In calculus courses, instructors often use the end-of-section problems in a textbook in homework assignments or other course assessments. As a result, these problems influence the teaching and learning of calculus. In this study, we examine the levels of cognitive demand of these problems in a mainstream calculus textbook and classify them within…

In this paper, we will describe the experience of the authors in using process-oriented guided inquiry learning (POGIL) in calculus at four institutions across the USA. We will briefly examine how POGIL compares to and fits in with other kinds of inquiry-based learning approaches. In particular, we will first discuss the unique structure of a…

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…

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,…

We delineate some types of structured practice (modeling, requests, feedback, and space-making) that help students learn to pose appropriate questions and to initiate exploration of those questions. Developing skills requires practice, so we suggest ways to embed structured practice into existing class sessions. Including structured practice is…

As STEM educators, we know it is beneficial to train students to think critically and mathematically during their early mathematical lives. To this end, the author teaches the College Algebra/Precalculus course in a flipped classroom version of an inquiry-based learning style. However, the techniques described in this paper can be applied to a…

Using technology to enhance the classroom environment can have a tremendous impact on student learning, as well as on an instructor's teaching. This paper describes one instructor's transition from traditional chalkboard lectures to a fully technological presentation of content. After carefully reviewing the literature, clicker technology was…

Our particular flavor of inquiry-based learning (IBL) uses mathematical discourse, conversations, and discussions to empower students to deepen their mathematical thinking, building on strengths of students in the humanities. We present an organized catalog of powerful questions, discussion prompts, and talk moves that can help faculty facilitate…

Inquiry-based learning is a topic of growing interest in the mathematical community. Much of the focus has been on using these methods in calculus and higher-level classes. This article describes the design and implementation of a set of inquiry-based learning activities in a Math for Liberal Arts course at a small, private, Catholic college.…

Through a series of six guided classroom discoveries, students create, via targeted questions, a definition for deciding when two sets have the same cardinality. The program begins by developing basic facts about cardinalities of finite sets. Extending two of these facts to infinite sets yields two statements on comparing infinite cardinalities…

A significant issue mathematics instructors face is how to cover all the material. Mathematics teachers of all levels have some external and internal pressures to "get through" all the required material. The authors define "the coverage issue" to be the set of difficulties that arise in attempting to cover a lengthy list of topics. Principal among…

In my Calculus classes I encourage my students to actively reflect on course material, to work collaboratively, and to generate diverse solutions to questions. To facilitate this I use peer instruction (PI), a structured questioning process, and i-clickers, a radio frequency classroom response system enabling students to vote anonymously. This…

Major changes in mathematics pedagogy include writing as pedagogy and the role of community in learning. The classroom community is naturally extended by the use of online discussion boards. In this article several models for student use of online discussion boards that have been successfully used to promote mathematical discourse are presented.…