I desire some of you lot to pass some fourth dimension past times playing the games – or solving the mathematics problems – at
MATIKA.IN (English, play now)The website includes the interactive environments or "schemes" which are generally equivalent to everything that pupils exposed to Hejný's method, the most prominent orthodox constructivist pedagogy method nosotros know inwards Czech Republic these days, receive got to solve during their 8 years inwards the simple school.
Helpfully enough, the website is available inwards English linguistic communication (and 7 other languages). It was coded past times Andrej Probst, a immature guy doing these things equally a charity (who is generally independent of the motion that spreads the method, equally far equally I tin say), in addition to is used past times schools inwards Czechia, Slovakia, in addition to Hungary.
In Czechia, at that spot are some 4,100 simple schools. 800 of them instruct mathematics past times Hejný's method. 200 of those receive got learned virtually the website in addition to at to the lowest degree recommend it to the kids.
The website is some straightforward JavaScript but it looks fresh in addition to helpful, partly because of illustrations past times kovidesign. I sure believe that such websites should live on used inwards conventional mathematics pedagogy – in addition to inwards other subjects, too.
You volition know basically everything virtually Hejný's method – in addition to what it tries to instruct the firstly graders (who acquire 7 years old sometimes during the schoolhouse year)... in addition to upward to the 8th graders (14 years old sometimes during that final year) – if you lot pass an hr with this website, in addition to if you lot larn virtually the broader philosophy of the method: The instructor ever wants the kids to experience good, she never teaches whatever theory, at that spot aren't whatever textbooks with theory, they don't larn whatever full general rules or formulae, at that spot are just textbooks with similar exercises, kids ever piece of work collectively in addition to right each other, they must larn everything past times themselves.
The instructor is sidelined in addition to ideally eliminated. This is inwards no way a Czech or Slovak (Hejný is mixed) invention. Americans may retrieve the math wars of the 1990s that were generally virtually some radical constructivists' efforts to eliminate teachers or instructors from the math education.
It would live on empty-headed to denounce every work on the page. After all, many of them are analogous to exercises that almost any approach to mathematics pedagogy has to include. So inwards my eyes, most of the lethal defects inwards the method are inwards the philosophy (including the relative truth in addition to the thought that kids should totally ignore everything that was invented past times other people); inwards the excessive repetition of these puzzles nosotros volition speak over inwards detail; in addition to specially inwards the things that are missing for the same argue (the fourth dimension is already spent on the repetition of the standardized childish puzzles).
But sometimes, fifty-fifty what is included is just painful.
OK, render 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 agency to add together or subtract one, respectively. So you're supposed to solve things similar "10 leftarrow leftarrow leftarrow" in addition to write "7" equally the answer, OK? Without the jargon, the kids exercise some "Abelian grouping theory". They retrieve their province (location) earlier the steps, in addition to after the steps (adding or subtracting one). That tin aid them to internalize the numbers in addition to mayhap larn virtually null in addition to negative numbers, too.
I receive got naught against this method past times itself – equally something that belongs "somewhere". I just think that the amount of fourth dimension they (and much older kids) pass past times marching in addition to stepping is orders of magnitude higher than it should be. The kids are effectively trained to add together larger numbers, similar 8+9, past times thinking virtually 17 steps, i past times one, inwards their heads. The method is absolutely obsessed with the elimination or delaying whatever memorization – such equally the fact that 8+9=17. You just can't acquire besides far inwards mathematics if you're scared of similar "memorizable results" that dramatically speed upward some problems or "trivialize" a degree of tasks. In some sense, all the mightiness of mathematics is virtually this speedup in addition to trivialization!
OK, firstly graders do consider some conventional add-on in addition to subtraction of the minor numbers, after all.
Then they bargain with pyramids. There's a triangle of boxes, similar inwards Pascal's triangle, in addition to the elevation box or bottom box is assigned the let on that is the total of 2 boxes underneath or inwards a higher house this box. You fill upward inwards the missing integers. Again, there's naught incorrect virtually this variety of a puzzle if it appears inwards isolation. What's incorrect is the obsession with this work – which, inwards Hejný's classrooms, becomes basically 1/10 of all of mathematics of the simple school.
