In this paper, we briefly introduce three theoretical frameworks for mathematical identity and why they matter to practitioners teaching undergraduate mathematics courses. These frameworks are narrative identities, communities of practice, and figured worlds. After briefly describing each theory, we provide examples of how each framework can be…

This article makes a case for introducing moving averages into introductory statistics courses and contemporary modeling/data-based courses in college algebra and precalculus. The authors examine a variety of aspects of moving averages and draw parallels between them and similar topics in calculus, differential equations, and linear algebra. The…

We present our analysis of 254 Calculus I final exams from U.S. colleges and universities to identify features of assessment items that necessitate qualitatively distinct ways of understanding and reasoning. We explore salient features of exemplary tasks from our data set to reveal distinctions between exam items made apparent by our analytical…

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…

We offer an analysis of calculus assessment items that highlights ways to evaluate students' application of important meanings and support their engagement in generative ways of reasoning. Our central aim is to identify characteristics of items that require students to apply their understanding of key ideas. We coordinate this analysis of…

Historically, math education at the high school and introductory college levels has focused on computational skills. With the advancement of computational technologies, problemsolving and other higher-order thinking skills should become the focal point. This article discusses ways in which the University of Minnesota has integrated problem-solving…

In calculus, related rates problems are some of the most difficult for students to master. This is due, in part, to the nature of the problems, which require constructing a nuanced mental model and a solid understanding of the function. Many textbooks present a procedure for their solution that is unlike how experts approach the problem and elide…

Several publishing companies now offer competing brands of online homework (OHW) software. Students rarely have a voice in choosing their required course materials, even though, ultimately, they will be asked to use and pay for these systems. In this study, Calculus I students each tested three different OHW systems. The students were then asked…

Storytelling has proved to be an effective way of passing on information from one generation to the next. Whether the information relates to history, culture, health, or morality, the story provides a framework so that complex ideas can be better understood and retained. In this article, we consider the role of narrative in the design and…

How should our students think about external forcing in differential equations setting, and how can we help them gain intuition? To address this question, we share a variety of problems and projects that explore the dynamics of the undamped forced spring-mass system. We provide a sequence of discovery-based exercises that foster physical and…

We describe the use of student-generated video content to assess students' engagement with, and understanding of, problem-solving skills. In this framework, students are tasked with using technology to create videos that show them working through, and explaining solutions to, challenging calculus exercises. The videos are then posted online,…

The Corvette Problem is a nonroutine investigation, one that provides an authentic context to explore many interrelated calculus ideas. In our years of sharing this problem with students and colleagues, we have found additional topics from the calculus curriculum to have rich interpretations in this environment. The Corvette Problem contextualizes…

In this paper, we will discuss a type of activity that lets the mathematics classroom become a place where students can learn more than mathematical content. Specifically, students can learn interdisciplinary skills in the area of communication (both verbal and non-verbal) and debate skills that are often taught in communication or English…

This paper uses the lens of a calculus student to examine different solutions to a weekly puzzler from the radio show "Car Talk," hosted by Tom and Ray Magliozzi. The puzzler describes an automobile that is traveling 75 miles per hour and is 75 miles from its destination. The trip is completed by traveling 1 mile at 75 miles per hour, 1…

Research has shown that the types of tasks assigned to students affect their learning. Various authors have described desirable features of mathematical tasks or of the activity they initiate. Others have suggested task taxonomies that might be used in classifying mathematical tasks. Drawing on this literature, we propose a set of task types that…