In other words, the game of sudoku is remarkably similar to the calculations mathematicians do. Why is it so difficult to teach mathematics then?
When I was in high school, we once had a visit of a few Chemistry students from the local university. They were visiting high schools all over the city to inspire young students to apply for university-level chemistry education. Their promo event went like this: they showed us chemicals. That was it. They mixed chemicals of various colours, and created colourful smoke, steam, and liquids in test tubes of various shapes. And then they went home.
This sort of presentation inspires non-scientists and gives the wrong idea of what science is! You should become a Chemist if you love doing Chemistry, not if you love the end product. Doing Chemistry means: these are your starting chemicals, and this is your equipment – which methods and in which order will you do them to arrive at the desired chemical end product? That is what scientists love, that is what gets you the best brains in the class; the colourful chemicals attract non-scientists.
And if you think about it, the same goes for mathematics: “there is an infinite amount of prime numbers” – may be true, may not be true, but let’s find out, let’s either prove or disprove it, we shall see for ourselves. That is exciting. The knowledge of the mathematical proof is much more exciting than the knowledge whether the statement was true or false.
And that is not what we teach, we teach answers, because answers can be graded. I sincerely believe that if the greatest mathematical minds there ever were were born today, they would be disgusted by today’s mathematical education, and would go on to pursue other fields. It is said that Gauss was a rather annoying pupil because he always finished early during the math classes, so his desperate teacher, in an attempt to keep him occupied for the rest of the class, told him to sum all numbers from 1 to 100 when he was 6 years old – he immediately came up with n*(n+1)/2. This formula is what some high school students need to make a “cheat sheet” for and hide it into their shirt sleeves during an exam, because they have not developed the skills needed to derive this formula by themselves – to them it is just a series of symbols.
So, to answer the original question, why is it so difficult to teach mathematics then? – I think the education of mathematics will not change until we find a way how to put an exam grade on mathematical creativity – which is something you can not grade. All we can grade is the “hard work” – reducing a fraction, deriving/integrating a function, all the tasks that no mathematician really enjoys, because it is calculation, not math. So, the only thing we could probably do, is to rename the high school subject from “mathematics” to “calculation”, as it used to be named just a few decades ago in the Czech Republic.
Sorry if I seem way to passionate about this subject matter. It is because I work at a photonics laboratory as a physicist, but the more I do this job, the more I regret I did not study mathematics instead. It was not my fault though – mathematics on the high school level is the most mind-numbing discipline imaginable. It was not until I went to the university when I first encountered what mathematics is about, but that was too late, I was already on Physics. As insulting to scientists as it may sound, from my experience in a modern high-tech physics lab, I notice that all sciences are nothing but subsets of mathematics. I choose the word “subset” very carefully – mathematics is without any doubt the broadest science (not a natural science though!) that defines pure reasoning, true wisdom in its purest form, that the human brain is capable of. Natural sciences, such as physics, take a part of mathematics and put it into the context of atoms. Not the other way around. It is clear that mathematics was always steps ahead than technology – it can not obviously be the other way around.
Please, feel free to disagree with me, I would be happy to be wrong here, as the modern state of math education is sadly not a happy topic. I have heard someone say that a major revolution in math education came during the Cold War, as both the Soviet Union and the US started training mathematicians as “soldiers” – it was believed it would be the mathematicians who would build the best nuclear bombs and win the war, not the infantry soldiers. That is when the strict military-like math grading was enforced. But I found no evidence to prove or disprove this explanation.