Abstract
Is it possible that a meeting of mathematicians and primary school teachers will be productive? This question became intriguing when one professor of mathematics initiated a professional development course for practicing primary school teachers, which he taught alongside a group of mathematics Ph.D. students. This report scrutinizes the uncommon meeting of these two communities, who have very different perspectives on mathematics and its teaching. The instructors had no experience in primary school teaching, and their professed goal was to deepen the teachers’ understanding of the mathematics they teach, while teachers were expecting the course to be pedagogically relevant for their teaching. Surprisingly, despite this mismatch in expectations, the course was considered a success by teachers and instructors alike. In our study, we analyzed a lesson on division with remainder for teachers of grades 3–6, taught by the professor. The framework used for the data analysis was mathematical discourse for teaching, a discursive adaptation of the well-known mathematical knowledge for teaching framework. Our analysis focuses on the nature of the interactions between the parties and the learning opportunities they afforded. We show how different concerns, which might have hindered communication, in fact fueled discussions, leading to understandings of the topic and its teaching that were new to all the parties involved. The findings point to a feasible model for professional development where mathematicians may contribute to the education of practicing teachers, while they are gaining new insights themselves.
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Notes
Translation taken from ICMI (2008).
In Israel, professional development is usually conducted by professional teacher educators or experienced teachers.
DWR is briefly mentioned in the context of the long division algorithm and may be mentioned again in the related context of dividing polynomials, a procedure required in Calculus.
The words in Hebrew are nearly interchangeable (“haluka” and “hiluk”), both based on the root “helek” (meaning “part”). For example, the action "hilakti" may be interpreted either as "I divided" or as "I distributed".
Although this view may seem somewhat dichotomist, and as such an oversimplification, we found it useful to relate to each community's main strength. Taking into account that mathematicians seldom have substantial teaching experience in primary classrooms, and that most primary teachers are not graduates of university mathematics departments, we feel that the dichotomy is justified.
See Section 4 in the lesson graph, “Appendix”.
Hebrew distinguishes between dividing by and dividing into in much the same way as English does.
See Section 6 in the lesson graph, “Appendix”.
See Sections 6 and 8 in the lesson graph, “Appendix”.
Or _:3 = 7(_:3) if we accept Rick’s new notation.
See Section 9 in the lesson graph, “Appendix”.
Such a shift from talk about process to talk about object is called by Sfard (2008) reification of the process.
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Acknowledgments
This research was supported by the Israel Science Foundation (Grant No. 615/13). The authors wish to thank Abraham Arcavi for many helpful suggestions.
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Appendix: Lesson graph of the DWR session
Appendix: Lesson graph of the DWR session
See Table 1.
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Cooper, J., Karsenty, R. Can teachers and mathematicians communicate productively? The case of division with remainder. J Math Teacher Educ 21, 237–261 (2018). https://doi.org/10.1007/s10857-016-9358-7
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DOI: https://doi.org/10.1007/s10857-016-9358-7