I don't know what kind of mind finds it easy to go from such a "concrete" example to the abstract, but I don't have one. Now let's consider taking just the fir

I don't know what kind of mind finds it easy to go from such a "concrete" example to the abstract, but I don't have one. Now let's consider taking just the first sentence on wikipedia for the Klein Four group: > In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. Starting with the abstract definition, now it's clear what we're talking about. The problem with math education generally tends to be that the teacher knows the concept that the students are meant to learn, but the students don't, and thus logical jumps which appear obvious to the teacher are not at all so to the student. Concrete examples which the teacher thinks clearly illustrate the concept just add noise that the student must cut through. Consider for example the pattern 1, 2, 3, 5, 7... what is the sequence I am trying to convey? I bet you're thinking the next number is 11, but you would be wrong. It's 9. The sequence is n+1 = floor((n + n+2)/2). Of course there are an infinite number of patterns that would fit the data. Expecting you to pick the specific one of those infinite I had in mind would be absurd. So to is the number of potential truths in mathematics infinite - expecting a student to get the one a teacher has in mind from a finite number of concrete examples is likewise woefully impractical.