Getting students to dig deeper into rich problems

So I was going to participate in the #MTBoS Challenges, but then, life happened. I did write a blog post responding to the first challenge, and even though I'm not participating in the full scope of challenges, I'd like to post what I can. So here is what I wrote in response to the question "What is one of your favorite open-ended/rich problems?  How do you use it in your classroom?"

Even though we were directed to write about a favorite rich problem, I’m going to write about a problem that is the most recent one that I’ve done with a class because I’m having some issues with the way it’s worked out and would love some feedback on how to make it better.

The problem that I recently gave my 2-year Algebra students (these are 8^th graders in the second year of a 2-year course that covers a standard Algebra 1 curriculum) was the first one in the Integrated Math Program, Year 1 book, called Broken Eggs. This was students’ first problem of the week, in which they are to write up their problem-solving process and justify their thinking and solution, if they find one. In this problem, you are told that a number of eggs when put into groups of 2, 3, 4, 5, or 6 always had one egg left over, but fit perfectly into groups of 7. Students were asked to determine whether there was only one unique solution or whether there were many possible solutions, and if so, how they were connected to each other.

Full problem:

Broken Eggs POW by Anna

I gave students 20 minutes in class to start working on the problem (5 minutes alone, then 15 minutes with their group members) and a week and a half to complete their write-ups. I also met with students individually who were struggling.

I thought that the problem was a good one with which to start as it could be approached from a variety of angles and would encourage for the looking of patterns. However, the results were pretty disappointing. Most kids were only able to find the first solution and did so using brute force (writing out the multiples of seven and testing each one to see how it divided by 2, 3, 4, 5, and 6). Quite a few kids just looked at numbers that weren’t evenly divisible rather than looking for a specific remainder. Almost no one found any other solutions and not a single student found a pattern between the solutions. Almost no students even attempted to find one. So the problem just turned into one that required some organization to keep track of things, but almost no algebraic thinking. So basically, the result was a lot of annoying calculations with little payoff.

I am trying to think about what I could have done differently to encourage students to keep going and to notice patterns that would make their work easier. Having students share strategies maybe would have helped to disseminate some of the shortcuts that a few of the students discovered, but not ones that no one figured out. I think that part of the tension for me is that I want open problems to really be about students’ thinking and approaches, but also be a learning opportunity that stretches them past their current abilities and into something more advanced, and I don’t know how to do that without giving hints or telling kids to change their approach. Basically, I want them to learn and be stretched mathematically, but have it be organic and come as an extension of their own thinking rather than a top-down approach where I direct them.

Part of the issue also is that kids are mostly to work on problems of the week outside of class so I’m not getting much of a chance to see their thinking before they turn them in to me. So another change that I’m thinking of doing is having students turn in a “rough draft” that I can give them feedback on or that we can confer about in person before they complete their write-up. There is a thin line between pushing a student’s thinking and directing it onto a predetermined path that takes away from the problem’s openness and richness, and I am still navigating how to do this in an optimal way. Suggestions welcome!