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Learning math? Think like a cartoonist

162 points by freeCandy 5y ago

chrisweekly 5y ago
Just want to say that https://BetterExplained.com https://BetterExplained.com has long been one of my favorite sites; IMHO the more math teachers and students know about it the better. The author's ability to make potentially daunting subjects comprehensible, even intuitive, is a rare and powerful gift.
Zhyl 5y ago
I think the emphasis on this title is 'Learning' as the principles described don't only apply to maths and very much shouldn't be the end point for understanding a concept. Reducing a concept to a 'cartoonish' essence is really the secret to getting an intuition for a new concept, to allow the learner to 'get' it for the first time. The reason is that when one sees something for the first time, its value or meaning isn't apparent. It's just shapes on a page. In other subjects it might be words on a page or a list of events without any throughline or narrative. Reducing something to a 'cartoon' allows you to focus on the orthogonality - the thing that that concept does that can't be done elsewhere and the reason why you should pay attention and find somewhere in your mind palace to put this new thing. But I would emphasise that while the ability to understand something well enough to reduce something to its bare essence for a given audience requires a high level of understanding and mastery, the thing the article skims over somewhat is that for the audience this is only the first step and should be re-enforced and expanded upon almost immediately. BetterExplained as an outlet mostly focus on the pop-explanations sector which is underserved in Mathematics (with 3Blue1Brown and Numberphile doing a lot of heavy lifting) but as a general philosophy of learning, this article needs a bit more development.
necovek 5y ago
In a nutshell, whatever this article suggests, just don't. This oversimplifies to a point where all learners learn the same way. They don't. I don't even see the wider jaw in that cartoonist depiction, and I'd never recognize the man from it: a good cartoon will amplify features of someone, but this is a completely non-existant feature of the face being drawn. If anything, the guy has overly oval face compared to an average face. So basically, based on the wrong premise, it happily leads you to a wrong conclusion. A better take-away would be to attempt to recognize multiple ways to learn something, and make an effort to see what works best for any single individual. If you can't afford that (too time consuming, thus too expensive, to cater to each individual student), choose what you optimize for: having most kids learn to a particular (likely lower) standard, or having "most-compatible" (eg. in maths, those who kinda already have the mathematical, algorithmic, abstract mind) get the best of their talent. But you will be compromising either way.
- savingsPossible 5y ago
  The model recursively self justifies. That is: all models are wrong, some are useful. Including this one. Have you ever taught a class? I have, and I absolutely need simplified (aka wrong, but the right kind of wrong) models of learning to be able to operate at all.
  - necovek 5y ago
    I haven't taught a class, but I don't think that discredits my opinion at all. Why do you think it does? I never said you don't need simplification, but the way article poses it is disingenuous at best. Eg. look at the example of multiplication they give and how it's "technically correct" to call multiplication mostly reducing numbers. Bollocks. Neither "making things bigger" nor "reducing numbers" is technically correct: neither applies. The natural approach to learning multiplication is to work with objects, and thus natural numbers. That's why it's called "multiplication": you make multiples of something. You can easily introduce negative numbers as being in debt for X things, and in that sense, multiplication still only increases your debt. Negative numbers are a shorthand for "you owe me this". The next step is to introduce rational numbers, which are parts of something. That's pretty clear too. Then you introduce multiples of parts of something, and it all still makes sense (quarter of a quarter is now something slightly different). Particularly inclined students will "feel" that you are now entering somewhat abstract "multiplication". Similar goes for real and complex numbers though that requires a bit of leap of faith. This is a proper way to teach multiplication. It starts simple, yet it's never wrong. The premise of the article makes it a come up with a completely unnecessary and untrue statement: "People generally multiply positive numbers greater than 1, so multiplication makes things larger." People mostly multiply natural numbers: this is what drives the intuition and naming of the operation (technically, this still falls under article's statement, but that one is unnecessarily technical in allowing only numbers greater than 1, and allowing rational and real numbers, though I am not sure where complex numbers fit in ;-)). By focusing on technicalities, the article misses the natural way to simplify things which are never wrong. Mathematics today is beautifully built from very simple concepts up. Depending on the level you are teaching at, it requires different levels of suspension of disbelief.
thissiteb1lows 5y ago
Agamus 5y ago
How many other 'mathy' people with aphantasia have trouble 'thinking like a cartoonist'? (And how many people with aphantasia are naturally 'mathy'?) Read Blake Ross' post for a better sense of why this method simply does not work for many people who are more comfortable with abstraction.
- mathstuf 5y ago
