The research study this article is based on aims to implement research knowledge to teaching, that is, the concept of critical aspects and dimensions of variation used in the variation theory. To do this, the researchers worked with willing teachers to explore how to make mathematics teaching more effective. This paper illustrates how teachers…

Building on the claim that teachers need to know how innovative instructional approaches work to be able to adapt them to their own classrooms, design research is presented as a research method that aims to offer exactly that kind of information. We elaborate design research that aims to develop a local instruction theory--a theory about the…

A process for helping students in the elementary grades develop their own algorithms for subtraction with carrying is described. Pupils choose their own times and ways to move from manipulative materials to written notation. (MP)

Examination of models for representing integers demonstrates that formal operational thought is required for establishing the operations on integers. Advocated is the use of many models for introducing negative numbers but, apart from addition, it is recommended that operations on integers be delayed until the formal operations stage. (JP)

It is noted that defining subtraction independently of addition through use of physical objects can help one to illustrate the non-commutativity of this operation. Lack of awareness of the importance of the order of the digits may be a primary cause of errors in compound subtraction. (MP)

The purpose of this study was to analyze the relationship of a child's performance in solving addition and subtraction problems using concrete embodiments compared to his performance using the symbolic mode. A random sample of 144 second grade children were tested. It was found that children proficient in regrouping addition and subtraction…

This study was designed to follow up earlier work on mapping instruction. The two main goals were to: (1) test the effectiveness of mapping instruction as a general cure for "buggy" subtraction algorithms, and (2) explore two alternative explanations of how this new form of instruction works. It was hypothesized that mapping cures bugs…

This report opens by asking how operations in general, and addition and subtraction in particular, are characterized for elementary school students, and examines the "standard" instruction of these topics through secondary schooling. Some common errors and/or "sloppiness" in the typical textbook presentations are noted, and…

Prior research on learning has been linked to instruction by the derivation of general principles of instructional design from learning theories. However, such design principles are often difficult to apply to particular instructional issues. A new method for relating research on learning to instructional design is proposed: Different ways of…

Suggests that children enter school with a great deal of informal intuitive knowledge of mathematics that can serve as the basis for developing much of the formal mathematics of the primary school curriculum. Describes a research-based model of children's thinking that teachers can use to interpret, transform, and reframe their informal or…

The Proceedings of PME-XI has been published in three separate volumes because of the large total of 161 individual conference papers reported. Volume I contains four plenary papers, all on the subject of "constructivism," and 44 commented papers arranged under 4 themes. Volume II contains 56 papers (39 commented; 17 uncommented)…