7 ms·

I'm afraid that this is because academic math is often taught and tested in a way that rewards memorization rather than understanding. Here's Richard Feynman's

I'm afraid that this is because academic math is often taught and tested in a way that rewards memorization rather than understanding. Here's Richard Feynman's take on the problem: https://v.cx/2010/04/feynman-brazil-education https://v.cx/2010/04/feynman-brazil-education

1970-01-01 2y ago
Feynman was correct for science in school, however arithmetic is fundamental and maybe one level above the root of all mathematics. Children should be able to do most of it via mental lookup tables and apply that knowledge on paper. For some reason, they can't.
- freeone3000 2y ago
  Why should they? We have the tables in our pockets at literally all times; doing arithmetic without it might be useful, or a bit faster sometimes, but is hardly an essential skill.
  - deleted 2y ago
    [deleted]
  - 1970-01-01 2y ago
    "Will you have enough fuel to make it over the mountain range?" "My phone is just rebooting" the pilot replied.
    - freeone3000 2y ago
      The plane has an onboard calculator for this :)
    - BeFlatXIII 2y ago
      These are the idiotic contrived examples that made me dismiss every lecture on the value of hard work my K–12 teachers gave.
  - BeFlatXIII 2y ago
    For catching when you keyed something in incorrectly.
  - somenameforme 2y ago
    It builds a numeric intuition. When you repeat something enough, it begins to do itself - you gain a subconscious mastery. Think about yourself as you read these words. Imagine if you were looking at the letters and actually trying to sound out each word, consciously thinking about each words meaning, and then finally trying to piece together the meaning. You'd spend 5 minutes reading a sentence or two, and oh God help yo if tere ws a tpyo. Instead it all just flows without you even thinking about it, even when completely butchered. And that sort of flow is, I think, obtainable for most of anything. But 100% for certain for numbers. Somebody who doesn't gain an intuitive understanding of basic arithmetic will have an extremely uncomfortable relationship with any sort of math, which mostly just means they'll avoid it at all costs, but you can't really. I don't even mean STEM careers, but everything from cooking (especially baking) to construction and generally an overwhelming majority of careers make heavy use of mathematical intuition in ways you might not consider, especially if you're already on good terms with numbers.
    - em-bee 2y ago
      that's why montessori math is so impressive. it starts with counting out beads one at a time by the hundreds until they have internalized that. then they get beads on a stiff wire 10 at a time, and repeat the process counting them out up to a 1000. and so forth until eventually they hold in their hands blocks of 1000 beads glued together in a cube, and only after they have internalized that the beads get replaced with more abstract woodblocks and sticks. and all that happens in the first year or so at the age of 3.
    - corimaith 2y ago
      It's a bit ironic to be saying this in the context of HN, the tech stack is built up on layers of abstraction that little of us have mastery of. If we made it so that only people who mastered assembly could be considered "real programmers" we'd get nowhere, certainly not to build modern web applications or video games.
      - milesrout 2y ago
        People that have a poor understanding of the stack under them almost invariably produce very poor software. Video games are an area where in fact good software is still produced, mostly because the people working on the cores of games DO know (at least how to read) assembly language. Modern web applications on the other hand so basically the same things we were doing with computers 20 years ago but consume 100x the resources to do so. No "modern web application" comes anywhere near the quality of Word or Excel 2003.
  - em-bee 2y ago
    so every time i go shopping i have to type all the prices of what i buy into my phone and also have the calculator connect to my bank account and not only make sure i have enough savings, but also tell me that i am not spending more than my average for weekly groceries? and when doing that i need to make sure to not make any typos because my lack of numeric intuition won't allow me to recognize where i made a mistake. and i also won't be able to tell if an item is overpriced. nor will i recognize a bargain unless it is marked with a big colorful sticker.
