The common core’s Standards for Mathematical Practice ask students to explain their thinking, model, and communicate precisely—and that often means showing how they worked though a problem.
Along those lines, Reich, an education researcher, wrote about project in which he, math teacher Michael Pershan, and a group of undergraduate education students at MIT examined 4th graders’ solutions to math problems to try to glean what they were thinking. Pershan is also the founder of a website called Mathmistakes.org, which is devoted to seeing where student problem-solvers go wrong.
The process went like this: Reviewers were asked to 1) notice salient details about the student work; 2) observe patterns among answers to the same problem; 3) hypothesize about what students understand, and 4) come up with possible interventions.
For instance, the reviewers asked themselves questions like, why did students draw some fractions as circles and some as rectangles? (As Reich wrote, “unless you grew up with six brothers and sisters, it’s basically impossible to divide a circle into even sevenths.”) And why did many students determine that 3/7 and 2/5 were equal based on their drawings?
The answers weren’t always clear. “We know more than we did before starting the exercise, but we still can’t say for sure what students are thinking,” wrote Reich. “What the exercise suggests, more than anything else, is that the very close examination of student work reveals a rich complexity in their thinking, a complexity that raises as many questions as answers.” He emphasizes the importance of collaborating on this kind of work with other teachers.
As I wrote last year, another way teachers may be able to get insight into what math students are thinking is through screencasting. A screencasting app lets students create a video of themselves solving a problem on a tablet. They can write and draw on the virtual whiteboard, and talk through the process as they do it, and it captures both their writing and narration. Sure, reviewing them would be time consuming for teachers, but not any more than one-on-one conferences with every student.
A version of this news article first appeared in the Curriculum Matters blog.