Productive learning of algebra is supported when students reflect on multiple strategies, compare them and discuss the rationale behind and relative merits of particular strategies. Comparison and Discussion of Multiple Strategies (CDMS) is an instructional approach designed to support these processes in math classrooms. In the current study, 16…

A first-year algebra student's curiosity about factorials of negative numbers became a starting point for an extended discovery lesson into territory not usually explored in secondary school mathematics. In this article, the authors, math teachers in Massachusetts, examine how to solve for factorials of negative numbers and discuss how they taught…

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…

This paper examines whether using "From Here to There!" ("FH2T:E"), a dynamic game-based mathematics learning technology relates to improved early algebraic understanding. We use student log files within FH2T to explore the possible benefits of student behaviors and gamification on learning gains. Using in app measures of…

The purpose of this study was to compare students' performances in a freshmen level quantitative reasoning course (QR) under three different instructional models. A cohort of 155 freshmen students was placed in one of the three models: needing a prerequisite course, corequisite (students enroll simultaneously in QR course and a course that…

The Better Math Teaching Network (BMTN) is looking to transform high school math instruction in New England and beyond. Through this network, researchers and practitioners are working together to make high school Algebra I classes more student centered. This report outlines findings from the first year of this developmental evaluation of the…

On the Partnership for Assessment of Readiness for College and Careers (PARCC) assessments, the attainment of performance level 4 is intended to indicate college readiness or being "on track" to college and career readiness. Students who achieve Level 4 should have a 0.75 probability of attaining at least a C in entry-level,…

Galileo dropped cannonballs from the leaning tower of Pisa to demonstrate something about falling bodies. Gauss was a giant of mathematics and physics who made unparalleled contributions to both fields. More contemporary (and not a person), the Green Monster is the left-field wall at the home of the Boston Red Sox, Fenway Park. Measuring 37 feet…

In this work we examined the content of approximately 950 mathematics questions used in measuring achievement among 4th, 5th and 6th graders in the US in an attempt to determine how the current standardized examinations, which were developed to reflect new national curricular mandates, compared to the old measures used for the same purpose. Using…

Recent federal legislation, such as the "No Child Left Behind Act" of 2001 and the "Education Sciences Reform Act (ESRA)" of 2002, has insisted that educational evaluations use rigorous research designs with quantitative outcome measures. In particular, the Institute of Education Sciences (IES) at the U.S. Department of…

This Framework builds upon the foundation of the 2010 Massachusetts Curriculum Framework for Mathematics as well as versions of the Massachusetts Mathematics Framework published since 1995. The current Framework incorporates improvements suggested by Massachusetts educators after six years of experience in implementing the 2010 standards in their…

In the United States, students' learning experiences around proof is generally concentrated in the domain of high school geometry with a focus on its verification function. Thus, providing students with a limited conception of what proof entails and the role that it plays in performing mathematics. Moreover, there is a lack of U.S.-based studies…

Comparison is a powerful tool that has been shown to improve learning in a variety of domains. In both laboratory studies and small-scale classroom studies, having learners compare and contrast worked examples has been shown to reliably lead to gains in students' knowledge. Comparison is also integral to "best practices" in mathematics…

Education researchers are increasingly working in practice-based partnerships in order to direct their research efforts toward important problems of practice. We argue for the creation of an infrastructure to support routine and sustained interaction among researchers, practitioners, and designers in order to make partnership efforts more…

This study investigated longitudinal pathways leading from early spatial skills in first-grade girls to their fifth-grade analytical math reasoning abilities (N = 138). First-grade assessments included spatial skills, verbal skills, addition/subtraction skills, and frequency of choice of a decomposition or retrieval strategy on the…