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Organizers |
Revisiting Felix Klein's "Elementary Mathematics from an Advanced Standpoint"
by
Bill Barton
The University of Auckland
A century ago, in 1908, Felix Klein's lectures on mathematics for secondary teachers were first published (in German). This comprehensive view of the field challenged both teachers and mathematicians to consider, from a perspective sensitive to both mathematical rigour and pedagogical practice, the relationship between mathematics as a school subject, and mathematics as a scientific discipline. The intervening 100 years have witnessed many changes in mathematics the crises in Foundations, the advent of computing, emergence of new fields, and resolutions of some major mathematical challenges. These, as well as changes in the economic environment, have provoked major change and challenges for school mathematics. While Klein's writing remains a valuable source insight, it seems timely to revisit this terrain by linking the topics and approaches of senior secondary or undergraduate mathematics with the field of mathematics. This is an important challenge for both mathematicians and mathematics educators.
This presentation will put up for discussion the idea of a writing project, possibly a joint project between IMU and ICMI, to revisit this work in a contemporary context.
Date received: November 1, 2007
Readiness for first-year mathematics studies: Management, placement and prognosis
by
Patricia Cretchley
University of Southern Queensland
There is a growing need for more careful placement of students in first-year university mathematics studies in Australia, and perhaps elsewhere. Widening access to tertiary education brings us increasing numbers of students for whom school mathematics grades are not indicative of preparedness. And online enrolment distances them from early academic counseling. As a result, many enter under-prepared and soon find themselves in difficulty. University response is variable. Support strategies for those at risk have increased, but uptake is often disappointing. Voluntary self-diagnostic skills-testing on entry is a common stimulus, but has long been viewed as inadequate, alone. Mandatory skills-testing is routinely practiced in some universities, and on the increase. But typical entry-skills tests have proved poor predictors of success in one-semester studies. And academics report limited success with at-risk students, many needing more time than a semester allows.
In this talk, I offer findings from recent studies, raise for discussion our moral, academic and ethical responsibilities towards such students, emphasise the need for entry testing to reach higher levels of prognosis of readiness for mathematics studies, and propose strategies for doing so. In particular, I turn the spotlight on dynamic testing (``teach and test”) and attitude testing. And I present findings on data on students' self-assessment of their ability to learn mathematics.
Date received: November 12, 2007
Teaching proofs in mathematics
by
David Easdown
School of Mathematics and Statistics, University of Sydney
One of the most difficult learning thresholds for students of mathematics is the concept of proof. The difficulty manifests itself in several ways: (1) appreciating why proofs are important; (2) the tension between verification and understanding; (3) proof construction. Students entering university are often very adept at performing sophisticated algorithms and calculations. However they tend to have very little experience with mathematical proofs even though these are central to verifying mathematical facts and building a corpus of reliable knowledge. For many, proof technique is an exceedingly difficult hurdle to overcome and has all of the hallmarks of a threshold concept, in the sense of Meyer and Land (2003, 2005). The ability to understand and construct proofs is transformative, both in perceiving old ideas and making new and exciting mathematical discoveries. The most inspiring mathematical proofs are integrative and almost always expose some hidden counter-intuitive interrelations. And of course they are troublesome: it can take a long time, even years, for students to learn to appreciate proofs and to develop sufficient technique to write their own proofs with confidence. However when the moment comes, the eureka effect can be irreversible and students are well on the way to becoming maths `addicts'. This talk will introduce some ideas and issues surrounding teaching proofs and introducing proof technique in the classroom.
References:
Meyer, J.H.F. and and Land, R. (2003) Threshold concepts and troublesome knowledge: linkages to ways of thinking and practising within the discipline. In Rust, C. (ed.) Improving Student Learning: Improving Student Learning Theory and Practice – Ten Years On. Oxford: Oxford Centre for Staff and Learning Development.
Meyer, J.H.F. and and Land, R. (2005) Threshold concepts and troublesome knowledge (2): epistemological considerations and a conceptual framework for teaching and learning. Higher Education 49, 373-388.
Date received: September 7, 2007
Where have all the mathematicians gone?
by
Derek Holton
University of Otago
There is a feeling that the numbers of maths majors being graduated in the world is diminishing. I was asked to chair a Survey Team for the maths education conference ICME 11 to look into this. I will cover in my talk, the ways that we are going about collecting data and what we are finding. It turns out that nothing is clear. For instance a country may find that its numbers are decreasing but individual university maths departments are booming. We look into these cases and try to make some suggestions that might be useful for all maths departments.
Discussion will be more than welcome.
