Virginia Mathematics Teacher Spring 2017

sequences that children move through as they construct their notions of composite and iterable units on their way to developing multiplicative thinking. We highlighted the important role of skip counting as both an indicator of, and a way to foster, children’s development of composite units. We recommend that teachers continue to emphasize the use of skip counting as a way to develop students’ multiplicative thinking beyond just the connection with multiplication facts. Prematurely deemphasizing skip counting may cause children to focus only on the idea that 5 × 3 is the result of their additive counting by 3s, whereas continued use may afford them the opportunity to develop the more sophisticated notion that these 3s represent composite units, and that 5 × 3 represents 5 times as much as 3, a composite of composite units. At the same time, it is important for teachers to recognize that the proficient use of skip counting does not necessarily imply that children have developed multiplicative thinking. By being aware of the characteristic ways of thinking associated with each of the different number sequences teachers are better able to provide appropriate scaffolding to move children through the stages. As an example, we feel that the intentional and selective use of manipulatives, both with elementary school students and middle school students who have yet to develop composite or iterable units, could provide invaluable support for the students’ mathematical development. References Olive, J. (2001). Children’s number sequences: An explanation of Steffe’s constructs and an extrapolation to rational numbers of arithmetic. The Mathematics Educator , 11 (1), 4-9. Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge . New York, NY: Springer. doi 10.1007/978-1-4419-0591-8 Steffe, L. P., & Cobb, P. (1988). Construction of arithmetical meanings and strategies . New York, NY: Springer. Ulrich, C. (2015). Stages in constructing and coordinating units additively and multiplicatively (Part 1). For the Learning of Mathematics, 35 (3), 2–7 .

Ulrich, C. (2016a). Stages in constructing and coordinating units additively and multiplicatively (Part 2). For the Learning of Mathematics, 36 (1), 34–39 . Ulrich, C. & Wilkins, J. L. M. (2016a). Using written work to assess stages in sixth-grade students’ construction and coordination of arithmetic units . Manuscript submitted for publication. Ulrich, C. & Wilkins, J. L. M. (2016b). Using student written work to investigate stages in sixth-grade students’ ways of operating with numbers . Paper presented at the American Educational Research Association, Washington, DC. Virginia Department of Education (2009). Mathematics standards of learning for Virginia public schools . Richmond, VA: Commonwealth of Virginia Board of Education. Available online at: http:// www.doe.virginia.gov/testing/sol/ standards_docs/mathematics/index.shtml

Jesse L.M. Wilkins Professor of Math Education Virginia Tech Wilkins@vt.edu

Katy Ulrich Assistant Professor Virginia Tech culrich@vt.edu

Virginia Mathematics Teacher vol. 43, no. 2

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