Teaching general problem-solving skills is not a substitute for teaching math [pdf] (2010)(ams.org)
ams.org
Teaching general problem-solving skills is not a substitute for teaching math [pdf] (2010)
https://www.ams.org/notices/201010/rtx101001303p.pdf
238 comments
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.
> 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! ;)
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! ;)
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.
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.
This description confounds two independent variables: "active vs passive learning" and "direct vs unguided instruction."
The studies you refer to are demonstrating that active/unguided is superior to passive/direct.
But the full picture is that active/direct > active/unguided > passive/direct. (I didn't include passive/unguided here because I'm not sure it's possible to create such a combination.)
Other studies -- that only manipulate one variable at a time -- support this big picture.
The studies you refer to are demonstrating that active/unguided is superior to passive/direct.
But the full picture is that active/direct > active/unguided > passive/direct. (I didn't include passive/unguided here because I'm not sure it's possible to create such a combination.)
Other studies -- that only manipulate one variable at a time -- support this big picture.
Well, sure. But very few formal educational settings are purely active/unguided. Unfortunately passive/direct is much more common.
To me though the more interesting result isn’t really about pedagogy, it’s that people’s (undergrad physics students, in the case of the specific study I’m thinking of) subjective impressions of the effectiveness of instruction are unreliable.
To me though the more interesting result isn’t really about pedagogy, it’s that people’s (undergrad physics students, in the case of the specific study I’m thinking of) subjective impressions of the effectiveness of instruction are unreliable.
> subjective impressions of the effectiveness of instruction are unreliable
Yes, common finding in studies that explore subjective vs objective measurements of learning under conditions involving "desirable difficulties":
https://en.wikipedia.org/wiki/Desirable_difficulty
Yes, common finding in studies that explore subjective vs objective measurements of learning under conditions involving "desirable difficulties":
https://en.wikipedia.org/wiki/Desirable_difficulty
> (I didn't include passive/unguided here because I'm not sure it's possible to create such a combination.)
Its possible, we call the end result LLM. It isn't very effective though as we can see from the result and how much learning it took.
Its possible, we call the end result LLM. It isn't very effective though as we can see from the result and how much learning it took.
Anecdotally, teaching in a manner that forces students to discover a key or difficult concept on their own is a way to weed out those who "can" from those who "will", if you get my meaning.
My undergraduate math professor was like that, and he was pretty brutal, but by the end of the 2nd semester it was pretty clear who was going to end up majoring in something to do with math and who wasn't. From a pure selection standpoint, this makes sense to me. On the other hand, for those who "won't" it can make the experience pretty miserable.
My undergraduate math professor was like that, and he was pretty brutal, but by the end of the 2nd semester it was pretty clear who was going to end up majoring in something to do with math and who wasn't. From a pure selection standpoint, this makes sense to me. On the other hand, for those who "won't" it can make the experience pretty miserable.
Imo, the need to weed out is counterproductive from a societal perspective. Imagine if in military conscription they weeded out everybody who didn't want to be there. They'd probably fall short of their service requirements quickly. In the same way, if America wants to bridge the supposed gap in math from Asia, it's not a matter of who is willing, it's a matter of whether they can teach or not.
Well, in a conscription scenario you don't weed out everyone who doesn't want to be there. That's ... what makes it conscription. In the AVF (All Volunteer Force) we do in fact weed out people who don't want to be there, and the relative pressure of that weed-out process increases the more elite the unit is that we're talking about. The state of military recruiting in the United States is the worst it's ever been, or close to it, but that is unrelated to that process described above since the problem is upstream from basic training.
I'm probably confusing people with my use of the word "will" in this context, since it can mean several things in English. What I'm really saying is that those who have the actual aptitude "will derive complex concepts on their own, and will be likely to pursue further their math education". It's already difficult to identify those people when they're young enough, and even harder if you teach math in a "lowest common denominator" approach, which is essentially what the American strategy is (with notable exceptions that probably just prove the rule).
I'm probably confusing people with my use of the word "will" in this context, since it can mean several things in English. What I'm really saying is that those who have the actual aptitude "will derive complex concepts on their own, and will be likely to pursue further their math education". It's already difficult to identify those people when they're young enough, and even harder if you teach math in a "lowest common denominator" approach, which is essentially what the American strategy is (with notable exceptions that probably just prove the rule).
Lots of fields have a 'weed out' class early on. I majored in CS, and it essentially weeded out all those that had no real interest in the field but had thought they'd like it because it paid well or they wanted to make video games. Those sorts of classes don't necessarily need to be overly hard, because the people who 'get it' won't struggle much and those who don't will find it hard regardless. Although I imagine in math specifically, even those who get it might need to struggle a bit.
I think there's a lot to unpack here. Teaching someone how to write a for loop is easy and can done in a straightforward way, but teaching them when it's best to use, and getting them to under why, is different. Even further, getting them evaluate novel situations, apply it correctly and be able to communicate why they did it that way is another thing.
At what point would you say they've actually acquired the skill?
At what point would you say they've actually acquired the skill?
Many of the drivers I encounter seem to have been given cars (or trucks) and then left to their own devices.
+100 for "Please, Just Work More Examples, I Swear It Helps".
I don't have nearly as impressive a backstory as you do here, but I did apply spaced repetition to my abstract algebra class in my math minor a few years back. I didn't do anything fancy, I just put every homework problem and proof into Anki and solved/rederived them over and over again until I could do so without much thinking.
