[HN Gopher] Teaching general problem-solving skills is not a sub...
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Teaching general problem-solving skills is not a substitute for
teaching math [pdf] (2010)
Author : JustinSkycak
Score : 171 points
Date : 2024-07-06 15:07 UTC (7 hours ago)
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(TXT) w3m dump (www.ams.org)
| Icy0 wrote:
| > There is no body of research based on randomized, controlled
| experiments indicating that such teaching leads to better problem
| solving.
|
| I'm sorry but one don't exactly come across randomized controlled
| experiments in teaching very often... not to even mention ones
| that are well designed... so this isn't saying much.
| ziofill wrote:
| Maybe you haven't had reasons to come across such research
| before, but rest assured there's plenty of it
| https://acrl.ala.org/IS/instruction-tools-resources-2/pedago...
| Icy0 wrote:
| You seem to have linked a collection of general research on
| teaching and learning, which I am aware of exists. I'm
| talking about randomized controlled trials, where you assign
| a group of students to receive the intervention and another
| group to not receive it, and if it's single- or double-
| blinded, without them and/or the researchers being aware of
| which group they are in. Even writing this brings up
| logistical questions about how you might get a reliable
| research result doing this for teaching (instead of, say,
| medicine, where it's easy to fool a patient into thinking a
| placebo is the drug).
| csa wrote:
| > Maybe you haven't had reasons to come across such research
| before
|
| No op, but I've "come across" a lot of education research. By
| "come across", I mean I've read so much that it makes my eyes
| bleed.
|
| There is some good research that yields interesting and
| compelling results. Rare, but out there. Usually by an
| individual researcher and maybe with a team. Almost never by
| a school of education of significant size or by (almost?) any
| specific field in education.
|
| Results in education are challenging to replicate by a
| different researcher in a slightly different context, and
| studies are often trivially easy to replicate and come out
| with a competing/contrary conclusion by controlling a
| variable that the original researcher mentioned but did not
| control for (e.g., motivated subjects versus unmotivated
| subjects).
|
| Additionally, much research in education is not well-
| designed, or is well-designed but on a relatively meaningless
| topic. There is a lot of touchy-feely research out there
| (like the idea that folks can learn math with just problem
| solving skills), and folks p-hack the hell out of data to
| support their a priori conclusions. It's a smart thing to do
| to maximize funding and/or visibility in academic journals,
| but it is absolutely irresponsible in the quest for "truth"
| and knowledge, which one would hope our education researchers
| would want (n.b.,they largely don't).
| epgui wrote:
| What it is saying is that we need to stop acting as if, or
| believing that, this knowledge is solid.
| JustinSkycak wrote:
| This is only one piece within a larger argument. You need to
| read on to understand what the rest of the argument is.
|
| The form of the argument is this: there is no direct evidence
| for X, but there is a mountain of circumstantial evidence
| supporting "not X", so therefore, almost certainly, "not X."
|
| X = "we can teach students how to solve problems in general,
| and that will make them good mathematicians able to discover
| novel solutions irrespective of the content"
| Icy0 wrote:
| Nice to see a response from you!
|
| I have read the rest of the argument. However, my take upon
| reading it is that this is just one more contribution in a
| back-and-forth argument about every aspect that has been
| studied in math education. Despite the fact that this was
| published in 2010, the landscape in 2024 very much points to
| "it's unclear" as the answer to "is [anything] effective?",
| at least for me, unfortunately.
| JustinSkycak wrote:
| > the landscape in 2024 very much points to "it's unclear"
| as the answer to "is [anything] effective?", at least for
| me, unfortunately.
|
| Interesting. Not sure if you saw the following post from a
| couple months ago, but if not, you may wish to check it
| out:
|
| Which cognitive psychology findings are solid that I can
| use to help students? -
| https://news.ycombinator.com/item?id=40348986
| Icy0 wrote:
| I did! On MESE first, then on Hacker News.
|
| Usually when there's a replication crisis, people talk
| about perverse incentives and p-hacking. But there's 2
| things I want to mention that people don't talk as much
| about:
|
| - Lack of adequate theoretical underpinnings.
|
| - In the case of math education, we need to watch out for
| the differences in what researchers mean by "math
| proficiency." Is it fluency with tasks, or is it ability
| to make some progress on problems not similar to worked
| examples?
| JustinSkycak wrote:
| > Is it fluency with tasks, or is it ability to make some
| progress on problems not similar to worked examples?
|
| That's an interesting point. Ideally students would have
| both. My impression is that the latter is far less
| trainable, and the best you can do is go through enough
| worked examples, spread out so that every problem in the
| space of expected learning is within a reasonably small
| distance to some worked example.
|
| I.e., you can increase the number of balls (worked
| examples with problem-solving experiences) in a student's
| epsilon-cover (knowledge base), but you can't really
| increase epsilon itself (the student's generalization
| ability).
|
| But if you know of any research contradicting that, I'd
| love to hear about it.
|
| > Lack of adequate theoretical underpinnings.
|
| If you have time, would you mind elaborating a bit more
| on this?
|
| My impression is that general problem-solving training
| falls into the category of lack of adequate theoretical
| underpinnings, but I doubt that's what you mean to refer
| to with this point.
| Icy0 wrote:
| > That's an interesting point. Ideally students would
| have both. My impression is that the latter is far less
| trainable, and the best you can do is go through enough
| worked examples, spread out so that every problem in the
| space of expected learning is within a reasonably small
| distance to some worked example.
|
| I simply mean that researcher team A will claim a
| positive result for method A because their test tested
| task fluency, while team B will claim a positive result
| for method B because their test tested ability to wade
| through new and confusing territory. (btw, I think
| "generalization ability" is an unhelpful term here. The
| flip side to task fluency I think more of as debugging,
| or turning confusing situations into unconfusing
| situations.)
|
| > If you have time, would you mind elaborating a bit more
| on this?
|
| I don't know what good theoretical underpinnings for
| human learning looks like (I'm not a time traveler), but
| to make an analogy imagine chemistry before the discovery
| of the periodic table, specifically how off-the-mark both
| sides of arguments in chemistry must have been back then.
|
| > My impression is that general problem-solving training
| falls into the category of lack of adequate theoretical
| underpinnings, but I doubt that's what you mean to refer
| to with this point.
|
| By the way, I see problem solving as a goal, not as a
| theory. If your study measures mathematical knowledge
| without problem solving, your tests will look like
| standardized tests given to high school students in the
| USA. The optimal way to teach a class for those tests
| will then be in the style of "When good teaching leads to
| bad results" that Alan Schoenfeld wrote about in regards
| to NYC geometry teachers.
| 23B1 wrote:
| 99% of math can be outsourced to either a machine, or to the rare
| and precious jewel of a human that enjoys it.
|
| But at this point it's basically vestigial knowledge - like
| balancing a checking account by hand. Good to understand the
| underlying principles of personal finance - but almost nobody
| keeps a checkbook anymore.
| frogeyedpeas wrote:
| if you can quickly outsource a solution of P=NP for me I'd love
| it. Surely that's as simple as balancing a checkbook.
|
| I'm not saying everyone needs to know math but its hardly
| "vestigial knowledge".
| LtWorf wrote:
| I use math to decide which bus to take to get home.
| dinkumthinkum wrote:
| Maybe that lack of knowledge explains why so many people are
| broke.
|
| Honestly, this is a pretty weird take to see on "Hacker News".
| This place sure has changed a lot.
| jeremyt wrote:
| What kind of math skills are you talking about that people
| lack that causes them to not have any money?
| frogeyedpeas wrote:
| I'd say if someone can't do calculus based statistics then
| a lot of high earning career paths (ex: machine learning,
| data science, actuary, quant) are not available to them.
|
| That doesn't mean you won't be rich. It's just some of the
| lowest hanging fruit are not an option.
| WalterBright wrote:
| Yah, failure to understand statistics is a big risk
| financially.
