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It's a shame he went to all the trouble of avoiding conventional education (for the wrong reasons), but then still ended up going down an execution rather than

It's a shame he went to all the trouble of avoiding conventional education (for the wrong reasons), but then still ended up going down an execution rather than concept focused route. As soon as you start memorizing operations and treating math more like a narrow grind only approached through arbitrary problems solved primarily through computation or use of rote application of poorly understood technique, you lose the ability to understand math as anything other than a human attempting to be a computer. The author expresses his misintuition of variables - there are plenty of simple examples and thought exercises which can inform the intuition. Later he expresses getting mixed up dealing with fractions - this is a clear demonstration that a fundamental understanding of fractions was skipped in order to grind out solutions. Modern math education is spectacularly guilty of going too far too fast. Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything. If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary.

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As_You_Wish 4y ago
Right. >Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything. This is so true. I was exactly like the author. I could not understand any math beyond algebra. Trigonometry was way beyond my ability. But one day, I found an old trig book lying around when I went to visit my parents - it was one of my sibling's books when she took it. I picked it up and had a thought - all those people who DID understand trig and higher math were not that much smarter than me. I mean, yeah, the super smart ones were, but I'm just talking about the regular students. So I got a bug to show myself that if they could do it, then I could do it. My approach that I would do is to read the first section of the first chapter until I understood it completely and not move on until I did. I would not even move on from a sentence, or indeed a word or term, until I understood exactly what was happening. I would look all over the internet, watch youtube videos on that one single concept, or that one single word, until I understood, and I was absolutely adament about not moving on until I understood. Word by word, sentence by sentence, section by section, I rolled on. I wanted to completely completely completely understand the very very basics of trig. It was slow going, as you might imagine. Super slow. Well........all of a sudden, about 4 or 5 chapters into the trig text, my mind exploded. It was blown apart by a blinding light. In one single instant, I swear, one instant, I knew all of trigonometry. I got it. All of trig. One single instant. It was probably one of the most singular moments in my life. So I then started paging through the entire rest of the trig book, and I knew it all. I swear. I would just glance over the pages and know what everything was about. Now, I was NOT an expert, by any stretch of the imagination, I'm not claiming I became a savant. No. I'm just saying I "got" it. As you said, I has the exponential results. So exponential, it was instantaneous. >If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary. This is it exactly. You describe it perfectly. I did not continue on, because I actually had other things in my life that I had to move on to. But it was great because I knew then that math was easy. No doubt I could do the exact same with any math class, no doubt in my mind whatsoever. For whomever wants to do great in almost any area of study, study HARD on those first 3-5 chapters. They set the tone for the rest of the course. If you study hard on them, they remainder of the course is trivial, with some things here and there that you need help with, but for the most part it is simple.
- lake_vincent 4y ago
  Very well said! It is true that once you grok this technique, you are basically unstoppable. It does make for slow progress at first, but it's worth it. Edit: I went to grad school for math, and I got there by pretty much doing what you described!
  - DennisP 4y ago
    Ok you two have inspired me. I've been wanting to learn math (and relearn college math from decades ago) and I always get bogged down and quit. I'm gonna try it your way.
    - lake_vincent 4y ago
      I am very glad to hear that! Enjoy :) I recommend the Art of Problem Solving community as a great place to start. I have no affiliation with them, they're just awesome for math learners of all levels.
  - Viliam1234 4y ago
    The problem with using the right techniques is that there are essentially three kinds of techniques: a) fast at the beginning, then gradually slow down, e.g. rote memorization; b) slow at the beginning, then gradually speed up, e.g. deep understanding; c) slow at the beginning, then remain slow forever, e.g. doing something chaotic and stupid. The problem is that sometimes you have incompetent people who use the third kind of method. Then, when things blow up, people start paying close attention to the speed, and reject any method that is slow at the beginning, because they suspect it would be the same story. (In other words: premature optimization, technical debt.)
- WalterBright 4y ago
  I've had a couple moments like that in my education. I've described it as stumbling about in a dark room, and suddenly the lights came on. One was when I struggling to learn how a computer worked. It was just a collection of arbitrary facts to me, more and more, until suddenly -- shazaam! And the rest was obvious and trivial. Thanks for your version. I enjoyed reading it!
2OEH8eoCRo0 4y ago
> “Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”
- sfpotter 4y ago
  Many other comments here don't seem to have spent much time reflecting on why his niece said this. Speaking as an actual math teacher, this is a very important thing for students to try to come to grips with. Memorization and "learning without understanding" have a bad rep, but memorization is a tremendously valuable skill. Instead of thinking of "learning by memorization" as doing a disservice to learning math, consider that learning math might increase your capacity for memorization as a byproduct. This skill carries over to so many others, but I think its importance gets glossed over as a result of it being simple to look things up using the internet. Another thing to think about: if you have something memorized (e.g., an identity, a theorem, a formula...), then you can think about it when you're walking around. When you have many of them memorized, you'll be able to think about how they relate to each other. The more you have memorized, the fewer stumbling blocks there will be to trip over when you're mulling over a problem or trying to figure something out. There's a lot of apocrypha out there about how if you're learning math, you shouldn't try to memorize things. You should instead try to "pick it up". This elides two important details: 1) there's selection bias at work---many people who are extremely good at math have minds like steel traps, remembering many things after seeing them only a couple times, 2) the expectation is that you spend a significant amount of time drilling and working, to give yourself an opportunity to pick it up. Now, why did his niece say this? When you're learning the basics like this, becoming "comfortable with formal manipulation" is really a matter of running a large number of experiments. You try this and that manipulation and get comfortable with them over time. You apply a formula in error and observe the results. You puzzle over what happened. Over time, you discover the logical framework which holds everything together. Asking "why" at this point (when you're learning very basic math!) is like putting the cart before the horse. You would receive an answer that you would be in no position to understand. Hence, "be like a child": play around, make some mistakes, try to understand what's going on. As you grow, you will become more sophisticated in your approach and better able to shift the balance so that you can attempt to simultaneously understand more of what you're learning.
zozbot234 4y ago
There's nothing wrong with practicing formal manipulations: these are just "the rules of the game" and are 100% rigorous. Now, if you want to know "why these rules and not others?", they just happen to follow from basic properties like associativity, commutativity, distributivity etc. There's not much else to it. The author's "misintuition" of variables is not critically important; variables can literally just be arbitrary symbols, and substitution understood separately. But the other "rules" are far more relevant.
- tankenmate 4y ago
  The one thing that gave me deep insights into algebra was discovering Peano's axioms.[0] I had been taught the associative, transitive, and distributive laws in 6th grade at school, but discovering the axioms really opened my eyes in high school; especially under what circumstances they could and couldn't be applied. The last penny to drop was to realise that where you have numbers (e.g. x) you could also substitute formulas, provided you adhered to the rules of when the axioms could be applied. Obviously for similar reasons this also applies to matrices (e.g. when can matrices be inverted (additive and multiplicative), and hence why matrix addition is commutative, but multiplication is not). Don't get me started on subtractive anti-commutativity vs divisive anti-commutativity. Another thing that bothers me is when helping the my own kids with maths I came to realise that their teachers don't know the Peano axioms, so how the heck can my kids even stand a chance at properly learning algebra. [0] https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms
