[HN Gopher] I rewired my brain to become fluent in math (2014)
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I rewired my brain to become fluent in math (2014)
Author : ColinWright
Score : 245 points
Date : 2024-04-26 08:44 UTC (14 hours ago)
(HTM) web link (nautil.us)
(TXT) w3m dump (nautil.us)
| DarkNova6 wrote:
| Wow. That author sure loves to talk about herself. I kept
| reading, but the whole article feels like an overdrawn
| introduction without payoff.
|
| If you want to know how you can become better in math and rewire
| your brain to be math compatible I'm afraid you will be none the
| wiser after reading this.
| eviks wrote:
| Maybe one needs another brain rewiring job to become fluent in
| learning from blogs like this
| chankstein38 wrote:
| Or the author needs to rewire their brain again to learn to
| write in a way that actually teaches lol
| DarkNova6 wrote:
| [deleted]
| dimitrios1 wrote:
| Quite dissapointing, because I found the authors book "A Mind
| For Numbers" to be good.
| dailykoder wrote:
| >If you want to know how you can become better in math and
| rewire your brain to be math compatible [...]
|
| ... then you just have to do it. And keep working on it, even
| though it feels awful
| DarkNova6 wrote:
| No shit
| keybored wrote:
| I skipped the purely biographical paragraphs. Some important
| parts:
|
| - Memorization and rote practice are important for learning,
| not just the current Zeitgeist (2014) of "understanding"
| without the former. This becomes the foundation that allows you
| to focus on higher-level things like understanding and applying
| formulas.
|
| - Experts develop "memory chunks" which allows for example
| chess masters to draw on thousands of different past games,
| openings, variations.
| jacobolus wrote:
| Memorization happens naturally through repeat exposure, and
| works better if this is exposure in some meaningful context
| rather than through cramming via flash cards or whatever. The
| best kind of practice is practice that you are motivated to
| do because is inherently interesting. The less "rote" you can
| make this the better. The author's example of thinking deeply
| about every aspect of the meaning of the formula F = MA is
| not at all a "rote" approach.
|
| For a primary school example, if you can solve basic
| arithmetic problems in service of a fun and challenging logic
| puzzle, that is more motivating than solving a page of
| arithmetic problems one after the other.
|
| More generally, while mathematics certainly requires putting
| in time and actually doing the work of thinking a whole lot
| about a variety of hard things in the service of solving hard
| problems, very little of that is memorization per se.
|
| > _In the United States, the emphasis on understanding
| sometimes seems to have replaced rather than complemented
| older teaching methods that scientists are--and have been--
| telling us work with the brain's natural process to learn
| complex subjects like math and science._
|
| The older teaching method also sucked. The new one is
| marginally better.
|
| In my opinion, the single most important thing primary school
| math education should be teaching is how to attack and solve
| nontrivial word problems. Unfortunately we did not have any
| of that before, and still do not have any now. Cf.
| https://cs-web.bu.edu/faculty/gacs/toomandre-com-
| backup/trav...
| kaitai wrote:
| It's important that we marry viewpoints when looking at
| what we call "meaningful". I spend a lot of time right now
| with a 6/7 yr old. This child does not care about meaning.
| This child likes to solve arithmetic problems one after
| another. It's motivating -- it's like a game -- the kid
| gets them right and gets a thrill. Word problems? Well, the
| math ability is outstripping the reading ability right now
| so 2000 x 2000 x 2000 is way more fun than reading some
| stupid sentence about tulips.
|
| I've spent plenty of time on math (PhD in algebraic
| geometry) and educating people, and for sure when I taught
| college freshmen and master's students I spent a lot of
| time challenging folks to engage their minds, spirits, and
| intellect. At the same time, we have to admit there is a
| stage of childhood where kids just love memorization and
| facts. Dino facts, shark facts, math facts, Pokemon facts,
| My Little Pony facts, whatever. Let's not force kids to
| reckon too much with meaning when they're in the facts for
| facts sake stage -- and once they've got their impressive
| facts list, they'll make sense of the meaning much more
| easily, as discussed in the article and here!
| DarkNova6 wrote:
| The thing is her definition of ,,understanding" isn't
| actually ,,understanding" but rather surface level intuition.
|
| Personally I only accept ,,understanding" once I can explain
| it and reuse it in a different context. But I am not self
| centric enough to deny that there absolutely are plenty of
| people who love to memorize without having an abstract
| understanding. And they are doing just fine.
| zero-sharp wrote:
| I never really know what people mean by "rewiring your brain".
| You just have to spend a lot of time studying it. The hard part
| as an adult is probably making the time, especially if your
| work is already mentally taxing.
| joelfried wrote:
| Ok, I'll bite.
|
| Thoughts flow through my brain like electrons through wires,
| both at a speed I cannot truly comprehend. The paths of my
| thoughts, the way words connect together, the emotions they
| evoke, the feelings that are associated with - these are all
| malleable. I have been working to rewire myself away from
| pessimism and towards optimism for years now. It's not easy,
| and sometimes I fall back into old patterns. As years have
| passed, though, I've found the new pathways easier, the new
| roads getting more familiar. My thoughts have previously
| wanted to go one way, and I spent time yanking at them to go
| a different way. I spent enough time at it that I now on good
| days more naturally go they way I want to go.
|
| It's not just sitting and repeating a single thing over and
| over again, it's working with your own natural inclinations
| so that you can recognize when you experience X and naturally
| reach for Y that perhaps Z is what your preference really is,
| upon reflection. If you can practice that enough then, in the
| moment, you can sometimes find yourself not following the old
| pathways but the new.
|
| Rewiring seems like a pretty good analogy as when you rewire
| a house you work hard -- it is dirty, dusty work. Pulling
| wire is hard and thankless because when you're done you cover
| it all up and, if you're lucky, it works! Then at the end
| you're . . . back in a state where nobody but you knows any
| different what is happening behind the walls. Things work,
| and others might have no idea anything changed at all. But
| you put in the hard work and you know how things actually
| work on the inside now and it's exactly how you want it, not
| how it was before.
|
| If you've got a better analogy, I'd love to hear it.
| chankstein38 wrote:
| Right?! I want to know "How they rewired their brain to be
| fluent in math" but so far all I've seen is a bunch of talking
| about how great they are. This article sucks.
| CyberDildonics wrote:
| Most people just call it practice.
| criddell wrote:
| There's a chance that the author isn't the person responsible
| for the title. I searched for rewire on the page and the only
| instance is in the title.
|
| The author, Barbara Oakley, has a free Coursera course that
| is pretty good:
|
| https://www.coursera.org/learn/learning-how-to-learn
| vunderba wrote:
| This was my experience with the article which could be
| consolidated down to a single sentence: "Manipulate and play
| with concepts you intend to internalize, rather than relying on
| rote memorization".
