Saturday, April 21, 2018

Tractor, paint, and what kids should learn about pi

Days ago, Czech kids who are 14-15 years old were trying to pass their high school entry exams designed by CERMAT, a centralized institution producing exams for schools. The most difficult problem was an exercise involving a tractor and a tube of paint in mathematics.

This El Risitas parody got over 100,000 views. El Risitas' German counterpart, Adolf Hitler, was just a little bit less successful.

Zetor Major

The problem is the following:
A tractor ran over a tube of paint. The tube exploded, paint was all around, and the tractor was leaving a mark on the road every 252 centimeters. What is the height of the center of the contaminated tractor's wheel?
Many people who follow the education of mathematics agreed that it was an easy enough, well-chosen, yet "somewhat nontrivial" problem that the good enough kids really should be able to solve. Just to be sure, the solution is \(252\,{\rm cm} / 2\pi \approx 40.16\,{\rm cm}\).

Tons of kids whined and claimed that it was harder than a year ago – and it was like a problem in an entry exam for a university. Oh, really?

OK, there are roughly 2-3 steps one must do or 2-3 "types of skill" needed to solve the problem:
  1. One must be able to translate the real world situations to the mathematical language. In particular, here we must realize that the distance between the marks is the circumference of a circle while the height is the radius
  2. One must know the formula for the circumference, \(\ell = 2\pi r\)
  3. Perhaps, if it's nontrivial, one must be able to invert that formula.
OK, where do kids learn these things? Concerning the first step, I believe it's mostly about the innate aptitude although some training surely helps. One must realize that messy things in the real world correspond to mathematical idealizations. The distance between the paint marks is the circumference; to realize that, one must have some true, real-world-connected understanding of the circumference. Similarly for the radius.

Some of these problems in which kids connect the mathematical propositions to the properties of real world situations must be trained at school, I think. On the other hand, the school can never cover "all possible real-world applications" of the mathematical concepts because their number is basically unlimited. It's possible that even after several examples, many kids just won't know how to apply the mathematical methods in a new, slightly different situation. The examples of real-world problems should strip the kids of the fear to "try to connect the dots". However, once the fear is gone, additional real-world problems may be useless. At that moment, some kids will know how to connect the dots and some won't.

(I think that the very fact that "something bad has happened to the tube of paint" came as a source of laughter as well as shock for most of the kids because they're basically not led to any real problem-solving. Instead, they're brainwashed that under the glorious leadership of the European Union, problems belong to the history and we will never face any new problems such as a broken tube of paint.)

Concerning the second point, I consider the formula \(\ell = 2\pi r\) to be the most fundamental and simplest relationship involving \(\pi\). Wikipedia tells you that \(\pi\) was "originally" defined as the ratio of the circumference and the diameter of a circle. It has other definitions, we learn from the English Wikipedia but not the Czech one.

At any rate, this is clearly the simplest intuitive situation in which \(\pi\) appears. The circumference of the unit circle \(2\pi\) is also the periodicity of \(\sin x\), \(\cos x\), \(\exp(ix)\), and related functions. With this definition, you may derive many other formulae that include \(\pi\) – volumes of balls and stuff like that (which are harder because they are higher-dimensional) as well as formulae that are helpful for the numerical calculation of \(\pi\).

I strongly believe that the memorization of basic formulae such as the circumference of a circle does belong to the classrooms. On top of that, children must obviously be able to do things such as the reversal of the circumference formula; one may consider this reversal to be a simple example of an equation.

Fine. So a curious kid is intrigued by the simplicity of the circle and by the nontrivial number that appears as the ratio – it's cool that \(\pi\approx 3.141592653589793238462643383279\dots\) is so universal and everyone who is sane would end up with the same digits – infinitely many of them. When I was 8, I memorized these 30 digits after the decimal point from a popular book because I still found it cool and important. When I was 10, I no longer thought it was terribly important to remember digits of \(\pi\) but I memorized additional 70 (to bring the total to 100 digits after decimal point). At that time, my motivation was to have fun while impressing some people who thought it was a superhero skill while it seemed easy enough for me. People always ask: Is it periodic? Is there some system? I assure you that there is no system, at least not a "simple enough" to calculate the digits in real time.

OK, define \(\pi\) so that \(2\pi\) is the circumference of the unit circle. Try to measure its numerical value. How do you get other formulae that include \(\pi\)?

What is the area of a circle? (If people weren't sloppy, they would insist on the phrase "area of a disk" and Czech mathematics teachers typically do insist on that!) Well, you may approximately divide the area of a circle to very thin triangles with a single vertex at the center of the circle and two nearby vertices at the circumference. When they're really thin, their combined area is arbitrarily close to the area of the disk. On the other hand, you may calculate the area of a triangle as \(S=w r/2\), one-half of the rectangle. But the sum of widths \(\sum w = \ell = 2\pi r\) is the circumference, so the sum of areas of rectangles is \(\sum S = (r/2) \sum w = \pi r^2\) because the factors of two cancel.

