[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.