In his 1976 book, "Proofs and Refutations," Lakatos presents a collection of case studies to illustrate methods of mathematical discovery in the history of mathematics. In this paper, we reframe these methods in ways that we have found make them more amenable for use as a framework for research on learning and teaching mathematics. We present an…

This research report describes the findings of a study on teachers' ways of interpreting student responses to tasks involving equivalent expressions. The teachers in this study were engaged in model-eliciting activities designed to promote the development of their knowledge and reveal their models (or interpretations) of their students' algebraic…

We designed and implemented curriculum intended to be used by students in an algebra-based introductory physics laboratory course. Our curricular goal was to foster, through observations in the lab, a coherent framework in students' understanding of general principles presented in the introductory mechanics course, while addressing known student…

Here is the only reference book you will ever need for teaching primary school mathematics and statistics. It is full of exciting and engaging snapshots of excellent classroom practice relevant to "The New Zealand Curriculum" and national mathematics standards. There are many fascinating examples of investigative learning experiences,…

This research explored the experiences of students in a school district that limited early access to the study of algebra and to inform education policymakers of the impact of such tracking policies on the lives and futures of the students. Quantitative analysis had already yielded a snapshot of inequities deriving from the policies surrounding…

Engaging all students in grade-level mathematics is often a challenge for teachers, especially in diverse urban classrooms. Too often in her role of supporting the development of effective practices with middle school teachers, the author observes traditional classroom instructors presenting information and demonstrating procedures. In this…

This study examined the effect of foster care experience and characteristics on educational outcomes. The typical strategy in examining the effect foster care has on educational outcomes is to compare the educational achievement of youth with foster care experience to that of their peers or to national norms. This strategy fails to take selection…

This is a summary of the May 2009 EdSource report, "Algebra Policy in California: Great Expectations and Serious Challenges." The report focuses on California's policies regarding mathematics, in particular state standards for when students should take Algebra I. It also provides a comprehensive look at state data related to both student…

To ensure that all high school graduates are prepared for the opportunities and challenges that await them, states have increasingly been focused on aligning their end-of-high school expectations with the demands of the real world. In 2005, no state had aligned their expectations with real world demands; now 29 states have adopted college- and…

As the 21st century's first decade draws to a close, America's public school students are participating in online academic classes in increasingly large numbers. More than one million K-12 public school students took online courses in the 2007-2008 school year, according to an estimate in a recent Sloan Consortium survey. This represents a 47…

The purpose of this study was to determine the competence of beginning California K-6 teachers in basic algebra topics, and to investigate their level of math anxiety and attitudes toward math. The sample for this study was beginning California elementary teachers in the Bay Area of California. An algebra content assessment and a self-report…

This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…

This article presents an analysis of how students' dispositions toward learning shift and change over time through their engagement with particular classroom practices and identifies classroom practices that support the development of more productive dispositions. An analysis of 4 purposefully selected students from 2 different classrooms reveals…

Algebra uses symbols for generalizing arithmetic. These symbols have different meanings and interpretations in different situations. Students have different perceptions about these symbols, letters and signs. Despite the vast research by on the students' difficulties in understanding letters in Algebra, the overall image that emerges from the…

Each year, well over a million students take college algebra and related courses. Very few of these students take the courses to prepare for calculus, but rather because they are required by other disciplines or to fulfill Gen Ed requirements. The present article discusses what the current mathematical needs are in most of those disciplines,…