Thursday, August 23, 2012

White Paper on Problem Solving: What I did last year

In the first part of this series I defined problem solving as struggling over difficult problems, and I tried to work out what the pedagogical benefits of struggle are. In this post I want to turn a critical eye to the way that I pulled this off last year. Hopefully, I'll end up with a better strategy for this coming year.

But first, here's another cat with a Rubik's cube.

Cats: the silent killers.

What I did last year:
Last fall, two things happened that lead to a change in my classroom.
  1. I noticed that students enjoyed working on my “Warm Up” problems.
  2. I read the book “Drive” by Dan Pink.

What happened as a result is that I put solving problems at the center of my classroom. I felt empowered by how my students enjoyed spending more class time solving problems, and because I read “Drive” I had a framework for understanding why they liked it. I was giving them more autonomy, and people like having choices about how they work.

Here is the sort of thing that I was putting in front of my students with regularity last year:

How proud should I feel of this work?  There are certainly some things that are going right here.
  • This problem set starts with a “Warm Up” section that brings together different problems that connect to the new problem being solved in the “Important Stuff.”
  • The problem set is designed so that students can work on it on their own, and with groups. Kids prefer that to a lecture or an activity that occurs with the whole group. I’ll get more face time with students who are struggling with these ideas, which is also a good thing.
  • The problem set also starts with a question that is pretty concrete and easy, so that the tricky, abstract stuff is built on top of a firm starting point.
But there are also some serious problems lurking under the surface. In particular, I’m doing much of the intellectual work for the students just by sequencing these questions the way that I do. Students knew to expect that there would be a relatively straight line running through the problem set leading to an insight. The result was that a major component of a student’s struggle over a problem was an attempt to figure out the connection between sequences of questions.

Take question 4 in the above document. Just from the sequence of questions students are likely to infer that the way to find the angles whose sine is 0.43 by using the arcsine function to find one angle, and then to find the other using the unit circle.  This kills any chance of multiple approaches to the problem, reduces the difficulty of the problem by many factors, and doesn’t give students a chance to search their memories for a helpful approach.

(I also think that, while it’s admirable that the “Warm Up” contains problems from different topics that connect to the new one, it’s probably a mistake to corner them off into their own section.)

If you see more issues with my approach, please give a shout in the comments.

There’s a reason why I made this problem set the way that I did, though. I was nervous that students wouldn’t be able to figure out the actual new problem of the day on their own. And that’s true. Kids would have gotten frustrated if I just gave them the difficult problem without any context. But the solution that I came up with was to make the problem much, much easier. I need to figure out different ways to keep kids from getting frustrated by difficult problems.

That’s what I’ll write about in the next post. 


  1. I struggle with this a lot too. You want to activate the prior knowledge that relates to the day's topic, but you're so right that you're doing much of the heavy lifting for them when you build an easy progression from question to question. What if you started the year with giving the students all of these supports, but gradually, took it away? You could give them the difficult problem first and ask what other concepts it relates to or reminds them of and get them to bring up the pieces that you would normally put in the warm-up. Another option is to maybe assign some "scaffolding"-type problems as part of the previous night's homework, but mixed in with other stuff so it's not as obvious which questions sequence where, but some of those prior concepts will have been activated anyway. I'm looking forward to your next post - getting kids to approach & persevere with difficult problems is one of my main goals as well.

    1. I'm worried that if I move casually into giving less support to kids, they'll wake up one day and realize that they've been given a harder job without us having talked about it.

    2. What is the danger in that? Isn't that really what the entire year is about? They come to us and one level and hopefully by the time they leave, they are at a whole new level. We never tell them, "hey, we are moving onto the next unit and the topic is more difficult than those in the previous unit." We just progress them and they tend to accept it.

  2. It sounds like they're working independently a lot? Putting them in groups is the simplest way to make harder problems possible. And group work is what we do in the real world, right? For the hardest parts, like starting a new topic, work as a group with the whole class so you can direct the flow of the discussion as much as needed at the moment. Sometimes they'll surprise you with an insight or question, and other times they'll be like deer in the headlights, and you'll be able to adjust on the fly before they get frustrated. When I lecture, if you can call it that, it's very interactive. Mostly I ask questions.

    All the questions above are abstract, in the sense that they don't relate to any realistic scenario. It's easier to understand, especially a new concept, when it's in the context of some problem or question that occurs in real life. The 3-act and WCYDWT type of activities have really helped me. I used to just relate the new concept to some realistic scenario, but now I can present the realistic scenario and help the students develop the concept themselves.

    Kelly Holman (for some reason the site won't let me enter a Name/URL)

    1. What I've done in the past is given them the choice of working alone or in groups. Maybe I should change that? Say, "OK, now would be a good time to work with a partner if you're stuck." But if you're not stuck? That seems like it would get logistically messy very quickly.

      And, you're right, I need to be doing a lot more 3-Act style questions than I have been. I see how they could help.

    2. I'd be more inclined to give them problems that are difficult & elaborate enough to need groups, and just start out that way. In the past you've been inclined to provide scaffolding & sequencing, and you're frustrated with that. The group can provide that, and it's better coming from each other than from you. A) when they provide ideas and direction, they learn to take risks and how to think of paths to follow B) when their classmate gives an idea, they have to evaluate it because it's not coming from an expert.

      When I was in college for computer programming, partners were usually optional. But *everybody* wanted a partner, because you'd never figure out how to solve the problem in time if you had only yourself to talk to. The prof was cryptic & evasive, by design. You want problems that are deep & wide enough to make groups necessary.

      I read of a curriculum out of California, can't remember anything else about it, that took away ALL the scaffolding. Right off the bat, they gave kids a challenge problem that you'd normally see at the end of a section, and the kids learned to throw ideas around and figure it out. I wouldn't be inclined to go that far every day, but it challenged my expectations.

    3. I agree with much of what is being said; however, I would question why you have to explicitly state whether or not students can work in groups. I find it helpful to have group work as being more of a normal functioning structure of my classroom. Granted some students choose a more independent approach and then confer afterward, but I see it as a choice based on learning style.

      Second point, scaffolding is important, I agree, but scaffolding really should be differentiated. I agree with Kelly that a lot of scaffolding and assistance can come from the group; however, that doesn’t mean you don’t have a role as well. When you scaffold up front, it seems as though you assume every student needs help and with the same exact things. What would happen if you held onto the scaffolding until a group could not resolve an issue on their own. Not only would they have a chance to think and make some connections themselves, you would also be able to tailor your assistance to their needs.

      Lastly, I love Kelly’s point regarding using a non-abstract problem that students can dive right into and then move to abstraction later. In my experience, students find far more entry points to an unfamiliar problem if it is in such a form.