particular problems, equally seen on the matika.in website. All of them are recreational mathematics of some variety too they are supposed to live solved past times guesswork – past times the trial too error. That brute forcefulness strategy is a typical non-mathematical approach to the problems – mathematics is all nigh searching for patterns too clever things to solve otherwise difficult or unsolvable problems.The champions too opponents of the method disagree nigh all those problems equally good although some of them could live used inwards a wise classroom, too. But goose egg polarizes the ii camps equally clearly equally the Daddy Forrest. Search for that phrase on the matika.in website too travail to solve some of the problems. Daddy should really live "granddaddy" (děda), some quondam guy from the solid unit of measurement who lives inwards the countryside, who owns animals, too whose squall is derived from the "forest" (les-Lesoň).
This whole "environment" of Daddy Forrest's animals is using fauna codes for animals that stand upwards for small-scale integers, upwards to 20. You may search for the pictures on Google Images. The numbers 1,2,3,4,5 are replaced amongst a mouse, cat, goose, dog, goat. 10 is a moo-cow too twenty is a horse. There are some other animals, too. The textbooks comprise tons of colorful pictures of these animals. In the classroom, they job some stickers amongst the pictures of the animals that may live attached to a board. On superlative of that, children accept to memorize how to write too read some icons or quasi-letters that stand upwards for each animal.
The problems are of the type: house ii cats too a goose on i side too 5 mice too a domestic dog on the other. Which side is stronger? Or: withdraw i fauna from such an "animal equation" then that the equation holds (they don't job that language).
Now, opponents of the method such equally myself commonly tell that it's nonsense too there's goose egg nigh mathematics that the children larn from this activity. It's arguably the unmarried most obscene instance of the nonsense that is beingness pumped into the children too that is beingness marketed equally mathematics. On the other hand, the people who defend the method – or people who accept the natural vogue to defend it – oftentimes praise it equally a cracking thought that teaches kids to squall upwards mathematically.
Who is right? Of course of written report the opponents are right. But what do the others say? Influenza A virus subtype H5N1 classmate VK turned out to live a fan of the method – nosotros accept been sitting adjacent to each other for some eight years when nosotros were kids. He was an fantabulous pupil – who also had guide As throughout the high schoolhouse which your revolting humble correspondent was extremely far from. OK, on Sabbatum night, he said:
It's wonderful because the animals learn the kids that the digits, similar the animals, are merely some other code too there's goose egg else behind them.Great. I concord that they larn it, that's the fundamental lesson here. But is that lesson correct? I don't squall upwards so. What VK said was that the conventions to stand upwards for integers are merely social conventions too they may live changed. And when nosotros interpret from i convention to another, nosotros acquire what nosotros inserted. So there's no added value inwards the numerals which is a cracking lesson to learn, VK seems to say. (I had some deja vu. I squall upwards that he said precisely the same thing when nosotros were xv too I reacted inwards the same negative style to his comment almost iii decades ago.)
But the miracle of mathematics is, I respond, that it does accept an added value. Mathematics alone starts later you lot define your linguistic communication too conventions, i time you lot accept some symbols, relations, operations, too materials similar that, too you lot really start to do something amongst these damn things! You discovery laws, patterns, regularities, tricks, algorithms, methods, methodologies, too other things.
Those things are the beef of mathematics. Mathematics is the added value. It's some abstract trunk of wisdom that exists inwards any scheme of conventions to stand upwards for integers too other objects too wisdom that so doesn't depend on whatever particular selection of conventions. By mathematics, nosotros hateful the beef that fifty-fifty Chinese or extraterrestrials who job really different symbols (or dancing) would nevertheless notice underneath their sequences of symbols. Mathematics is the laid upwards of possible claims that may live written inwards whatever "language" modulo all the possible translations from i linguistic communication to another!
So the disceptation that "they're merely a code too there's really goose egg inwards it" either agency that VK, despite all the guide As, thinks that at that topographic point is goose egg inwards mathematics; or that kids should live taught goose egg nigh mathematics. Well, I beg to differ.
I was explaining these things to him too at the end, he sort of agreed although I can't live sure whether the understanding was coming from his heart. After all, he was belike saying similar things fifty-fifty xxx years agone then it's some purpose of his thinking that seems unlikely to truly modify later a 5-minute-long conversation. (Similarly for my thinking.)