|
| I would _think that would be obvious_ to anyone that merely
| memorizing something like 'f=ma' would be meaningless without
| deliberate attempts at application (both theoretical and
| practical).
|
| There was a kludgy attempt at tying the study of foreign
| languages to STEM, but it just amounted to, _everything_ is
| ultimately a craft. You have to practice to perfect it.
| keybored wrote:
| Oh, I didn't connect the dots before after I read where's she's a
| professor at. She's one of the instructors in Learning How To
| Learn https://www.coursera.org/learn/learning-how-to-learn
| JabavuAdams wrote:
| Yeah, I had the same realization. Loved that course.
| xchip wrote:
| Articles that start with "I" are usually about bragging and hence
| disappointing.
| JabavuAdams wrote:
| What did you think of _this_ article?
| xchip wrote:
| Too much bragging and disappointing
| ykonstant wrote:
| I know, right?
| ltbarcly3 wrote:
| Some people just won't ever be good at math. There are people who
| have basically no visual intuition and can't predict what an
| object will look like when rotated in 3 dimensions. It may be
| possible to practice this but someone who does poorly at this
| untrained is unlikely to ever reach even median level ability.
| There are techniques you can learn to perform this task in a
| multiple choice test situation but they don't train your brain to
| be better at visualizing 3d rotations at all.
|
| This is not to say that object rotation is a proxy for
| mathematical ability, but just a demonstration that there are
| cognitive tasks which people vary widely in ability on when
| untrained and which we don't have an effective way to train, at
| least at an arbitrary age. Lots of Math seem to be this way.
| JabavuAdams wrote:
| Possibly true, but "good enough" is within reach of many more
| people than realize it. Engineering level calculus is not
| particularly high-level math, but will get you a long way in
| terms of applications.
| yura wrote:
| > It may be possible to practice this but someone who does
| poorly at this untrained is unlikely to ever reach even median
| level ability. There are techniques you can learn to perform
| this task in a multiple choice test situation but they don't
| train your brain to be better at visualizing 3d rotations at
| all.
|
| Any source for this claim? I'd like to read more about it.
| zero-sharp wrote:
| I know you said object rotation isn't a proxy for mathematical
| ability. But I just want to add that not all intuition is
| geometric intuition. And not all math is even geometric.
|
| Of course, there isn't a switch that gets flipped which
| suddenly makes you "good at math". But still, I think most
| people with an interest can study it, _learn something_ , and
| make _some_ progress. I 'm happy playing a sport, but I know
| what separates me from an athlete and I know I won't ever be a
| professional athlete.
| keiferski wrote:
| I wish mathematics education would incorporate more history and
| philosophy. Personally, I was never great at math in school, less
| because of aptitude and more because I just found it boring and
| disconnected from the things I found interesting as a kid.
|
| Years later, I've been slowly trying to "catch up" with my
| mathematical knowledge, and I find myself the most interested in
| topics that relate to the lives of mathematicians (and how events
| impacted their work) and to the philosophy of mathematics. I
| didn't get any of this in school math classes, which focused
| purely on calculations and formulas.
|
| I had the same experience with accounting, as well: boring in
| isolation but fascinating when connected to the history of double
| entry accounting in Italy, global trade from the 1500s onwards,
| and so on.
| eXpl0it3r wrote:
| But aren't you then teaching philosophy and history instead of
| mathematics?
|
| I mean it can be a great story hook to start off with a
| subject, but in the end isn't math about the calculations,
| formulas and proofs?
| layer8 wrote:
| The history and philosophy is about giving context and
| meaning to the calculations, formulas and proofs.
| keiferski wrote:
| I don't think that's true at all and would consider the
| history and deeper structures (philosophy) of something to be
| integral to that topic. In fact, I think the (very modern)
| rigid division of subjects into separate categories is part
| of the problem. Intellectuals from a few centuries ago would
| find it absurd that we consider the liberal arts and
| mathematics to be entirely opposite types of things.
| glitchc wrote:
| Maybe your true love is history, not accounting or math.
| keiferski wrote:
| Well, the goal here is to help people who _don 't_ like math
| learn to like it, or at least find it interesting enough to
| learn.
|
| I think that people get slotted into the _likes math_ or
| _doesn 't like math_ category quite early in life, often
| because they don't have an on-ramp to finding it interesting.
| They then spend the rest of their education avoiding it
| because "they're not a math person," when in many cases they
| probably just needed a bit more
| context/history/philosophy/something interesting to get them
| on the right track.
| mettamage wrote:
| Similar in my case. I'm beginning to like math just now because
| after years of software engineering I'm seeing commonalities in
| math and software engineering. Math feels like "creating a
| software systems for numbers" where I also am the compiler at
| the same time.
|
| From an intellectual perspective (different than my engineering
| perspective), I'd label mathematics as quantitative philosophy.
| I like philosophy too.
| itronitron wrote:
| Yeah, the big lie of mathematics education is that everything
| currently known just appeared fully formed inside
| mathematicians heads because they are mathematicians gifted
| with mathematical thinking.
|
| The reality is that every advance in mathematics evolved after
| decades of people crunching numbers (typically for some
| engineering application) with an earlier form of math until
| they worked out the advance. I think students would have more
| confidence in their own ability if they understood that the
| mathematical innovators were equally stumped for long periods
| of time.
| msteffen wrote:
| I have to say, I actually loved this article. Especially
| "Understanding doesn't build fluency; instead, fluency builds
| understanding."
|
| I love math and majored in it in college. The rest of my family
| is all scientifically inclined, but I think found/find math
| itself opaque and somewhat intimidating. I remember my brother
| asking me at one point how one would ever find, for example, the
| Pythagorean theorem intuitive. The author's quote is the response
| I wish I had. The Pythagorean theorem becomes intuitively true
| not when you have some deep insight about Euclidean space, but
| when, on seeing a right triangle, three proofs of it spring
| instantly to mind. Which happens after a lot of practice.
|
| FWIW I think it's appropriate that the author talks about herself
| a lot. She's trying to explain the subjective, cognitive
| experience of going from math-phobia to math mastery over her
| career. She can't explain that without talking about her
| background and her perception of the process from inside her
| head.
| msteffen wrote:
| This actually calls to mind this great talk by Grand Sanderson
| (the YouTuber behind 3blue1brown):
| https://youtu.be/z7GVHB2wiyg?si=jcUtUo-TT3ycpTpD
|
| That talk is about something superficially different--ego in
| math--but on reflection, I think the desire to look smart
| actually really does set one up for success in math in the
| particular way that the OP article describes.
|
| When you just want to look smart, you don't care whether you
| know something because you thought of it or because you read it
| in a book. You just care that you can show off what you know
| and solve problems easily. So you voraciously read and memorize
| and try to accumulate a massive mental database of facts to
| show off. Then at the end you find you're actually good at the
| thing.
