What Are TeacherResearchers Learning about Their Practice?The Math Forum's Bridging Research and Practice Project:

K. Ann Renninger, Swarthmore College
Annie Fetter, The Math Forum Art Mabbott, Seattle School District 
Harrah's Copper Room Research Presession NCTM Annual Meeting, Las Vegas, Nevada April 20, 2002, 11 a.m.  12:30 p.m. 
Math Forum at NCTM 2002  Math Forum  Student Center  Teachers' Place
The Math Forum's [http://mathforum.org/] Bridging Research and Practice (BRAP) Project is a collaboration of teachers and Math Forum staff who have been studying the use of discourse as a basis for encouraging students' mathematical thinking in classrooms and online. Findings from this collaboration are reported in an interactive online videopaper a multimedia document that includes video clips from the teachers' classroom work with their students and allows for ongoing discussions about its content.
The videopaper, Encouraging Mathematical Thinking: Discourse Around a Rich Problem [http://mathforum.org/wrap/brap], builds on research about mathematics learning (e.g., Resnick, 1988; Schoenfeld, 1992) and the type of discourse that facilitates mathematics learning (e.g., Ball, 1993; Cobb, 1995; Stigler & Hiebert, 1999).
Over time these questions shifted to include:
We worked together over a threeyear period using both technology (emails, webbased discussions, links to articles, pages, etc.) and facetoface workshops. We
Now that the project is technically completed, our practice has changed. We each
Ball, D. L. (1993). "Halves, pieces, and twoths: Constructing representational contexts in teaching." In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of Research (pp. 157196). Hillsdale, NJ: Lawrence Erlbaum Associates.
Cobb, P. (1995). "Mathematical learning and small group interaction: Four case studies." In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in classroom cultures (pp. 25129). Mahwah, NJ: Lawrence Erlbaum Associates.
Resnick, L. B. (1988). "Treating mathematics as an illstructured discipline." In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 3260). Hillsdale, NJ: Lawrence Erlbuam Associates.
Schoenfeld, A. H. (1992). "Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics." In D. A. Grouws (Ed.) Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 334370). New York: Macmillan.
Stigler, J. W. & Hiebert, J. (1999). The teaching gap: Best ideas from the world's teachers for improving education in the classroom. New York: Free Press.
Susan Stein describes how she learned what was causing her student difficulty:
One of my seventh graders did not complete his homework several times during the first two weeks of school. The first time he said he didn't understand the question, the next time he said he forgot, and the third time he again said he "didn't get it." I was becoming concerned and finally pinned him down to meet with me.The two problems he "didn't get" were very similar. Both were pretty openended and involved four different, simple sketches of graphs showing only quadrant I, with no numbers or other identifiers on the axes. The graphs were all different shapes, a straight line going up, a curve that increased more at the start and then tapered off, a line with a negative slope, an archshaped curve. The two variables being compared were named in the body of the question. The first one compared age and height; the second compared selling price per Tshirt and profit. In each case the question asked which representation best showed the relation between the two variables, and expected an explanation/justification of the student's reasoning. Also in both cases, there were several "right" answers possible.
It took a while to get Chen to explain what he meant by "not getting it." I asked him to tell me what the different graphs showed. For the height and age one, he could describe pretty well how the straight (positive) line showed that height increased as age increased, the one that tapered off showed that you grew quickly when you were young, but slowed down, even stopped as you got older, etc. In fact, Chen understood each graph in both problems. With no specific prompting from me (except to ask, "what does this one mean?"), Chen seemed to me to "get it" very well.
So, what was the trouble? The trouble was that Chen understood very well. He was able to justify each of the four choices as being reasonable representations of the relation between age and height. "So," he asked, "how do I know which one is right?" What he didn't "get" was how to answer a question with more than one right answer. Helping Chen recognize that he understood the mathematical ideas allowed him to gain confidence in his own ability to evaluate when his ideas were "right" and learn how to demonstrate his knowledge.
Much can happen as a result of the mere expectation that the student has more to say. Whether we wait silently or ask questions, we create space for the student to figure out a way forward. We can ask for a description of what the problem asks. We can involve other students in explaining what they understand. We can ask a question like, "What part do you understand?" instead of, "What don't you understand?" We can help students articulate which parts of a problem make sense, in order to help them move from "I don't get it" to "I know this much, but I'm stuck on this part." When students start from what is already known, they are no longer totally lost.
 From, Encouraging Mathematical Thinking: Discourse Around a Rich Problem [http://mathforum.org/brap/wrap/]