Curriculum development in university mathematics: where mathematicians and education collide

This snapshot looks at educational aspects of the design of curricula in mathematics. In particular, we examine choices textbook authors have made when introducing the concept of the completness of the real numbers. Can significant choices really be made? Do these choices have an effect on how people learn, and, if so, can we understand what they are

Mathematical question spaces

It is uncontroversial to assert that learning mathematics is only effective when it is an active process on the part of the learner. Setting questions is a ubiquitous technique to engage students, and answering such questions constitutes a large proportion of the activity they undertake. Indeed, asking students questions is a central part of all theories of learning. This paper examines in detail the process of randomly generating versions of mathematical questions for CAA. In doing this we examine not only a single mathematical question, but how such questions are linked together into coherent structured schemes. Two important pragmatic reasons are often cited by colleagues for wishing to generate a random sequence of questions. • Randomly generated questions may reduce plagiarism • Distinct but equivalent questions may be used for practice Even if giving each student a distinct problem sequence reduces plagiarism, professional experience unfortunately demonstrates it is not eliminated. However, some students are well aware of the potential benefits of collaborative learning, possibilities for which are traditionally hard to provide in the mathematics classroom. As one student commented in their feedback evaluations: "The questions are of the same style and want the same things but they are subtly different which means you can talk to a friend about a certain question but they cannot do it for you. You have to work it all out for yourself which is good." Notice here the student voices the opinion that the questions "want the same things but they are subtly different". In this paper we address exactly this issue, by examining equivalent mathematical problems in some detail

Towards a shared research agenda for computer-aided assessment of university mathematics

In this article we describe our plan to develop a shared research agenda for computer-aided assessment of university mathematics, drawing on input from the community of mathematics education researchers and university teachers interested in this topic. Such an agenda will help to establish a programme of research aligned with practical concerns, which would contribute to both theoretical and practical development. As well as describing the process that we will follow, we provide three illustrative examples of use-inspired research questions that have arisen in our own teaching of university mathematics

Computer algebra based assessment of mathematics online

In this paper, we investigate computer algebra based assessment of mathematics online, with a focus on undergraduate students. Introducing a computer algebra system to assist in marking allows a paradigm shift from teacher-provided answers to student-provided answers. The Computer Algebra Based Learning and Evaluation (CABLE) system is presented as an open source infrastructure for mathematical learning objects. Features of the CABLE system, including the modular design, database structure, learning object specification and learning object contextualisation are described and areas for future work are identified

Systems of controlled functional differential equations and adaptive tracking

An adaptive servomechanism is developed in the context of the problem of approximate or practical tracking (with prescribed asymptotic accuracy), by the system output, of any admissible reference signal (absolutely continuous and bounded with essentially bounded derivative) for every member of a class of controlled dynamical systems modelled by functional differential equations

Tracking with prescribed transient behaviour

Universal tracking control is investigated in the context of a class S of M-input, M-output dynamical systems modelled by functional differential equations. The class of systems encompasses a wide variety of nonlinear and infnite-dimensional systems and contains - as a prototype subclass - all finite-dimensional linear single-input single-output minimum-phase systems with positive high-frequency gain.

Comparing expert and learner mathematical language: A corpus linguistics approach

Corpus linguists attempt to understand language by statistically analyzing large collections of text, known as corpora. We describe the creation of three corpora designed to enable the study of expert and learner mathematical language. Our corpora were formed by collecting and processing three different genres of mathematical texts: mathematical research papers, undergraduate-level textbooks, and undergraduate dissertations. We pay particular attention to the method by which our corpora were created, and present a mechanism by which LaTeX source files can be easily converted to a form suitable for use with corpus analysis software packages. We then compare these three different types of mathematical texts by analyzing their word frequency distributions. We find that undergraduate students write in remarkably similar ways to textbook authors, but that research papers are substantially different. These differences are discussed