|
| I remember a rich man interviewed on TV who said he got
| his start making money in high school by running gambling
| games. He understood statistics while the other kids did
| not, and although the game was fair, he cleaned up
| regularly.
|
| Take a walk through a Vegas casino, and you'll see
| legions of people who do not understand statistics and
| pay a heavy price for that.
| pirocks wrote:
| > you'll see legions of people who do not understand
| statistics and pay a heavy price for that.
|
| At the risk of stating the obvious, and not adding to the
| conversation, I think we all know that people putting
| their life savings into slot machines aren't doing so
| because they don't understand expected value. They may or
| may not understand that they are going to lose all their
| money, but they are gambling because they are
| addicted/have some kind of mental health problem.
| Knowledge of statistics doesn't really affect things for
| problem gambling.
|
| As for those putting modest amounts of money into
| gambling, most of them will tell you that card games/etc.
| are fun, and are therefore worth it.
| mquander wrote:
| Many people lack the numeracy to understand basic ideas
| about money and finance, which directly results in them
| getting scammed by banks, brokerages, credit card
| companies, and various hucksters.
| 23B1 wrote:
| And yet mathematics has been a mandatory topic in public
| schools for at least a century if not longer.
|
| We also don't teach car repair, or hunting, or sewing, or
| cooking much anymore either, not because we don't need
| those things but because those high-friction tasks have
| been highly optimized to the point of being background
| noise.
| dinkumthinkum wrote:
| Meta-adaptations and "kink-shaming" about math ... I
| don't know if I walked into some parody of the Silicon
| Valley show or if this is some kind of weird AI bot,
| either way I guess this is the shape of things to come
| ...
| 23B1 wrote:
| _I 'm not like the other girls_
|
| (commenting on the internet _can_ be fun you know ;-) )
| dinkumthinkum wrote:
| I think I am making an obvious point that normal people
| understand. A lot of people have trouble understanding the
| concept of spending less or significantly less than they
| earn. This is another it seems like HN has changed a lot.
| The idea what I'm saying is controversial is pretty
| hilarious and sad at the same time.
| fragmede wrote:
| What's controversial about that is that's simply not
| possible for low wage earners. I'd hazard a guess that it
| doesn't affect high wage readers here, but if you're
| making minimum wage in a HCOL area, _not_ buying big
| screen TVs and _not_ buying luxury cars and shoes isn 't
| enough to make ends meet if you also want to save
| anything for retirement.
|
| That or everyone else is an idiot, but I've found that
| mindset is only good for feeling smug about yourself and
| underestimating people, so let's assume it's not that and
| try to find something else.
| magicalhippo wrote:
| There's broke and there's broke.
|
| One of my in-laws bought a house in the suburbs. She kept
| her low-wage job downtown, despite pay being average for
| her vocation, and she could easily get a job closer to her
| new home.
|
| So now she has a long commute, and decided to get a petrol
| car.
|
| Despite knowing very well that petrol is taxed heavily and
| electricity is cheap here in Norway, petrol cars have
| significantly higher road tax and congestion charge than
| EVs here. The distance she needs to commute is well within
| what even a first-gen Leaf could do during winter, so she
| had plenty of EV options.
|
| She also knew the job had no parking for employees, so she
| has to park at a public parking facility, which downtown
| costs a fortune.
|
| Basic math skills shows that between the petrol, the road
| tax and congestion charge and the parking, that car is
| costing her half her daily paycheck each time she goes to
| work.
|
| Didn't take long after she got the car till she started
| complaining she was "broke each month".
| 23B1 wrote:
| > this is a pretty weird take to see on "Hacker News". This
| place sure has changed a lot.
|
| I mean I've been in 'tech' for ~25 years. The simple fact is
| that technology is a meta-adaptation whose primary purpose to
| make life easier and more enjoyable for humans. I'm not kink-
| shaming math lovers or anything.
| elefanten wrote:
| In many cases, you need to understand the concepts to conceive
| of applying them beneficially to a problem at hand, whether you
| apply them or outsource the application.
| vundercind wrote:
| I think this goes too far, but I do think math is... overrated?
| Kinda? Overrated isn't exactly the word I want, but there's
| definitely something weird going on.
|
| 98% (I'm being generous) of people can no longer work almost
| math past early algebra and maybe a handful of finance-related
| plug-in-the-numbers formulas by age 35 because, assuming they
| ever learned any, they have _never_ used it, so it's gone by
| then. And that state of things seems to be entirely Ok. Like,
| if they needed it, they'd have _used it_ and the many of them
| who could once at least kinda work with calculus, or what have
| you, wouldn't have lost that skill.
|
| Meanwhile, I've not found the "it teaches problem solving
| skills, that's why it's important even if you never use 80% of
| it outside of school" thing to really hold. Maybe for the kinds
| of courses math majors take in college, I dunno, but not for
| the rest. If it does teach any, they don't seem to generalize
| well for almost all people who learn them, and the rest, I
| think that's more about who they are than that they took some
| math courses.
|
| Ultimately, it's not clear to me that if we taught quite a bit
| less math to most kids and even college students, anything bad
| would happen.
|
| I think there are probably ways to approach math in primary and
| secondary school, and maybe also math courses for undergrads
| who have a small load of math courses anyway, that would temper
| its evident uselessness quite a bit--namely, a laser-focus on
| applications past the very earliest grades--but most math
| majors seem to want math education to go exactly the opposite
| way. Maybe they're right and I'm wrong, I dunno.
| WalterBright wrote:
| I used calculus a lot in my mechanical engineering job.
|
| As a programmer, not. But as a programmer, I use a different
| kind of math (such as 2s complement arithmetic, boolean
| logic, floating point math, vectors, graph math, etc.) all
| the time.
|
| Knowing math has blocked many attempts by salesmen,
| contractors, bankers, etc., from ripping me off. If I didn't
| know math, I never would have even realized that my
| tailfeathers had been plucked. As for "anything bad would
| happen", bad things probably happened to you that you were
| not aware of.
|
| An anecdote: years ago, it used to be popular to run 30
| minute seminars on TV called (my version) "Get Rich In Real
| Estate Using Scams". I recall one that bragged about making a
| quick $10,000. I figured it was a con, and so watched the
| show carefully, noting each transaction. And yes, it did net
| a $10,000 score for the person. But how it worked was through
| a confusing combination of transactions meant to obfuscate
| what was actually happening. The key in it was getting your
| mark to accept a bond that would be worth $XXXX in the future
| while you got the $XXXX today. In essence, it was exploiting
| the mark's failure to understand the concept of current value
| vs future value. The beauty (if you could call it that) was
| there was nothing illegal about this.
|
| With my math knowledge, it stunk from the outset, even though
| it took me a while to find the dead rat. Just like with my
| knowledge of physics, when it was posted on HN that electric
| cars were 90% efficient, that set me off immediately, and
| sure enough, there was a rat corpse in it. (The actual
| efficiency is 60% on a good day.) I was shocked at the well-
| educated people who bought that article hook, line, and
| stinker.
|
| (The cake topper on that one was the author was a ski
| instructor!)
| mamcx wrote:
| > it's not clear to me that if we taught quite a bit less
| math to most kids and even college students, anything bad
| would happen.
|
| My grandparents never passed grade 4 of primary school.
|
| They _absolutely_ *crushed* anybody, including university
| people with supposedly strong math inclination, in math and
| solving skills. Basic arith and probably a bare-acquired
| intuition of some algebra.
|
| They could do everything in their heads, buy things, make
| deals, and could dance around most people with riddles and
| stuff like that some were math-related. I remember one of
| them that around 15 people of later generations were trying
| to solve (like for a week), and only one did it. (remember,
| there was large family and friends, I have 6 uncles)
|
| Even those around 80-90 years old still crush it.
|
| No, they were not savants. Other grandparents of that
| generation were like that.
|
| And their sons could do better than grandsons. I need
| machines to help me. And I was the #1 in school.
