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ABSTRACT
I present a viable learning trajectory for prospective elementary teachers’ number sense development with a focus on whole-number place value, addition, and subtraction. I document a chronology of classroom mathematical practices in a Number and Operations course. The findings provide insights into prospective elementary teachers’ number sense development. These include the role of standard algorithms and their relationship to the evolution of classroom mathematical practices that involve reasoning flexibly about number composition, sums, and differences.
Notes
1 Heirdsfield and Cooper (2004) studied the processes of accurate flexible and accurate inflexible mental calculators. The focus here is the distinction between flexibility and inflexibility.
2 It is certainly possible for the standard algorithms, or mental analogues thereof, to be used with understanding. It is also possible for students to learn to use nonstandard strategies without understanding. However, this has not been typical, historically, in school mathematics in the United States, which has tended to emphasize the standard algorithms.
3 Markovits and Sowder (Citation1994) defined nonstandard with no reformulation specifically in terms of a student using a “left-to-right process.” However, there are strategies that prospective elementary teachers use, namely aggregation strategies, that are nonstandard and without reformulation, but are not accurately characterized as left-to-right processes. I view both right-to-left and left-to-right processes as Transition strategies because both involve separating numbers into tens and ones and computing place-value-wise. Furthermore, individuals who use one of these often use the other. They may choose between a right-to-left or left-to-right process depending on whether or not regrouping is necessary. For prospective elementary teachers, right-to-left and left-to-right processes seem to be closely related.
4 Toulmin (1958/Citation2003) also discussed rebuttal and qualification as elements of an argument, but these are not relevant to the present discussion since they do not tend to be used in analyses of classroom mathematical practices (e.g., Stephan & Rasmussen, Citation2002).
5 Izsák, Tillema, and Tunç-Pekkan (Citation2008) studied students’ learning in a rather traditional, teacher-centered classroom. For their purposes, they adapted the definition of classroom mathematical practice to account for the different set of norms that existed in the class.
6 Note that the research question is concerned with how collective activity evolved in the classroom that was studied. I do not claim that similar improvements in PTs’ number sense could not be accomplished through a different instructional sequence and associated learning trajectory—only that the trajectory documented here appears to be a viable one for the purpose.
7 Claims that lacked both data and warrant were ignored; a claim alone does not constitute an argument.
8 Specifically, these are mathematical arguments. Nonmathematical arguments were not included in the analysis. Note that what counts as mathematical depends on the course topic and on the researchers’ interests. For example, in the context of the mathematics content course, claims concerning how to name mental computation strategies were considered mathematical.
9 This definition differs from that of Cobb and Yackel (Citation1996) in that a CMP is defined in terms of a set of mathematical ideas, rather than a single idea. For Stephan, Cobb, and Gravemeijer (Citation2003), who did earlier work on classroom mathematical practices, the construct corresponded to a single taken-as-shared idea.
10 For reasons of length and clarity, the evolution of collective activity is described at the level of classroom mathematical practices. The details of the as-if shared ideas that constitute the CMPs, including the particular criteria that each satisfied, are provided in Appendix A.
11 This height problem was adapted from Thompson (Citation1993). It appears in the course textbook (Sowder, Sowder, & Nickerson, Citation2014).
12 Thanheiser (Citation2009) highlighted the importance of reasoning flexibly about reference units (e.g., viewing one hundred as a number of tens and a number of ones) in her analysis of PTs’ conceptions of multidigit numbers.
13 In terms of collective activity, I am not claiming that all or most of the PTs in the class used the strategy of shifting the difference themselves. Rather, Valerie's argument was founded upon as-if-shared ideas.
14 In terms of being familiar with, and often dependent upon, the standard arithmetic algorithms, PTs may be similar to other student populations. A major difference is that elementary PTs, given their career path, take courses that involve revisiting elementary mathematics from a new perspective. These courses would likely be beneficial to a much wider range of students, but it is not typical for other students to take such courses.