Date received: October 28, 2007
From lessons to lectures: NCEA mathematics and first year performance
by
Alex James
University of Canterbury
Coauthors: Clemency Montelle, Phillipa Williams
In 2005, students entered the University of Canterbury with the new NCEA school qualifications for the first time. We analyse the relationship between NCEA Level 3 Mathematics with Calculus qualifications of incoming students and their results in the core first-year mathematics papers at Canterbury. These findings are used to investigate the suitability of this new qualification as a preparation for tertiary mathematics and to revise and update entrance recommendations for students wishing to succeed in their first-year mathematics study.
Date received: October 25, 2007
Interactive visualization in advanced university mathematics
by
Matthias Kawski
Arizona State University
Interactive visualization tools have become routine equipment to facilitate inquiry based learning in secondary and entry-level university mathematics courses. Such tools allow the learner to actively participate in the discovery process, develop ownership, and, we argue, they can help build deep conceptual roots.
Using the Vector Field Analyzer II (VFA II), a powerful free JAVA applet, we demonstrate how such tools and approach can readily be adpated to even proof-oriented advanced undergraduate and graduate classes. Even more critical at this level is the ability to perform experiments with virtual zero start up-costs.
The VFA II was originally designed for visualizing the curl and divergence, the integral theorems of vector calculus, and to integrate vector calculus with the first course on differential equations. We will report on successfully using this tool in complex analysis and graduate level differential equations courses for topics such as Poincare-Bendixson theory, omega-limit sets, variational equations, and the stable manifold theorem.
Date received: October 31, 2007
Secondary Mathematics from an Advanced Standpoint
by
William McCallum
University Distinguished Professor of Mathematics, Institute for Mathematics and Education, Department of Mathematics, The University of Arizona
The courses in the mathematics major are often oriented towards graduate school. Prospective high school teachers need courses that are oriented towards the mathematics they will teach. High school algebra, often seen as a mechanical subject, provides rich opportunities for reasoning and interpretation. Simple problems in high school geometry can be connected to advanced research in algebraic geometry. In this talk we will consider ways in which advanced topics in mathematics can provide a deeper understanding of high school mathematics, and make recommendations for a university curriculum for prospective high school teachers.
Date received: November 11, 2007
Three Attributes of Tertiary-level Mathematical Education to One's Society and its Advancement of Science
by
G. Arthur Mihram
Princeton, NJ
Coauthors: Danielle Mihram, Center for Excellence in Teaching, University of Southern California
Tertiary-level mathematical education is as valuable for its context as for its content: It provides future citizens/leaders with mental discipline, and provides future citizens/scientists with training in the mental tool basic to the advancement of science. First, the ancient Greeks recognised that training in mathematics provides leaders with minds more likely to be more disciplined for sorting issues political, for striving for impeccably logical conclusions. Second, science (and scientific politics) is/are to search for the very explanation for (i.e., for the truth about) any particular naturally occurring phenomenon. Our mathematics is itself a language, not a science; yet, it is a, but not the only, language that a scientist might use for his/her explanation/model (e.g., C Darwin or Nobel Laureate K Lorenz). Thirdly, any advancement of human knowledge is a result of an analogy made with some knowledge which we [Mankind] had established earlier: Polya remind us that our mathematics is well-suited to educate students (future leaders/scientists) in the use of analogy-making, as per the challenge to prove a conjecture in geometry class.
Date received: October 17, 2007
Online learning resources for engineering students: Do they work?
by
Mark Nelson
School of Mathematics, University of Wollongong, Australia
Coauthors: Anne Porter, Elahe Aminifar, Richard Caladine
The basic mathematical abilities of first-year engineering students have been in steady decline over many years. To counter this electronic learning resources have been developed for a first-year service course. These learning resources consist of mathematical problems with worked solutions. The worked solutions are available either in a static format, or as a video in which a solver goes through the problem explaining their reasoning.
We compare the performance of students taking the course in 2007, when these resources were available, to those taking the course in 2004, when resources were not available. In week one of the course students take a basic skills test. Analysis of this test shows that the two cohorts had equivalent base-line skills. A comparison of the performance of the students in 2004 and 2007 shows that the new learning resources improved students outcomes over virtually all assessment tasks.
Date received: October 30, 2007
A Flexible, Extensible Online Testing System for Mathematics
by
Tim Passmore
University of Southern Queensland, Toowoomba Qld., Australia
Coauthors: Leigh Brookshaw and Harry Butler
An online testing system developed for entry-skills testing of first-year university students in algebra and calculus is described. The system combines the open-source computer algebra system MAXIMA with PHP scripts and XML configuration files to parse student answers, which are entered using standard mathematical notation and conventions. The answers can involve data structures like lists, variable-precision-floating-point or integer numbers and algebra, which allows more sophisticated testing designs than the multiple-choice, or exact-match, paradigms common in other systems. Experience using the system and ideas for further development are discussed.
Date received: August 28, 2007