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…

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…

We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor's use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor's simultaneous use of various forms of embodiment…

In this article, the authors describe an algebra 2 class activity, called the Exponent Maze, that had students practice many mathematics problems, display their creativity, and work together collaboratively by creating their own mazes. The authors discuss how they introduced the activity, what the students created, and how creating mazes can be…

The notion of function is central in all of the secondary curriculum, and indeed functional models appear in almost all higher education that is based on mathematics. However, in secondary education, functions usually appear in restricted and somewhat sterile forms. In this (mostly theoretical) paper, we present a proposal -- exemplified by a…

In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for 2×2 matrices. Suggested for…

In the third of a series, the author describes how symmetrical quintic functions can be expressed in three equivalent formats--like other symmetrical functions. Quadratic functions have a line of symmetry through their vertex and cubic functions have 1800 rotational symmetry around their point of inflection. However, polynomial functions of degree…

Making sense of algebraic expressions in context is multifaceted and extends beyond representing real-world situations using algebra. As part of Project MathTalk, the authors created videos that feature a pair of Grade 9 algebra 1 students coming to understand algebraic expressions in context. The students found three aspects of making sense of…

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some…

Human-driven carbon emissions have resulted in increased levels of dissolved carbon dioxide in the Earth's oceans. This dissolved carbon dioxide reacts with water to form carbonic acid, which impacts ocean acidity as well as the solubility of carbonate-containing compounds, with far-reaching impacts on marine ecosystems and the human communities…

In this article, the author will demonstrate how gestures--often used in combination with offloading information on scratch paper--can be used to help students with difficulties in mathematics succeed with secondary level algebra. The authors will provide key examples from three studies of the algebra learning of students with mathematics…

Seeds of Algebraic Thinking comes from the Knowledge in Pieces (KiP) perspective of learning. KiP is a systems approach to learning that stems from the constructivist idea that people learn by building on prior knowledge. As people experience the world, they acquire small, sub-conceptual knowledge elements. When people engage in a particular…

Our aim is to contribute to the body of research on learning trajectories (LTs) in mathematics by making visible a process for articulating a hypothetical learning trajectory implicit in a widely adopted, reform-based, middle-grades mathematics curriculum. In doing so, we highlight considerations, decisions, and challenges we faced as part of this…

To a student in introductory physics, using vectors is at best an exercise in bookkeeping. A two-dimensional kinematics problem effectively doubles the number of equations that a student must know, and invites the student to memorize factoids: "The horizontal motion is constant," "Gravity is only in the y-direction," etc. Force…

Basis vectors e[subscript alpha] play a useful role in special and general relativity. In particular they allow an expansion of the vectorial spacetime interval dl along infinitesimal curvilinear coordinate differences dl = e[subscript alpha]d[Xi][superscript alpha]: (thus the definition e[subscript alpha] = dl/d[Xi][superscript alpha]). In this…