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A Local Instruction Theory for the Development of Number Sense

Susan D. Nickerson, Ian Whitacre

2010
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Mathematical Thinking and Learning
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Gravemeijer's (1999 Gravemeijer's ( , 2004 construct of a local instruction theory suggests a means of offering teachers a framework of reference for designing and engaging students in a set of sequenced, exemplary instructional activities that support students' mathematical development for a focused concept. In this paper we offer a local instruction theory to guide the design of a set of instructional activities in support of the development of number sense. We make explicit the goals,
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... t the goals, assumptions, underlying rationale, and related instructional activities and provide examples from a mathematics content course for preservice elementary teachers. In this way, we contribute to an elaboration of the construct of local instruction theory. Externally-developed local instruction theories are indispensable for reform mathematics education (Gravemeijer, 2004 , p. 108). Gravemeijer's (1999 , 2004 construct of a local instruction theory, which was developed in the context of design research, suggests a means of offering teachers a framework of reference for designing and engaging students in a set of sequenced, exemplary instructional activities that support students' mathematical development of a focused concept. Gravemeijer (2004) described a local instruction theory with regards to learning goals, instructional activities, and the role of tools and imagery in the envisioned learning route. Gravemeijer illustrated how Realistic Mathematics Education can be used to develop a local instruction theory for an instructional sequence for addition and subtraction up to 100. Like Gravemeijer, we see a local instruction theory as indispensable to the design of instruction. The paucity of examples in the literature suggested to us that the field might benefit from further illustration and elaboration of the construct of local instruction theory. In this paper, we offer a local instruction theory in support of the development of number sense. This theory evolved out of a review of the literature, together with the results of a classroom teaching experiment, and subsequent iterations of teaching the course and fleshing out our analysis. We first discuss the differences between local instruction theory and hypothetical learning trajectory (Gravemeijer, 1999 (Gravemeijer, , 2004 Simon, 1995) ; next we discuss the literature on number sense. Within the context of our design research in a class for preservice elementary teachers, we provide an example of a local instruction theory to illustrate the relationship between the local instruction theory and resulting hypothetical learning trajectories. A colleague asked us whether we see a local instruction theory as being made up of a sequence of hypothetical learning trajectories, like Correspondence should be sent to 228 NICKERSON AND WHITACRE so many bricks in a row. We do not. Rather, we see a local instruction theory (LIT) as undergirding and informing particular hypothetical learning trajectories (HLTs). A hypothetical learning trajectory consists of learning goals for students, planned instructional activities, and a conjectured learning process in which the teacher anticipates the collective mathematical development of the classroom community and how students' understanding might evolve as they participate in the learning activities of the classroom community (Cobb, 2000; Cobb & Bowers, 1999; Simon, 1995) . Hypothetical learning trajectories have been described for a number of teaching experiments in diverse areas, such as linear measurement, equivalence of fractions, and statistics (cf. The construct of a hypothetical learning trajectory allows for a range of grain sizes. Simon and Tzur (2004) described an HLT in support of students' understanding the quantitative relationships between a fraction and an equivalent fraction whose denominator is a multiple of the original fraction. Gravemeijer, Bowers, and Stephan (2003) described a much broader HLT for the design of early-grade linear measurement instruction in a teaching experiment with two sequences: linear measurement and flexible arithmetic. Our own notion of HLT involves a smaller grain size, comparable to that of Simon and Tzur (2004) . Some of the difficulties that people have distinguishing between HLT and LIT may stem from this issue of grain size. According to Gravemeijer (2004) , a local instruction theory refers to "the description of, and rationale for, the envisioned learning route as it relates to a set of instructional activities for a specific topic" (p. 107). In Gravemeijer's (1999) view, there are two important differences between an LIT and an HLT: (1) an HLT deals with a small number of instructional activities, while an LIT encompasses a whole sequence; (2) HLTs are envisioned within the setting of a particular classroom, whereas an LIT comprises a framework, which informs the development of HLTs for particular instructional settings. Thus, the distinction between LIT and HLT is two-fold. One distinction is the duration of the learning process and the other is the situatedness in a particular classroom. In this report, we describe an LIT, clearly distinguishable from an HLT, that both informed and was informed by a teaching experiment. The aim of the classroom teaching experiment was to foster preservice teachers' development of number sense with a particular focus on flexible mental computation (Heirdsfield & Cooper, 2004) and computational estimation. BACKGROUND

doi:10.1080/10986061003689618
fatcat:nvhxndwdtfgctclcya63lv3rw4