I ended up walking out with a perfect score on the 2 hour final - in about 15 minutes. Most of the problems were totally novel things I had never seen before, but the fluency I gained in the weeks prior just unlocked something in me. A lot of the concepts of group actions, etc. have stuck with me to this very day, heavily informing my approach to software engineering. Great stuff.
I don't have nearly as impressive a backstory as you do here, but I did apply spaced repetition to my abstract algebra class in my math minor a few years back. I didn't do anything fancy, I just put every homework problem and proof into Anki and solved/rederived them over and over again until I could do so without much thinking.
I ended up walking out with a perfect score on the 2 hour final - in about 15 minutes. Most of the problems were totally novel things I had never seen before, but the fluency I gained in the weeks prior just unlocked something in me. A lot of the concepts of group actions, etc. have stuck with me to this very day, heavily informing my approach to software engineering. Great stuff.
Great story! This is exactly the kind of thing that we see all the time at Math Academy, that I saw in the classes I taught, and that many MA users report experiencing -- but unfortunately, lots of people find it counterintuitive and have a hard time understanding/believing it until they experience it firsthand.
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).
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.
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.
Is that really fundamental? Maybe for studies in pathological real functions.
But in realistic functions relevant to our actually universe, these pathological cases aren't important.
But in realistic functions relevant to our actually universe, these pathological cases aren't important.
I would argue that understanding the pathological behavior in something is critical to developing an accurate intuition for it, yes. These cases don't show up often, but when it comes to having a good sense of smell for when part of a proof is flawed, it really helps to have that olfactory memory.
Aside from that, understanding counterexamples teaches you to understand the definitions and theorems better. Which matters for proving future results.
Fractals and Dirac delta functions both have somewhat pathological properties at times, and both pop up as approximations of real systems in physics.
I would personally not consider it fundamental either though, more of a “let’s cross that bridge when we get to it” problem.
I would personally not consider it fundamental either though, more of a “let’s cross that bridge when we get to it” problem.
Most of the stuff we learned in college is not useful except conceptually. There is almost never a need to use it in real life
> 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.edu/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 - https://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/instructionalproducts/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.
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.edu/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 - https://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/instructionalproducts/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.
The problem with most of this research is that high IQ and low IQ people learn differently, but aren't separated out in the studies.
Most research makes assumptions like these without laying them out. It might explain none of their work can be reproduced.
> 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.
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.
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 Poincaré 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 möbius 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.
Two I remember were:
- In an early geometry course there was a problem to prove/determine something described in terms of the Poincaré 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 möbius 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.
Differential equations seems to be a particularly bad offender here
I think that’s a problem with differential equations as a subject. The only ones we know how to solve are special cases. Solving them in general is an open problem.
I think that’s a problem with differential equations as a subject. The only ones we know how to solve are special cases. Solving them in general is an open problem.
I always liked systematic methods, including analytic (e.g. laplace transform) and numerical approaches (e.g. runge-kutta.)
My education was basically made of worked examples. What was missing was WHY they were worked THAT way. The thought process of the person solving the problem was missing. Yes, intermediate steps were all there - and still no answer to "why go in THAT direction - from the onset?".
It's not "problem solving", it's the deeper understanding of a discipline which makes the experienced practionner go one way rather than the other.
It's not "problem solving", it's the deeper understanding of a discipline which makes the experienced practionner go one way rather than the other.
The importance of worked examples in helping students understand new topics
The downside of teaching using worked examples is that it teaches only one problem solving skill to students: mimicking.
Many students will look only at examples in the textbook and happily ignore definitions, theorems, and proofs. They don't know whether the strategy they picked works, only that it worked on a similar looking problem.
Sure, when (good) teachers explain the example they do go through the effort of referring to the definitions and theorems, but that is not necessarily what the students remember.
Many students will look only at examples in the textbook and happily ignore definitions, theorems, and proofs. They don't know whether the strategy they picked works, only that it worked on a similar looking problem.
Sure, when (good) teachers explain the example they do go through the effort of referring to the definitions and theorems, but that is not necessarily what the students remember.
They skip definitions, theorems, and proofs for good reason too. Students are spending a lot of time and money they don't have to get a degree and have an obligation to work efficiently. With a limited amount of time and energy they would actively hurting themselves by focusing on things that aren't graded. In general I've found that teachers grade quite harshly on things you could have memorized, and find little value in understanding or institution.
Your grades doesn't matter as much as your understanding does, for most people. In some cases your grades will be the deciding factor, but in most cases it is worth more to you to get a better understanding and worse grades.
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?
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?
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!
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!
Would observe that math is something one should not learn by memorization … rather should try to get to the essence and build it up naturally while at the same time get some sense and intuition of things … so that one would become a natural …
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.
That's almost exactly how people did the trivium, to lay the ground for what follows. I think people are trying to say memorising is critical, understanding follows, not leads.
There's a reason that "trivia" comes to us from trivium.
Once, in medieval times, rote memorization was a large part of the education process. This system was renowned for not producing flexible or imaginative thinkers. Medieval times produced very significant mathematicians, notably and mathematical advances actually detoured around medieval Europe, going from ancient Rome and Greece to India and the Muslim world and returning to Italy with the Renaissance.
I think people are trying to say memorising is critical, understanding follows, not leads. Sure, I think those posters framing things this way are incorrect. Problem solving may be something one learns and remembers but rote memorization, as in the medieval trivium, isn't significant part of such learning.
Once, in medieval times, rote memorization was a large part of the education process. This system was renowned for not producing flexible or imaginative thinkers. Medieval times produced very significant mathematicians, notably and mathematical advances actually detoured around medieval Europe, going from ancient Rome and Greece to India and the Muslim world and returning to Italy with the Renaissance.