  - cratermoon 2y ago
    I suggest learning to use a slide rule and abacus.
  - Ekaros 2y ago
    And soon those things will be able to read any text shown to them. And speak it out. So why teach kids to read at all. It can all be done by voice.
  - gsich 2y ago
    It is an esssential skill.
- throwway120385 2y ago
  No, it goes beyond that. There's "arithmetic," the applied usage of addition, multiplication, subtraction, and division to permute numbers, and then there's Arithmetic, the set of theorems and axioms that give rise to that system of applied arithmetic. Memorization only works for the applied part, and children aren't usually taught that there is a system of reasoning behind those rules. Without that, no amount of mathematical dexterity in pushing symbols across a page will help them understand anything past the 100 level, and sometimes not even that. I also think there's a huge undercurrent of resistance from adults to having children learn that system of reasoning because adults don't understand why it's useful, and in my experience when people don't understand something they dismiss it. Edit: A nice example of another axiomatic system that might be more approachable is Euclid's Elements, in which five postulates are used to develop a system of geometry using an unmarked straightedge and a collapsible compass that you could, if you were careful, use to build bridges and other large buildings.
  - 1970-01-01 2y ago
    The study was limited to cities in India. We shouldn't put much weight into this applying globally.
    - throwway120385 2y ago
      I mean I remember seeing this first-hand student teaching Mathematics 13 years ago in the US. They got to me having never seen any of that stuff, and the curriculum attempted to provide a good education in mathematics. But the staff and the way the whole system is structured is to skip all of that and memorize the single rule you need to know to get through the test. So it's all done by rote and the only time you find out how you've been cheated is when you try to go through Calculus. And I remember that was how we learned everything when I was a kid, and the teachers chose not to do anything else. I also remember from my math ed curriculum one of the professors joking about the elementary education students complaining about having to learn middle school math from the college perspective. So I think portions of this apply here. I've also seen carpenters apply trigonometry very effectively to do things like cuts for roofs and stair jacks, so there's certainly a lot of truth to people learning maths by occupation and not in a formal setting, and I think part of it is the formal setting.
derbOac 2y ago
I had a similar reaction. I had a lot of reactions, and found the paper interesting. First, it reminded me of something a stats professor said in grad school: "there are two kinds of mathematicians, those who are good at arithmetic, and those who are not." He was speaking as someone who identified with the latter. I can't tell if this is something related to this domain of math in particular or something broader. My guess is it's something broader. I have colleagues (speaking as a professor) who have complained about admitted students who come in with very high grades and test scores, but who can't actually reason independently very well and despair when they are not "told exactly how to respond" on tests and whatnot. You have to be careful because sometimes these complaints hide bad teaching, but I think this is a common sentiment, and I've seen articles written about similar sentiments at other places. The paper touches on a lot of issues, like applied versus abstract concepts, generalizability of learning, "being a good student" versus actual cognitive ability, learning how to take tests versus learning concepts, the difficulty of measuring cognitive and academic ability, and the fallibility of measuring complex human attributes in general.
- zdragnar 2y ago
  Even in lower education, as a student I hated word problems. Partly, I just wanted to be told what equation to solve. In retrospect, though, I think a lot of it was the framing. It was always presented as some variation of short exposition followed by a question. The question was usually framed as an outside observer asking for some fact about the story. Think of the classic "A train leaves station A headed west at 6:30 traveling at 30 miles an hour. A second train leaves another station at 7 traveling 50 miles an hour. When do they pass each other?". There's no problem here to solve. Who cares when they pass each other? Why do we care? Sure, a little exposition helps build up analysis and application skills, but it doesn't actually offer much in the way of engagement.