  - erehweb 4y ago
    People understood algebra just fine before Peano came up with this axioms.
    - tankenmate 4y ago
      The point of the Peano axioms is that it codified all the basic rules of Algebra in 8 (now 9) rules.
auggierose 4y ago
Ever heard of new math? https://www.americanheritage.com/whatever-happened-new-math-0 https://www.americanheritage.com/whatever-happened-new-math-... I agree with you, especially if you are in your 60s and have a math professor as your teacher, one would think a new math approach works best!
- bryanrasmussen 4y ago
  sure https://www.youtube.com/watch?v=UIKGV2cTgqA https://www.youtube.com/watch?v=UIKGV2cTgqA I used to listen to that several times a day when I was a kid.
  - auggierose 4y ago
    Wow, great animations! I only read about the song in the article, I never actually listened to it until now, thanks.
  - Benobba 4y ago
    I was disappointed by this. It’s just a caricature of only one side - It’s the easiest form of come. Almost things that are innovative will not be best on the first iteration. Why not offer (funny) ideas on how to improve it?
    - auggierose 4y ago
      You could read the article I linked, if you want to know why new math is problematic. It's not that easy to improve things. And who said that comedy needs to be balanced?
- deleted 4y ago
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- Viliam1234 4y ago
  If you know the history of education, the "new math" thing is a sad story of a mind-blowing stupidity that keeps hurting math students for decades, because it divided most people into two camps that keep promoting two different wrong ideas. Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people. If we skip the medieval theories, one of the relatively modern ones was called "associationism". The theory was that human mind is basically a set of associations. We are born with zero associations; as we observe the world, we learn to associate this with that; and after many years we have learned to associate things properly and now we are smart adults. For extra nuance, some of us form new associations faster than others, probably for biological reasons; that is what intelligence is. Anyway, associations are all there is. Building an education theory on associanism is quite easy. You need a teacher who understands the subject (has the correct associations, a lot of them). Then the teacher stands in front of the classroom and keeps talking. The more he talks, the more associations the students can make. That's all there is. -- The order of lessons is not relevant; ultimately, after the teacher mentions everything, all associations will be properly connected into one large network; until then, you have to memorize. You don't wait until the students "understand", that would be a waste of time (there is no such thing as "understanding", you either have the right associations or you don't); the more you keep talking, the more associations the students can make. Of course you can (and should) repeat the facts, that's how the associations are deepened. But after a while, if some students don't get it, they are just hopeless: they had the opportunity to make the right associations, and yet they failed. It is their fate to remain farmers. This is a bit of a strawman, and yet many teachers follow this method intuitively, even without knowing the underlying theory. And their students complain that they don't get it. And the teachers reply that yeah, some students are just talented and some are not, "the camel has two humps", et cetera. In psychology, the next step after associationism was Piaget's "genetic epistemology". Where "genetic" is an adjective for "genesis", not the DNA. In modern language, we would probably call it "developmental epistemology", i.e. the study of the ontogenetic origins of understanding. The revolutionary approach was to watch how kids actually learn, rather than trying to shoehorn everything into a simplistic framework. One of the interesting findings was that kids actually do not make linear progress from "zero associations" to "correct understanding", but the process often takes a detour through a phase of magical thinking or some other kind of wrong understanding. Instead of "no opinion -> correct opinion", it is often "no opinion -> wrong opinion -> correct opinion". There are specific examples, not important now. Also, Piaget got some things wrong; this was later improved by Vygotsky. The important thing is the idea that the child is making mental models of the world. It is not just associations floating in a vacuum; the child has a paradigm, and tries to fit the new knowledge in that paradigm, and sometimes it doesn't work and the paradigm changes into a better one. An educational theory built on this, originally called "constructivism", says that the teacher should not just keep saying random true facts, but also check that the students have the right models. This is achieved on one hand by making the models explicit, saying the facts in proper order, putting them in the right context... and on the other hand by checking the students' models, finding the problems and fixing them. If you find out that many students keep making the same mistake, you should adjust your way of teaching accordingly: make it obvious at the beginning that it is X not Y, maybe change the order of lessons so that making the correct model becomes easier. Keep checking the students' models regularly, because the sooner you find the mistake, the easier it is to fix it. Etc. This is what many good tutors do intuitively, because if you teach 1:1, there is more interaction, and it is easier to catch the mistakes right when they happen and ask "why did you do this?" You do not wait until the student makes the same mistake hundred times to declare him a failure without talent; you notice when the mistake happens for the first time and keep "debugging" until the mistake is fixed. This is easy to do when tutoring; more difficult to make it scale to a classroom full of kids. And then... there is another thing, I do not know if it has a proper name in psychology, but "postmodernism" is what some people use (and other people object to this usage)... the "edgy" idea that knowledge transfer is actually impossible, everyone lives in their own different reality, trying to teach something is an oppression, if only we left the kids alone they would reinvent the civilization and make it much better (Rousseau's "Emile"). -- For stupid political reasons ("there are exactly two sides of the story, not more, not less", "the enemy of my enemy is my friend"), these people are typically associated with the constructivists, because they both oppose rote memorization. But although the opponent may be the same, the proposed solutions are quite different ("teaching better" vs "not teaching at all"). To increase the confusion, the educational theory build on this was called "radical constructivism", misleadingly suggesting that this might be "something like the famous Piaget, only much more so", when if fact it is something completely different. The kids taught using this philosophy are left alone to reinvent the math... and fail predictably! Or sometimes they are taught dozen different methods how to do addition (bonus point if the method was used by some indigenous population, because, you know, "noble savage", doesn't matter if the specific method only works for adding 7+8 and 8+9), hoping that this will kickstart their math thinking so now they will develop the rest of the math independently. Predictably, that also never happens. So, how is this related to the "new math"? The curriculum of the "new math" was based exactly on this "radical constructivist" thought, except the authors did not emphasise the "radical" part enough and often just called it "constructivism". So you had the math curriculum that didn't work at all, and was a complete disaster. And when finally people got angry and returned to the traditional math education, the lesson everyone remembered was that "constructivism has been debunked". So now, whenever someone proposes to teach math in a way that emphasizes understanding over memorization, the kneejerk reaction is "haha, that sounds like constructivism... yeah, we tried that but that didn't work at all", plus a link to some web page that criticizes "new math". And if you try to explain how this is completely unrelated to Piaget, you are dismissed with "yeah right, the true constructivism has never been tried, comrades, hahaha". And then, ironically, people keep quoting the Feynman's story about how actually understanding physics is better than mere memorization that "light is waves". And the helpless (Piagetian) construstivist is like "yeah guys, that's exactly what I was trying to tell you all the time", but no one cares, and when it comes back to adding some understanding to the lessons, someone inevitably comes with the condescending "haha, but constructivism has been debunked" and a link to Ten Facts Why New Math Sucks. Possible solution: perhaps the brand of "constructivism" has been thoroughly poisoned, and we need to reinvent it and call it "Feynmanism". Then you can go and say "nope, I am totally not proposing a constructivist curriculum, that has already been tried and debunked, haha, what I am proposing instead is the Feynmanist curriculum", and then people will go online and say "wow, I hated math at school, but then we got a new teacher who used this new Feynmanist method, and the math finally started make sense and now I love it".
