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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 student

by SOTGO 2y ago

Anecdotally I have found this to be the case for the students I tutor. When I introduce a new topic I always start with worked examples, and I find that students are able to learn much more effectively when they have a reference. Poor pedagogy is also one of my biggest gripes with my undergraduate math program too, where the professors and textbooks often included too few worked problems and proofs, and the ones they did include were not very useful. What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. Differential equations seems to be a particularly bad offender here, since I've had the same issue with the examples in many texts.

JustinSkycak 2y ago
> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. Amusingly, many people think the solution to this is "abandon worked examples and focus exclusively on trying to teach general problem-solving skills," which doesn't really work in practice (or even in theory). That seems to be the most common approach in higher math, especially once you get into serious math-major courses like Real Analysis and Abstract Algebra. What actually works in practice is simply creating more worked examples, organizing them well, and giving students practice with problems like each worked example before moving them onto the next worked example covering a slightly more challenging case. You can get really, really far with this approach, but most educational resources shy away from it or give up really early because it's so much damn work! ;)
- kiba 2y ago
  Teaching a skill directly is known to be more a more efficient way of learning rather than force students to try to discover it on their own.
  - smogcutter 2y ago
    Interestingly, there have been studies that show that students lectured to feel like they’ve learned more, and self-report that they have, while students learning the same material in self-guided labs report feeling like they’ve learned less but perform better on assessments.
    - 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
- 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.
magicalhippo 2y ago
> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. That was the bane of my University degree. "And, since our function f happens to be of this form, all the difficult stuff cancels out and we're left with this trivial stuff" and then none of the problems have these "happy accident" cancellations and you're none the wiser on how to proceed. The statistics book we used was an especially egregious offender in this regard.
- dan-robertson 2y ago
  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.
golemiprague 2y ago
[dead]
will1am 2y ago
The importance of worked examples in helping students understand new topics
- aaplok 2y ago
  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.
  - vsuperpower2021 2y ago
    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.
    - Jensson 2y ago
      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.
      - vsuperpower2021 2y ago
        [flagged]
chongli 2y ago
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.
- musicale 2y ago
  I always liked systematic methods, including analytic (e.g. laplace transform) and numerical approaches (e.g. runge-kutta.)
creer 2y ago
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.

by SOTGO 2y ago

JustinSkycak 2y ago

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

kiba 2y ago
Teaching a skill directly is known to be more a more efficient way of learning rather than force students to try to discover it on their own.
- smogcutter 2y ago
  Interestingly, there have been studies that show that students lectured to feel like they’ve learned more, and self-report that they have, while students learning the same material in self-guided labs report feeling like they’ve learned less but perform better on assessments.
  - 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
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.

magicalhippo 2y ago

> What I found especially frustrating was when a worked example solved a special case with a unique approach, and the general case required a much more involved method that wasn't explained particularly well. That was the bane of my University degree. "And, since our function f happens to be of this form, all the difficult stuff cancels out and we're left with this trivial stuff" and then none of the problems have these "happy accident" cancellations and you're none the wiser on how to proceed. The statistics book we used was an especially egregious offender in this regard.

dan-robertson 2y ago
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.

golemiprague 2y ago

[dead]

will1am 2y ago

The importance of worked examples in helping students understand new topics

aaplok 2y ago
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.
- vsuperpower2021 2y ago
  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.
  - Jensson 2y ago
    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.
    - vsuperpower2021 2y ago
      [flagged]

chongli 2y ago

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.

musicale 2y ago
I always liked systematic methods, including analytic (e.g. laplace transform) and numerical approaches (e.g. runge-kutta.)

creer 2y ago

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.