The session scheduled after Alan Schoenfeld’s talk at the workshop I attended this week was called “The challenge of assessing mathematical proficiency: A student interview.” What I didn’t realize until the workshop started was that it would be a live interview. Deborah Ball started the session by explaining that she would soon be joined at the front of the room by Andrea, a local sixth-grade student. The purpose of the interview would be to learn in some detail about Andrea’s mathematical thinking.
Before they started, Deborah requested that audience members do our best to disappear by staying quiet and even minimizing gestures and facial expressions. Then she asked Andrea to try to pretend that we weren’t there. (We were able to see Andrea and Deborah’s work by way of a document projector.) After some general questions Deborah started the mathematical conversation by asking Andrea, “Have you been working with fractions this year? Could you write an example of a fraction? What is that fraction?” Thus began a fascinating hour-long conversation. This was not a long oral quiz; Deborah often asked “How do you know…?” and “How did you decide…?” Once or twice she said “I’ve asked other sixth graders this question, and some of them say… What do you think they’re thinking when they say that?” By the end, we had a clear picture of Andrea’s understanding of fractions and the number line, as well as her approach to new problems.
I’d love to spy on Andrea’s math class. She consistently responded with complete and accurate sentences, and she has a rich mathematics vocabulary; I suspect her teacher has something to do with that. But the bigger lessons for me are these: (1) student understanding is complex and nuanced, (2) a skilled interviewer can elicit responses that reveal at least some of the complexities, but (3) it takes time. After Andrea left, a high school teacher said that she would love to do this with each of her students, but it’s clearly not possible. Deborah Ball’s response was that her intent was not to suggest that every teacher do the same sort of interview with every student. Others pointed out that the questioning techniques could be used in a classroom setting.
Something else struck me as we were discussing what we’d just seen. Yes, we got real insight into Andrea’s mathematical proficiency, but that proficiency is not static. In fact we watched it grow a little during that hour. At one point Deborah, after asking a few preliminary questions, asked Andrea if she could write down all three-digit numbers that use only the digits 4, 5, and 6, and using each digit exactly once. Andrea produced all six of them; Deborah asked “How do you know you have all of them?” She asked whether 454 or 46 would be solutions, and Andrea explained exactly why they wouldn’t be. Next came a question about three people running a race. Again, there were preliminary questions; eventually Andrea showed the six orders in which the three could have finished. It was evident that at first she didn’t see that it was essentially the same problem as the previous one, but in the interview wrap-up, she indicated that she now saw the connection. This suggests a bunch of questions to consider, but here’s just one: what effects, positive and not, do various assessments have on student learning? Do students come out of an interview or a test with a different understanding of the mathematics? What about their understanding of their own proficiencies?