| syndicatedjelly wrote:
| > When you just want to look smart, you don't care whether
| you know something because you thought of it or because you
| read it in a book. You just care that you can show off what
| you know and solve problems easily. So you voraciously read
| and memorize and try to accumulate a massive mental database
| of facts to show off. Then at the end you find you're
| actually good at the thing.
|
| What should one do instead, in order to avoid merely
| "looking"/"sounding" smart?
| kaiwen1 wrote:
| Just do math. A student driven merely by the pleasure of
| doing math without concern for external validation is
| lucky. But if external validation is a driver, that's lucky
| too. In both cases, math gets learned.
| edanm wrote:
| Just in case this isn't a typo - his name is Grant, not
| Grand.
| UncleOxidant wrote:
| He is doing some grand work.
| rustybolt wrote:
| "Young man, in mathematics you don't understand things. You
| just get used to them."
| JadeNB wrote:
| Often attributed to von Neumann.
| runlaszlorun wrote:
| Is it not?
| kian wrote:
| >>> The Pythagorean theorem becomes intuitively true not when
| you have some deep insight about Euclidean space, but when, on
| seeing a right triangle, three proofs of it spring instantly to
| mind.
|
| To be honest, this sounds like orienting one's self in the
| 'space of mathematics'. Is it not possible that, just like one
| can navigate by landmarks (proofs) or by the space itself (deep
| understanding), that there are in fact two roads to intuition
| in mathematics, of which ones is practice and fluency, and the
| other is deep insight and understanding?
| zora_goron wrote:
| I commented on this on HN a couple months ago, but I had a
| similar conclusion regarding the value of memorization when I
| joined med school after studying computer science in undergrad
| and grad.
|
| It took me a while to buy in to high-volume memorization as a
| learning technique (especially coming from CS, where memorizing
| facts is not a huge emphasis). After a while though, I started
| recognizing how the quick recall encouraged by the system
| enhanced my understanding of concepts vs replacing it (I wrote
| about this a couple years ago [0]).
|
| [0] https://samrawal.substack.com/p/on-the-relationship-
| between-...
| 2OEH8eoCRo0 wrote:
| That's my takeaway from _The Shallows_ by Nicholas Carr.
| Knowing how to derive information or where to find information
| doesn 't mean you _know_ the information and knowing the
| information is necessary to form higher level associations in
| the mind.
| JonChesterfield wrote:
| Huh. That's an interesting premise. I think I would split it
| though - knowing the basis well enough to derive results is
| probably fine for later deduction, knowing where to find the
| information is definitely useless for it.
| nescioquid wrote:
| The argument is actually more about insight. You can't have
| insights about things you don't already have in your head.
| Insight in this context is noticing some new relationship
| between two facts.
|
| There was a slang term "refrigerator thoughts" that
| described someone staring into the refrigerator while
| thinking of a show plot and realizing there is a plot hole,
| a disturbing implication, etc. In any case, that's an
| example of insight. Hopefully we have these spontaneous
| realizations about less trivial things as well.
|
| The more stuff you've got in your head, the more insights
| you'll have, which drive more questions whose
| investigations puts yet more stuff in your head.
|
| I don't think you can split it. The critical piece is
| actually getting information into your head in the first
| place.
| anon291 wrote:
| I have sung the praises of memorization since I was a kid,
| and yet the attitude that, with the internet, memorization is
| no longer required because we have access to unending
| knowledge, seems oh so prevalent. One wonders what these
| advocates must think... do they claim to know Norwegian
| because they can use Google to translate at any particular
| instant, for example? One wonders what life might be like for
| people so confident in their non-existent abilities.
| Aerbil313 wrote:
| Related thread a while back: "Learning Is Remembering":
| https://news.ycombinator.com/item?id=32982513
| bilsbie wrote:
| How did he do it? I read half and gave up.
| downrightmike wrote:
| TL:DR
| dguest wrote:
| I'd like to know what education reformers would say to the
| subtitle
|
| > Sorry, education reformers, it's still memorization and
| repetition we need.
|
| It seems needlessly confrontational, and misses what the article
| is about. The article asserts that practice and repeated _use_ of
| math is important. I don 't think it's really suggesting that we
| should go back to how math was taught in the US 40 years ago.
|
| But maybe I'm just out of touch with math education reform: In
| high school I was graded on how fast I could do matrix
| multiplication, and thought matrices were kinda stupid. Then I
| learned about linear algebra and coupled oscillators in college
| and thought they were awesome.
|
| So I'd assumed the educational reform was about removing the busy
| work from math and focusing on what you'd actually use it for. Am
| I wrong?
| detourdog wrote:
| I had a same experience with you but there was no algebra at
| art school. I learned why math was interesting from trying to
| figure out how a computer works. I know I have huge gaping
| blind spots but I can use math for what I need. I'm now trying
| to avoid math and only using the shapes of math.
| caddemon wrote:
| I didn't even know what a matrix was until I got to college,
| and I went to a supposedly well-ranked high school and was on
| the higher level math track. This was about 10 years ago. I
| think some of the education reform might've removed concepts
| altogether instead of actually improving the presentation?
| Though I suppose in order to present things well you might need
| to cut back on the total number of topics covered, I'm not sure
| I'd describe what I did learn as well-presented either.
| deltarholamda wrote:
| I remember hearing an interview with a woman who was doing
| charter school-type work in the UK (IIRC), where most of her
| students were thought of as "underperforming".
|
| She was successful because they emphasized drills, lots of
| drills, and more drills.
|
| Teachers and students _hate_ drills. Teachers, because they 're
| tedious to grade, and students because they're boring. But they
| work. It's no different than doing the same Super Mario Bros.
| level again and again until you time your jumps just right.
|
| I've often thought that gamification of drills would be a great
| way to get kids to learn their math facts or whatever, but
| there seems to be an allergy to doing this in the US education
| system. What the US education system seems to be addicted to is
| moving from one hype/fad to the next, as that's where the money
| trough seems to be.
| barfbagginus wrote:
| Teach category theory, and if you can't, then don't bother
| teaching math at all.
|
| Because you'll be teaching the awful wasteful rote math that
| everyone hates and can't use, instead of the nice universal
| stuff that lets you transfer intuition and see how ideas
| communicate between different knowledge domains far beyond what
| was traditionally seen as math.
|
| The 20th century gave us real new ideas in math. But our
| primary math education is still 200 years out of date. Until
| that changes, math education will remain a deadening cargo cult
| that throws away far more human potential than it develops.
| vundercind wrote:
| The rote stuff's about all I've ever actually managed to find
| a use for, as an adult.
