This El Risitas parody got over 100,000 views. El Risitas' High German counterpart, Adolf Hitler, was precisely a footling chip less successful.
Zetor Major
The work is the following:
A tractor ran over a metro of paint. The metro exploded, pigment was all around, too the tractor was leaving a grade on the route every 252 centimeters. What is the summit of the oculus of the contaminated tractor's wheel?Many people who follow the pedagogy of mathematics agreed that it was an piece of cake enough, well-chosen, yet "somewhat nontrivial" work that the expert plenty kids genuinely should live able to solve. Just to live sure, the solution is \(252\,{\rm cm} / 2\pi \approx 40.16\,{\rm cm}\).
Tons of kids whined too claimed that it was harder than a twelvemonth agone – too it was similar a work inward an entry examination for a university. Oh, really?
OK, at that topographic point are roughly 2-3 steps ane must do or 2-3 "types of skill" needed to solve the problem:
- One must live able to interpret the existent basis situations to the mathematical language. In particular, hither nosotros must realize that the distance betwixt the marks is the circumference of a circle piece the summit is the radius
- One must know the formula for the circumference, \(\ell = 2\pi r\)
- Perhaps, if it's nontrivial, ane must live able to invert that formula.
Some of these problems inward which kids connect the mathematical propositions to the properties of existent basis situations must live trained at school, I think. On the other hand, the schoolhouse tin give the sack never comprehend "all possible real-world applications" of the mathematical concepts because their lay out is basically unlimited. It's possible that fifty-fifty afterwards several examples, many kids precisely won't know how to apply the mathematical methods inward a new, slightly dissimilar situation. The examples of real-world problems should strip the kids of the fearfulness to "try to connect the dots". However, in ane lawsuit the fearfulness is gone, additional real-world problems may live useless. At that moment, some kids volition know how to connect the dots too some won't.
(I think that the rattling fact that "something bad has happened to the metro of paint" came every bit a source of laughter every bit good every bit stupor for most of the kids because they're basically non led to whatever existent problem-solving. Instead, they're brainwashed that nether the glorious leadership of the European Union, problems belong to the history too nosotros volition never confront whatever novel problems such every bit a broken metro of paint.)
Concerning the 2nd point, I consider the formula \(\ell = 2\pi r\) to live the most fundamental too simplest human relationship involving \(\pi\). Wikipedia tells y'all that \(\pi\) was "originally" defined every bit the ratio of the circumference too the diameter of a circle. It has other definitions, nosotros acquire from the English linguistic communication Wikipedia but non the Czech one.
At whatever rate, this is clearly the simplest intuitive province of affairs inward which \(\pi\) appears. The circumference of the unit of measurement circle \(2\pi\) is also the periodicity of \(\sin x\), \(\cos x\), \(\exp(ix)\), too related functions. With this definition, y'all may derive many other formulae that include \(\pi\) – volumes of balls too materials similar that (which are harder because they are higher-dimensional) every bit good every bit formulae that are helpful for the numerical calculation of \(\pi\).
I strongly believe that the memorization of basic formulae such every bit the circumference of a circle does belong to the classrooms. On top of that, children must evidently live able to do things such every bit the reversal of the circumference formula; ane may consider this reversal to live a simple instance of an equation.
Fine. So a curious kid is intrigued past times the simplicity of the circle too past times the nontrivial lay out that appears every bit the ratio – it's cool that \(\pi\approx 3.141592653589793238462643383279\dots\) is too then universal too everyone who is sane would halt upwards amongst the same digits – infinitely many of them. When I was 8, I memorized these thirty digits afterwards the decimal request from a pop bulk because I all the same establish it cool too important. When I was 10, I no longer thought it was terribly of import to recollect digits of \(\pi\) but I memorized additional lxx (to convey the total to 100 digits afterwards decimal point). At that time, my motivation was to receive got fun piece impressing some people who thought it was a superhero science piece it seemed piece of cake plenty for me. People ever ask: Is it periodic? Is at that topographic point some system? I assure y'all that at that topographic point is no system, at to the lowest degree non a "simple enough" to calculate the digits inward existent time.
