Previously: PBL Part 4: Exercises, Problems and Projects
In May 2011, I was a student at the Stanford Teacher Education Program. I had just accepted a position for the fall at a start-up charter school, where we would be starting with a 9th grade class only and I would be a founding math teacher; classes were to be completely heterogeneous. Deborah Loewenberg Ball, Dean of Michigan’s School of Ed and God in the math education world, happened to be on the Stanford campus one day, and one of my classmates convinced her to hang out with us for 30 minutes. 12 people attended, so it was a pretty intimate setting. Naturally, I asked her the one question that I had been thinking about since I accepted my job; the one that had been keeping me up at night in excitement and anxiety: “If you had a totally blank slate of a school, how would you design your 9th grade math curriculum for heterogeneous learners?”
Her response went something like this: “Pick 20 to 30 really good problems for the year. Have your kids work on them together. Let them struggle some and really dive into the problems, no less than a day on a problem. Have them present these problems – either orally or in writing. Structure your class around that – attacking, solving and presenting math problems.”
I didn’t follow her advice, at least that year. I built the 9th grade curriculum (in partnership with others) around the CA state standards, implemented Standards-Based Grading, tried my hand at heavily differentiated content lessons and discovery activities, used Khan Academy exercises as a tool for procedural practice, and sprinkled in some problems and a couple capstone projects. One day, I’ll write a post (or 7) documenting all I learned that year – I’ve spent a lot of time thinking about the ups and downs we had and the opportunity costs of all those choices we made. For the purpose of this post, I’ll say that the math problems generally felt the most valuable to me, though I had difficulty articulating why. Looking back now, it is clear that the language I lacked could be found in the Common Core Math Practice Standards, which were barely on my radar at that point. Problems gave students an opportunity to problem solve, to debate, to argue, to reason, to prove and so on. I think this is what D-Ball had hoped for me and my class.
As mentioned in earlier posts, the curriculum in my charter organization is now entirely project-based. 9th grade math has 8 projects for the year, some of which I’ve written about already. I’ve been pretty pleased with all 8, but I’m again thinking about the opportunity cost. How many times do we want students to see unfamiliar problems and have to persevere in the problem solving process (Math Practice Standard 1)? How many times do we want them to construct arguments and critique others in the context of a math problem (Standard 3)? Etc. etc. The answer is always: as much as possible.
Certainly, something is lost when condensing a 3-week project to a few days. For instance, I couldn’t imagine doing the population project (where students produced a long report, analyzing demographic trends in a developing country to predict its future population) in just a few days; the data analysis alone, before any predictions were happening, took many days. And the likelihood of completing a task with a proud product is much higher when spending weeks on it.
But many 3-week projects I encounter are actually problems dressed up as projects; tasks that are perplexing and require cognitive skills, but are stuffed with lard (e.g. detours that make the problem longer but amount to busy work or don’t directly work towards answering the driving question; or time to perfect the fonts and get the Powerpoint slides just right). Those are problems – valuable problems, most of the time – but not projects; they shouldn’t have to masquerade as something they’re not.
A Modest Proposal
Both projects and problems are opportunities to learn and assess math practice standards. Have a curriculum that includes both. Don’t make a problem into a project, and vice versa. Cut out lard. Maximize the ratio: (Instructional value) / (Minutes expended)