Students must leave elementary school with a firm understanding of fractions and decimals in order to make satisfactory progress in advanced mathematics. Firm understanding includes manipulation of rational numbers using different models of representation such as area and length. Research has demonstrated that a graduated sequence such as…

"That was the day I decided I was bad at math." Countless times, preservice and in-service teachers make statements such as this after sharing vivid memories of learning multiplication facts. Timed tests; public competitive games, such as Around the World; and visible displays of who has and has not mastered groups of facts still…

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about…

TWO STRATEGIES FOR TEACHING MULTIPLE-DISCRIMINATION TASKS WERE REPORTED. THE "MULTIPLE CONCEPT" PRESENTS SIMPLE DESCRIPTIONS OF SEVERAL RELATED CONCEPTS AT THE BEGINNING OF INSTRUCTION. INCREASINGLY COMPLEX MATERIAL PERTAINING TO THESE CONCEPTS IS THEN GRADUALLY INTRODUCED. THE "SINGLE CONCEPT" PRESENTS ONE CONCEPT AT A TIME,…