.Do students learn better when challenged (specifically, in math education)? Originally posted on Math.SE,
but it was suggested cogsci.SE would be a more suitable venue.

I'm aware of two publications that have trickled onto the radar screens
of non-specialists:


Fortune Favors the Bold by Diemand-Yauman &amp; Oppenheimer
Designing Effective Multimedia for Physics Education
 by Derek Muller
(TED talk)


The former focuses on the impact of "disfluency" on retention and success when processing written material, and the latter seems to focus on learning in the 
case where preexisting misconceptions need to be corrected (physics).

I'm interested to hear about similar research results in the learning of
abstract concepts, specifically mathematics.

The motivation described in the original question on Math.SE is the
fact that many mathematicians seem to hold the (painful, to students) 
belief that forcing students to struggle (in some sense) is beneficial.
I've quoted multiple examples of this attitude from various sources in 
the original question, you are welcome to have a look.

Does current research support the belief that (in mathematics) difficult
learning is better learning?
This might be in terms of retention, long-term achievements, motivation, etc'.

Update 

I've posted an expanded form of the question on matheducators.SE beta: https://matheducators.stackexchange.com/questions/875/

Still hoping for helpful answers from the cogsci.SE community.
Let me begin by saying that the answer is nowhere near as simple as you or I would like it to be. There are several reasons for this, but the main reason is that there are myriad ways that students can struggle through the material.

I became interested in this subject when I was a graduate teaching assistant in the Industrial Engineering program at Iowa State. One of the classes that I taught dealt with the concept of measurement variation, which inescapably includes several important statistical concepts, which to me (having had four advanced statistics/methods courses) were painfully obvious; however, my students had no hope whatsoever.  When lab report grading time came around, I was very disappointed to see that even my brightest students completely missed the point of the experiment, &lt;sarcasm&gt; despite my obviously brilliant instruction &lt;/sarcasm&gt;.

So, I set out to do research and conduct experiments into exactly why my students had trouble with this material.  I've written two papers on the subject (with a third in-work).  Here are some of my general findings:


The learning process depends upon the learner (pre-existing knowledge and conceptions), the material (type and difficulty), and the teaching method. These three interact in a complex way, as several researchers have explored. The most comprehensive set of literature on this can be found by searching Cognitive Load Theory.
Problem-based learning (which is what @JohnYetter describes in his answer) has been proven to be a highly effective teaching method for science and math when executed properly.  Proper execution of PBL requires that students are familiar with the method, and my studies have shown that properly-designed scaffolding (things that help guide the process and point out key concepts) is a significant, essential component.
Lecture does not appear to have any effect on learning (at least for measurement variation and the related concepts), and may actually harm the learning process (this is what I am studying further right now).


The theories agree that the reason PBL is effective is that it forces students to make the necessary schematic connections to build their knowledge. The intent of the process is to take novices and begin to develop their knowledge base such that it functions more like an experts' knowledge base (there are proven differences between how novices and experts explore problems).  Along with this, I think that you need to have a particular type of student, one who has self-discipline and doesn't give up. Such students are well-suited for engineering and science; students who don't possess these abilities or who are not capable of rapidly adapting find themselves on the way over to a non-technical degree (so my studies of engineering students only automatically exclude this population of students, which is another piece I need to study further).