| WalterBright wrote:
| Failure to understand math leads to a lifetime of poor
| financial decisions. I've found this to be consistent in my
| experience.
|
| For a small example, ever watch "Shark Tank" on TV? The sharks
| are constantly throwing out ROI, valuations, percentages,
| interest rates, and it's clear the sharks understand the math
| behind it implicitly, and how each of those numbers relates to
| the other numbers.
|
| With the rapid fire back-and-forth with the acolyte, it's clear
| they're at a severe disadvantage if they cannot keep up. If the
| acolyte were to whip out a calculator, it's pretty clear that
| would be "no deal".
| 23B1 wrote:
| > Failure to understand math leads to a lifetime of poor
| financial decisions
|
| Failure to understand a few basic mathematical principles
| leads to a lifetime of poor financial decisions, but the
| underlying math is something a 10 year old can handle - as
| can anyone with a calculator.
| WalterBright wrote:
| A calculator is of little use to someone who does not
| understand mathematical principles. I've seen that in
| action many times.
| xanderlewis wrote:
| Why are there always so many commenters on here who don't seem
| to understand the difference between 'mathematics' and
| 'arithmetic'?
| 23B1 wrote:
| Why are there always so many commenters on here who don't
| seem to understand the difference between "hacker news" and
| the "real world"?
|
| Outsourcing low-value/high friction tasks is the whole point
| of technology.
| ibash wrote:
| This matches up with programming too.
|
| You can teach software engineering in school. But you become an
| expert by reading source code and seeing the many ways to solve a
| problem.
|
| An expert can intuit a solution because of pattern matching. And
| their argument is that math is the same.
| Jtsummers wrote:
| It's the "$10,000 for knowing which screw to turn" problem. A
| non-domain-expert (but good general problem solver) could
| eventually come up with the solution, but they'll take longer.
| They have to work out a solution either by trial & error (most
| common) or from first principles (very rare). Either way takes
| longer than letting an expert look at it and pull a solution
| seemingly out of thin air, when the reality is it's the decade
| or decades of experience looking at similar problems that they
| draw from.
| dfee wrote:
| > But you become an expert by reading source code and seeing
| the many ways to solve a problem.
|
| More so by iteratively building, at least so for me.
| orthopodvt wrote:
| So there is such a thing as "knowledge". Learning problem solving
| skills in the absence of subject-matter knowledge is simply a
| Markov walk exercise.
|
| edit - corrected spelling
| thundergolfer wrote:
| This is the central claim of E.D Hirsch's _Why Knowledge
| Matters_ [1] book on educational reform. Hirsch is perhaps best
| known for coining the term "cultural literacy" in his book of
| the same name.
|
| A little while back I wrote about cultural literacy in the
| software industry, following the lead of Hirsch's book.[2]
|
| 1. https://hep.gse.harvard.edu/9781612509525/why-knowledge-
| matt...
|
| 2.
| https://thundergolfer.com/software/culture/2024/01/14/comput...
| xqcgrek2 wrote:
| 10000 hours to mastery is what it takes, this is not new news
|
| However, everyone wants shortcuts, specially recent generations
| with short attention spans
|
| Do your 10k hours conscientiousnessly in a specific domain and
| you're automatically at a huge advantage in the current market
| spacecadet wrote:
| 10,000 hours also isn't that long in the grand scheme of
| things. I remember when I broke 10,000 in the arts, then 10,000
| writing code, and so on, I dont even keep track now.
| kragen wrote:
| '10000 hours to mastery' is incoherent bullshit. 10000 hours to
| mastery of programming? to mastery of programming numerical
| methods? to mastery of programming gauss-seidel elimination? to
| mastery of programming sparse gauss-seidel elimination on
| vector supercomputers? to mastery of programming sparse gauss-
| seidel elimination on vector supercomputers for fluid
| mechanics? if your first 10000 hours were in c++, do you need
| another 10000 hours to get to mastery of programming sparse
| gauss-seidel elimination on vector supercomputers for fluid
| mechanics in fortran? at most one of these can be correct for a
| given level of mastery and for a given person (and that's not
| even getting into variations between people)
|
| like most things gladwell made up, it sounds good at first but
| falls apart the moment you think about it for a second
| paulpauper wrote:
| It is wrong and was debunked. Some people need far fewer than
| 10k and others never become good no matter how long. There so
| much variability it is useless as a heuristic.
| frogeyedpeas wrote:
| This comes down to the old saying "everything is memorization at
| the end of the day".
|
| Some people literally memorize answers. Other folks memorize
| algorithms. Yet other folks memorize general collections of
| axioms/proofs and key ideas. And perhaps at the very top of this
| hierarchy is memorizing just generic problem solving
| strategies/learning strategies.
|
| And while naively we might believe that "understanding is
| everything". It really isn't. Consider if you are in the middle
| of a calculus exam and need to evaluate $7 \times 8$ by
| calculating $7+7+7+7...$ and then proceed to count on your
| fingers up to 56 because even $7+7$ wasn't memorized. You're
| almost certainly not going to make it past the first problem on
| your exam even though you really do understand exactly whats
| going on .
|
| Similar things are true for software engineering. If you have to
| stackoverflow every single line of code that you are attempting
| to write all the way down to each individual print statement and
| array access it doesn't fucking matter HOW well you understand
| whats going on/how clear your mental models are. You are simply
| not going to be a productive/useful person on a team.
|
| At some point in order to be effective in any field you need to
| eventually just KNOW the field, meaning have memorized shortcuts
| and paths so that you only spend time working on the "real
| problem".
|
| To really drive the point home. This is the difference between
| being "intelligent" versus "experienced".
| throwuxiytayq wrote:
| > And perhaps at the very top of this hierarchy is memorizing
| just generic problem solving strategies/learning strategies.
|
| I'm not sure this counts as memorization. I don't even think
| you can _really_ "memorize" high level learning and problem
| solving strategies, even when explained by an expert. You kind
| of have to re-discover them internally. And then, there are
| people who "memorized" the explanation and are completely
| unable to put it into practice because to them it's just a word
| sequence, instead of an internalized change to the way you
| perceive and work with problems.
| frogeyedpeas wrote:
| You absolutely can. I remember struggling with some problems
| on AOPS and then reading in a book "always consider smaller
| $n$ when dealing with a problem that is difficult because of
| large $n$" and ever since then that habit has stuck. Whenever
| I have a problem thats hard and involves numbers and i'm
| stuck I just remember to ask "what if the numbers were
| smaller? what do we do then?"
|
| If that isn't memorizing something and making a new habit as
| a kid then I don't know what memorizing means.
|
| Said another way, the ability to remember to "____" when
| dealing with a problem of type "___" is what I mean by
| "memorize".
| throwuxiytayq wrote:
| > Whenever I have a problem thats hard and involves numbers
| and i'm stuck I just remember to ask "what if the numbers
| were smaller? what do we do then?"
|
| I think you underestimate the amount of internalized
| understanding of the "unblock yourself on a difficult
| problem by solving a simpler version of it" strategy that
| you possessed or unlocked at learn-time which allowed you
| to notice its effectiveness. Isn't the sentence more of an
| easily-retrievable mnemonic for a concept that's much more
| complicated (than just the information transferred by
| language) and requires a particular background to recognize
| how useful it is?
| nickpsecurity wrote:
| They're called heuristics in problem-solving literature. Both
| heuristics and meta-heuristics have been used in planning
| software. Heuristics from one system are sometimes reused in
| another system. So, you can memorized generic, problem-
| solving strategies.
|
| I don't know how much human brains do in that area vs non-
| memorization approaches. Ive read about how practicing
| rational, problem solving in specific domains to bake those
| heuristics into one's intuition for faster responses. Most of
| us have done that, too. Any type of intuitive, problem
| solving probably involves memorization for that reason.
| mhh__ wrote:
| I think the antidote is driving education as a journey through
| the great questions of history.
|
| What was Newton trying to do? What Faraday investigating?
| Darwin? Smith? Marx? Descartes and so on.
