids. He was an excellent student – who also had straight As throughout the high school which your revolting humble correspondent was extremely far from. OK, on Saturday night, he said:

It's wonderful because the animals teach the kids that the digits, like the animals, are just another code and there's nothing else behind them.
Great. I agree that they learn it, that's the key lesson here. But is that lesson correct? I don't think so. What VK said was that the conventions to represent integers are just social conventions and they may be changed. And when we translate from one convention to another, we get what we inserted. So there's no added value in the numerals which is a great lesson to learn, VK seems to say. (I had some deja vu. I think that he said exactly the same thing when we were 15 and I reacted in the same negative way to his comment almost three decades ago.)

But the miracle of mathematics is, I respond, that it does have an added value. Mathematics only starts after you define your language and conventions, once you have some symbols, relations, operations, and stuff like that, and you actually start to do something with these damn things! You discover laws, patterns, regularities, tricks, algorithms, methods, methodologies, and other things.

Those things are the beef of mathematics. Mathematics is the added value. It's some abstract body of wisdom that exists in any scheme of conventions to represent integers and other objects and wisdom that therefore doesn't depend on any particular choice of conventions. By mathematics, we mean the beef that even Chinese or extraterrestrials who use very different symbols (or dancing) would still find underneath their sequences of symbols. Mathematics is the set of possible claims that may be written in any "language" modulo all the possible translations from one language to another!

So the statement that "they're just a code and there's really nothing in it" either means that VK, despite all the straight As, thinks that there is nothing in mathematics; or that kids should be taught nothing about mathematics. Well, I beg to differ.

I was explaining these things to him and at the end, he sort of agreed although I can't be sure whether the agreement was coming from his heart. After all, he was probably saying similar things even 30 years ago so it's some part of his thinking that seems unlikely to genuinely change after a 5-minute-long conversation. (Similarly for my thinking.)

When I was 17, I read "Surely You're Joking Mr Feynman" and it was the first time when I was exposed to the story about his father who taught small Richard that the names of birds don't constitute knowledge. But I am pretty sure that my opinions about these basic matters were the same long before I was exposed to the Feynman phenomenon.

You know, Daddy Forrest's animals are just another "language". The translation from one system to write numerals to another is analogous to the translation from one language to another. Just like in the case of the birds, you don't learn a damn thing about the bird by that translation! Those are just words. And as the well-known proverb says: The more languages you know, the more time you have wasted with some humanities junk. ;-) So similar things should be taught at language classes – or classes that focus on these conventions should be considered analogous to classes of languages! And those are not mathematics. They are really inferior in comparison with mathematics, every mathematically thinking person agrees, but even if you work hard to be diplomatic, you should appreciate the difference between mathematics classes and language classes.

Now, the animals are a particularly stupid system to write integers. There's some similarity to the Roman numerals – except that the Roman numerals are much more clever than Forrest's animals. You may write things like MMXVIII in Roman numerals – it's not such a bad way to write 2018. But some numbers are much worse. I guess that the numbers with "8" in it are the most complicated ones: 888 is written as DCCCLXXXVIII which is pretty bad. Nearby numbers are represented by Roman numerals that may have a very different length – which is a big disadvantage relatively to the decimal system, I think. Lots of things are more awkward and less systematic if expressed by Roman numerals.

But the Roman numerals only use a few letters. Three is III. You just write I thrice. You don't need to memorize that three mice is equal to a goose. You're just adding lines. And when there are too many of them, e.g. five or ten, you replace them with V or X. Such emergent symbols are used for powers of ten or "five times powers of ten" which makes it rather easy to convert between decimal and Roman numerals. On top of that, the Roman numerals allow you to subtract so that IX is nine to save some space.

But even Roman numerals, while more intelligent than the animals, are pretty low-brow. I think that small children – even first-graders – can learn Roman numerals. I surely did learn them when I was in the kindergarten. Kids may add somewhat bigger numbers when they become third-graders. But there's really nothing in it. It is a very special skill that doesn't lead to many interesting ramifications.

It's a coincidence that we use the decimal system and we could use other systems. I guess that this is the point that VK is very excited about. I agree that this point is valid. We could use a base-8 or base-16 (hexadecimal) system to write integers. Everything would still work. But this is just one correct conceptual proposition about our "mathematical culture". It isn't useful elsewhere. In the same way, the conversion from base-8 to base-10 or even to base-7 isn't useful for anything so it may be fun if you can do it but there's no point in teaching it to every kid. (BTW Feynman specifically expressed the same opinion in the chapters about his work in the textbook committee. On the other hand, Hejný's method also tries to teach the kids to use the binary code, in the Biland environment.)

