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I've often come across a view on teaching/learning math to the effect that forcing students to struggle rather then "spoon-feeding them" (as they put it) makes for much more able students in the long run. a combination of "what doesn't kill you..." and "tough love".

I have friends who have been borderline traumatized by college level lecturers who subscribed to this view (or were just jaded, heartless butchers) and they tend to strongly disagree. Still, I keep hearing and reading this claim and more often then not from respected and enviably capable mathematicians past and present.

Here are some examples. The first is by Lebesgue.

When I was a rather disrespectful student at the Ecole Normale we used to say that 'If Professor Jordan has four quantities which play exactly the same role in an argument he writes them as $u$, $A''$, $\lambda$ and $e_{3}'$ Our criticism went a little too far but, nonetheless, we felt clearly how little Professor Jordan cared for the commonplace pedagogical precautions which we could not do without, spoiled as we were by our secondary schools. <...> Professor Jordan's only object is to make us understand the facts of mathematics and their interrelations. If he can do this by simplifying the standard proofs, he does so; <...> But he never goes out of his way to reduce the reader's trouble or compensate for the reader's lack of attention.

Another (extreme) example is the approach of R.L moore and the so called "Moore Method" of teaching, still alive and kicking in various degrees of severity.

Here's an excerpt from P.R. Halmos' autobiography "I want to be a mathematician":

Can the mathematician of today be of any use to the budding mathematician of tomorrow? Yes. We can point a student in the right direction, put challenging problems before him, and thus make it possible for him to "remember" the solutions. Once the solutions start being produced, we can comment on them, we can connect them with others, and we can encourage their generalizations. Almost the worst we can do is to give polished lectures crammed full of the latest news from fat and expensive scholarly journals and books—that is, I am convinced, a waste of time. You recognize, I am sure, that I am once more advocating something like the Moore method. Challenge is the best teaching tool there is, for arithmetic as well as for functional analysis, for high-school algebra as well as for graduate-school topology.

Lastly, here's a quote from the preface to Mathematics Made Difficult. Although the book is written tongue-in-cheek I believe the following passage is ultimately uttered in earnest:

there is no doubt that an absolute ignoramus (not a mere qualified ignoramus, like the author) may be become slightly confused on reading this book. Is this bad? On the contrary, it is highly desirable. <..misleading redaction...> it is hoped that this book may help to confuse some uninitiated reader and so put him on the road to enlightenment, limping along to mathematical satori. If confusion is the first principle here, beside it and ancillary to it is a second: pain. For too long, educators have followed blindly the pleasure principle. This over-simplified approach is rejected here. Pleasure, we take it, if for the initiated; for the ignoramus, if not precisely pain, then at least a kind of generalized Schmerz

To put this into the form of a concrete question, what does modern research say about the effectiveness of this approach, is it justified? Or are mathematicians testifying to it's benefit perhaps just a manifestation of selection bias?

Update

As suggested, I've asked the question seperately on cogsci.SE https://cogsci.stackexchange.com/questions/5921/

Update Two relevent papers:

Still hoping to hear of more specific research.

Update

Also asked on the new math educators SE: https://matheducators.stackexchange.com/questions/875/

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  • $\begingroup$ It depends on your audience and intent. $\endgroup$ – copper.hat Mar 6 '14 at 18:25
  • $\begingroup$ It usually does. Do you know of any authoritative research results on the question? $\endgroup$ – user133266 Mar 6 '14 at 18:27
  • $\begingroup$ en.wikipedia.org/wiki/Reinforcement_learning ;-) $\endgroup$ – user76568 Mar 6 '14 at 18:27
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    $\begingroup$ Micromanaging the difficulty level of the course is probably just as bad as micromanaging the content of the course (trying to control what everyone takes away from the course). Mathematics is self-selecting in the sense that serious students who are comfortable with any given level of external difficulty can easily occupy themselves with more difficult problems. The only way to do it wrong as a teacher is to shut down their personal initiatives. $\endgroup$ – Joshua Pepper Mar 6 '14 at 18:28
  • $\begingroup$ It is unclear to me how much of this question is specific to mathematics. Perhaps you should consider asking the question at cogsci.stackexchange.com/help/on-topic where there may be more experts who are familiar with the "modern research" concerning the "effectiveness of this [learning] approach". $\endgroup$ – Willie Wong Mar 7 '14 at 9:04
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Learning math is difficult; it takes work. Euclid stated this quite nicely: "There is no royal road to geometry." What should teachers do? Encourage, prod, and guide students into doing that work. Obviously, requiring too much or too little isn't good. Any decent teacher will try to find a good balance.

Of course, a teacher can pointlessly 'make things difficult.' For example, one of my professors is Chinese, and it sure would make learning math a lot harder for me if he taught in Chinese. The purpose of using words (and symbols) is to convey information. Whatever an instructor does cover should be as clear as possible. But sometimes, it is better not to say everything. For this reason, a 'hint' can be much better than a complete solution.

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  • $\begingroup$ Thanks, that makes sense. I was hoping for an answer that cites actual studies on learning, specifically in a math education context. $\endgroup$ – user133266 Mar 7 '14 at 16:46
  • $\begingroup$ Right; sorry I don't have that. It would be nice if someone did. $\endgroup$ – Andrew Kelley Mar 7 '14 at 23:03

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