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I am curious if you guys learn more from Hard problems, or do you learn more from easy problems.

I heard that you learn a lot more from doing harder problems, because you have to learn all of the easier mathematical processes along the way.

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    $\begingroup$ Problem too hard, student gets nowhere, student learns nothing. Problems too easy, student gets bored and stops doing them. We need to find the right middle ground: easy enough that the student can make progress, hard enough that it is not boring. Another twist is, of course, that this middle ground may differ from one student to another. One-on-one instruction can get the level right. Maybe in the future we will use AI to do that one-on-one instruction, but currently (when we use actual people to teach) it is too expensive. $\endgroup$ – Gerald Edgar Oct 31 '19 at 14:11
  • $\begingroup$ This is a great way of thinking about the right approach. Thank you @GeraldEdgar $\endgroup$ – LifeBeyondTheClouds Oct 31 '19 at 14:13
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    $\begingroup$ @GeraldEdgar: That should be an answer. $\endgroup$ – Ben Crowell Oct 31 '19 at 15:37
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I think a lot of people here and in research mathematics overemphasize the value of harder problems, discovery learning, etc. It appeals to them since they are the tip of the spear in intellect. And it's more intrinsically interesting to someone who already knows the stuff (them).

Also many of them have never done sports coaching and learning the benefit of easy, progressive training, with only very occasional all out efforts (mostly for building emotional fortitude, not skill). You can learn a LOT by imitation and practice. In fact our brains are probably wired for it--look at how kids learn. Of course the real analysis jocks don't get that since they think in terms of legalistic formal frameworks and subconsiously think this is how the human animal (and it is an ANIMAL) learns.

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    $\begingroup$ P.s. I think this question is in the pedagogy center of interest for math educators, despite (perhaps) having been asked with a student point of view. I would hope that teachers are interested in how students learn best. Specifically, teachers have a huge impact in terms of recommending drill problems, selecting appropriate texts, etc. $\endgroup$ – guest Oct 31 '19 at 18:48
  • $\begingroup$ I see the impact. This is super important to think about. $\endgroup$ – LifeBeyondTheClouds Nov 1 '19 at 3:16
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    $\begingroup$ This seems like a poor answer to me. The first paragraph is an ad hominem directed at unspecified Bad People. (You know who you are). The second paragraph is based on a weak analogy with sports training coupled with what I think is a false statement about sports training. (I don't think it's valid to lump together math with sports, nor am I convinced that it's valid to lump all sports together. The sports that I'm serious about are running and rock climbing. I think the claim about easy practice may be 50% true and 50% false for climbing, and 0% true for running.) $\endgroup$ – Ben Crowell Nov 1 '19 at 4:23
  • $\begingroup$ I'm rather stating the obvious, but teaching math is quite different from "sports coaching". I do not see why something that works in sports coaching carries over automatically to math education. As it stands I disagree with this answer and find it poorly supported. $\endgroup$ – YiFan Dec 2 '19 at 0:58
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One way to address this issue is with more open-ended problems that have a "low floor" (easy entry) and "high ceiling" (hard aspects that can be addressed but don't need to be).

Sometimes this comes in a series of related problems. See the Julia Robinson Mathematics Festival activities for many great examples (click on 'activities library' for a pdf).

Other times there can be one problem that a group of students can work on. One example that comes to mind is the "squareness" problem described here (on my blog). Henri Picciotto discusses this idea in the context of "rich activities" on this blog page. And that term takes us to a wealth of rich problems, called rich starting points (RISP) in the UK, at this site.

Most of these examples are K12. But these concepts can also be used at higher levels.

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  • $\begingroup$ You mean like Exploding Dots? $\endgroup$ – kcrisman Nov 1 '19 at 16:23
  • $\begingroup$ That is definitely one good example. $\endgroup$ – Sue VanHattum Nov 1 '19 at 17:23
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I don't think that "hard" and "easy" are the right categories when it comes to answer the question: "What kind of problems do you learn most from?" Indeed "hard" and "easy" are really subjective and many people might find "hard" the "easy" problems I learn most from and vice versa.

Since I guess that purpose of the question is giving students problems from which they can learn most, I think a better distinction is between:

  • Imperative problems: Find $x$... Prove that... Check the condition...

and

  • Question problems: Is it true that...? Does exist ...? Can we compute ...?

Althought both categories of problems are important, I believe that one learns much more from the second one (which, unfortunately, unsurprisingly, is also the more neglected in tests and exercises).

In fact, on the one hand, when solving imperative problems, the target is clear from the beginning and usually the student has only to apply some algorithmic methods in the right way. What he learns is applying this methods better and better.

On the other hand, when solving question problems, neither the target nor the methods are clear from the beginning, and the student has to make many tries and guesses. What he learns is developing a better intuition of the subject.

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  • $\begingroup$ This is super informative. I didn't think about questions in this way. Solving question problems sounds preferable. $\endgroup$ – LifeBeyondTheClouds Nov 1 '19 at 20:31

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