OK, define \(\pi\) too then that \(2\pi\) is the circumference of the unit of measurement circle. Try to mensurate its numerical value. How do y'all acquire other formulae that include \(\pi\)?
What is the expanse of a circle? (If people weren't sloppy, they would insist on the phrase "area of a disk" too Czech mathematics teachers typically do insist on that!) Well, y'all may unopen to separate the expanse of a circle to rattling sparse triangles amongst a unmarried vertex at the oculus of the circle too 2 nearby vertices at the circumference. When they're genuinely thin, their combined expanse is arbitrarily unopen to the expanse of the disk. On the other hand, y'all may calculate the expanse of a triangle every bit \(S=w r/2\), half of the rectangle. But the heart of widths \(\sum w = \ell = 2\pi r\) is the circumference, too then the heart of areas of rectangles is \(\sum southward = (r/2) \sum w = \pi r^2\) because the factors of 2 cancel.
The bulk of a ball, \((4/3)\pi r^3\), may live obtained past times the integral of areas of disks if y'all piece the ball horizontally. The surface of the ball is \(4\pi r^2\) which y'all may obtain past times differentiating \((4/3)\pi r^3\) amongst honour to \(r\); the factor of \(1/3\) cancels. Equivalently, y'all may acquire the bulk from the surface past times adding \(1/3\) which is the same \(\int_0^1 dx\,x^2=1/3\) that y'all consider inward the bulk formulae for cones etc. too it has the same reason. You may cutting the bulk of a ball to many "pyramids".
Even though I no longer considered the digits of \(\pi\) to live rattling important, I continued to play amongst \(\pi\) a lot. So a Commodore 64 computer program that I wrote inward the 6510 machine code – to motion into fast – was able to calculate upwards to 40,000 digits using Machin's formula inward ane week. The fourth dimension required for the calculation scaled similar \(N^2\) where \(N\) is the lay out of digits. I wrote the total code – which needed to compute powers, multiply, divide, add together – on the newspaper too it almost straightaway worked when I transferred it to C64 using an assembler that I wrote, too. It's non 100% clear to me whether I could genuinely do the same now. Well, I would in all probability say that I don't receive got the motivation to bear witness it in ane lawsuit to a greater extent than but unless I do it again, I don't really know whether I genuinely tin give the sack do it.
Also, the bulk of the \(K\)-dimensional unit of measurement ball may live integrated to obtain the bulk of the \((K+1)\)-dimensional one. I did it, tried to limited the outcome every bit a multiple of a ability of \(\pi\). And it did work. However, the concise formulae seemed to piece of work differently for fifty-fifty too strange \(K\), respectively. It took some fourth dimension to realize that ane may unify the 2 formulae if ane extrapolates the factorial to half-integer (or whatever existent or complex numbers), starting from \((-1/2)! = \sqrt{\pi}\).
The nicest derivation of the bulk of the \(K\)-dimensional ball uses the \(K\)-dimensional Gaussian integral. On ane hand, the higher-dimensional Gaussian integral may live easily calculated every bit a ability (product) of the one-dimensional ones. On the other hand, it may live calculated inward spherical coordinates too then that the integral reduces to the surface times a one-dimensional integral that is easily converted to the Euler integral for the Gamma component division (a factorial).
OK, at that topographic point are lots of other funny too sometimes unexpected contexts where nosotros run into \(\pi\). For example, \(1+1/2^2+1/3^2+\dots = \pi^2/6\). That's \(\zeta(2)\) too \(\zeta(2k)\) is a rational multiple of a ability of \(\pi\) for whatever positive integer \(k\). Those may live computed from some norms of periodic but locally polynomial functions. That's also a lot of fun. As college freshmen, nosotros did a challenger how far nosotros tin give the sack acquire without whatever calculator. I think I got to \(\zeta(14)\). ;-)
It's obvious that at some point, the interests of the people diverge too most of the materials is uninteresting too likewise academic for a large bulk of people which may include most of engineers or men who utilisation mathematics to earn billions of dollars every bit investment gurus.
