We provide a gentle alert that the standard definitions of basic algebraic structures--such as that of a group--contain aspects that may be questionable to students, and discuss what instructors could do to minimise the questionability.

We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier…

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…

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…

This paper introduces the basic knowledge of integral factors of first-order ordinary differential equations and Lie symmetry analysis. It then discusses the principle of constructing an integral factor of the first-order ordinary differential equation by the Lie symmetric method. Finally, it presents some examples to show the process of…

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…

Inverse and injective functions are topics in most college algebra courses. Yet, current materials and course structures may not afford students' conceptual understanding of these important ideas. We describe how students' work with digital activities, "techtivities," linking two different looking graphs that represent relationships…

Four variations of the Koch curve are presented. In each case, the similarity dimension, area bounded by the fractal and its initiator, and volume of revolution about the initiator are calculated. A range of classroom exercises are proved to allow students to investigate the fractals further.

Optimization is introduced to teaching in secondary school, focussing on functions of one variable with the first derivative criterion for critical points and monotonicity of functions. The second derivative criterion is also used to determine the nature of the critical points. Similarly, in university-level teaching, algebraic work using criteria…

The National Council of Teachers of Mathematics (NCTM) promotes creating resources that build procedural fluency from conceptual understanding through intentionally sequenced tasks that draw on students' prior knowledge and move from simple, concrete representations to more complex and abstract representations (Boston et al., 2017). Liljedahl…

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…

Any quadratic function has a line of symmetry going through its vertex; any cubic function has 1800 rotational symmetry around its point of inflection. However, polynomial functions of degree greater than three can be both symmetrical and asymmetrical (Goehle & Kobayashi, 2013). This work considers algebraic conversions of symmetrical quartic…

Incorporating open-ended real-world tasks enhances students' access to mathematical and real-life knowledge and maximizes learning experiences. This article examines how a secondary mathematics teacher used an open-ended problem on distributing pay raises to teach mathematical concepts and fairness to students. The teacher's actions exemplify…

Problems involving the composition of mixtures are common in chemical practice and are thus part of introductory Chemistry courses at the early undergraduate level. However, they are often perceived by students as a rather obscure matter, which may be due to poor familiarity with algebraic manipulations. Furthermore, to increase the distaste of…

Our bodies help us learn, and our bodies may know something before we can articulate it. Embodied explorations are body-based learning experiences where learners have opportunities to make sense of and convey ideas with their bodies. These embodied explorations were taught at a suburban public high school, in the southern United States, that…