Learning to teach to reason: reasoning and proving in mathematics teacher education

jeppe Skott, Dorte Moeskcer Larsen &

Camilla Hellsten 0stergaard


Two recurrent recommendations for school mathematics and mathematics teacher education form the background to the pilot study that we present in what follows. First, it is one aspect of current reform efforts in school mathematics that students need to become involved in genuine mathematical activity. Irrespectively of whether this is phrased as process standards (NCTM, 2000), threads of mathematical proficiency (National Research Council, 2001), mathematical practices (Common Core State Standards Initiative, 2010), or competencies (OECD, 2000), it includes an element of mathematical reasoning and proving (R&P). Forms of R&P, then, have been promoted as a significant part of reform efforts for the last couple of decades.

Second, recommendations for mathematics teacher education (MTE) increasingly emphasize issues that are specific to the profession. This is so in suggestions that academic mathematics does not suffice as teachers’ content preparation (Ball, Thames & Phelps, 2008; Rowland, Turner, Thwaites & Fluckstep, 2009). It is also apparent in the explicit emphasis on the tasks of teaching in the college- based parts of programmes. This involves a shift “from a focus on what teachers know and believe to a greater focus on what teachers do” (Ball & Forzani, 2009, p. 503), and it includes for instance organizing productive classroom communication, selecting and developing challenging tasks and assessing student work for formative purposes. And the professional emphasis in MTE shows in suggestions that teaching-learning processes in MTE should model those envisaged for school mathematics, if reform recommendations are to materialize in school (Lunenberg, Korthagen & Swennen, 2007).

Between them, the two recommendations outlined here indicate not only that R&P should be part of MTE; they also suggest that prospective teachers should work with R&P in ways that resemble the ones recommended for school students.

R&P, however, is notorious for the difficulties it creates for students at all levels, including prospective teachers. In this chapter we present the background and pilot to an intervention study that is to alleviate these problems in the case of Danish teachers for the primary and lower secondary levels. The study, Reasoning and Proving in Teacher Education (RaPiTE), is in line with other studies in the field in the sense that we suggest that MTE needs to be close to the challenges of the profession. However, based on previous studies of practising teachers, we suggest not taking this recommendation too far. These studies indicate that there is a need to coordinate the recommendation to professionalize teacher education with a concern for the disciplinary practice of R&P. The rationale of the pilot is to address the questions of whether this dual emphasis on school mathematics and academic mathematics is important also in the education of prospective teachers, and if our interpretation of how such a balance may be conceived is warranted. We address these questions in what follows. The results of the pilot suggest an affirmative answer.

To make our point, we begin by introducing the notion of R&P leading to a presentation of our use of the term. We then discuss recent educational scholarship on R&P in school mathematics and MTE, including the problems that many students face with R&P and suggestions for how to address them. We then present our framework. Patterns of Participation (PoP), and elaborate on the approach we take in RaPiTE, before describing the organization, methods and results of the pilot. We finish with a discussion of our main concern: does it make sense to balance academic and professional issues in mathematics teacher education on R&P, and what may it mean to do so?

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