[HN Gopher] Learning math? Think like a cartoonist
___________________________________________________________________
Learning math? Think like a cartoonist
Author : freeCandy
Score : 150 points
Date : 2022-04-01 08:48 UTC (3 days ago)
(HTM) web link (betterexplained.com)
(TXT) w3m dump (betterexplained.com)
| Zhyl wrote:
| 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.
| chrisweekly wrote:
| Just want to say that 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.
| Agamus wrote:
| 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 wrote:
| 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 wrote:
| 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.
| mathstuf wrote:
| > I consider myself on the opposite end of the spectrum
| from someone who suffers (is blessed?) from aphantasia
|
| One distinct downside is that I'm queasy around _other_
| people 's blood or gore. I can't watch visceral movie
| scenes for too long or visit some science exhibits (that
| I'd otherwise enjoy) because of it (at least this is my
| suspicion). My problem is that because I can't _visualize_
| a discussion about, say, amputation, my mind grafts the
| description _onto me_ and things go downhill from there.
| However, I was able to watch some small procedures on
| myself just fine because I could _see_ that it was OK
| (though I still cannot watch my own blood being drawn).
|
| > 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.
|
| Yeah, I don't have anything to compare that to. I recall
| "gists" of things through feelings and notions rather than
| anything I'd call "sensory". I cannot recalls scents or
| tastes all that well and even sounds get muddled up very
| easily (I have a _very_ hard time finding a rhythm, melody,
| or other patterns out of music though I can usually pick
| out instruments while listening; "tap out this song"
| detectors never worked well for me because I would tap
| nearly every note and lyric syllable, not some "core"
| pattern from it).
|
| Some interesting side effects of this include a blank stare
| while I think about a list of items when someone asks
| something like "what is your favorite food or drink?". The
| answer sometimes changes depending on whether I remember
| "everything" too. Even "what do you think about your
| travels to X?" brings it up because I have to first conjure
| up the feelings associated with that travel, remember the
| big events in that timeline, and then put it into words for
| the answer (hoping I didn't miss anything).
|
| > What I'm wondering is, how different really are our
| methods for reasoning about mathematical things, especially
| the shape rotation/unfolding you mentioned.
|
| The first thing I notice is that I do not "unfold" the
| possibilities at all; I turn it into a logic puzzle. I
| focus on one face that is in "most" of the images and look
| for contradictions with the net by focusing on one or two
| vertices of that face. Some rules come out of this like
| "two steps in one axis is an opposing face" and if you see
| those two symbols on one possibility, it's obviously not a
| candidate. It also works fine for the harder versions where
| the symbols _on_ the face are right, but rotated
| incorrectly to the facing edge. This usually gets you down
| to one candidate and you can move on to the next one while
| your hand is filling in the bubble. I suspect (but have no
| actual experience with it as most of such tests are on
| cubes) this scales up well for even octahedra and
| dodecahedra (but starts to fall apart trying to keep the
| order /orientation of the 5 items around a vertex for an
| icosahedron or when the possibilities do not share many
| symbols from their view because that greatly slows down the
| method). Giving "adversarial" nets where the answer is
| actually based on two distant legs of the net that I need
| to actually mentally fold multiple edges also likely
| stymies my strategy because I have to think "star circle
| square" as I walk the edges and cannot just "snap" a
| picture of the sequence as a gestalt as easily.
|
| I took part in a research experiment recently where they
| have some 3d shapes made of cubes rotated around a vertical
| axis and ask "are these two objects the same or mirror
| images of each other". I obviously don't have the answer
| key, but I felt that I did really well _except_ when mirror
| images were rotated 180deg from each other. I don 't know
| if I have an explanation for that (usually I realized my
| mistake right after answering).
|
| > If that were the case, I'd argue aphantasia is an
| advantage, as you're not bothering your brain with
| unnecessary work.
|
| I think a major difference is in working memory. What you
| might be able to "freeze frame" as a collection, I have to
| extract out the distinct parts and remember them more
| individually. Sometimes I can "find" a pattern and gestalt
| that way which tends to stick quite well (like I do for my
| wife's phone number or a friend's phone number I saw a
| pattern in once he got it), but most things are just rote
| memorization.
|
| For example, "star square circle" could just be "decreasing
| side counts" and just using recognition to determine
| membership later versus "star circle square" which is more
| likely to be 3 working items. If they are also colored,
| that doubles the work for me right away as they otherwise
| don't have _any_ color association to me.