Pyramids proceed upward to the 6th grade. The only progress is that i apace gets from 3-level pyramids to 4-level pyramids; occasionally, you lot demand to subtract because some other numbers inwards the boxes are missing, non the sums; in addition to the 6th graders add together non just integers but numbers similar 7.9 – i digit after the decimal point. Is that the appropriate progress after iv years of playing with this exercise? I don't think so.
Triangles are virtually the same matter equally pyramids except that to a greater extent than numbers are missing. That would acquire into besides difficult or ambiguous hence you're frequently given the listing of numbers that are missing – inwards the incorrect order. Or you're told some extra status that the missing numbers satisfy. So all these problems halt upward beingness "solved" past times brute force, only past times trying all possible permutations or minor integers that tin live on filled somewhere. That's how completely uninformed people may solve Sudoku – after all, it's almost the same thing. But there's no mathematics inwards it because mathematics only starts i time you lot detect some cleverness to bargain with the simple objects around you.
Triangles are a flake less straightforward than pyramids hence they proceed upward to the 6th grade. Six years of repeating the same variety of problems that you're supposed to solve almost alone past times brute force, without learning whatever methods, formulae, tricks, anything. You know, mathematics has a lot of levels of abstraction – everyone who is unopen to mathematics must totally know what I am talking about. The kids inwards Hejný's classroom ever remain on the Blue Planet flooring mentally; they only acquire to a greater extent than experienced with the Blue Planet floor.
Snakes in addition to spider webs (they facial expression similar similar problems). Boxes are connected with arrows. Each oriented arrow agency "plus 3 equals" or "times 4 equals" in addition to you lot receive got to fill upward inwards the let on to the boxes or to the arrows. Thank God, that surround only continues through the 3rd grade. Spider webs are a flake to a greater extent than complex in addition to "two-dimensional", snakes are generally one-dimensional. In spider webs, only add-on is trained, it seems, in addition to it continues through the 5th grade. In the spider spider web above, only 3 colored "additive arrows" are used.
Fill inwards the numbers hence that the total of 3 neighbors is ever 8. Obviously, the dominion only plant for 3x1 or 1x3 neighbors, non for L-shape neighbors around a corner.
Neighbors. Another puzzle similar to pyramids or triangles but the numbers are inwards a rectangle in addition to kids memorize some particular weather condition on neighbors that has to live on truthful inwards all the problems of this kind. You may consider that the exercises move increasingly arbitrary in addition to kids are memorizing an increasing let on of nonsensical rules that receive got naught to do with mathematics or its application anywhere inwards the world. Neighbors proceed to the 3rd grade.
Fairground (well, I think that the right translation is an exhibition hall but who cares). You move along a path inwards a 3 past times 3 foursquare in addition to the digits 1,2,...9 successively appear on your path. 2 or 3 digits are there, you lot demand to consummate the remaining 7 or 6. Influenza A virus subtype H5N1 straightforward puzzle but it "enriches" the kid inwards a management that has piffling to do with what I telephone outcry upward mathematics. Like inwards most similar problems, i just tries the possibilities past times brute strength (it's unremarkably trivial).
Buses. There are 7 people inwards the double-decker earlier i stop. Then 3 people acquire into or acquire out. How many people are inwards the double-decker afterwards? Through several stations, you lot demand to hold rails of the let on of people inwards the bus. The firstly graders receive got virtually 10 people inwards the bus. The exercise appears inwards the 6th degree equally good – in addition to the progress? The 6th graders receive got eighteen people inwards the double-decker or so. Wow, what an impressive progress inwards just v years.
Word problems. John has 10 dumplings, Anne has to a greater extent than past times 2, how much is 2+2? The reply is 22.
Now, to twine the firstly grade, at that spot are lots of "games" that are variations of Don't acquire angry, buddy, a German linguistic communication children's game pop inwards Czech Republic where you lot curl a die in addition to advance your slice along a path. Sometimes you lot may receive got to solve some problems at some positions. Shockingly, these games proceed upward to the 8th grade. When I was a kid, no i pretended that "we were learning mathematics" when playing games similar Don't acquire angry, buddy. ;-) But this is an instance of the full general goal: to sell ordinary games that kids play exterior schoolhouse equally "mathematics".