I think people are trying to say memorising is critical, understanding follows, not leads. Sure, I think those posters framing things this way are incorrect. Problem solving may be something one learns and remembers but rote memorization, as in the medieval trivium, isn't significant part of such learning.
Understanding leads to memorization, not the other way around. Chess masters can easily remember chess positions, but they didn't practice that it comes from fluency. Same with math, memorization comes by itself when you have grokked the subject well enough to make memorization trivial.
The arrow doesn't go one way. Understanding and fluency go hand in hand. When you have all of the theorems and techniques at your fingertips, understanding actually is easier. And vice versa.
But "understanding" on its own doesn't allow you to reproduce a textbook full of theorems, you have to actually study and memorize at least a bit.
But "understanding" on its own doesn't allow you to reproduce a textbook full of theorems, you have to actually study and memorize at least a bit.
I'm not a mathematician so if this is an axiom of the ontology of meaning in maths, I have to accept it but if we are using the words in their commonly understood meaning you're completely wrong. I have memorised the first line of "I've got a lovely bunch of coconuts" in swedish and i assure you I have no underlying understanding of Swedish.
(For jor can inte far opp min kokosnutt btw)
(For jor can inte far opp min kokosnutt btw)
Yeah, I doubt the "common meaning" really covers it. To extend your example: I would bet (although I know no Swedish beyond "skål!"), that the manner in which you say those words would instantly make apparent (to a Swedish speaker) that you do not speak Swedish.
I think the commenters need to agree on a more-nuanced set of terms to reach agreement on this.
I think the commenters need to agree on a more-nuanced set of terms to reach agreement on this.
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.
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.
Absolutely, the point about memorization being a skill that can be improved with practice is so simple yet not understood by many
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? :)
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? :)
LOL, when I read just the first paragraph of your comment, I immediately thought of computer graphics.
There are other examples. I wouldn't be motivated to analyze some filter circuit's transfer function if it's not related to guitar somehow.
If you are someone who is primarily about Making Stuff, this will resonate.
I think some of the academics in math are not Make Stuff people; they can get motivated by the math itself. Or, well, maybe they are Make Stuff people, but what hey make is the math itself. Their application for something is, oh, I need that to prove this other thing in some structure I'm making.
I've experienced the Make Stuff motivation playing with just math. For instance, in high school, I independently came up with double and triple integration along multiple a, and used that to work out the volumes of common solids (easily verifiable to be right). I was thinking, I'm following this cool idea where we integrate along one variable, to get a formula which we integrate along another; will that work?
There are other examples. I wouldn't be motivated to analyze some filter circuit's transfer function if it's not related to guitar somehow.
If you are someone who is primarily about Making Stuff, this will resonate.
I think some of the academics in math are not Make Stuff people; they can get motivated by the math itself. Or, well, maybe they are Make Stuff people, but what hey make is the math itself. Their application for something is, oh, I need that to prove this other thing in some structure I'm making.
I've experienced the Make Stuff motivation playing with just math. For instance, in high school, I independently came up with double and triple integration along multiple a, and used that to work out the volumes of common solids (easily verifiable to be right). I was thinking, I'm following this cool idea where we integrate along one variable, to get a formula which we integrate along another; will that work?
Make Stuff I think is one.
But probably, greed/money would be the other.
Now, that's probably a whole other conversation, given the propensity that "capitalism" or whatever one wants to call it is pretty much dedicated to you and I getting this wrong consistently, but hey.
But probably, greed/money would be the other.
Now, that's probably a whole other conversation, given the propensity that "capitalism" or whatever one wants to call it is pretty much dedicated to you and I getting this wrong consistently, but hey.
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.
Oh, of course, they exist. But my guess is that's the extreme minority, and I was making suggestions in the realm of "We seem to be doing math badly in general population education, what to do about it?"
Part of the problem with math is that it's often taught using examples that are so trivial that they don't require the techniques being taught. If harder examples aren't introduced pretty early, students are left with the idea that they don't need to learn the techniques. You don't really need know any real techniques to figure out that 2x+4=8, you can kinda just figure out that 4+4=8, so x is 2, and if you can't figure that out intuitively, you can just guess and check a few numbers until you have the correct answer. If the entirety of algebra 1 or pre-algebra is taught with similar examples, students will pass the standardized test but not have any of foundations necessary for further math classes and actually won't even have the ability the solve even slightly more complicated versions of the same sorts of problems.
It's decently rare, but the people who actually like mathematics tend to become mathematicians. Hence they become the spokespeople for mathematics, as well as the experts who are consulted about how to teach and structure curicula. They also become maths teachers.
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.
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.
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.
> the majority of professional mathematicians, at least in academia, look down on applied mathematics.
Not true.
Not true.
I mean, this sort of generally describes a lot of academia.
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.
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.
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.
What is g-loaded?
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)
2) What is g-"loaded"?
This is a good summary: 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
1) What is "g"?
"g" is "general intelligence." IQ is a specific measurement of g.
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/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
You can take this "g" stuff more or less seriously, depending on how much of a math/stat background you have:
http://bactra.org/weblog/523.html
http://bactra.org/weblog/523.html
Math progression looks roughly like the following:
1. "Concrete" math, where you learn how to manipulate mathematical constructs, usually guided by worked examples. Little proof involved. (up to advanced HS / junior college level)
2. Proof driven math, use of worked examples becomes more rare (undergrad math)
3. Highly abstract math, where worked examples are more or less entirely abandoned (grad school math)
The vast majority of world will never be exposed to math beyond (1), and even people in the STEM field will only be limited to (2). You almost need to study math at a high level, or something very adjacent to math, in order to reach (3).
