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I’ve read a couple of texts by Paul Lockhart. His opinion (which I support) is that what is taught in high school as math isn’t math, as in what a mathematician

by wkjagt 3y ago

I’ve read a couple of texts by Paul Lockhart. His opinion (which I support) is that what is taught in high school as math isn’t math, as in what a mathematician does. As an example: Pythagoras proved the relation between the lengths of the sides of a right triangle. The journey towards finding the answer to this is math. Or even exploring why this relation even exists based on his findings is math. The answer to the question is expressed as a2+b2=c2. What we’re asking students to do is just to take that formula and apply it hundreds of times without going into the actual math behind it. This, he says, isn’t math at all. The whole “math” step is skipped. He compares it to if for example we were teaching music in school without ever touching a musical instrument, and just learning how to read and write notes. Or teaching art without ever touching a pencil or brush. The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art. His text that explains this most clearly is “A Mathematician’s Lament”. It’s a great read and not very long.

j7ake 3y ago
Society thinks of math as technical and functional, rather than beautiful. The math used in jobs are also to “get things done” rather than for the sake of their beauty. A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”. Considering how much society throws money at STEM, rather than art and music, it makes sense society wants a return in their investment in terms of training practical people.
- TeMPOraL 3y ago
  I disagree somewhat with both you and GP here: > Society thinks of math as technical and functional, rather than beautiful. The math people suck at is the technical and functional part, just as much as the beautiful parts. I'd frame the issue like this: there's the journey of arriving at the result, there's wielding the result, and then there's applying it to a very particular problem. Schools are teaching mostly the last part - applying a^2+b^2=c^2 to a set of contrived problems that are meant to exercise your ability in arithmetic, variable substitution, and simple symbolic transformation. They do not teach people how to wield the formula - as in, how can you realize on your own that any particular real-life challenge calls for application of that tool (or of mathematics in general). That's the middle part that's directly valuable, in the immediate term. That's the difference between a carpenter, and a person who figured out how to use a hammer to drive a nail into a wall. The beauty and journey part? That's study of history and art of inventing new types of hammers and other impactors, and perhaps inventing your own. Even further upstream of the "knowing how to use it to build something" part, but neither is taught in schools.
- ndriscoll 3y ago
  I tend to not like A Mathematician’s Lament for being so lofty, but there's a more straightforward reason to care about "pure" math: students complain that math "doesn't make sense" or is a bunch of random difficult to remember rules, which is absurd. More than any other field, math has logical reasons for everything. You can figure out pretty much anything you'll encounter in school by just thinking about it; you don't need to have any external knowledge from having done some experiment or being told some fact. On the flip side, if you do use physical intuition, you'll usually go down the right path (for anything pre-university, or even most undergraduate level material). Regardless of how beautiful music is, teaching how to read sheet music without actually listening to music is stupid because you're not teaching what the symbols even mean. Of course no one will understand what you're talking about.
- edanm 3y ago
  > A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”. I agree with this, but I want to point out that the few "pure mathematicians" are not the only ones who can enjoy the beauty of math, similar to how the few classical musicians (or classical music experts) aren't the only ones who can enjoy classical music. In fact, looking at the success of various YouTube math channels, or Science channels, etc, there are clearly orders of magnitude more people interested in material on Science and Math than there are people who actively engage in these fields professionally. (This is true of probably every field, btw.)
tiznow 3y ago
"The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art." This is more or less what I've found as an adult who is trying to improve their math skills. Like 90% of what I learned in school was never really applied, I couldn't tell you why I learned them, just that I was told they would be essential for my later life if I chose certain paths.
ecshafer 3y ago
This is very similar to what I was coming to post. There is very little mathematics taught in k-12 in the US. There are almost no proofs taught, there is a lack of mathematical thinking taught. My classes personally were done as a series of rote memorization steps to perform before putting it into the calculator. The mathematical logic, proof, rigor, and how to create a model is what is needed and that is missing. The specific courses taught don't matter as much if they aren't taught correctly.
- fatnoah 3y ago
  > The mathematical logic, proof, rigor, and how to create a model is what is needed and that is missing. This is interesting to me. My experience was the opposite. So much time in my honors math in high school was focused on proofs and deriving them, that developing and intuitive understanding of the material and how to apply the math to solve problems was secondary.
  - ndriscoll 3y ago
