The clinical interview
Understanding how students think about mathematics requires careful listening — and may yield surprising results.
Before you can help your students develop their own mathematical understanding, it’s important to understand how they already think about math. Do they have a strong number sense, or do they rely on memorized procedures, floundering when faced with unfamiliar problems? Are they more comfortable with numerical abstractions or with geometric representations?
To demonstrate to participants at LEARN NC’s May 2001 Problem-Based Mathematics workshop how students think about math — and how to get a handle on their own students’ thought processes — presenter Grayson Wheatley interviewed an eighth-grade student from a local school. The student generally received A grades in math and considered herself to be a good math student. The clinical interview revealed, however, that although she had a firm grasp on certain common procedures, she was not comfortable taking on new types of problems, and tended to look for the "right" procedure before or instead of developing an understanding of a problem. As Professor Wheatley noted, "This is a child who has learned well what she was asked to learn. She tries to use the language" of mathematics. But she had not been taught to make connections between concepts or to apply what she learned to new situations.
During the course of the interview, Professor Wheatley gave the student a series of mathematical problems. Some she was asked to do in her head; for others she had pencil and paper. Because the goal of the interview was to understand how this student already thought about math, Professor Wheatley avoided giving her hints, leading her to a solution, or even using her struggles as a "teachable moment." (On a few occasions, however, he gently redirected her to avoid unnecessary frustration.) It is important to note that a clinical interview is, at least in the short run, for the benefit of the teacher rather than the student. In the long run, however, understanding the thought processes students bring with them into your classroom will help you teach more effectively.
The following materials are broken up into seven sections, by problem. For each problem the student was given, there are one or more video clips, a transcript, the text of the problem (with illustrations where appropriate), a copy of the student’s work, and comments from Professor Wheatley about what he and the student were thinking and why he asked particular questions. The video clips are in QuickTime format; PDF files require the free Adobe Acrobat Reader for viewing.