| auto wrote:
| Thanks for the in depth writeup, this is all super
| interesting, and lo and behold it sounds like the answer
| to what I was poking at was "it depends" and "there is
| nuance", which is honestly super neat. Just more fuel for
| the fire of "the human brain is fantastically complex and
| we know so little about it", the fact that the two of us
| could have such different experiences and mechanisms,
| that ultimately serve similar purposes; move a human
| being through their lives, while keeping them alive and
| "well".
| Agamus wrote:
| I agree - I find it to be incredibly valuable in almost all
| contexts.
|
| Someone linked to Ross' post in this 'Tech Times' article: ht
| tps://www.techtimes.com/articles/154165/20160427/mozilla-f...
|
| The original Facebook post is presumably here:
| https://www.facebook.com/notes/blake-ross/aphantasia-how-
| it-...
|
| When I read his original post and first learned about
| aphantasia, realized that most people have a functioning
| 'mind's eye', it was one of the weirdest days of my life.
| mathstuf wrote:
| I don't know if I could classify it as "incredibly valuable
| in almost all contexts". I have no idea what it's like to
| have such imagery for comparison. I dream, but cannot
| remember them beyond tiny details. I've had those nights
| where I wake up really wanting to remember a dream, but
| it's like trying to hold water in my palm. However, I also
| know when my eyes are closed because it's just pitch black
| (though I can tell if there's a bright light and some
| direction), so I know when dreaming is over quite easily.
|
| I wonder if some of my love of word play comes from being
| able to just "reorient" around a concept easily. I can also
| read upside down or in a mirror (but not if both transforms
| are applied) at reasonable speed. The former was quite
| handy in groupwork in school because I could just sit on
| the other side of the table or desk and follow along
| instead of crowding around.
|
| > When I read his original post and first learned about
| aphantasia, realized that most people have a functioning
| 'mind's eye', it was one of the weirdest days of my life.
|
| The strangest thing I found is that, with hindsight, the
| clues are everywhere. Functioning "mind palaces",
| eyewitness testimony, the ability to even slightly help a
| sketch artist. I remember being in a yoga class and they
| told us to imagine "three circles around yourself". I was
| thinking of the circles orbiting like moons; apparently
| everyone else had them as concentric circles centered on
| themselves. I could easily flip between them, but others
| thought it weird how I "saw" it at first.
| djmips wrote:
| Everyone here, including the original poster has, I believe,
| made a mistake of thinking the article meant to literally
| visualize a problem like a cartoonist. The cartoon caricature
| is only meant by analogy. Possibly a blind spot for those with
| aphantasia, but the thesis of this article isn't centered
| around literally visualizing something but capturing some
| salient points, perhaps exaggerated and not technically correct
| as an initial way of starting to explain a mathematical
| concept. It's perhaps unfortunate that the analogy chosen was
| difficult to understand or sidetracked people that have trouble
| with the process of making a cartoon caricature.
| ginnungagap wrote:
| I have aphantasia and I'm doing a PhD in math.
|
| I draw on paper when I'm trying to visualize something, but the
| area I work in is quite abstract and very often the images are
| huge oversimplifications that give some intuition, but on which
| one shouldn't rely too much.
|
| I agree that the suggested method doesn't sound like a good
| approach at all to me
| eternityforest wrote:
| 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.
| goldenkey wrote:
| 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 wrote:
| Sigh...
| xg15 wrote:
| 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 wrote:
| 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 wrote:
| True, but I haven't seen profiling results actually applied
| back to the actual code file, line-by-line.
| ar_imani wrote:
| 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 wrote:
| 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 wrote:
| From the HN guidelines:
|
| > Don't be snarky.
| necovek wrote:
| 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 wrote:
| 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 wrote:
| 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.
| t43562 wrote:
| 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.
| woolion wrote:
| 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
| Konigsberg 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.
| jsmith45 wrote:
| Yeah, I must say I prefer the math teaching approach of:
|
| 1. Explain some problem that is difficult to solve using other
| techniques. Ideally the problem should be interesting, even if
| potentially somewhat abstract.
|
| 2. Propose a technique to solve it, without requiring full
| rigor, but that allows for intuitively feeling that the
| technique is probably valid.