But nosotros all the same receive got the circumference of a circle – too perhaps the slightly less simple formulae for the ball. I do think that those would belong to some "basic culture" of our civilization fifty-fifty if they weren't useful. But they are useful. Just 2 weeks ago, my begetter needed to calculate how much the bird of H2O changes inward the swimming puddle when y'all add together some 200 liters or something similar that. You genuinely demand such things if y'all do some theoretical or practical things inward your life.
The people who defend option methods (e.g. Hejný's method) to learn mathematics often say that they don't know the formula for the bulk of a ball, they don't demand it, they tin give the sack derive it or Google it. You know, when I was a teenager, I was often making similar statements – but ordinarily inward non-mathematical contexts. If y'all tin give the sack derive something or observe something inward books, it's a waste product of fourth dimension to memorize the answers. It turns us into dull machines.
Yes, I all the same believe it's rattling often the case. But at that topographic point are some of import "buts".
Concerning derivations, ane should enquire whether it's genuinely the instance that "you tin give the sack derive it" assuming that y'all don't know anything. The circumference of the circle involving \(\pi\) is a spectacular example. The circumference of the unit of measurement circle provides us amongst "the definition" or at to the lowest degree "the simplest definition" of \(2\pi\). If y'all don't know the formula for the interference, it basically agency that y'all don't know what \(\pi\) is. It follows that y'all can't maybe live able to "derive" whatever contention that involves \(\pi\) at all! So your promises that "you tin give the sack derive it" are demonstrably lies, precisely similar when politicians hope things that they can't maybe fulfill. I think that the people who purchase these promises are rattling similar folks who purchase the contention "but I tin give the sack derive it".
Even if at that topographic point is no impenetrable mathematical hurdle that prevents y'all from deriving something, it is all the same truthful that inward most cases, people solely say that they tin give the sack derive something but inward practice, they can't because they're precisely non skillful enough. When they verbalize likewise much too hope likewise much – but do likewise footling piece of work – they halt upwards heavily underestimating the form of difficulties that often halt the people's attempts to derive something. "Too much verbalize (and promises), likewise footling work" non solely creates people who can't genuinely do of import plenty things. It also creates people who don't receive got plenty honour for those who tin give the sack do these things.
From a social perspective, I often think that the 2nd outcome is fifty-fifty to a greater extent than toxic than the offset one. Some people solely claim "Yes, nosotros can" (for example, "Yes, nosotros tin give the sack derive some formula") but they genuinely cannot do anything too they haven't done anything useful inward their life. (Yes, I can... confirm that I would in all probability include Barack Obama to that category.) Some other people who genuinely tin give the sack do these things are downgraded to some subordinates of those who solely talk, promise, too marketplace things – too that's precisely wrong.
Instead of deriving things, y'all may observe the formulae through Google. And the Internet search engines – too also Wikipedia where they often Pb y'all – are immensely helpful tools for those who know what they're doing. But they may also live an extremely evil main for those who don't know what they're doing.
One obvious tendency that we're seeing is that kids too students etc. are looking for the lastly answers on Google. They receive got all the engineering scientific discipline that would let them to human face for the simple facts that "cannot live derived" – too the users could "derive" the right conclusion themselves. But that's non happening. Most users want the lastly answers direct – too then that they don't demand to think at all. An extreme cast of this civilisation is the habit to copy-and-paste whole essays from the Internet, of course.
I think that technology, Google, too Wikipedia shouldn't live blamed. Google, Wikipedia, too other services may also live used too then that people human face for the "pieces" too they do their ain conclusions too derivations. Instead, the ideology that "we don't demand cognition because nosotros tin give the sack Google things out" is the actual culprit of this trend.
