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As someone who did a BA and PhD in math, went through high school geometry proofs, taught geometry proofs to math students and math education students: Geometr

• lupire 5y ago
  That's like saying teaching arithmetic algorithms is bad because you can do arithmetic in your head. You have to know what logical rigor is before you can take defensible shortcuts.
• cperciva 5y ago
  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.
• jjoonathan 5y ago
  Agreed. Personally I think we should swap our high school curriculum: Geometry -> Proofy Linear Algebra Trig -> Computational Linear Algebra I think the heydey of Trig was in the age of navigation and the heydey of Geometry was in the age of machines. Meanwhile, today, linear algebra has supplanted both of them, runs the world besides, and it seems to only be getting started.
  • monocasa 5y ago
    Rather than geometry or linear algebra, I'd love to see logic and set theory taught that early as more than a passing topic.
    • lstodd 5y ago
      Can only up this. Not teaching set theory is just an abomination. And logic. That year of Prolog taught me so much.
  • lstodd 5y ago
    Which would be utterly useless. Math, and specifically Geometry aren't about useful skills, they are about understanding the world.
    • jjoonathan 5y ago
      Linear algebra has geometry flat out beat in that arena, too. This isn't to say that Geometry doesn't help you learn about the world, but it is to say that Linear Algebra should be prioritized above Geometry if your goal is to understand the world. I can't remember a physics class where LA didn't pay heavy dividends -- in particular, differential equations are the language of physics, and you'd be hard pressed to find either an analytical or computational diff eq technique that didn't have a core of linear algebra. Meanwhile, the last time Power of a Point gave me dividends was on math contests.
      • lstodd 5y ago
        LA is a very useful tool. But imagine that you never heard of concepts behind it. It'd just become a more-or-less useful artefact. And in no time it would devolve into a useless ... thing so to say, a decoration, as you lose knowledge how to apply it.
• taeric 5y ago
  I'm curious on how you structure proofs, then? That is, how do you recommend?
• jacobmartin 5y ago
  It is interesting to me that you didn't encounter problems with the rote geometric method before abstract algebra. I went to a so-so public school in the American South, so I don't think I exactly had a stellar introduction to math or even school/academics in general, but beginning in trigonometry, we had to structure our proofs with more words and 'connective tissue,' and less explicit axiomatization/symbolization. Calculus continued this trend, and by the time I took linear algebra (at a local university while I was still in high school), the proofs were exactly like the proofs I would write throughout college. But there were several 'stepping stones' away from the original geometric proof. Though admittedly I only got an SB in Math, I think geometric proofs were an okay introduction to the idea of the proof and of logic. Of course they are not representative of professional math. There are many things we learn as an introduction which don't turn out to bear that much resemblance to the more advanced form. I don't write 5-paragraph essays with clear thesis sentences anymore either. But it helped me to learn to write that way as a way of clarifying that I was stating my arguments effectively. Sometimes, for my homework, I tried to state very explicitly in my proofs which axiomata I was using, both as an intellectual exercise and to see, e.g., where I might have gotten my logic backwards or where I used the axiom of choice (which constantly surprised me). I wouldn't turn these overexplicit proofs, but it helped to clarify that I was doing things correctly. If anything, when I was in college, the trend was away from 'informal' language. When I took, e.g., Discrete Math, for more complicated proofs, we were encouraged to write out a DAG for the dependencies of the proofs/lemmata so that it was exceedingly clear what depended on what and that we had in fact proven what we set out to prove.
  • akomtu 5y ago
    What are those "geometric proofs"? I've never seen this term before. Geometry is really just numbers in disguise, e.g. a circle is really numbers that define its center and its radius, and sometimes those numbers are defined implicitly, e.g. when its a highly complex curve that's not reducible to simple circles and lines. In this sense, all proofs are about symbols and numbers, geometry is only there to guide our intuition.
