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Prospective and in-service teachers’ perspectives about launching a problem

Abstract

Launching a problem is critical in a problem-based lesson. We investigated teachers’ perspectives on the use of a problem that was analogous to the one provided during a launch. Our goal was to identify teachers’ underlying assumptions regarding what should constitute a launch as elements of the practical rationality of mathematics teaching. We analyzed data from four focus groups that consisted of prospective (PST) and in-service (IST) teachers who viewed animated vignettes of classroom instruction. We applied Toulmin’s scheme to model the arguments that were evident in the transcriptions of the discussions. We identified nine claims and 13 justifications for those claims, the majority of which were offered by the ISTs. ISTs’ assumptions focused on reviewing, providing hints, and not confusing students, whereas PSTs’ assumptions focused on motivation and student engagement. Overall, the assumptions were contradictory and supported different strategies. The assumptions also illustrated different stances regarding how to consider students’ prior knowledge during a launch. We identified a tension between ensuring that students could begin a problem by relying on the launch and allowing them to struggle with the problem by limiting the information provided in the launch. This study has implications for teacher education because it identifies how teachers’ underlying assumptions may affect their decisions to enable students to engage in productive struggle and exercise conceptual agency.

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Notes

  1. In our work, the terms “launch” and “task setup” refer to the moment during a lesson when a teacher introduces a problem. We use the terms interchangeably but with a preference for “launch” because this was the term that we used during the focus group sessions.

  2. We use the term “activity” for the launch similar to Leinhardt and Greeno (1986) who identified “activity structures” in a lesson and to Lemke (1990, p. 215–218) who identified “activity types” in science classrooms.

  3. Berlak and Berlak (1981) established several “control dilemmas” including time, operations, and standards. The teachers’ control of time refers to their authority in determining the temporality of classroom activities such as the duration of students’ work. The teachers’ control of the standards refers to the establishment of parameters for students’ work. Although these two other control dilemmas are related to the launch of a problem because one can envision that teachers have to make decisions about the duration of the launch and the explicitness of the standards for students’ work in the launch, we decided to focus our work on the control of operations.

  4. The theme of the need for providing a demonstration also surfaced in Argument A when the participants stated that they were looking for an object to visualize the revolution themselves. However, since the focus was on their understanding of the resulting solid and not of the effect on the students’ visualization, we coded those comments as related to Argument A instead of Argument D. It is possible that, for the participants, their inability to visualize the solid without a demonstration may be a justification for why they feel that the launch must include a demonstration. However, in Argument D, they voiced justifications that were aligned with their students.

  5. Using Toulmin’s scheme, the statement that “in a problem-based lesson, the students should work by themselves” can be the backing of the warrant “the launch should be about the students’ prior knowledge.” However, to be consistent with our analysis, we classified these two statements as “justifications.” We include these ideas in our analysis of the assumptions.

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Acknowledgments

This research was supported in part by a Campus Research Board Award from the University of Illinois at Urbana-Champaign granted to the first author. A poster with preliminary results was presented at the 2013 American Educational Research Association annual meeting, San Francisco, CA.

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Correspondence to Gloriana González.

Appendices

Appendix 1

See Table 6.

Table 6 Description of animated vignettes shown in focus group sessions prior to the selected interval

Appendix 2

Focus Group Session Protocol until the Discussion of the Launch with the Analogous Problem.

Part I: Work on the problem

  1. 1.

    (10 min) Introductions and logistics of the session.

  2. 2.

    (30 min) Work on the SOR problem individually, in pairs, or in groups. Resources available: pencil, paper, and a calculator.

  3. 3.

    (30 min) Presentation and discussion of solutions to the SOR problem.

Part II: Overview of the session

  1. 4.

    Moderator refers to problem-based instruction in the NCTM standards.

  2. 5.

    Moderator describes the goals for the session and asks the participants to consider the following:

    1. a.

      What might a lesson using the SOR problem look like in a middle grades classroom?

    2. b.

      We will watch a series of animated vignettes showing one possible way for a teacher to implement a lesson around the SOR problem.

    3. c.

      The vignettes are meant to show not best practices but examples of what a teacher could do in a class.

    4. d.

      We will be showing you snapshots of three different moments in a lesson (i.e., launch, exploration, and summary).

    5. e.

      The purpose of showing the vignettes is to provoke discussion about your ideas for implementing a lesson around the SOR problem.

  3. 6.

    The moderator shows the L1-E1-S1 and asks the following questions:

    1. f.

      What do you think about the teacher in the lesson?

    2. g.

      What do you think about the students in the lesson?

    3. h.

      What would you do the same?

    4. i.

      What would you do differently?

    5. j.

      Other thoughts/reactions?

  4. 7.

    Moderator introduces alternatives for launching the problem and asks:

    1. k.

      What should the teacher review before working on a problem about solids of revolution?

    2. l.

      We will watch three different alternatives for launching the lesson. As you watch these alternatives, consider what you would do the same or differently.

  5. 8.

    Moderator shows L2, L3, and finally L4. After showing the launches, the moderator asks:

    1. m.

      How do L2, L3, and L4 compare?

    2. n.

      Would you use any of these launches?

    3. o.

      In what ways might you add to or change these launches?

Appendix 3

See Table 7.

Table 7 Evaluation stances with examples from data set

Appendix 4

See Table 8.

Table 8 Coding of transcription from February 16, 2012, interval 27

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González, G., Eli, J.A. Prospective and in-service teachers’ perspectives about launching a problem. J Math Teacher Educ 20, 159–201 (2017). https://doi.org/10.1007/s10857-015-9303-1

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Keywords

  • Problem-based instruction
  • Tensions
  • Mathematics teaching
  • Prior knowledge
  • Task setup
  • Launch
  • Visualization tasks
  • Toulmin’s argument scheme