To understand relationships between students' quantitative reasoning with fractions and their algebraic reasoning, a clinical interview study was conducted with 18 middle and high school students. Six students with each of 3 different multiplicative concepts participated. This paper reports on the 6 students with the most basic multiplicative…

This research report examines how two groups of bilingual algebra students made connections among representations while solving a non-routine generalization problem. Using a socio-cultural orientation to mathematics learning, together with a semiotic lens on students' joint mathematical activity, this report details the type of connections among…

This study investigates cognitive difficulties and mathematical ideas that are related to making connections among representations. A three-week intervention was designed and implemented to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to connection of representations in algebra. This study…

Third- through fifth-grade students participating in a classroom teaching experiment investigating the impact of an Early Algebra Learning Progression completed pre- and post-assessment items addressing their abilities to engage in functional thinking. We found that after a sustained early algebra intervention, students grew in their abilities to…

This paper reports findings from the second year of a two-year study designed to develop approaches to teaching algebra in years 9 and 10. The aim of the research was to explore and develop teaching approaches to assist students to acquire a conceptual understanding of algebra, and to document the impact of these approaches on student outcomes.…

The importance of using real-life contexts in teaching mathematics is emphasised in many policy and curriculum statements. The literature indicates using contexts to teach mathematics can be difficult and few detailed exemplars exist. This article describes the use of real-life contexts in one New Zealand Year 11 algebra lesson. Data included a…

In higher education, many high-enrollment introductory courses have evolved into "gatekeeper" courses due to their high failure rates. These courses prevent many students from attaining their educational goals and often become graduation roadblocks. At the authors' home institution, general chemistry has become a gatekeeper course in which…

This paper compares lessons on linear equations from the same curriculum materials taught by two teachers of different levels of mathematical knowledge for teaching (MKT). The analysis indicates that the mathematical quality of instruction in these two classrooms appears to be a function of differences in MKT. Although the two teachers were…

This study examined the effects of types of representations, constructivist teaching approaches, and student engagement on middle school algebra students' procedural knowledge and conceptual understanding. Data gathered from 16 video lessons and algebra pretest/posttests were used to run three multilevel structural equation models. Symbolic…

This project was a response to the changes in the Washington State math assessments for high school students. The creation of an exit exam for Washington State high school students and the expectation that they pass it to graduate has placed tremendous pressure on high schools that struggled with low passing rates on the Washington State math…

In this study, it is explained how to use Escher's tessellations in teaching group concept which is one of the most abstract concepts in mathematics. MC Escher's monohedral tessellations provide detailed study in an undergraduate course in abstract algebra. This study attempts to provide useful visual references for the students on learning some…

This article examines the ellipsograph of Archimedes, also known as the locus problem of Franciscus van Schooten, and related mechanisms. We not only solve the algebraic system explicitly, but we also reverse engineer the problem and find configurations that provide a particular solution. Using the modern techniques of polynomial Grobner Bases we…

Curriculum materials' developers typically assume the existence of certain general social-educational classroom practices and norms. Conversely, the current study addresses the effects of context on curriculum enactment, focusing on the interrelations between teacher, class and curriculum materials. Each of the two case studies presented herein…

Linear algebra is a standard undergraduate mathematics course. This paper presents an overview of the design and implementation of an inquiry-based teaching material for the linear algebra course which emphasizes discovery learning, analytical thinking and individual creativity. The inquiry-based teaching material is designed to fit the needs of a…

The purpose of this study was to validate the development and proposal of what the authors call STEMs (Standards Tests to Evaluate Mastery) and have defined them as calibrated classroom assessments that increase student motivation and provide authentic evaluation of student learning. Theoretical and empirical research on classroom assessment and…