|
| 3. Show that the technique also works on other different
| problems.
|
| 4. Show show some flaws (or limits of generality) of the
| technique. Propose fixes for the flaw, and/or better outline
| where the technique viable.
|
| 5. Now either prove the technique's validity more rigorously
| (need not be a perfect proof, especially if a full proof
| requires far more complicated mathematics) or extend this
| technique to be able to solve more (but eventually coming back
| to proving the validity).
|
| This sort of approach is has been used in educational videos
| like some of threeblueonebrown's on youtube, and I've also seen
| this used as a faux historical development of algebra and high
| school calculus in "Algebra: the easy way", and "Calculus: the
| easy way", which did a great job of motivating the development
| of most of the ideas in those courses. I'm not familiar with
| similar texts for higher level mathematics (which would have a
| different tone, since they would be targeting adults, not
| teens/children), but surely some must exist, right?
|
| This is infinitely better than the all too common higher level
| math textbook approach of:
|
| 1. Here is some unfamiliar theorem, that you might not even
| really understand, and certainly have no clue of the relevance.
|
| 2. Now here is how to prove the theorem. (Which you might be
| able to follow, but you probably don't really care about right
| now).
|
| 3. Now we finally explain what the theorem is, and hint at (but
| may fail to show why) it might be useful.
| thesuitonym wrote:
| Unfortunately what we typically end up with when learning
| math is:
|
| 1. Explain some problem that is difficult to solve using
| other techniques. The problem is painfully abstract and
| completely absurd.
|
| 2. Propose a technique to solve it. Never mind how we got to
| this technique or why it makes sense, just apply this
| equation and you'll get the answer. Memorize the numbers and
| symbols, that's all that matters.
|
| 3. Show that the technique also works on different (but
| really the same) problems.
|
| 4. Exam time! Here's a problem that looks nothing like the
| ones you've seen so far, but use the magic technique you
| memorized! You did memorize it right? Even though you didn't
| understand it at all?
|
| 5. This class is over so you will never use that technique
| again--even though it's generally useful, but since you never
| learned how to actually apply it or _why_ it 's useful,
| you'll forget it the day the exam is finished.
|
| 6. Math is great! Why don't people like math?!
| xg15 wrote:
| Fully agreed!
|
| I'm also puzzled why so little math courses seems to involve
| the actual history of an idea. Often, an idea is presented in a
| vacuum or - even worse - confusing a formal proof with an
| explanation of purpose.
|
| In my opinion, that's about as effective for teaching
| mathematical concepts as teaching vi by giving you a copy of
| the source code.
|
| I think explaining the historical context of an idea and the
| problems the authors _back then_ were trying to solve with it
| would be much more valuable for understanding than just the
| bare proof.
| macrolocal wrote:
| If you're interested in ring theory, for example, Kummer's
| 1844 manuscript on "ideal complex numbers" clarified a bunch
| of undergraduate algebra for me. Overviews:
|
| https://projecteuclid.org/journalArticle/Download?urlid=bams.
| ..
|
| https://arxiv.org/pdf/1108.6066.pdf
| agumonkey wrote:
| I found a math book aimed at math grads going into official
| sub-college teaching positions, and was shocked about the
| historical context provided. Every idea had 2 or 3 variants
| that differed in origin. To me it would have made math 2x
| more fun for a lot of pupils. But none of it made it into our
| classrooms (lack of time ? a dusty pedagogical approach maybe
| ..). It's probably vital for most people that don't have
| their neurons naturally properly aligned to a subject to have
| various informations and point of views about the problems
| and goals.
| Jtsummers wrote:
| IME, you will get that information if you get a math degree.
| You are less likely to get it, unless you read the textbook
| (that is, not from the teacher) from any K-12 or non-major
| math classes because that's not their focus (for better or
| worse). But often the motivational cases for the math are
| present in high school and non-major math course textbooks.
| Just not part of the class because that's not what you're
| going to be evaluated on.
| neel_k wrote:
| 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.
| n4r9 wrote:
| Even without the context, there's something to be said for a
| formalised approach. When I was in undergrad there was a
| lecture course given by a notoriously aloof and formal
| lecturer. One of the other - more popular - lecturers decided
| to give an "understandable" alternative to the course at the
| same time of the day. Myself and a few others were in the 5%
| that continued with the official schedule. Those notes were
| hard as hell to work through, but once you understood
| something, you REALLY understood it.