But it should be mentioned that one part of why worked examples diminish as you work your way up, is that you're kind of expected to make your own examples - meaning that you can take highly abstracted mathematical constructs/objects, and relate them to something tangible.
Some people have no problem learning math that way, while others struggle. I personally struggled to learn math without any examples, so getting my mind into graduate level math was rough.
Luckily there are so many resources to higher-level math, these days. You're not bound to a handful of "bibles" that are filled with "... is left as an exercise for the reader"
1. "Concrete" math, where you learn how to manipulate mathematical constructs, usually guided by worked examples. Little proof involved. (up to advanced HS / junior college level)
2. Proof driven math, use of worked examples becomes more rare (undergrad math)
3. Highly abstract math, where worked examples are more or less entirely abandoned (grad school math)
The vast majority of world will never be exposed to math beyond (1), and even people in the STEM field will only be limited to (2). You almost need to study math at a high level, or something very adjacent to math, in order to reach (3).
But it should be mentioned that one part of why worked examples diminish as you work your way up, is that you're kind of expected to make your own examples - meaning that you can take highly abstracted mathematical constructs/objects, and relate them to something tangible.
Some people have no problem learning math that way, while others struggle. I personally struggled to learn math without any examples, so getting my mind into graduate level math was rough.
Luckily there are so many resources to higher-level math, these days. You're not bound to a handful of "bibles" that are filled with "... is left as an exercise for the reader"
[Ed. finding these resources is left as an exercise for the reader]
The "theorem/proofs" are worked examples in this context since that is what mathematicians do all day. The ubiquitous "existence proof" is just about showing an object satisfying a property exists without actually giving an example.
Higher math is a big exercise in shifting symbols around. If you don't have an intrinsic motivation to solve puzzles you will hate higher math.
Higher math is a big exercise in shifting symbols around. If you don't have an intrinsic motivation to solve puzzles you will hate higher math.
As a professional mathematician, I strongly disagree with the claim that "higher math" abandons worked examples. Any course or book that does not devote a significant amount of time to examples is a bad course or book.
Even Grothendieck, who was famously known for thinking very abstractly and avoiding examples, was motivated by concrete questions (e.g., the Weil conjectures) coming from concrete examples. To me, and most other mathematicians, the whole point of mathematics is to do examples, and theory building or any other abstract nonsense should be motivated by the desire to better understand or unify examples.
Even Grothendieck, who was famously known for thinking very abstractly and avoiding examples, was motivated by concrete questions (e.g., the Weil conjectures) coming from concrete examples. To me, and most other mathematicians, the whole point of mathematics is to do examples, and theory building or any other abstract nonsense should be motivated by the desire to better understand or unify examples.
What the higher math classes don't tell you is that you are supposed to study those worked examples an practice problems on your own.
It is challenging for learners who benefit from examples
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.
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.
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.
> 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.
More so by iteratively building, at least so for me.
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.
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.
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
edit - corrected spelling
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...
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...
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.
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.
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.
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.
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.
> 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.
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.
> 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 packages 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
A little calculus yields the solution.
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.
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 packages 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))
A little calculus yields the solution.
y(t) = y(0) b exp(bkt) /
( y(0)( exp(bkt) - 1) + b))
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.
Interesting, but I still think most problems like that are solvable via Excel. Put some formulas in some cells, tweak some variables until you find a way to maximize something. Possibly use graphs or pivot tables or other advanced features to help if needed.
Once you’ve figured out the solution, then you build a pretty graph for the BoD proving it. Make sure to keep the spreadsheet around as evidence.
Sure, for some applications of calculus
can use just discrete steps. That is,
instead of the calculus dy/dt just use
something like (y)dt.
Then, for the arithmetic, some code can be short and, compared with cells in a spreadsheet, easier and with more control over the time steps, e.g., in Rexx with cf for customer fraction:
Adjust some of the constants in the program and can get more output, say, for each month, day, etc. The code above uses 10 steps per day.
For more, someone could use the calculus solution and compare.
In a sense, for the FedEx problem and the assumptions about what was driving the growth, the calculus solution is a smooth version of the somewhat more appropriate discrete time version.
But when I did the calculation at FedEx, my best source of arithmetic was an HP calculator in which case the calculus solution was a lot easier.
Of course, this FedEx calculation was just one example and there are many others.
My view from 10,000 feet up is that in business, at times some math can be an advantage if not the work of a steady job.
If some math is an advantage, then that advantage tends to go to the owners of the business. If a mathematician wants to get paid for some math they have in mind, maybe they should start a business and be the owner.
Then, for the arithmetic, some code can be short and, compared with cells in a spreadsheet, easier and with more control over the time steps, e.g., in Rexx with cf for customer fraction:
Say ' ==== Growth ===='
Say ' '
Say ' Customer'
Say ' Year Fraction'
max_years = 5
steps_per_year = 10 * 365
cf = 1 * ( 1 / 100 )
year = 1
k = 1 * ( 1 / 2000 )
Do Forever
Do i = 1 To steps_per_year
cf = cf + k * cf * ( 1 - cf )
End
Say Format(year,9) Format(100*cf,10,2) || '%'
If year = max_years Then Leave
year = year + 1
End
==== Growth ====
Customer
Year Fraction
1 5.89%
2 27.97%
3 70.66%
4 93.73%
5 98.93%
Adjust some of the constants in the program and can get more output, say, for each month, day, etc. The code above uses 10 steps per day.
