Contributions from students should extend beyond factual answers and contribute to the growth of mathematical ideas (NCTM, 2000). Achieving these ambitious aims requires that teachers posed questions that challenge and extend student thinking. However, teachers’ tendency to reduce challenging problems to routine procedures, to select inappropriate tasks for their goals, and to shift the focus to merely finding a correct answer all contribute to lowering a task’s cognitive demand, the level of student thinking needed to solve a task (Stein, Smith, Henningsen, & Silver, 2000). Teacher questions have the potential to maintain cognitive demand, lower it to executing of procedures, or raise it to exploring concepts.

       According to Elder and Paul (1998) “If we want to engage students in thinking through content we must stimulate their thinking with questions that lead them to further questions” (p. 298). Such questions require an understanding of both content and students not likely possessed by novice teachers. Thus, mathematics teacher educators need theories about how novice teachers formulate questions in planning lessons so that novices can be helped to develop this skill. In particular, this study asks:

1. What alignment exists between preservice secondary mathematics teahcers’ (PSTs) planned questions and their identified lesson goals?

2. How can PSTs’ planned questions be characterized in terms of cognitive demand?

       Six PSTs were invited to participate in this study based on their methods course instructor’s assessment of their ability to critically reflect on their teaching practice. This small group was appropriate for an exploratory qualitative study because it allowed for extensive analysis of each participant’s data. To assess changes in the planned questions, the three participants who accepted were interviewed twice, during their first methods course and at the end of student teaching. The first interview asked them to reflect on a recently completed methods course lesson plan assignment; the second interview asked them to reflect on a lesson planned for their student teaching. Interviews were audio recorded and lesson plans discussed were collected. 

       Interview transcripts were coded to focus on questions and goals for the lesson. Interview and lesson quotes were used to compose a goal statement for each participant’s lessons. Questions were categorized by type and were correlated to cognitive demand levels identified by Stein, Smith, Henningsen, and Silver (2000): memorization, procedures without connections, procedures with connections, and doing mathematics. Question type categories included: open, reflective (Beckmann, Rubenstein, & Thompson, 2009), guiding, probing (Sahin, 2007), leading (Nicol 1999), and specific for questions only requiring recall of specific facts. Participant narratives were composed to highlight prominent themes in their planned questions.

       Results will be presented for one PST that exemplified struggles experienced by participants. At the beginning of the study, Katy was a junior in her first year of her teacher education program. As an undergraduate female in her early 20’s who was a life-long resident of her school’s state, Katy was a typical student for her program. In her first lesson plan she failed to state the goal of the lesson and only list activities the students would complete. She did not enumerate the mathematical concepts that students should take away from the activity; however, in the interview she explained the lesson was designed “to show them they have a really strong bias” when trying to select a random sample and how to “integrate the terminology into the activities.” She employed specific questions to address procedural goals about correctly applying terminology and these questions established a cognitive demand of memorization. Katy used more powerful questions to support her sophisticated goals about bias, often asking students to identify a specific interesting result (“Which is closer?”) and then to reflect on a justification (“Why? What are some possible reasons?”). In these instances Katy was able to elevate the cognitive demand to doing mathematics.

       During student teaching Katy developed long-term goals that included students’ discovery of concepts. However, as she gained more experience with students, she tried to find a comfortable middle ground of semi-discovery because “as much as problem solving is good, there’s some things that you just have to be told.” In the lesson that was the focus of her second interview, she described the goal as getting the students to understand different rates “and how long it takes to travel a certain amount of distance” by using a formula to make calculations. This tension between telling with her procedural goals for the lesson and her desire for students to discover mathematical relationships was reflected in her questions. She rarely planned probing questions to extend students’ thinking beyond obtaining correct student solutions, explaining, “If they have an answer, then they know how to do it and I don’t need to go over it as much.” However, she did anticipate student difficulties, and, rather than correcting students, she suggested probing and guiding questions to help students identify and rethink errors, establishing her cognitve demand as procedures with connections. She described her planning saying, “I think about what they’re going to have problems with, and put enough thought into it that I can figure out, ‘How can I make this a little bit more manageable for them?’”

       Though Katy’s questions and goals aligned well, her procedural goals led her to employ questions that lowered the cognitive demand and to minimize questions in response to correct solutions. Only in considering incorrect student responses did her planned questions maintain or raise the cognitive demand. The results here indicate that just exposure to students is not enough; preservice teachers need structured experiences to let them investigate how students think about mathematical concepts.

       This presentation is appropriate for the interactive paper session because it will allow for discussions about the broader topic of teacher education. After 15-minute presentations of this study and two others, attendees will engage in round table discussions about how teacher education programs can support teacher development in questioning and lesson planning that maintains cognitive demand. This session will address professional learning by engaging attendees in discussing what activities will help teachers learn to maintain cognitive demand.

 References

Beckmann, C. E., Rubenstein, R. N., & Thompson, D. R. (2009, February). Developing prospective teachers’ ability to ask questions to support thinking. Paper presented at Association of Mathematics Teacher Educators National Conference, Orlando, FL.

Elder, L. & Paul, R. (1998). The role of Socratic questioning in thinking, teaching, and learning. The Clearing House, 71, 297–301.

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, Virginia: Author.

Nicol, C. (1999). Learning to teach mathematics: Questioning, listening, and responding. Educational Studies in Mathematics, 37, 45–66.

Sahin, A. (2007). Teachers’ classroom questions. School Science and Mathematics, 107, 369–370.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A case for professional development. Teachers College Press: New York.