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…

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.

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and…

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…

Integer addition and subtraction entail significant conceptual shifts for many students, and teachers often offer rules intended to help mitigate this challenge. When these rules are not connected to sense making, a misremembered rule can function as a barrier to student learning. In this article, the authors highlight the work of an expert…

In this paper, we consider mathematical competitions for pre-university students, such as the "International Mathematical Olympiad" (IMO) and many national and regional Olympiads following a similar model. The problems proposed in these contests must be solvable by 'elementary' methods (i.e., without using calculus) and belong…

The algebraic group Spin[subscript 3 × 3] arises from spinning collections of the numbers 1-9 on a 3×3 game board. The authors have been using this group, as well as a corresponding online application, to introduce undergraduate students to core concepts in group theory. We discuss the benefits of using this deceptively simple, toy-like puzzle in…

In an undergraduate analysis course taught by one of the authors, three prompts are regularly given: (i) What do we know? (ii) What do we need to show? (iii) Let's draw a picture. We focus on the third prompt and its role in helping students develop their confidence in learning how to construct proofs. Specific examples of visual models and their…

In this paper, we focus on two particularly problematic concepts in teaching mathematics: the complex unit i and angles. These concepts are naturally linked via De Moivre's theorem but are independently misused in numerous contexts. We present definitions, notations, and ways of speaking about these terms from mathematics education that are not…

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

Solving problems involving absolute value is one of the hardest topics for students learning in elementary algebra course. This is an important topic in student's mathematics life since absolute value functions are important example of a function which is continuous on the real line but not differentiable at the origin. A deep understanding of…

In this article we outline the role evidence and argument plays in the construction of a framing theory for Proof Based Teaching of basic operations on natural numbers and integers, which uses tiles to physically represent numbers. We adopt Mariotti's characterization of a mathematical theorem as a triple of statement, proof and theory, and…

This chapter considers psychological and neuroscience research on how people understand the integers, and how educators can foster this understanding. The core proposal is that new, abstract mathematical concepts are built upon known, concrete mathematical concepts. For the integers, the relevant foundation is the natural numbers, which are…

In this article, a mathematics teacher educator presents an activity designed to pique the interest of prospective secondary mathematics teachers who may doubt the value of learning abstract algebra for their chosen profession. Herein, he contemplates: what "is" intended by the widespread requirement that high school mathematics teachers…