For more, someone could use the calculus solution and compare.
In a sense, for the FedEx problem and the assumptions about what was driving the growth, the calculus solution is a smooth version of the somewhat more appropriate discrete time version.
But when I did the calculation at FedEx, my best source of arithmetic was an HP calculator in which case the calculus solution was a lot easier.
Of course, this FedEx calculation was just one example and there are many others.
My view from 10,000 feet up is that in business, at times some math can be an advantage if not the work of a steady job.
If some math is an advantage, then that advantage tends to go to the owners of the business. If a mathematician wants to get paid for some math they have in mind, maybe they should start a business and be the owner.
Above gave the calculus solution in
algebra and also values from discrete
arithmetic with (10)(365) steps per year.
For the arithmetic for the calculus solution, in Rexx,
For the arithmetic for the calculus solution, in Rexx,
y.0 = ( 1 / 100 )
b = 1
k = ( 1 / 2000 )
Do t = 1 To 5
t1 = t * (10 * 365 )
e1 = RxCalcExp( b * k * t1, 16 )
y.t = ( y.0 * e1 ) / ( y.0 * ( e1 - 1 ) + b )
Say Format(t,9) Format(100*y.t,6,2) || '%'
End
==== Growth ====
Customer Fraction
Year Calculus Discrete
1 5.90% 5.89%
2 27.99% 27.97%
3 70.68% 70.66%
4 93.73% 93.73%
5 98.93% 98.93%
I got better at math after programming for a few years.
Maybe, we need alternative approaches, to make the topic more interesting.
Maybe, we need alternative approaches, to make the topic more interesting.
Everybody gets better in every (related) field after years of experience.
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.
> 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.
>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.
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.
I think that's a common experience!
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
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
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.
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
'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
like most things gladwell made up, it sounds good at first but falls apart the moment you think about it for a second
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.
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.
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.
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.
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.
* 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?
* Or the way you can make endless fun puzzles you can solve on a rainy day?
That's a separate question-- how (especially earlier) math classes should grab student engagement and encourage all students to progress as much as possible.
Re: your point: I've said often that I think most math classes should be roughly equal amounts of
* Exploration/play/intuition-building
* Rote practice of problems, facts, specific problem solving or symbolic manipulation techniques.
* Rigor / careful explanation
(The exact balance may shift a bit, from younger students doing the first two a bit more and older students doing the latter a lot more).
I think if you do this better, more of the populace will happily make it past Algebra 2. But still, even so, others will still get stuck in real understanding somewhere between Algebra I and Geometry, and I think we need to decide what to do then so as to best serve those students.
Re: your point: I've said often that I think most math classes should be roughly equal amounts of
* Exploration/play/intuition-building
* Rote practice of problems, facts, specific problem solving or symbolic manipulation techniques.
* Rigor / careful explanation
(The exact balance may shift a bit, from younger students doing the first two a bit more and older students doing the latter a lot more).
I think if you do this better, more of the populace will happily make it past Algebra 2. But still, even so, others will still get stuck in real understanding somewhere between Algebra I and Geometry, and I think we need to decide what to do then so as to best serve those students.
> 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!
Mathematicians RIP.
Math texbook: Here are worked examples after the chapter. Worked by you. Enjoy your homework!
> 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.
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.
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"
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"
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.
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.
> 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
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
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?
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?
> 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.
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.
> 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.
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.
What it is saying is that we need to stop acting as if, or believing that, this knowledge is solid.
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...
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).
> 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).
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).
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
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
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This is a response to what's called "Math Equity". Search for that.
Also see “Oppression Olympics”.
Teaching math is not a substitute for teaching general problem-solving skills.
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
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
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, =)
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, =)
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".
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".
> 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.
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.
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.
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.
Knowing a topic includes instant recall of a key body of knowledge.
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.
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.
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.
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.
I’m ABD in math. It was 30 years ago that I decided to not get a Ph.D. because I realized that I was never going to be decent at research. In the last 30 years I have forgotten a great of mathematics. It is no longer true that I know Galois Theory. I used to know it and I know the basic idea behind and I believe I can fairly easily relearn it. But right now I don’t know it.
That's wild, we all use AES cipers w/ TLS/HTTPS everyday - and Galois fields are essential to AES - but few people understand how HTTPS works.
The field is probably onto post-AES, PQ algos where Galois Theory is less relevant; but back then, it seemed like everyone needed to learn Galois Theory, which is or isn't prerequisite to future study.
The problem-solving skills are what's still useful.
Perhaps problem-solving skills cannot be developed without such rote exercises; and perhaps the content of such rote exercises is not relevant to predicting career success.
The field is probably onto post-AES, PQ algos where Galois Theory is less relevant; but back then, it seemed like everyone needed to learn Galois Theory, which is or isn't prerequisite to future study.
The problem-solving skills are what's still useful.
Perhaps problem-solving skills cannot be developed without such rote exercises; and perhaps the content of such rote exercises is not relevant to predicting career success.
> 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.
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.
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.
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.
You can only think using memory.
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.
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.
> 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.
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.
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".
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".
> 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?
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?
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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agree. how else do famous unsolved math problems eventually get solved?
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 Poincaré conjecture, etc.
Search.
> 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.
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.
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.
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.
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.
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.
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.
> 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.
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.
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.
Like, once you learn a programming language, you already know the syntax for 90% of all languages.
