I’ve been thinking a lot about that question the last few weeks. I teach math to three different “ability groupings” of kids, and yet in many ways they have similar sets of reactions in grappling with a difficult problem.

When they first hear the problem, there generally is interest and excitement, especially if I have chosen a good problem. (By the way, by “problem”, I don’t mean a routine exercise, but rather something that requires them to think, to use what they have previously learned but in a different way.) Students generally start talking among themselves about what they might do, or they raise their hands to ask clarifying questions and to propose a plan of attack. At this point, things have a really good feel– students are engaged, and they are anticipating solving a thorny problem if they just make a sincere effort. And indeed, that is often how things go– students spend a few minutes trying an approach or two, make a mini-breakthrough, and then solve the problem.

But often, after 5-10 minutes of thinking, students find themselves at an apparent dead end. As a teacher in a department that really values letting students come to their own understanding of material as much as possible, this is a particularly crucial moment. If I walk over and just tell them what to do next, then the students don’t get to make sense of things themselves, and being able to do that is one of the most powerful experiences mathematics can offer. Having the teacher “tell you the answer” is, conversely, usually deflating, even if subtly so.

I have certainly “punted” in this situation before, and in effect just gave in and, in the nicest way I could, essentially said what to do. Those are certainly not my proudest teaching moments, although at the time I feel like I am saving the class from the greater danger of a kind of almost existential despair. I think all of us have had moments like those students, feeling defeated by a problem and not knowing what to do next. As a math teacher, while I want my students to struggle some, I am acutely aware that too much struggle without concommitant success is just depressing.

When I have been more successful, I have used a range of strategies, but one of the most effective for me is to ask students to look at a highly simplified form of the problem. I asked my 9th grade class very early on this year a question about the minimum number of non-overlapping triangles which a 2008-sided polygon (with each vertex of each triangle also a vertex of the polygon) can be divided, and most of them found it bewildering. Even after 5 or 10 minutes of talking with each other and drawing pictures, most were focused on how intractable a 2008-sided polygon was, and how the answer would clearly be very large.

But when I acknowledged how overwhelming the problem seemed, and talked with them therefore about trying simpler cases first — like a 3, 4, or 5-sided figure– and seeing what conclusions they could draw– they were able to make progress at last. Of course, one might say that I gave them the crucial key to solving the problem, and I would be inclined to agree. But crucially, they did not perceive it that way– they still had to try out the simpler cases, and recognize the pattern, and then explain why the pattern was there. They still had a lot of problem solving to do.

So I tend to judge whether an “intervention” on my part has been successful or not based on how the students feel at the end of the problem. Do they look excited and pleased with their work, as they did at the end of the triangle problem? Or do their shoulders slump and they look sort of detached? How they feel is not a trivial concern at all; their willingness to tackle the next challenge is completely dependent upon their reaction to the last one, in my experience.

What strategies have you used when your students are stuck?

## 3 Comments

Yes, habits of mind hints are great when they work! Trying simpler cases is probably the one I use the most, too, and that proves to be most fruitful to kids. It’s one that they might not have thought to use themselves, and yet it’s pretty clear what it means to try simpler cases — kids immediately have something to do.

As Arnaldo would probably agree, “Tinker” is usually not recommended when kids get to that point. They’ve probably already been tinkering and by now are frustrated with how wide-open the problem seems.

Another issue that affects what kind of hint I will give students is how confident I am that they’ll eventually be able to get it if they just sit with it a while longer. I certainly have more perspective than them in this area, and since I know the solution to the problem I can, in my encouragement, provide them with some implicit information about how close they are. If I see that they’re totally barking up the wrong tree and they’re tired, I have no shame about giving a substantive hint about content. If they seem to me to be giving up too quickly, I’m much more likely to say something like, “draw a picture; you’ll get it.”

I could not agree more that the overall goal is to have students come away with the feeling that they have grappled with the problem and gotten somewhere. It doesn’t matter if they’ve had help; the point is that somewhere in there they’ve connected some of their own dots, and if those connections feel significant to them then they’re both developing a love of problem-solving and getting better at it.

How familiar this story feels! I’ve been in this situation many times as a teacher, and now am helping teachers grapple with it as their coach.

One question I use as a touchstone is: what are our learning goals right now? What are the students focused on learning (not what are they trying to do or solve, but what are they trying to figure out more about)? Also, what was my intention in choosing this problem?

If I am trying to teach, and students are trying to figure out about, problem-solving and habits of mind, then I need them to make the decisions. I can help them identify where they are stuck and why, but they need to experience coming up with what they can do next and seeing if it works or not.*

If we are trying to figure out some mathematical relationships, for example, the relationship between number of sides and number of vertex angles, and students’ habits of mind are getting in the way, then I’ll provide them problem-solving support so they can explore the math more deeply.

In each case I think the reflection & generalizing that happens after they solve are really where the learning happens. Can students identify what it was that got them unstuck? How would they describe the strategy/habit? Does it remind them of anything they’ve done before? When might they use it again?

*The habits of mind in this problem are “solve a simpler problem” and “get unstuck,” right? In the Math Forum Problem Solving & Communication Activity Series (http://mathforum.org/pow/support/activityseries) the routine we use around simpler problem is: “What makes this problem hard?” and “What could you change about the problem to make it simpler?” For Get Unstuck we use a series of diagnostic questions that partners can ask themselves.

If I were using this problem to teach either one of those routines, I’d start by having students think about times they got unstuck or figures out how to do something by starting with a simpler version. How did it work? What was effective? What wasn’t? What was hard about it? We’d make a list of “things to do when you’re stuck” or “ways to try a simpler problem.”

Then I’d give them the tasks and turn them loose. As they needed support I’d point them to our lists and offer other suggestions from my experience. If the suggestions worked, I’d add them to the growing list of “ways to try a simpler problem” and “things to do when you’re stuck.”

I’d also return to some touchstone questions/diagnoses, like “what makes this problem hard?”

I am a student, and I’m trying to figure out how long I should struggle with a problem before I just look at a solution? I finally made the realization that spending time with a problem, even when I feel stuck, is much more beneficial than just knowing an answer to a question. I have seen an improvement in my thinking skills by following this tactic. I have developed confidence in myself, and I feel that given enough time, I can solve any problem. But by enough time, I could mean hours, days, maybe weeks. I have actually spent an entire week on a problem without looking at the solution, and when I arrived at the correct answer the feeling I got was extremely rewarding.

That is my cause for success, but also my downfall. I spend too much time trying to understand something perfectly, and hammering away at a single problem. In that I learn one concept, my peers are able to finish a few concepts. Granted, the one concept I do learn, I will know it very thoroughly, and be able to explain all the details about it. But when it comes to test time, my peers often get better grades than me because they “a jack of all trades but a master of none”.

If I had all the time in the world, I’d be a great academic, but since that’s not the case, I’m trying to figure out what the *right* amount of struggling is.