A Local Instruction Theory for the Development of Number Sense

Abstract

Gravemeijer's (1999Gravemeijer'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, 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, , 2004Simon, 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 inst...

Investigating the improvement of prospective elementary teachers’ number sense in reasoning about fraction magnitude

“…A local instruction theory (LIT) consists of ''the description of, and rationale for, the envisioned learning route as it relates to a set of instructional activities for a specific topic'' (Gravemeijer 2004, p. 107). In a previous article, we describe our LIT for the development of number sense with a focus on whole-number mental computation (Nickerson and Whitacre 2010). Broadly, the envisioned learning route involves students moving from dependence on standard algorithms to reasoning flexibly about numbers and operations.…”

“…Broadly, the envisioned learning route involves students moving from dependence on standard algorithms to reasoning flexibly about numbers and operations. More specifically, our LIT is organized around three goals: (1) students capitalize on opportunities to use number-sensible strategies for problem-solving situations both inside and outside the classroom; (2) students draw on deep, connected knowledge of number and operations to develop a repertoire of number-sensible strategies; and (3) students reason with models to build on their understanding and flexibly use numbersensible strategies (Nickerson and Whitacre 2010). The envisioned learning route is one in which students move from relying on standard algorithms to developing and using their own nonstandard strategies.…”

“…To help address the problem of PTs' mathematical preparation, we previously designed a local instruction theory for the development of number sense (Nickerson and Whitacre 2010). Our previous research showed that PTs involved in a classroom teaching experiment developed improved number sense, particularly in the area of whole-number mental computation (Whitacre and Nickerson 2006).…”

Traveling down the road: from cognitive neuroscience to mathematics education … and back

“…In a next research step, these cognitive characteristics should be reconciled with principles from both domain-general (Van Merriënboer & Kirschner, 2007) and domain-specific (Nickerson & Whitacre, 2010) theories of instructional design. The effectiveness of these environments should be evaluated in various phases of empirical research (Sloane, 2008), using appropriate scientific methods, such as smallscale intervention studies followed by larger randomized controlled trials, before they can be applied in the classroom.…”

The Mathematics Education I and II Courses’ Effect on Teacher Candidates’ Development of Number Sense

“…It has been found that although teacher candidates' tendencies to standard computational techniques continue after training, number sense can be improved with proper training (Kaminski, 2002;Markovits & Sowder, 1994). Moreover, the research reports that a significant improvement in the number sense of teacher candidates was observed when they attend the courses that focus on the development of number sense (Nickerson & Whitacre, 2010;Whitacre, 2007;Whitacre & Nickerson, 2006).…”