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…

Students' proof abilities were explored in the context of an inquiry-based learning (IBL) approach to teaching an introductory proofs course. IBL is a teaching method that puts the responsibility for proof on students and focuses on student discussion and exploration. Data collected from each of the 70 participants included a portfolio consisting…

Connecting and comparing across student strategies has been shown to be productive for students in elementary and secondary classrooms. We have recently been working on a project converting such practices from the K-12 level to the undergraduate classroom. In this paper, we share a particular instantiation of this practice in an abstract algebra…

In this note, I argue for and discuss my experiences with explicitly incorporating principles of critical and creative thinking in a transitions course which serves mathematics, mathematics education, and statistics majors. I describe several specific assignments and classroom tasks designed to enhance critical and creative thinking. I also…

Students often view their role as that of a replicator, rather than a creator, of mathematical arguments. We aimed to engage our students more fully in the creation process, helping them to see themselves as legitimate proof creators. In this paper, we describe an instructional activity (i.e., the "group proof activity") that is…

In undergraduate mathematics, deductive reasoning plays important roles in teaching and learning various ideas, and is primarily characterized by the concept of logical implication. This comes up whenever conditional statements are applied, i.e., one checks if a statement's hypotheses are satisfied and then makes inferences. In calculus, students…

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…

For many of my students, Real Analysis I is the first, and only, analysis course they will ever take, and these students tend to be overwhelmed by epsilon-delta proofs. To help them I reordered Real Analysis I to start with an "Analysis Boot Camp" in the first 2 weeks of class, which focuses on working with inequalities, absolute value,…

This paper offers instructional interventions designed to support undergraduate math students' understanding of two forms of representations of Calculus concepts, mathematical language and graphs. We first discuss issues in students' understanding of mathematical language and graphs related to Calculus concepts. Then, we describe tasks, which are…

As teachers of mathematics we encourage our students to ask good questions, and we strive to help our students find and understand answers to these questions. This journey can be made more meaningful if students conclude by reflecting on their learning process. If we find careful questioning and reflection important, we should include such…

This article describes a system of specifications-based grading used in an introduction to proofs course. The system was introduced to address two issues that arose in the course: how to spend less time grading and how to encourage use of feedback. We describe the implementation of the system and the results on grading and on students.

This paper describes, as an alternative to the Moore Method or a purely flipped classroom, a student-driven, textbook-supported method for teaching that allows movement through the standard course material with differing depths, but the same pace. This method, which includes a combination of board work followed by class discussion, on-demand brief…

In this paper I discuss my experience in using the inverted classroom structure to teach a proof-based, upper level Advanced Calculus course. The structure of the inverted classroom model allows students to begin learning the new mathematics prior to the class meeting. By front-loading learning of new concepts, students can use valuable class time…

Calculating Laplace transforms from the definition often requires tedious integrations. This paper provides an integration-free technique for calculating Laplace transforms of many familiar functions. It also shows how the technique can be applied to probability theory.

How can we teach inquiry? In this paper, I offer practical techniques for teaching inquiry effectively using activities built from routine textbook exercises with minimal advanced preparation, including rephrasing exercises as questions, creating activities that inspire students to make conjectures, and asking for counterexamples to reasonable,…