OK, hence lots of these problems receive got appeared inwards the firstly grade! So the firstly degree could receive got been OK. Kids aren't learning whatever "real mathematics" inwards the firstly grade, anyway? Indeed, I think that the method becomes increasingly indefensible if you lot facial expression at the older kids. You just consider that there's no progress.
OK, you lot acquire to the 2d graders. What is added?
Dog handlers. One counts how many dogs an possessor has in addition to how many eyes the dogs have. I could solve 99% of the problems straight off but I just couldn't figure out what I should add together into a 4 past times 4 tabular array with xx inwards the lower right corner. The problems looks totally underdefined to me. I suspect that they're adding eyes in addition to dogs – in all likelihood because they didn't receive got plenty apples in addition to oranges. ;-)
There's some "multiplication square", it's similar the snakes or spider webs with the operations' beingness multiplication in addition to some special rules. Again, this type of puzzle – with the exact same geometry of the foursquare in addition to basic rules – is taught betwixt 2nd degree in addition to 5th grade.
Biland. The kids are basically learning to convert minor plenty integers to the binary form. It continues to the 3rd grade. It's an exceptional instance inwards which i could combat that besides immature kids are exposed to some material. But maybe it's OK. Binary code may live on of import for some portions of information processor scientific discipline but it's a classic "dead end" that leads nowhere.
Parquettes. Cover some rectangle with several prescribed tetris-like or smaller pieces. Is that genuinely mathematics?
The symbols "ice cream, cat, peace, camel, in addition to bird" genuinely announce "mouse, cat, goose, dog, goat".
The most shocking addition: daddy Forrest. It should genuinely live on granddaddy but OK. Kids receive got to memorize that 1,2,3,4,5, [??], 10, xx should live on written equally mouse, cat, goose, dog, goat... [sheep, ram], cow, horse. See this video past times a bunch of fifth-graders explaining the values. The kids receive got to memorize special icons for every brute – a novel organisation to write digits in addition to beyond. On matika.in, this continues upward to the 4th graders but classes receive got been seen where 5th graders even hence do it – after all, the video I just linked to was created past times 5th graders.
(I think it's no coincidence that the icons facial expression similar to the letters inwards Glagolitic script, the firstly alphabet used on the Czech territory since 863 when it was brought hither past times St Cyril in addition to Methodius who constructed it artificially fifty-fifty though they could receive got only added some accents inwards a higher house Greek letters to enrich the Greek alphabet in addition to acquire into usable here. OK, that's basically what Russians did when they invented the Cyrillic script – in addition to nosotros did the same with the Latin alphabet. The Glagolitic letters were rather stupid, excessively symmetric symbols. Should all the schoolkids inwards Czech or mathematics classes larn the Glagolitic script again? Would it convey something to them? Maybe if a few kids larn such things, it's fine, but equally a mandatory stuff for everybody?)
Every kid who is at to the lowest degree inwards a higher house the average inwards mathematics must know that this is just land retarded in addition to has naught to do with mathematics. But inwards Hejný's method, tons of problems are beingness solved where you lot receive got to exercise the animals in addition to their icons instead of regular numbers. "Mouse plus caprine animal equals truthful cat plus goose": erase i brute hence that it's true. You receive got to erase the mouse. But what do you lot precisely larn past times this game based on the assignment of bogus values to brute species? Clearly, kids who dearest to memorize nonsense may dearest it but smarter kids suffer. They may receive got literally physical obstacles that preclude them from memorizing similar bogus stuff. I was sure similar that in addition to I've heard virtually kids who endure inwards such contexts, too. The smart kids are clearly the "villains" who must live on punished.
Games, board games everywhere. Don't acquire angry, buddy.
Third grade.
Indian multiplication is added. You exercise an ancient, alternative method to calculate 85*23. Should that alternative genuinely supercede the normal way how nosotros calculated such things? The Indian multiplication isn't just some curiosity that appears inwards i or 2 classes. It's an surround studied repeatedly betwixt 3rd in addition to 6th degree equally if it were a foundation of mathematics! Just to live on sure, they also larn the normal method to multiply larger numbers. There's some quasi-conventional partition with residual equally well. As far equally I tin say, the ancient Indian method only differs from the normal modern i past times "details" in addition to the teaching of 2 methods may only confuse the kids.
