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Every time I read such lucid explanations of math like this I'm filled with resentment for the math instruction I received in junior high and high school in the

Every time I read such lucid explanations of math like this I'm filled with resentment for the math instruction I received in junior high and high school in the U.S. There's a real sense of 'play' that these explanations evoke and that make thinking about numbers and their relationships genuinely fun and interesting. That sense of play was entirely absent from my early math education. It was all "These are the rules. Memorize them for the quiz tomorrow." I can't help feeling like my math upbringing was akin to a child being raised by parents who speak their own made-up language. Integrating with the rest of the normal-language-speaking world is anxiety inducing and filled with challenges that may never be completely overcome.

lambdasquirrel 3y ago
I used to think this was just restricted to pre-college math, and to East-Asian parents. After I had erstwhile-parent types infantilize me over my adult hobbies, I realized, you need a creative person in order to facilitate creative discovery. But it takes guts to try to learn something the "right" way. Imagine going through a whole semester, never once learning what was going to be on the AP test. And it takes some stature to teach it too. Imagine doing a writing class, where there absolutely must not be any rules or criteria for grading, where you really must say, I'll know it when I see it, but there really are going to be some yeses and nos. What do you want to produce? People who can come up with new solutions to new problems on their own, or people who do cookie-cutter rules? How do you keep doing that throughout your life, long after school? People say they want the former until they come up against the reality of uncertainty, the possibility that there might not actually be any answers, and then they go right back to memorizing and teaching rules. They go back to justifying things the way they are.
- wizzwizz4 3y ago
  > Imagine going through a whole semester, never once learning what was going to be on the AP test. This is how I learn mathematics – mostly because I'm empirically incapable of doing otherwise. I will achieve a lower grade than I “could”, and miss out on many opportunities as a result.
- anonymouskimmer 3y ago
  There are a few colleges/universities that use non-grade evaluations. Evergreen State uses a narrative evaluation: https://www.evergreen.edu/registration/transcript-format https://www.evergreen.edu/registration/transcript-format Really smart people, and sometimes even just moderately smart people, coast along as A, B, or C students while learning what they want and just doing the minimum of the requirements to coast along. And then a bunch of other people have academic-like side-interests that they pursue in their free time.
jjtheblunt 3y ago
I think it’s very common to have uninspiring math teachers, but it’s not uniformly the case across the United States. That site, for example, is at Caltech. The math teachers I had in suburban Chicago in public schools were superlative.
mmmmpancakes 3y ago
The ability to communicate math this way is honestly rare. It comes from a combination of deep understanding, long experience in communicating math, and a certain level of "culturing" that is specific to the academic experience. Among the best, Feynman was singular in his ability to communicate math and physics. In other words, don't be so hard on the teachers who were disappointing in comparison to the stellar examples you see from top mathematical communicators. What you're reading is quite rare and, while education quality could certainly improve, its not fair to expect this of a 5th grade teacher who covers 5 topics in a day. Even for the best, developing this type of material takes time and thought that a school teacher probably does not have.
- melagonster 3y ago
  but government hired professors to design content of mathematics textbooks for students. there are some problem...
  - mmmmpancakes 3y ago
    Disagree. The purpose of a textbook and a lecture is very different. A good textbook can be a helpful resource for teaching and lecturing, but it is not sufficient to guarantee high quality math education. Conversely, a good educator who deeply understands the material can deliver fantastic education without a good textbook. Claiming that profs writing bad textbooks is the cause of poor quality in class math instruction is absurd.
    - mlyle 3y ago
      > Claiming that profs writing bad textbooks is the cause of poor quality in class math instruction is absurd. It certainly doesn't help. The tendency to pile more and more into standards, and then to have haphazard treatment in the textbooks, with problems that don't make sense... isn't great. Stick a new teacher in the classroom, and they're going to run their book's recommended pacing and content. And even a veteran is probably going to lean on the book a lot in a pinch. And, your course needs to fit together with 2 other teachers who are too likely to be running the absurd pacing and content in the courses before and after yours. The rushed pace leaves no choice but to devote a huge fraction of the time to procedural knowledge. The net result doesn't serve anyone: the top students are left unchallenged and without the context and enrichment that could let them really grow. The bottom students are in painful struggle. And the middle are perpetually slightly confused, learning specific tools that they'll immediately forget when the unit completes.
starcraft2wol 3y ago
Math teaching selects for people who do their homework everyday. This is very different from what actual math selects for.
- tux3 3y ago
  People who can memorize and apply rules. I never did any homework, and that worked great until undergrad, where the strategy of reading things on my phone all day instead of listening to anything in class stopped working quite so well. I'm not very good at proofs, but following arbitrary rules provided with no motivation? Decades of experience!
- cafard 3y ago
  Doing one's homework is an excellent way of exposing and filling gaps in one's understanding. I'm more or less adequate at math, but I'd be better if I had done more of my homework.
  - starcraft2wol 3y ago
    No doubt. But ranking math skills, being especially consistent in studies/homework is maybe number 5. The problem I am pointing out is math teachers often only have that 1.
  - monadINtop 3y ago
