What Are Teacher-Researchers Learning about Their Practice?

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).

Whose questions are addressed in the study?

• How do I know if I am right?
• How can I help students to take responsibility for learning?
• What are some alternative ways to organize my pedagogy to make it possible for students to learn?

Over time these questions shifted to include:

• What kind of data you need in order to show that you had made a difference (e.g., changed grades, student comments, etc.)?
• What types of questions make a difference to student learning?
• How might we most effectively share our videopaper with others?

Methods

We worked together over a three-year period using both technology (emails, web-based discussions, links to articles, pages, etc.) and face-to-face workshops. We

• Discussed (a) research articles and chapters, (b) our classroom practice, and (c) videotapes of ourselves teaching the same nonroutine challenge problem, the cylinder problem.
• Worked with our own students on the cylinder problem.
• Observed and described students' work with the cylinder problem.
• Wrote reflections about (a) what we read and its applications to practice, and (b) our practice and its links to the research.

Results

• The extremes that plague debates in math education (e.g., algorithms vs. student understanding; back-to-basics vs. problem solving) are complementary.
• Discourse can be a tool to help us carry the question, Am I right? Discourse makes thinking public and creates an opportunity to negotiate meaning.
  • Many approaches can be used to enable students to develop their mathematical thinking. These form a continuum, ranging from direct approaches in which the leader provides an answer, a demonstration, or a leading question, to less direct approaches that encourage students to articulate their thinking or to reflect inwardly on their questions and insights.
  • Even though there is no one way to facilitate students' mathematical thinking, there may be more effective ways for a given teacher to meet the strengths and needs of his or her students.
  • Listening is critical. Good questioning (by either the teacher or the student) requires good listening to what has gone before. We find that we constantly need to be listening and by listening we continue to learn.
• Reflecting on the questions of this paper outside of class makes it more likely that we attend to them during class and are positioned to know what we might do about them.

Now that the project is technically completed, our practice has changed. We each

• have a language with which to talk about our work with students that is different than it was before.
• are interested in research and share articles and our applications of them with each other.
• are mindful that our goal is to enable students to question.
• recognize that our job is to work to understand what students understand, acknowledge this, and use questions, examples, tasks, etc. that will lead them to question.

References

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. 157-196). 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. 25-129). Mahwah, NJ: Lawrence Erlbaum Associates.

Resnick, L. B. (1988). "Treating mathematics as an ill-structured discipline." In R. I. Charles & E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 32-60). 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. 334-370). 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.

Appendix

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 open-ended 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 arch-shaped 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 T-shirt 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/]