If you learn the music notes – or if you learn some bizarre new way to write the notes – you haven't composed or played any music yet. You're surely far from being a Beethoven. In the same way, by playing with some strange codes for small integers, you haven't done any real mathematics yet. Music and mathematics is in the patterns.

OK, do the problems of the type "which animal do you remove for the goose, horse, crocodile, and five mice to be as strong as a cow, skunk, hamster, and three cats" teach the kids to think mathematically? Well, they teach something. It's some rudimentary arithmetic problem expressed in an unusual language with lots of unusual symbols that the kids won't use anywhere when they leave the school – and they use it nowhere in other classes at the same school, either.

But one question is how do the kids actually solve these problems and how they're expected to solve the problems?

Well, you can always convert all the animals to mice (a mouse is one). So you just draw lots of lines (well, the correct icon for a mouse is that ice cream) and in that way, you may compare which of the sums on the two sides of the equation or inequality is larger and by how much. I actually think that this reduction to "lots of ones" and the conversion of any problem to "addition of one and subtraction of one" is what they actually want the kids to do in their heads. This interpretation is also supported by the "staircase" environment – kids march and all addition and subtraction is reduced to individual steps, i.e. to the repeated addition or subtraction of one.

This does teach something but as soon as you need to work with many numbers or larger numbers, it is a catastrophically inefficient way to do the sums, right?

So in practice, the kids must memorize the sums. Just like you memorize that 2+3=5 – there are not too many things of this importance – the kids probably do the same thing and they effectively memorize almost the same set of identities but in an unusual language. So in this case, they memorize cat+goose=goat. Well, they don't really use "plus". To make their environment even more offensive, they write "cat goose = goat" with pictures.

At the end, the only thing they learn is addition and subtraction of small integers using a very awkward artificial "language" where most of the kids' energy is probably consumed by the memorization of the distracting animals and their icons – which is clearly the non-mathematical (language-like) portion of the process. And this language that they spend so much time with is completely arbitrary, stupid, and useless in their future.

It's just bizarre that the defenders of the method criticize the memorization of definitions of mathematical concepts, formulae, identities, rules, theorems, and algorithms. As far as I can see, these "templates" – which may be applied and generalized in so many ways – are clearly the most useful things that the kids should memorize. What is stupid is to force the kids to memorize lots of isolated facts and factoids, especially artificially invented ones, that aren't good for anything except for themselves.

You know, it's surely easier to memorize isolated facts – because there's nothing conceptually hard about them – but that's exactly what makes them not very useful. If one memorizes some things that may be applied or generalized in many ways, that's a gem – even if the kid doesn't immediately get what's going on. But it's a point in the knowledge space that the kid may rely upon.

When you memorize a list of Egyptian pharaohs (their names only), it's useless because the only question where this knowledge may be useful is the question "what is the list of Egyptian pharaohs" (or some "subsets" of this question). On the other hand, if you learn an algorithm to divide numbers or solve a set of two linear equations, that may be applied in infinitely many situations – not only with infinitely many numbers that define the exact problem but also in infinitely occupations and activities that these occupations may face.

So at the end, I think it's fair to say that those who promote the retarded games with Forrest's animals are those who haven't really understood the power of mathematics at all. Also, they probably dislike the very suggestion that mathematics is powerful and they prefer the kids to memorize useless isolated facts and factoids – because they're better at this mindless activity, too!

It's very important for those who appreciate the power and importance of mathematics – for the human wisdom, science, and very important engineering and other occupations – to fight for the continued presence of "our understanding" in the education process. If all kids in a nation are trained to play with these animals throughout most of their "mathematics" classes, and if they're led to think that this is a good way to use their brains, the nation is going to become a nation of idiots who can't do most of the things that we associate with the advanced civilization.



P.S.: This guide, on page 3/7, claims that the animals are a propedeutics (preparation) for variables, conversion of units, and equations. I think that they suck in all three cases.

They're not really variables because the animals are said to have constant values, and if they were not constant, nothing is left at all. The kids don't learn anything such as \((a+b)^2 = a^2+2ab+b^2\) which would survive if the animal values were not constant.

Second, they're bad preparations for the conversion of units because the ratios are unnaturally rational numbers and they never seem to use any "rule of three", direct proportionality.

Third, they are surely some primitive cases of equations except that there are no variables in them and kids learn no nontrivial methods to deal with equations.

So one may say that the exercises with the animals just "vaguely resemble" these mathematical concepts in certain ways but the similarity is so vague and has so many "buts" that the experience gained from the games with the animals may make it harder, not easier, for the kid to understand the actual mathematics because the details aren't really right and things therefore become confusing if the kid is trying to learn the pieces of mathematics properly.

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