.Do students learn better when challenged (specifically, in math education)?

Question

^{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 & 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.

Answer 1

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, <sarcasm> despite my obviously brilliant instruction </sarcasm>.

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:

1. 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.
2. 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.
3. 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).

Answer 2

I have an anecdotal answer with regard to learning Physics. I sat in on a colloquium where a Physics professor discussed his experience with a course that was taught completely through experimentation. Students had to derive their learning of Physics completely through semi-guided experiments, and no lecture. In the beginning of the course, the professor reports that students found the process very tedious. However, as they got the knack for experimentation, they enjoyed it more, and retained their learning exceptionally.

In this case, I think that struggle can lead to better learning. If you personally derive a concept from first hand experimentation, you are very likely to remember it. In that case, you are not learning science, you are learning to be a scientist. That distinction between learning about versus learning to be almost always deepens understanding, and consequently recall.

On the other hand, this does not apply to struggle for struggle sake. It seems likely that there are some teaching methods that require students to struggle, and actually work. Yet there may be other methods where the struggle is fruitless.

Answer 3

Robert Bjork calls this desirable difficulties. That is, students seems to learn best when they are required to encode and retrieve information. Some examples of desirable difficulties include: testing, spacing/interleaving, generating information, changing studying environments, etc. In the long run, these seem to promote long term learning.

Answer 4

Norman Triplett's research on social facilitation shows, that people tend to be better in simple tasks or physical activity (cyclists were faster) in presence of others. However, in cases such as solving mathematical tasks, quantity of tasks solved increases, while the quality decreases (social loafing). The answer therefore depends on whether you use "be better than all these people" challenge or "show yourself you're the best" challenge. Most of the researches, however, show that motivating or challenging students causes rather stress and tension than better results.