sirsinsalot 4y ago
I left school at 15, never to return to education, in part because of how maths was taught. There was never an explanation of the _why_ of things. It was very frustrating, and the teaching was very poor. I ended up a programmer, in part, because I figured if teachers weren't teaching, i'd teach myself. This has been a good strategy for me, however there's serious holes in my pure maths knowledge.
- gavinray 4y ago
  Lol, I've been kicked out of class for asking the teacher to explain why something works in math here in the US I think it was long division and lattice multiplication in elementary school Doing math by drawing numbers in predefined shapes so that it magically worked out was the most ludicrous thing I'd ever seen "Because that's how it works." Wasn't really a satisfactory answer lol.
  - 2OEH8eoCRo0 4y ago
    One of my friends in HS was against calculators, and also very stubborn, and got in trouble for asking how to do sin, cos, tan functions without a calculator.
    - nobodywasishere 4y ago
      Good old taylor series, of course!
      - reacweb 4y ago
        Sine Tables Charts is easier if you can bring them to exam.
        hgomersall 4y ago
        Just read off the axis on a unit circle. If you need more precision, draw a bigger circle.
      - 4y ago
WalterBright 4y ago
When I remarked yesterday that engineering curriculums that emphasized memorization of engineering formulas rather than understanding how to derive them were inferior, I was dismissed as arrogant and egotistical :-)
- mmmpop 4y ago
  By whom? An instructor/professor or a student/engineering graduate?
  - WalterBright 4y ago
    It's yesterday in my comment history, you shouldn't have any trouble finding it. I prefer people interpret it for themselves as I said my piece in it.
    - BeetleB 4y ago
      Can't speak for the others, but you may come across as too one sided. I had your mindset, and one of the reasons I started getting left behind in grad school whereas others didn't was my stubborn refusal to memorize anything and insisting only on understanding and deriving as needed. Once you go deep enough, you'll often find yourself relying on N random theorems you learned some courses ago to solve a problem, and those who had memorized them were more likely to solve the problem than someone like me who happened to forget the theorem existed in some book and is trying to rederive everything from scratch. Of course, those who mostly or only memorize perform the worst. The other issue is in my other comment: https://news.ycombinator.com/item?id=32388595 https://news.ycombinator.com/item?id=32388595 Basically, most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform. So some level of "take it on faith and memorize a few items" are needed. There are other examples of this. Also, having studied under many top class physicists, I can tell you that a significant number of them cannot derive much of the mathematics they use, despite being wizards in applying the techniques.
      - WalterBright 4y ago
dkarl 4y ago
> this is a clear demonstration that a fundamental understanding of fractions was skipped in order to grind out solutions Leaving aside that you concluded this from a one-sentence story about solving a single problem, it's funny that the idea that a "clear understanding of fractions" can make the skills easy to learn is something that virtually every trained teacher has believed for the last (I'm not sure how many) decades and yet nobody alive today is reporting that it happened that way for them. It's possible that every future teacher getting certified to teach politely pretends to believe these things, but secretly wishes to spend their career torturing children, so they abandon the teaching of concepts as soon as they step into a classroom. Or maybe they go through the motions of teaching kids to understand the meaning behind fractions, but they intentionally do it wrong and laugh with each other about it in the teacher's lounge. That's one explanation. Another explanation is that this opposition of "understanding" to "execution" and "concepts" to "skills" is an incomplete way of looking at things that has limited application to how people learn but isn't a panacea, and people still struggle with skills no matter how much you try to prepare them with concepts. I think it does a vast disservice to anyone when we assume that their job performance would be radically improved by the first cliché we ever learned about their field. If the idea is so magical, and we know that it has permeated the field of education for generations, where are the generations of students who remember math as being "trivial, intuitive, beautiful, clever?" Different educational philosophies have come and gone, emphasis has shifted, so where's the generation of parents saying math was much easier for them than for their kids? Where's the PTA meeting where all the parents say, "We think there's something wrong with the way math is taught here, because we all remember fractions being trivial and intuitive, but our kids are finding them hard?" Maybe teachers aren't such idiots that they consistently overlook the value of the most cherished and shopworn truism their field has produced in its history?
- thesuitonym 4y ago
  I'll start my reply with this: I'm not a teacher, nor am I an understander of maths, but whenever I talk to teachers at any level, and especially mathematics teachers, they lament that they want to teach differently, but due to time constraints, or predetermined lesson plans, or any number of reasons, they simply can't. They have to teach in a way that gets students enough knowledge to pass the test, but only those that seem to have a natural aptitude end up really learning anything. The rest discard the information as soon as it's no longer necessary. I don't think it's a disservice to teachers to say it's a better way of teaching, because they say it themselves. It's just a model that only really works one on one. It can't be done in a room full of 30 bored children, and especially in a room of 400 university sophomores who are ready to get out of class to go to a party.
jorvi 4y ago
Yes. The clearest example of this for me was the fact that many things I was taught in high school were taught to me as an arbitrary concept first, without the accompanying visual relation. Being taught a² + b² = c² with the triangle and accompanying squares drawn out much much later is the best example I can give.
jjice 4y ago
> ...still ended up going down an execution rather than concept focused route. This is something that made me really disappointed when I realized it in school. It was always a step by step process taught to me, and I loved math. I took my AP Calculus class in my senior year of high school and my teacher (fantastic man) showed us how the formula for a derivative is derived. I was blown away. We had just learned and it were told to memorize it the previous year and I had just accepted it. After that, math changed for me. Everything had a reason that was connected to everything else. If you understood where it came from, you could do even more than if you were just given a formula to follow. How beautiful. I realized that I had 5 years (assuming we only cared to do so after algebra) of match education that could have been better taught if done conceptually. I don't blame teachers. You need to get everyone to pass your class for state requirements, so you try to streamline it to get this year done, and not prepare for the future. Also, most math teachers I had did it as a job, not because they loved it, and there's nothing wrong with that. When I had that calculus class during my senior year though, I realized that that man loved math. Hard but fair. I learned so much and that teacher taught me how to develop a work ethic. I wish meta-learning and a more concept based approach could be applied to high school courses (at least looking back). I understand why they aren't, but man would I love to see how it would play out.
- jonathanlydall 4y ago
  When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer. I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”. Which was such a thoroughly disappointing response and I hated trigonometry since it was mostly about memorisation rather than applying logic from understanding. About 10 years later I did a math course through the UK’s Open University and the text book taught trigonometry through explaining the unit circle. It made me so happy to finally understand and see the actual logic as opposed to punching numbers into a calculator, but I was also sad and a bit upset the unit circle was not taught in high school.
  - 4y ago
jrmg 4y ago
This is addressed in the article - just after the ‘(7/2) / 2 = 7’ misunderstanding: [I called] Deane Yang, my friend who is a mathematics professor at NYU. “The way you remember procedures is you remember why,” he said. “Because?” “Because people learn math as a collection of procedures,” he said. “When things get difficult, they’re lost, and math becomes religion class. The teacher says what’s right and wrong, and for all you know math came out of the sky, and some prophet told you how to do it, and it’s just blind belief then. Interesting that he wants to know _why_ knowing why helps too! I do wonder if this is covered more in-depth in his book.
rawgabbit 4y ago
I agree. He tried to memorize his way through Algebra. It is hard to understand his mindset because of the stream of consciousness writing. But he appears to be confused with definitions versus equality. E.g., By definition (a+b)^2 is (a+b)(a+b). It is not just "equal".