|
| One of the exam questions was conveniently targeted at one of
| the lectures from the more difficult course. I think it was
| proving that A5 is simple by considering the rotations of a
| dodecahedron.
| jacobolus wrote:
| I disagree. An overly dry and formal style (definition,
| definition, lemma, theorem, corollary, definition, lemma,
| lemma, theorem, ...) does not make students " _really_
| understand" the material. It just focuses students on low
| level details of formal definitions and symbolic
| manipulation and gives a lot of practice regurgitating
| /performing those, often at the expense of knowing the
| purpose or meaning of the subject. Low-level details are
| certainly essential, but the only way to _really_
| understand is to figure out what the formalism is for (what
| problems does it solve), grapple with the possibility space
| of definitions and theorems (if we picked this alternate
| definition, would that also get us where we want?), figure
| out how topics and structures relate to each-other, spend
| some time doing personal explorations, and build up mental
| models of what the definitions _mean_ , not just their
| formal content.
|
| A too-dry mathematics course/book is like a screenwriting
| course where you focus on snappy dialog and details of the
| setting but never talk about the plot or themes of the
| story.
| n4r9 wrote:
| I don't disagree. What we found was that, starting from
| the lecture notes and a few examples, getting to the
| point where we could complete the exercises meant that we
| had to do most of the above figuring-out for ourselves.
| And because we did it ourselves rather than have it laid
| out for us, the learning was more established.
| bsedlm wrote:
| > a sufficiently motivated reader could learn it without
| having to learn it master-apprentice style
|
| if that's the case, I would say they failed.
|
| however, what they accomplished would certainly help jog the
| memory of somebody who knew the material once upon a time.
|
| maybe it's a bit like looking at a zip file directly and
| uncompressing the contents on the fly in your head?
| (something about 'understanding' as a compression scheme)
| woolion wrote:
| 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?
| thissiteb1lows wrote:
| photochemsyn wrote:
| 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?
| feoren wrote:
| > 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 wrote:
| 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'.
| feoren wrote:
| > You seem to be specifically looking to find a frame in
| which the author is incorrect
|
| Actually I read the article because I love the idea, and
| agreed with everything he said up until halfway through, when
| it completely veered off into nonsense-land with awful
| examples. I have no idea why you've decided you know my
| motivation for criticizing the article. I'm criticizing it
| because it is bad!
|
| > The model definitely worked for me when I first encountered
| it and it worked for many of my early friends.
|
| Unfortunately it doesn't sound like either of us really have
| data on this, your anecdotes notwithstanding. But I have to
| admit I really wonder whether you're (a) remembering your
| elementary-school mental state correctly, and (b) correct
| about your friends' mental states as well. After all, you
| claimed to know my motivation for writing my comment, so
| maybe you're not as much of a mind-reader as you think. And
| it's really hard to remember the way you thought about
| something N decades ago!
|
| Maybe you're thinking of multiplication as being "repeated
| addition"? Great! That's a good cartoon of multiplication.
| That's a foothold into the concept that doesn't lead you
| astray.
|
| The reason it's a good cartoon is that it is correct --
| _technically correct_ -- for a sensible subset of numbers,
| and it extends naturally to other concepts. Multiplying 7 by
| 0.5 really is analogous to "adding 0.5 sevens together";
| multiplying 7 by -2 is analogous to "adding -2 sevens
| together". And it's useful if you've been needing to do lots
| of addition and you're tired of hitting the + button on your
| calculator over and over. It's immediately useful because
| addition is useful, and it makes repeated addition easier!
|
| But "it makes things bigger" gives you none of this power --
| none whatsoever. It just gives you a weird, bad mental image
| of it inflating a number into a bigger number ... somehow.
| It's not correct even for the Whole Numbers -- multiplying by
| 1 does not make anything bigger! You don't even need 0 for it
| to be wrong! And if you multiply by 0.5 or 17lbs or i,
| there's absolutely no way of stretching "making it bigger"
| into an analogy that's at all useful. It's just bad, bad,
| bad, bad. _Completely different_ from "multiplication is
| repeated addition"!
|
| The rule "it makes things bigger, except when it doesn't" is
| exactly what people _hate_ about math. A weird, opaque non-
| definition that isn 't even true most of the time.
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