The authors describe their approach to teaching a course on finite fields and combinatorial applications, including block designs and error-correcting codes, using a hybrid of lectures and active learning. Under the discussed classroom model, there are two lecture days and two discovery-based discussion days each week. Discussions center around…

Through galleries of graphs and short filmstrips, this paper aims to sharpen students' eyes for visually recognizing continuous functions. It seeks to develop intuition for what visual features of a graph continuity does and does not allow for. We have found that even students who can work correctly with rigorous definitions may not be able to…

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 paper, I analyze the impact of culture and metaphor on cryptology education. I will compare and contrast the historically grounded metaphors of cryptology-is-warfare and encryption-is-security to a set of counter-metaphors: cryptology-is-privacy and encryption-is-communication. Using this explicit understanding of conceptual metaphor, I…

Mathematics education faculty working in mathematics departments are commonly involved in teaching undergraduate mathematics courses, teaching and supervising postgraduate students of mathematics education, and developing mathematics teachers (pre-service and/or in-service). This paper proposes a new elective course that can be taken by either…

In the precalculus curriculum, logistic growth generally appears in either a discrete or continuous setting. These actually feature distinct versions of logistic growth, and textbooks rarely provide exposure to both. In this paper, we show how each approach can be improved by incorporating an aspect of the other, based on a little known synthesis…

In this note, we describe a large-scale transition to an active learning format in first-semester calculus at the University of Florida. Student performance and attitudes are compared across traditional lecture and flipped sections.

This paper presents ideas for constructing mathematics homework assignments based on lessons from the cognitive science of learning and memory. In particular, we focus on two popular techniques from the field: spaced practice and mixed practice. The paper describes the techniques and supporting research, and then discusses various straightforward…

Preservice teachers commonly struggle with topics related to fractions, ratio, and proportion. The use of double number lines can facilitate the teaching in these areas and provide autonomy and flexibility in mathematical thinking. Moreover, double number lines support many topics in the curriculum, including unit rate, convenience scaling, and…

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…

We outline the structure of a term-long graduate course on teaching undergraduate mathematics. We analyze reflections written by the graduate students participating in the course to understand how they experienced the course and how the course impacted their beliefs about teaching. We find that students enjoy the course and find it to be useful.…

We discuss an innovative course model used in first-year calculus courses at a large North American research university, which we refer to as the "Blended Model." We argue that our model takes the best features of lectures, flipped classrooms, and traditional recitations, and gathers them into a cohesive, robust structure that benefits…

According to the MAA Instructional Practices Guide, deep learning requires student engagement with content both inside and outside the classroom. Hence, mathematics instructors must work to engage students in the learning experience. Higher education research often equates engaged learning and active learning, but what does engaged learning mean…

This paper describes a course designed to introduce students to mathematical thinking and a variety of lower level mathematics topics using baseball while satisfying the goals of quantitative reasoning. We give suggestions for sources, topics, techniques, and examples so any mathematics teacher can design such a course to fit their needs. The…

Experienced provers employ a host of skills when assessing the validity of a justification, often without names for those skills. This paper offers an introduction to a lens called Toulmin analysis that can help make sense of this process. Then this paper describes both an in-class module to help students learn to apply Toulmin analysis and…