ISUME3

• Travel
• Accommodation

Lara AlcockWatching students construct proofs

While at Rutgers University in the USA I collected video data of students attempting proofs of the kind one might see in an introductory course on proof-based mathematics. I have now turned some of this into subtitled video material that also shows what the students are writing in real time. This seminar will use this material, with a focus on two different students working on a task about upper bounds. It will provide an opportunity to observe their proof attempts, to discuss their interactions with definitions and examples, and to identify their reasoning strengths as well as their weaknesses. It will also provide an opportunity to reflect upon our expectations of our own students and upon what kinds of skills they need to develop in order to become effective at constructing and understanding mathematical proofs.

Elena NardiProof by Mathematical Induction: conveying a sense of the domino effect (or: don't worry, you are not assuming what you are supposed to be proving...)

In a series of several recently completed studies I and Paola Iannone have engaged mathematicians from six universities in the UK in focused group interviews in which we invited them to discuss the teaching and learning of mathematics at the undergraduate level. Each interview focused on a student learning theme seen in the research literature as seminal and discussion of the theme in each interview was initiated by a Dataset. Datasets consisted of: a short literature review and bibliography; samples of student data (e.g.: students' written work, interview transcripts, observation protocols); and, a short list of issues to consider. In Amongst Mathematicians: the teaching and learning of mathematics at university level (due by Springer in Spring 2007) I present the discussion in these interviews in the slightly unconventional format of a fictional, yet strictly data-grounded, dialogue between two characters: M, a mathematician, and RME, a researcher in mathematics education. In this session I would like to discuss a sample of the book. I have chosen an Episode that focuses on the difficulties students face in their encounter with Proof by Mathematical Induction and on certain pedagogical practices that their lecturers may wish to consider towards facilitating this encounter. In the session, following a short introduction of the studies and the book, ISUME3 participants will be invited to consider and discuss: a mathematical problem that involves Proof by Mathematical Induction; one or two student responses to the problem; pedagogical issues emerging from these responses; and, excerpts of the dialogue between M and RME concerning these issues.

Tom Carroll and Kieran Mulchrone Mathematics e-learning initiative at UCC: progress report

We will describe a web-based project to tackle knowledge and skills deficits of first year university students. The project, funded by the President's Fund at UCC, uses a mix of Web-Mathematica and internet-based resources. The project is currently in the start-up phase and the presentation will therefore take the form of a progress report.

Thérèse Dooley Children's reasoning about the infinite divisibility of fractions in the context of the 'racetrack problem'

From the time man began to consider the world in which he lived, questions about infinity surfaced. These questions centred on time and space, e.g., Did the world always exist? Would it go on forever? What happened if one kept travelling in a particular direction? Was the universe finite? Zeno's paradoxes concern the difficulties and contradictions inherent in infinity. Of chief interest to this paper is his paradox of motion known as the racetrack or dichotomy paradox. What is suggested in this paradox is that one can never reach the end of a racetrack for in order to do so, one has first to reach the half-way mark, then the halfway mark of the remaining half, then the halfway mark of the next part and so on ad infinitum. To him, the equivalence between a finite process (the journey from the beginning to the end of the track) and the infinite process (1/2 + 1/4 + 1/8 + ...) seemed impossible.

Research shows that children in senior primary grades have a well developed understanding of the infinity of natural numbers and of the density of rational numbers on an interval of the number line. In other research it has been found that student judgement of infinite divisibility of physical entities such as matter and space precedes judgement of the infinite divisibility of number. In this presentation, an account is given of the way in which a group of pupils, aged 10-11 years, reasoned about the "racetrack problem". Most of these children, although unsure about accepted fraction terminology, seemed to be able to deal with the continuous division of fractions. They did not seem to think of time and motion as infinitely divisible and their understanding of the infinite divisibility of length was less well developed than that of the infinite divisibility of number. In follow-up written work, it was apparent that some children had given consideration to the inherent paradox of the problem.

Preliminary findings of this research indicate that discussion about infinity can facilitate the development of children's mathematical thinking and serve as an entry point to many of the topics that appear on the primary curriculum, e.g., ordering of fractions, fraction symbolism, division of fractions, decimals and the subdivisions of measurement. More importantly, it may give children a sense of the enthrallment and wonder that draw mathematicians to their subject.

Patricia EatonBridging the Gap - Transition from School to University Mathematics

Research shows that students often struggle with making the transition from school mathematics to university mathematics both in terms of content and style of teaching. This paper investigates the views and opinions of 1st Year undergraduates studying Calculus and Analysis, their perceptions of this transition and their insights into why there appears to be a "learning gap". It then looks at the efficacy of an online self-help style program to ease this transition and examines how students made use of this program throughout their first semester.

Sharon FlynnSupporting Mathematics for Computing Students

The last number of years has seen a fall in the entry standards of students entering degree programmes in Computing, and our students are struggling. The study of mathematics plays a foundational role in the education of a Computing student, and yet it is seen as an imposition by many. This presentation will consider a number of issues including: what maths is appropriate for Computing; who should be teaching it, and when; what support is available. The material is based on 11 years experience of teaching undergraduate Information Technology students, and 4 months in the Centre for Excellence in Learning and Teaching at NUI Galway.

Kevin JenningsOne to One Tutorials in a Virtual Interactive Classroom - some comments

With a view to providing some form of drop-in maths support service for evening students who cannot avail of the daily opening hours, the UCD Mathematics Support Centre ran a series of pilot tutorials on a group of volunteer learners from the Access to Science maths programme. We used commercial web-conferencing software for this trial. I will speak about the software, its advantages and limitations in facilitating an effective tutorial in maths, the volunteer students and, most importantly, their perceptions and comments on the software. I will present some sensible tips for best practice and open a discussion on future development of suitable software and potential applications of current software for teaching/communicating mathematics effectively.

Maria MeehanDefine, State and Prove: assessing student learning in an introductory advanced mathematics course

Those among us who have taught introductory courses in advanced mathematical topics such as Analysis or Abstract Algebra, are most likely familiar with the experience of grading end-of-term exam papers and not being able to shake the feeling that some students have learned material off by heart in order to regurgitate it in response to the inevitable "State and Prove" questions. What have these students actually learned? Should we be assessing these courses differently? If so, what impact will this have on our teaching? [I should mention that answers to these questions will most likely not be given in this talk!]

Much has been written in the research literature in the last 20 years on the difficulties students face in making the transition to courses in advanced mathematics. In this talk, I will discuss initiatives that have been adapted from the literature and introduced in a second year introductory Analysis course in UCD with the following aims:

• To encourage students to develop a conceptual understanding of the main concepts in Analysis, and intuitively understand the relationships between them;
• To encourage students to formally describe the main concepts in Analysis, and rigorously explain the relationships between them.
The initial impact these initiatives have had on the assessment of the course will be discussed.

Brien NolanUsing Case Studies in a Maths Tutor Training Programme

Abstract: We describe the development and use of training materials for a mathematics tutor training programme at DCU. This involved the preparation of Case Studies for Maths Tutor Training, following the model pioneered in mathematics tutor education by Solomon Friedberg of Boston College. We discuss how the Case Studies element was implemented in the general Tutor Training Programme in DCU, and the feedback from trainee tutors on this and other aspects of the training programme both before and after their first weeks of classroom practice.

[HTML]