The pilot study
The pilot study takes place at a prestigious college in Denmark. The prospective teachers have all performed fairly well in secondary school, and according to curricular documents they have worked with mathematical reasoning both in primary and secondary school. At the college, they need to specialize in teaching either Grades 1-6 or 4-9, and in teaching either Danish or mathematics. All research participants in RaPiTE are to teach Grades 4—9, and they are all among the 35% of the prospective teachers (from now on, teachers), who specialize in mathematics.
There are two phases in the pilot study. The first phase consists of a questionnaire that the prospective teachers received on the day of their first mathematics class at the college (for more detail, see the next section). The second phase, which is the focus of the present chapter, consists of three parts. First, the participants attended a short teaching-learning sequence on R&P at the college; second they were expected to work with R&P with their students in their first practician; and third, they were to present and analyse video clips on R&P from their practician when back at the college. Fifty-seven prospective teachers participated in the first phase, 31 in the second.
Phase 2 of RaPiTE was included as part of the first of four compulsory courses on mathematics and mathematics education for teachers specializing in the subject. The bulk of the course is on numbers and algebra and on students’ learning of these topics. The teachers are expected to spend 275 working hours on the course, including 70 lessons in classrooms with a teacher educator. The practi- cum is not normally linked to the course in mathematics, and the learning goals for the practicum are phrased in general terms concerning, for instance, classroom organization and building and maintaining productive relationships with students. In RaPiTE we sought to bridge the usual divide between the mathematics course and the practicum.
Organization and methods
The two phases of the pilot study provide very different settings for the participants, and we do not expect the questionnaire from the first phase and the observations of the teaching-learning sequence from the second to shed light on relatively stable and context-independent mental constructs. Also, we do not assume any causal relation between responses to the questionnaire and teachers’ contributions to classroom practice. At best, the questionnaire allows us to understand how the teachers react discursively to R&P in a setting in which they are not challenged by other concerns that may emerge in classroom interaction. From a PoP perspective, it is an empirical question whether teachers orient themselves towards such a discourse as they engage with their students in the classroom. However, if teachers face significant problems with R&P in the questionnaire, we consider it unlikely that they engage proficiently with these processes when teaching. In other terms, it seems a necessary but by no means a sufficient condition for their proficient engagement with R&P in the classroom that they are able to deal with these processes in the questionnaire.
As mentioned earlier, the research participants filled out a questionnaire at the very beginning of the academic year. The questionnaire consists of open items on why they decided to go into teaching, why they chose to specialize in mathematics and what their general experiences are with school mathematics. They are also asked about specific experiences with R&P (e.g. “Describe how you felt about reasoning and proofs in mathematics”) and to comment on situations from school mathematics with an element of mathematical reasoning, including situations with students working with R&P (Larsen, 0stergaard & Skott, 2018).
As mentioned earlier, the present chapter focuses on the second phase of the pilot. The first part of this phase is a 12-lesson teaching-learning sequence on R&P organized as two sessions of six 45-minute lessons. This sequence was not taught by the authors of this chapter, but the second and third authors planned it and developed the teaching-learning materials. The intentions and the contents were discussed in detail with the colleague, who taught the sequence.
In the sequence the teachers are introduced to R&P and to different arguments for why to engage with R&P in school mathematics. More specifically, it was discussed why R&P is important in its own right and what one may expect in terms of student learning; how R&P may support the students’ understanding of other contents; what the character, advantages and disadvantages are of different types of argumentation, including possible transitions from empirical to deductive reasoning; and what the differences are between definitions, axioms and theorems in mathematics. Also, it was discussed how one may seek to create learning environments that support students’ learning of R&P. This discussion was based, for instance, on a video recording of school students making and justifying conjectures about a number pattern in a sequence of geometric figures. This leads to discussions about the quality of the students’ arguments and how students may be supported in developing them further. Subsequently, the teachers become involved in all three parts of the R&P cycle, for instance as they work on a version of the third task mentioned previously on making “trains” of equal length out of Cuisenaire rods (cf. section 5). As part of this, they are to make geometrical or number theoretical justifications for their claims. Finally the teachers discuss comments from school students, who have previously worked on the same task. One of these reads:
If I am to make two trains of equal length the sum must be even. If the sum is odd, I would have one left over. If the sum is even there could be other problems. . . . We do not know if it is sufficient that the sum is even.
After these sessions on mathematical R&P, the teachers form eight groups of three or four, each group going on a two-week practicum in a middle or lower secondary school (Grades 4—6 and 7-9, respectively). In the practicum each group normally teaches 8—10 lessons of mathematics a week as well as other subjects for approximately 4—6 lessons. As part of the pilot study, the students are before and during their practicum to (1) plan for their students’ involvement in R&P; (2) video record each other’s teaching and (3) select one video clip from the practicum in which the students are particularly involved in R&P. After the practicum, and as the last part of the second phase of the pilot study, the teachers discuss the video clips and the inherent potentials for and problems with R&P in a whole-day session. In what follows, we focus on the teachers’ response to the last requirement and on the subsequent discussion.
The sequences on R&P at the college before and after the practicum (parts 1 and 3 of the second phase) were video-recorded and transcribed. Like the responses to the questionnaire, the transcripts were analysed with no predeveloped set of codes, using coding procedures inspired by grounded theory (Charmaz, 2006). The initial coding and categorization of the data was first done independently by the second and third authors. The coding included word-byword, line-by-line, incident-to-incident and in Vivo coding, and memo-writing was used in all phases. Subsequently, codes and categories were compared and discussed among all authors, and inconsistencies were resolved.
The analysis resulted in categories on (1) teachers’ reasons for selecting the specific video clips; (2) the character of R&P in the teachers’ discussions of these processes; (3) school students’ learning of R&P and their learning of other contents from R&P; (4) the general significance of R&P in school mathematics and (5) “blackboard-talk”, that is, whole-class teaching as it relates to student learning in general and to R&P in particular.