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:

Geometry proofs are a terrible way of introducing rigorous mathematical proofs to students. Seriously. I cannot overstate how misleading they are. I remember my first abstract algebra class in college, after the first HW I got called in by the TA because I tried to structure my proofs like I learned in geometry class - a sequence of symbols and references to hard-coded lists of axioms and prior deductions. I thought that's what proofs are, but they are not, at least not as humans do higher math. High school geometry tries to distill this process down to a symbol manipulation game that students can memorize and regurgitate, and in the process loses the essence of the thing it was meant to capture in the first place.

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.

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.

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.

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

From Texas, and can absolutely confirm. I was one of my area’s top Number Sense (mental arithmetic) scorers in middle school and high school, and made it to the state MathCounts finals two years in middle school, and always got high A’s in my math classes. I squeaked by on the IB Math exam, which should have been a sign that something was off about my math education.

I did ok in college Cal I and II (I’d been warned by older friends that our high school calculus class was a joke) but hit the wall in Cal III and ended up dropping Diff Eq before I could actually fail it. I barely passed Discrete Math by memorizing a bunch of proofs.

Turns out, I’m really great at memorizing arithmetic tricks and wasn’t actually that good at math.

“The remainder had parents who taught this kind of thinking from a much younger age.”

Fortunately for our child, my husband was a success in Germany’s math education system, and will therefore have the lead on how our kid is taught to think about numbers - I’ll just give the English words for things.

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.

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.

Exactly. You can't teach people that sometimes their intuition is wrong if they don't have any intuition to begin with.

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.

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.

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.

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.

High school students are frequently taught a highly routinized form of proof in Geometry class in the US: https://www.cuemath.com/geometry/geometrical-proofs/ That is what it seems the parent was referring to and what I was referring to.

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.

> I think the heydey of Trig was in the age of navigation and the heydey of Geometry was in the age of machines.

Sorry, but this is crazy. If you're in physics, electrical engineering, mechanical engineering, or civil engineering[1], then trigonometry appears in so many problems - including ones that have little to do with triangles and geometry.

Calc II is heavy on trigonometry for a good reason. So many integrals that show up in the "real world" are solvable if you know your trig identities. These show up in semiconductor theory, mechanics, electromagnetics, quantum mechanics, etc.

Trig is the one thing I'm glad I was taught well in high school. Used it all the time for over a decade.

[1] And probably many other disciplines.

Trigonometry has little to do with triangles. It's more appropriately titled circleometry.

That's a clever insight. Though the measurement of circles is done using triangles, so maybe there is an even cleverer more Germanic name? Circulotrigometry.

Rather than geometry or linear algebra, I'd love to see logic and set theory taught that early as more than a passing topic.

Can only up this. Not teaching set theory is just an abomination. And logic. That year of Prolog taught me so much.

Which would be utterly useless.

Math, and specifically Geometry aren't about useful skills, they are about understanding the world.

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.

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.

Yes, Geometry makes the funnest puzzles :-)

Believe me, it pains me to suggest bumping it back. Introductory proofy linear algebra just hits a quadruple home run: it's astonishingly useful, it's getting moreso, it's simple enough to remain age appropriate, and it can serve as an intro to proofs. As much as I enjoy geometry puzzles, I just don't see how they compete with that.

For the true fans, of course, the answer today & in this hypothetical future is still to learn both.

> 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.

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.

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.

I'm curious on how you structure proofs, then? That is, how do you recommend?

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).

While both are clearly important, if we only have so many hours in the day I'd rather a high school student have a grasp on basic stats than basic formal proofs.

As a data point of one (heh there's that stats again...) my two high school geometry classes 15 years ago never touched any proofs. I have no clue if they still teach that way.

Edit: just noticed that I'm replying to someone who is far more knowledgeable about math than me. I stand by my point though, people like Colin can seek that out in more advanced classes.

I had proofs in HS geometry way back when. I also hated it and it was the worst I ever did in a math class K-12.

"Lies, damned lies, and statistics"

Proof-based high school geometry is an absolute joke and should be excised as soon as possible. There is no better way to ruin someone's enthusiasm for math while teaching them nothing useful... imo.

https://www.maa.org/external_archive/devlin/LockhartsLament.... makes the best case for this.

It is in geometry one can experience a spectrum of evidence from a picture (animated or not, interactive or not) to a line-to-line modus ponens proof. Not to mention the historical value that Euclid invented a literary genre that is now called "proofs". Good luck teaching kids "more useful" stuff like data science and quantum computing. High school teachers don't understand these topics and are more likely to teach them ineffectively. Now that's harmful and no one can blame geometry anymore.

A somewhat similar point is expressed in Underwood Dudley's article "What is Mathematics For?" from the May 2010 Notices of the AMS [1].

Briefly, he argues that we greatly exaggerate how much math you actually need in ordinary life and in most jobs that people usually think of as needing math (and what you do actually need can be learned on the job), but nevertheless learning math is worthwhile and important because of what it teaches you about how to think.

It's hard to imagine now, but back in the early days of the US similar debates played out over teaching arithmetic in schools. Most people didn't need more than counting and simple addition and subtraction, so why make everyone learn any more than that? Those few who meeded more could learn it outside of school.

[1] http://www.ams.org/notices/201005/rtx100500608p.pdf

I got nothing out of my high school geometry class but loved my foundations class in college. I would have been a lot better off just going straight to real proofs instead of those annoying pseudo-proofs. But I don't know, other people probably had different experiences and got a lot out of geometry.

At least at my school, there was a special geometry class that included proofs ("formal geometry"). Other than the handful of few people who bothered to take that class, there was very little way in the way of proofs in the geometry class that 98% of people took (ok, I tested out of geometry so I don't really know what was in the curriculum).

This makes me sad. High school Geometry without proofs is indeed not worth keeping. (Except maybe as a two week discussion of triangles and right angles so that students understand what they are when they hit trig.)

I have to agree wholeheartedly. The actual content in a classical geometry class is mostly related to logic; axiom, proof, theorem, inference, etc.

If you remove all of that, you can embed actual geometric content into other classes as you go, much like probability and statistics is currently integrated across multiple years.