Cards. You exercise some cards with digits to create a let on that is closest to some other i in addition to stuff similar that. You tin do diverse things with digits in addition to cards with digits but is that genuinely the immature mathematician's way of dealing with digits? It reminds me of the cave men who flora the Škoda car. ;-)
Cycle route. Pick segments inwards a map hence that the route doesn't intersect itself. An OK puzzle, the relationships with mathematics is limited. Hours ago, you lot could detect YouTube videos from such classrooms with a wheel route. The instructor was doing literally naught in addition to the kids didn't seem to know what they were doing. Just an hr of chaos.
Fourth grade.
ABCD – algebrograms. A,B,C stand upward for digits, you're given some constraints, detect the value of A,B,C. As inwards other cases, a normal work when used inwards isolation. Here it is trained inwards the 4th in addition to 5th grade. It's non besides much fourth dimension but it's even hence a bigger exposure to this variety of puzzle than what I detect appropriate. (We encountered it inwards the 5th grade, too.) In principle, exercises similar that Pb to a laid of equations with many variables. In practice, the kids don't solve them equally equations but equally a combinatoric work that needs brute strength to examine a finite let on of possible answers.
Area. They're calculating areas except that inwards all cases, it seems just counting the integer value of some boxes inwards a grid. If some geometry doesn't lucifer into a grid (2D grid or a 3D grid of cubes), the kids in all likelihood larn naught virtually it at all.
I receive got already covered over 1/2 of the environments or templates of the problems. It goes on in addition to on. Mostly the same environments that were explained inwards a higher house are repeated throughout the simple school, some of them sometimes end, others appear. But where do the kids get?
You consider some slightly advanced stuff – primes in addition to factorization – inwards the 7th grade. But the 7th degree in addition to 8th degree also instruct kids to do "digit sums". Now, digit sums are a rattling immature activity to do with a number. It's precisely similar the cave men who flora the car. Unless you lot desire to larn whether something is divisible past times 9, you lot don't do digit sums because it's genuinely a rattling unnatural performance from a mathematical viewpoint. (They don't larn the rules virtually the divisibility past times 9, I think. Remember they don't larn whatever rules that they couldn't uncovering past times themselves.)
Influenza A virus subtype H5N1 shocking combination is that 8th graders are supposed to play "Don't acquire angry, buddy", in addition to it's decorated past times tasks to compute the digit sums. This is genuinely retarded.
Eighth graders acquire the project to solve a laid of linear equations with 2 unknowns. I am non certain how they tin all of a precipitous do it, without whatever theory in addition to with the childish problems they did before. There's also a project to solve a quadratic equation. But I think that they just never larn anything such equally the full general formula – or what the discriminant is. So they just approximate the possible solutions in addition to the problems are constructed hence that the roots are 7 in addition to 8 hence there's a adventure for them to approximate correctly, after all. If the roots were to a greater extent than complicated, they would receive got no chance.
And we're talking virtually 8th graders.
I can't believe this. When I was a 3rd grader, I heard virtually the full general solution to the quadratic equation, it intrigued me, in addition to I derived the full general formula, in addition to was somewhat happy virtually it. But most kids never derive such things. They even hence demand a huge amount of things that someone else was able to derive in addition to discover; in addition to that they can't derive or uncovering but they may larn how to exercise it. It's just completely sick to demand – in addition to Hejný's method demands it – that kids may only live on taught what they uncovering themselves. It's precisely equally stupid equally to demand that people tin only drive cars that they receive got made themselves. The greatness of of import inventions in addition to discoveries is exactly inwards their mightiness to live on used past times lots of people who weren't able to invent or uncovering them.
The 8th graders' magic additions, magic equations, in addition to magic foursquare don't intellectually surpass the problems that already appeared inwards the firstly grade. It's the same variety of standardized schemes, environments, at that spot are no novel ideas, no novel levels of abstractions, no edifice on the previous insights. So the pupils larn basically naught deep during these subsequent 7 years. You know, if it were mathematics, they should larn lots of formulae, rules to logically think, methods to show in addition to disprove things, numerous particular proofs, algorithms to describe in addition to calculate areas in addition to convert completely novel problems in addition to situations to variables in addition to equations. They just larn naught of the sort.
