2 ms·

I would agree that proofs as introduced in high school Geometry have more in common with formal axiomatic logic than the vast majority of "real-world" mathemati

I would agree that proofs as introduced in high school Geometry have more in common with formal axiomatic logic than the vast majority of "real-world" mathematical proofs; but a large part of that is just clumsy notation. The fundamental concepts 1. We have axioms which are things we accept without proof because we all agree that they're obvious; 2. Everything else should follow logically from things which come earlier; 3. Don't skip steps! are extremely important and apply regardless of the field you're in.

joshuamorton 5y ago
But geometry is sort of badly suited to teaching this because it's very hard to justify. We're good at thinking geometrically so things look "obvious" unless you do bad things like explicitly mislabel images (I remember some geometric problems where clearly unequally drawn line segments were labeled as equivalent). Like you end up having to reteach proof in university discrete math or stats classes anyway! Using motivations that we know to be true (or can see empirically are true) but don't feel trivial because "duh those lines are the same length" is a lot more compelling. You actually feel like you've learned something, not just a dumb formalism for something you could already intuit (and I mean if that's all proof does, let you formalize things you already know, what's the point?) You're of course not the audience of these changes, because I expect that my last statement made you gag a little bit. But that's the way a lot of people are, and we shouldn't restrict ourselves to teaching the way they did 2500 years ago[0] unless there's still good reason to do it that way. [0]: I'll admit that I think "look they were able to prove these things in this way 2500 years ago prior to the conceptualization of zero" is a compelling way to teach this stuff, but we don't do it that way either.
- cperciva 5y ago
  I disagree: It's precisely when things are obvious that it's useful to teach formal proof. You can't teach the critical part of "don't skip steps" if nobody is even tempted to skip steps.
  - joshuamorton 5y ago
    Why? "Do as I say without clear motivation except that I'm telling you it's supposed to be done this way" is maybe the worst way to teach. Is that justification going to keep a group of 15 year olds engaged? It certainly didn't for me.
    - darawk 5y ago
      I think the point is to teach people that apparently obvious things are not always actually true. This is extremely important in math as you move on, as i'm sure you know. Many "obviously true" things are not actually true when you get really serious about formalization. Teaching kids that "seems true to me" isn't good enough is a critical step along that path.
      - cperciva 5y ago
        Exactly. You can't teach people that sometimes their intuition is wrong if they don't have any intuition to begin with.
        joshuamorton 5y ago
        I see this in theory, but I'm having a hard time thinking of geometric problems that violate intuition and that are appropriate to teach people who have no calculus and limited algebraic skills (admittedly this limits statistical problems too). I'm actually curious if you have any good examples, because I don't. And I think that's where my objection comes in: if you spend the entire first year just developing that intuition but forcing a rote procedure, you'll have lost the majority of people already. Proof isn't arithmetic and we shouldn't teach it in the same way. Sure develop the process on the easy method first, but then show people why it's important to not skip steps.
  - amalcon 5y ago
    I think a large part of the problem is just that it's poorly integrated into the larger curricula[0]. By the time a typical student reaches geometry class, they have incorrectly learned that math consists of memorizing things, pattern matching onto things they've memorized, and making at most minor adjustments. The kind of logical inference and objective-based reasoning that is necessary for a rigorous proof is covered more in writing class than in math. Some of the kids I've seen adjust well to it have dabbled in software and pattern-matched proofs onto that (which has its own pitfalls, but better than nothing). The remainder had parents who taught this kind of thinking from a much younger age. For those not so lucky, they need to spend probably about a semester un-learning what math is before they can really start. [0]- In the arbitrary parts of the US that I know anything about
- AlanYx 5y ago
  One advantage of geometric proofs is that they seem to be more accessible to some kinds of learners (visual/spatial thinkers) than proofs of purely symbolic results.

joshuamorton 5y ago

But geometry is sort of badly suited to teaching this because it's very hard to justify. We're good at thinking geometrically so things look "obvious" unless you do bad things like explicitly mislabel images (I remember some geometric problems where clearly unequally drawn line segments were labeled as equivalent). Like you end up having to reteach proof in university discrete math or stats classes anyway! Using motivations that we know to be true (or can see empirically are true) but don't feel trivial because "duh those lines are the same length" is a lot more compelling. You actually feel like you've learned something, not just a dumb formalism for something you could already intuit (and I mean if that's all proof does, let you formalize things you already know, what's the point?) You're of course not the audience of these changes, because I expect that my last statement made you gag a little bit. But that's the way a lot of people are, and we shouldn't restrict ourselves to teaching the way they did 2500 years ago[0] unless there's still good reason to do it that way. [0]: I'll admit that I think "look they were able to prove these things in this way 2500 years ago prior to the conceptualization of zero" is a compelling way to teach this stuff, but we don't do it that way either.

cperciva 5y ago
I disagree: It's precisely when things are obvious that it's useful to teach formal proof. You can't teach the critical part of "don't skip steps" if nobody is even tempted to skip steps.
- joshuamorton 5y ago
  Why? "Do as I say without clear motivation except that I'm telling you it's supposed to be done this way" is maybe the worst way to teach. Is that justification going to keep a group of 15 year olds engaged? It certainly didn't for me.
  - darawk 5y ago
    I think the point is to teach people that apparently obvious things are not always actually true. This is extremely important in math as you move on, as i'm sure you know. Many "obviously true" things are not actually true when you get really serious about formalization. Teaching kids that "seems true to me" isn't good enough is a critical step along that path.
    - cperciva 5y ago
      Exactly. You can't teach people that sometimes their intuition is wrong if they don't have any intuition to begin with.
      - joshuamorton 5y ago
        I see this in theory, but I'm having a hard time thinking of geometric problems that violate intuition and that are appropriate to teach people who have no calculus and limited algebraic skills (admittedly this limits statistical problems too). I'm actually curious if you have any good examples, because I don't. And I think that's where my objection comes in: if you spend the entire first year just developing that intuition but forcing a rote procedure, you'll have lost the majority of people already. Proof isn't arithmetic and we shouldn't teach it in the same way. Sure develop the process on the easy method first, but then show people why it's important to not skip steps.
- amalcon 5y ago
  I think a large part of the problem is just that it's poorly integrated into the larger curricula[0]. By the time a typical student reaches geometry class, they have incorrectly learned that math consists of memorizing things, pattern matching onto things they've memorized, and making at most minor adjustments. The kind of logical inference and objective-based reasoning that is necessary for a rigorous proof is covered more in writing class than in math. Some of the kids I've seen adjust well to it have dabbled in software and pattern-matched proofs onto that (which has its own pitfalls, but better than nothing). The remainder had parents who taught this kind of thinking from a much younger age. For those not so lucky, they need to spend probably about a semester un-learning what math is before they can really start. [0]- In the arbitrary parts of the US that I know anything about
AlanYx 5y ago
One advantage of geometric proofs is that they seem to be more accessible to some kinds of learners (visual/spatial thinkers) than proofs of purely symbolic results.