greymalik 4y ago
What’s the alternative? How does a neophyte go about cultivating the mindset and understanding you describe?
- rawgabbit 4y ago
  Unfortunately, I don't have an easy answer. My son recently graduated from my high school and I noticed that while his school used Pearson's website to drill him on the mechanics. My son struggled with word problems precisely because he did not have the fundamental concepts. For example, I argued that division is more intuitive when we multiply using fractions. - Whole numbers have an implicit denominator of one. E.g., 7 is equal to 7/1. - It is more intuitive to think of division as multiplying by fractions. E.g., 7/2 is equal to (7/1) x (1/2). We multiply the numerators together. And multiply the denominators together. - Where this is helpful is when we deal with complex arithmetic like (7/2)/2. We have 7 pizzas. Divide by 2. And divide by 2 again. If you break it into fractions, this is equal to (7/1) x (1/2) x (1/2). The numerator is 7x1x1. While the denominator is 1x2x2. Putting it together, it is (7x1x1) / (1x2x2) which is equal to 7/4. This discussion usually leads to the question of what is multiplication (again)? I argued that multiplication can be thought of as a shortcut for addition or can be thought of as set theory. I have two bags of marbles and each bag contains 10 marbles each. How many marbles do I have? You can either write 10+10 (multiplication as a shortcut for addition) or you can write 2x10 (set theory). Taking it back to division. I have two bags of marbles and each bag contains 10 marbles and I have to share my marbles with my sibling (divide by two), how many marbles do I have?. You can write either (10+10)/2 or you can write 2 x 10 x (1/2).
- mattwilsonn888 4y ago
  This person describes an experience similar to the one which inspired my mindset. https://news.ycombinator.com/threads?id=As_You_Wish https://news.ycombinator.com/threads?id=As_You_Wish
crispyambulance 4y ago
> ...still ended up going down an execution rather than concept focused route. Everyone will say, of course, that "concepts matter". But the reality is that there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables. One can't move on to new concepts in math until the previous dependent concepts have been mastered. Mastery means practice, practice means drills, and all of that is boring but necessary. The best we can do is to make the mathematics more meaningful by tying it to problem solving. At some point many students will experience an "ah ha" realization and be further stimulated, while for some others math will always be meaningless drudgery. For most it will be something in-between.
- bslorence 4y ago
  > there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables. Sure, but imagine learning multiplication tables without having any idea what it means to "multiply"; literally just memorizing sequences of symbols, without ever looking at piles of coins or whatever. Multiplication is so basic that this is hard to imagine, but I'm sure that, say, difference-of-squares rules feel like this to most beginning algebra students -- and for most people that probably never changes. I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.
  - BeetleB 4y ago
    > I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college. Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretation is merely a manifestation of it that just happens to work in some domain. As an example, the formula is equally valid for complex numbers, but I doubt you'd get there from Euclid's. No doubt some geometric interpretation can be found for that as well, but then I'd pick some other algebraic field where it's true and you'd have to search yet again for a geometric interpretation.
stakkur 4y ago
I agree, but the fundamental problem is how most western math education is structured top-down to be a goals-driven curriculum. In other words, teaching to the test. In the US, it’s a sprint towards meeting goals tied to government funding (“core curriculum “, etc.) And so many get left behind in that sprint, often feeling they’re ‘not smart enough’ or ‘not a math person’.
photochemsyn 4y ago
I think an introductory course would ideally hop back and forth between the explicit execution of specific examples and the understanding of abstract concepts. As you say the concepts are very important - for example, the author never mentioned the associative, distributive, or commutative rules which apply to all cases of addition and multiplication (but not necessarily to subtraction and division). Those are universally applicable, and indeed understanding them is important for moving on to higher maths (group theory etc.) However, without doing a fair amount of ditch-digging, abstract general theories of ditches might be hard to grasp. For a programming example, consider data structures - lists, trees, graphs, and so on. The abstract concept is clearly what allows larger constructs to be made, i.e. entire programs, but if you've never actually implemented a linked list in any particular programming language, and try to, the result is almost certainly going to be a buggy disaster. The optimal math education might be one where for every hour the instructor spends on the conceptual approach, the student then spends about three hours on the execution approach as applied to specific examples of those concepts (ideally with a teaching assistant around to help if the student gets stuck).
shon 4y ago
This is insightful and rings true for me as well. A good blend of theory and practical knowledge is important to get the most out of any education. Do you have any recommendations for learning math this way?
- mattwilsonn888 4y ago
  This person describes an experience similar to the one which inspired my mindset. One other thing to note is that the order you might learn math, even the 'bundles' of math confined to the standard classes, is somewhat arbitrary. If you take on the freedom, you are responsible for the direction. That being said the linear path from arithmetic to Calculus or Linear Algebra, or Vector Calculus (to combine the two) will give you very strong fundamentals and chops that will apply broadly. It feels like once you get some understanding of Vector Calculus you can branch off wherever you want, though many advanced topics past that don't strictly require it, learning to derive concepts in that subject is excellent preparation. All that said, there are plenty of interesting topics not involving Calculus or Linear Algebra. I really recommend what As_You_Wish has said as being honest with your own understanding and developing deep rigor, even across just one chapter or one sentence, will be your guide. Once you get deeply acquainted with one area, even if just a tiny area like "I finally understand trig functions," you understand better where you can jump off to next. https://news.ycombinator.com/threads?id=As_You_Wish https://news.ycombinator.com/threads?id=As_You_Wish
User23 4y ago
> you lose the ability to understand math as anything other than a human attempting to be a computer I get what you're saying and it's important, but I also disagree with how you're framing it. It's desirable to get to the point where you can automatically do purely mechanical symbol transformations as you work through a problem, thus saving your higher reasoning for the conceptual problem of searching for which transformations will lead to a solution. Frankly, I find the continuing concern with "intuition" with respect to purely mechanical transformations to be rather medieval. It is slowly changing though as the influence of computing science osmoses back into more traditional mathematics. Using Mathematica or similar tools makes it abundantly clear how distinct the mechanical and conceptual challenges are.
- mattwilsonn888 4y ago
  I don't believe layers should not be built on top of to the point of rarely, or even never thinking about the inner workings again. I just believe that when building those layers, having an intimate relationship in your mind with how they work and relate to each other will have positive effects, even after you start executing steps without regard for their deeper concepts. That's the case even for just the 'engineer' archetype, who uses math strictly as a tool. For any person doing theoretical work, discovery is often preceded by a fundamental understanding and tweaking of the most basic tools - so I believe the mindset is useful in both cases but borderline necessary for the second.
bowsamic 4y ago
I disagree. My favourite lecturer for general relativity said the important thing is fluency, which comes from repetition. It’s great to know everything, but you should be able to do the basics, such as calculus or algebra, very quickly and without thinking. You will need that brain power for the new stuff. The best way to gain mathematical fluency is by lots of practise.
- Viliam1234 4y ago
  You can understand something first, and practice for the speed later. Understanding does not prevent practice. I know the result of 8×8 immediately, and I don't think I would ever forget it. But, hypothetically speaking, if I ever made a mistake here, I have the option to slow down and verify the result. If I wouldn't have the option, there is a risk that I would make a mistake and then keep making the same mistake forever, because once the lessons are over, there would be nothing to correct me.
ikrenji 4y ago
it's way easier to understand the concept if you see it repeated many times in action though...

