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We identify and document 3 cognitive obstacles, 3 cognitive affordances, and 1 type of integer understanding that can function as either an obstacle or affordance for learners while they extend their numeric domains from whole numbers to include negative integers. In particular, we highlight 2 key subsets of integer reasoning: understanding or knowledge that may, initially, interfere with one's learning integers (which we call cognitive obstacles) and understanding or knowledge that may afford progress in understanding and operating with integers (which we call cognitive affordances). We analyzed historical mathematical writings related to integers as well as clinical interviews with children ages 6-10 to identify critical, persistent cognitive obstacles and powerful ways of thinking that may help learners to overcome obstacles.

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We interviewed 40 students each in grades 7 and 11 to investigate their integer-related reasoning. In one task, the students were asked to write and interpret equations related to a story problem about borrowing money from a friend. All the students solved the story problem correctly. However, they reasoned about the problem in different ways. Many students represented the situation numerically without invoking negative numbers, whereas others wrote equations involving negative numbers. When asked to interpret equations involving negative numbers in relation to the story, students did so in two ways. Their responses reflect distinct perspectives concerning the relationship between arithmetic equations and borrowing/owing. We discuss these findings and their implications regarding the role of contexts in integer instruction.

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...