|
| Everything is connected and there is something interesting for
| everyone, we just don't try.
| bitshiftfaced wrote:
| Nah, there's such a thing as creative thinking, idea
| generation, and connecting existing ideas in new ways. I
| wouldn't mind a coder that has to look at stack overflow a lot
| but is able to figure out a new method to do something better.
| frogeyedpeas wrote:
| You absolutely would never hire a coder that needs to google
| "how to access an array by index" every-time they need to
| access an index of an array.
|
| You can say a politically correct answer like "i don't care
| how they do it, as long as they get it done" but such a coder
| will DEFINITELY take months to finish what might take someone
| else hours.
|
| Such a coder might still be able to suggest new methods to do
| something better and if there job description was
| "organizational optimizer" perhaps thats fine but as soon as
| you also expect software output out of this person you will
| quickly realize that you take for granted how valuable
| someone that has fully memorized a bunch of fundamentals up
| to and including some problem strategies truly is.
| youerbt wrote:
| That makes no sense to me. If this coder has to access
| array by index twenty times a day, then he is going to
| remember it, eventually, no? If is it rare that he has to
| do it, then why memorize it?
|
| You really think there is more value in remembering how to
| do something in some arbitrary, shitty, programming
| language than understanding the concept of doing it? With
| understanding the idea you can do it in any language, at
| any time, it is just a few seconds away.
| mrmetanoia wrote:
| It makes no sense because it indeed makes no sense.
| People who successfully solve realworld problems
| understand concepts and ideas and how to apply them, they
| understand how to iterate and extrapolate.
|
| I've met too many people who can do a specific thing but
| actually have no idea what's going on for the GP's logic
| to hold any water at all.
| kiba wrote:
| It's not about the value in remembering syntax. It's the
| value in being able to recall a concept from memory.
|
| Memory is a key part of learning. Understanding is great
| for learning new concepts, but you want to already know a
| concept. That way lies knowledge and experience.
| paulpauper wrote:
| agree. how else do famous unsolved math problems eventually
| get solved?
| brigadier132 wrote:
| Search.
| math_dandy wrote:
| It varies, but it often comes down to deep expertise
| combined with creativity, years of toil, and standing on
| the shoulders of giants. Cf. Fermat's Last Theorem, bounded
| gaps between primes, the Weil conjectures, the Poincare
| conjecture, etc.
| chmod775 wrote:
| > Consider if you are in the middle of a calculus exam and need
| to evaluate $7 \times 8$ by calculating $7+7+7+7...$ and then
| proceed to count on your fingers up to 56 because even $7+7$
| wasn't memorized. You're almost certainly not going to make it
| past the first problem on your exam even though you really do
| understand exactly whats going on.
|
| This is not a counterexample because exams aren't an end goal.
| The process of filling out exams isn't an activity that
| provides value to society.
|
| If an exam poorly grades a student who would do great solving
| actual real-world problems, the exam is wrong. No ifs. No buts.
| The exam is wrong because it's failing the ultimate goal:
| school is supposed to increase people's value to society and
| help figure out where their unique abilities may be of most
| use.
|
| > Similar things are true for software engineering. If you have
| to stackoverflow every single line of code that you are
| attempting to write all the way down to each individual print
| statement and array access it doesn't fucking matter HOW well
| you understand whats going on/how clear your mental models are.
| You are simply not going to be a productive/useful person on a
| team.
|
| If their mental models are truly so amazing, they'd make a
| great (systems) architect without having to personally code
| much.
| sim04ful wrote:
| I can't totally agree with your counter-counter example. Most
| non trivial problems are time bound, deadline exist, and no
| matter how well ingrained you are in first principles
| thinking you won't be useful if it takes months to come up
| with a solution.
| frogeyedpeas wrote:
| > Re: "this is not a counter example because exams aren't an
| end goal..." for any end goal with a set end time there are
| habits that need to be second nature and information that one
| needs to know in order to achieve that goal. If you lack
| those habits and don't know those facts it's going to be very
| hard to achieve that goal.
|
| I used the example of a calculus test and not being able to
| do addition. But this really could be any example. It could
| have even been a Wide Receiver failing to read the play thats
| happening quickly enough despite being physically fit enough
| to execute the right play in hindsight.
|
| >Re: they'd make a great (systems) architect...
|
| But you wouldn't hire them as a programmer. My sentence was
| biased in the sense that "team" meant "team of software
| engineers". You would hire them for a different job sure.
|
| Also good mental model here just means "Always knowing and
| being able to clearly articulate what I need to accomplish
| next to write my code". It doesn't even mean they are good at
| designing systems but lets go with that example anyways
| below:
|
| The Architect version of this is that they perhaps have
| perfectly clear mental models of exactly how to code
| (memorizing very obscure language shortcuts and syntactic
| sugar and writing very clear code when they know what to
| build) but they cannot for the love of god think critically
| about what a design should be BEFORE they implement it far
| enough to reach a major issue.
|
| And you would rightly say "well I would never hire that guy
| as an architect but I might have hired them as a programmer
| thats led by more senior folks". At the end of the day you
| are only hiring people for the parts of their mental models
| that are useful.
|
| And the ability to clearly recall facts about that their
| domain is basically the fundamental detail here.
| 1659447091 wrote:
| I agree with you that memorization is an optimization for
| getting daily task done (maybe not as optimal when novel
| solutions are needed; understanding/mental model might win
| out here). But we have tools to help take the load off
| memorization. The person that `understands` addition not as
| 7 + 7 but as incrementing a number a certain amount of
| times can use a calculator to solve the problem in a more
| efficient way.
|
| I would probably not make a developer who had great mental
| models but lacked coding chops my first hire. Nor the
| programmer that could make code do amazing things but can
| not grasp the domain model. I would, however, probably
| consider them(the mental model one) the 100th to clean up
| backlogged bug fixes, and the code whiz to implement the
| more technically difficult backend niche
| feature/optimization. As much as it pains me to say it,
| github copilot chat works surprisingly well IF you can give
| it a clear concise description of the model and
| expectations. Then someone with an excellent mental model
| can create the smaller lego pieces and put it together,
| minimal coding required. Not only for the popular
| languages, I play with it from time to time using clojure.
| skhunted wrote:
| To know something includes speed of regurgitation. Consider a
| trauma surgeon. You want them to know, off the top of their
| head, lots of stuff. You don't want them taking their time
| and looking things up. You don't want them redefining
| everything from first principles each time they perform
| surgery.
|
| Knowing a topic includes instant recall of a key body of
| knowledge.
| westurner wrote:
| Maybe survey engineers with a first order derivative
| question and a PDE question n years after graduation with
| credential?
|
| CAS and automated tests wins again.
|
| A robosurgeon tech that knows to stop and read the docs and
| write test assertions may have more total impact.
| 1659447091 wrote:
| I would say knowing and understanding is not necessarily
| the same. In this example the surgeon having both
| understanding and memory/knowing is best/required. If I had
| to pick between the two, I want the one that understands my
| particular trauma, even if that means they have give
| instructions for someone else or a machine to performing
| it.
|
| I think an example closer to the above posts would be: If I
| needed cpr or defibrillation, I would much prefer a
| paramedic be next to me and make that call and performance
| than a med student or a defibrillator manufacture's
| electrical engineer.
| pfortuny wrote:
| You can only think using memory.
| FredPret wrote:
| Maybe understanding is simply having memorized a handy
| instantiation of the relevant concept
| drewcoo wrote:
| > This comes down to the old saying "everything is memorization
| at the end of the day".
|
| I certainly don't remember hearing that!
| lo_zamoyski wrote:
| > At some point in order to be effective in any field you need
| to eventually just KNOW the field, meaning have memorized
| shortcuts and paths so that you only spend time working on the
| "real problem".
|
| Yes, there is a "habitus" to mastery. It becomes you, or you
| become it, so to speak.
|
| But pedagogically speaking, I think what people miss is that
| you can't really use or think about something you don't
| remember.