    I suppose the commonality is just that primary/secondary math education is often bad, but doing proofs without intuition seems to be completely missing the point. A decent proof should mostly be the intuition. Once you've learned the basic techniques, you just say "by induction X", "evidently X", etc. to wave away routine calculations, and mostly stick to the "meat" of the argument, with explicit calculations for technical steps only if something is tricky.
  - ChainOfFools 3y ago
    > honors math This may or may not be comparable to what is referred to as high school math in the general sense. I don't know how your curriculum worked, but presumably the students who are taking honors math had already demonstrated a capable, if not above average faculty for the minimum level of rote memorization necessary to simply get the basic arithmetic and algebraic symbol manipulation right most of the time. This would leave class time and homework time available to explore the matrix in which these manipulations are grounded, since "how" they work is taken to be table stakes. But the vast majority of students don't have the extra bandwidth available, due to competing pressures in other areas of life, or simply insufficient short-term memory capacity, to get past "how to" with enough time left over for "but why" even if they have a natural curiosity and interest in the question. Beside irony told this is that the more one understands why, the easier how it becomes to memorize. But finding this balance seems to be a very unique mixture for students who are not extraordinarily capable in the ladder, there is a huge advantage to being able to build a working toolchain of mechanical spellcasting to reliably obtain repeatable, testable results so that you can be productive when postulating why the spells work at all. Essentially it helps to be capable.of mimicking an ersatz computer when you need to develop human intuition fpr computation. but computers are getting so good at thinking like a human that humans who best think like computers are going to be less prized than they have been in, say, the 30s, 40s, 50s and so on, when the premises that are still baked into our current pedagogical sensibilities were formed.
nonameiguess 3y ago
I'm not an educator or a pundit and won't express any personal opinion on what the purpose of primary math education should be, but as far as I can tell, the purpose as it stands in both K-12 and college undergraduate education if you're not a math major is not to teach math. It's to teach analytic and computational techniques necessary for research and practice in business, finance, economics, sciences, and engineering. They're not trying to get you to understand what a continuous function is or why it matters. They're trying to teach you just enough about how to compute compound interest that getting an amortization schedule out of a spreadsheet won't be incomprehensible magic. That said, as Temporal said, even this isn't necessarily done well. It's entirely possible that teaching students to derive the techniques from first principles based on actually understanding math might lead to better outcomes. But it would also be quite difficult to do and I'm not sure existing K-12 teachers are even qualified to do it. To Lockhart's credit, he walked the walk and actually taught children at a primary school. How many similarly qualified people are willing to do that?
raxxorraxor 3y ago
Rather than just being able to solve a problem, people need to be able to create the questions that beg the answers. Physics class helped me understand differential calculus because I was able to create problems that I could then solve with tools learned in math. There are also some mathematical tools that or often quite simple but tremendously help with solving equations. Extending a formula with neutral or inverse elements for example. This is the same in almost all fields in math, but it is rarely though as a set of tools. That said, if studies like PISA are to be believed, the problems are much more fundamental.
raydev 3y ago
> The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. I'm skeptical that the kids who struggle with memorization will suddenly be able to write proofs with ease. I'd argue proof-writing is much harder to wrap one's head around than being forced to remember when to apply a2+b2=c2.
- edanm 3y ago
  I think this is a false dichotomy of what it means to be "smart" or "good at math/studies". Someone can jbe bad at arithmetic, but good at writing proofs. Hell, plenty of mathematicians say they are bad at arithmetic! More importantly in the context of education, someone can be completely unmotivated by arithmetic, but be more motivated by something else, leading to different amounts that they apply themselves, leading to different outcomes.
  - raydev 3y ago
    > Someone can jbe bad at arithmetic, but good at writing proofs I'm not saying it's impossible, just that the median child/teen isn't going to demonstrate this. I don't think it's debatable that proofwriting is harder to teach, and requires more time and effort from both students and teachers.
- strikelaserclaw 3y ago
  proofs are actually what made me get interested in math, proofs allowed for such a range of self expression and creativity that i was hooked even though they usually required more thought. Compare this to the studying the myriad of integration techniques which i found very boring. My wife on the other hand seemed to enjoy rote mechanical memorization. I guess the fundamental question is, long after most students graduate high school or college and forget most of the math they learned, what are they supposed to keep from the many years of mathematical education? If you ask me, it is the ability to reason that they should keep and proof based math teaches you exactly that.
  - raydev 3y ago