|
| It's useful daily. Pre-algebra is useful fairly often (even
| if I weren't a programmer, plugging numbers into a formulas
| and basic graphing are very handy, quite often). Trig I think
| I managed to use once, but only because I didn't know the
| right way (if you find yourself using trig on a minor home
| project you're probably missing some trick or standard or
| something that lets you _not_ do that--I suspect it was the
| case then)
|
| That's... about it. Stats, kinda, but mostly looking up the
| formula for the thing I want and plugging in the numbers,
| which barely counts.
| anon291 wrote:
| Linear algebra is probably one of the most useful
| mathematics there is and not commonly taught in high
| school.
| zzzbra wrote:
| This actually is a fairly controversial stance in education, so
| there's reason to be combative about it. A lot of education
| tends to emphasize meeting students where they are to the point
| that it can completely subsume the irreducibility of complexity
| when confronting some knowledge conceptually; one _can_ be
| better served instead by attempting to memorize some bulky,
| impenetrable abstraction and instead make sense of it through
| its application. A lot of knowledge only becomes clearer when
| one forges ahead with a dim appreciation of what is being
| articulated but the confidence, willingness, and (most
| crucially) feedback mechanism for testing it out anyway.
| cjs_ac wrote:
| On UK Teacher Twitter, there are two factions that disagree
| noisily but civilly: the 'trads' and the 'progs'. Both factions
| have educational psychology research to back up their claims:
| the progs lean mainly on research from big institutions, often
| international; the trads often do research in their own
| classrooms. The trads' pupils seem to do better in the UK's
| public examinations, but that might be an artefact of those
| exams.
|
| It's also worth noting that half of all teachers leave the
| profession leave within five years. Forty years thus represents
| about eight generations of teachers being trained with their
| own biases, refining their ideas in classroom practice, and
| then training the next generation. On top of this are the
| cycles of trendiness in the various schools of thought in
| educational psychology, as well as the varying policy platforms
| of governments. Educational practice from four decades is less
| historical and more archaeological.
| abdullahkhalids wrote:
| When I was teaching university level mathematics, there were
| many people in my classes who couldn't do fast matrix
| multiplication or even fast multiplication. The inevitable
| result was that they had to constantly drop from the higher
| level of abstraction that we were trying to learn, to the lower
| level abstraction of arithmetic, and failing to learn the
| higher level of abstraction.
|
| On this forum, I will say, imagine your object-oriented
| programming language can build all sorts of abstractions but
| can't multiply numbers. So every time you have to multiply
| numbers in your algorithm, you have to instead write a few
| lines of assembly code that do the same thing. How much
| efficiency would you lose.
|
| Just practice multiplying numbers.
| anon291 wrote:
| You naturally memorize that which you are exposed to, but to
| say that that means we should discourage memorization in favor
| of purely exposure (which is the current status quo AFAICT), is
| completely misguided.
|
| Yes, you will almost certainly memorize anything with enough
| exposure, but targeted memorization is also useful, if the
| former's not going fast enough.
| johngossman wrote:
| I think it's ironic that when we rewire a virtual neural network,
| we call it training and the field is called by machine learning,
| but when humans learn something, train themselves, they call it
| rewiring their brain. At some point the reductionist language
| just obscures the point and makes it less accessible
| keiferski wrote:
| One wonders: will the robo-humanoids of the future use
| biological metaphors to describe their electronic bodies?
| syndicatedjelly wrote:
| What does accessible mean in this context?
| johngossman wrote:
| I was actually thinking about a phrase like "it activated my
| amygdala" instead of "it made me anxious" et al, where you
| have to know some neuroscience, hence less accessible to the
| general populace. Another example is "updated my priors"
| instead of "learned some new information" or "changed my
| mind."
| keybored wrote:
| I agree. I really dislike technical-sounding jargon that
| really are just replacements for feeling-talk.
|
| Use scientific terms for science. Use normal words for your
| own experience unless you really were hooked up to a
| machine and were measured in some way.
| Funes- wrote:
| I've always loved quick calculation games, learning mathematical
| principles and concepts, applied mathematics, probability theory,
| etcetera. However, I developed huge mathematical anxiety
| throughout all of high-school, because math class followed an
| appalling, horrifying routine: the teacher would randomly pick
| any of us to forcibly go up to the blackboard to solve an
| exercise from the previous lesson's homework, especially when we
| hadn't done it, and we would get badly scolded in a humilliating
| manner when we couldn't solve it, getting some laughs or hurtful
| remarks from our classmates, as well. When we were done, we would
| get more homework, and so the cycle repeated. The actual teaching
| always took less than ten minutes, if it took place at all.
|
| Teachers were so utterly disparaging, it became an extremely
| stressful experience. It undermined our ability to focus in the
| first place, so not instantly getting whatever was being "taught"
| induced more fear, which made you lose even more focus. It was a
| terribly negative feedback loop.
|
| Later on, I started reading math books on my own and realized
| that not only I wasn't bad at it, but what kind of motherfuckers
| were those so-called teachers, and how clueless they were in
| pedagogical terms.
| ivan_ah wrote:
| Thanks for sharing.
|
| I talk to a lot of people about math and many of them (adults)
| have intense math anxiety. I always wonder what kind of trauma
| could have led to this, because I assumed the I-suck-at-math is
| such a private feeling, at worst maybe your parents might see
| your grades and scold you about them. None of this is super
| traumatic, I thought...
|
| But the perspective about public shaming by the teacher and
| other students piling on is much more intense, so I can see how
| some people really don't want _anything_ to with math in later
| life. Those motherfucker teachers indeed!
| toss1 wrote:
| OK, the article convinced me that repetition intentionally
| focused on the full range how to use and not use each small
| component such as a word, formula term, or concept is the key.
|
| Great. So where are some books, programs, or apps that will help
| us do exactly this? Not impressed that the article focused more
| on their personal journey and provided no recommendations on how
| to follow it.
|
| Anyone here have recommendations on apps that might help? One
| HNer a few years ago posted a great little web app to practice
| rapid addition/subtraction, etc. which I used daily to noticeable
| benefit until it disappeared. Of course something working up in
| complexity from there would be good too.
| constantcrying wrote:
| When taking mathematics classes in university I always noticed an
| enormous gap between what I _thought_ I understood compared to
| how confusing the problems were. I am glad the author mentions
| that phenomenon.
|
| For (the few) students who actually understood the subject the
| problems are just busywork, for those who didn't it is the most
| important part of the learning process. There is exactly one way
| to understand mathematics, which is actually doing it. This can
| be many things, but actually solving problems is an important
| part. I believe that problems should be interesting, but repeated
| recall definitely is important as well.
| the__alchemist wrote:
| To me, it seems that there are two general categories of things
| referred to as "math": A: the one used in this article: What
| people generally refer to as math. What's used by engineers,
| (most) scientists, etc. B: The one used by math majors and
| mathematicians. This type is abstract, contains things domains
| that end in "theory".
|
| My question: Do you think an approach like in the article is
| possible to learn Math B? I have tried several times,
| unsuccessfully. I'm proficient in most domains of Math A.