The consequences of this tendency are obvious too evidently negative: Kids too people don't know how to independently think, they can't select handle of fifty-fifty the most self-evident mistakes, pranks, hoaxes, manipulation, too and then on. So this tendency should live slowed downward or stopped. Kids should live demanded to genuinely know things too derive things – instead of repeating the oft untrue promises that "yes, they can". Kids should live held responsible for these promises – precisely similar politicians.
Computers too other machines are doing lots of piece of work that solely humans could do inward the past. The spectrum of activities inward which humans may live replaced amongst machines is growing every year. But the humans are all the same the ultimate bosses of the machines. It agency that sure "key decisions" or the "big painting thinking" must live done past times the humans. The machines may do rattling sophisticated things but pretty much past times definition, all these things should live considered "basically mechanical or automatized" because they are beingness done past times "automatons". Unless y'all want to literally give-up the ghost a slave of the machines too then that the machines may command y'all – too then far, no machines fifty-fifty "want" this role but it may modify – y'all but receive got to hold some abilities that save your role every bit someone who has the province of affairs nether control. You must ideally live able to do what the subordinates too machines are doing for you, albeit less efficiently – to a greater extent than realistically, y'all must receive got at to the lowest degree some clue what's happening inside.
If I human face at the progress, I tin give the sack concur that it's much less of import for kids to know how to compute the foursquare root of a lay out on the newspaper – the calculators receive got already made these too other things much less important. But ane of the reverse examples is the circumference of a circle. I don't consider anything inward the technological progress that would justify the children's reduced cognition of similar basic formulae involving \(\pi\).
So I believe that kids should all the same memorize the formulae for the circumference too expanse of a circle too the surface too the bulk of a ball or a sphere. The mathematically oriented kids should also acquire familiar amongst the proofs that demo that it's the same \(\pi\) inward all these formulae – that the formulae may live derived from each other, if y'all wish. But at that topographic point should naturally live many kids who know the right formulae but don't know all the proofs. Knowing all the proofs underlying your mathematical cognition is a corking matter too a sign that you're a existent mathematician who has his mathematical cognition nether control. But on top of that, these formulae are fundamental too often useful which is why it's all the same amend for rattling many people to know these answers fifty-fifty if they don't know all the proofs too derivations.
Hejný's method is the extreme instance of an ideology that says that kids shouldn't live taught anything inward mathematics that they haven't derived themselves. Even though I receive got ever been extreme inward the want to derive – inward ways that could live called "heuristic" past times others but I had reasons to live 100% sure almost the derivations, anyway – all the things I knew, I all the same couldn't avoid "knowing" lots of things for a long fourth dimension earlier I could derive them or sympathise all their connections to other insights. Well, I all the same don't know all the connections. There are many connections too some of them are unknown past times anybody inward the world.
It seems obvious to me that the pct of the mathematical cognition whose root too implications aren't quite understood must live much higher for kids too people who aren't mathematically oriented. The claim that "the kid should solely know what it derives" must hateful that they don't know likewise much themselves. If they knew enough, they would realize how much each of us has to stand upwards on the "shoulders of giants". We but couldn't alive a similar civilized life – every bit individuals too every bit a monastic tell – if nosotros couldn't utilisation the inventions, discoveries, too the results of piece of work past times others – including the generations who lived inward the past. These things that are needed for our decent modern lives also include the mathematical discoveries that were made past times others. The thought that people are to a greater extent than self-sufficient when it comes to mathematical discoveries too derivations than when it comes to the production of cars, smartphones, lunches, movies, or pop music is precisely a delusion – a delusion that arises from the grouping think of uncultural people who dislike mathematics too who persuade each other that mathematics is unimportant too worthless.
It's not. It's precious. And what others receive got establish inward mathematics is rattling of import too precious, too. You can't supersede it amongst some uncontrolled playing of the kids.
And that's the memo.