Documenting the Process of a Prospective Elementary Teacher’s Flexibility Development: Scaffolded Strategy Ranges and Sociomathematical Norms for Mental Computation

Abstract

This article contributes to the research literature concerning prospective elementary teachers’ mathematical thinking and learning with a focus on flexibility. We present a case study of a prospective elementary teachers’ development of flexibility in mental addition and subtraction during a Number and Operations course. Building upon the construct of strategy ranges, we introduce scaffolded strategy ranges, which describe the sets of strategies that people use given the opportunity to solve a task in multiple ways. Like many prospective elementary teachers, Brandy initially appeared inflexible in mental addition and subtraction. In fact, her unscaffolded strategy ranges were limited to just the mental analogs of the standard algorithms. However, Brandy’s scaffolded strategy ranges revealed greater potential for flexibility. Furthermore, the way of reasoning that appeared in Brandy’s scaffolded strategy ranges (a) influenced her interpretations of nonstandard strategies that she encountered in the Number and Operations course and (b) foreshadowed the direction in which her flexibility would develop.

Acknowledgment

The first author thanks Susan D. Nickerson for her guidance throughout the dissertation process. We also thank Paula Levin and the anonymous reviewers for their helpful feedback on previous versions of this article.

Notes

1 As this work focuses on prospective elementary teachers, the vast majority of whom are female, we use female pronouns as generic.

2 Star and Seifert (2006, p. 284) used “alternative ordering tasks” to promote flexible equation solving. Students were asked to solve the same algebraic equations in more than one way. Over time, those students developed improved flexibility in their approaches to equation solving, whereas students in a control group did not. In this study, in contrast to the work of Star and Seifert (2006), scaffolded alternatives tasks were used as a research tool to investigate PTs’ scaffolded strategy ranges, rather than being part of an instructional intervention. Scaffolded strategy ranges are important because they reveal PTs’ prior knowledge that is relevant to their flexibility development. They enable researchers to see what the individual can do in her ZPD, which indicates learning that is within reach (Vygotsky, 1978).

3 For the purposes of the previous analysis, arguments were coded according to their components, following Toulmin’s (1958/2003) model. The details of that coding are not particularly relevant to this analysis, as only the repeated-use criterion was applied; see the following.

4 These important points relate to matters of classroom mathematical practice, which are described in detail elsewhere (Whitacre, 2017).

5 Children in the United States often use known doubles to derive less familiar arithmetic facts (Fuson et al., 1997). Brandy’s strategy seems to be a more sophisticated version of this approach. We expect that 78 + 78 = 156 was not a known fact for her, but was one that she was able to compute readily.

6 We recognize that readers may question the appropriateness of this name, given that borrow may imply that something is to be returned later, as in end compensation, which is not the case here. This is the term that was used in the class. Consistent with the norms of class, the PTs had considerable influence over the choice of such names. As a result, the names chosen sometimes departed from those that the authors or instructor might have preferred. At the same time, we also note that there was no apparent misunderstanding of “Borrow to Build.” This seems to have been a case of a nonmathematical term used a bit loosely; however, this loose usage did not seem to reflect confusion about the mathematical features of the strategy.

7 This clip is part of the Integrating Mathematics and Pedagogy (IMAP) searchable collection of video clips (Philipp, Cabral, & Schappelle, 2005).