NCTM Denver 2013: Saxe et al's Engagement in Mathematical Discussion: Linking Practices and Outcomes

Research Pressession - Wednesday, April 17, 8:30 am

Geoffrey B. Saxe - University of California, Berkeley
Maryl Gearhart - University of California, Berkeley
Ronli Diakow - University of California, Berkeley
Nicole Leveille Buchanan - University of California, Berkeley
Jennifer Collett - University of California, Berkeley
Bona Kang - University of California, Berkeley
Kenton De Kirby - University of California, Berkeley
Marie Le - University of California, Berkeley
Discussant: Deborah Loewenberg Ball - University of Michigan

This group from Berkeley presented their findings from the use of Learning Mathematics through Representations, or LMR. LMR is a research-based curriculum unit for the teaching of integers and fractions in the elementary grades. Despite only being 19 lessons long (at the time of their study), it carefully attended to students' definitions of number, unit intervals, subintervals, and early fraction sense. When compared to similar coverage by Everyday Math, LMR produced significantly higher learning gains at all stages.

I'd have been hard-pressed to take a worse picture than this.

The group did find variability in LMR results. After checking for curriculum coverage differences and not finding anything significant, they developed measures for both content and participation. The group found that communication was key, especially for the lowest-achieving students.

This research group also did intensive classroom observation and video collection. They looked at interesting teacher moves designed to disrupt student thinking in ways that elicited student protest, where students became motivated to express their understandings of concepts in ways that corrected the teacher's intentional mistakes. The group paid specific attention to the trajectory of student understanding about unit intervals. Over six weeks, with pre-, interim, and post-assessments, they showed how students' understanding of the fraction 8/7 grew (for most students) over time.

Saxe's anthropological approach to studying shifts in understandings over time add some theoretical nuance to this work, examining the semi-durability of ideas as they are reproduced and altered. This kind of detail is often lacking in research, but can provide some key insights about teaching and learning.

The Discussant, Deborah Ball, began her comments with "Wow." I think that says a lot. She commented on the project at a meta-level, about the project itself, and "being greedy," she asked questions about what else we can get out of this body of work. Ball appreciated the connectedness of the project, both to other projects and across the history of Saxe's work. She also appreciated how the work was situated across all aspects of instruction, including teaching, curriculum, and class discussion. Ball called the work "programmatic" in the ways it carefully broke down the issues of the study and carefully applied the right methods, the care taken with definitions, and the depth with which instruction was analyzed. Ball also asked about "correcting the teacher," wondering more specifically what they perceive that move/strategy to be, and how it might not fit into either direct or dialogic instruction. For her "greedy" questions, Ball asked:

  1. What are you learning about teaching?
  2. What are you learning about the challenge of "drop-in" curriculum?
  3. What are you learning about the assessment of student learning?
  4. What did you learn about who was talking in class? What were the relationships with social or identity markers? What are you learning about the use of problems that weren't situated in the real-world of the students?
  5. How can the rest of us learn how to do this kind of programmatic work?

To answer #4, the nature of the work and issues with Human Subjects precluded them from collecting demographic information about students. In the interest of time, the panel decided to take most of the other questions under considerations while allowing time for Q&A from the audience. The most interesting answer in the Q&A was in regards to the availability of the LMR curriculum. Saxe said they tried getting it published commercially, but commercial publishers want to sell K-5 series of textbooks, not a 19-lesson replacement unit. So instead, the group is planning to post all the materials online and make them free for teachers to use in their classrooms.

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