As you ask your readers for “difficulties and paradoxes of their early mathematical experiences” let me tell a big one I had.
When I was first introduced to “proofs” (by the way of elementary arithmetic and geometric theorems), I did not understood a lot of things.
A) I did not understand why there was a need for a proof. I had already learned a lot of things without any kind of proof.
- I had learned that letter “a” was written in a certain way; nobody “had proved” me that, they just tell me “that’s the way to do it”.
- I had learn that there were four seasons thru the year, but nobody ever prove me that.Why bother whith a proof.? “Just tell me the way it is.”
B) I did not understood the proof itself. There were several problems with that:
B1) I did not understood what is “logic”.
Why were there some things that could be done and others that could not be done? The basis of theorem proving is to use axioms or other theorems. Why was it that way? Most of the time it all looked to me like “pulling rabbits out of a hat”; some times there was an axiom that solved the problem, some time there was not.
B2) I could not “build it just by myself”
I learned several proofs, but I had to memorise them, there was no way for me to “build it just by myself”.
B3) “Several steps” and “What was proved and what was to be proved”
- Later on I was able to understand each step of a proof, but could not see that the “whole proof” was “a lot of small steps put together”.
- Still later, I was able to “follow a reasoning” (to see how several steps contribute to the final result), but there still was a problem. For example: I remember having a lot of trouble with the proof that the square root of two is not a rational number. I was able to follow the reasoning, but what was failing was my understanding of what a rational number is. So I had learned a proof, but that proof had no meaning at all.
B4) “Get a feeling”
There was another problem in many of these theorems: I could not “feel them”. At the time I just thought nothing about the “feeling” of a theorem. But now, many years later, I think this a very important step in understanding, at least for myself. Whenever I read something new for me (mostly mathematics or physics) I start trying to get a feeling of any theorem, before looking at its proof. Of course, I do not get it right away, but I keep reading the result of the theorem until I am able to give it some sense. The biggest help I have found to get this “feeling” is to read the same theorem on several books; every one of them helps a little to “clear the fog”.
For example: when I learned Euclid’s theorem on the infinity of prime numbers, I learned the proof, but I could not “feel it”. My “reasoning” was: whenever I multiply two integers I get another one. So when I have a lot of prime numbers, say one million, I can multiply them in an infinite number of ways, using any one of them as many times as I choose, so I will be able to build an infinity of numbers. So Euclid’s theorem “was not reasonable”.
You may use (or not use) the previous text in any way you like, just say that it was send to you by ELOT.