  I have aphantasia. For reference, until 2019 or so I thought things like mental imagery and mind palaces were just metaphors and my memory for such things was poor (even after seeing Sherlock put his mind palace to screen). As you can tell from my username, I am mathematically inclined (4th in the state in MathCounts, 2nd place team, a 5 on the AP Calc AB exam in 10th grade, and 5 on the BC exam the next year despite not having a math class that year[1]). I'm actually decent at the folding nets problems ("which cube matches this unfolded pattern?") as well, but I suspect my methods just don't match how those that can visualize and manipulate those visualizations do it. As for the article at hand, there's no real difference to me in terms of what I get out of it (the cartoon gives me a slight smile, but good caricatures usually do) as I only remember the "gist" of such images anyways. I can easily recognize it as the Married with Children actor, but I couldn't begin to verbally describe it in any useful way for even a cartoonist to make without looking at it at the same time (barring "it's a well-known actor": "it's a male face…surprised or 'deer in the headlights', some stubble if any facial hair, short messy hair"). I feel like on some level the aphantasia it is helpful. Namely when expanding some concept such as "equality" to "equivalence" to allow such things as "sum of all natural numbers is -1/12" or modular equivalence relations because I can just replace "is" with "can be treated as" without much mental work. Someone who has a visual representation of such things may have trouble "uprooting" such understandings to plant them in richer soil (a metaphor for which I have no trouble using despite not having a picture of such a thing handy). Do you have a link to the Ross post? [1] Having finished Honors Calc in 10th, I instead tripled up on science classes my junior year and went to a local college for math my senior year.
  - auto 5y ago
    This is interesting, and I'm curious if I could pick your brain a bit. I don't have the math accolades you do, but got through all of grades school math (up thru Calculus) with ease, as just "one of those kids who are good at math", and got my B.Sc. in math without much fuss. I consider myself on the opposite end of the spectrum from someone who suffers (is blessed?) from aphantasia; I can muster up not only a mental image of just about anything, but really recreate any sensory experience (although it has a dullness to it) in my head. This was obviously a huge part of math for me growing up, and as a professional SWE for over a decade now, something I still lean on heavily. What I'm wondering is, how different really are our methods for reasoning about mathematical things, especially the shape rotation/unfolding you mentioned. Do I get to the solution via the mental visualization of the process, or perhaps do you and I perform a similar non-imaged subconscious process of unfolding, but I just "check my work" in a way. If that were the case, I'd argue aphantasia is an advantage, as you're not bothering your brain with unnecessary work. This is all handwavey speculation of course, but I've always found this stuff super interesting.
ar_imani 5y ago
I'm tired of these kinds of titles. They are like news headlines and nothing but a way that maybe it worked for someone. as the adage states: "there is no royal way for learning."
- thesuitonym 5y ago
  If only there were some kind of way the author could elaborate on the title. Some kind of body of content. An article, if you will. But alas, such a thing does not exist, so we must judge everything based on a single sentence.
  - BeetleB 5y ago
    From the HN guidelines: > Don't be snarky.
woolion 5y ago
I like the approach, even though in practice I would not necessarily follow the suggested presentation. From my experience, most math courses start with the "dead" book definitions. Which is not the way mathematics are built: people start with intuitive but incorrect ideas, the idea is proven to be useful, and only then made correct. The definitions end up coming from refinements to avoid contradictions arising from edge-cases. This was a common reproach to the Bourbaki group who wanted to formalize mathematics into an almost computer-digestible form; I think it was Grothendieck who said they were "embalming mathematics". This is absolutely not a good way to learn mathematics, although it is the lowest effort to come up with. I generally think it is much better to go through what do we want to achieve, how it came to be, etc. Which you can generally find with presentation of the Seven Bridges of Königsberg problem, some game mathematics (Rubik's cube, etc), or "You could have invented..." type articles (the famous one being for "spectral sequences"). Although there should be different approaches for different learning types. But this is another problem, which is more linked to the (bad) educational structure.
- neel_k 5y ago
  It's worth understanding the context Bourbaki arose in. An entire generation of French mathematicians was turned to bits of blood and gristle in the trenches of World War I, and so French mathematicians in the 1920s and early 1930s faced an acute shortage of teachers who were current with modern mathematics. The premise of the Bourbaki effort was to write everything down in enough detail that a sufficiently motivated reader could learn it without having to learn it master-apprentice style -- because too many potential masters were dead.
  - woolion 5y ago