| (Differential equations, linear algebra etc, symbol manipulation,
| geometry, and how tho apply them to practical problems).
|
| Math B seems, in contrast, beyond me. There is a programming
| analogy: Math B is like Haskell, or pure functional programming,
| which also is as ungraspable to me. I am wondering if maybe this
| is partially genetic, partially something you have to learn at an
| early age. Or maybe it takes a formal learning path.
| parpfish wrote:
| I loved math type A, so I majored in it.
|
| Once you're fully ensconced in the major, it pivots into type
| B. And it turns out that I hate type B but slogged through it
| with medium-good grades.
|
| looking back on it now, I've come to like type B and wish I
| could go retake those classes with my current perspective.
|
| I think my original distaste was largely due to what felt like
| a bait-n-switch: start out majoring in something you like and
| are good at, but then pull the rug out and switch to something
| completely different
| jacobolus wrote:
| The real problem is that "type B", despite being much more
| important of an activity to learn (for mathematics or any
| other kind of technical problem solving) is almost entirely
| ignored in primary/secondary education.
| abdullahkhalids wrote:
| The fact that intro math classes don't do proofs (Type B) is
| because of the same pressure from people who only want to do
| Type A.
|
| Due to internal changes in my uni, for the first time, my
| freshmen year, the math department taught proper proof-based
| Calculus 101 (from Apostle of all books) to all majors. Then
| the engineers and biologists complained so much, they had to
| cut out a lot of proofs from Calculus 102. There were even
| more complaints, so by second year, there were hardly any
| proofs in the core math courses. In a few years, the calculus
| courses had become devoid of proofs.
|
| Some unis have separate intro courses for math majors, but
| it's very difficult to offer them in the current economic
| climate.
| parpfish wrote:
| I think Proof vs non-proof is part of it, but it's mostly
| related to level of abstraction.
|
| You can do proofs for calculus, probability, or logic and
| still feel like you're working with the types of problems
| you do in type A math.
|
| But once you start doing proofs in modern algebra or
| topology you're doing things with abstract objects that
| seem to exist for the amusement of mathematicians that look
| down on "applied math"
| mythhabit wrote:
| Abstract math (type B) is a very rigorous discipline that
| underpins the other kind used by engineers (type A). Type A is
| indeed learned by repetition along with understanding. It is
| very important to simply do the math to become better at it and
| understand what you can expect from your calculations.
|
| Type B on the other hand far more about understanding. You will
| never understand the theory of a mathematical space and how to
| apply it, by simple repetition. That is a far more theoretical
| and creative endeavour. You need to learn it and apply it to
| understand it. I suppose you could call the process of applying
| it some kind of repetition, but in my opinion the insights
| comes from applying it to concepts you already know.
|
| A formal learning path is a very good idea, because people with
| more knowledge know what order you can progress in, in such a
| way that you actually apply your knowledge in a natural way and
| build on previous learnings. And it is definitely a huge help
| that teachers can help you guide your learning when you are
| stuck.
| abdullahkhalids wrote:
| Proofs in abstract algebra, for example, require the ability
| to _quickly_ and _correctly_ manipulate symbols on paper
| (using already discovered rules /lemmas/theorems).
|
| The repetitive practice is in this manipulation of symbols.
| It takes a long time and deliberative practice to learn this
| skill. You just practice by doing symbol repetition in
| different contexts, instead of doing the same thing over and
| over again like multiplication tables, because your symbol
| manipulation abilities have to be general [1].
|
| If you try to teach, you will quickly discover that there is
| a wide difference in this ability for math majors by their
| final years. And the students who have poor symbol
| manipulation abilities inevitably struggle at the higher
| level concept application, because they keep making mistakes
| in the symbol manipulations and having to redo it.
|
| [1] Contrast the training of 100m sprinters (multiplication
| table), who only run 100m on a fixed track that they will
| eventually race on, and the training of cross country runners
| (symbol manipulation), who practice on a variety of routes,
| because their races are on different routes.
| _xerces_ wrote:
| I think there is another one, Math C that involves day-to-day
| mental arithmetic which I am terrible at despite being good at
| Math A and holding engineering degrees. There might also be
| another element of Math C which is a feel for numbers and lets
| you know if an estimate or the value staring at you on the
| calculator screen makes sense or if it is obviously wrong.
|
| I tie my poor mental arithmetic skills partly to never properly
| learning multiplication tables, at least not all of them and
| perhaps something lacking in my brain which also means I have a
| terrible sense of direction.
|
| Yet, when it comes to symbol manipulation where the numbers
| don't matter until the very end, then I am good at that.
| Buttons840 wrote:
| > I tie my poor mental arithmetic skills partly to never
| properly learning multiplication tables
|
| I thought this too.
|
| When you're young the multiplication table seems like a
| daunting thing to memorize, but after graduating university,
| it doesn't seem so bad.
|
| So I went back and memorized my times tables using Anki. It
| was pretty easy, but ultimately changed very little and I
| easily forget them if I stop practicing.
|
| I've come to realize that not mastering the times tables were
| a symptom, not a cause, of my learning difficulties.
| Modified3019 wrote:
| I've something similar
|
| For whatever reason, 6x7, 6x8, 7x7 and 7x8 are a persistent
| hole in my ability to memorize. Sure I can temporarily
| memorize them, but they shortly evaporate back into the
| void and I'll have return to quickly having to calculate
| them out again.
|
| I've also got this thing where I get mixed up between
| verbal "eleven" and "twelve". They sound different, but at
| the same time somehow sound just similar enough that the
| boundary that should exist around them as symbols never
| properly formed. I have to pause and manually match the
| number to the sound, every time. What's especially funny to
| me is I have no such problem distinguishing between _onze_
| and _douze_ from French, which I only know a few words of
| and certainly never hear in real life.
|
| I'd like to think the first problem I'd eventually fix if I
| was using those multiples constantly, but I'm not so sure
| because the second problem definitely doesn't improve.
| jacobolus wrote:
| I wonder if it would help to remember that etymologically
| "eleven" comes from "one left" (as in, I counted the
| first ten and there was still one more) and "twelve"
| comes from "two left".
|
| As for the others, I think remembering these in several
| different ways is stickiest. For example, you might think
| of 7*7 = (5 + 2)2 = 25 + 2*10 + 4 or perhaps 7*7 = (6 +
| 1)2 = 36 + 2*6 + 1 or 7*7 = 7*(10 - 3) = 70 - 21. If you
| already know 7*7, then 6*8 = (7 - 1)(7 + 1) = 49 - 1. You
| can try computing 7*8 by repeatedly doubling: 7, 14, 28,
| 56. Etc.
| DataDaoDe wrote:
| I like the shove it to the nearest 10 approach. It makes
| a lot of calculations much simpler b/c they can be
| transformed to a simple multiplication by 10 and a
| addition or subtraction or two.
