The fifth-grade curriculum presents many opportunities to use students' prior mathematical knowledge as a way to bridge new and more difficult mathematical ideas. In this article, the authors document an area tiling task given to fifth-grade students to connect aspects of area measurement covered in earlier grades to grade-level standards such as:…

The importance of mathematical reasoning is unquestioned and providing opportunities for students to become involved in mathematical reasoning is paramount. The open-ended tasks presented incorporate mathematical content explored through the contexts of problem solving and reasoning. This article presents a number of simple tasks that may be…

Designing a mathematically worthwhile task is critical for promoting students' reasoning. To improve task design skills, teachers often engage in collaborative lesson planning activities such as lesson study. However, to learn from the process of lesson study, it is important for teachers to notice productively the concepts, students' confusion…

The author wants her students to see any new mathematics--fractions, negative numbers, algebra--as logical extensions of what they already know. This article describes two students' efforts to make sense of their conflicting interpretations of 1/2 × -6, both of which were compelling and logical to them. It describes how discussion, constructing…

We present a reconceptualized framework for opportunity to learn (OTL) in school mathematics that builds on previous conceptualizations of OTL and includes features related to both quantity (i.e., time) and quality. Our framework draws on existing literature and on our own observational research of mathematics teaching practices. Through the…

Designed to provide intermediate grade students with experience using fractions and mixed numbers, the PLATO sequence described in this manual includes interactive models for students to work with, as well as lessons for review, practice, or experience. These lessons are intended to be integrated with the classroom mathematics instruction, and…

A project developed and tested articulation and instructional models for an electronics program in secondary schools. Two nontraditional methods of instruction were used: (1) computer-assisted video and (2) video, laboratory, and recitation. Electronic theory was taught at the student's home high school; laboratory exercises were completed at…