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Teaching general problem-solving skills is not a substitute for teaching math [pdf] (2010)

238 points by JustinSkycak 2y ago

Icy0 2y ago
> 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 2y ago
  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/pedagogy/a-selected-list-of-journals-on-teaching-learning/ https://acrl.ala.org/IS/instruction-tools-resources-2/pedago...
  - Icy0 2y ago
    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 2y ago
    > 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).
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    - JustinSkycak 2y ago
      I would agree there are a lot of problems with a lot of education research. Many purported findings do not replicate or are otherwise impossible to replicate. However, there are also many findings that are actually legit. As you say, they're rare, but there are enough of them to paint a surprisingly complete picture when you pull them together. Discussed at length a couple months ago here: https://news.ycombinator.com/item?id=40348986 https://news.ycombinator.com/item?id=40348986
- epgui 2y ago
  What it is saying is that we need to stop acting as if, or believing that, this knowledge is solid.
- JustinSkycak 2y ago
  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 2y ago
    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 2y ago
      > 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 https://news.ycombinator.com/item?id=40348986
      - Icy0 2y ago
        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?
23B1 2y ago
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.
- deleted 2y ago
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- frogeyedpeas 2y ago
  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 2y ago
    I use math to decide which bus to take to get home.
- dinkumthinkum 2y ago
  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 2y ago
    What kind of math skills are you talking about that people lack that causes them to not have any money?
    - frogeyedpeas 2y ago
      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.
ibash 2y ago
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 2y ago
  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 2y ago
  > 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.
- lupire 2y ago
  Is computer science involved in anyway?
  - ibash 2y ago
    I meant software engineering as inclusive to computer science. Computer science/software engineering as taught in school gives a lot of foundational and theoretical understanding. But to apply and practice that knowledge,” then
orthopodvt 2y ago
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 2y ago
  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-matters/ https://hep.gse.harvard.edu/9781612509525/why-knowledge-matt... 2. https://thundergolfer.com/software/culture/2024/01/14/computing-cultural-literacy/ https://thundergolfer.com/software/culture/2024/01/14/comput...
frogeyedpeas 2y ago
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 2y ago
  > 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 2y ago
    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".
SOTGO 2y ago
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 2y ago
  > 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 2y ago
    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 2y ago
      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.
xqcgrek2 2y ago
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 2y ago
  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 2y ago
  '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 2y ago
  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.
- MiscCompFacts 2y ago
  It’s not just “10000 hours”, it’s deliberate practice that’s necessary for mastery. You must first identify the expert skills before practicing and work problems in those domains where you are lacking. Here’s a video on this exact subject: https://youtu.be/WbUOY9ioIqw?si=a60ILLODGDEU7_CA https://youtu.be/WbUOY9ioIqw?si=a60ILLODGDEU7_CA
cschmidt 2y ago
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 2y ago
  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.
  - stogot 2y ago
    What is g-loaded?
    - JustinSkycak 2y ago
      Two questions here: 1) What is "g"? "g" is "general intelligence." IQ is a specific measurement of g. https://en.wikipedia.org/wiki/G_factor_(psychometrics) https://en.wikipedia.org/wiki/G_factor_(psychometrics) 2) What is g-"loaded"? This is a good summary: https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4tfc https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4... "it's the degree to which that test correlates with g. A relatively high g-loaded test will have a higher correlation with g, meaning that performance on the test is more indicative of g than performance on a test that is less g-loaded. Often greater complexity or how much mental manipulation a test requires results in a higher g-loaded test. In contrast, higher difficulty (as measured by the percentage of people who fail) does not always mean higher g-loading. For instance, tests of reasoning are generally more g-loaded than tests of rote memorization even when the tests are of equal difficulty." - oscarjeff on Reddit
Silamoth 2y ago
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.
xanderlewis 2y ago
> 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 2y ago
  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 2y ago
  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 2y ago