    Yes people who follow the rules and do well in "school maths" are very likely to also do well and succeed in "higher mathematics". Unfortunately, this selection misses (most?) children who may not be well suited to "school maths" - for whatever reason. But these children may succeed just as well in "higher mathematics". Two anecdotes: (1) June Huh dropped out of high school and stagnated for 6 years in university. In his 6th year, he ran into the fields medal Heisuke Hironaka. It was only then his "slow thinking" and deep creative insight (perhaps the things that hindered him in "school maths" type courses?) proved to be fruitful in higher mathematics. June Huh now has a fields medal. (https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/ https://www.quantamagazine.org/june-huh-high-school-dropout-...) (2) I was frequently in trouble at school and underachieved relative to my predicted grades. I resented the route learning and arbitrariness of "school mathematics". Due to some miracle I'm currently working towards a PhD in theoretical physics, in the mathematics department of a top university, and I also spend about 90% of my free time working through various advanced maths textbooks for fun. Turns out I'm quite suited for thinking about higher mathematics, despite not being particularly well disposed for school. If my school experience was different, I probably would have done a PhD in pure mathematics instead.
krsrhe 3y ago
Usually when people say this, it's because learning something for the first time is harder than learning something you already know.
jodrellblank 3y ago
> "Every time I read such lucid explanations of math like this I'm filled with resentment for the math instruction I received in junior high and high school in the U.S. There's a real sense of 'play' that these explanations evoke and that make thinking about numbers and their relationships genuinely fun and interesting." You could wonder why you feel this way. If there is a sense of play which makes it interesting, and you are reading it and enjoying it today, why isn't that enough? What is the resentment about - you haven't missed out on the interesting math explanation - it's right here, you're reading it and enjoying it. (This is the work of cognitive behavioural therapy - "I read a math thing which I found interesting, but instead of feeling elevated, happy, awed in the presence of brilliance, greatful that I stumbled upon such a thing when I could have gone my whole life not knowing about it, I instead jumped to feelings of resentment about things which happened many years ago, leaving me in a bad mood. I wonder what in my head made that connection and why?" Ref: a different person I was replying to in a different thread who was claiming that CBT is about colouring-in while handwaving problems away or waiting for acceptance that life sucks to bestow itself upon you).
- deleted 3y ago
  [deleted]
- plonk 3y ago
  This isn’t about enjoying the explanation now, it’s about the academic success and open doors one could have had if their teachers had been better communicators. Math doesn’t get studied in a vacuum, it’s the biggest selection criterion for STEM education.
  - jodrellblank 3y ago
    You could be right, that’s one possible explanation, but that comment didn’t read like “I resent my highschool math teacher because I don’t earn enough in a STEM job” to me.
    - harry8 3y ago
      Upset because of all that was missed out on. The realisation of what might have been. Could I have been a decent mathematician if the opportunity was not missed while I was totally unaware of its existence? Obviously there can be no definitive answer to that question. And it’s also ok to be a little resentful of having been tortured for many hours with something that clearly could have been a lot of fun.
jwells89 3y ago
This was an issue with my high school math classes too, but the other thing that made for something of a stumbling block for me was the total absence of and refusal to produce practical examples of the math being taught in action. The way my mind works, in order to grasp a concept well enough to be able to actively use it, I need to see it in action by way of a non-contrived, realistic example. The way any math past basic arithmetic tends to be taught in abstract dramatically slows acquisition. It’s a quality that cuts both ways. It doesn’t work well with a lot of traditional academics, but it enables me to self-teach highly applied subjects like programming with little friction.
jibe 3y ago
Push the language analogy further. In a sense, a child really I s raised by parents speaking a made-up language. It just happens to be a locally shared made-up language. Like math, you brute force language learning. Imitation and memorization. Only after years, or decades, can you go from basic language understanding to appreciating the beauty of words, poetry, literature. I don’t think there is a short cut in language or math. You have to go through the unfun multiplication table, spelling test phase to build the foundation for higher level appreciation l.
User23 3y ago
I think you’d really enjoy the book Visual Complex Analysis.
devchix 3y ago
I feel the same way. My college years math was a clueless grind (to me) and I don't think I made the connection to the real world until long after school was behind me. There's a very good site called BetterExplained, where I go to when I need to review some concept. I wish I had that when I was in school.
photochemsyn 3y ago
I think this paragraph in particular should be explained to all introductory students in any STEM-related field. It sets the stage for a basic yet profound understanding of what mathematics is all about: > "To discuss this subject we start in the middle. We suppose that we already know what integers are, what zero is, and what it means to increase a number by one unit. You may say, “That is not in the middle!” But it is the middle from a mathematical standpoint, because we could go even further back and describe the theory of sets in order to derive some of these properties of integers. But we are not going in that direction, the direction of mathematical philosophy and mathematical logic, but rather in the other direction, from the assumption that we know what integers are and we know how to count." Beyond that, an understanding of algebraic concepts in terms of how equations can be manipulated from one form to another (and of the rationale for why one can) might benefit from the addition of the concepts of distribution and commutation, which are not included in this lecture (although the important basic idea of a successor is).
deleted 3y ago
[deleted]

