This paper is a practical how-to guide to help you start using active learning or to have greater success and more fun with it. We categorize active learning techniques as Think, Pair, Share, Composite, Group, Move, or Lead and discuss how to implement activities in each category, along with advice on creating engaging, effective, and equitable…

In this paper, we discuss mathematical modeling opportunities that can be included in an introductory Differential Equations course. In particular, we focus on the development of and extensions to the single salty tank model. Typically, salty tank models are included in course materials with matter-of-fact explanations. These explanations miss the…

The Euclidean Discus Toss is an active and tactile learning activity that models the extended Euclidean algorithm with a frisbee relay. The extended Euclidean algorithm involves both iterative and recursive programming and is regularly taught throughout the mathematics and computer science curricula. The Euclidean Discus Toss invites students to…

In this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students' prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as…

In this paper, we invite the reader to think deeply about professional development opportunities for faculty. The focus of this professional development is to improve teaching and learning by including more inquiry in the mathematics classroom. Building on theories of teaching and learning, we develop the idea of "teaching procedures"…

The typical trigonometry, precalculus, or calculus student might not agree that logarithms are hot stuff, but we drew motivation from chili peppers to help students get a better taste for logarithms. The Scoville scale, which ranges from 0 to 16,000,000, has been the sole quantitative metric to measure the pungency (spiciness) of peppers since its…

Nationwide, many students fail to complete the key mathematics courses that are required for most STEM majors, including Precalculus. This paper describes the rationale, implementation, and impact of the redesign of Precalculus at one regional west coast institution. Prior to the redesign, pass rates in Precalculus were modest (75.8% of all…

With knowledge in hand that teaching mathematics through active engagement leads to better learning outcomes for students, the field of mathematics is now tasked with researching effective facilitation of professional development for faculty. Our university embarked on an initial year of active learning professional development for mathematics…

Team-Based Learning (TBL) is a cooperative learning strategy blending elements of flipped learning, inquiry-based learning, and problem-based learning. Although used quite frequently in other disciplines, use of this strategy in mathematics has been limited. In this article, we describe how TBL can be implemented in math courses with adherence to…

Faculty professional development is an important lever for change in supporting instructors to adopt research-based instructional strategies that engage students intellectually, foster learning-supportive attitudes and habits of mind, and strengthen their persistence in mathematics. Yet the literature contains few well-rationalized models for…

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 introduces lessons designed to incorporate applications to teaching high school mathematics in an undergraduate Discrete Mathematics course. Because many prospective high school teachers do not take courses that are specifically designed for teachers, providing materials aimed at teacher preparation that can be easily integrated into…

Over the past 9 years, we have implemented, developed, and refined a novel student-centered teaching method called Extreme Apprenticeship in university mathematics context. In this paper, we report the theoretical orientation behind the method, as well as its practical implementation. We also reflect on our experiences gained from using it in our…

The Mathematics Department at Cornell University has recently secured a grant from the University to implement systemic change in how we teach courses that reach students at critical transition points in their mathematical development. In this article, we report on the changes made to our large multi-section first-semester calculus course in order…

We describe a method that instructors can use to grade tests and assignments that can be used in follow-up inquiry-based activities in class to review material. The grading process is essentially a sorting of student assignment/test responses based on the instructor's conception of what makes responses acceptable. The instructor can assign a grade…