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?

What a great description! Thanks PSB!

This is the kind of assessment that has a chance of really getting at the mathematical proficiency of students, and precisely because of learning experience described at the end of the post. Learning mathematics ought to include learning how to learn mathematics, how to see mathematical connections and draw mathematical conclusions. This is not to set aside entirely the need for facility with basic arithmetic or algebraic techniques, though–as Liping Ma argues in her book–even that facility is best when at its base is a command of ideas that makes each computation an opportunity to seek the most efficient method. But it is clear that there is a crying need for more learning how to learn in American mathematics classrooms. I agree with the teachers that performing this ind of assessment would be an enormous strain on resources. On the other hand, it would radically alter the incentives for the teaching of mathematics in positive ways.

I agree that at this moment this level of work with the students is over the top, but how about with the teachers? Couldn’t we arrange for teachers to have this kind of experience themselves, and get tot he point where they are able to do this on a far more anecdotal basis with their students? Five minutes a day with four different students would only take half a period, and a teacher might go through an entire class in a week or two.

I still worry a lot about the culture of teaching in public schools. Teachers need to be learners, but there is not enough inquisitiveness about the subject itself amongst the teachers I know. (This is not meant to be a blanket statement, just a general pattern.) Teachers who are seeing new connections can help their students do the same, and then offer them assessments that enable us to know who is learning to learn and who needs more help.