The volume of a ball, \((4/3)\pi r^3\), may be obtained by the integral of areas of disks if you slice the ball horizontally. The surface of the ball is \(4\pi r^2\) which you may obtain by differentiating \((4/3)\pi r^3\) with respect to \(r\); the factor of \(1/3\) cancels. Equivalently, you may get the volume from the surface by adding \(1/3\) which is the same \(\int_0^1 dx\,x^2=1/3\) that you see in the volume formulae for cones etc. and it has the same reason. You may cut the volume of a ball to many "pyramids".

Even though I no longer considered the digits of \(\pi\) to be very important, I continued to play with \(\pi\) a lot. So a Commodore 64 program that I wrote in the 6510 machine code – to make it fast – was able to calculate up to 40,000 digits using Machin's formula in one week. The time required for the calculation scaled like \(N^2\) where \(N\) is the number of digits. I wrote the full code – which needed to compute powers, multiply, divide, add – on the paper and it almost immediately worked when I transferred it to C64 using an assembler that I wrote, too. It's not 100% clear to me whether I could actually do the same now. Well, I would probably say that I don't have the motivation to try it again but unless I do it again, I don't really know whether I actually can do it.

Also, the volume of the \(K\)-dimensional unit ball may be integrated to obtain the volume of the \((K+1)\)-dimensional one. I did it, tried to express the result as a multiple of a power of \(\pi\). And it did work. However, the concise formulae seemed to work differently for even and odd \(K\), respectively. It took some time to realize that one may unify the two formulae if one extrapolates the factorial to half-integer (or any real or complex numbers), starting from \((-1/2)! = \sqrt{\pi}\).

The nicest derivation of the volume of the \(K\)-dimensional ball uses the \(K\)-dimensional Gaussian integral. On one hand, the higher-dimensional Gaussian integral may be easily calculated as a power (product) of the one-dimensional ones. On the other hand, it may be calculated in spherical coordinates so that the integral reduces to the surface times a one-dimensional integral that is easily converted to the Euler integral for the Gamma function (a factorial).

OK, there are lots of other funny and sometimes unexpected contexts where we encounter \(\pi\). For example, \(1+1/2^2+1/3^2+\dots = \pi^2/6\). That's \(\zeta(2)\) and \(\zeta(2k)\) is a rational multiple of a power of \(\pi\) for any positive integer \(k\). Those may be computed from some norms of periodic but locally polynomial functions. That's also a lot of fun. As college freshmen, we did a contest how far we can get without any calculator. I think I got to \(\zeta(14)\). ;-)

It's obvious that at some point, the interests of the people diverge and most of the stuff is uninteresting and too academic for a big majority of people which may include most of engineers or men who use mathematics to earn billions of dollars as investment gurus.

But we still have the circumference of a circle – and perhaps the slightly less elementary formulae for the ball. I do think that those would belong to some "basic culture" of our civilization even if they weren't useful. But they are useful. Just two weeks ago, my father needed to calculate how much the level of water changes in the swimming pool when you add some 200 liters or something like that. You really need such things if you do some theoretical or practical things in your life.

The people who defend alternative methods (e.g. Hejný's method) to teach mathematics often say that they don't know the formula for the volume of a ball, they don't need it, they can derive it or Google it. You know, when I was a teenager, I was often making similar statements – but usually in non-mathematical contexts. If you can derive something or find something in books, it's a waste of time to memorize the answers. It turns us into boring machines.

Yes, I still believe it's very often the case. But there are some important "buts".

Concerning derivations, one should ask whether it's actually the case that "you can derive it" assuming that you don't know anything. The circumference of the circle involving \(\pi\) is a spectacular example. The circumference of the unit circle provides us with "the definition" or at least "the simplest definition" of \(2\pi\). If you don't know the formula for the interference, it basically means that you don't know what \(\pi\) is. It follows that you can't possibly be able to "derive" any statement that involves \(\pi\) at all! So your promises that "you can derive it" are demonstrably lies, just like when politicians promise things that they can't possibly fulfill. I think that the people who buy these promises are very similar folks who buy the statement "but I can derive it".

Even if there is no impenetrable mathematical hurdle that prevents you from deriving something, it is still true that in most cases, people only say that they can derive something but in practice, they can't because they're just not skillful enough. When they talk too much and promise too much – but do too little work – they end up heavily underestimating the kind of difficulties that often stop the people's attempts to derive something. "Too much talk (and promises), too little work" not only creates people who can't really do important enough things. It also creates people who don't have enough respect for those who can do these things.

