John: During class, I have them work on one challenging problem at a time. They work. I walk around and listen. They are encouraged to try it themselves first, then discuss their work with their neighbors. I do not give hints or show methods to solve the problem. They know that they have been given a problem, not an exercise. That is, I expect that it will take time to solve it, it is related to what we are working on but is not a copy of other problems they have been asked to work on. I do not ask them to do the problem, but insist that they work on it. Every fifteen seconds, or so, I walk by and observe progress. If you want them to become persistent you must provide situations where persistence is the only way. At some point during the class, we discuss solutions that various students have proposed. If it is a worthy problem, one that requires persistence, there will be several ways to approach it. When the problem has been solved, they have learned the content for tonight's homework. That is the intent of the problem, to teach the new lesson by having them figure out what the next thing is. Then I give them another problem. I never give them a worksheet nor do I give notes. We spend class time working on worthy problems. They learn the content, they learn how to solve problems, they learn "grit" and most of all, they love it.
It takes a while for them to get used to the concept that I will not "show them how to do the problem before I ask them to do it." but once they do, they are actually learning math.
PJ.: One lesson I learned about this is that it's hard to learn more than one thing at a time. So a good problem to use to teach grit is one in which the actual skills embedded in the problem aren't that difficult, but maybe they have to think through a lot of different possibilities, go down a blind alley, or something. It helps a lot if the problem has someplace fairly obvious to start (even if it's ultimately the wrong place) and if there's some obvious way to check that your answer is correct. Many math contest problems require a lot of grit but fail on these two counts, because there's basically only one thing to do, and the "problem" is seeing what that one thing is. On the other hand, problems like the camel & bananas problem (one version + solution here), or the farmer with the broken eggs (look here), or the locker problem (look here) can be good places to start.
In terms of classroom practice, I'm not as hands-off as John, but I think it's really important NOT to scaffold the problem, especially in problem-specific ways. Scaffolding the problem by asking leading questions just leads the students, and teaches them that they needed your help to do the problem, which isn't what you want them to learn. Following Polya and my friend Doug, I have a series of nudges I tend to give in these situations, and I choose the problem with those nudges in mind. (That is, I'm thinking about which general strategies I want kids to apply.) There aren't many of these nudges, and I use the questions a lot, to the point that I can start asking students "What do you think I'm going to tell you to do?" A not-totally-exhaustive list, in no particular order:
- Try a simpler case.
- Try several simpler cases, and look for a pattern.
- What are the conditions? Can you state ______ as a mathematical sentence?
- What's the unknown?
- Look for congruent triangles/similar triangles. [I teach a lot of geometry]
- Try chasing angles. [Same]
- Reduce the problem to finding a point. [Same]
- Try dropping one condition and satisfying the others.
- Can you satisfy even one of these conditions?
- What's a related problem? Can you transform this problem into that one?
- Use algebra
John: I think it is more what we choose to emphasize when we try to get other people to understand what we are trying to do. I believe teachers tell kids too much so I emphasize that I do not tell them anything. The things you listed are certainly questions I ask my students , as well as things like, can you prove it, can you do it another way, does your answer make sense, does your answer agree with the answers others got, perhaps you should draw a bigger picture.....
[After this, another teacher wrote in about Carol Dweck's work on mindset, and fourth teacher said that she regularly assigns her kids hard problems to work over a week or two; students don't ask her for help unless they're "totally stuck". I wrote back the following pair of replies:]
P.J.: Two brief extra thoughts:
- It turns out that kids actually respond when they are taught the science behind cognitive development -- that in itself can be a way to change their mindset -- which is maybe surprising?
- I actually encourage my students to see me relatively early on (before they're totally stuck), because I worry that their peers will give them too much help. But I have a small enough roster that I don't have to worry about being overwhelmed.