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Teaching with proofs

CaptainBlack

Well-known member
Jan 26, 2012
890
In online maths fora we often see posts like the first post in this thread from kanderson.

When I was in secondary education and then an undergraduate we were never taught about poofs, rather we saw them and produced our own. When I was first at university we were recomended Polya, but it was already too late there was lttle in "How to Solve It" that was not familiar. The only thing I learned about proof after leaving secondary education was a research technique: "If you can't prove what you want, prove what you can" and that I had to manufacture myself to resolve a crisis while writing up my thesis.

So my question is, when did picking up methods of proof become a problem (assuming it is)

CB
 

caffeinemachine

Well-known member
MHB Math Scholar
Mar 10, 2012
834
In online maths fora we often see posts like the first post in this thread from kanderson.

When I was in secondary education and then an undergraduate we were never taught about poofs, rather we saw them and produced our own. When I was first at university we were recomended Polya, but it was already too late there was lttle in "How to Solve It" that was not familiar. The only thing I learned about proof after leaving secondary education was a research technique: "If you can't prove what you want, prove what you can" and that I had to manufacture myself to resolve a crisis while writing up my thesis.

So my question is, when did picking up methods of proof become a problem (assuming it is)

CB
Few years back when I first encountered the problem that "In a party 6 friends meet, prove that one can find 3 mutual friends or 3 mutual strangers among them" I found out that Pigeon-Hole principle lies at the heart of the solution. Although pigeon-hole principle is a pretty natural thing in itself, it was not obvious to me that the problem could be solved using PHP. Then many Olympiad problems had their solutions starting with "Consider the smallest...", and suddenly the problem became trivial. So this was another proof technique I had to assimilate-The Extremal Principle. Induction seemed to be natural in many cases but in the beginning I used to find it tricky 'on what I should apply induction'.
But now, after so many years, I am comfortable with method of proof I see.
 

kanderson

Member
Jul 7, 2012
13
Lol sorry, its not that its hard, I just think the books I get are pretty bad at explaining or dont have as much excercises.
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,052
Lol sorry, its not that its hard, I just think the books I get are pretty bad at explaining or dont have as much excercises.
I think CB has a legitimate point and wasn't picking on you, just your thread prompted him to make his post. The first time proofs were introduced to me was in 8th grade geometry and they weren't brought up again until Linear Algebra in college. When brought they were more like "watch and repeat" forms, which doesn't lead to a deep understanding. I wish proofs were introduced in high school curriculum to some degree everywhere.