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I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physi

by inbox 9y ago

I wish I came across the so called proof-based math before calculus and trigonometry. It would have grabbed me instantly. High school math (and especially physics) classes would leave me with very uneasy feeling that something crucial is left unmentioned, something important is swept under the rug and something important is hidden for whatever reason. Turns out I was wanting for proofs, but couldn't articulate it - I just sensed that something was off :)

jessaustin 9y ago
Your high school didn't teach geometry?
- adamnemecek 9y ago
  Geometry proofs aren't real proofs.
  - yequalsx 9y ago
    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.
    - eximius 9y ago
      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.
  - jessaustin 9y ago
    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...
    - deleted 9y ago
      [deleted]
  - TheRealPomax 9y ago
    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.
    - adamnemecek 9y ago
      Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy type theory) but that's neither here nor there.
      - TheRealPomax 9y ago
        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."
  - patta54 9y ago
    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.
- analog31 9y ago
  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.
  - PhantomGremlin 9y ago
    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.
    - jessaustin 9y ago
      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?
  - madengr 9y ago
    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.
  - deleted 9y ago
    [deleted]
gumby 9y ago
Brewster Kahle (of Alexa and Internet Archive fame) went this way: he and his wife decided that school wasn't right for their younger son so the two of them (father and son) worked through Euclid together.

by inbox 9y ago

jessaustin 9y ago

Your high school didn't teach geometry?

adamnemecek 9y ago
Geometry proofs aren't real proofs.
- yequalsx 9y ago
  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.
  - eximius 9y ago
    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.
- jessaustin 9y ago
  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...
  - deleted 9y ago
    [deleted]
- TheRealPomax 9y ago
  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.
  - adamnemecek 9y ago
    Ofc, geometrical proofs are technically algebraic proofs (due to Homotopy type theory) but that's neither here nor there.
    - TheRealPomax 9y ago
      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."
- patta54 9y ago
  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.
analog31 9y ago
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.
- PhantomGremlin 9y ago
  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.
  - jessaustin 9y ago
    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?
- madengr 9y ago
  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.
- deleted 9y ago
  [deleted]

gumby 9y ago

Brewster Kahle (of Alexa and Internet Archive fame) went this way: he and his wife decided that school wasn't right for their younger son so the two of them (father and son) worked through Euclid together.