| jltsiren wrote:
| I'd say memorization and building expertise are orthogonal.
|
| Expertise is lossy intuitive reasoning. It's pattern
| recognition based on practice and experience. Then there is
| logical reasoning based on memorized facts, which is a fallback
| mechanism people use when they don't have the necessary skills.
| It usually fails, because it's inefficient, it doesn't scale,
| and it doesn't generalize.
|
| Sometimes memorization is necessary, but it's often not the
| actual point. When kids are asked to memorize the
| multiplication table, they are not really supposed to memorize
| it. They are supposed to build a mental model for multiplying
| numbers without resorting to first principles or memorized
| answers. Then if your model can calculate 7 * 8, you can also
| use it to calculate 7e10 * 8e11, even if you haven't memorized
| that specific fact.
| brigadier132 wrote:
| > It's pattern recognition based on practice and experience
|
| This is arguably another form of memorization. Magnus Carlson
| is the best Chess player in the world because he memorizes
| everything without effort.
| iwsk wrote:
| When kids are asked to memorize the multiplication table,
| they are actually supposed to memorize it.
| tsimionescu wrote:
| The multiplication table doesn't have patterns, or it only
| has a few. You really do need to remember all of the 100
| results. I know what 7*8 is, and I know the rules for
| exponents, so I can compute 7e10*8e11. But I can't "deduce"
| what 7*8 is by any rule, it's just a fact I remember. I have
| certainly not added 7 to itself 8 times in decades.
| LouisSayers wrote:
| > I can't "deduce" what 7*8 is by any rule
|
| But you can break this into a different problem knowing
| that 2^3 = 8, and doing 7*2*2*2.
|
| This isn't as fast but is in a way more useful because
| while 7*8 is fairly easy to remember you're not going to
| remember 17*8 etc but you can problem solve it fairly
| quick.
|
| There are other ways of seeing the multiplication table as
| well. For example 9 times something can be thought of as
| 9*x = 10*x-x.
|
| I never learnt these, but simply realised over time that
| there are different approaches to doing calculations.
| Dylan16807 wrote:
| > But you can break this into a different problem knowing
| that 2^3 = 8, and doing 7*2*2*2.
|
| Doing that multiplication all the way through is super
| slow. When they said "can't" they meant in an effective
| sense, since they did mention repeated addition as an
| option. And that's not an effective way to get there.
|
| > There are other ways of seeing the multiplication table
| as well. For example 9 times something can be thought of
| as 9*x = 10*x-x.
|
| Yes, you can do that one. But that's just about the only
| fast trick there is.
| wizzwizz4 wrote:
| > _If you have to stackoverflow every single line of code that
| you are attempting to write all the way down to each individual
| print statement and array access_
|
| Then you may be a perfectly adequate programmer. This, what,
| _doubles_ the length of time it takes to type out the program?
| _Triples_? Typing out the program is not what takes the time!
|
| I've just spent a couple of days writing a plugin in a language
| I _don 't know_. (The system documentation spends _two
| paragraphs_ explaining how hard it is to solve the problem I
| solved.) Yes, I had to look up _absolutely everything_
| (including basic language syntax - repeatedly), and that was
| really annoying, but most of my time and effort went into
| figuring out _how_ to do the thing.
| kiba wrote:
| You already have programming knowledge that you can use to
| leverage toward that task. For a complete beginner, such a
| project might be a non-starter.
|
| Like, once you learn a programming language, you already know
| the syntax for 90% of all languages.
| TeMPOraL wrote:
| Memorization is caching. You need it because otherwise you'd be
| too slow at anything, but you can't possibly memorize
| everything, and the whole point of understanding is so you
| don't have to. And like with any caching, the more you store,
| the more it costs to maintain it, and the longer the lookups
| become. If you want to cram a lot of stuff into it, you may
| need to start doing _actual, expensive work_ - e.g. spaced
| repetition - to keep it all.
|
| AS for memorizing generic problem solving strategies - I don't
| think it's about not memorizing, but rather that understanding
| comes through examples, and if you learn high-level stuff
| without actually applying it in practice, and experiencing the
| process, then you haven't actually learned the high-level
| stuff, you just think so, and will parrot the description
| without comprehending it.
| bbor wrote:
| I love this long detailed conversation with many people jumping
| in, and 0 references to philosophers of the mind... gee guys, I
| wonder how we could crack this code? Even the paper itself
| cites one cognitive psychologist then moves on! A bit of
| relatable intellectual arrogance from us SWEs/mathematicians, I
| think -- we are "on top of the world" right now.
|
| FWIW I think you in particular are exactly right. I always
| think of Schopenhauer's quote, and I think any software
| engineer might appreciate it: human memory isn't storing items
| received from the world in a database, it's more like folding
| creases into a napkin so that it naturally tends to fall into
| that shape in the future. In other words: remembering an event
| is equivalent to developing the skill of imagining a
| scene/dataframe that relates to that event.
|
| In specific math terms: math is a collection of intellectual
| tools building on one another. You can certainly practice the
| ability to apply tools in new situations, but if you don't also
| practice the ability to recall the tools themselves, it's
| useless.
| fallingknife wrote:
| But is that actually what human memory is like? AFAIK nobody
| actually understands the internals. The "philosophers of the
| mind" who claim to know are the ones guilty of arrogance, not
| those who don't cite them.
| bbor wrote:
| Well, we should collect some evidence and write a book! If
| we did, it would be filed into the philosophy of mind
| section, I believe ;)
|
| We don't know everything, but we have more evidence than
| "it's a black box" - in fact, that's basically the
| scholastic / Aristotlean view that was conquered by our
| friends Bacon, Hume and Kant a few hundred years ago.
| superposeur wrote:
| To support your point, I think the role of memory in creative
| work is highly underrated.
|
| I've seen up close a few people who could fairly be described
| as "most creative researchers in the world" (in my field at
| least) according to metrics such as h-index and Nobel prizes.
| It always strikes me how essential exceptional memory is to
| what they do -- they have detailed, immediate recall of the
| problems in their field and, to the extent this recall is no
| longer present, then they are no longer producing
| groundbreaking work. Their recall of facts outside the field is
| often nothing special.
|
| Imagination, creativity, intelligence all seem to _rely_ on
| memory in order to operate.
| greentxt wrote:
| "everything is memorization at the end of the day"
|
| Only somebody who has never thought about or studied human
| cognition would memorize such a thing. ;)
|
| But in all seriousness, memory isn't even memory isn't just
| memorization. Much of it is attention, some would even say
| attention is all you need. ;)
|
| In all seriousness though, arguably, reducing the human mind
| down to a single dimension like "recall" (or attention) while
| ignoring other dimensions like emotion, creativity and so on is
| probably good evidence that human cognition is neither simple,
| nor unidimensional, for some of us humans at least. Ymmv
| SOTGO wrote:
| Anecdotally I have found this to be the case for the students I
| tutor. When I introduce a new topic I always start with worked
| examples, and I find that students are able to learn much more
| effectively when they have a reference. Poor pedagogy is also one
| of my biggest gripes with my undergraduate math program too,
| where the professors and textbooks often included too few worked
| problems and proofs, and the ones they did include were not very
| useful. What I found especially frustrating was when a worked
| example solved a special case with a unique approach, and the
| general case required a much more involved method that wasn't
| explained particularly well. Differential equations seems to be a
| particularly bad offender here, since I've had the same issue
| with the examples in many texts.
| JustinSkycak wrote:
| > What I found especially frustrating was when a worked example
| solved a special case with a unique approach, and the general
| case required a much more involved method that wasn't explained
| particularly well.
|
| Amusingly, many people think the solution to this is "abandon
| worked examples and focus exclusively on trying to teach
| general problem-solving skills," which doesn't really work in
| practice (or even in theory). That seems to be the most common
| approach in higher math, especially once you get into serious
| math-major courses like Real Analysis and Abstract Algebra.