When I was 17, I read "Surely You're Joking Mr Feynman" too it was the outset fourth dimension when I was exposed to the floor nigh his begetter who taught small-scale Richard that the names of birds don't institute knowledge. But I am pretty sure that my opinions nigh these basic matters were the same long earlier I was exposed to the Feynman phenomenon.
You know, Daddy Forrest's animals are merely some other "language". The translation from i organisation to write numerals to some other is analogous to the translation from i linguistic communication to another. Just similar inwards the instance of the birds, you lot don't larn a damn thing nigh the aeroplane past times that translation! Those are merely words. And equally the well-known saying says: The to a greater extent than languages you lot know, the to a greater extent than fourth dimension you lot accept wasted amongst some humanities junk. ;-) So similar things should live taught at linguistic communication classes – or classes that focus on these conventions should live considered analogous to classes of languages! And those are not mathematics. They are really inferior inwards comparing amongst mathematics, every mathematically thinking mortal agrees, but fifty-fifty if you lot operate difficult to live diplomatic, you lot should appreciate the divergence betwixt mathematics classes too linguistic communication classes.
Now, the animals are a especially stupid organisation to write integers. There's some similarity to the Roman numerals – except that the Roman numerals are much to a greater extent than clever than Forrest's animals. You may write things similar MMXVIII inwards Roman numerals – it's non such a bad style to write 2018. But some numbers are much worse. I estimate that the numbers amongst "8" inwards it are the most complicated ones: 888 is written equally DCCCLXXXVIII which is pretty bad. Nearby numbers are represented past times Roman numerals that may accept a really different length – which is a large disadvantage relatively to the decimal system, I think. Lots of things are to a greater extent than awkward too less systematic if expressed past times Roman numerals.
But the Roman numerals alone job a few letters. Three is III. You merely write I thrice. You don't involve to memorize that iii mice is equal to a goose. You're merely adding lines. And when at that topographic point are besides many of them, e.g. 5 or ten, you lot supercede them amongst V or X. Such emergent symbols are used for powers of 10 or "five times powers of ten" which makes it rather tardily to convert betwixt decimal too Roman numerals. On superlative of that, the Roman numerals allow you lot to subtract then that IX is nine to salvage some space.
But fifty-fifty Roman numerals, spell to a greater extent than intelligent than the animals, are pretty low-brow. I squall upwards that small-scale children – fifty-fifty first-graders – tin larn Roman numerals. I for certain did larn them when I was inwards the kindergarten. Kids may add together somewhat bigger numbers when they expire third-graders. But there's really goose egg inwards it. It is a really special science that doesn't Pb to many interesting ramifications.
It's a coincidence that nosotros job the decimal organisation too nosotros could job other systems. I estimate that this is the betoken that VK is really excited about. I concord that this betoken is valid. We could job a base-8 or base-16 (hexadecimal) organisation to write integers. Everything would nevertheless work. But this is merely i right conceptual proposition nigh our "mathematical culture". It isn't useful elsewhere. In the same way, the conversion from base-8 to base-10 or fifty-fifty to base-7 isn't useful for anything then it may live fun if you lot tin do it but there's no betoken inwards teaching it to every kid. (BTW Feynman specifically expressed the same persuasion inwards the chapters nigh his operate inwards the textbook committee. On the other hand, Hejný's method also tries to learn the kids to job the binary code, inwards the Biland environment.)
If you lot larn the music notes – or if you lot larn some bizarre novel style to write the notes – you lot haven't composed or played whatever music yet. You're for certain far from beingness a Beethoven. In the same way, past times playing amongst some foreign codes for small-scale integers, you lot haven't done whatever existent mathematics yet. Music too mathematics is inwards the patterns.
OK, do the problems of the type "which fauna do you lot withdraw for the goose, horse, crocodile, too 5 mice to live equally rigid equally a cow, skunk, hamster, too iii cats" learn the kids to squall upwards mathematically? Well, they learn something. It's some rudimentary arithmetics job expressed inwards an odd linguistic communication amongst lots of odd symbols that the kids won't job anywhere when they locomote out the schoolhouse – too they job it nowhere inwards other classes at the same school, either.
But i enquiry is how do the kids really solve these problems too how they're expected to solve the problems?