  - singleshot_ 2y ago
    > There's no problem here to solve. Who cares when they pass each other? Why do we care? Not trying to “but acktually” you, however, this is more or less how I calculate the optimal time to take a pit stop in a lap-based auto race. I have a little spreadsheet widget that I made to be able to plug the numbers in, but the problem is simply stated: If old tires decrease my speed, and making a pit stop takes time, when should I stop. Agreed that elementary school word problems are dumb, though.
deleted 2y ago
[deleted]
almostgotcaught 2y ago
i hate when people quote random celebrities as authoritative on any topic, let alone as a counterpoint to actual authorities (google the authors of this study). Edit: hn is just as anti-intellectual as any other place these days but y'all style yourselves as intelligentsia because your celebrities are special. I'll repeat: check out the qualifications of the authors of this study and compare them to Feynman's on this subject. Any reasonable person would conclude that comparing them is exactly like comparing Kim Kardashian and Feynman's on QED.
- gjstein 2y ago
  Feynman is no random celebrity. In addition to be a renowned physicist, his famous "Feynman Lectures" and his thoughts on pedagogy are similarly legendary.
  - BeetleB 2y ago
    The Feynman Lectures are great at giving you an intuitive understanding, but is no substitute for the regular curriculum. You don't find many people who read only the Feynman Lectures who can then go on to solve physics problems well. You do find many who read the regular textbooks and who can.
    - codelieb 2y ago
      You have to bear in mind that the lectures in The Feynman Lectures on Physics were only one third of an introductory physics course, the other parts being recitation sections (in which homework problems, quizzes and tests were given and discussed), and labs. Lecture attendance was optional - many people prefer reading to listening - but the recitation sections and labs were mandatory, because they were considered much more important. Nobody learns physics from just reading lectures.
- wizzwizz4 2y ago
  In this case, Richard Feynman is just writing about his personal experiences of a well-known phenomenon. https://profkeithdevlin.org/wp-content/uploads/2023/09/lockhartslament.pdf https://profkeithdevlin.org/wp-content/uploads/2023/09/lockh... ("Lockhart's Lament") would perhaps be a better reference, but nearly anyone who's been through the education system would be able to tell you this.
mvieira38 2y ago
This tracks nearly 1 to 1 with my experience growing up in Brazil. Nice to see someone so accomplished pointing it out, thanks for sharing
blackeyeblitzar 2y ago
I wonder if this is true for all cohorts. There are a lot of children who are just fundamentally not intelligent, and deal with math classes by basically memorizing things and repeating them without real understanding. But for children who are understanding what they’re learning, I would expect academic learning to translate to other things.
wahern 2y ago
Applied math in daily life, such as making change, also involves a lot of memorization, it's just that the person doesn't realize they're committing various formulas and equalities to memory. The problem with proficiency in, e.g., making change, is that it doesn't carry over to higher-order math, logic, and reasoning. Arithmetic as taught in elementary school is attempting to achieve two things at once: proficiency in applied arithmetic, and foundational number theory (e.g. commutativity). My mother was a waitress and emphasized skills like making change. While she never articulated the rules, I ended up developing many of the mental arithmetic techniques that (I later discovered) Isaac Asimov discussed in Quick and Easy Math: https://archive.org/details/QuickAndEasyMath-English-IsaacAsimov https://archive.org/details/QuickAndEasyMath-English-IsaacAs... But I never developed an appreciation for number theory until it was too late--i.e. after high school. I did get into philosophy and logic during high school, but the connection (theoretical and applied) between the two didn't click until later. Sadly (or not?), proficiency in mental arithmetic has become much less common even among waiters, clerks, etc, at least where they don't deal in cash directly and without the aid of a register. And professional mathematicians have always humble-bragged about their impoverished mental arithmetic skills. So maybe we should drop the pretense that we're attempting to teach applied mathematics in the early years and admit the purpose is to lay theoretical foundation for higher-order math, applied and theoretical.