    In that case, that means they have entirely failed. The very few people I've met who actually bothered with the books only made fun of how horrible they were to read and understand. I never even did open one myself! I always admired the principle though, because you can reduce everything to pure logic (and even in some cases you can "brute-force" formalism to obtain new results). Which makes me think category theory kind of fills this role in a better way?
t43562 5y ago
I think the problem I have with maths is that at the point where I start to realise that the simplifications are wrong, I lose trust in every area of my understanding and want to give up. I can't be comfortable feeling that there's a missing hole in my mental model. OTOH, it doesn't work at all to give exhaustive mathematical proof because it doesn't construct a model in my head. So I don't know what the solution is really. To take complex numbers for example: the idea that -1 has a square root seemed like utter bullshit as it contradicts other models one has had drummed into one. If you look at it as a means to an end, however, it's a useful little bit of machinery that can help to make other problems easier to solve (e.g. converting a differential equation to a quadratic). That makes sense and explains why we're doing it whereas talking about SQRT(-1) starts one off with the feeling that everything is nonsense.
goldenkey 5y ago
Multiplication only makes numbers smaller if we restrict ourselves to starting with a positive number. If we instead start with a negative number, multiplication will make the number bigger. If we allow for any integer, multiplication on average, will neither be biased one way or another. It's a bit of a carny trick to hide the bias in the selection of only positive numbers, and then act like multiplication itself has the issue.. This leads me to believe the author doesn't know math very well and their learning techniques are not efficient or worth giving credence to. If you give an analogy to explain a concept, it shouldn't be fast one, it should be honest!
- djmips 5y ago
  Sigh...
photochemsyn 5y ago
When we talk about 'learning mathematics' it's important to recognize a huge difference between learning how to use mathematical discoveries/inventions, and learning how to develop mathematics from scratch, via the whole theorem -> proof -> new theorem route that defines 'pure mathematics'. Most people are simply not going to get a lot out of learning the latter method and will indeed be turned off by it, much to the disappointment of the professional mathematicians (i.e. most college professors in maths). I'd guess > 95% of people taking higher maths courses are not going to ever develop new proofs - but they will use what they've learned in other areas, such as physics, biostatistics, finance, etc. Essentially we just take it on faith that the mathematicians got their proofs right, and we gratefully use the fruits of their labor. (They're all quite mad, those mathematicians, if you ask me) Now, when you first learn how to apply maths to things like physical problems, this is where tyhe cartoons, or 'simple approximations neglecting complex factors' becomes really important to learning. You don't want to try to include friction when first examining falling weights and springs and pendulums through a physical viewpoint, for example. Later on, when you get that job with SpaceX, understanding friction in depth will be critically important, but if you don't start with the simple cartoon approximations, it'll be way too much to comprehend. However, this probably wouldn't work for the real mathematicians. They've got their axioms, then from the axioms they develop proofs, then from those proofs they develop more proofs - there's no approximation or simplification involved in all that, is there?
- necovek 5y ago
  The only problem with this dualism is that it really isn't so. The way mathematics is developed is the same way it should be learned and applied. Your geometry class today does not start with Euclid's axioms, but instead by developing intuition for concepts in the simplest way possible (2D plane, 1D lines), playing around with them, and then formalizing them (which means defining them rigorously to the very last detail, to ensure eg. your concept of a straight line is always a straight line), and generalizing what you learned about them (stating it as theorems). The same is with numbers: I've yet to hear of people starting their arithmetic training with Peano's axioms, yet it's common to teach numbers by pointing at one object and moving another one next to it, and claiming that's now a "two"; adding another one to that gives you a "three" (successor() function, right?). Sure, most primary education skips this playing around with concepts, and fails to demonstrate failings of not being precise enough, but they are all trivial to demonstrate. But all of those are what is exactly used in formal/pure math development, and exactly the same processes are used when applying mathematical insights anywhere else. There is no difference at all! The level of rote memorization and combinatorial explosion rises as you go into more advanced math with more abstract concepts to deal with (which is what I think people really struggle with, since it's a different language, and then intelligence does come into play as well as you start dealing with bigger complexity), but I like to say that math is the simplest form of expression of the human brain, but no simpler. It has well-defined boundaries of our comprehension too (we need to assume some things are true without proving them, aka axioms and base terms). The biggest problem of all math instruction today is that it's mostly performed by those not understanding the simplicity of it, and thus the only thing they can do is transfer their misunderstanding on to their students. This includes parents who teach their kids basic concepts like basic geometric shapes or numbers. Another big problem is that people simply fail to realize how much of rote memorization is required for understanding mathematics. Being a different language, you have to learn all the "new" words and grammar to be able to effectively apply it, and people usually discount that as not important, and how only understanding mathematics is hard because of the abstract nature.