|
| What actually works in practice is simply creating more worked
| examples, organizing them well, and giving students practice
| with problems like each worked example before moving them onto
| the next worked example covering a slightly more challenging
| case. You can get really, really far with this approach, but
| most educational resources shy away from it or give up really
| early because it's so much damn work! ;)
| kiba wrote:
| Teaching a skill directly is known to be more a more
| efficient way of learning rather than force students to try
| to discover it on their own.
| smogcutter wrote:
| Interestingly, there have been studies that show that
| students lectured to _feel like_ they've learned more, and
| self-report that they have, while students learning the
| same material in self-guided labs report feeling like
| they've learned less but perform _better_ on assessments.
| catgary wrote:
| Eh, I think that's setting students up for failure once they
| enter graduate studies or more open ended problems that don't
| come from a problem bank. Productive struggle is a perfectly
| valid approach to teaching, it's just less pleasant in the
| moment (since the students are expected to struggle).
| nrr wrote:
| This is true (i.e., the struggle is productive) only if the
| struggle allows for students to develop the intuition of
| the subject required for synthesis.
|
| Even then, before you get to that point, you have to prime
| students for it. Throwing them into the deep end without
| teaching them to float first will only set them up to
| drown. This does typically mean lots of worked motivating
| (counter-)examples at the outset.
|
| It's a big reason why we spent so long on continuity and
| differentiability in my undergraduate real analysis class
| and why most of the class discussion there centered on when
| a function could be continuous everywhere but nowhere
| differentiable. Left to our own devices and without that
| guidance, our intuition would certainly be too flawed for
| such a fundamental part of the material.
| JustinSkycak wrote:
| > Productive struggle is a perfectly valid approach to
| teaching
|
| Is this supported by research though? As I understand it,
| for students (not experts), empirical results point in the
| opposite direction.
|
| One key empirical result is the "expertise reversal
| effect," a well-known phenomenon that instructional
| techniques that promote the most learning in experts,
| promote the least learning in beginners, and vice versa.
|
| It's true that many highly skilled professionals spend a
| lot of time solving open-ended problems, and in the
| process, discovering new knowledge as opposed to obtaining
| it through direct instruction. But I don't think this means
| beginners should do the same. The expertise reversal effect
| suggests the opposite - that beginners (i.e., students)
| learn most effectively through direct instruction.
|
| Here are some quotes elaborating on why beginners benefit
| more from direct instruction:
|
| 1. "First, a learner who is having difficulty with many of
| the components can easily be overwhelmed by the processing
| demands of the complex task. Second, to the extent that
| many components are well mastered, the student will waste a
| great deal of time repeating those mastered components to
| get an opportunity to practice the few components that need
| additional practice.
|
| A large body of research in psychology shows that part
| training is often more effective when the part component is
| independent, or nearly so, of the larger task. ...
| Practicing one's skills periodically in full context is
| important to motivation and to learning to practice, but
| not a reason to make this the principal mechanism of
| learning."
|
| ^ from Radical Constructivism and Cognitive Psychology
| (Anderson, Reder, & Simon, 1998) - https://www.andrew.cmu.e
| du/user/reder/publications/98_jra_lm...
|
| 2. "These two facts -- that working memory is very limited
| when dealing with novel information, but that it is not
| limited when dealing with organized information stored in
| long-term memory -- explain why partially or minimally
| guided instruction typically is ineffective for novices,
| but can be effective for experts. When given a problem to
| solve, novices' only resource is their very constrained
| working memory. But experts have both their working memory
| and all the relevant knowledge and skill stored in long-
| term memory."
|
| ^ from Putting Students on the Path to Learning (Clark,
| Kirschner, & Sweller, 2012) -
| https://files.eric.ed.gov/fulltext/EJ971752.pdf
|
| And some other references:
|
| * Why Minimal Guidance During Instruction Does Not Work: An
| Analysis of the Failure of Constructivist, Discovery,
| Problem-Based, Experiential, and Inquiry-Based Teaching - h
| ttps://www.tandfonline.com/doi/pdf/10.1207/s15326985ep4102_
| ...
|
| * Should There Be a Three-Strikes Rule Against Pure
| Discovery Learning? The Case for Guided Methods of
| Instruction - https://app.nova.edu/toolbox/instructionalpro
| ducts/ITDE_8005...
|
| Intuitively, too: in an hour-long session, you're going to
| make a lot more progress by solving 30 problems that each
| take 2 minutes given your current level of knowledge, than
| by attempting a single challenge problem that you struggle
| with for an hour. (This assumes those 30 problems are
| grouped into minimal effective doses, well-scaffolded &
| increasing in difficulty, across a variety of topics at the
| edge of your knowledge profile.)
|
| To be clear, I'm not claiming that "challenge problems" are
| bad -- I'm just saying that they're not a good use of time
| until you've developed the foundational skills that are
| necessary to grapple with those problems in a productive
| and timely fashion.
| magicalhippo wrote:
| > What I found especially frustrating was when a worked example
| solved a special case with a unique approach, and the general
| case required a much more involved method that wasn't explained
| particularly well.
|
| That was the bane of my University degree. "And, since our
| function f happens to be of this form, all the difficult stuff
| cancels out and we're left with this trivial stuff" and then
| none of the problems have these "happy accident" cancellations
| and you're none the wiser on how to proceed.
|
| The statistics book we used was an especially egregious
| offender in this regard.
| dan-robertson wrote:
| I think often the reason this happens is that the chosen
| examples[1] are just more advanced topics in disguise. Eg
| maybe you are given some group with a weird operation and
| asked to prove something about it, and the hidden thing is
| that this is a well-known property of semi-direct products
| and that's what the described group is.
|
| Two I remember were:
|
| - In an early geometry course there was a problem to
| prove/determine something described in terms of the Poincare
| disc model of the hyperbolic plane. The trick was to convert
| to the upper half-plane model (where there was an obvious
| choice for which point on the boundary of the disc maps to
| infinity in the uhp). There I was annoyed because it felt
| like a trick question, but the lesson was probably useful.
|
| - in a topology course there was a problem like 'find a space
| which deformation-retracts to a mobius strip and to an
| annulus. This is easy to imagine in your head: a solid torus
| = S1*D2 can contain an embedding of each of those spaces into
| R3. I ended up carefully writing those retractions by hand,
| but I think the better solution was to take the product space
| and apply some theorems (I think I'm misremembering this -
| product space works for an ordinary retraction but for the
| deformation retraction I don't think it works. I guess both
| retract to S1 and you could glue the two spaces together
| along that, or use the proof that homotopy equivalence <=>
| deformation retracts from common space, but I don't think we
| had that). I felt less annoyed at missing the trick there.
|
| [1] I'm really talking about exercises here. I don't really
| recall having problems with the examples.
| will1am wrote:
| The importance of worked examples in helping students
| understand new topics
| cschmidt wrote:
| The "worked example effect" they talk about it interesting. The
| idea that you learn best from worked examples lines up with my
| experience. However, it seems like higher math abandons this
| completely. So many math textbooks are just in "theorem, proof"
| form, with almost no examples or even motivation.
| JustinSkycak wrote:
| This is one reason why so many people struggle with higher
| math. Textbooks & classes are typically not aligned (and often,
| are in direct opposition) to decades of research into the
| cognitive science of learning.
|
| Not saying that higher math would be "easy" if taught properly.
| Just that many more people would be able to learn it, than are
| currently able to learn it.
|
| Higher math is heavily g-loaded, which creates a cognitive
| barrier for many students. The goal of guided/scaffolded
| instruction is to help boost students over that barrier. Of
| course, the amount of work it takes to create a textbook
| explodes with the level of guidance/scaffolding, so in practice
| there's a limit to the amount of boosting that is feasible,
| especially if the textbook is written entirely by a single
| author... but most textbooks don't even come close to the
| theoretical limit for a single author, much less the
| theoretical limit for a team of content writers.
