The study of mathematics is often described as the science of patterns (Resnik 1981), and the cognitive tasks involving the search, recognition, and assessment of patterns are fundamental to learning mathematics. Patterning activities lead students to a better understanding of dependent relations among representations, create a transparent way for…

This paper describes the conceptions about complex numbers that a group of university students has, these were built from the application of an activity sequence centered on these numbers. This sequence is based on the APOS theory, some aspects of semiotic representation theory, and the use of digital technology. Particularly, both the general…

The purpose of this article is to present and discuss two recommended sequences of learning the areas of polygons, starting from the area of a rectangle. Exploring the algebraic derivations of the two sequences reveals that both are valid teaching progressions for introducing the area formula for various polygons. Further, it is suggested that…

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, to…

Introducing new concepts to learners in an order of increasing complexity appears to be beneficial for learning, but typically introduction of concepts does not always adhere to this principle. We examined whether introducing new algebra concepts in a contrasted manner or in an order of increasing complexity instead of a different more typical…

In this study I explore the construction of algebraic structure sense through engaging students in task sequences designed to elicit the development of competencies related to this understanding. The primary task-design principle employed in this study is strategic variation, in which certain components of items within task sequences vary while…

This article examines the process of shifting to a "Physics First" sequence in science course offerings in three school districts in the United States. This curricular sequence reverses the more common U.S. high school sequence of biology/chemistry/physics, and has gained substantial support in the physics education community over the…

The Tennessee Department of Education explored course enrollment patterns in an effort to better understand in which courses students are enrolling and whether course enrollment policies and procedures are promoting students' interests. This report focuses on eighth grade Algebra I enrollment, which can propel students to take more rigorous math…

Students benefit from learning multiple procedures for solving the same or related problems. However, past research on comparison instruction has focused on comparing multiple formal procedures. This study investigated whether the benefits of comparing procedures extend to comparisons that involve informal and formal procedures. We also examined…

This review investigates effective interventions for teaching algebra to students with learning disabilities and evaluates the complexity and alignment of skills with the Common Core State Standards in math. The review includes the results of 10 experimental and 5 single-subject designs (N = 15) producing a moderate overall effect size (g = 0.48).…

As an alternative to looking solely at linear functions, a three-lesson learning progression developed for Year 6 students that incorporates triangular numbers to develop children's algebraic thinking is described and evaluated.

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…

The technology principle in the "Principles and Standards for School Mathematics" (NCTM, 2000) states that technology plays an important role in how teachers teach mathematics and in how students learn mathematics. The purpose of this sequential explanatory mixed methods study was to examine the impact of interactive applets on students'…

Competence in algebra is linked to access to higher education, employment in better-paying jobs, and, increasingly, the ability to earn a high school diploma. For many students with learning disabilities, developing proficiency in algebra represents a challenging, but necessary goal. Teachers of students with learning disabilities need access to…

Structuring the algebra course to provide a link between a student's existing knowledge and the new topic being presented is discussed. Developing relationships among topics is suggested through examples describing a mathematical problem and effective teaching approaches. Stress is placed on teachers reflecting on mistakes pupils are likely to…