deleted 4y ago

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As_You_Wish 4y ago

Right. >Fundamental understanding gives exponential results yet most courses try and get the axioms and basics out of the way as fast as possible, working through basic proofs or derivations is cute sideshow, if anything. This is so true. I was exactly like the author. I could not understand any math beyond algebra. Trigonometry was way beyond my ability. But one day, I found an old trig book lying around when I went to visit my parents - it was one of my sibling's books when she took it. I picked it up and had a thought - all those people who DID understand trig and higher math were not that much smarter than me. I mean, yeah, the super smart ones were, but I'm just talking about the regular students. So I got a bug to show myself that if they could do it, then I could do it. My approach that I would do is to read the first section of the first chapter until I understood it completely and not move on until I did. I would not even move on from a sentence, or indeed a word or term, until I understood exactly what was happening. I would look all over the internet, watch youtube videos on that one single concept, or that one single word, until I understood, and I was absolutely adament about not moving on until I understood. Word by word, sentence by sentence, section by section, I rolled on. I wanted to completely completely completely understand the very very basics of trig. It was slow going, as you might imagine. Super slow. Well........all of a sudden, about 4 or 5 chapters into the trig text, my mind exploded. It was blown apart by a blinding light. In one single instant, I swear, one instant, I knew all of trigonometry. I got it. All of trig. One single instant. It was probably one of the most singular moments in my life. So I then started paging through the entire rest of the trig book, and I knew it all. I swear. I would just glance over the pages and know what everything was about. Now, I was NOT an expert, by any stretch of the imagination, I'm not claiming I became a savant. No. I'm just saying I "got" it. As you said, I has the exponential results. So exponential, it was instantaneous. >If the fundamentals are well understood, the applications feel trivial, intuitive, beautiful, clever - when the fundamentals are taken for granted, one word describes the relationship to the rest of the material: arbitrary. This is it exactly. You describe it perfectly. I did not continue on, because I actually had other things in my life that I had to move on to. But it was great because I knew then that math was easy. No doubt I could do the exact same with any math class, no doubt in my mind whatsoever. For whomever wants to do great in almost any area of study, study HARD on those first 3-5 chapters. They set the tone for the rest of the course. If you study hard on them, they remainder of the course is trivial, with some things here and there that you need help with, but for the most part it is simple.

lake_vincent 4y ago
Very well said! It is true that once you grok this technique, you are basically unstoppable. It does make for slow progress at first, but it's worth it. Edit: I went to grad school for math, and I got there by pretty much doing what you described!
- DennisP 4y ago
  Ok you two have inspired me. I've been wanting to learn math (and relearn college math from decades ago) and I always get bogged down and quit. I'm gonna try it your way.
  - lake_vincent 4y ago
    I am very glad to hear that! Enjoy :) I recommend the Art of Problem Solving community as a great place to start. I have no affiliation with them, they're just awesome for math learners of all levels.
- Viliam1234 4y ago
  The problem with using the right techniques is that there are essentially three kinds of techniques: a) fast at the beginning, then gradually slow down, e.g. rote memorization; b) slow at the beginning, then gradually speed up, e.g. deep understanding; c) slow at the beginning, then remain slow forever, e.g. doing something chaotic and stupid. The problem is that sometimes you have incompetent people who use the third kind of method. Then, when things blow up, people start paying close attention to the speed, and reject any method that is slow at the beginning, because they suspect it would be the same story. (In other words: premature optimization, technical debt.)
WalterBright 4y ago
I've had a couple moments like that in my education. I've described it as stumbling about in a dark room, and suddenly the lights came on. One was when I struggling to learn how a computer worked. It was just a collection of arbitrary facts to me, more and more, until suddenly -- shazaam! And the rest was obvious and trivial. Thanks for your version. I enjoyed reading it!

2OEH8eoCRo0 4y ago

> “Here is some advice,” she said firmly. “I get it that you try to put things into a framework that you can understand. That’s fine, but at first, until you become comfortable with the formal manipulation, you have to be like a child.”

sfpotter 4y ago
Many other comments here don't seem to have spent much time reflecting on why his niece said this. Speaking as an actual math teacher, this is a very important thing for students to try to come to grips with. Memorization and "learning without understanding" have a bad rep, but memorization is a tremendously valuable skill. Instead of thinking of "learning by memorization" as doing a disservice to learning math, consider that learning math might increase your capacity for memorization as a byproduct. This skill carries over to so many others, but I think its importance gets glossed over as a result of it being simple to look things up using the internet. Another thing to think about: if you have something memorized (e.g., an identity, a theorem, a formula...), then you can think about it when you're walking around. When you have many of them memorized, you'll be able to think about how they relate to each other. The more you have memorized, the fewer stumbling blocks there will be to trip over when you're mulling over a problem or trying to figure something out. There's a lot of apocrypha out there about how if you're learning math, you shouldn't try to memorize things. You should instead try to "pick it up". This elides two important details: 1) there's selection bias at work---many people who are extremely good at math have minds like steel traps, remembering many things after seeing them only a couple times, 2) the expectation is that you spend a significant amount of time drilling and working, to give yourself an opportunity to pick it up. Now, why did his niece say this? When you're learning the basics like this, becoming "comfortable with formal manipulation" is really a matter of running a large number of experiments. You try this and that manipulation and get comfortable with them over time. You apply a formula in error and observe the results. You puzzle over what happened. Over time, you discover the logical framework which holds everything together. Asking "why" at this point (when you're learning very basic math!) is like putting the cart before the horse. You would receive an answer that you would be in no position to understand. Hence, "be like a child": play around, make some mistakes, try to understand what's going on. As you grow, you will become more sophisticated in your approach and better able to shift the balance so that you can attempt to simultaneously understand more of what you're learning.

zozbot234 4y ago

There's nothing wrong with practicing formal manipulations: these are just "the rules of the game" and are 100% rigorous. Now, if you want to know "why these rules and not others?", they just happen to follow from basic properties like associativity, commutativity, distributivity etc. There's not much else to it. The author's "misintuition" of variables is not critically important; variables can literally just be arbitrary symbols, and substitution understood separately. But the other "rules" are far more relevant.

tankenmate 4y ago
The one thing that gave me deep insights into algebra was discovering Peano's axioms.[0] I had been taught the associative, transitive, and distributive laws in 6th grade at school, but discovering the axioms really opened my eyes in high school; especially under what circumstances they could and couldn't be applied. The last penny to drop was to realise that where you have numbers (e.g. x) you could also substitute formulas, provided you adhered to the rules of when the axioms could be applied. Obviously for similar reasons this also applies to matrices (e.g. when can matrices be inverted (additive and multiplicative), and hence why matrix addition is commutative, but multiplication is not). Don't get me started on subtractive anti-commutativity vs divisive anti-commutativity. Another thing that bothers me is when helping the my own kids with maths I came to realise that their teachers don't know the Peano axioms, so how the heck can my kids even stand a chance at properly learning algebra. [0] https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms
- erehweb 4y ago
  People understood algebra just fine before Peano came up with this axioms.
  - tankenmate 4y ago
    The point of the Peano axioms is that it codified all the basic rules of Algebra in 8 (now 9) rules.

auggierose 4y ago

Ever heard of new math? https://www.americanheritage.com/whatever-happened-new-math-0 https://www.americanheritage.com/whatever-happened-new-math-... I agree with you, especially if you are in your 60s and have a math professor as your teacher, one would think a new math approach works best!

