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…

Calculus instructors struggle to teach infinite series, and students have difficulty understanding series and related concepts. Four instructional strategies, prominently used during the calculus reform movement, were implemented during a 3-week unit on infinite series in one class of second-semester calculus students. A description of each…

This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking.…

Instructors are constantly struggling to help students understand mathematical concepts as well as the relevance of mathematics to the real world. In calculus, students possess misconceptions of the limit concept. "Pushing the Limit" refers to a semester-long calculus class project that required students to read about, interview calculus…

This article is a reflection on an elementary pre-service teacher's intuitive idea offered as a mistaken solution strategy for counting matchsticks. It shows the difficulties in responding to flawed lines of intuitive reasoning arising in the classroom setting. A number of learning environments for teacher professional development are suggested…

I discuss my experience with teaching an advanced undergraduate Real Analysis class using both lecturing and the small-group guided discovery method. The article is structured as follows. The first section is about the organizational and administrative components of the class. In the second section I give examples of successes and difficulties…

Many students, when they take an elementary differential equations course for the first time, bring with them misconceptions from numerical methods that they had learnt in their calculus courses, most notable of which concerns the mesh width in using a numerical method. It is important that we strive to dispel any of these misconceptions as well…

One of the most publicly celebrated figures in mathematics teaching during the last century was Jaime Escalante (dramatized in the film "Stand and Deliver"). The Escalante story aptly fulfills three prevailing myths about effective teachers: the Teacher is self-made, the Teacher is the expert, and everything depends on the Teacher. These…

This paper describes writing assignments from a modern algebra course which were used to foster the development of proof-writing skills. Each assignment consists of proofs and mathematical work supposedly written by fictional characters, where the proofs are designed to illustrate common misconceptions, difficult points, and typical writing flaws.…

This paper describes a sequence of lessons from two Calculus I classes for teaching the epsilon-delta definition of a limit. In these lessons, the author elicited students' misconceptions and perceptions of this definition through a reading/writing lesson and then used these student ideas to design a lesson aimed at addressing these misconceptions…

We present a design for a junior level numerical methods course that focuses on a series of five open-ended projects in applied mathematics. These projects were deliberately designed to present many of the ambiguities and complexities that appear any time we use mathematics in the real world, and so they offered the students a variety of possible…