From a social perspective, I often think that the second consequence is even more toxic than the first one. Some people only claim "Yes, we can" (for example, "Yes, we can derive some formula") but they actually cannot do anything and they haven't done anything useful in their life. (Yes, I can... confirm that I would probably include Barack Obama to that category.) Some other people who actually can do these things are downgraded to some subordinates of those who only talk, promise, and market things – and that's just wrong.

Instead of deriving things, you may find the formulae through Google. And the Internet search engines – and also Wikipedia where they often lead you – are immensely helpful tools for those who know what they're doing. But they may also be an extremely evil master for those who don't know what they're doing.

One obvious trend that we're seeing is that kids and students etc. are looking for the final answers on Google. They have all the technology that would allow them to look for the elementary facts that "cannot be derived" – and the users could "derive" the right conclusion themselves. But that's not happening. Most users want the final answers directly – so that they don't need to think at all. An extreme form of this culture is the habit to copy-and-paste whole essays from the Internet, of course.

I think that technology, Google, and Wikipedia shouldn't be blamed. Google, Wikipedia, and other services may also be used so that people look for the "pieces" and they create their own conclusions and derivations. Instead, the ideology that "we don't need knowledge because we can Google things out" is the actual culprit of this trend.

The consequences of this trend are obvious and obviously negative: Kids and people don't know how to independently think, they can't catch even the most self-evident mistakes, pranks, hoaxes, manipulation, and so on. So this trend should be slowed down or stopped. Kids should be demanded to actually know things and derive things – instead of repeating the frequently untrue promises that "yes, they can". Kids should be held responsible for these promises – just like politicians.

Computers and other machines are doing lots of work that only humans could do in the past. The spectrum of activities in which humans may be replaced with machines is growing every year. But the humans are still the ultimate bosses of the machines. It means that certain "key decisions" or the "big picture thinking" must be done by the humans. The machines may do very sophisticated things but pretty much by definition, all these things should be considered "basically mechanical or automatized" because they are being done by "automatons". Unless you want to literally become a slave of the machines so that the machines may command you – so far, no machines even "want" this role but it may change – you simply have to keep some abilities that preserve your role as someone who has the situation under control. You must ideally be able to do what the subordinates and machines are doing for you, albeit less efficiently – more realistically, you must have at least some clue what's happening inside.

If I look at the progress, I can agree that it's much less important for kids to know how to compute the square root of a number on the paper – the calculators have already made these and other things much less important. But one of the opposite examples is the circumference of a circle. I don't see anything in the technological progress that would justify the children's reduced knowledge of similar basic formulae involving \(\pi\).

So I believe that kids should still memorize the formulae for the circumference and area of a circle and the surface and the volume of a ball or a sphere. The mathematically oriented kids should also get familiar with the proofs that show that it's the same \(\pi\) in all these formulae – that the formulae may be derived from each other, if you wish. But there should naturally be many kids who know the correct formulae but don't know all the proofs. Knowing all the proofs underlying your mathematical knowledge is a great thing and a sign that you're a real mathematician who has his mathematical knowledge under control. But on top of that, these formulae are fundamental and often useful which is why it's still better for very many people to know these answers even if they don't know all the proofs and derivations.

Hejný's method is the extreme example of an ideology that says that kids shouldn't be taught anything in mathematics that they haven't derived themselves. Even though I have always been extreme in the desire to derive – in ways that could be called "heuristic" by others but I had reasons to be 100% certain about the derivations, anyway – all the things I knew, I still couldn't avoid "knowing" lots of things for a long time before I could derive them or understand all their connections to other insights. Well, I still don't know all the connections. There are many connections and some of them are unknown by anybody in the world.

It seems obvious to me that the percentage of the mathematical knowledge whose origin and implications aren't quite understood must be much higher for kids and people who aren't mathematically oriented. The claim that "the kid should only know what it derives" must mean that they don't know too much themselves. If they knew enough, they would realize how much each of us has to stand on the "shoulders of giants". We simply couldn't live a similar civilized life – as individuals and as a society – if we couldn't use the inventions, discoveries, and the results of work by others – including the generations who lived in the past. These things that are needed for our decent modern lives also include the mathematical discoveries that were made by others. The idea that people are more self-sufficient when it comes to mathematical discoveries and derivations than when it comes to the production of cars, smartphones, lunches, movies, or popular music is just a delusion – a delusion that arises from the group think of uncultural people who dislike mathematics and who persuade each other that mathematics is unimportant and worthless.

It's not. It's precious. And what others have found in mathematics is very important and precious, too. You can't replace it with some uncontrolled playing of the kids.

And that's the memo.

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