xg15 5y ago
I think the bit about the definition of multiplication could be applied to code as well: A source file necessarily contains all possible logic flows, no matter if a flow represents the main operation, an obscure special case or parts that are completely unused or practically unreachable. This can easily hide the "gist" of a function among heaps of clutter. I think some tool would be interesting that took a code traces or coverage information and generated a view of a code file where each line has a font size dependant on how often that line was executed. Ideally, "important" lines would end up with a large font, while special cases, error checks etc would become "fine print". I'd be curious if this would help making codebases easier to understand.
- WanderPanda 5y ago
  That is exactly what profilers do. Xcode gives you a heatmap of what percentage of the execution time has been spent in a particular function (at least in ~2013 when I used this function the last time)
  - xg15 5y ago
    True, but I haven't seen profiling results actually applied back to the actual code file, line-by-line.
feoren 5y ago
> We agree that multiplication makes things bigger, right? Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! > Imaginary numbers let us rotate numbers. Kind of ... it's useful to visualize multiplication by imaginary numbers as analogous to rotation, but this makes it sound like imaginary numbers are useful because they let us "rotate numbers", like that's something we always wanted to do. This is neither the essence of nor the impetus behind imaginary numbers. > The number e is a little machine that grows as fast as it can No no no! We happen to use e as an exponential base a lot because it's convenient, but it's exponentiation, not e, that grows quickly! f(x) = 4^x grows faster than e^x does; e^-x shrinks; e^0 stays where it is ... this is a horrible intuition to have! > The Pythagorean Theorem explains how all shapes behave (not just triangles) It works on other shapes because you draw triangles in those shapes ... These are awful, awful intuitions about math. We should not take advice about how to learn math from someone who so deeply struggles with basic concepts about it! Edit: this article is so bad that I just have to add another example: > Euler’s Formula makes a circular path. But you just said that e is a little machine that grows as fast as it can! How can it just spin in a circle!? His own intuitions are completely inconsistent! Besides, Euler's Formula has no free variables ...
- twojacobtwo 5y ago
  You seem to be specifically looking to find a frame in which the author is incorrect, which is basically a sign of the problem he's getting at in the article - the inability to allow ideas to be reduced to 'cartoonish' principles in order to facilitate a different (more engaging?) method of learning math, at the expense of being "technically correct". All your points seem to be making the same type of argument, so I'll just address the first one from my perspective, with the author's points in mind. > Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! The model definitely worked for me when I first encountered it and it worked for many of my early friends (a couple of whom are now accredited mathematicians). The other cases surrounding multiplication revealed themselves naturally, as I encountered multiplication by 0, multiplication by negatives, etc. (exceptions/cases noted by the author). The basic model gave me a way to think about how I could put the tool (multiplication, in this case) to use without being overwhelming and my self-driven discovery of the unexpected parts made me much more interested in what else I might be missing than if someone had given me a full, technically correct definition from the get-go. The key is that this is specifically aimed at those who are learning, not those who are 'learned'.
eternityforest 5y ago
I don't see what's wrong with "Multiplication copies things". The fact that "copying" can be a continual scale from 10 copies to 1 copy down to half a copy and zero copies captures most of the behavior that a non-math person working as a programmer(One of the main uses for math, right next to using it to learn other math and becoming a math-heavy coder) is interested in. The main actual uses of multiplication before you get to advanced stuff are things like volume controls, total price of N units, area of a square, etc, that are pretty much copying. Except Ohm's law. That doesn't quite fit intuitive models of math exactly, and when you actually go to use it you usually wind up wanting to know power dissipation which is nonlinear in a very bizzare multi-variable way with some IRL use cases, since you're usually looking at systems as a whole.
nagonago 5y ago
Mathematics aside, am I the only one who finds this style of cartoon caricature unappealing? I find that these kinds of caricatures often bear little resemblance to the actual person, as well as being simply unpleasant to look at. In this instance particularly: > Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect. I honestly don't understand this point. Looking at the original image, his head and jaw seem perfectly average to me. I understand that caricaturists love to exaggerate features, but why praise them for amplifying a feature that does not really exist?