|
| 1. 6[?]7 = (6[?]10) - (6[?]3) = 60 - 18 = 42
|
| 2. 7[?]7 = (7[?]10) - (7[?]3) = 70 - 21 = 49
|
| 3. 13[?]19 = (13[?]20) - (13[?]1) = 260 - 13 = 247
|
| 4. 58[?]61 = (58[?]60) + (58[?]1) = 3480 + 58 = 3538
|
| If we go up another order of magnitude, then the system
| starts really grinding to a halt though tbh :)
| card_zero wrote:
| The squares of primes become memorable if you've ever
| tried searching for primes in your head. That's because
| 7*7 is the smallest product of prime factors that are all
| larger than or equal to 7: in other words, you can check
| for the primality of numbers smaller than that by testing
| for division by 2, 3, or 5 only, because they must divide
| by one of those or be prime.
|
| Because of this pointless mental exercise it also sticks
| in my mind that 11 squared is 121 and 13 squared is 169
| (though the presence of 69 helps with that one).
| dleink wrote:
| I'm in the same boat. The multiplication tables for me in
| that zone are constructs from other principles. :) So,
| Fives and Nines are easy and I can derive the Sixes
| Sevens and Eights from those. It's definitely extra
| steps. I think I'm reasonably good at the sort of mental
| arithmetic described in another post, Those particular
| operations just remain as symbols until I absolutely need
| to define them more precisely. I don't have a problem
| with 11s and 12s, but 5s and Rs trip me up.
|
| The way that our brains process symbols is fascinating.
| If anyone out there has any literature or reading on
| this, I'd be interested. Especially, as related to
| ADHD/Autism.
| dkarras wrote:
| Very similar story. Never managed to memorize
| multiplication table. Can do it, but it vanishes. By that
| I mean the "tricky" pairs but I don't know what is tricky
| about them. Been programming computers for close to 30
| years, do lots of math but multiplication table is still
| tricky to me.
|
| Been playing guitar for 20+ years, can't memorize the
| note names on some frets.
|
| Studied music in college, I still need to count lines
| sometimes when reading sheet music, besides some
| reference points I can't seem to memorize the locations
| of notes on the staff.
|
| Not like I have a general memorization problem. I am good
| with human languages, programming languages. Have very
| good working memory etc. But some things just stump me.
| card_zero wrote:
| The Hitchhiker's Guide to the Galaxy (see start of
| chapter 32) lets me remember that "what do you get when
| you multiply six by seven?" was a proposed Ultimate
| Question for The Answer. I couldn't remember it until I
| started remembering it in that context.
| rahimnathwani wrote:
| If anyone else here wants to memorize multiplication facts,
| this is great: https://mathigon.org/multiply
|
| If you want to practice division as well, check out Zetamac
| or (if you don't want to be timed) my simpler tool:
| https://math.twilam.com/
| llm_trw wrote:
| The secret is that you can convert most type B math into type A
| by looking at steps in a proof as rules in a term rewriting
| system where the terms are mathematical expressions.
|
| I've not found a book that makes this point completely
| explicitly, but most of those which cover sequent calculus get
| you half way there.
|
| The rest of type B math is intuition which lets you guess at
| new conjectures and how to get you from the assumptions that
| you've made and the conjecture that you want to prove
| efficiently.
| boothby wrote:
| In high school, I got a D in first semester calculus, and
| declared myself "done" with math. Up until that point, I had
| used a calculator as a crutch, but calculus required symbolic
| manipulation that could not be faked. My dad's influence was
| stronger than my mom's -- she was fearless, but he frequently
| spoke of how "bad at math" he was. And that was an easy out. I
| was just taking after my dad, "bad at math!"
|
| Around that time, I went from noodling around with programming,
| to taking it seriously. I learned a bunch of programming
| languages, and landed a web development job straight out of
| school. I wasn't just done with math, I was done with school,
| too!
|
| After a few years of that, I got bored with web dev, and
| decided I'd rather try my hand at engineering of some sort. I
| enrolled in community college, and quickly discovered that all
| of the engineering courses had... math prerequisites. So I bit
| the bullet, and for the first time, _applied myself_. Turns out
| that I wasn 't _intrinsically_ bad at math; I just hadn 't been
| sufficiently motivated! I was paying my own way, so I ended up
| taking a job in the tutoring center. As I transferred to
| university, I found myself taking more and more of these math
| "prerequisites" and not following through on the engineering
| courses. I matriculated as a math major, and today I've got a
| PhD in math.
|
| In my mid-20s, I didn't even believe that I could be Math A
| person. But I got good at that stuff, for the sake of
| engineering! And then I went straight through to Math B (and,
| almost amusingly, forgot most of those Math A skills -- watch
| out, unused skills get rusty!)
|
| I actually credit my programming experience for the
| intermediate transition from my "bad at math" late teens to my
| "willing to try Math A" mid-20s. Programming taught me to think
| rigorously, and abstractly. So I must push back on the notion
| that this is intrinsic to a person, and must be learned at an
| early age: I wasn't doing Math B until after 25 when my brain
| was supposedly fully mature. And while I did have the benefit
| of a formal education, I would assert with some confidence that
| the relevant detail there was that I was in a cohort of
| students who were working together, beholden to homework
| deadlines and exams -- because math is _hard_ and it 's really
| easy to get demoralized without that external reinforcement.
| kian wrote:
| I have also found that programming is the gateway drug to
| Math B. Thanks to Functional Programming and Type Theory I
| eventually found may way into Abstract Algebra, Topology, and
| Category Theory... Wish I had time to go back and study these
| with a mentor, though!
| nine_k wrote:
| While at it: pure functional programming is very easy to grasp.
| You should just think about programming as of not tinkering
| with the state, not altering things, but as of producing
| outputs from inputs.
|
| Say, analog electronics mostly works in the pure functional
| domain. An amplifier does not try to change the input signal.
| Instead, it produces a more powerful output signal, following
| the shape of the input signal. A tone generator in a musical
| instrument does not try to make a key on the keyboard vibrate.
| Instead it produces a sound signal according to the key pressed
| (which note and what velocity).
|
| The simplest way to try practical pure functional programming
| is to connect a few Unix processes via pipes:
| cat somefile.py | egrep '^def \w+' | wc -l
|
| The above is a pure function compositon, as a map-reduce
| pipeline, in point-free style. (Yay, buzzwords.) It counts top-
| level functions in a Python file.
|
| But how to achieve something like updating with that? By
| looping the output back to the input, and switching o the "next
| version" once it's computed. Conway's game of Life looks like
| an ultimate "update in place" thing. But it's purely
| functional, too: the new state of the map is completely
| computed based on the previous state if the map. Then the new
| map is seen as "the current map". Similarly, frames in a drawn
| animation do not change, but they are shown at the same place
| one after another, giving the impression of motion and change
| of "the same" picture.
