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they are cutting geometry a bit, which doesn't make as much sense today as it did 100 years ago when far more people grew up to be farmers or ranchers. The poi

they are cutting geometry a bit, which doesn't make as much sense today as it did 100 years ago when far more people grew up to be farmers or ranchers. The point of high school Geometry was never to compute areas of fields. Trigonometry is far more useful for that anyway. The point of high school Geometry is that it is the first introduction to rigorous mathematical proof -- and has been, ever since Euclid's Elements.

ThinBold 5y ago
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.
showerst 5y ago
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.
- jelliclesfarm 5y ago
  "Lies, damned lies, and statistics"
- ghaff 5y ago
  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.
ivanech 5y ago
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.
heyitsguay 5y ago
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.
- 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.
cozzyd 5y ago
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).
- cperciva 5y ago
  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.)
  - andrewprock 5y ago
    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.
ajkjk 5y ago
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.pdf https://www.maa.org/external_archive/devlin/LockhartsLament.... makes the best case for this.
tzs 5y ago
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 http://www.ams.org/notices/201005/rtx100500608p.pdf

ThinBold 5y ago

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.

showerst 5y ago

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.

jelliclesfarm 5y ago
"Lies, damned lies, and statistics"
ghaff 5y ago
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.

ivanech 5y ago

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.

heyitsguay 5y ago

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.

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.

cozzyd 5y ago

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).

cperciva 5y ago
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.)
- andrewprock 5y ago
  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.

ajkjk 5y ago

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

tzs 5y ago

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 http://www.ams.org/notices/201005/rtx100500608p.pdf