Source: Journal of Mathematics Teacher Education, Volume 20, Issue 2, p. 159–201 (2017)
(Reviewed by the Portal Team)
In this study, the authors aimed investigated to identify elements that constitute the practical rationality of mathematics teaching.
Specifically, they focused on the assumptions that participants made regarding what should constitute the launch of a problem-based lesson.
The authors hypothesized that different assumptions may lead to tensions and dilemmas when launching a problem.
Methods
The participants were 6 secondary mathematics in-service teachers (ISTs) and 10 pre-service teachers (PSTs).
The pre-service teachers were seniors pursuing their undergraduate degree in a secondary mathematics teacher education program during the semester prior to their student teaching experience.
The authors analyzed data from four focus groups that consisted of prospective and in-service teachers who viewed animated vignettes of classroom instruction.
Discussion
The findings revealed that there were differences between PSTs' assumptions about the launch and ISTs' assumptions.
Several PSTs supported the assumption that the launch is a crucial moment for motivating students to solve the problem.
Furthermore, the PSTs thought that the assumption that the launch should promote student engagement is associated with the goal of motivating students, but it specifically addresses the avowed difficulty of students’ limited attention spans.
In contrast, ISTs uniquely argued that assumptions about the launch that supported the notion that teachers should establish explicit connections with students’ prior knowledge and support students’ success in solving the problem, such as providing hints.
The authors claim that this assumption considers the launch’s aim to make explicit the prior knowledge that students will need to solve the problem.
Hence, the authors argue that the assumption that a launch should include a review may be embedded in the practical rationality of mathematics teaching.
The authors also note that the assumption that teachers provide hints for solving the problem during the launch and that these hints are directly connected to students’ work may be another strategy for ensuring that students solve the problem.
The authors found that a majority of both ISTs and PSTs agreed with three other assumptions regarding what should constitute a launch.
First, both groups agreed that teachers should preview the work necessary to solve the problem during the launch.
This assumption asks teachers to identify a fundamental idea in the problem and to highlight that idea during the launch.
Second, both groups mentioned that the launch is an opportunity for teachers to clarify the key concepts of the problem.
Finally, both groups of participants supported the assumption that teachers should not disclose the procedures for solving the problem in the launch.
The authors also 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.
Furthermore, the participants’ claims revealed two different perspectives with regard to their considerations of students’ prior knowledge in relation to the launch.
Several participants assumed that it is important for teachers to help students develop an understanding of ‘‘revolution’’ during the launch.
However, several participants assumed that teachers should review the prior knowledge that students need to solve the problem during the launch.
The authors gave as an example the problem-based instruction, which intensifies teaching tensions and dilemmas at the start of a lesson.
The authors argue that different assumptions regarding what constitutes a launch pull teachers in different directions and support different courses of action.
Hence, the justification that students should be working by themselves may be a core idea that changes the nature of the launch and influences teachers’ decisions about how to scaffold students’ work on problems.
Conclusion
The authors conclude that the manner in which teachers set up a problem can reduce the opportunities for high-level mathematical reasoning.
Hence, they argue that the launch is important for teachers to maximize student engagement and mathematical reasoning.
They also note that teachers’ decisions about launching a problem can enable students to exercise conceptual agency.
They mention one implication of this study for teacher education, which is that discussions about launching a problem should include analyses of the implicit assumptions that underlie the PSTs’ and ISTs’ perspectives.