    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 2y ago
      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.
Joel_Mckay 2y ago
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 2y ago
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 2y ago
This is a response to what's called "Math Equity". Search for that.
- dinkumthinkum 2y ago
  Also see “Oppression Olympics”.
motohagiography 2y ago
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 2y ago
  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.
  - Kim_Bruning 2y ago
    * How about showing kids the intrinsic beauty that can be found in parts of it? * Or the way you can make endless fun puzzles you can solve on a rainy day?
jrm4 2y ago
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 2y ago
  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 2y ago
    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.
  - lupire 2y ago
    > the majority of professional mathematicians, at least in academia, look down on applied mathematics. Not true.
  - jrm4 2y ago
    I mean, this sort of generally describes a lot of academia.
- rocqua 2y ago
cyberax 2y ago
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 2y ago
  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 2y ago
    I think I was having the same issue
- kaashif 2y ago
  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 2y ago
    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.
  - joe_the_user 2y ago
    What do you mean by memorizing? Of course, a mathematician remembers things but no mathematician learns a field by first attain the ability to recite the theorems in a book without understanding.
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k__ 2y ago
I got better at math after programming for a few years. Maybe, we need alternative approaches, to make the topic more interesting.
- will1am 2y ago
  I think that's a common experience!
- obscurette 2y ago
  Everybody gets better in every (related) field after years of experience.
- Jensson 2y ago
  We can't make kids spend a few extra years, school is already long as it is. > Maybe, we need alternative approaches, to make the topic more interesting. You realize kids only have around a single full time year to learn math, if you add up all 12 school years? They don't have the time you did when you started practicing programming.
  - Suppafly 2y ago
    >You realize kids only have around a single full time year to learn math, if you add up all 12 school years? Probably not even if you consider that basically half of every school year is review and relearning what some of the students forgot from the previous year. Rich people have the right idea with having private schools and private tutors for everything. We really need to democratize individualized education more. We are getting there somewhat with inclusion of technology based learning but we still aren't really allowing students to reach their full potential.
Ozzie_osman 2y ago
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.
- joe_the_user 2y ago
  I depends what you mean by "memorization". All learning requires memory so with an expansive definition all learning is "memorization". But if one means pure rote memorization, I think the value depends very much on the field. Writing English requires knowledge of the spellings of words since English spellings are fairly arbitrary. A student can benefit from memorizing multiplication table to 10 but they'd do better learning principles than memorization multiplication up 100 or 1000. And many of principles, terminologies and rules of thumb are best remembered in-context. One thing to consider is that "memory training" approaches can be effectively used to remember long arbitrary sequences of data (the arrangement of a deck of playing cards or whatever) through adding colorful/memorable (but arbitrary) associations. But such methods are seldom actually used by practitioners of memory intensive fields. Usually such practitioners need to recall facts and ideas in context and so they achieve a high recall naturally, by associations facts and ideas with each other.
akasakahakada 2y ago
> There is now a large body of evidence showing that studying worked examples is a more effective and efficient way of learning to solve problems than simply practicing problem solving without reference to worked examples (Paas & van Gog, 2006). Mathematicians RIP. Math texbook: Here are worked examples after the chapter. Worked by you. Enjoy your homework!
dventimi 2y ago
Teaching math is not a substitute for teaching general problem-solving skills.
kazinator 2y ago
Math is full of specialized problem-solving skills. Like, "oh, this seems to fit the pattern where we can integrate by parts". If you don't know that, you don't even know that you don't know that; it's an unknown unknown. General problem-solving skills aren't a substitute for special skills. General problem-solving skills have limits; one of the outcomes of general problem solving is the conclusion "I don't know how to solve this; it may require someone with special skills". Without properly honed general skills, you may waste time avoiding this correct conclusion (among other mistakes). General skills let you undestand what the problem actually is, what a solution looks like, and whether you are getting closer.
trash_cat 2y ago
I have a math disability called dyscalculia. Which basically means "number blindness". Math is about manipulating numerosities, which has nothing to do with problem solving.

