4 ms·

> 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

> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. Amusingly, many people think the solution to this is "abandon worked examples and focus exclusively on trying to teach general problem-solving skills," which doesn't really work in practice (or even in theory). That seems to be the most common approach in higher math, especially once you get into serious math-major courses like Real Analysis and Abstract Algebra. What actually works in practice is simply creating more worked examples, organizing them well, and giving students practice with problems like each worked example before moving them onto the next worked example covering a slightly more challenging case. You can get really, really far with this approach, but most educational resources shy away from it or give up really early because it's so much damn work! ;)

kiba 2y ago
Teaching a skill directly is known to be more a more efficient way of learning rather than force students to try to discover it on their own.
- smogcutter 2y ago
  Interestingly, there have been studies that show that students lectured to feel like they’ve learned more, and self-report that they have, while students learning the same material in self-guided labs report feeling like they’ve learned less but perform better on assessments.
  - JustinSkycak 2y ago
    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.
    - Jensson 2y ago
      > (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.
    - smogcutter 2y ago
      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.
      - JustinSkycak 2y ago
        > 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 https://en.wikipedia.org/wiki/Desirable_difficulty
- remarkEon 2y ago
  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.
  - corimaith 2y ago
    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.
    - remarkEon 2y ago
      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).
  - Suppafly 2y ago
    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.
- makeitshine 2y ago
  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?
- musicale 2y ago
  Many of the drivers I encounter seem to have been given cars (or trucks) and then left to their own devices.
catgary 2y ago
Eh, I think that’s setting students up for failure once they enter graduate studies or more open ended problems that don’t come from a problem bank. Productive struggle is a perfectly valid approach to teaching, it’s just less pleasant in the moment (since the students are expected to struggle).
- nrr 2y ago
  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.
  - lupire 2y ago
    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.
    - nrr 2y ago
      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.
      - abnry 2y ago
        Aside from that, understanding counterexamples teaches you to understand the definitions and theorems better. Which matters for proving future results.
hiAndrewQuinn 2y ago
+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.
- JustinSkycak 2y ago
  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.

kiba 2y ago

Teaching a skill directly is known to be more a more efficient way of learning rather than force students to try to discover it on their own.

smogcutter 2y ago
Interestingly, there have been studies that show that students lectured to feel like they’ve learned more, and self-report that they have, while students learning the same material in self-guided labs report feeling like they’ve learned less but perform better on assessments.
- JustinSkycak 2y ago
  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.
  - Jensson 2y ago
    > (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.
  - smogcutter 2y ago
    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.
    - JustinSkycak 2y ago
      > 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 https://en.wikipedia.org/wiki/Desirable_difficulty
remarkEon 2y ago
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.
- corimaith 2y ago
  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.
  - remarkEon 2y ago
    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).
- Suppafly 2y ago
  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.
makeitshine 2y ago
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?
musicale 2y ago
Many of the drivers I encounter seem to have been given cars (or trucks) and then left to their own devices.

catgary 2y ago

Eh, I think that’s setting students up for failure once they enter graduate studies or more open ended problems that don’t come from a problem bank. Productive struggle is a perfectly valid approach to teaching, it’s just less pleasant in the moment (since the students are expected to struggle).

nrr 2y ago
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.
- lupire 2y ago
  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.
  - nrr 2y ago
    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.
    - abnry 2y ago
      Aside from that, understanding counterexamples teaches you to understand the definitions and theorems better. Which matters for proving future results.

hiAndrewQuinn 2y ago

+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.

JustinSkycak 2y ago
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.