| will1am wrote:
| It is challenging for learners who benefit from examples
| Silamoth wrote:
| I feel this article's argument is weak, largely for one key
| reason: They don't clearly define anything. Their references
| might clarify some things, but not all. They argue against
| "general problem-solving strategies" with a reference to Polya,
| but they don't provide a clear definition of what these
| strategies entail. How broad is the set of strategies they're
| arguing against? What are some examples of such strategies? I'd
| like something beyond two sentences on Polya.
|
| Furthermore, what audience and level of mathematics education are
| we discussing? The goals (and hence appropriate metrics of
| success) are certainly different for high schoolers targeting
| non-STEM careers vs. engineering undergrads vs. math grad
| students. The authors reference "aspiring mathematicians" and
| "domain specific mathematical problem-solving skills", indicating
| they're arguing about education for math majors, or at least
| students in STEM fields. In that case, the argument is somewhat
| meaningless - who's arguing math majors shouldn't learn math-
| specific skills? But, as I understand it, the argument for
| general problem-solving skills is that students outside of math
| don't actually need many specific math skills. Instead, math is a
| vessel for teaching logic, reasoning, and problem-solving skills.
| Then again, this might not be the type of problem-solving the
| authors are referencing - as I said above, it's not very clear.
|
| On a similar note, they cite evidence that studying worked
| examples is more effective than "general problem-solving
| strategies", citing an "improvement in subsequent problem-solving
| performance" without explaining how this performance is measured.
| If students are tested on specific problem types, of course
| they'll perform better when taught strategies for those specific
| problem types. But it's not clear that this is meaningful. For
| STEM majors, sure, solving specific problems is a skill worth
| cultivating. But for most students, solving specific problems
| isn't as important as learning logic, reasoning, and general
| problem-solving skills. In my anecdotal experience tutoring math,
| students tend to just memorize strategies for specific problem
| types instead of learning transferable logic and reasoning skills
| because that's what's tested. I'd be curious to see which method
| of learning facilitates better performance on a more general
| problem-solving test of some sort.
|
| Now, I'm not an education researcher or an educator of any sort.
| But I am passionate about good STEM education, especially in
| math. I genuinely feel that math education fails most students,
| at least here in America. If I'm being generous, this article is
| a well-intentioned but poorly-executed argument for effective
| math education strategies. If I'm not being so generous, this
| article advocates for the status quo in math education that
| forces students to slog through years of math classes for little
| discernible benefit. Either way, it's a disappointing article
| with a poorly-explained thesis.
| zeroimpl wrote:
| > Furthermore, what audience and level of mathematics education
| are we discussing?
|
| I wonder this too, I think they might mean university-level as
| well. For younger audiences, I feel one of the biggest problems
| for most people to understand math is they don't understand why
| any of it is relevant. If educators can make it seem more like
| teaching general problem solving abilities, that will likely
| improve the overall acceptance and lead to better overall math
| skills as a result.
|
| As a specific example, our high-school math curriculum taught a
| lot of calculus, but framed it incorrectly as being a useful
| tool that people would use. Eg as if a business man would write
| down an equation for their revenue based on inputs, and then
| take the derivative to compute the maximum. I'm assuming they
| told students this to try and get them motivated, but it
| clearly was a lie since everybody knows you could just plot a
| graph and look at it to find the maximum. If they instead were
| honest that the point of learning calculus was to help with
| understanding more advanced concepts in
| math/engineering/science, while also being a valuable learning
| tool for general problem solving, I think that would have been
| a better result.
| graycat wrote:
| > As a specific example, our high-school math curriculum
| taught a lot of calculus, but framed it incorrectly as being
| a useful tool that people would use.
|
| One day at FedEx the BoD (board of directors) was concerned
| about the future of the company and as part of that wanted an
| estimate of the likely growth of the company.
|
| In the offices there were several efforts, free-hand, wishes,
| hopes, guesses, what the marketing/selling people thought,
| etc., and none of those efforts seemed to be objective or
| with a foundation or rationality.
|
| We knew the current revenue. We could make an okay estimate
| of revenue when all the airplanes were full. So, the problem
| was essentially to interpolate over time between those two
| numbers.
|
| For the interpolation, how might that go? That is, what, day
| by day, would be driving the growth? So, notice that each day
| current customers would be shipping packages, and customers
| to be would be receiving them and, thus, learning about FedEx
| and becoming customers. That is, each day the growth would be
| directly proportional to (1) the number of current customers
| creating publicity and (2) the number of customers to be
| receiving that publicity.
|
| So, for some math, let t be time in days, y(t) the revenue on
| day t, t = 0 for the present day, and b the revenue when all
| the planes were full. Then for some constant of
| proportionality k, we have y'(t) = k y(t)
| (b - y(t))
|
| where y'(t) = dy/dt the calculus first derivative of y(t)
| with respect to t.
|
| A little calculus yields the solution.
| y(t) = y(0) b exp(bkt) / ( y(0)( exp(bkt) -
| 1) + b))
|
| Seeing how the growth goes for several values of k, pick one
| that seems reasonable. Draw the graph and leave it for the
| BoD.
|
| That was a Friday, and the BoD meeting started at 8 AM the
| next day, Saturday.
|
| First thing at the meeting, two crucial BoD members asked how
| the graph was drawn. For several hours, no one had an answer.
| The two members gave up on FedEx, got plane tickets back to
| Texas, returned to their rented rooms, packed, and as a last
| chance returned to the BoD meeting. FedEx was about to die.
|
| I did all the work for the graph, the idea, calculus,
| arithmetic (HP calculator), but didn't know about the BoD
| meeting. Someone guessed that I did know about the graph, and
| I got a call and came to the meeting. The two crucial BoD
| members were grim, standing in the hallway with their bags
| packed, and their airline tickets in their shirt pockets.
|
| I reproduced a few points on the graph, and FedEx was saved.
|
| So, some math saved a business.
| xanderlewis wrote:
| > The superiority of chess masters comes not from having acquired
| clever, sophisticated, general problem-solving strategies but
| rather from having stored innumerable configurations and the best
| moves associated with each in long-term memory.
|
| I guess that's why we don't seem to hire chess players as
| generals or... really, anything else. Being good at chess --
| whilst it clearly necessitates a certain level of intelligence --
| is basically just _being good at chess_. The cultural image of
| the great chess player being a deep thinker doesn 't seem to line
| up with the evidence. I find it particularly interesting that,
| with very rare exception, none of the world's best chess players
| seem to go on to contribute anything intellectual other than
| their chess games.
| yazzku wrote:
| And I suppose this is supported by current evidence too, where
| grandmasters have been beaten by computers, which hold more
| long-term memory, have memorized more moves, and can enumerate
| the state tree more deeply. Rote state space exploration,
| nothing intellectual.
| bee_rider wrote:
| It is actually pretty remarkable, you'd think given the
| automatic reputation advantage that a chess grandmaster gets as
| a serious deep thinker, at least one would have managed to work
| that into a political career.
| TeMPOraL wrote:
| Politics is all about soft skills; being _really good_ at
| anything hard pretty much makes you unemployable there,
| because you 'll come across as a weird nerd.
| Spivak wrote:
| This is the D&D / video game fallacy -- that being really
| good at hard things means you had forgo points in other
| skills. It should be encouraging and liberating that this
| isn't true and you can be smart (in multiple fields),
| athletic, artistic, charismatic, a social butterfly, and
| everything in between.
| xanderlewis wrote:
| Definitely true, but sinking large amounts of time into
| learning very technical things in huge detail can often
| involve long periods in isolation during which one's
| social skills are likely to atrophy.
|
| Also, for some, being a 'social butterfly' is perfectly
| possible (with some effort) but is boring. This tends to
| be true the more into 'hard things' you are. Chatting to
| people about banality isn't hard, so it isn't
| interesting.
| xanderlewis wrote:
| Kasparov might have had a chance -- if Russia wasn't the way
| it is.
| bee_rider wrote:
| True! He's like the exception that makes the rule, and
| through coincidence, ended up not even really an exception.