Here's a collage of some of my favourite bits. (I've inlined some functions, but this is nearly as readable as the actual code.)
open Option Parse Scan;
(maybe embedded --| minus -- name >> (fn arg => ((if ! testing then #2 arg |> Output.writeln else (); #1 #> curry getOpt) arg (implode [])))) -- command_name "supply" >> (op ^)
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"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
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
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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.
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.
> This comes down to the old saying "everything is memorization at the end of the day".
I certainly don't remember hearing that!
I certainly don't remember hearing that!
Maybe understanding is simply having memorized a handy instantiation of the relevant concept
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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.
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.
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.
> 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.
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.
> 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.
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.
> Yes, you can do that one. But that's just about the only fast trick there is.
I dunno about that. For division, anyway, there's a bunch of fast tricks that give you a decent approximation (i.e. decent precision, maybe to the nearest integer)
Someone recently was surprised that I worked out the VAT (Value Added Tax, 15%) on a very large number in a few seconds. It's because its 10% of the number plus `(10% of the number)/2`.
It's easy to get 10% of any number. It's easy to halve any number. It's a fast trick because there's two easy operations.
There's a bunch of similar "tricks": 1%, 10%, 25% and 50% are fast to calculate in your head (at most 2 easy operations, like `(half of half of N)`). Then you either add or subtract N. Or you multiply N by 2.
At most three easy operations gives you 1%, 2%, 4%, 5%, 10%, 11%, 12%, 14%, 15%, 20%, 21%, 24%, etc
To someone who doesn't know how you are getting the answer it might seem like you are a human calculator because you get so many of those quickly, and they don't see the ones you don't do in 3 easy operations (say, 13%, which is 10% + 1% + (1% * 2)).
IOW, it looks like a very impressive trick, but it isn't.
I dunno about that. For division, anyway, there's a bunch of fast tricks that give you a decent approximation (i.e. decent precision, maybe to the nearest integer)
Someone recently was surprised that I worked out the VAT (Value Added Tax, 15%) on a very large number in a few seconds. It's because its 10% of the number plus `(10% of the number)/2`.
It's easy to get 10% of any number. It's easy to halve any number. It's a fast trick because there's two easy operations.
There's a bunch of similar "tricks": 1%, 10%, 25% and 50% are fast to calculate in your head (at most 2 easy operations, like `(half of half of N)`). Then you either add or subtract N. Or you multiply N by 2.
At most three easy operations gives you 1%, 2%, 4%, 5%, 10%, 11%, 12%, 14%, 15%, 20%, 21%, 24%, etc
To someone who doesn't know how you are getting the answer it might seem like you are a human calculator because you get so many of those quickly, and they don't see the ones you don't do in 3 easy operations (say, 13%, which is 10% + 1% + (1% * 2)).
IOW, it looks like a very impressive trick, but it isn't.
> For division, anyway
Did you not see the neighbor comment where I made it extra clear I'm talking about times tables in particular?
There are tons of math tricks! But the multiplication table of the numbers between 1 and 10 is mostly rote.
Did you not see the neighbor comment where I made it extra clear I'm talking about times tables in particular?
There are tons of math tricks! But the multiplication table of the numbers between 1 and 10 is mostly rote.
Multiplication and division by 2 and 5 correspond to each other. Multiplication by 11 is shift-and-add. There are many such tricks.
The first one doesn't help with remembering times tables. The second one is outside the 1-10 times table we were talking about.
6 is 2×3, so 6×5 = 30. Likewise, 17 = 2×8 + 1, so 5×17 = 85.
I completely disagree. First of all, at the time children learn the multiplication table, they definitely don't know the concept of exponentiation. Secondly, 7*2*2*2 is not some immediately obvious shortcut.
Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.
Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.
> Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table
You've actually just proved my point - you used a method of breaking down the problem into a different problem and then solving it rather than simply memorising.
If you give the same question to multiple people there will be numerous ways different people use to go about solving it.
As an example, I might solve this by doing
20*8 = 160 3*8 = 24 160 - 24 = 136
Or 10*8 = 80 7*8 = 56 80+56 = 136
And I might apply different tools like the one I originally mentioned within these calculations. I know that 80+20 is 100 and so "borrow" 20 from 56, so that I can easily add 100 and 36 together.
These ways of calculating happen in your mind very quickly if this is how you get used to calculating.
You've actually just proved my point - you used a method of breaking down the problem into a different problem and then solving it rather than simply memorising.
If you give the same question to multiple people there will be numerous ways different people use to go about solving it.
As an example, I might solve this by doing
20*8 = 160 3*8 = 24 160 - 24 = 136
Or 10*8 = 80 7*8 = 56 80+56 = 136
And I might apply different tools like the one I originally mentioned within these calculations. I know that 80+20 is 100 and so "borrow" 20 from 56, so that I can easily add 100 and 36 together.
These ways of calculating happen in your mind very quickly if this is how you get used to calculating.
Sure, but all of those work for numbers higher than 10, and all assume you know the multiplication table by heart. The multiplication table (the result of multiplying every number between 1 and 10 with each other) is something you have to memorize. You can get away with memorizing only some of these results and computing the others based on them, but it's basically impossible to do any more complex arithmetic if you don't know most of it by rote memorization.
> Also, learning multiplication with numbers higher than 10 still relies on knowing the multiplication table. 17*8 is 7*8=56, hold the 5, 1*8 + 5 = 13, so 136.
