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.
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.
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.
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).
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?