lambdasquirrel 3y ago

I used to think this was just restricted to pre-college math, and to East-Asian parents. After I had erstwhile-parent types infantilize me over my adult hobbies, I realized, you need a creative person in order to facilitate creative discovery. But it takes guts to try to learn something the "right" way. Imagine going through a whole semester, never once learning what was going to be on the AP test. And it takes some stature to teach it too. Imagine doing a writing class, where there absolutely must not be any rules or criteria for grading, where you really must say, I'll know it when I see it, but there really are going to be some yeses and nos. What do you want to produce? People who can come up with new solutions to new problems on their own, or people who do cookie-cutter rules? How do you keep doing that throughout your life, long after school? People say they want the former until they come up against the reality of uncertainty, the possibility that there might not actually be any answers, and then they go right back to memorizing and teaching rules. They go back to justifying things the way they are.

wizzwizz4 3y ago
> Imagine going through a whole semester, never once learning what was going to be on the AP test. This is how I learn mathematics – mostly because I'm empirically incapable of doing otherwise. I will achieve a lower grade than I “could”, and miss out on many opportunities as a result.
anonymouskimmer 3y ago
There are a few colleges/universities that use non-grade evaluations. Evergreen State uses a narrative evaluation: https://www.evergreen.edu/registration/transcript-format https://www.evergreen.edu/registration/transcript-format Really smart people, and sometimes even just moderately smart people, coast along as A, B, or C students while learning what they want and just doing the minimum of the requirements to coast along. And then a bunch of other people have academic-like side-interests that they pursue in their free time.

jjtheblunt 3y ago

I think it’s very common to have uninspiring math teachers, but it’s not uniformly the case across the United States. That site, for example, is at Caltech. The math teachers I had in suburban Chicago in public schools were superlative.