Interestingly, I do those less efficiently:
Interestingly, I do those less efficiently:
17 * 8
= (10 * 8) + (7 * 8)
= (80) + (56)
= (80) + (50 + 6)
= 130 + 6
Of course there is a pattern: (n+1)x = nx+x. Your brain can learn it just fine, and then it can multiply numbers without having to burden your slow inefficient symbolic reasoning machinery with rules and facts.
How is that pattern useful for replacing memorization of the multiplication table? 7*8 = 6*8 + 8 - fine. I still need to memorize what 6*8 is, or go through the extraordinarily slow process of expanding this recursively: definitely not an option in school.
It's a linear function your brain can learn. Your brain, not the conscious you. A lot of learning is about bypassing the slow inefficient consciousness that thinks it's in charge.
In sports and other physical activities, you don't memorize the right moves. You practice them until you can do them automatically. The same approach also works with cognitive activities.
In sports and other physical activities, you don't memorize the right moves. You practice them until you can do them automatically. The same approach also works with cognitive activities.
If that were how people learned the multiplication table, you would see that people take longer to come up with the result of 9*9 than it takes them to compute 3*5. I have never seen anyone work this way, and so I believe it's far more likely people just remember the results in a table. "9*9=81" is simply a fact you have stored, not a computation you perform, even subconsciously.
Edit: I should also note that it's pretty well known people learn arithmetic as symbol manipulation and not some higher order reasoning. The reason this is pretty well established is that historically, the switch from Roman numerals to Arabic numerals led to a huge flurry of arithmetic activity, because it was so much easier to do arithmetic with the new symbols. If people had learned by subconsciously calculating the underlying linear functions and not through symbolic manipulation, the switch would have been entirely irrelevant. Yet for most mathematicians in Europe at the time, doing 27*3 was much easier than doing XXVII*III.
Edit: I should also note that it's pretty well known people learn arithmetic as symbol manipulation and not some higher order reasoning. The reason this is pretty well established is that historically, the switch from Roman numerals to Arabic numerals led to a huge flurry of arithmetic activity, because it was so much easier to do arithmetic with the new symbols. If people had learned by subconsciously calculating the underlying linear functions and not through symbolic manipulation, the switch would have been entirely irrelevant. Yet for most mathematicians in Europe at the time, doing 27*3 was much easier than doing XXVII*III.
People are more than their conscious minds. A neural network can compute a linear function with a small domain without having to store each case separately.
I never memorized the multiplication table, because I found it boring and unnecessary. When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.
Some other things I had to memorize. Those were usually lists of arbitrary names with no apparent logic behind them. And if I didn't need them often enough, they never became more than lists of random facts. For example, I often can't tell the difference between sine and cosine without recalling the memorized definitions.
Or, to give another example, Finnish language has separate words for intercardinal directions (such as northeast). Usually when I need one of them, I have to iterate over the memorized list, until I find the name for the direction I had in mind. Similarly, I had to iterate over the six locative cases in Finnish grammar whenever I needed a name for one of them.
I never memorized the multiplication table, because I found it boring and unnecessary. When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.
Some other things I had to memorize. Those were usually lists of arbitrary names with no apparent logic behind them. And if I didn't need them often enough, they never became more than lists of random facts. For example, I often can't tell the difference between sine and cosine without recalling the memorized definitions.
Or, to give another example, Finnish language has separate words for intercardinal directions (such as northeast). Usually when I need one of them, I have to iterate over the memorized list, until I find the name for the direction I had in mind. Similarly, I had to iterate over the six locative cases in Finnish grammar whenever I needed a name for one of them.
Whether it happens consciously or unconsciously, computation takes time. So, if your theory that the brain computes the results instead of remembering them were true, it should take measurably longer to compute 9*9 than it takes to compute 2*3. I am certain that doesn't happen for me, but it could be measured for others as well.
> When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.
This is prefectly explained by some results becoming memorized as you see them more and more, and makes no sense if your unconscious mind were computing things. If your brain was computing these results unconsciously because it had learned the function to apply, it should have come up with results automatically for any (small) multiplication. That it didn't, and you had to consciously do the computation for some numbers, is pretty clear proof that you slowly memorized the same multiplication table, but only filled it in gradually.
Overall I'm not advocating for the importance of cramming the multiplication table. I'm just saying that people who want to do mental arithmetic, or even pen-and-paper arithmetic, can only realistically do it if and when they learn the multiplication table by heart. And, that the reason the multiplication table is taught to children is strictly to have them memorize it so that they can do arithmetic without a calculator at realistic speeds.
> When I had to multiply numbers, some answers just appeared automatically, while I could calculate the rest quickly enough. Over time, more and more answers would appear magically, until I no longer had to calculate at all.
This is prefectly explained by some results becoming memorized as you see them more and more, and makes no sense if your unconscious mind were computing things. If your brain was computing these results unconsciously because it had learned the function to apply, it should have come up with results automatically for any (small) multiplication. That it didn't, and you had to consciously do the computation for some numbers, is pretty clear proof that you slowly memorized the same multiplication table, but only filled it in gradually.
Overall I'm not advocating for the importance of cramming the multiplication table. I'm just saying that people who want to do mental arithmetic, or even pen-and-paper arithmetic, can only realistically do it if and when they learn the multiplication table by heart. And, that the reason the multiplication table is taught to children is strictly to have them memorize it so that they can do arithmetic without a calculator at realistic speeds.
From my point of view, what happened with the multiplication table was practice without memorization, while the word lists were memorization without practice. Two different approaches to learning with two different outcomes.