- adsteel_ 2y ago
  I only learned this in adulthood, too. I had to discover and memorize those shortcuts. Now I know that some of the "brilliant" math students I went to high school with had simply learned these skills the rest of us didn't.
RAdrien 2y ago
I agree. Academic math is taught as a set of rote rules or steps. The focus is not on intuitive understanding. I was taught the usual method of long division and carrying all by rote. Only later on in my academic life did I work out on my own why it works as it does.
- lupire 2y ago
  This hasn't been true in most of the USA for decades. A common failure more is for students to forget something and then claim they were never taught it. Arguably the should have been taught it more thoroughly.
  - esseph 2y ago
    This is still the way my kids in middle and high school are taught today.
- milesrout 2y ago
  Long division is taught to 8 year olds. That is not "academic maths" it is primary school arithmetic. When I was taught long division I was told how it worked, although it is fairly obvious if you think about it for a few seconds anyway.
dkarl 2y ago
That take has been accurate in specific times and places, but it is also blindly repeated in contexts where it's simply not true. All the teachers I encountered in my time in the US public school system were trained in the progressive spirit expressed by Feynman, with a strong bias against rote learning and in favor of a conceptual, understand-based approach. My math education in school both encouraged and rewarded figuring things out, which was good for me because I hated memorization and was always bad at it. Despite that, the criticism that school rewards memorization and doesn't teach critical thinking is still the only criticism I ever heard about the education I received. It's the standard thing that well-meaning people say. Which I think is a shame. When virtually every teacher in the system is trained in the progressive approach to education, and most of them sincerely believe in it and do their best to practice it, only to have the entire society turn around and claim that they are actually implementing ideas that nobody has believed in in a century, must be incredibly discouraging.
- carabiner 2y ago
  This is why American student math performance on PISA is excellent: https://i0.wp.com/www.edwardconard.com/wp-content/uploads/2023/12/2022-PISA-Math-.png?fit=1092%2C878&ssl=1 https://i0.wp.com/www.edwardconard.com/wp-content/uploads/20...
- LegionMammal978 2y ago
  Yeah, it seems almost as if a lot of people look back at their experiences of U.S. schools in the 90s, and assume that the schools in 2025 must not have changed one iota. While obviously I can't speak for every school district (nor can anyone), many of the criticisms I've heard seem disconnected from the schools I'm familiar with. (Not that they aren't subject to newer criticisms!) Is my personal experience a big outlier, or are people just extrapolating from the past? It makes me worry that school districts will greatly overcorrect in their efforts to ward off the old criticisms.
  - zmgsabst 2y ago
    This is a general problem with social change.
HellDunkel 2y ago
Made me laugh because I have no higher education but was able to answer the polarization question.
emursebrian 2y ago
I struggle when trying to solve math problems without context. I learned enough trigonometry to pass the final exams in high school, but I didn't REALLY understand it until I took a graduate-level graphics programming class. Some people enjoy the process of solving equations and math problems. For me, it's a tough process. Unless I have a tangible goal, I struggle to visualize the problem. Starting with basic algebra, it would be more effective if mathematics were paired with some practical problems. Computer graphics, engineering, construction, finance and the analysis of data would be good areas to do this in because it's exactly where you'd need said math!