mmmmpancakes 3y ago

The ability to communicate math this way is honestly rare. It comes from a combination of deep understanding, long experience in communicating math, and a certain level of "culturing" that is specific to the academic experience. Among the best, Feynman was singular in his ability to communicate math and physics. In other words, don't be so hard on the teachers who were disappointing in comparison to the stellar examples you see from top mathematical communicators. What you're reading is quite rare and, while education quality could certainly improve, its not fair to expect this of a 5th grade teacher who covers 5 topics in a day. Even for the best, developing this type of material takes time and thought that a school teacher probably does not have.

melagonster 3y ago
but government hired professors to design content of mathematics textbooks for students. there are some problem...
- mmmmpancakes 3y ago
  Disagree. The purpose of a textbook and a lecture is very different. A good textbook can be a helpful resource for teaching and lecturing, but it is not sufficient to guarantee high quality math education. Conversely, a good educator who deeply understands the material can deliver fantastic education without a good textbook. Claiming that profs writing bad textbooks is the cause of poor quality in class math instruction is absurd.
  - mlyle 3y ago
    > Claiming that profs writing bad textbooks is the cause of poor quality in class math instruction is absurd. It certainly doesn't help. The tendency to pile more and more into standards, and then to have haphazard treatment in the textbooks, with problems that don't make sense... isn't great. Stick a new teacher in the classroom, and they're going to run their book's recommended pacing and content. And even a veteran is probably going to lean on the book a lot in a pinch. And, your course needs to fit together with 2 other teachers who are too likely to be running the absurd pacing and content in the courses before and after yours. The rushed pace leaves no choice but to devote a huge fraction of the time to procedural knowledge. The net result doesn't serve anyone: the top students are left unchallenged and without the context and enrichment that could let them really grow. The bottom students are in painful struggle. And the middle are perpetually slightly confused, learning specific tools that they'll immediately forget when the unit completes.

starcraft2wol 3y ago

Math teaching selects for people who do their homework everyday. This is very different from what actual math selects for.

tux3 3y ago
People who can memorize and apply rules. I never did any homework, and that worked great until undergrad, where the strategy of reading things on my phone all day instead of listening to anything in class stopped working quite so well. I'm not very good at proofs, but following arbitrary rules provided with no motivation? Decades of experience!
cafard 3y ago
Doing one's homework is an excellent way of exposing and filling gaps in one's understanding. I'm more or less adequate at math, but I'd be better if I had done more of my homework.
- starcraft2wol 3y ago
  No doubt. But ranking math skills, being especially consistent in studies/homework is maybe number 5. The problem I am pointing out is math teachers often only have that 1.
- monadINtop 3y ago
  Yes people who follow the rules and do well in "school maths" are very likely to also do well and succeed in "higher mathematics". Unfortunately, this selection misses (most?) children who may not be well suited to "school maths" - for whatever reason. But these children may succeed just as well in "higher mathematics". Two anecdotes: (1) June Huh dropped out of high school and stagnated for 6 years in university. In his 6th year, he ran into the fields medal Heisuke Hironaka. It was only then his "slow thinking" and deep creative insight (perhaps the things that hindered him in "school maths" type courses?) proved to be fruitful in higher mathematics. June Huh now has a fields medal. (https://www.quantamagazine.org/june-huh-high-school-dropout-wins-the-fields-medal-20220705/ https://www.quantamagazine.org/june-huh-high-school-dropout-...) (2) I was frequently in trouble at school and underachieved relative to my predicted grades. I resented the route learning and arbitrariness of "school mathematics". Due to some miracle I'm currently working towards a PhD in theoretical physics, in the mathematics department of a top university, and I also spend about 90% of my free time working through various advanced maths textbooks for fun. Turns out I'm quite suited for thinking about higher mathematics, despite not being particularly well disposed for school. If my school experience was different, I probably would have done a PhD in pure mathematics instead.