162 points by freeCandy 5y ago

Just want to say that https://BetterExplained.com https://BetterExplained.com has long been one of my favorite sites; IMHO the more math teachers and students know about it the better. The author's ability to make potentially daunting subjects comprehensible, even intuitive, is a rare and powerful gift.

Zhyl 5y ago

I think the emphasis on this title is 'Learning' as the principles described don't only apply to maths and very much shouldn't be the end point for understanding a concept. Reducing a concept to a 'cartoonish' essence is really the secret to getting an intuition for a new concept, to allow the learner to 'get' it for the first time. The reason is that when one sees something for the first time, its value or meaning isn't apparent. It's just shapes on a page. In other subjects it might be words on a page or a list of events without any throughline or narrative. Reducing something to a 'cartoon' allows you to focus on the orthogonality - the thing that that concept does that can't be done elsewhere and the reason why you should pay attention and find somewhere in your mind palace to put this new thing. But I would emphasise that while the ability to understand something well enough to reduce something to its bare essence for a given audience requires a high level of understanding and mastery, the thing the article skims over somewhat is that for the audience this is only the first step and should be re-enforced and expanded upon almost immediately. BetterExplained as an outlet mostly focus on the pop-explanations sector which is underserved in Mathematics (with 3Blue1Brown and Numberphile doing a lot of heavy lifting) but as a general philosophy of learning, this article needs a bit more development.

necovek 5y ago

In a nutshell, whatever this article suggests, just don't. This oversimplifies to a point where all learners learn the same way. They don't. I don't even see the wider jaw in that cartoonist depiction, and I'd never recognize the man from it: a good cartoon will amplify features of someone, but this is a completely non-existant feature of the face being drawn. If anything, the guy has overly oval face compared to an average face. So basically, based on the wrong premise, it happily leads you to a wrong conclusion. A better take-away would be to attempt to recognize multiple ways to learn something, and make an effort to see what works best for any single individual. If you can't afford that (too time consuming, thus too expensive, to cater to each individual student), choose what you optimize for: having most kids learn to a particular (likely lower) standard, or having "most-compatible" (eg. in maths, those who kinda already have the mathematical, algorithmic, abstract mind) get the best of their talent. But you will be compromising either way.

savingsPossible 5y ago
The model recursively self justifies. That is: all models are wrong, some are useful. Including this one. Have you ever taught a class? I have, and I absolutely need simplified (aka wrong, but the right kind of wrong) models of learning to be able to operate at all.
- necovek 5y ago
  I haven't taught a class, but I don't think that discredits my opinion at all. Why do you think it does? I never said you don't need simplification, but the way article poses it is disingenuous at best. Eg. look at the example of multiplication they give and how it's "technically correct" to call multiplication mostly reducing numbers. Bollocks. Neither "making things bigger" nor "reducing numbers" is technically correct: neither applies. The natural approach to learning multiplication is to work with objects, and thus natural numbers. That's why it's called "multiplication": you make multiples of something. You can easily introduce negative numbers as being in debt for X things, and in that sense, multiplication still only increases your debt. Negative numbers are a shorthand for "you owe me this". The next step is to introduce rational numbers, which are parts of something. That's pretty clear too. Then you introduce multiples of parts of something, and it all still makes sense (quarter of a quarter is now something slightly different). Particularly inclined students will "feel" that you are now entering somewhat abstract "multiplication". Similar goes for real and complex numbers though that requires a bit of leap of faith. This is a proper way to teach multiplication. It starts simple, yet it's never wrong. The premise of the article makes it a come up with a completely unnecessary and untrue statement: "People generally multiply positive numbers greater than 1, so multiplication makes things larger." People mostly multiply natural numbers: this is what drives the intuition and naming of the operation (technically, this still falls under article's statement, but that one is unnecessarily technical in allowing only numbers greater than 1, and allowing rational and real numbers, though I am not sure where complex numbers fit in ;-)). By focusing on technicalities, the article misses the natural way to simplify things which are never wrong. Mathematics today is beautifully built from very simple concepts up. Depending on the level you are teaching at, it requires different levels of suspension of disbelief.

thissiteb1lows 5y ago

Agamus 5y ago

How many other 'mathy' people with aphantasia have trouble 'thinking like a cartoonist'? (And how many people with aphantasia are naturally 'mathy'?) Read Blake Ross' post for a better sense of why this method simply does not work for many people who are more comfortable with abstraction.