|
| In general, our Universe may be seen as a purely functional
| computation: its next state is a function of its past states,
| and the past is immutable.
| dndn1 wrote:
| I like your conviction Re "functional programming is very
| easy to grasp".
|
| Many won't but I agree in the purest (sorry) sense.
|
| There is no scattered changing state. I think we all learned
| input-function-output as a construct in maths class?
|
| Spreadsheets (sans-VBA) is arguably the most prolific
| programming language and simplest, being used by people who
| do not recognise they are programming. Felienne Hermans gave
| a good talk on this subject in GOTO 2016.
|
| Spreadsheets have numerous shortfalls though, and "real"
| functional programming languages make it difficult to not
| feel intimidated: in my experience, but this is getting much
| better.
|
| [1] is a game of life in calculang, functional language I'm
| developing, where for all it's verbosity at least I hope the
| rules and development over generation (g) can be reasoned
| with (sans-state!).
|
| Not very practical but can show calculang
| computation/workings as it progresses and as parameters
| change - things that are easy for FP and otherwise
| intractable, and which further help with reasoning.
|
| But, a big challenge is to be approachable (not
| intimidating), and I'm trying to make that better. I think it
| helps enormously to be focused on numbers as calculang is,
| and not general programming.
|
| [1] https://6615bc99ad130f0008ecc588--calculang-
| editables.netlif...
| tines wrote:
| I think the OP was trying to say that type theory is
| difficult, not that the kind of "no mutable state" idea is
| difficult.
| DataDaoDe wrote:
| I was just thinking about this the other day. Personally, I
| think that math falls into two categories, though I think I
| would distinguish them differently from you (If I'm
| understanding you correctly). Its kind of like the difference
| b/t the hammer maker and the carpenter, the producer and the
| consumer. For me, mathematics (the kind you research and which
| is abstract and theoretical) is largely in the hammer maker
| camp. We'll call this math X, these guys are creating and
| polishing tools (aka in analysis providing proofs and arguments
| for why the real numbers can be considered complete or that a
| derivative actually can be taken on a given class of
| functions).
|
| Then there is Math "Y". This is all the guys who use those
| things the X guys are selling, the proverbial hammers they have
| produced. They assume the X guys did their work correctly and
| that when they use the products they've bought i.e. the rules,
| theorems and strategies developed by the X guys, to solve a
| particular equation or problem, the answer is correct. For
| example, they assume the limit of the sum of two polynomial
| functions on the reals is equivalent to the sum of the limits
| of those functions - they don't care about all the nitty gritty
| details and justifications - the X guys figured all that out
| for them. They Y guys are much more concerned with figuring out
| how to get the rocket into space or ensure the skyscraper is
| soundly built.
|
| I would say from my experience, very little of mathematics
| education is in the X camp, I'm not saying this is a bad thing
| though, perhaps it is similar to the fact that most programmers
| are not compiler programmers or programming language creators
| :)
| anon291 wrote:
| Um... yes, it's even more important in math B to be able to
| have, at your fingertips, all the theorems related to Ideals,
| Rings, Groups, Categories, Topologies, etc. This is why I re-
| read my math textbooks from time to time. You always miss some
| theorems, and they're often key to higher-level understanding.
| analog31 wrote:
| I studied both A and B. In college, I declared a double major
| in math and physics. Then I went to grad school in physics.
|
| Granted, it was one brain (mine) studying both subjects, so it
| should not be shocking that I learned both in the same way. Of
| course I practiced lots of problems and derivations in my
| physics class, but I also practiced and memorized lots of
| proofs in my upper level (i.e., more theoretical) math classes.
|
| And truth be told, maybe even in my liberal arts courses as
| well. Thanks to programming, I got really good at typing.
| Thanks to owning a personal computer (one of the first at my
| college) I started writing and re-writing a lot. Repetition and
| practice even got me through those courses.
|
| It was simply mercenary at the time, not wanting to waste time
| during exams recalling the easy stuff gave me more time to
| think about the hard stuff. But I think it did help me in the
| long run. I still use a lot of that stuff today, at age 60,
| though it's certainly more computer-aided than it was back
| then.
| InPanthera wrote:
| Poorly written article on wats with the website asking needing to
| store cookies just to read ?
| hosh wrote:
| I've come to understand learning math as:
|
| 1. First gain the intuition
|
| 2. Second develop rigorous understanding
|
| 3. Third gain fluency from repetition
|
| Repetition without understanding, or even intuition, is not
| really going to help.
|
| There are game-based learning that really gets to the fluency
| part.
| comfortabledoug wrote:
| How do you gain intuition without repetition?
|
| I learned math by blindly following algorithms. Over time i
| gained intuition from seeing how variations in the input
| changed the output. The deeper understanding kind've slipped in
| there...I don't know how. I don't think you can build real
| understanding without rigorous practiced repetition.
| hnthrowaway0328 wrote:
| I figured from learning proofs back in high school that there
| are a limited number of approaches one can tackle a
| Mathematical proof problem, at least for textbook problems. You
| just have to be familiar with each of them, that is to work on
| many problems of the same method, to obtain an intuition.
|
| Of course this probably does not help with very tough questions
| that require out of the box thoughts, but I think it still
| helps.
| wuj wrote:
| I didn't grasp the similarity between human and machine learning
| until I took a machine learning class at Berkeley. Listening to
| lectures or reading the textbook is akin to fitting a machine
| learning model: You receive just enough information to begin
| understanding, but not enough to master it. True mastery requires
| you to validate your knowledge through homework, quizzes, exams,
| iteratively tuning your approach based on feedback. Without this
| practice, similar to a machine, your learning is merely
| overfitting to training data - you might handle familar problems
| well, but struggle with new challenges.
| MrDrMcCoy wrote:
| When I read "memorization and repetition", the first thing that
| springs to my mind was being unsuccessfully forced to learn
| multiplication tables in my youth. I have learned over time that
| I'm simply incapable of memorizing something that I either don't
| understand to a certain depth or see as unmoored from obvious
| utility. Even when I comprehend and see the use for something,
| it's still hard to remember without practice through usage.
|
| I think she does mean "use and practice" when she says
| "memorization", which is fine, but I think that phrasing could
| lead education in a direction that would be worse for people with
| memory issues like mine.
| vundercind wrote:
| Heh--all that early memorization-heavy arithmetic represents a
| very high proportion of the applicable value I get out of my
| decade-and-a-half of mathematics education. Some _weeks_ I bet
| it's nearly all the value.
|
| (Not that a kid can necessarily see that, it's just funny
| because that stuff is about as useful as it gets)
| djeastm wrote:
| Meanwhile I have a B.S in Mathematics and I'm still just as bad
| at it as I ever was. I paired it with Computer Science in a
| Double Major and thankfully I am much more comfortable with that.
