Undergraduate Analysis Course Research Articles

Demystifying Proofs Through Structured Interaction: A Case Study of One Instructor’s Teaching in an Undergraduate Analysis Course

The concept of proof is central to every undergraduate mathematics curriculum, but it is also a difficult concept for university students to understand and for university instructors to teach. Prior studies have contributed useful research knowledge about the nature and sources of undergraduate students’ difficulties with proof, but they have paid less attention to the development of effective instructional practices to enhance student learning of proof. As a step towards this direction, this paper makes a contribution to the limited body of case studies of promising or potentially effective teaching practices in undergraduate mathematics by reporting a case study of an instructor’s teaching, locally characterized as “effective,” in an undergraduate analysis course at a leading Indian university. The instructor did not deviate from the so called “Definition-Theorem-Proof” (DTP) format that is followed in most proof-oriented university mathematics courses, but her teaching presented a set of features that, collectively, form a rather innovative teaching practice at the undergraduate level. Specifically, our case study shows that, even in the context of a rather crowded university classroom, proof can be demystified for students through structured interaction between the instructor and the students, that is, an interactive, conversational style of proof instruction invoking the participation of students. This is based on a solid foundation in symbolic logic at the very outset and on a significant time investment in definitions being explained in depth using informal language, visual aids, and real-life analogies. In addition to contributing to existing images of potentially effective teaching practices in undergraduate mathematics, this paper draws attention to the Indian educational context that has had little representation in international forums of mathematics education research thus far.

Let’s Draw a Picture

In an undergraduate analysis course taught by one of the authors, three prompts are regularly given: (i) What do we know? (ii) What do we need to show? (iii) Let’s draw a picture. We focus on the third prompt and its role in helping students develop their confidence in learning how to construct proofs. Specific examples of visual models and their impact on student work are presented.

Comparison of some tests for convergence of positive series

In this survey, aimed at professors and students of undergraduate analysis courses, two hierarchies of tests are considered. The first is originated from the ratio test and the second from the root test. The test construction techniques are exposed and relations between the tests in each family are discussed. The examples of convergent and divergent series clarify the range of application for each of the introduced tests and situations when they are not conclusive. The behaviour of the partial sums of these series is illustrated geometrically and the level of complexity of each series is evaluated in terms of the rate of its convergence or divergence.

Applications of full covers in real analysis

In this paper we briefly introduce the reader to the concept of full covers and indicate how it can be used to prove theorems in an undergraduate analysis course. The technique exposes the student to the idea of covering an intervalTa; bU with a collection of sets and then extracting from this collection a subcollection that partitionsTa; bU. As a consequence, the student is furnished with a unifying thread that ties together and simplifies the proofs of many theorems.

The aberrancy of plane curves

The aberrancy of a plane curve is a property of the curve that is invariant under both translation and rotation. It provides a numerical measure for the non-circularity of the curve at each point of the curve. (Recall that curvature gives an invariant numerical measure of nonlinearity.) The concept of aberrancy has been around for two centuries, but it has received very little attention. We hope to stir a little interest in the concept by presenting four different derivations of the formula for aberrancy. This may appear to be redundant, but there are some good reasons for doing so. First of all, the only derivation for aberrancy in the literature is a bit confusing and makes a few unjustified assumptions. Secondly, the derivations we present are all significantly different from each other and involve some interesting ideas in elementary real analysis. Finally, although the basic idea behind each derivation is simple, the details can become extremely messy unless a proper path is chosen. We encourage the reader to find one or two approaches of particular interest and use the concept of aberrancy as an extended problem set in an undergraduate analysis course or even a further mathematics class in an English school.

Infinite processes in elementary mathematics How much should we tell the children?

Mathematics at all levels depends on infinite processes. This is a “fact of life”. But, like those other “facts of life”, this one has traditionally been hushed up—especially when teaching elementary mathematics to beginners. Of course, in our liberated age, adult mathematicians are no longer embarrassed by infinite processes, and even discuss them openly. But it is not easy to see the connection between the way they discuss infinite processes (as in, say, an undergraduate analysis course) and those parts of elementary mathematics which depend on infinite processes, and which should provide the soil in which the beginner’s seedling intuitions about infinite processes can take root. So even today most children, adolescents, and their teachers must either make do with the traditional smutty versions, or work things out for themselves.