1970-01-01 2y ago

Feynman was correct for science in school, however arithmetic is fundamental and maybe one level above the root of all mathematics. Children should be able to do most of it via mental lookup tables and apply that knowledge on paper. For some reason, they can't.

freeone3000 2y ago
Why should they? We have the tables in our pockets at literally all times; doing arithmetic without it might be useful, or a bit faster sometimes, but is hardly an essential skill.
- deleted 2y ago
  [deleted]
- 1970-01-01 2y ago
  "Will you have enough fuel to make it over the mountain range?" "My phone is just rebooting" the pilot replied.
  - freeone3000 2y ago
    The plane has an onboard calculator for this :)
  - BeFlatXIII 2y ago
    These are the idiotic contrived examples that made me dismiss every lecture on the value of hard work my K–12 teachers gave.
- BeFlatXIII 2y ago
  For catching when you keyed something in incorrectly.
- somenameforme 2y ago
  It builds a numeric intuition. When you repeat something enough, it begins to do itself - you gain a subconscious mastery. Think about yourself as you read these words. Imagine if you were looking at the letters and actually trying to sound out each word, consciously thinking about each words meaning, and then finally trying to piece together the meaning. You'd spend 5 minutes reading a sentence or two, and oh God help yo if tere ws a tpyo. Instead it all just flows without you even thinking about it, even when completely butchered. And that sort of flow is, I think, obtainable for most of anything. But 100% for certain for numbers. Somebody who doesn't gain an intuitive understanding of basic arithmetic will have an extremely uncomfortable relationship with any sort of math, which mostly just means they'll avoid it at all costs, but you can't really. I don't even mean STEM careers, but everything from cooking (especially baking) to construction and generally an overwhelming majority of careers make heavy use of mathematical intuition in ways you might not consider, especially if you're already on good terms with numbers.
  - em-bee 2y ago
    that's why montessori math is so impressive. it starts with counting out beads one at a time by the hundreds until they have internalized that. then they get beads on a stiff wire 10 at a time, and repeat the process counting them out up to a 1000. and so forth until eventually they hold in their hands blocks of 1000 beads glued together in a cube, and only after they have internalized that the beads get replaced with more abstract woodblocks and sticks. and all that happens in the first year or so at the age of 3.
  - corimaith 2y ago
    It's a bit ironic to be saying this in the context of HN, the tech stack is built up on layers of abstraction that little of us have mastery of. If we made it so that only people who mastered assembly could be considered "real programmers" we'd get nowhere, certainly not to build modern web applications or video games.
    - milesrout 2y ago
      People that have a poor understanding of the stack under them almost invariably produce very poor software. Video games are an area where in fact good software is still produced, mostly because the people working on the cores of games DO know (at least how to read) assembly language. Modern web applications on the other hand so basically the same things we were doing with computers 20 years ago but consume 100x the resources to do so. No "modern web application" comes anywhere near the quality of Word or Excel 2003.
- em-bee 2y ago
  so every time i go shopping i have to type all the prices of what i buy into my phone and also have the calculator connect to my bank account and not only make sure i have enough savings, but also tell me that i am not spending more than my average for weekly groceries? and when doing that i need to make sure to not make any typos because my lack of numeric intuition won't allow me to recognize where i made a mistake. and i also won't be able to tell if an item is overpriced. nor will i recognize a bargain unless it is marked with a big colorful sticker.
- cratermoon 2y ago
  I suggest learning to use a slide rule and abacus.
- Ekaros 2y ago
  And soon those things will be able to read any text shown to them. And speak it out. So why teach kids to read at all. It can all be done by voice.
- gsich 2y ago
  It is an esssential skill.
throwway120385 2y ago
No, it goes beyond that. There's "arithmetic," the applied usage of addition, multiplication, subtraction, and division to permute numbers, and then there's Arithmetic, the set of theorems and axioms that give rise to that system of applied arithmetic. Memorization only works for the applied part, and children aren't usually taught that there is a system of reasoning behind those rules. Without that, no amount of mathematical dexterity in pushing symbols across a page will help them understand anything past the 100 level, and sometimes not even that. I also think there's a huge undercurrent of resistance from adults to having children learn that system of reasoning because adults don't understand why it's useful, and in my experience when people don't understand something they dismiss it. Edit: A nice example of another axiomatic system that might be more approachable is Euclid's Elements, in which five postulates are used to develop a system of geometry using an unmarked straightedge and a collapsible compass that you could, if you were careful, use to build bridges and other large buildings.
- 1970-01-01 2y ago
  The study was limited to cities in India. We shouldn't put much weight into this applying globally.
  - throwway120385 2y ago
    I mean I remember seeing this first-hand student teaching Mathematics 13 years ago in the US. They got to me having never seen any of that stuff, and the curriculum attempted to provide a good education in mathematics. But the staff and the way the whole system is structured is to skip all of that and memorize the single rule you need to know to get through the test. So it's all done by rote and the only time you find out how you've been cheated is when you try to go through Calculus. And I remember that was how we learned everything when I was a kid, and the teachers chose not to do anything else. I also remember from my math ed curriculum one of the professors joking about the elementary education students complaining about having to learn middle school math from the college perspective. So I think portions of this apply here. I've also seen carpenters apply trigonometry very effectively to do things like cuts for roofs and stair jacks, so there's certainly a lot of truth to people learning maths by occupation and not in a formal setting, and I think part of it is the formal setting.

derbOac 2y ago

I had a similar reaction. I had a lot of reactions, and found the paper interesting. First, it reminded me of something a stats professor said in grad school: "there are two kinds of mathematicians, those who are good at arithmetic, and those who are not." He was speaking as someone who identified with the latter. I can't tell if this is something related to this domain of math in particular or something broader. My guess is it's something broader. I have colleagues (speaking as a professor) who have complained about admitted students who come in with very high grades and test scores, but who can't actually reason independently very well and despair when they are not "told exactly how to respond" on tests and whatnot. You have to be careful because sometimes these complaints hide bad teaching, but I think this is a common sentiment, and I've seen articles written about similar sentiments at other places. The paper touches on a lot of issues, like applied versus abstract concepts, generalizability of learning, "being a good student" versus actual cognitive ability, learning how to take tests versus learning concepts, the difficulty of measuring cognitive and academic ability, and the fallibility of measuring complex human attributes in general.

