Students often instrumentally use variables and unknowns without considering the variational thinking behind them. Using parameters to modify the coefficients or unknowns in equations or systems of linear equations (without altering their structure) involves consciously incorporating variational thinking into problem-solving. We will test the…

This paper analyses learners' common errors in simplifying algebraic problems. 102 Grade 10 learners from three rural schools in South Africa participated in the study. Following a quantitative approach, content analysis of learners' responses to algebraic tests revealed that while learners commit several errors in algebraic problems, encoding and…

While many aspects of the teaching and learning of specific advanced mathematics courses have been studied, limited work has examined mathematical themes like sameness or its instantiations across disciplines. In this paper, we explore algebraists' collective example space for mathematical sameness. We used qualitative methods to analyze survey…

This work gathers in one place what is pertinent about the connections between metric coefficients, basis- and unit vectors in a four-dimensional relativistic manifold. Some of this material can be found scattered elsewhere; its collection into one place reveals connections that either are not known or are obscure, for example that the metric…

On 24 June 1994 at Fairchild Air Force Base, during practice for an air show, a low-flying B-52H aircraft banked its wings vertically and crashed. Emphasizing the activity of modeling drag and gravity, these notes examine the possibility of recovery with several models. First, with algebra, historical data lead to a model where in a free fall near…

In this article, the authors share their favorite "Construct It!" activity, which focuses on rate of change and functions. The initial approach to instruction was procedural in nature and focused on making use of formulas. Specifically, after modeling how to find the slope of the line given two points and use it to solve for the…

The Miller-Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography.

QR decomposition is widely used for solving the least squares problem. However, existing materials about it may be too abstract for non-mathematicians, especially STEM students, and/or require serious background in linear algebra. The paper describes theoretical background and examples of GNU Octave compatible MATLAB scripts that give relatively…

This article aims to problematize the role of programming in mathematics education. Problematizing involves acknowledging a deeper complexity in the understanding of programming than originally perceived and questioning its assumed value in mathematics education. This approach entails identifying its underlying paradigmatic assumptions in relation…

This study analyzes the mathematical knowledge to teach early algebra exhibited by pre-service early childhood education teachers, from the perspective of the Mathematical Knowledge for Teaching (MKT) model. The research adopts a mixed exploratory-descriptive methodological approach, based on the application of the MKT-early algebra questionnaire…

Developing both conceptual and procedural knowledge is important for students' mathematics competence. This study examined whether U.S. Grade 9 general education mathematics teachers' self-reported use of concept-focused instruction (CFI) and procedure-focused instruction (PFI) were associated differently with ninth graders' algebra achievement…

This study used the Motivated Strategies for Learning Questionnaire (MSLQ) to assess the motivation and learning strategies of students in a self-paced developmental mathematics class. A total of 194 undergraduate students participated in this study. Descriptive statistics, correlation coefficients, and regression analysis were conducted to assess…

Background: Many students face difficulties with algebra. At the same time, it has been observed that fraction understanding predicts achievements in algebra; hence, gaining a better understanding of how algebra understanding builds on fraction understanding is an important goal for research and educational practice. Objectives: However, a wide…

The ability to solve equations and inequalities is necessary for success in algebra. However, reasoning biases and misconceptions may create barriers for students to build knowledge of algebraic symbols and their values. This study investigated whether students' errors when solving equations and inequalities could be attributed to their tendency…

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new…