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Mathematics education researchers have long pursued—and many still pursue—an ideal instructional model for operations on integers. In this chapter, I argue that such a pursuit may be futile. Additionally, I highlight that ideas of relativity have been overlooked; and, I contend that current uses of translation within current integer instructional models do not align with learners’ inventions. Yet, conceptions of relativity and translation are essential for making sense of integers as directed quantities. I advocate for drawing on learners’ unique conceptions and actions about directed number in developing instructional models. Providing evidence of student work from my research, I illustrate the powerful constructions of relativity and translation as students engage with directed quantities.


Originally published in Wessman-Enzinger N.M. (2019) Integers as Directed Quantities. In: Norton A., Alibali M. (eds) Constructing Number. Research in Mathematics Education. Springer, Cham.

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