mathstuf 5y ago
I have aphantasia. For reference, until 2019 or so I thought things like mental imagery and mind palaces were just metaphors and my memory for such things was poor (even after seeing Sherlock put his mind palace to screen). As you can tell from my username, I am mathematically inclined (4th in the state in MathCounts, 2nd place team, a 5 on the AP Calc AB exam in 10th grade, and 5 on the BC exam the next year despite not having a math class that year[1]). I'm actually decent at the folding nets problems ("which cube matches this unfolded pattern?") as well, but I suspect my methods just don't match how those that can visualize and manipulate those visualizations do it. As for the article at hand, there's no real difference to me in terms of what I get out of it (the cartoon gives me a slight smile, but good caricatures usually do) as I only remember the "gist" of such images anyways. I can easily recognize it as the Married with Children actor, but I couldn't begin to verbally describe it in any useful way for even a cartoonist to make without looking at it at the same time (barring "it's a well-known actor": "it's a male face…surprised or 'deer in the headlights', some stubble if any facial hair, short messy hair"). I feel like on some level the aphantasia it is helpful. Namely when expanding some concept such as "equality" to "equivalence" to allow such things as "sum of all natural numbers is -1/12" or modular equivalence relations because I can just replace "is" with "can be treated as" without much mental work. Someone who has a visual representation of such things may have trouble "uprooting" such understandings to plant them in richer soil (a metaphor for which I have no trouble using despite not having a picture of such a thing handy). Do you have a link to the Ross post? [1] Having finished Honors Calc in 10th, I instead tripled up on science classes my junior year and went to a local college for math my senior year.
- auto 5y ago
  This is interesting, and I'm curious if I could pick your brain a bit. I don't have the math accolades you do, but got through all of grades school math (up thru Calculus) with ease, as just "one of those kids who are good at math", and got my B.Sc. in math without much fuss. I consider myself on the opposite end of the spectrum from someone who suffers (is blessed?) from aphantasia; I can muster up not only a mental image of just about anything, but really recreate any sensory experience (although it has a dullness to it) in my head. This was obviously a huge part of math for me growing up, and as a professional SWE for over a decade now, something I still lean on heavily. What I'm wondering is, how different really are our methods for reasoning about mathematical things, especially the shape rotation/unfolding you mentioned. Do I get to the solution via the mental visualization of the process, or perhaps do you and I perform a similar non-imaged subconscious process of unfolding, but I just "check my work" in a way. If that were the case, I'd argue aphantasia is an advantage, as you're not bothering your brain with unnecessary work. This is all handwavey speculation of course, but I've always found this stuff super interesting.

ar_imani 5y ago

I'm tired of these kinds of titles. They are like news headlines and nothing but a way that maybe it worked for someone. as the adage states: "there is no royal way for learning."

thesuitonym 5y ago
If only there were some kind of way the author could elaborate on the title. Some kind of body of content. An article, if you will. But alas, such a thing does not exist, so we must judge everything based on a single sentence.
- BeetleB 5y ago
  From the HN guidelines: > Don't be snarky.

woolion 5y ago

I like the approach, even though in practice I would not necessarily follow the suggested presentation. From my experience, most math courses start with the "dead" book definitions. Which is not the way mathematics are built: people start with intuitive but incorrect ideas, the idea is proven to be useful, and only then made correct. The definitions end up coming from refinements to avoid contradictions arising from edge-cases. This was a common reproach to the Bourbaki group who wanted to formalize mathematics into an almost computer-digestible form; I think it was Grothendieck who said they were "embalming mathematics". This is absolutely not a good way to learn mathematics, although it is the lowest effort to come up with. I generally think it is much better to go through what do we want to achieve, how it came to be, etc. Which you can generally find with presentation of the Seven Bridges of Königsberg problem, some game mathematics (Rubik's cube, etc), or "You could have invented..." type articles (the famous one being for "spectral sequences"). Although there should be different approaches for different learning types. But this is another problem, which is more linked to the (bad) educational structure.

neel_k 5y ago
It's worth understanding the context Bourbaki arose in. An entire generation of French mathematicians was turned to bits of blood and gristle in the trenches of World War I, and so French mathematicians in the 1920s and early 1930s faced an acute shortage of teachers who were current with modern mathematics. The premise of the Bourbaki effort was to write everything down in enough detail that a sufficiently motivated reader could learn it without having to learn it master-apprentice style -- because too many potential masters were dead.
- woolion 5y ago
  In that case, that means they have entirely failed. The very few people I've met who actually bothered with the books only made fun of how horrible they were to read and understand. I never even did open one myself! I always admired the principle though, because you can reduce everything to pure logic (and even in some cases you can "brute-force" formalism to obtain new results). Which makes me think category theory kind of fills this role in a better way?

