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I have found that when I give problems that require multiple steps or ideas to solve, students often give up quickly and come to office hours begging for hints. Sometimes I break up such problems into steps, with hints for each step; this enables the students to solve them, but (as a colleague pointed out) has the unintended consequence of teaching them (or, rather, reinforcing the lesson they learned in high school) that all solvable problems can be solved in 5 minutes. What are some ways to teach students to keep plugging away at a problem even if they don't seem to be making progress?

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  • $\begingroup$ i'm sure other members will be able to cite specific articles to give more background, but there has been a lot of research on involving relevant contexts of the students lives into math class and how it can help with performance and perseverance. While I know it is not possible for every single exercise, are there any ways that you can work student interests into at least one problem a class? I can't speak for undergrad but this is my go-to technique for my HS algebra and geometry classes when they start to get bored with the material and give-up easily $\endgroup$ – celeriko Dec 16 '15 at 0:22
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    $\begingroup$ One keyword if you want to comb through some of the literature is grit. As a side-note, perhaps what you want is students to stick with problems, but also to monitor their own approaches: It can be good to "keep plugging away at a problem" even if they aren't making progress, but another important skill is to examine the strategy (or strategies) being used. It may be that the lack of progress is not due to a lack of effort, but rather a poor choice of approach! See Schoenfeld on metacognition for further remarks in this general area. $\endgroup$ – Benjamin Dickman Dec 16 '15 at 0:52
  • $\begingroup$ I may be totally wrong, but my gut reaction is that this is a character trait, and that a character trait can't be instilled by a school or a teacher -- especially not for students of college age, whose personalities are already formed. $\endgroup$ – Ben Crowell Feb 8 '16 at 2:50
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I believe that my students can learn best when they display perseverance (sometimes called grit). I'll discuss how I teach, model, and encourage mathematical perseverance.

Tell a story about perseverance

When I speak to students who are struggling, especially if they failed the class before, I tell them the story of Ella, one of my algebra students. When Ella took my algebra class for the first time, she struggled with math and eventually gave up on passing the class. Later, she took the class with me a second time. I must confess that I had my doubts--but she came to me and said "this time is going to be different." I told her that I was glad to hear it, but she must have heard the doubt in my voice, because she felt the need to explain: "I have a notebook," she said. "In it I'm going to write down every single problem, carefully and step-by-step." Sure enough, whenever I saw her doing her homework (which was on the computer, by the way--no notebook needed), she carefully wrote down every problem step-by-step. As the weeks went by, she worked through all the sections that she struggled with before, eventually acing material that had caused her to quit. I'm delighted to report that Ella passed the algebra class the second time with an A.

How can we can get more students to do display such perseverance? How can we get students to learn from their mistakes and struggles and do better next time?

Communicate the value of hard work

Sometimes in mathematics, we worship genius and intellect. With good reason: the math we study and love is the result of mathematical insights from great minds like Newton and Gauss. We too easily forget that mathematical insight is also the fruit of hard work--Newton didn't invent calculus in a single moment of brilliant insight, rather he spent years of study developing calculus.

There's strong evidence that telling students they are smart is actually very detrimental to their education. If we give students the idea that success in math depends on their intelligence, we're telling them that their success comes down to how lucky they got in the genetic lottery. Although intelligence doubtless has a strong effect on students' success in math, we should encourage students to focus on what they can change--how they work--and compliment them on their hard work when they succeed.

Model perseverance

I sometimes insert errors into my lectures--sometimes even intentionally--to make sure they are paying attention, but also to demonstrate how to deal with mistakes.

In an algebra class, you might write $x^2-4 = (x-2)^2$. Keep going as if nothing is wrong--they should catch you soon. Ask them: why this is false? How should we have proceeded?

In a calculus class, begin "solving" $\lim_{x \to 0} \frac{x^2}{e^x}$ using repeated applications of L'Hopital's rule. You students probably won't realize that L'Hospital's rule doesn't apply, so you may get all the way to "solution". Then tell them there is a mistake somewhere in your work. Ask them to find it and then discuss the conditions for applying L'Hopital's rule.

When I ask students questions, I occasionally write what they say on the board even if it's wrong. I'll play along a little while until another student catches the mistake. Then I'll ask: who's right? How do we know?

This TED talk conclusively proves it

If you'd like a very nice introduction to the research on grit and its important role in fostering student, check out Angela Lee Duckworth's TED Talk on the subject.

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  • $\begingroup$ Edison's say that "Genius is 1% inspiration and 99% perspiration" is often overlooked... $\endgroup$ – vonbrand Dec 23 '15 at 11:48
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For teaching perseverance, I would recommend works of Barbara Oakley.

She doesn't advocate working on a problem 24/7 but rather using smart perseverance techniques when you switch between two modes of your brain: focused and diffuse. The first mode allows you to stick to your methodology while the second one - lets you explore other methodologies when the current one doesn't work or not efficient.

She also explains the science behind the worst enemy of perseverance - procrastination, and gives tips and tricks for preventing it, for example the Pomodoro technique.

The last but not least in her list of efficient learning techniques is sleep - something that we often neglect in our struggles to finish our tasks no matter what.

Below you can find online resources about her methods and much more:

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Take a look at Pólya's classic "How to Solve It" (Princeton University Press, 1945). It gives hands-on advice on strategies on how to solve complex problems, mostly in the setting of problems understandable/solvable by high school students.

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