The explanation of proofs I got in school was terrible. I had no idea that you could actually reduce everything to one of the applications of the axioms until I found metamath's proof explorer.
Of course, it's too much work for most normal purposes, and in school they accepted whatever random breakdown people used inconsistently and never explained why.
Actually understanding that it wasn't about convincing anyone so much as having a chain of reasoning going all the way back to the axioms was something of a revelation for me.
Fortunately proofs were still part of the curriculum when I was in school. Typically, geometry focused on proofs. When my kids took geometry, proofs took a back seat to problems. This also got us to the point where the college math curriculum has to include a class on how to do proofs.
The idea the proofs are the heart and soul of mathematics is an unfortunate unforced error, and will lead to the death of the professions now that machines are better at making proofs.
And if your job is to train people to become mathematicians, that is absolutely what you should be doing.
The explanation of proofs I got in school was terrible. I had no idea that you could actually reduce everything to one of the applications of the axioms until I found metamath's proof explorer.
Of course, it's too much work for most normal purposes, and in school they accepted whatever random breakdown people used inconsistently and never explained why.
Actually understanding that it wasn't about convincing anyone so much as having a chain of reasoning going all the way back to the axioms was something of a revelation for me.
Fortunately proofs were still part of the curriculum when I was in school. Typically, geometry focused on proofs. When my kids took geometry, proofs took a back seat to problems. This also got us to the point where the college math curriculum has to include a class on how to do proofs.
The idea the proofs are the heart and soul of mathematics is an unfortunate unforced error, and will lead to the death of the professions now that machines are better at making proofs.
or memorize a few