t43562 5y ago

I think the problem I have with maths is that at the point where I start to realise that the simplifications are wrong, I lose trust in every area of my understanding and want to give up. I can't be comfortable feeling that there's a missing hole in my mental model. OTOH, it doesn't work at all to give exhaustive mathematical proof because it doesn't construct a model in my head. So I don't know what the solution is really. To take complex numbers for example: the idea that -1 has a square root seemed like utter bullshit as it contradicts other models one has had drummed into one. If you look at it as a means to an end, however, it's a useful little bit of machinery that can help to make other problems easier to solve (e.g. converting a differential equation to a quadratic). That makes sense and explains why we're doing it whereas talking about SQRT(-1) starts one off with the feeling that everything is nonsense.

goldenkey 5y ago

Multiplication only makes numbers smaller if we restrict ourselves to starting with a positive number. If we instead start with a negative number, multiplication will make the number bigger. If we allow for any integer, multiplication on average, will neither be biased one way or another. It's a bit of a carny trick to hide the bias in the selection of only positive numbers, and then act like multiplication itself has the issue.. This leads me to believe the author doesn't know math very well and their learning techniques are not efficient or worth giving credence to. If you give an analogy to explain a concept, it shouldn't be fast one, it should be honest!

djmips 5y ago
Sigh...

photochemsyn 5y ago

When we talk about 'learning mathematics' it's important to recognize a huge difference between learning how to use mathematical discoveries/inventions, and learning how to develop mathematics from scratch, via the whole theorem -> proof -> new theorem route that defines 'pure mathematics'. Most people are simply not going to get a lot out of learning the latter method and will indeed be turned off by it, much to the disappointment of the professional mathematicians (i.e. most college professors in maths). I'd guess > 95% of people taking higher maths courses are not going to ever develop new proofs - but they will use what they've learned in other areas, such as physics, biostatistics, finance, etc. Essentially we just take it on faith that the mathematicians got their proofs right, and we gratefully use the fruits of their labor. (They're all quite mad, those mathematicians, if you ask me) Now, when you first learn how to apply maths to things like physical problems, this is where tyhe cartoons, or 'simple approximations neglecting complex factors' becomes really important to learning. You don't want to try to include friction when first examining falling weights and springs and pendulums through a physical viewpoint, for example. Later on, when you get that job with SpaceX, understanding friction in depth will be critically important, but if you don't start with the simple cartoon approximations, it'll be way too much to comprehend. However, this probably wouldn't work for the real mathematicians. They've got their axioms, then from the axioms they develop proofs, then from those proofs they develop more proofs - there's no approximation or simplification involved in all that, is there?

necovek 5y ago
The only problem with this dualism is that it really isn't so. The way mathematics is developed is the same way it should be learned and applied. Your geometry class today does not start with Euclid's axioms, but instead by developing intuition for concepts in the simplest way possible (2D plane, 1D lines), playing around with them, and then formalizing them (which means defining them rigorously to the very last detail, to ensure eg. your concept of a straight line is always a straight line), and generalizing what you learned about them (stating it as theorems). The same is with numbers: I've yet to hear of people starting their arithmetic training with Peano's axioms, yet it's common to teach numbers by pointing at one object and moving another one next to it, and claiming that's now a "two"; adding another one to that gives you a "three" (successor() function, right?). Sure, most primary education skips this playing around with concepts, and fails to demonstrate failings of not being precise enough, but they are all trivial to demonstrate. But all of those are what is exactly used in formal/pure math development, and exactly the same processes are used when applying mathematical insights anywhere else. There is no difference at all! The level of rote memorization and combinatorial explosion rises as you go into more advanced math with more abstract concepts to deal with (which is what I think people really struggle with, since it's a different language, and then intelligence does come into play as well as you start dealing with bigger complexity), but I like to say that math is the simplest form of expression of the human brain, but no simpler. It has well-defined boundaries of our comprehension too (we need to assume some things are true without proving them, aka axioms and base terms). The biggest problem of all math instruction today is that it's mostly performed by those not understanding the simplicity of it, and thus the only thing they can do is transfer their misunderstanding on to their students. This includes parents who teach their kids basic concepts like basic geometric shapes or numbers. Another big problem is that people simply fail to realize how much of rote memorization is required for understanding mathematics. Being a different language, you have to learn all the "new" words and grammar to be able to effectively apply it, and people usually discount that as not important, and how only understanding mathematics is hard because of the abstract nature.