bryanrasmussen 4y ago
sure https://www.youtube.com/watch?v=UIKGV2cTgqA https://www.youtube.com/watch?v=UIKGV2cTgqA I used to listen to that several times a day when I was a kid.
- auggierose 4y ago
  Wow, great animations! I only read about the song in the article, I never actually listened to it until now, thanks.
- Benobba 4y ago
  I was disappointed by this. It’s just a caricature of only one side - It’s the easiest form of come. Almost things that are innovative will not be best on the first iteration. Why not offer (funny) ideas on how to improve it?
  - auggierose 4y ago
    You could read the article I linked, if you want to know why new math is problematic. It's not that easy to improve things. And who said that comedy needs to be balanced?
deleted 4y ago
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Viliam1234 4y ago
If you know the history of education, the "new math" thing is a sad story of a mind-blowing stupidity that keeps hurting math students for decades, because it divided most people into two camps that keep promoting two different wrong ideas. Theories of education are often based on some psychological theory -- you have a theory how people think in general, and you use it to design a process to teach people. If we skip the medieval theories, one of the relatively modern ones was called "associationism". The theory was that human mind is basically a set of associations. We are born with zero associations; as we observe the world, we learn to associate this with that; and after many years we have learned to associate things properly and now we are smart adults. For extra nuance, some of us form new associations faster than others, probably for biological reasons; that is what intelligence is. Anyway, associations are all there is. Building an education theory on associanism is quite easy. You need a teacher who understands the subject (has the correct associations, a lot of them). Then the teacher stands in front of the classroom and keeps talking. The more he talks, the more associations the students can make. That's all there is. -- The order of lessons is not relevant; ultimately, after the teacher mentions everything, all associations will be properly connected into one large network; until then, you have to memorize. You don't wait until the students "understand", that would be a waste of time (there is no such thing as "understanding", you either have the right associations or you don't); the more you keep talking, the more associations the students can make. Of course you can (and should) repeat the facts, that's how the associations are deepened. But after a while, if some students don't get it, they are just hopeless: they had the opportunity to make the right associations, and yet they failed. It is their fate to remain farmers. This is a bit of a strawman, and yet many teachers follow this method intuitively, even without knowing the underlying theory. And their students complain that they don't get it. And the teachers reply that yeah, some students are just talented and some are not, "the camel has two humps", et cetera. In psychology, the next step after associationism was Piaget's "genetic epistemology". Where "genetic" is an adjective for "genesis", not the DNA. In modern language, we would probably call it "developmental epistemology", i.e. the study of the ontogenetic origins of understanding. The revolutionary approach was to watch how kids actually learn, rather than trying to shoehorn everything into a simplistic framework. One of the interesting findings was that kids actually do not make linear progress from "zero associations" to "correct understanding", but the process often takes a detour through a phase of magical thinking or some other kind of wrong understanding. Instead of "no opinion -> correct opinion", it is often "no opinion -> wrong opinion -> correct opinion". There are specific examples, not important now. Also, Piaget got some things wrong; this was later improved by Vygotsky. The important thing is the idea that the child is making mental models of the world. It is not just associations floating in a vacuum; the child has a paradigm, and tries to fit the new knowledge in that paradigm, and sometimes it doesn't work and the paradigm changes into a better one. An educational theory built on this, originally called "constructivism", says that the teacher should not just keep saying random true facts, but also check that the students have the right models. This is achieved on one hand by making the models explicit, saying the facts in proper order, putting them in the right context... and on the other hand by checking the students' models, finding the problems and fixing them. If you find out that many students keep making the same mistake, you should adjust your way of teaching accordingly: make it obvious at the beginning that it is X not Y, maybe change the order of lessons so that making the correct model becomes easier. Keep checking the students' models regularly, because the sooner you find the mistake, the easier it is to fix it. Etc. This is what many good tutors do intuitively, because if you teach 1:1, there is more interaction, and it is easier to catch the mistakes right when they happen and ask "why did you do this?" You do not wait until the student makes the same mistake hundred times to declare him a failure without talent; you notice when the mistake happens for the first time and keep "debugging" until the mistake is fixed. This is easy to do when tutoring; more difficult to make it scale to a classroom full of kids. And then... there is another thing, I do not know if it has a proper name in psychology, but "postmodernism" is what some people use (and other people object to this usage)... the "edgy" idea that knowledge transfer is actually impossible, everyone lives in their own different reality, trying to teach something is an oppression, if only we left the kids alone they would reinvent the civilization and make it much better (Rousseau's "Emile"). -- For stupid political reasons ("there are exactly two sides of the story, not more, not less", "the enemy of my enemy is my friend"), these people are typically associated with the constructivists, because they both oppose rote memorization. But although the opponent may be the same, the proposed solutions are quite different ("teaching better" vs "not teaching at all"). To increase the confusion, the educational theory build on this was called "radical constructivism", misleadingly suggesting that this might be "something like the famous Piaget, only much more so", when if fact it is something completely different. The kids taught using this philosophy are left alone to reinvent the math... and fail predictably! Or sometimes they are taught dozen different methods how to do addition (bonus point if the method was used by some indigenous population, because, you know, "noble savage", doesn't matter if the specific method only works for adding 7+8 and 8+9), hoping that this will kickstart their math thinking so now they will develop the rest of the math independently. Predictably, that also never happens. So, how is this related to the "new math"? The curriculum of the "new math" was based exactly on this "radical constructivist" thought, except the authors did not emphasise the "radical" part enough and often just called it "constructivism". So you had the math curriculum that didn't work at all, and was a complete disaster. And when finally people got angry and returned to the traditional math education, the lesson everyone remembered was that "constructivism has been debunked". So now, whenever someone proposes to teach math in a way that emphasizes understanding over memorization, the kneejerk reaction is "haha, that sounds like constructivism... yeah, we tried that but that didn't work at all", plus a link to some web page that criticizes "new math". And if you try to explain how this is completely unrelated to Piaget, you are dismissed with "yeah right, the true constructivism has never been tried, comrades, hahaha". And then, ironically, people keep quoting the Feynman's story about how actually understanding physics is better than mere memorization that "light is waves". And the helpless (Piagetian) construstivist is like "yeah guys, that's exactly what I was trying to tell you all the time", but no one cares, and when it comes back to adding some understanding to the lessons, someone inevitably comes with the condescending "haha, but constructivism has been debunked" and a link to Ten Facts Why New Math Sucks. Possible solution: perhaps the brand of "constructivism" has been thoroughly poisoned, and we need to reinvent it and call it "Feynmanism". Then you can go and say "nope, I am totally not proposing a constructivist curriculum, that has already been tried and debunked, haha, what I am proposing instead is the Feynmanist curriculum", and then people will go online and say "wow, I hated math at school, but then we got a new teacher who used this new Feynmanist method, and the math finally started make sense and now I love it".

sirsinsalot 4y ago

I left school at 15, never to return to education, in part because of how maths was taught. There was never an explanation of the _why_ of things. It was very frustrating, and the teaching was very poor. I ended up a programmer, in part, because I figured if teachers weren't teaching, i'd teach myself. This has been a good strategy for me, however there's serious holes in my pure maths knowledge.

