# Angle measure, quantitative reasoning, and instructional coherence: an examination of the role of mathematical *ways of thinking* as a component of teachers’ knowledge base

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## 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.

## Keywords

Mathematical knowledge for teaching Ways of thinking Quantitative reasoning Instructional coherence Angle measure## Notes

### 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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