My kids are in high school right now. I was getting my daughter psyched up for geometry, by promising her that she'd get to do proofs.
The geometry class completely glossed over proofs. It was much more oriented towards solving problems. I don't know if it was because of standardized testing, but I have my suspicions. Fortunately, my daughter worked on the proofs herself, outside of class.
I was saddened for many reasons, one of which is that lots of people I've talked to -- especially women -- loved high school geometry because of the proofs. That was where math came alive.
I hated proofs in geometry, which explains why I got A's in all my college math classes except linear algebra, specifically because of those damn proofs.
Same deal with my kids in high school, proofs were barely covered.
When I took Geometry in high school about 45 years ago, proofs were essential, and IIRC they dominated the curriculum. And that was good, because it taught logical thinking. Geometry was my favorite course in high school.
This background was incredibly useful to me in both hardware design and in computer programming.
This was also my experience about 25 years ago. Many classmates were not fond of proofs, so perhaps the move away from them is just giving the customers what they want?
They most certainly are real proofs. However, they are a different _kind_ of proof if you mistakenly believe that only algebra can yield proofs. Provided you have the algebraic rules for the type of geometry you are working with, any geometric proof can be expressed as algebraic proof and vice versa, but the trick is to appreciate that depending on what needs to be proven, one can represent in a single step what the other takes many tedious pages of step upon step upon step. And that goes both ways of course.
You don't get the luxury of hand waiving: the two are equivalent, and thus any rigorous proof in one has an equivalent proof in the other. In acknowledging this, you accepted your original claim was false.
So what you really meant here was:
"Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy type theory). I must have had a brain fart when I implied that one could be more, or less real than the other. That made no sense."
Why do you believe this? I think they are. They provide a nice example of using logic and axioms. I don't think one will be able to have a proof based course other than geometry before calculus. Students just aren't mathematically mature enough for that.
It very much depends how it is taught. When I took it, it was very focused on using only the building blocks you were given. Step one: apply theorem 1.2b; step two: apply theorem 1.4a; etc. It was barely more than a search through the space of operations given. The statements proven felt trivial and the proofs needlessly convoluted and rigid.
Whereas in calculus and algebra and analysis and number theory, the proofs often had different paths to prove them or different constructions/descriptions and the things we proved felt substantial.
By that measure, most proofs seen in early undergraduate years are also not real (e.g. the proofs given for the fundamental theorem of calculus). The impression one gets from GP is that they needed to see some proofs in high school. If they had taken geometry with me in Mr. Schardt's class they would have gotten their fill...
They certainly are. Also, geometric proofs are considered the origin of all mathematical proofs. They might seem crude, but open up to any chapter in Elements and you'll see many difficult problems solved using those 5 axioms.