| tsimionescu wrote:
| Gary Kasparov was actually involved in Russian politics,
| maybe he would have had some small chance of a career if
| Putin hadn't quickly quashed him.
| techostritch wrote:
| Kasparov seems to be a respected public intellectual or at
| least it's debatable which is more than you can say for most
| others (though maybe that's the exception that proves the
| rule).
| xanderlewis wrote:
| I think he's a thoughtful and wise man anyway; it doesn't
| seem to have much connection to chess.
| vkou wrote:
| Is that because he is actually a genius, or is it because he
| has a platform to talk from, and he talks from it?
|
| Because I find that there's a very wide range among 'well-
| regarded (by some) public intellectuals.' Some of them say
| things worth thinking about. Many others, not so much, the
| only noteworthy thing about them is that they stand on a
| soapbox.
| JustinSkycak wrote:
| To be clear, this is not just chess. To quote the paper:
|
| "[these] results have been replicated in a variety of
| educationally relevant fields, including mathematics (Sweller &
| Cooper, 1985)."
|
| Now, I would agree that I wouldn't want to hire a mathematician
| as a general (on the basis of their being a mathematician), for
| the same reason that you wouldn't want to hire a chess player
| as a general (on the basis of their being a chess player).
|
| I just want to emphasize that this applies to math too.
| xanderlewis wrote:
| I'm pretty sure mathematics is slightly more general of a
| subject than chess...
| kiba wrote:
| It is hard, though not impossible to generalize expertise.
| Joel_Mckay wrote:
| Difficult to determine absolute value, as unambiguous isomorphism
| manifests the same ideas in many specializations.
|
| While some solutions may prove sub-optimal, a refinement process
| by its very nature emulates a reductionist goal without the
| confines of abstract contextual dependency or impossible to
| implement/prove rigorous meanings.
|
| I never understood which approach was superior for practical
| application, or obfuscation of delusional wishful thinking.
|
| Have a wonderful day, =)
| kragen wrote:
| a different way to look at this is that, when we find a way to
| generalize a problem-solving skill, we call it math
|
| if you can throw a spear and hit the mammoth, that's a problem-
| solving skill. but when we learned a technique that can calculate
| the trajectory of the spear, the effect of the timing of
| jupiter's rising and setting on mars's, and the penetration depth
| of a baseball into the water, that's math
| Animats wrote:
| This is a response to what's called "Math Equity". Search for
| that.
| dinkumthinkum wrote:
| Also see "Oppression Olympics".
| motohagiography wrote:
| there is a constant tension in any field between fox/hedgehog,
| breadth-first/depth-first competence where in their own contexts
| they are absolutely correct about the supriority of their
| approaches, but quite wrong outside of it. we could frame these
| in a general category of Endian conflicts, where depth-firsts
| think the breadth-firsts are handwavey bullshitters, and breadth-
| firsts have a finite amount of patience for depth-first's
| concrete thinking and denial of abstractions.
|
| I often state I don't know anything about math as if there's a
| python library and a wikipedia page that's usually enough for my
| purposes, and then use a kind of profane math to do stuff instead
| of the sacred math that seems mostly to be about arguing and
| telling people what is impossible. Learn math for real, it's
| admirable and useful, and maybe someone will hire you to turn
| their handwavings into something someone wants.
| mlyle wrote:
| Yes, but since an effort to educate students in math is
| universal, it's worth thinking about the different outcomes we
| want:
|
| A. To create top flight mathematicians who can push the
| frontiers of the field forward. Arguably not a whole lot of
| what we do in K-12 and the first couple of years of college
| isn't _really_ aimed at this for the most part, since there is
| such a strong applied math push and the proofs stuff we teach
| in K-12 is broken.
|
| B. To create people competent enough in math to be engineers
| and scientists. Most math systems are pretty squarely aimed at
| this.
|
| C. To create people competent enough to live a life which
| tangentially touches mathematics (even if they are in a field
| like _most of_ finance or accounting or whatever, the amount of
| mathematics they will do is limited). Here, I think we go
| pretty far off: getting a person just _barely_ through Algebra
| 2 or trig doesn 't serve them well; you'd be better off
| teaching them first and foremost not to be scared of
| mathematical reasoning, about general problem solving ("look,
| you can just hold up the shape and rotate it!" "we can figure
| out the length of the board with a compass!"), and
| strengthening their general arithmetic and lower math skills.
|
| I think we need to diversify out from path "B" to do both "A"
| and "C" better.
| jrm4 wrote:
| The older I get the more I believe (realize) the issue with math
| really is 100% skin-in-the-game. When they're young, I suppose
| you can force memorization on them, but very quickly: If an
| individual has no immediate percieved use for the math, they're
| not going to want or need to learn it. Simple as that.
|
| This _really_ hit me as someone who did the overachievey college
| math. None of it sticks with me at all unless I can think about
| "what it's for."
|
| Corollary: When I was a kid, we didn't have the thing we have now
| which strikes me as the CLEAR USE CASE -- video game development;
| such a no-brainer for me.
|
| X Y algebra? Oh, you mean making a rainbow in Minecraft? :)
| tsimionescu wrote:
| And yet, the vast majority of math research serves no direct
| purpose, and the majority of professional mathematicians, at
| least in academia, look down on applied mathematics.
| borroka wrote:
| You can't take the extreme tail end of the distribution of
| interests, aptitudes and abilities, that is, people who
| pursue an academic career in mathematics, for the entire
| distribution of people who are taught or need to use
| mathematics at some point in their lives.
|
| Twenty years ago, when I was in college, I remember a
| classmate had problems installing the particular software we
| needed to use. The teacher told her that the only solution
| would be to install Linux on her laptop. All the other
| students had managed to install that software on their
| Windows laptops. The teacher was either one step ahead or 25
| steps behind.
| rocqua wrote:
| I don't think that's true for everyone. Your math PhDs and
| enthusiasts appreciate math as beautiful in and of itself. The
| disconnect might be that they forget many others do want skin
| in the game, and that makes the teachers not understand what
| the students need.
| cyberax wrote:
| The other often-overlooked point is that _memorization_ _itself_
| is a skill. You get better at remembering stuff as you keep
| practicing.
|
| And it doesn't necessarily have to be math. You can also train
| yourself by memorizing poetry, Chinese characters, foreign
| language words, and so on. And somehow all of these activities
| are getting sidelined in the modern education. After all, what
| use is memorization when you can always look up the answer on a
| phone?
| analog31 wrote:
| Indeed, learning how to memorize is how I finally got my stride
| in math. I was already good at proofs and problem solving. But
| constantly having to dig for stuff was hobbling me.
| will1am wrote:
| I think I was having the same issue
| kaashif wrote:
| A lot of people don't seem to understand that fluency in
| problem solving comes partially from memorization.
|
| Memorizing all of the theorems you need, proofs, and a diverse
| set of examples is going to make it substantially easier to
| approach new problems.
|
| I've heard it from people conducting interviews, when we're
| discussing what we want from candidates: "I'm not looking for
| memorization, I'm looking for problem solving!" - if you've
| memorized 1000 problems, you'll be better at problem solving
| than if you didn't!
| magicalhippo wrote:
| While it's true that memorization can help improve your
| skill, it's not a given.
|
| There are lots of folk who can remember all sorts of details
| but never seem to be able to figure out how to put the pieces
| together.
| will1am wrote:
| Absolutely, the point about memorization being a skill that can
| be improved with practice is so simple yet not understood by
| many
| k__ wrote:
| I got better at math after programming for a few years.
|
| Maybe, we need alternative approaches, to make the topic more
| interesting.
| will1am wrote:
| I think that's a common experience!
| Ozzie_osman wrote:
| People like to dismiss memorization because you can only use it
| to solve very simple problems, but someone once gave me the
| analogy that to "you can't write a symphony without having
| memorized all the notes first", and I've found that to be a great
| analogy. By memorizing the simple stuff, you can tackle the hard
| stuff.
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