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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Sign up to join this communityI 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.
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.
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.
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:
and
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.