zdragnar 2y ago
Even in lower education, as a student I hated word problems. Partly, I just wanted to be told what equation to solve. In retrospect, though, I think a lot of it was the framing. It was always presented as some variation of short exposition followed by a question. The question was usually framed as an outside observer asking for some fact about the story. Think of the classic "A train leaves station A headed west at 6:30 traveling at 30 miles an hour. A second train leaves another station at 7 traveling 50 miles an hour. When do they pass each other?". There's no problem here to solve. Who cares when they pass each other? Why do we care? Sure, a little exposition helps build up analysis and application skills, but it doesn't actually offer much in the way of engagement.
- singleshot_ 2y ago
  > There's no problem here to solve. Who cares when they pass each other? Why do we care? Not trying to “but acktually” you, however, this is more or less how I calculate the optimal time to take a pit stop in a lap-based auto race. I have a little spreadsheet widget that I made to be able to plug the numbers in, but the problem is simply stated: If old tires decrease my speed, and making a pit stop takes time, when should I stop. Agreed that elementary school word problems are dumb, though.

deleted 2y ago

[deleted]

almostgotcaught 2y ago

i hate when people quote random celebrities as authoritative on any topic, let alone as a counterpoint to actual authorities (google the authors of this study). Edit: hn is just as anti-intellectual as any other place these days but y'all style yourselves as intelligentsia because your celebrities are special. I'll repeat: check out the qualifications of the authors of this study and compare them to Feynman's on this subject. Any reasonable person would conclude that comparing them is exactly like comparing Kim Kardashian and Feynman's on QED.

gjstein 2y ago
Feynman is no random celebrity. In addition to be a renowned physicist, his famous "Feynman Lectures" and his thoughts on pedagogy are similarly legendary.
- BeetleB 2y ago
  The Feynman Lectures are great at giving you an intuitive understanding, but is no substitute for the regular curriculum. You don't find many people who read only the Feynman Lectures who can then go on to solve physics problems well. You do find many who read the regular textbooks and who can.
  - codelieb 2y ago
    You have to bear in mind that the lectures in The Feynman Lectures on Physics were only one third of an introductory physics course, the other parts being recitation sections (in which homework problems, quizzes and tests were given and discussed), and labs. Lecture attendance was optional - many people prefer reading to listening - but the recitation sections and labs were mandatory, because they were considered much more important. Nobody learns physics from just reading lectures.
wizzwizz4 2y ago
In this case, Richard Feynman is just writing about his personal experiences of a well-known phenomenon. https://profkeithdevlin.org/wp-content/uploads/2023/09/lockhartslament.pdf https://profkeithdevlin.org/wp-content/uploads/2023/09/lockh... ("Lockhart's Lament") would perhaps be a better reference, but nearly anyone who's been through the education system would be able to tell you this.

mvieira38 2y ago

This tracks nearly 1 to 1 with my experience growing up in Brazil. Nice to see someone so accomplished pointing it out, thanks for sharing

blackeyeblitzar 2y ago

I wonder if this is true for all cohorts. There are a lot of children who are just fundamentally not intelligent, and deal with math classes by basically memorizing things and repeating them without real understanding. But for children who are understanding what they’re learning, I would expect academic learning to translate to other things.

