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For me it seem to be obvious that exercises and exercise courses are important for undergraduates to learn mathematics and skills like finding a proof or writing it down. Was there any research showing that exercises are necessary for math education (undergraduate education, math majors) or that they are very effective?

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    $\begingroup$ My own personal research seems to indicate that students who actually do their own homework do better in the course. Despite ethical considerations, I even have a group of placebo students, these students tell themselves they study, but, in reality they just talk to friends in study groups. Weirdly, doing math is needed for understanding math. I also have a group who has neither placebo nor actual effort, these try to survive on hope alone. It usually goes worst for the third category. $\endgroup$ – James S. Cook Aug 17 '16 at 20:00
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    $\begingroup$ I might quibble that the usual notion of "exercise(s)" is far too stylized to serve the purposes it claims... unless success is measured exactly and only by more-of-the-same, which is usually the case. Surely what we really want is (prolonged) engagement, and having assigned exercises is a caricature of that. I'd claim that mathematics is about phenomena, or human mock-ups of phenomena (whether "pure" or "applied" math), and our struggles to understand them. The typical contrived end-of-chapter exercises do not reliably do this (from what I've seen of myriad textbooks at many levels)... $\endgroup$ – paul garrett Aug 17 '16 at 21:37
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    $\begingroup$ Is there any research showing that exercises are necessary to become a plumber, an athlete, an artist… What I am trying to say is, that mathematics is "doing something" and to learn to do something you must do it. $\endgroup$ – Dirk Sep 1 '16 at 17:39
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    $\begingroup$ Can you learn to ride a bike by watching someone else ride a bike? $\endgroup$ – mweiss Sep 2 '16 at 2:36
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    $\begingroup$ Oh, I'd agree that "doing the thing" is important to learn how to ... do the thing. But genuine mathematics does not much resemble the exercises in textbooks. $\endgroup$ – paul garrett Sep 4 '16 at 4:15
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From my own experiences as a undergrad math major, I would say that good, quality exercises are one of the best tools for teaching math to people. Paul Halmos was right: "The only way to learn mathematics is to do mathematics."

I myself, got by for a long time on just showing up to class and paying attention in college math classes (I even aced vector calculus doing that; grade was all exams, so I didn't do any of the homework). But eventually I got the point in math where that stopped working (and did rather poorly in Advanced Linear Algebra because of it; it was also a rather challenging senior-level class and that was my first year of university after having been at a community college). By my senior year, I was actually working though the homework, and I was acing all of my advanced math classes. I remember the material I actually worked on far better than the material I didn't actually work though (I don't even know vector calculus very well these days, but I can still do challenging topology proofs even though it's been 3 years since I took that class).

So what are good, quality exercises? In my opinion, the best kinds of exercises and the ones that present a problem, maybe a new thing related to what they've been learning, and then guide them in exploring that new thing via some prodding questions designed to make the student examine its relation to the things they've been learning about.

For example, suppose you were teaching algebra students about associativity and commutativity. A good, quality exercise, might be to introduce 2x2 matrices and their addition and multiplication operations (definition in problem style; as is common in advanced math texts) and then ask them to show that those operations are associative. Then ask if they are commutative and to explain their answer.

Questions like this are great for several reasons: They force the student to comprehend the definitions presented, as well as requiring them to understand the ideas they have been learning. It requires basic problem solving skills, and implicitly forces them to learn about the new object they are being introduced to (great way to stimulate curiosity in mathematics students). It forces them to work from definitions towards what they want to show, which emphasizes the necessity for justification and the reasoning behind a result.

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I keep rather copious statistics on student performance in my classes, but these statistics are unpublished and not peer-reviewed. In my community college remedial algebra courses, doing the assigned homework exercises is, among 40 factors assessed, the single highest-correlated item with the test average, final exam score, and end-of-semester grade ($R^2$ = 0.20, 0.10, 0.15 respectively; N = 149 over the last year-and-a-half). This is self-reported by students (Likert 5-point scale), and despite the homework not being collected nor directly worth any points in the class.

This effect is not so pronounced in my sophomore-level statistics class ($R^2$ = 0.02, 0.02, 0.03 on the same measures; N = 208). I might guess that more competent students are able to "see" the principles and concepts immediately in class, whereas weaker students are more dependent on an extended series of rote exercises to develop the expected skills.

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