Oh, absolutely, you learned it through practice instead of pure memorization, which overall tends to be a better strategy. But just like chess masters tend to mostly learn game positions (through practice, note trying to do rote memorization) and not some advanced logic, I am pretty sure you learned the specific results and did not learn some subconscious algorithm deeper than "lookup 9*5 in the stored table".
When kids are asked to memorize the multiplication table, they are actually supposed to memorize it.
> 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.
This is arguably another form of memorization. Magnus Carlson is the best Chess player in the world because he memorizes everything without effort.
It's memorization in the same sense as an LLM is a database. You can think like that if you prefer, but I don't find it a useful perspective.
> 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.
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.
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.
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.
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.
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.
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.
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.
It is fundamentally true, though, because you "level up" those skills through concentrated effort, which requires focus and time, both of which are finite resources. It may feel you can improve on all without sacrificing something, but that just means you're operating far from Pareto frontier - i.e. you're not particularly good at anything.
> It is fundamentally true, though, because you "level up" those skills through concentrated effort, which requires focus and time, both of which are finite resources.
I dunno. My experience is that it's true for some fields, such as videogames/sports.
What I've found is that people who have true expertise in a field (excluding videogames/sports) are generally competent in a number of other fields. The characteristics required to become an expert oil painter or an expert in applied mathematics (for example) are focus, concentration and the ability to recognise new patterns as patterns, and then apply them!
IOW, someone who is an actual master in a certain field should easily become at least competent in other things that they try.
I dunno. My experience is that it's true for some fields, such as videogames/sports.
What I've found is that people who have true expertise in a field (excluding videogames/sports) are generally competent in a number of other fields. The characteristics required to become an expert oil painter or an expert in applied mathematics (for example) are focus, concentration and the ability to recognise new patterns as patterns, and then apply them!
IOW, someone who is an actual master in a certain field should easily become at least competent in other things that they try.
Kasparov might have had a chance — if Russia wasn’t the way it is.
True! He’s like the exception that makes the rule, and through coincidence, ended up not even really an exception.
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.
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).
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.
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.
I think he’s a thoughtful and wise man anyway; it doesn’t seem to have much connection to chess.
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.
"[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.
I’m pretty sure mathematics is slightly more general of a subject than chess…
It is hard, though not impossible to generalize expertise.
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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.
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.
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.
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".
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".
> 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.
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.
A calculator is of little use to someone who does not understand mathematical principles. I've seen that in action many times.
> but the underlying math is something a 10 year old can handle – as can anyone with a calculator.
The calculator is just a tool to get the precision needed. If you understand the concepts, you can get a rough idea of what '17%' is and whether that ballpark is acceptable.
You whip out the calculator only when you need the precision.
The calculator is just a tool to get the precision needed. If you understand the concepts, you can get a rough idea of what '17%' is and whether that ballpark is acceptable.
You whip out the calculator only when you need the precision.
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.
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.
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!)
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!)
> 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.
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.
Why are there always so many commenters on here who don't seem to understand the difference between 'mathematics' and 'arithmetic'?
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.
Outsourcing low-value/high friction tasks is the whole point of technology.
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".
I'm not saying everyone needs to know math but its hardly "vestigial knowledge".
I use math to decide which bus to take to get home.
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.
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.
Honestly, this is a pretty weird take to see on “Hacker News”. This place sure has changed a lot.
What kind of math skills are you talking about that people lack that causes them to not have any money?
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".
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".
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.
That doesn't mean you won't be rich. It's just some of the lowest hanging fruit are not an option.
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.
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.
> 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.
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.
Many of these people claim to have a "system" which will enable them to win. I've talked with some of them. None of them I've spoken to had money. Coincidence?
Watch people at the slots. Do they look like they're having fun? Not to me.
Personally, I've gambled a few times. Lost money. I don't like losing money, it is not entertaining to me in the slightest.
Tell me about people who play the lottery, picking their "lucky numbers". It's sad.
Watch people at the slots. Do they look like they're having fun? Not to me.
Personally, I've gambled a few times. Lost money. I don't like losing money, it is not entertaining to me in the slightest.
Tell me about people who play the lottery, picking their "lucky numbers". It's sad.
> Many of these people claim to have a "system" which will enable them to win. I've talked with some of them. None of them I've spoken to had money. Coincidence?
The original thread was about how statistics education will not cause people of gambling. Of course people almost always lose money gambling, except for very rare exceptions, but that doesn't really have anything to do with my point that people spending meaningful amounts of money on gambling are addicted. Addicts aren't going to just tell you that they gamble, because they are addicted(maybe some will but not in general).
> Personally, I've gambled a few times. Lost money. I don't like losing money, it is not entertaining to me in the slightest.
Some people could probably say the same thing about video games, but nobody disputes that some people enjoy video games.
The original thread was about how statistics education will not cause people of gambling. Of course people almost always lose money gambling, except for very rare exceptions, but that doesn't really have anything to do with my point that people spending meaningful amounts of money on gambling are addicted. Addicts aren't going to just tell you that they gamble, because they are addicted(maybe some will but not in general).
> Personally, I've gambled a few times. Lost money. I don't like losing money, it is not entertaining to me in the slightest.
Some people could probably say the same thing about video games, but nobody disputes that some people enjoy video games.
Blackjack can be beaten (i.e. Ed Thorpe)
Or are these "systems" for slots?
Or are these "systems" for slots?
I read Thorpe's book. Vegas changed they way they operated to defeat it.
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.
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.
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.
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 …
I'm not like the other girls
(commenting on the internet can be fun you know ;-) )
(commenting on the internet can be fun you know ;-) )
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.
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.
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.
> 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.
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.
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