krsrhe 3y ago

Usually when people say this, it's because learning something for the first time is harder than learning something you already know.

jodrellblank 3y ago

> "Every time I read such lucid explanations of math like this I'm filled with resentment for the math instruction I received in junior high and high school in the U.S. There's a real sense of 'play' that these explanations evoke and that make thinking about numbers and their relationships genuinely fun and interesting." You could wonder why you feel this way. If there is a sense of play which makes it interesting, and you are reading it and enjoying it today, why isn't that enough? What is the resentment about - you haven't missed out on the interesting math explanation - it's right here, you're reading it and enjoying it. (This is the work of cognitive behavioural therapy - "I read a math thing which I found interesting, but instead of feeling elevated, happy, awed in the presence of brilliance, greatful that I stumbled upon such a thing when I could have gone my whole life not knowing about it, I instead jumped to feelings of resentment about things which happened many years ago, leaving me in a bad mood. I wonder what in my head made that connection and why?" Ref: a different person I was replying to in a different thread who was claiming that CBT is about colouring-in while handwaving problems away or waiting for acceptance that life sucks to bestow itself upon you).

deleted 3y ago
[deleted]
plonk 3y ago
This isn’t about enjoying the explanation now, it’s about the academic success and open doors one could have had if their teachers had been better communicators. Math doesn’t get studied in a vacuum, it’s the biggest selection criterion for STEM education.
- jodrellblank 3y ago
  You could be right, that’s one possible explanation, but that comment didn’t read like “I resent my highschool math teacher because I don’t earn enough in a STEM job” to me.
  - harry8 3y ago
    Upset because of all that was missed out on. The realisation of what might have been. Could I have been a decent mathematician if the opportunity was not missed while I was totally unaware of its existence? Obviously there can be no definitive answer to that question. And it’s also ok to be a little resentful of having been tortured for many hours with something that clearly could have been a lot of fun.

jwells89 3y ago

This was an issue with my high school math classes too, but the other thing that made for something of a stumbling block for me was the total absence of and refusal to produce practical examples of the math being taught in action. The way my mind works, in order to grasp a concept well enough to be able to actively use it, I need to see it in action by way of a non-contrived, realistic example. The way any math past basic arithmetic tends to be taught in abstract dramatically slows acquisition. It’s a quality that cuts both ways. It doesn’t work well with a lot of traditional academics, but it enables me to self-teach highly applied subjects like programming with little friction.

jibe 3y ago

Push the language analogy further. In a sense, a child really I s raised by parents speaking a made-up language. It just happens to be a locally shared made-up language. Like math, you brute force language learning. Imitation and memorization. Only after years, or decades, can you go from basic language understanding to appreciating the beauty of words, poetry, literature. I don’t think there is a short cut in language or math. You have to go through the unfun multiplication table, spelling test phase to build the foundation for higher level appreciation l.

User23 3y ago

I think you’d really enjoy the book Visual Complex Analysis.

devchix 3y ago

I feel the same way. My college years math was a clueless grind (to me) and I don't think I made the connection to the real world until long after school was behind me. There's a very good site called BetterExplained, where I go to when I need to review some concept. I wish I had that when I was in school.

photochemsyn 3y ago

I think this paragraph in particular should be explained to all introductory students in any STEM-related field. It sets the stage for a basic yet profound understanding of what mathematics is all about: > "To discuss this subject we start in the middle. We suppose that we already know what integers are, what zero is, and what it means to increase a number by one unit. You may say, “That is not in the middle!” But it is the middle from a mathematical standpoint, because we could go even further back and describe the theory of sets in order to derive some of these properties of integers. But we are not going in that direction, the direction of mathematical philosophy and mathematical logic, but rather in the other direction, from the assumption that we know what integers are and we know how to count." Beyond that, an understanding of algebraic concepts in terms of how equations can be manipulated from one form to another (and of the rationale for why one can) might benefit from the addition of the concepts of distribution and commutation, which are not included in this lecture (although the important basic idea of a successor is).

deleted 3y ago

[deleted]