gavinray 4y ago
Lol, I've been kicked out of class for asking the teacher to explain why something works in math here in the US I think it was long division and lattice multiplication in elementary school Doing math by drawing numbers in predefined shapes so that it magically worked out was the most ludicrous thing I'd ever seen "Because that's how it works." Wasn't really a satisfactory answer lol.
- 2OEH8eoCRo0 4y ago
  One of my friends in HS was against calculators, and also very stubborn, and got in trouble for asking how to do sin, cos, tan functions without a calculator.
  - nobodywasishere 4y ago
    Good old taylor series, of course!
    - reacweb 4y ago
      Sine Tables Charts is easier if you can bring them to exam.
      - hgomersall 4y ago
        Just read off the axis on a unit circle. If you need more precision, draw a bigger circle.
    - 4y ago

WalterBright 4y ago

When I remarked yesterday that engineering curriculums that emphasized memorization of engineering formulas rather than understanding how to derive them were inferior, I was dismissed as arrogant and egotistical :-)

mmmpop 4y ago
By whom? An instructor/professor or a student/engineering graduate?
- WalterBright 4y ago
  It's yesterday in my comment history, you shouldn't have any trouble finding it. I prefer people interpret it for themselves as I said my piece in it.
  - BeetleB 4y ago
    Can't speak for the others, but you may come across as too one sided. I had your mindset, and one of the reasons I started getting left behind in grad school whereas others didn't was my stubborn refusal to memorize anything and insisting only on understanding and deriving as needed. Once you go deep enough, you'll often find yourself relying on N random theorems you learned some courses ago to solve a problem, and those who had memorized them were more likely to solve the problem than someone like me who happened to forget the theorem existed in some book and is trying to rederive everything from scratch. Of course, those who mostly or only memorize perform the worst. The other issue is in my other comment: https://news.ycombinator.com/item?id=32388595 https://news.ycombinator.com/item?id=32388595 Basically, most undergrad engineering curricula will not teach you the math needed to properly understand the Laplace transform. So some level of "take it on faith and memorize a few items" are needed. There are other examples of this. Also, having studied under many top class physicists, I can tell you that a significant number of them cannot derive much of the mathematics they use, despite being wizards in applying the techniques.
    - WalterBright 4y ago

dkarl 4y ago

> this is a clear demonstration that a fundamental understanding of fractions was skipped in order to grind out solutions Leaving aside that you concluded this from a one-sentence story about solving a single problem, it's funny that the idea that a "clear understanding of fractions" can make the skills easy to learn is something that virtually every trained teacher has believed for the last (I'm not sure how many) decades and yet nobody alive today is reporting that it happened that way for them. It's possible that every future teacher getting certified to teach politely pretends to believe these things, but secretly wishes to spend their career torturing children, so they abandon the teaching of concepts as soon as they step into a classroom. Or maybe they go through the motions of teaching kids to understand the meaning behind fractions, but they intentionally do it wrong and laugh with each other about it in the teacher's lounge. That's one explanation. Another explanation is that this opposition of "understanding" to "execution" and "concepts" to "skills" is an incomplete way of looking at things that has limited application to how people learn but isn't a panacea, and people still struggle with skills no matter how much you try to prepare them with concepts. I think it does a vast disservice to anyone when we assume that their job performance would be radically improved by the first cliché we ever learned about their field. If the idea is so magical, and we know that it has permeated the field of education for generations, where are the generations of students who remember math as being "trivial, intuitive, beautiful, clever?" Different educational philosophies have come and gone, emphasis has shifted, so where's the generation of parents saying math was much easier for them than for their kids? Where's the PTA meeting where all the parents say, "We think there's something wrong with the way math is taught here, because we all remember fractions being trivial and intuitive, but our kids are finding them hard?" Maybe teachers aren't such idiots that they consistently overlook the value of the most cherished and shopworn truism their field has produced in its history?

thesuitonym 4y ago
I'll start my reply with this: I'm not a teacher, nor am I an understander of maths, but whenever I talk to teachers at any level, and especially mathematics teachers, they lament that they want to teach differently, but due to time constraints, or predetermined lesson plans, or any number of reasons, they simply can't. They have to teach in a way that gets students enough knowledge to pass the test, but only those that seem to have a natural aptitude end up really learning anything. The rest discard the information as soon as it's no longer necessary. I don't think it's a disservice to teachers to say it's a better way of teaching, because they say it themselves. It's just a model that only really works one on one. It can't be done in a room full of 30 bored children, and especially in a room of 400 university sophomores who are ready to get out of class to go to a party.

jorvi 4y ago

Yes. The clearest example of this for me was the fact that many things I was taught in high school were taught to me as an arbitrary concept first, without the accompanying visual relation. Being taught a² + b² = c² with the triangle and accompanying squares drawn out much much later is the best example I can give.

jjice 4y ago

> ...still ended up going down an execution rather than concept focused route. This is something that made me really disappointed when I realized it in school. It was always a step by step process taught to me, and I loved math. I took my AP Calculus class in my senior year of high school and my teacher (fantastic man) showed us how the formula for a derivative is derived. I was blown away. We had just learned and it were told to memorize it the previous year and I had just accepted it. After that, math changed for me. Everything had a reason that was connected to everything else. If you understood where it came from, you could do even more than if you were just given a formula to follow. How beautiful. I realized that I had 5 years (assuming we only cared to do so after algebra) of match education that could have been better taught if done conceptually. I don't blame teachers. You need to get everyone to pass your class for state requirements, so you try to streamline it to get this year done, and not prepare for the future. Also, most math teachers I had did it as a job, not because they loved it, and there's nothing wrong with that. When I had that calculus class during my senior year though, I realized that that man loved math. Hard but fair. I learned so much and that teacher taught me how to develop a work ethic. I wish meta-learning and a more concept based approach could be applied to high school courses (at least looking back). I understand why they aren't, but man would I love to see how it would play out.

jonathanlydall 4y ago
When learning trigonometry in high school (in South Africa, ‘98), we were taught how to use the correct operation for a particular situation and then use a calculator to get the correct answer. I asked the teacher where the numbers which were spat out by the calculator came from and was only told “in my day we didn’t have calculators and had to lookup the answers from a table on a book!”. Which was such a thoroughly disappointing response and I hated trigonometry since it was mostly about memorisation rather than applying logic from understanding. About 10 years later I did a math course through the UK’s Open University and the text book taught trigonometry through explaining the unit circle. It made me so happy to finally understand and see the actual logic as opposed to punching numbers into a calculator, but I was also sad and a bit upset the unit circle was not taught in high school.
- 4y ago

jrmg 4y ago

This is addressed in the article - just after the ‘(7/2) / 2 = 7’ misunderstanding: [I called] Deane Yang, my friend who is a mathematics professor at NYU. “The way you remember procedures is you remember why,” he said. “Because?” “Because people learn math as a collection of procedures,” he said. “When things get difficult, they’re lost, and math becomes religion class. The teacher says what’s right and wrong, and for all you know math came out of the sky, and some prophet told you how to do it, and it’s just blind belief then. Interesting that he wants to know _why_ knowing why helps too! I do wonder if this is covered more in-depth in his book.

rawgabbit 4y ago

I agree. He tried to memorize his way through Algebra. It is hard to understand his mindset because of the stream of consciousness writing. But he appears to be confused with definitions versus equality. E.g., By definition (a+b)^2 is (a+b)(a+b). It is not just "equal".

greymalik 4y ago

What’s the alternative? How does a neophyte go about cultivating the mindset and understanding you describe?

