The ways teachers understand mathematics that is useful for teaching, known as mathematical knowledge for teaching (MKT), has been a center of attention for the research and policy communities. A teacher's MKT has been shown to be a significant factor in predicting student outcomes and positively related to the quality of mathematics instructional…

We review the history and previous literature on radical equations and present the rigorous solution theory for radical equations of depth 2, continuing a previous study of radical equations of depth 1. Radical equations of depth 2 are equations where the unknown variable appears under at least one square root and where two steps are needed to…

In a recent submission to "The Physics Teacher," we related how trigonometric identities can be used to find the extremes of several functions in order to solve some standard physics problems that would usually be considered to require calculus. In this work, the functions to be examined are polynomials, which suggests the utilization of…

Mathematical models that are constructed through modeling activities should be appropriate for the situation at hand. In this study, we seek to monitor the modeling routes of different learners as well as their modeling sub-competencies in order to learn how these are related to the semiotic characteristics of the resulting mathematical models.…

George Spencer-Brown's Laws of Forms was first published in 1969. In the fifty years since its publication, it has influenced mathematicians, scientists, philosophers, and sociologists. Its influence on mathematics education has been negligible. In this paper, I present a brief introduction to the theory and its philosophical underpinnings. And I…

The purpose of this research is to classify the mathematical modelling problems produced by pre-service mathematics teachers in terms of the number of variables and to determine the mathematical modelling skills and mathematical skills used in solving the problems in each class. The current study is a qualitative research and the data was analyzed…

We describe how we supported middle-school students developing meanings for quadratic growth by bringing together three theoretical framings and designing a task sequence grounded in that framework. Specifically, this framework was assembled from the research on students' quantitative and covariational reasoning, Realistic Mathematics Education,…

Reflection on the theme of the paper makes one to remember Sir Snow's 1959 lecture on The Two Cultures in which Snow had advanced the thesis that the sciences and the humanities had become split into two cultures and had argued that this division had become a major handicap to solving the world's problems. Mathematics education itself has always…

This study focused on investigating the ability of 58 pre-service mathematics teachers (PSMTs) to construct-evaluate-refine mathematical conjectures and proofs. The PSMTs enrolled in a three-credit mathematics education course that offered various opportunities to engage with mathematical activities including constructing-evaluating-refining…

The purpose of this study was to examine "pre-service science teachers'" (PSTs) development of multiple levels of representation and written arguments using an immersive approach to argument, the "science writing heuristic," in general chemistry laboratory I and II courses. Fifty PSTs participated the study, 20 experiments were…

In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants' approaches to identifying the nth term were…

Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students' perceptions of the general binary operation via a two-phase study consisting of task-based…

This paper demonstrates how the transcendental number "e" may be arrived at by observing the discharge of a capacitor through a fixed resistor and then modelling the system using a simple step-wise procedure. The experimental phase makes use of the Arduino microcontroller, while simple modelling of the system is carried out by means of…

The aim of the study was to determine the arithmetic and algebraic problem-solving approaches of prospective mathematics teachers in the Department of Elementary Mathematics Education, and mathematics teachers in secondary schools. The study adopted mixed methods research design using exploratory sequential pattern. The participants of the study…

Few high school students associate mathematics with playfulness. In this paper, we offer a series of lessons focused on the underlying algebraic structures of the Rubik's Cube. The Rubik's Cube offers students an interesting space to enjoy the playful side of mathematics, while appreciating mathematics otherwise lost in routine experiences.