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Should mathematics exams be required at the end of high school?

In recent weeks, I was involved in various discussions about the education of mathematics in Czechia. One of the topics was the "playful" Hejný method (a long CZ thread) to teach mathematics to kids which may be fun and useful but it's simply not a legitimate replacement for mathematics as I define it.

Yesterday, someone asked me to solve one page of undergraduate problems in mathematical statistics. Compute the averages, variances and standard deviations, medians, quantiles, draw some histograms, use computer software to do a quadratic fit. And also compute the probability that you get all 4 kings out of 32 cards in a pile of 7. An hour of work. I did consider the problems nicely chosen and adequate for someone who should have background in any experimental science etc.

But they were taken from an exam (a take-home exam?) for mostly female students who want to get a bachelor degree and become nurses. That's tough because I do think that most nurses just can't do a big majority of these things. But the statistics course is mandatory and right now, unlimited nurses do need the bachelor degree. It looks like an anomaly: Ways to deal with a senior who urinated himself could be more useful for them than the calculations of the residual variance of a quadratic fit. ;-) Some lawmakers are preparing a reform that will allow nurses to work without the bachelor degree – the high school plus a year of a "higher school" will be enough. But it's not reality yet.

At the end, however, I have big sympathies for the instructor who is trying hard to convince the students to learn these things. If you asked me, I would probably agree that people with college degrees in science-related disciplines – and medicine is one of them – should be able to do most of these things, at least in principle. It's not possible for most people to know such things and again, I do agree that nurses shouldn't necessarily be "college-educated folks".

The mathematics instructor is universally hated by his students, of course. This is the level that primarily determines my emotions. I just couldn't support the students in their bitter jihad against the noble man. The fact that some soon-to-be-nurses are being pushed to learn things they don't need is one thing. But this guy was hired to teach college-level mathematical statistics and it's simply right to do it right. It's in no way insane to expect the college students majoring in a science-based discipline to know how to do these standard things after two semesters of statistics!

A bigger discussion is one about the required subject in the mandatory Czech final high school exam, the well-known "maturity exam" (Czech: maturita). Unless I misunderstand something, the required subjects are Czech language (which contains some literature etc.) and a foreign language. Mathematics isn't required now and only 25% of the students choose is as their optional subject. It should change since Spring 2021 if I understand well – but the decision may probably be reversed again.

Lots of essays have been written that argued in both directions. It's spectacularly obvious that the two sides may be described as those who "like and respect mathematics" and those who don't. Needless to say, the postmodern "guru" of the "playful" teaching of mathematics to kids, Mr Hejný, is on the side of the opponents of the required mathematics maturity exam – he is really a hater of mathematics itself although his fans love to obfuscate this key fact.

Numerous authors have made great points why it's appropriate to make the mathematics exam required. Mr Plzák was comparing mathematics and Czech. Our native tongue is so yummy (so far so good – our Czech teacher would always lovably talk about "červeňoučké jablíčko", emotionally decorated words for a "red apple" that is so boring in English, to explain the superiority of Czech) and you can use it everywhere... Wait a minute. 0.171% of the mankind speaks Czech and the percentage is unlikely to grow substantially. On the other hand, the whole Universe speaks the language of mathematics. You can be sure what this guy thinks and I agree with his witty arguments.

He also argued that many Czechs don't see that the increase of price of a bun from CZK 1.50 to CZK 2.00 – a big 33% increase – looks negligible to most Czechs who don't really understand the rule of three and related things in mathematics. And they may have problems to calculate mortgage, too.

Others have pointed out e.g. that Mr Hejný must fail to believe his "amazing" pedagogical method, otherwise he wouldn't be afraid of introducing the mandatory mathematics exam.

I am nicely surprised that Ms Kateřina Valachová, the current social democratic minister of education otherwise considered a puppet of the progressive NGOs, is promoting the required mathematics maturity exam. Sometimes, it has to happen that the least likely folks turn out to stand on the right side of some important enough question.

Opponents of the required mathematics exam like to say ludicrous things – e.g. that the maturity exam should reduce teenagers' obesity (?) instead of encouraging them to learn some wisdom such as mathematics. Great, you can prevent obese people from completing the high school but what is it good for? Will you sleep well? But in most cases, their tirades – just like the monologues of Mr Hejný and his fans – quickly deteriorate to the universal song of math haters: Mathematics is just terrorizing children because it's forcing them to memorize various algorithms instead of being creative and thinking logically.

What they fail to notice is that if someone doesn't ever learn or rediscover most of the basic mathematical algorithms, then it proves that she or he is not thinking logically and she or he is not creative. If you don't know how to solve a set of two linear equations at the end of the high school (or later), it's a very unflattering testimony about your logical thinking and creativity. There are many ways to solve these problems and many others but if you haven't been able to get any of them, it's too bad.

You may be completely ignorant about all actual formulae, identities, laws, algorithms that allow rational people to think about the world around them – and you may still call yourself a logical if not creative person. But in that case, you are just lying to yourself and other idiots around you. Some logical or rational or independent or even creative thinking may be a primary driver that allows people to reinvent rules, laws, algorithms, tricks to figure out various things. But it's not a goal by itself and if you don't learn how to deal with any actual and particular problems or questions in mathematics, it probably means that you didn't have enough logical thinking and creativity, after all.

This systematically repeated talk about creativity and logical thinking by people who self-evidently suck both at creativity and logical thinking annoys me greatly.

Children, teenagers, and even adults should be encouraged to educate themselves especially in the knowledge that is important for other things because many other things can be built upon that primary knowledge – and the knowledge of mathematics is a key example. Some people can't do it well and they shouldn't be automatically sent to gas chambers because of that. But they shouldn't be given amazing stamps and degrees, either. If someone hasn't mastered basics of the high school mathematics, he shouldn't be getting diplomas proving any "broad education" at the high school level.

Similarly, people who haven't mastered basic undergraduate statistics and similar things shouldn't be getting college or university diplomas at colleges and universities focusing on scientific subjects. It's too bad when so many people try to circumvent all these common sense rules. They are basically trying to make the schools and diplomas from the schools useless.

On the contrary, the knowledge that should be left to people's personal opinions – such as their opinions about the value of environmentalism etc. – should be completely avoided at schools. Also, answers that one may obtain by a 30-second Google search and they seem good enough shouldn't be covered at schools, either. Schools should focus on the construction of the skeleton of knowledge that the people can't learn themselves by a simple search or by their ordinary recreational activities.

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