Abstract
This paper reports findings from a study that establishes empirical support for Harel’s (Zentralblatt für Didaktik der Math 40:893–907 2008b) inclusion of mathematical ways of thinking as a component of teachers’ professional knowledge base. Specifically, we examined the role of quantitative reasoning (Smith and Thompson, in: Kaput, Carraher, Blanton (eds) Algebra in the early grades, Erlbaum, New York 2007; Thompson, in: A theoretical model of quantity-based reasoning in arithmetic and algebra, Center for Research in Mathematics & Science Education: San Diego State University 1990; Thompson, in: Hatfield et al (eds) New perspectives and directions for collaborative research in mathematics education, University of Wyoming, Laramie 2011) on the quality and coherence of an experienced secondary teacher’s instruction of angle measure. We analyzed 37 videos of the teacher’s instruction to characterize the extent to which he attended to supporting students in reasoning about angle measure quantitatively, and to examine the consequences of this attention on the quality and coherence of the meanings the teacher’s instruction supported. Our analysis revealed that the inconsistent meanings the teacher conveyed were occasioned by his lack of awareness of the conceptual affordances of students’ quantitative reasoning on their ability to construct coherent, meaningful understandings of angle measure. Our findings therefore support Harel’s notion that teachers’ mathematical ways of thinking constitute an essential component of their specialized content knowledge.
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Notes
PCK refers to the character of subject matter knowledge required for effective teaching, which includes an awareness of “the ways of representing and formulating the subject that make it comprehensible to others” (Shulman 1986, p. 9).
We define quantity below, as well as several other constructs within the theory of quantitative reasoning. For now it suffices to say that a quantity is a measurable attribute of an object and that quantitative reasoning involves reasoning about quantities and their relationships.
By “teachers’ attention to quantitative reasoning,” we are referring to teachers’ awareness of the conceptual affordances of students’ quantitative reasoning on their ability to construct coherent, meaningful understandings of specific mathematical ideas. We are also referring to teachers’ instructional actions motivated by this awareness.
It is important to note that iconic assignments of numerical values as measures of particular angles is not an instance of quantification (e.g., assigning 90° to a right angle, 180° to a line, and 360° to a circle) because such assignments are not based on an interiorized measurement process (Moore 2013).
We acknowledge that this definition of meaning necessarily implies a multiplicity of meanings conveyed.
“Incoherence” is often used in a denigratory sense. We stress that by defining coherence as a property of a system of cognitive schemes, our use of “incoherent” is as an explanatory construct, not a pejorative.
For ease of communicating, we henceforth use “students” instead of “our construction of a generalized Algebra II student.”
National Board Certification is a voluntary, peer-reviewed, and performance-based certification program for schoolteachers in the United States (http://www.nbpts.org/national-board-certification).
There are an odd number of classes because in one of David’s afternoon classes, he taught content unrelated to angle measure, sine, and cosine.
We reiterate that this is an important meaning for students to construct since one can deduce from it that any unit of angle measure must be proportional to the circumference of the circle centered at an angle’s vertex. However, this abstraction of the essential criterion that any unit of angle measure must satisfy is only available to those who have coordinated this way of understanding with the meaning of angle measure as the length of a class of subtended arcs.
See Moore et al. (2016) for similar findings about pre-service teachers’ meanings for the unit circle.
A comprehensive discussion of the implications of a quantitative understanding of angle measure for supporting students’ learning of other important trigonometry concepts is beyond the scope of this paper. We refer the interested reader to Thompson et al. (2007) for an analysis of the many ways in which conceptualizing angle measure quantitatively is a precondition to developing a coherent understanding of trigonometric functions.
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Acknowledgements
This manuscript is based on data used in the lead author’s doctoral dissertation. This research was supported by National Science Foundation Grant 943360412. Any conclusions and recommendations stated here are those of the authors and do not necessarily reflect official positions of the NSF. We thank Dr. Marilyn P. Carlson and Dr. Kevin Moore for providing feedback on a previous version of this paper. Finally, we thank SIGMAA on RUME for the opportunity to present a previous version of this paper at their 20th annual conference.
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Tallman, M.A., Frank, K.M. Angle measure, quantitative reasoning, and instructional coherence: an examination of the role of mathematical ways of thinking as a component of teachers’ knowledge base. J Math Teacher Educ 23, 69–95 (2020). https://doi.org/10.1007/s10857-018-9409-3
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DOI: https://doi.org/10.1007/s10857-018-9409-3