    • jacobmartin 5y ago
• maire 5y ago
  > Geometry proofs are a terrible way of introducing rigorous mathematical proofs to students You might be right, but they are the only high school math class that teaches proofs. If you take away geometry you are left with nothing. Having said that - any math concept can be memorized and regurgitated. My kids are in CA elementary math right now and it seems to be better than when I was a kid. The emphasis is on derivation rather than memorization. It might just be that geometry is better now than when you were a kid.
• BeetleB 5y ago
  For quite a while I thought I'd never be great at math because I struggled with geometry. Fortunately, I ignored that voice in my head and went on to do lots of advanced math (analysis, abstract algebra, computational methods), ranking in the top 10% of Polya, etc. I still suck at geometry. Geometry is a really poor representation of whether one can do proof based maths.
• OccamsRazr 5y ago
  It's funny, I also think Euclidean geometry is a bad way to introduce proofs, but my reason is different from what you just said. I think what you just said is primarily about "writing proofs", which is certainly important, but I'm not sure it's the fundamental issue with Euclidean geoemtry (and I know that not everyone learns Euclidean geometry and comes away from it writing proofs as you did). It sounds like you came out of your highschool Euclidean geometry experience with good logic skills but poor communication skills (?). The issue with Euclidean geometry as an entrance to proofs and logic is two-fold. First, the definitions and axioms of Euclidean geometry are incomplete in many ways. Many definitions, such as that of a line, cannot be understood from what is written in Euclid but require a significant amount of intution pumping to be able to know how to properly work with them. Thus, the fundamental definitions in Euclidean geometry are bad examples of what a definition in math should be, and we are already starting on very rocky footing. Moreover, the axioms, as they are presented in Euclid or common modern educational sources, are not sufficient to do what is claimed. For example, the proof of postulate 1 already has a logical gap because there is no axiom which guarantees the existence of a point lying at the intersection of the two circles one has drawn. Again, the argument relies crucially on a figure, which is the intuition pump used in all of Euclid, but the picture is not based on any of the axioms, so it is reinforcing a bad way of thinking about proofs that is intuitive and not at all focused on carefully using definitions and axioms. (Of course, one can fix Euclidean geometry to be rigorous by adding many extra axioms, but the resulting axiomatic system is much more complex and is not at all suitable for highschool students, except possibly the brightest.) The second, somewhat more minor, issue I have with Euclidean geometry, related to the first, is that the way arguments are phrased, and the use of figures to illustrate, often hides many logical steps that are only implicit. In particular, I am thinking of the implicit use of qunatifiers in statements that are proven in Euclidean geometry. This is an issue because as soon as one moves on to proofs in any other context encountered by students, e.g. in first year undergrad, it becomes much more difficult to do things correctly while only thinking about quantifiers implicitly. It would be much more beneficial, in my opinion, to have the first introduction to proofs be a topic that is much, much simpler (such as integers) where the focus can be purely on how statements are formed with quantifiers, how strategies of proof are determined based on the form of the statement and which quantifiers are involved, and how the (much shorter and simpler, and not requiring an intuition pump to use correctly) definitions and basic properties of that topic can be applied in a proof. The root of all of this is the importance in proofs of properly using the definitions and axioms. Students in highschool, except the most talented, just are not careful thinkers and will revert to their preferred lazy way of thinking (such as pictures and vague ideas) as soon as you give them the opening. In my experience, the only way to force students to understand how to properly prove things is to pull out the rug of their intuition, even briefly, for just long enough so that they learn how to do things without it. Then later, once they have properly adopted the mindset of using the definitions, you can let the intuition back in. For context, I don't know if any of what I just said reflects how Euclidean geometry is taught in, say, Caifornia. I only know this from the perspective of having tried to incorporate Euclidean geometry in an undergraduate proofs course (in Canada where Euclidean geometry hasn't been part of the grade school curriculum for some time).
• bradleyjg 5y ago
  If you wrote everything like a high school essay, you’d be a pretty terrible writer. In real life you don’t need a topic sentence for every paragraph and sometimes an infinitive needs splitting. Nonetheless, high school essay writing is a pretty good way station to good writing. Perhaps high school geometry proofs are similar.