We present a mathematical analysis that distinguishes two quantitative perspectives on ratios and proportional relationships: Multiple Batches and Variable Parts. We argue that (a) existing research on proportional relationships has addressed Multiple Batches but has largely overlooked Variable Parts, (b) Multiple Batches makes the co-variation…

In this study, I examined how 34 Spanish-speaking English language learners (ELLs) used their linguistic resources (English and Spanish) and language modes (oral and written language) to demonstrate their knowledge of proportional reasoning in a dual language mathematics assessment task. The assessment allows students to see the item in both…

One of the challenges of understanding the complexity of so-called reform mathematics instruction lies in the observational tools used to capture it. This article introduces a unique tool, drawing from commognitive theory, for describing classroom discussions. The Realization Tree Assessment tool provides an image of a classroom discussion,…

Algebra 1 is a gatekeeper course with profound implications for students' academic and professional success. This study examines the implications of teaching Algebra 1 through a standardized, fully online, asynchronous course developed by one of the largest online course vendors in the United States in alignment with Common Core State Standards…

The traditional technique for converting repeating decimals to common fractions can be found in nearly every algebra textbook that has been published, as well as in many precalculus texts. However, students generally encounter repeating decimal numerals earlier than high school when they study rational numbers in prealgebra classes. Therefore, how…

The current research around the Fibonacci's and Lucas' sequence evidences the scientific vigor of both mathematical models that continue to inspire and provide numerous specializations and generalizations, especially from the sixties. One of the current of research and investigations around the Generalized Sequence of Lucas, involves it's…

In this paper, we examine the interactions of mathematical abilities when 6 high achieving Swedish upper-secondary students attempt unfamiliar non-routine mathematical problems. Analyses indicated a repeating cycle in which students typically exploited abilities relating to the ways they orientated themselves with respect to a problem, recalled…

This paper provides a set of five indicators of CAS use, which provide a means to analyse a student's written solution to determine whether they used CAS for solving a given problem. These indicators provide a way for researchers and teachers to determine CAS use without requiring classroom observation or the explicit recording of extra…

The use of digital games as an instructional tool has garnered increasing attention in the education community. Empirical work supported by theory on the learning affordances of digital games allowed the game-based learning community to arrive at the consensus that digital games provide an excellent medium for the acquisition of skills and the…

Across community colleges in the United States, most students place into a developmental math course that they never pass. This can leave them without the math skills necessary to make informed decisions in major areas of social life and the college credential required for participation in growing sectors of our economy. One strategy for improving…

Elementary school students often exhibit a variety of conceptions associated with algebraic thinking that their teachers fail to recognize or understand. It is crucial that elementary school teachers possess knowledge of the variety of student conceptions and also have abilities to address varying states of conceptions. Otherwise, students who are…

The purpose of this research study was to understand how mathematics majors make sense of similar matrices, a linear algebra topic that is rich in internal and external connections, in a series of in-depth qualitative interviews in a technology-assisted (MATLAB) learning environment. The study was also an attempt to develop a cognitive theory in…

We present a short-term class project used in an introductory linear algebra course, designed to engage students in matrix algebra. In this activity, students responded to a survey of their pop culture tastes. Using the survey responses, they worked to design a series of matching algorithms, using matrices, with the goal of matching the students…

Rich mathematical modeling activities are crucial to giving students agency and making mathematics meaningful. Proportional reasoning and transitional algebraic reasoning are the primary topics in the prealgebra curriculum, so a need exists for meaningful modeling activities using proportional reasoning in addition to geometric modeling. In…

This dissertation study aims to understand the instructional interactions that foster students' generalizing. By analyzing video recorded lessons from 13 different Grade 3 classrooms in which an early algebra intervention is being implemented, activities related to generalizing and students' generalizing activities were identified, and the nature…