They just pass their 8 years with the same variety of puzzles that the firstly graders or kids inwards the kindergarten may straight off understand. They are just substituting somewhat larger numbers, somewhat larger animals to the collection of granddaddy Forrest's animals, add together fractions of the cast x/2 in addition to x/3 to the integers that were acquaint from the beginning, in addition to that's it. It's bad non only because they larn naught advanced; they don't fifty-fifty larn that something that is to a greater extent than advanced exists. So they halt upward thinking that the greatest mathematicians inwards the Earth are those who tin compute non with xx but 100 or to a greater extent than animals of granddaddy Forrest. ;-) Such kids just halt upward beingness intellectual cripples.
The progress is only non equivalent to the progress inwards a conventional classroom. But if I had to state what the alumni of Hejný's simple schoolhouse are roughly equivalent to, well, I would state that the old 8th graders from Hejný's classroom are roughly equivalent to the conventional 5th graders.
By constantly repeating these childish problems in addition to puzzles, it's sure the instance that numerous slower kids "get it" after 8 years, fifty-fifty though inwards the conventional classroom, a smaller fraction would "get it" after v years. But lots of kids who had As (and sometimes Bs) inwards the conventional classroom are genuinely learning much to a greater extent than advanced, mature stuff inwards mathematics. And kids with the same or similar skills at historic menstruum of 14-15 would live on basically non-existent if Hejný's method became dominant.
This would live on a catastrophe for the land because lots of occupations genuinely depend on the old kids who got As or at to the lowest degree Bs inwards mathematics inwards conventional classrooms – in addition to the noesis that they could receive got accumulated at the higher charge per unit of measurement that was expected. Kids who got somewhere – equally kids or adults – had to exercise lots of skills in addition to methods that they were able to reproduce but they just didn't precisely understand, specially non "instinctively", how they were derived or why they work.
If you lot slowed downward this whole progress past times a factor of two, in addition to if the high schools in addition to colleges adapted to the slowdown inwards the simple schools – in addition to obviously, they volition receive got to arrange in addition to dull downward equally good if Hejný's method becomes genuinely widespread at basic schools – it would hateful that the college alumni inwards 2025 volition live on literally equivalent to high schoolhouse alumni inwards 2010. So all the jobs that had expert reasons to demand academy diplomas inwards 2010 volition live on impossible to meaningfully fill. (Maybe, total professors inwards the novel Hejný organisation volition live on equivalent to the contemporary bachelors. And at that spot volition also live on a Hejný-Nobel prize which which volition live on on par with the contemporary masters of science.)
Sure, it's convenient for kids, specially those who had problem with normal mathematics; for their parents who are rich plenty to fund the kids fifty-fifty when they grow older in addition to the immediate happiness of the children is to a greater extent than of import than other things; in addition to for the teachers who don't genuinely receive got to do almost anything inwards Hejný's classroom. But it's non a expert way to pass the taxpayer money in addition to it's a threat for the time to come of the Czech economic scheme in addition to the intellectual credentials of the whole nation.
But please, live on to a greater extent than than gratis to accept the problems from Hejný's method in addition to spread them inwards your country, too. While I think that the blanket application of the method would live on rather devastating for every civilized nation, I also think that at that spot are ever pieces of the methods that you lot may alternative or helpfully exercise inwards your education.
Bonus: the levels of abstraction inwards mathematics
I receive got made it clear that the method ignores the characteristic of mathematics (and physics) that i needs to build on the previous insights, in addition to acquire familiar with increasingly abstract levels of abstraction. That makes mathematics (but also physics) unlike from mutual sense. It makes these subjects unlike from a laid of isolated insights – which is what many other subjects are all virtually (memorization is ever enough; in addition to skipping some classes unremarkably doesn't wound later).
What do I hateful past times these levels of sophistication?
One starts to larn minor integers. And addition. At some point, he learns numbers greater than 9. You tin write them using several digits. You may all of a precipitous do things with higher numbers or several digits at the same moment. Of course, they sort of acquire at that spot inwards that alternative classroom. They also empathize the generalization to negative numbers etc. although I experience that it's genuinely difficult to detect negative numbers inwards Hejný's books of exercises.
