The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under…

This study addresses first graders' conceptual knowledge exhibited in addition strategies in relation to instructional practices and teachers' beliefs in a suburban Alabama school. To address the purpose of the study, the central question was "What is the nature of the relationship between curricular practices and beliefs and students'…

To improve understanding of identities and inverses, and to provide a stronger foundation for future mathematics, Sandra Miles designed a two-part lesson that makes the relationship between identities and inverses explicit. This article illustrates Miles' teaching of the first part of the lesson, focused on addition and subtraction, and then gives…

The present study aimed to examine the opinions of in-and preservice middle school mathematics teachers about teaching four operations with integers through "number line" and/or "counters". Participants were 42 middle school mathematics teachers and 51 pre-service mathematics teachers. The data were collected through open-ended…

Although basing instruction on a learning trajectory (LT) is often recommended, there is little direct evidence to support the premise of a "LT approach"--that to be maximally meaningful, engaging, and effective, instruction is best presented one LT level beyond a child's present level of thinking. The present report serves to address…

These four chapters describe studies of using models for integer addition and subtraction. The models draw principally on the two grounding metaphors of object collection and motion along a path. A strength of all chapters is detailed analysis of how the models are and can be implemented and how they influence student's learning. Together the…

One of the factors associated with the less than positive mathematics performance of American students could be poorly designed textbooks that fail to facilitate the development of critical mathematical ideas. This study examined two reform-based textbooks (Go math! and Investigations) in reference to essential, mathematical big ideas emphasized…

In this paper, we report how 5-year-olds' arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children's ways of experiencing numbers' part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report…

There is a growing awareness that many children are not developing fast and accurate retrieval-based strategies for solving single-digit addition problems. In this study we individually assessed 166 third and fourth grade children to identify a group of children (called accurate-min-counters) who frequently solved simple single-digit addition…

This study aims to investigate the effect of history-of-mathematics activities on the development of mental computation of middle-grade students and to determine the students' views on the history of mathematics. This study was carried out as an action research project. The participants of the study consisted of 19 seventh-grade students studying…

This chapter describes instances of play within a teaching episode on integer addition and subtraction. Specifically, this chapter makes the theoretical distinction between integer play and playing with integers. Describing instances of integer play and playing with integers is important for facilitating this type of intellectual play in the…

When two third-graders collaboratively manipulated a multi-modal, digital learning device called TouchTimes (hereafter, TT), that introduces multiplication through visual, tangible and symbolic means, their thinking about quantity shifted from being additive to being multiplicative. In this study, I examine the children's interactions around/with…

The aim of this study was to investigate the effect of a precision teaching (PT) framework on the mathematical ability of students with intellectual and developmental disabilities. We also examined if students of moderate mathematical ability could perform as well as their peers with fewer difficulties with their math skills. Sixteen students…

Mixed numbers and fraction in general have been a problem for students not only in understanding the concept of itself, but also in understanding the operation of it, such as addition. Thus, supporting students in learning addition of mixed numbers is needed. The aim of this study is to develop a local instructional theory and materials related to…

This chapter focuses on the development of concepts that children draw on as they work toward understanding negative numbers. Framed from a conceptual change lens, I discuss different interpretations children have of minus signs, numerical order, numerical values, and addition and subtraction operations and how children draw on these varied…