by wkjagt 3y ago

j7ake 3y ago

Society thinks of math as technical and functional, rather than beautiful. The math used in jobs are also to “get things done” rather than for the sake of their beauty. A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”. Considering how much society throws money at STEM, rather than art and music, it makes sense society wants a return in their investment in terms of training practical people.

TeMPOraL 3y ago
I disagree somewhat with both you and GP here: > Society thinks of math as technical and functional, rather than beautiful. The math people suck at is the technical and functional part, just as much as the beautiful parts. I'd frame the issue like this: there's the journey of arriving at the result, there's wielding the result, and then there's applying it to a very particular problem. Schools are teaching mostly the last part - applying a^2+b^2=c^2 to a set of contrived problems that are meant to exercise your ability in arithmetic, variable substitution, and simple symbolic transformation. They do not teach people how to wield the formula - as in, how can you realize on your own that any particular real-life challenge calls for application of that tool (or of mathematics in general). That's the middle part that's directly valuable, in the immediate term. That's the difference between a carpenter, and a person who figured out how to use a hammer to drive a nail into a wall. The beauty and journey part? That's study of history and art of inventing new types of hammers and other impactors, and perhaps inventing your own. Even further upstream of the "knowing how to use it to build something" part, but neither is taught in schools.
ndriscoll 3y ago
I tend to not like A Mathematician’s Lament for being so lofty, but there's a more straightforward reason to care about "pure" math: students complain that math "doesn't make sense" or is a bunch of random difficult to remember rules, which is absurd. More than any other field, math has logical reasons for everything. You can figure out pretty much anything you'll encounter in school by just thinking about it; you don't need to have any external knowledge from having done some experiment or being told some fact. On the flip side, if you do use physical intuition, you'll usually go down the right path (for anything pre-university, or even most undergraduate level material). Regardless of how beautiful music is, teaching how to read sheet music without actually listening to music is stupid because you're not teaching what the symbols even mean. Of course no one will understand what you're talking about.
edanm 3y ago
> A very tiny fraction of population will ever be pure mathematicians. Most just need to know enough to “get things done”. I agree with this, but I want to point out that the few "pure mathematicians" are not the only ones who can enjoy the beauty of math, similar to how the few classical musicians (or classical music experts) aren't the only ones who can enjoy classical music. In fact, looking at the success of various YouTube math channels, or Science channels, etc, there are clearly orders of magnitude more people interested in material on Science and Math than there are people who actively engage in these fields professionally. (This is true of probably every field, btw.)

tiznow 3y ago

"The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. The same way we could totally miss out having students never discovering their artistic talents if art class was taught without ever actually doing any art." This is more or less what I've found as an adult who is trying to improve their math skills. Like 90% of what I learned in school was never really applied, I couldn't tell you why I learned them, just that I was told they would be essential for my later life if I chose certain paths.

ecshafer 3y ago

This is very similar to what I was coming to post. There is very little mathematics taught in k-12 in the US. There are almost no proofs taught, there is a lack of mathematical thinking taught. My classes personally were done as a series of rote memorization steps to perform before putting it into the calculator. The mathematical logic, proof, rigor, and how to create a model is what is needed and that is missing. The specific courses taught don't matter as much if they aren't taught correctly.

