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# Does overcoming difficulty independently make learning more effective, specifically in math education?

Originally posted on Math.SE, but it was suggested cogsci.SE would be a more suitable venue.

I've included several quotations from various mathematical figures to the effect that requiring students to overcome difficulties on their own ("not helping them" in a narrow sense) is a superior way to teach/learn mathematics.

There have been classes in my past where a lecturer flaunted this philosophy, when in fact students felt overwhelmed and discouraged, and the results of the course overall were quite the opposite of what its advocates suggest, and on multiple occasions.

So, what does current research have to say on the matter, and what are some of the insights it provides? Specifically:

1. What role do the students' innate abilities play? How do weaker/strong students respond?
2. Assuming the learning process is slower, what's the long-term effect? Is it better overall, or is depth of understanding directly exchanged for breadth?
3. Does age play a factor? For example, is it effective for adults but not children?

For the sake of completeness, I reproduce here the examples I included in the original question. First, it seems appropriate the mention the so-called "Moore Method" of teaching. Next, a quote by Lebesgue, certainly a very able mathematician:

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.

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.

Last, 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.

 asked Mar 7 '14 at 13:12 user4549