238 points by JustinSkycak 2y ago

Icy0 2y ago

> 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 2y ago
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/pedagogy/a-selected-list-of-journals-on-teaching-learning/ https://acrl.ala.org/IS/instruction-tools-resources-2/pedago...
- Icy0 2y ago
  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 2y ago
  > 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).
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  - JustinSkycak 2y ago
    I would agree there are a lot of problems with a lot of education research. Many purported findings do not replicate or are otherwise impossible to replicate. However, there are also many findings that are actually legit. As you say, they're rare, but there are enough of them to paint a surprisingly complete picture when you pull them together. Discussed at length a couple months ago here: https://news.ycombinator.com/item?id=40348986 https://news.ycombinator.com/item?id=40348986
epgui 2y ago
What it is saying is that we need to stop acting as if, or believing that, this knowledge is solid.
JustinSkycak 2y ago
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 2y ago
  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 2y ago
    > 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 https://news.ycombinator.com/item?id=40348986
    - Icy0 2y ago
      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?

23B1 2y ago

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.

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frogeyedpeas 2y ago
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 2y ago
  I use math to decide which bus to take to get home.
dinkumthinkum 2y ago
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 2y ago
  What kind of math skills are you talking about that people lack that causes them to not have any money?
  - frogeyedpeas 2y ago
    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.

ibash 2y ago

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 2y ago
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 2y ago
> 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.
lupire 2y ago
Is computer science involved in anyway?
- ibash 2y ago
  I meant software engineering as inclusive to computer science. Computer science/software engineering as taught in school gives a lot of foundational and theoretical understanding. But to apply and practice that knowledge,” then

orthopodvt 2y ago

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 2y ago
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-matters/ https://hep.gse.harvard.edu/9781612509525/why-knowledge-matt... 2. https://thundergolfer.com/software/culture/2024/01/14/computing-cultural-literacy/ https://thundergolfer.com/software/culture/2024/01/14/comput...

frogeyedpeas 2y ago

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 2y ago
> 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 2y ago
  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".

SOTGO 2y ago

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 2y ago
> 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 2y ago
  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 2y ago
    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.

xqcgrek2 2y ago

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 2y ago
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 2y ago
'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 2y ago
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.
MiscCompFacts 2y ago
It’s not just “10000 hours”, it’s deliberate practice that’s necessary for mastery. You must first identify the expert skills before practicing and work problems in those domains where you are lacking. Here’s a video on this exact subject: https://youtu.be/WbUOY9ioIqw?si=a60ILLODGDEU7_CA https://youtu.be/WbUOY9ioIqw?si=a60ILLODGDEU7_CA

cschmidt 2y ago

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 2y ago
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.
- stogot 2y ago
  What is g-loaded?
  - JustinSkycak 2y ago
    Two questions here: 1) What is "g"? "g" is "general intelligence." IQ is a specific measurement of g. https://en.wikipedia.org/wiki/G_factor_(psychometrics) https://en.wikipedia.org/wiki/G_factor_(psychometrics) 2) What is g-"loaded"? This is a good summary: https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4tfc https://www.reddit.com/r/cogsci/comments/j5pug9/comment/g7u4... "it's the degree to which that test correlates with g. A relatively high g-loaded test will have a higher correlation with g, meaning that performance on the test is more indicative of g than performance on a test that is less g-loaded. Often greater complexity or how much mental manipulation a test requires results in a higher g-loaded test. In contrast, higher difficulty (as measured by the percentage of people who fail) does not always mean higher g-loading. For instance, tests of reasoning are generally more g-loaded than tests of rote memorization even when the tests are of equal difficulty." - oscarjeff on Reddit

Silamoth 2y ago

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.

xanderlewis 2y ago

> 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 2y ago
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 2y ago
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 2y ago
  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 2y ago
    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.