| wuliwong wrote:
| > After all, I'd flunked my way through elementary, middle, and
| high school math and science.
|
| I have to question the veracity of this sentence. How could they
| possible keep progressing to the next grade if they keep failing
| math and science? I'm sure it is hyperbole but this doesn't seem
| like a great way to start off an article like this.
| the_sleaze9 wrote:
| D's get degrees
| wuliwong wrote:
| In my parlance a D is not flunking though. Flunking means you
| don't get credit for the class. But maybe not where she is
| from?
| aegypti wrote:
| It isn't hyperbole.
|
| You can fail math all year long, but achieve the bare minimum
| in the subject during your annual standardized test and pass to
| the next grade.
|
| Course credits/grades do not ~~affect~~ limit? progression in
| many/most? US school districts before high school.
| wuliwong wrote:
| Was that the case in the 1960s/1970s because that was when
| the author was in elementary through high school? I graduated
| high school in 1995 and what you are describing was not the
| case at my school nor any other school that I knew of at that
| time.
| dragonwriter wrote:
| I know of districts using some degree of social promotion,
| but I've never heard of one promoting on performance but
| using standardized tests alone instead of class grades or
| clearing a certain bar for both grades and standardized test
| as the performance criteria.
| aegypti wrote:
| Not social promotion, explicitly illegal in my state at
| least.
|
| School year F -> STAR test minimum + intervention or summer
| school D -> graduates
| wuliwong wrote:
| >It isn't hyperbole.
|
| At best you present a potential way that it might not be
| hyperbole.
| dragonwriter wrote:
| Social promotion.
| https://en.wikipedia.org/wiki/Social_promotion
| wuliwong wrote:
| Obviously, we don't know the author's actual experience but
| you think that it is more likely that she failed math and
| science continuously from elementary school through to high
| school and just kept getting socially promoted OR it is more
| likely that the author employed a bit of hyperbole and maybe
| was just generally a poor student in math and science
| throughout her childhood?
| wuliwong wrote:
| >students who have been reared in elementary school and high
| school to believe that understanding math through active
| discussion is the talisman of learning.
|
| I graduated high school in 1995. Was I too early? I don't
| remember high school math just being a bunch of discussions
| without problem solving.
| tndibona wrote:
| Question: After preparing and failing at a faang interview, it
| seems clear the only way to pass this is to look through leetcode
| and memorise graph search patterns, bfs and dfs of trees,
| recursion patterns, etc. Because if I truly use my natural
| problem solving techniques, it takes me hours to days to solve
| the leetcode problems. At least according to the article and the
| tech industry, memorisation is intelligence. I've always
| understood the subject and avoided memorising all my life. I'm
| suffering massive imposter syndrome at this point. A part of me
| is considering quitting my existing tech job voluntarily because
| maybe I'm not supposed be around the smarter people who cleared
| the interview. Is the tech industry interviewing right? I suppose
| I need help.
| vundercind wrote:
| 1) I've had a career in tech for--holy shit, way too long. I've
| never had an interview that was terribly close to a FAANG-type
| one. I've also never had a hiring process take more than a week
| from interview to offer.
|
| 2) I don't make FAANG money but I've consistently earned 2.5x+
| median for a person my age in my area (I'm in like a 3rd tier
| US city, couple million people, several tech headquarters, a
| couple of which are household names)
|
| 3) FAANG interviews are a _game_. The point of how awful it is
| and how prep-necessary is to selection-bias the pool so the
| vast majority of candidates are suitable, to select for only
| those who _want it bad enough_ (and /or have lots of free
| time), to make jumping to peer companies difficult to keep
| wages down (if that last weren't the case, they'd find a way to
| stop doing it for people who'd already passed it one or more
| times--the process is expensive) and finally to build _esprit
| de corps_ via hazing (hazing is very effective at that). Don't
| take it personally.
| tndibona wrote:
| Thanks mate. Needed to hear this.
| currymj wrote:
| most of the leetcode problems are based on material that a new
| grad CS major would have learned recently from taking a class
| that uses CLRS as the textbook. if this background is assumed
| then the key to success becomes building problem solving
| skills.
|
| if you haven't very recently taken a class that uses CLRS as
| the textbook, then it makes sense you would have to do more
| memorization and practice with those concepts.
| tombert wrote:
| I interviewed and worked at Apple, but it's extremely variable
| between teams to take this with a grain of salt.
|
| Basically, the things you need to get good at is remembering
| the runtime complexity of the main data structures, and to be
| familiar with the core structures that are available in
| whatever platform you'd be writing code in.
|
| I was working on a Java-heavy team, so the important thing was
| to remember the various Map types, PriorityQueues, Stacks,
| arrays, and being familiar on how the references in Java work.
| The algorithm stuff wasn't too hard once you have a somewhat
| intuitive understanding of all these structures and when
| they're useful.
|
| For example, one thing that they seemed to love in interviews
| was having you implement a least recently used cache,
| specifically an LRU cache where every operation is constant-
| time. Easiest way to do that is to build a doubly-linked list
| and have those point to a wrapper type inside a hashmap, it's
| not terribly hard but it does require familiarity on which data
| structures are useful. In this particular case, the rookie
| mistake it so try doing it with a minheap.
| mkehrt wrote:
| If you ever need to do that in the future in Java for
| whatever reason, Java actually has it already in the standard
| library--it's called LinkedHashMap.
| tombert wrote:
| Yeah, I think I knew that even at the time, but I suspect
| if my answer was just "import java.util.LinkedHashMap",
| they might have been a bit disappointed.
| anon291 wrote:
| A depth first search is exactly what it says it is. I'm not
| sure what there is to 'memorize' about it, but yes, if you need
| to memorize the acronyms, I would think that's important.
| dang wrote:
| Related:
|
| _I Rewired My Brain to Become Fluent in Math_ -
| https://news.ycombinator.com/item?id=33890921 - Dec 2022 (9
| comments)
|
| _I Rewired My Brain to Become Fluent in Math (2014)_ -
| https://news.ycombinator.com/item?id=13674101 - Feb 2017 (46
| comments)
|
| _The building blocks of understanding are memorization and
| repetition_ - https://news.ycombinator.com/item?id=12508776 -
| Sept 2016 (94 comments)
|
| _How I Rewired My Brain to Become Fluent in Math_ -
| https://news.ycombinator.com/item?id=8402859 - Oct 2014 (144
| comments)
|
| _How I Rewired My Brain to Become Fluent in Math_ -
| https://news.ycombinator.com/item?id=8400837 - Oct 2014 (6
| comments)
| SamPatt wrote:
| I started reading the article and it reminded me of a great book
| I read, A Mind For Numbers.
|
| Then I realized it was the same author!
|
| Definitely worth reading if you've been avoiding deepening your
| math understanding.
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