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Titles and Abstracts
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Lara Alcock
Watching 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.
Feedback.
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Elena Nardi
Proof 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.
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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.
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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.
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Patricia Eaton
Bridging 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.
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Sharon Flynn
Supporting 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.
Link:headrush.typepad.com/creating_passionate_users/2006/11/why_does_engine.html
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Kevin Jennings
One 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.
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Maria Meehan
Define, 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.
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Brien Nolan
Using 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.
Link: www.dcu.ie/~nolanb/casestudies.htm
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Ray Ryan and Sean Dineen
To Infinity and Beyond: Supervising PhD Students
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