[SOTGO]

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]

> 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]

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]

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]

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.

[smogcutter]

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]

> 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

[Jensson]

> (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.

[remarkEon]

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]

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]

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]

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]

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]

Many of the drivers I encounter seem to have been given cars (or trucks) and then left to their own devices.

[hiAndrewQuinn]

+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]

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.

[catgary]

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]

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]

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]

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]

Aside from that, understanding counterexamples teaches you to understand the definitions and theorems better. Which matters for proving future results.

[setopt]

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.

[nothercastle]

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

[JustinSkycak]

> 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.

[lupire]

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.

[vsuperpower2021]

Most research makes assumptions like these without laying them out. It might explain none of their work can be reproduced.

[magicalhippo]

> 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]

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.

[chongli]

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]

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

[creer]

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.

[will1am]

The importance of worked examples in helping students understand new topics

[aaplok]

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]

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]

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.