But the existent mathematics doesn't halt there. One learns that aside from integers, at that spot are fractions. Fractions already move almost outlawed inwards Hejný's classroom. And if fractions appear, they receive got minor integers inwards the denominator. The full general methods to add together fractions in addition to do other things are beingness delayed in addition to suppressed. We could telephone outcry upward fractions the 2d flooring of the mathematical skyscraper.
Rational numbers may live on extended to existent numbers. The Hejný kids larn something virtually existent numbers but they don't practise the decimal system. I suppose it's beingness assumed that they exercise calculators for everything. But calculators can't instruct them virtually diverse human relationship betwixt rational in addition to existent numbers, virtually solutions to simple problems that shouldn't require whatever calculators, in addition to hence on. I think that calculators should justify the reduction of fourth dimension that kids pass past times doing numerical calculation because it tin indeed live on done past times machines; but I even hence think that they should larn 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, live on my guest. Obviously, Hejný's method is meant to live on for simple schools hence there's no room for those advanced structures. But the kids don't larn "any" construction whose nature isn't obvious to the smart kids inwards the kindergarten. And that's a work because the kids aren't just ignorant virtually particular advanced things, they're ignorant virtually the existence of advanced things.
Now, extending the "set of numbers" is far from the only management how novel floors are beingness built inwards mathematics. In some sense, they're even hence some of the most trivial ways to extend in addition to generalize inwards mathematics.
Elementary schools should instruct variables, the mightiness of \(x\). It's genuinely a taboo. You tin detect \(x\) in addition to \(y\) inwards the 7th degree but none of these things are genuinely trained hence the kids can't acquire what it's expert for. One should larn how to manipulate with equations. How to empathize that 2 lines are precisely equivalent to each other. Sometimes, the manipulation only plant inwards i management in addition to i needs to depository fiscal establishment check the candidate results. Some of them volition work. This variety of logical thinking isn't trained.
Now, at that spot are full general formulae where you lot may substitute whatever 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 in addition to his disciples repeat. In fact, they fifty-fifty state that a kid who knows some formulae is an "intellectual parasite", I kid you lot not. But inwards reality, they're absolutely essential inwards mathematics inwards the normal pregnant of the word.
Aside from exercises designed for schools, mathematics may genuinely live on used inwards lots of situations inwards the existent Earth – to some extent, inwards all of them. This requires the kid to live on able to extract the mathematical description of the relevant real-world questions, larn how to overlook the distracting features of a specific province of affairs that receive got naught to do with the calculation, in addition to how to exercise the full general mathematical ways to solve the total of the problem. Again, this rattling philosophy is considered a taboo. The kids are never supposed to solve "problems of whatever novel kinds". The repetition of the standardized exercises or "environments" is where the kids should alive forever. In some sense, these environments are novel examples of the "safe spaces". The kids are beingness protected against the existent world!
In mathematics, at that spot are lots of full general formulae in addition to rules to solve classes of problems in addition to Hejný's kids aren't learning them at all. So they can't fifty-fifty start with the numerous levels of abstractions in addition to broader lessons that arise farther inwards that direction. What do I mean?
Well, some problems demand a lot of infinite to live on formulated but the solution is simple. When I was 6, my granddaddy asked me how much is 34+78-56-29-34+56+29-78. I don't retrieve the numbers, they may receive got been smaller. I got the right result, zero, in addition to he told me: But look, I tin acquire null easily, past times rearranging the terms. You tin detect pairs that cancel. My full general betoken is that the solution may frequently live on shorter than the formulation of the problem. Some things are trivial to solve if you're clever. And when you're learning mathematics, you lot should receive got the destination to live on to a greater extent than clever inwards this sense.
On the other hand, some problems, similar Fermat's Last Theorem, may live on easily formulated but it's incredibly difficult to solve them or detect a proof. Even if you lot don't know whatever genuinely mature examples of that, you lot should live on persuaded virtually the full general fact that solutions may live on extremely complicated in addition to it's sometimes OK to pass hours or centuries on a work written on i line. It's of import because people realize that mathematicians etc. cannot live on paid just for the let on of characters used to formulate a problem. It's needed for the people to receive got at to the lowest degree an adequate honour to some folks doing intellectual things – whose content isn't understandable to the public. There may live on some valuable total inwards that piece of work fifty-fifty if it seems that they're just solving something rattling simple.
