Klockan3 said:
You are delusional if you really believe in this. It might apply to a small minority but in my experience most actually likes the pseudo maths taught early better than the real maths taught later, people in general do focus on the real world and the further away something is from it the less people likes it.
Maybe I'm delusional, but until something different is attempted we really can't determine whether that is the case.
When people prefer "pseduo math" to "real math" I feel that this too is because they have never been taught how to understand math. They still see math as a mechanical activity, except now it's a mechanical activity together with a bunch of "tricks". People may memorize entire proofs word by word, or remember very specific tricks. A good student will when he encounters something he couldn't think of himself stop and ask himself what kind of motivation went into this and what the general thought behind it is, and then when later asked to reconstruct a proof he can reconstruct it on the fly because he understands all the techniques needed.
Take something such as integration by parts. People learn mnemonics for it, write it on cheat sheets, etc. and in my experience it's one of those formulas that people in calculus 1 are most scared of. However people should realize that it's really just the product rule of differentiation being integrated, and few people seem to have trouble with the product rule. If people just understood this they wouldn't need to remember a proof since it would be obvious.
Similarly in linear algebra people set great store by finding a formula for a linear transformation given the corresponding matrix, and conversely finding a matrix given a linear transformation. The important part is not the procedure, but that there is an isomorphism between the space of linear transformations F^n \to F^m and the m x n matrices with entries in F. Once you know this the procedure should be pretty obvious (for instance just evaluate at the n basis elements F to get the columns of the matrix). People are presented with this viewpoint, but due to exercises and exams testing them on the procedure not the theory behind it, often people only remember the procedure and not why it works or what's so important about it.
A math freshman at my college mentioned to me how he was confused on an algebra exam because question 1 said "Give an argument showing that ..." and question 2 said "Give a proof for the fact ...", but he was unsure exactly what to do. Basically he wondered how rigorous an argument has to be compared to a proof. The answer is of course that in math an argument and a proof is exactly the same thing. There is no such thing as a somewhat rigorous argument. There is a correct argument (aka a proof), incomplete arguments and motivation. I feel many people have this notion that a proof is something very mathematical and when they are asked to give one they start thinking what kind of proofs there are (contradiction, contraposition, direct, by cases, etc.) But ask the same person why tic-tac-toe can't be won against a good opponent, and he would just start experimenting and probably start with something like "well if I place my first piece in a corner, then, ..., if I place it in the center then, ..." this is actually a proof by cases, but it comes intuitively to them, and often a few arguments by contradiction are also used. People need to see math in the same intuitive manner, and not think of it as symbol manipulation even if at a technical level that's a valid viewpoint.
Actually I feel that in early grades analysis of various games is an excellent way to both excite young students and teach them what math is about, not what technical machinery has been introduced through the ages. Even people not interested in math seem to enjoy learning about a good game. For instance in the tic-tac-toe example there is a great deal of symmetry which will teach students how to exploit this kind of structure in an argument. It's also easy to miss explaining how you deal with a particular case and this kind of error will teach the students to think through carefully whether their proof really works.
Maybe your experiences differ, but in my opinion people enjoy a good thought-experiment as long as you remember not to call it a thought-experiment or math. They have learned that they do not enjoy math, but they can often enjoy a discussion which is essentially of a mathematical nature, but with the machinery removed.
Ask a senior in high school how to add two positive integers of 2-5 digits and he can probably do it and has probably done it hundreds of times, but ask him to explain why to procedure he uses is correct and he will probably not be able to give you a satisfactory answer. He will probably not even try, but what should be the case is that he should immediately start experimenting and thinking. Maybe start by thinking of the case where there are no carries, or maybe just consider 2-digit numbers. We need to add some creativity into math, because until college (and to some extent in many intro-college courses) it is severely lacking.
I got into trouble for not reading fiction books, which I didn't because I wanted to read books on geography, rocks and minerals, rockets and spacecraft , . . . . , and occasionally history, or other non-fiction.