fatnoah 3y ago
> The mathematical logic, proof, rigor, and how to create a model is what is needed and that is missing. This is interesting to me. My experience was the opposite. So much time in my honors math in high school was focused on proofs and deriving them, that developing and intuitive understanding of the material and how to apply the math to solve problems was secondary.
- ndriscoll 3y ago
  I suppose the commonality is just that primary/secondary math education is often bad, but doing proofs without intuition seems to be completely missing the point. A decent proof should mostly be the intuition. Once you've learned the basic techniques, you just say "by induction X", "evidently X", etc. to wave away routine calculations, and mostly stick to the "meat" of the argument, with explicit calculations for technical steps only if something is tricky.
- ChainOfFools 3y ago
  > honors math This may or may not be comparable to what is referred to as high school math in the general sense. I don't know how your curriculum worked, but presumably the students who are taking honors math had already demonstrated a capable, if not above average faculty for the minimum level of rote memorization necessary to simply get the basic arithmetic and algebraic symbol manipulation right most of the time. This would leave class time and homework time available to explore the matrix in which these manipulations are grounded, since "how" they work is taken to be table stakes. But the vast majority of students don't have the extra bandwidth available, due to competing pressures in other areas of life, or simply insufficient short-term memory capacity, to get past "how to" with enough time left over for "but why" even if they have a natural curiosity and interest in the question. Beside irony told this is that the more one understands why, the easier how it becomes to memorize. But finding this balance seems to be a very unique mixture for students who are not extraordinarily capable in the ladder, there is a huge advantage to being able to build a working toolchain of mechanical spellcasting to reliably obtain repeatable, testable results so that you can be productive when postulating why the spells work at all. Essentially it helps to be capable.of mimicking an ersatz computer when you need to develop human intuition fpr computation. but computers are getting so good at thinking like a human that humans who best think like computers are going to be less prized than they have been in, say, the 30s, 40s, 50s and so on, when the premises that are still baked into our current pedagogical sensibilities were formed.

nonameiguess 3y ago

I'm not an educator or a pundit and won't express any personal opinion on what the purpose of primary math education should be, but as far as I can tell, the purpose as it stands in both K-12 and college undergraduate education if you're not a math major is not to teach math. It's to teach analytic and computational techniques necessary for research and practice in business, finance, economics, sciences, and engineering. They're not trying to get you to understand what a continuous function is or why it matters. They're trying to teach you just enough about how to compute compound interest that getting an amortization schedule out of a spreadsheet won't be incomprehensible magic. That said, as Temporal said, even this isn't necessarily done well. It's entirely possible that teaching students to derive the techniques from first principles based on actually understanding math might lead to better outcomes. But it would also be quite difficult to do and I'm not sure existing K-12 teachers are even qualified to do it. To Lockhart's credit, he walked the walk and actually taught children at a primary school. How many similarly qualified people are willing to do that?

raxxorraxor 3y ago

Rather than just being able to solve a problem, people need to be able to create the questions that beg the answers. Physics class helped me understand differential calculus because I was able to create problems that I could then solve with tools learned in math. There are also some mathematical tools that or often quite simple but tremendously help with solving equations. Extending a formula with neutral or inverse elements for example. This is the same in almost all fields in math, but it is rarely though as a set of tools. That said, if studies like PISA are to be believed, the problems are much more fundamental.

raydev 3y ago

> The risk is that students think they’re bad at math, while in fact they’re bad at mindlessly memorizing and applying formulas. I'm skeptical that the kids who struggle with memorization will suddenly be able to write proofs with ease. I'd argue proof-writing is much harder to wrap one's head around than being forced to remember when to apply a2+b2=c2.

edanm 3y ago
I think this is a false dichotomy of what it means to be "smart" or "good at math/studies". Someone can jbe bad at arithmetic, but good at writing proofs. Hell, plenty of mathematicians say they are bad at arithmetic! More importantly in the context of education, someone can be completely unmotivated by arithmetic, but be more motivated by something else, leading to different amounts that they apply themselves, leading to different outcomes.
- raydev 3y ago
  > Someone can jbe bad at arithmetic, but good at writing proofs I'm not saying it's impossible, just that the median child/teen isn't going to demonstrate this. I don't think it's debatable that proofwriting is harder to teach, and requires more time and effort from both students and teachers.
strikelaserclaw 3y ago
proofs are actually what made me get interested in math, proofs allowed for such a range of self expression and creativity that i was hooked even though they usually required more thought. Compare this to the studying the myriad of integration techniques which i found very boring. My wife on the other hand seemed to enjoy rote mechanical memorization. I guess the fundamental question is, long after most students graduate high school or college and forget most of the math they learned, what are they supposed to keep from the many years of mathematical education? If you ask me, it is the ability to reason that they should keep and proof based math teaches you exactly that.
- raydev 3y ago