Let me render closer to the beginning. I talked virtually the extension of integers to rational numbers in addition to existent numbers. But there's also the increasing sequence of operations. You do addition, subtraction, multiplication, division. But already at the basic school, kids should larn something virtually exponentiation. Clearly, Hejný's classroom teaches naught similar the identities involving powers allow lonely logarithms.
But hence you lot receive got functions that are "not rational", starting with logarithms in addition to exponentiation. You demand to larn how they piece of work for non-integer exponents, how they're inverses of each other, how they're used to calculate the involvement inwards banks. And hence at that spot may live on special functions. And operations i may do with functions, starting with derivatives in addition to integrals. And differential equations etc. Again, most of these things don't belong to the simple schools. But the kids don't fifty-fifty larn whatever kids' version of the insight that increasingly non-obvious operations with numbers, generalized numbers, in addition to operations with operations (functions) may live on made in addition to at that spot may live on expert reasons for that. The kids inwards that alternative method don't genuinely empathize anything virtually the value of whatever of the stuff they learn. All they larn is some recreational mathematics that is only justified because the instructor wants the puzzles to live on solved.
One other branch of the skyscraper is the probability theory in addition to statistics. So the kids do some things inwards the direction. They curl die for an hr in addition to calculate how many times they got 6. But they don't larn whatever laws. It's forbidden for the instructor to tell them the laws. So they don't genuinely larn anything similar full general combinatoric numbers, rules for probabilities of independent things that multiply, allow lonely measure deviations, linear regression etc. An average individual cannot rediscover most of these things which is why no i is allowed to larn such things inwards Hejný's classroom.
The kids who receive got to rely on the management they're led to halt upward believing that it makes feel to let on granddaddy Forrest's animals, in addition to it doesn't brand feel to do what advanced mathematicians are doing. It would sure live on empty-headed to consider full general functions in addition to do operations with the functions – similar integration. The schoolhouse just teaches them incorrect ideas virtually what is meaningful, useful, in addition to natural to do; in addition to what isn't.
The method's focus on the brute strength is alone pathological for the kids' sentiment virtually "brain activities" inwards full general because the kids measures everything past times the amount of brute force. It's beingness assumed that every work is "solved" past times trying all possibilities for answers. When the let on of possible answers is infinite, e.g. when it is a generic existent number, the kids are screwed. But methods to acquire the answers right away – or at to the lowest degree faster, past times many, 100, or a googol of orders of magnitude – frequently exist. They should genuinely live on the content of mathematics pedagogy but they're treated equally taboos past times Hejný's method, too.
What genuinely makes me rather emotional is that I know lots of people who genuinely believe that rational reasoning doesn't work, physics in addition to scientific discipline don't work, in addition to mathematics is at most useful for the solution of some special puzzles inwards recreational mathematics. They misunderstand everything virtually mathematics in addition to scientific discipline in addition to why the whole Universe basically plant on the ground of rules that may live on mathematically formulated in addition to mathematically analyzed. And what makes me angry is that I experience that Hejný's method is designed to "confirm" these utterly idiotic misconceptions virtually the mightiness in addition to role of mathematics, mathematically formulated laws, in addition to mathematical reasoning inwards the whole world! While the puzzles may live on fine inwards isolation, the message conveyed inwards betwixt the lines is a laid of misconceptions that a individual who doesn't empathize mathematics at all may believe. Hejný's method is a political computer program to legitimize the stupid people's opinions virtually mathematics.
The kids halt upward beingness innumerate at all levels (starting from the multiplication tables in addition to ending with all the levels of abstraction sketched above) in addition to if they volition ever stand upward for a substantial fraction of the productive generation inwards a nation, the land volition cease to live on civilized. Maybe such a land volition even hence furnish the Earth with sufficiently many workers who may press some buttons inwards a factory. But it can't facial expression to live on alongside the nations with the highest gross domestic product per capita.
No comments:
Post a Comment