rawgabbit 4y ago
Unfortunately, I don't have an easy answer. My son recently graduated from my high school and I noticed that while his school used Pearson's website to drill him on the mechanics. My son struggled with word problems precisely because he did not have the fundamental concepts. For example, I argued that division is more intuitive when we multiply using fractions. - Whole numbers have an implicit denominator of one. E.g., 7 is equal to 7/1. - It is more intuitive to think of division as multiplying by fractions. E.g., 7/2 is equal to (7/1) x (1/2). We multiply the numerators together. And multiply the denominators together. - Where this is helpful is when we deal with complex arithmetic like (7/2)/2. We have 7 pizzas. Divide by 2. And divide by 2 again. If you break it into fractions, this is equal to (7/1) x (1/2) x (1/2). The numerator is 7x1x1. While the denominator is 1x2x2. Putting it together, it is (7x1x1) / (1x2x2) which is equal to 7/4. This discussion usually leads to the question of what is multiplication (again)? I argued that multiplication can be thought of as a shortcut for addition or can be thought of as set theory. I have two bags of marbles and each bag contains 10 marbles each. How many marbles do I have? You can either write 10+10 (multiplication as a shortcut for addition) or you can write 2x10 (set theory). Taking it back to division. I have two bags of marbles and each bag contains 10 marbles and I have to share my marbles with my sibling (divide by two), how many marbles do I have?. You can write either (10+10)/2 or you can write 2 x 10 x (1/2).
mattwilsonn888 4y ago
This person describes an experience similar to the one which inspired my mindset. https://news.ycombinator.com/threads?id=As_You_Wish https://news.ycombinator.com/threads?id=As_You_Wish

crispyambulance 4y ago

> ...still ended up going down an execution rather than concept focused route. Everyone will say, of course, that "concepts matter". But the reality is that there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables. One can't move on to new concepts in math until the previous dependent concepts have been mastered. Mastery means practice, practice means drills, and all of that is boring but necessary. The best we can do is to make the mathematics more meaningful by tying it to problem solving. At some point many students will experience an "ah ha" realization and be further stimulated, while for some others math will always be meaningless drudgery. For most it will be something in-between.

bslorence 4y ago
> there ARE points in a student's academic path where they HAVE TO memorize stuff and do rote operations like the multiplication tables. Sure, but imagine learning multiplication tables without having any idea what it means to "multiply"; literally just memorizing sequences of symbols, without ever looking at piles of coins or whatever. Multiplication is so basic that this is hard to imagine, but I'm sure that, say, difference-of-squares rules feel like this to most beginning algebra students -- and for most people that probably never changes. I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college.
- BeetleB 4y ago
  > I first encountered difference-of-squares in late middle school, memorized the procedure and used it handily through twelfth grade, and had no idea that there was a visualizable geometric basis to it until I read Book 2 of Euclid's Elements in college. Many mathematicians would disagree with your characterization. For them, the difference of squares is an abstract concept in algebra, and the geometric interpretation is merely a manifestation of it that just happens to work in some domain. As an example, the formula is equally valid for complex numbers, but I doubt you'd get there from Euclid's. No doubt some geometric interpretation can be found for that as well, but then I'd pick some other algebraic field where it's true and you'd have to search yet again for a geometric interpretation.

stakkur 4y ago

I agree, but the fundamental problem is how most western math education is structured top-down to be a goals-driven curriculum. In other words, teaching to the test. In the US, it’s a sprint towards meeting goals tied to government funding (“core curriculum “, etc.) And so many get left behind in that sprint, often feeling they’re ‘not smart enough’ or ‘not a math person’.

photochemsyn 4y ago

I think an introductory course would ideally hop back and forth between the explicit execution of specific examples and the understanding of abstract concepts. As you say the concepts are very important - for example, the author never mentioned the associative, distributive, or commutative rules which apply to all cases of addition and multiplication (but not necessarily to subtraction and division). Those are universally applicable, and indeed understanding them is important for moving on to higher maths (group theory etc.) However, without doing a fair amount of ditch-digging, abstract general theories of ditches might be hard to grasp. For a programming example, consider data structures - lists, trees, graphs, and so on. The abstract concept is clearly what allows larger constructs to be made, i.e. entire programs, but if you've never actually implemented a linked list in any particular programming language, and try to, the result is almost certainly going to be a buggy disaster. The optimal math education might be one where for every hour the instructor spends on the conceptual approach, the student then spends about three hours on the execution approach as applied to specific examples of those concepts (ideally with a teaching assistant around to help if the student gets stuck).

shon 4y ago

This is insightful and rings true for me as well. A good blend of theory and practical knowledge is important to get the most out of any education. Do you have any recommendations for learning math this way?

mattwilsonn888 4y ago
This person describes an experience similar to the one which inspired my mindset. One other thing to note is that the order you might learn math, even the 'bundles' of math confined to the standard classes, is somewhat arbitrary. If you take on the freedom, you are responsible for the direction. That being said the linear path from arithmetic to Calculus or Linear Algebra, or Vector Calculus (to combine the two) will give you very strong fundamentals and chops that will apply broadly. It feels like once you get some understanding of Vector Calculus you can branch off wherever you want, though many advanced topics past that don't strictly require it, learning to derive concepts in that subject is excellent preparation. All that said, there are plenty of interesting topics not involving Calculus or Linear Algebra. I really recommend what As_You_Wish has said as being honest with your own understanding and developing deep rigor, even across just one chapter or one sentence, will be your guide. Once you get deeply acquainted with one area, even if just a tiny area like "I finally understand trig functions," you understand better where you can jump off to next. https://news.ycombinator.com/threads?id=As_You_Wish https://news.ycombinator.com/threads?id=As_You_Wish

User23 4y ago

> you lose the ability to understand math as anything other than a human attempting to be a computer I get what you're saying and it's important, but I also disagree with how you're framing it. It's desirable to get to the point where you can automatically do purely mechanical symbol transformations as you work through a problem, thus saving your higher reasoning for the conceptual problem of searching for which transformations will lead to a solution. Frankly, I find the continuing concern with "intuition" with respect to purely mechanical transformations to be rather medieval. It is slowly changing though as the influence of computing science osmoses back into more traditional mathematics. Using Mathematica or similar tools makes it abundantly clear how distinct the mechanical and conceptual challenges are.

mattwilsonn888 4y ago
I don't believe layers should not be built on top of to the point of rarely, or even never thinking about the inner workings again. I just believe that when building those layers, having an intimate relationship in your mind with how they work and relate to each other will have positive effects, even after you start executing steps without regard for their deeper concepts. That's the case even for just the 'engineer' archetype, who uses math strictly as a tool. For any person doing theoretical work, discovery is often preceded by a fundamental understanding and tweaking of the most basic tools - so I believe the mindset is useful in both cases but borderline necessary for the second.

bowsamic 4y ago

I disagree. My favourite lecturer for general relativity said the important thing is fluency, which comes from repetition. It’s great to know everything, but you should be able to do the basics, such as calculus or algebra, very quickly and without thinking. You will need that brain power for the new stuff. The best way to gain mathematical fluency is by lots of practise.

Viliam1234 4y ago
You can understand something first, and practice for the speed later. Understanding does not prevent practice. I know the result of 8×8 immediately, and I don't think I would ever forget it. But, hypothetically speaking, if I ever made a mistake here, I have the option to slow down and verify the result. If I wouldn't have the option, there is a risk that I would make a mistake and then keep making the same mistake forever, because once the lessons are over, there would be nothing to correct me.

ikrenji 4y ago

it's way easier to understand the concept if you see it repeated many times in action though...