xg15 5y ago

I think the bit about the definition of multiplication could be applied to code as well: A source file necessarily contains all possible logic flows, no matter if a flow represents the main operation, an obscure special case or parts that are completely unused or practically unreachable. This can easily hide the "gist" of a function among heaps of clutter. I think some tool would be interesting that took a code traces or coverage information and generated a view of a code file where each line has a font size dependant on how often that line was executed. Ideally, "important" lines would end up with a large font, while special cases, error checks etc would become "fine print". I'd be curious if this would help making codebases easier to understand.

WanderPanda 5y ago
That is exactly what profilers do. Xcode gives you a heatmap of what percentage of the execution time has been spent in a particular function (at least in ~2013 when I used this function the last time)
- xg15 5y ago
  True, but I haven't seen profiling results actually applied back to the actual code file, line-by-line.

feoren 5y ago

> We agree that multiplication makes things bigger, right? Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! > Imaginary numbers let us rotate numbers. Kind of ... it's useful to visualize multiplication by imaginary numbers as analogous to rotation, but this makes it sound like imaginary numbers are useful because they let us "rotate numbers", like that's something we always wanted to do. This is neither the essence of nor the impetus behind imaginary numbers. > The number e is a little machine that grows as fast as it can No no no! We happen to use e as an exponential base a lot because it's convenient, but it's exponentiation, not e, that grows quickly! f(x) = 4^x grows faster than e^x does; e^-x shrinks; e^0 stays where it is ... this is a horrible intuition to have! > The Pythagorean Theorem explains how all shapes behave (not just triangles) It works on other shapes because you draw triangles in those shapes ... These are awful, awful intuitions about math. We should not take advice about how to learn math from someone who so deeply struggles with basic concepts about it! Edit: this article is so bad that I just have to add another example: > Euler’s Formula makes a circular path. But you just said that e is a little machine that grows as fast as it can! How can it just spin in a circle!? His own intuitions are completely inconsistent! Besides, Euler's Formula has no free variables ...

twojacobtwo 5y ago
You seem to be specifically looking to find a frame in which the author is incorrect, which is basically a sign of the problem he's getting at in the article - the inability to allow ideas to be reduced to 'cartoonish' principles in order to facilitate a different (more engaging?) method of learning math, at the expense of being "technically correct". All your points seem to be making the same type of argument, so I'll just address the first one from my perspective, with the author's points in mind. > Huh!? I have luckily never heard anyone try to explain multiplication that way before. It's a horrible mental model for multiplication! The model definitely worked for me when I first encountered it and it worked for many of my early friends (a couple of whom are now accredited mathematicians). The other cases surrounding multiplication revealed themselves naturally, as I encountered multiplication by 0, multiplication by negatives, etc. (exceptions/cases noted by the author). The basic model gave me a way to think about how I could put the tool (multiplication, in this case) to use without being overwhelming and my self-driven discovery of the unexpected parts made me much more interested in what else I might be missing than if someone had given me a full, technically correct definition from the get-go. The key is that this is specifically aimed at those who are learning, not those who are 'learned'.

eternityforest 5y ago

I don't see what's wrong with "Multiplication copies things". The fact that "copying" can be a continual scale from 10 copies to 1 copy down to half a copy and zero copies captures most of the behavior that a non-math person working as a programmer(One of the main uses for math, right next to using it to learn other math and becoming a math-heavy coder) is interested in. The main actual uses of multiplication before you get to advanced stuff are things like volume controls, total price of N units, area of a square, etc, that are pretty much copying. Except Ohm's law. That doesn't quite fit intuitive models of math exactly, and when you actually go to use it you usually wind up wanting to know power dissipation which is nonlinear in a very bizzare multi-variable way with some IRL use cases, since you're usually looking at systems as a whole.

nagonago 5y ago

Mathematics aside, am I the only one who finds this style of cartoon caricature unappealing? I find that these kinds of caricatures often bear little resemblance to the actual person, as well as being simply unpleasant to look at. In this instance particularly: > Technically, his head is an oval, like yours. But somehow, making his jaw wider than the rest of his head is perfect. I honestly don't understand this point. Looking at the original image, his head and jaw seem perfectly average to me. I understand that caricaturists love to exaggerate features, but why praise them for amplifying a feature that does not really exist?