Joel_Mckay 2y ago

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 2y ago

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 2y ago

This is a response to what's called "Math Equity". Search for that.

dinkumthinkum 2y ago
Also see “Oppression Olympics”.

motohagiography 2y ago

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 2y ago
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.
- Kim_Bruning 2y ago
  * How about showing kids the intrinsic beauty that can be found in parts of it? * Or the way you can make endless fun puzzles you can solve on a rainy day?

jrm4 2y ago

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 2y ago
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 2y ago
  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.
- lupire 2y ago
  > the majority of professional mathematicians, at least in academia, look down on applied mathematics. Not true.
- jrm4 2y ago
  I mean, this sort of generally describes a lot of academia.
rocqua 2y ago

cyberax 2y ago

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 2y ago
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 2y ago
  I think I was having the same issue
kaashif 2y ago
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 2y ago
  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.
- joe_the_user 2y ago
  What do you mean by memorizing? Of course, a mathematician remembers things but no mathematician learns a field by first attain the ability to recite the theorems in a book without understanding.

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k__ 2y ago

I got better at math after programming for a few years. Maybe, we need alternative approaches, to make the topic more interesting.

will1am 2y ago
I think that's a common experience!
obscurette 2y ago
Everybody gets better in every (related) field after years of experience.
Jensson 2y ago
We can't make kids spend a few extra years, school is already long as it is. > Maybe, we need alternative approaches, to make the topic more interesting. You realize kids only have around a single full time year to learn math, if you add up all 12 school years? They don't have the time you did when you started practicing programming.
- Suppafly 2y ago
  >You realize kids only have around a single full time year to learn math, if you add up all 12 school years? Probably not even if you consider that basically half of every school year is review and relearning what some of the students forgot from the previous year. Rich people have the right idea with having private schools and private tutors for everything. We really need to democratize individualized education more. We are getting there somewhat with inclusion of technology based learning but we still aren't really allowing students to reach their full potential.

Ozzie_osman 2y ago

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.

joe_the_user 2y ago
I depends what you mean by "memorization". All learning requires memory so with an expansive definition all learning is "memorization". But if one means pure rote memorization, I think the value depends very much on the field. Writing English requires knowledge of the spellings of words since English spellings are fairly arbitrary. A student can benefit from memorizing multiplication table to 10 but they'd do better learning principles than memorization multiplication up 100 or 1000. And many of principles, terminologies and rules of thumb are best remembered in-context. One thing to consider is that "memory training" approaches can be effectively used to remember long arbitrary sequences of data (the arrangement of a deck of playing cards or whatever) through adding colorful/memorable (but arbitrary) associations. But such methods are seldom actually used by practitioners of memory intensive fields. Usually such practitioners need to recall facts and ideas in context and so they achieve a high recall naturally, by associations facts and ideas with each other.

akasakahakada 2y ago

> There is now a large body of evidence showing that studying worked examples is a more effective and efficient way of learning to solve problems than simply practicing problem solving without reference to worked examples (Paas & van Gog, 2006). Mathematicians RIP. Math texbook: Here are worked examples after the chapter. Worked by you. Enjoy your homework!

dventimi 2y ago

Teaching math is not a substitute for teaching general problem-solving skills.

kazinator 2y ago

Math is full of specialized problem-solving skills. Like, "oh, this seems to fit the pattern where we can integrate by parts". If you don't know that, you don't even know that you don't know that; it's an unknown unknown. General problem-solving skills aren't a substitute for special skills. General problem-solving skills have limits; one of the outcomes of general problem solving is the conclusion "I don't know how to solve this; it may require someone with special skills". Without properly honed general skills, you may waste time avoiding this correct conclusion (among other mistakes). General skills let you undestand what the problem actually is, what a solution looks like, and whether you are getting closer.

trash_cat 2y ago

I have a math disability called dyscalculia. Which basically means "number blindness". Math is about manipulating numerosities, which has nothing to do with problem solving.