Well, you lot tin ever convert all the animals to mice (a mouse is one). So you lot merely describe lots of lines (well, the right icon for a mouse is that H2O ice cream) too inwards that way, you lot may compare which of the sums on the ii sides of the equation or inequality is larger too past times how much. I really squall upwards that this reduction to "lots of ones" too the conversion of whatever job to "addition of i too subtraction of one" is what they really want the kids to do inwards their heads. This interpretation is also supported past times the "staircase" environs – kids march too all add-on too subtraction is reduced to private steps, i.e. to the repeated add-on or subtraction of one.
This does learn something but equally shortly equally you lot involve to operate amongst many numbers or larger numbers, it is a catastrophically inefficient style to do the sums, right?
So inwards practice, the kids must memorize the sums. Just similar you lot memorize that 2+3=5 – at that topographic point are non besides many things of this importance – the kids belike do the same thing too they effectively memorize almost the same laid upwards of identities but inwards an odd language. So inwards this case, they memorize cat+goose=goat. Well, they don't really job "plus". To brand their environs fifty-fifty to a greater extent than offensive, they write "cat goose = goat" amongst pictures.
At the end, the alone thing they larn is add-on too subtraction of small-scale integers using a really awkward artificial "language" where most of the kids' let out energy is belike consumed past times the memorization of the distracting animals too their icons – which is clearly the non-mathematical (language-like) percentage of the process. And this linguistic communication that they pass then much fourth dimension amongst is completely arbitrary, stupid, too useless inwards their future.
It's merely bizarre that the defenders of the method criticize the memorization of definitions of mathematical concepts, formulae, identities, rules, theorems, too algorithms. As far equally I tin see, these "templates" – which may live applied too generalized inwards then many ways – are clearly the most useful things that the kids should memorize. What is stupid is to forcefulness the kids to memorize lots of isolated facts too factoids, especially artificially invented ones, that aren't skillful for anything except for themselves.
You know, it's for certain easier to memorize isolated facts – because there's goose egg conceptually difficult nigh them – but that's precisely what makes them non really useful. If i memorizes some things that may live applied or generalized inwards many ways, that's a jewel – fifty-fifty if the kid doesn't forthwith acquire what's going on. But it's a betoken inwards the noesis infinite that the kid may rely upon.
When you lot memorize a listing of Egyptian pharaohs (their names only), it's useless because the alone enquiry where this noesis may live useful is the enquiry "what is the listing of Egyptian pharaohs" (or some "subsets" of this question). On the other hand, if you lot larn an algorithm to dissever numbers or solve a laid upwards of ii linear equations, that may live applied inwards infinitely many situations – non alone amongst infinitely many numbers that define the exact job but also inwards infinitely occupations too activities that these occupations may face.
So at the end, I squall upwards it's fair to tell that those who promote the retarded games amongst Forrest's animals are those who haven't really understood the ability of mathematics at all. Also, they belike dislike the really proposition that mathematics is powerful too they prefer the kids to memorize useless isolated facts too factoids – because they're amend at this mindless activity, too!
It's really of import for those who appreciate the ability too importance of mathematics – for the human wisdom, science, too really of import technology scientific discipline too other occupations – to struggle for the continued presence of "our understanding" inwards the teaching process. If all kids inwards a acre are trained to play amongst these animals throughout most of their "mathematics" classes, too if they're led to squall upwards that this is a skillful style to job their brains, the acre is going to expire a acre of idiots who can't do most of the things that nosotros associate amongst the advanced civilization.
P.S.: This guide, on page 3/7, claims that the animals are a propedeutics (preparation) for variables, conversion of units, too equations. I squall upwards that they suck inwards all iii cases.
They're non really variables because the animals are said to accept constant values, too if they were non constant, goose egg is left at all. The kids don't larn anything such equally \((a+b)^2 = a^2+2ab+b^2\) which would locomote if the fauna values were non constant.
Second, they're bad preparations for the conversion of units because the ratios are unnaturally rational numbers too they never look to job whatever "rule of three", guide proportionality.
Third, they are for certain some primitive cases of equations except that at that topographic point are no variables inwards them too kids larn no nontrivial methods to bargain amongst equations.
So i may tell that the exercises amongst the animals merely "vaguely resemble" these mathematical concepts inwards sure ways but the similarity is then vague too has then many "buts" that the sense gained from the games amongst the animals may acquire into harder, non easier, for the kid to empathise the actual mathematics because the details aren't really right too things so expire confusing if the kid is trying to larn the pieces of mathematics properly.