(Or, we want to know when a student’s struggle is likely to be productive and we should just smile and listen instead of yammering away.)
Right away, we run into a serious problem. It’s frankly impossible to talk about this while ignoring the details of the child, the course and the classroom. Kids don’t universally enter school with a wealth of experience that would help them dive into this problem. (This is a sense in which this problem is quite different from addition and subtraction word problems.) We need to know, what relevant experiences will this imagined kid have had before working on this problem? That question, is essentially, a curricular one. What comes before this problem?
There’s no easy way to answer this, because there is no agreed on curricular sequence. We can imagine that students have experience finding the area of shapes by chopping up and rearranging shapes, but there’s no guarantee of this. (Kids have some everyday experiences with chopping up areas, but maybe not the kind that would help them with this.) We can imagine kids have thought about area formulas for different shapes and how they’re similar or different, but who knows if a class studied that.
If we can’t know the experiences that students are likely to draw on, then we’re unlikely to be able to plan out helpful feedback or hints. This speaks against the possibility of pursuing a hint-gathering project in all its generality. Boo.
We’re left with some options, both of which are pretty promising:
- Focus on particular curricular sequences. Make those curricular assumptions explicit. (e.g. My kids have experience chopping up and rearranging shapes. They also have experience with finding the area of right triangles and rectangles and some triangles. They know triangle area formula.) Craft feedback and hints that are particular to the course and the curriculum.
- Start with problems at the beginning of the development (with problems that students are very likely to be able to use their everyday experience to solve) rather than the problems at the end (the problems we eventually want our students to be able to solve). (e.g. What strategies do kids tend to use with Ramp Steepness Comparison problems? vs. What strategies do kids tend to use for right triangle trigonometry?) Outline student thinking, in general, for problems that children’s common experiences are highly likely to equip them to solve. These problems can’t just be accessible, they also need to be promising, in the sense that they have to yield mathematical thinking that can be directly applicable to problems that we eventually want our students to solve. (The strategies you develop from getting better at comparing the steepness of ramps can be directly applicable to finding missing sides of right triangles using trigonometry.)
The first option seems promising on a personal level. After all, I know my kids and my course and my classroom. I’ll be able to directly observe my students’ thinking, so I can articulate a bunch of learning trajectories that are consistent with my curriculum, and then adjust these in response to what I actually see in my students. I think this would make me a better teacher for my students, as I could also plan for feedback and hints in a way that is sensitive to my students’ likely thinking. (As a bonus, any teachers or departments that are working together could work on developing frameworks in this way.)
The big downside to that first option is that the work is unlikely to be helpful for teachers who don’t share my curricular assumptions. That’s a bit of a downer.
The second option seems better suited for coordinating a broader conversation around student thinking. Since we’re starting with problems that students can almost certainly solve without instruction (using everyday or very common school experiences and knowledge) we can all understand each other when we talk about how students will think about these problems. We can then show the way that students can develop in their thinking about these very accessible problems. To be useful, we’d have to show how these strategies are useful for the problems we eventually want our students to solve.
When it comes to area, what might this project look like?
- I use the CME Project’s Geometry text in my classes. I could look at the way units are sequenced there, along with what I suspect (based on experience) that my students will know from their previous courses. I can use this to detail the thinking that my students are likely to use on this problem. I can use my guess at the thinking that students will use to plan for feedback/hints in advance.
- The other thing I could do is identify a problem that most students are almost certainly able to work on productively using everyday knowledge (or strategies that they almost certainly would have from earlier coursework), but that can also lead to being able to find the area of parallelograms and other shapes. Then I could use student work and experience to outline how thinking might develop as students get a lot of experience with these sorts of problems. I’d want to draw a line from early strategies to advanced strategies, and show how these strategies can be applied to finding the area of a parallelogram.
- These projects could go together. If I look at the CME Geometry text, I might find problems that students could work on using everyday strategies. If I find problems that kids could make sense of using earlier experience, I might want to rewrite parts of my curriculum around them.
Long-winded, but I think that’s the plan.