wahern 2y ago

Applied math in daily life, such as making change, also involves a lot of memorization, it's just that the person doesn't realize they're committing various formulas and equalities to memory. The problem with proficiency in, e.g., making change, is that it doesn't carry over to higher-order math, logic, and reasoning. Arithmetic as taught in elementary school is attempting to achieve two things at once: proficiency in applied arithmetic, and foundational number theory (e.g. commutativity). My mother was a waitress and emphasized skills like making change. While she never articulated the rules, I ended up developing many of the mental arithmetic techniques that (I later discovered) Isaac Asimov discussed in Quick and Easy Math: https://archive.org/details/QuickAndEasyMath-English-IsaacAsimov https://archive.org/details/QuickAndEasyMath-English-IsaacAs... But I never developed an appreciation for number theory until it was too late--i.e. after high school. I did get into philosophy and logic during high school, but the connection (theoretical and applied) between the two didn't click until later. Sadly (or not?), proficiency in mental arithmetic has become much less common even among waiters, clerks, etc, at least where they don't deal in cash directly and without the aid of a register. And professional mathematicians have always humble-bragged about their impoverished mental arithmetic skills. So maybe we should drop the pretense that we're attempting to teach applied mathematics in the early years and admit the purpose is to lay theoretical foundation for higher-order math, applied and theoretical.

adsteel_ 2y ago
I only learned this in adulthood, too. I had to discover and memorize those shortcuts. Now I know that some of the "brilliant" math students I went to high school with had simply learned these skills the rest of us didn't.

RAdrien 2y ago

I agree. Academic math is taught as a set of rote rules or steps. The focus is not on intuitive understanding. I was taught the usual method of long division and carrying all by rote. Only later on in my academic life did I work out on my own why it works as it does.

lupire 2y ago
This hasn't been true in most of the USA for decades. A common failure more is for students to forget something and then claim they were never taught it. Arguably the should have been taught it more thoroughly.
- esseph 2y ago
  This is still the way my kids in middle and high school are taught today.
milesrout 2y ago
Long division is taught to 8 year olds. That is not "academic maths" it is primary school arithmetic. When I was taught long division I was told how it worked, although it is fairly obvious if you think about it for a few seconds anyway.

dkarl 2y ago

That take has been accurate in specific times and places, but it is also blindly repeated in contexts where it's simply not true. All the teachers I encountered in my time in the US public school system were trained in the progressive spirit expressed by Feynman, with a strong bias against rote learning and in favor of a conceptual, understand-based approach. My math education in school both encouraged and rewarded figuring things out, which was good for me because I hated memorization and was always bad at it. Despite that, the criticism that school rewards memorization and doesn't teach critical thinking is still the only criticism I ever heard about the education I received. It's the standard thing that well-meaning people say. Which I think is a shame. When virtually every teacher in the system is trained in the progressive approach to education, and most of them sincerely believe in it and do their best to practice it, only to have the entire society turn around and claim that they are actually implementing ideas that nobody has believed in in a century, must be incredibly discouraging.

carabiner 2y ago
This is why American student math performance on PISA is excellent: https://i0.wp.com/www.edwardconard.com/wp-content/uploads/2023/12/2022-PISA-Math-.png?fit=1092%2C878&ssl=1 https://i0.wp.com/www.edwardconard.com/wp-content/uploads/20...
LegionMammal978 2y ago
Yeah, it seems almost as if a lot of people look back at their experiences of U.S. schools in the 90s, and assume that the schools in 2025 must not have changed one iota. While obviously I can't speak for every school district (nor can anyone), many of the criticisms I've heard seem disconnected from the schools I'm familiar with. (Not that they aren't subject to newer criticisms!) Is my personal experience a big outlier, or are people just extrapolating from the past? It makes me worry that school districts will greatly overcorrect in their efforts to ward off the old criticisms.
- zmgsabst 2y ago
  This is a general problem with social change.

HellDunkel 2y ago

Made me laugh because I have no higher education but was able to answer the polarization question.

emursebrian 2y ago

I struggle when trying to solve math problems without context. I learned enough trigonometry to pass the final exams in high school, but I didn't REALLY understand it until I took a graduate-level graphics programming class. Some people enjoy the process of solving equations and math problems. For me, it's a tough process. Unless I have a tangible goal, I struggle to visualize the problem. Starting with basic algebra, it would be more effective if mathematics were paired with some practical problems. Computer graphics, engineering, construction, finance and the analysis of data would be good areas to do this in because it's exactly where you'd need said math!