New developments and trends in tertiary mathematics education: or, more of the same?

Abstract

This paper examines four developments, or trends, in tertiary mathematics education today: (1) technology as an engine driving pedagogical change; (2) the often difficult transition for students from secondary to tertiary mathematics; (3) the responsibilities of mathematicians for the mathematical preparation of future teachers, and (4) the potential impact on teaching of research into the teaching and learning of tertiary mathematics.

Notes

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A shorter version of this paper was presented at the 2004 ICME-10 Topic Study Group 3 (TSG3): New Developments and Trends in Mathematics Education at Tertiary Level.

Engelbrecht and Harding's intent was for designers of online course to use their system to evaluate the content and coverage of their own courses (Engelbrecht, personal communication), but the question remains.

The full CUPM Curriculum Guide 2004 can be found at www.maa.org/cupm/. The 22-page Executive Summary can be found at: http://www.maa.org/cupm/summary.pdf. The Curriculum Guide itself was a project four years in the making; it began by holding a series of workshops with ‘client disciplines’ and soliciting discussion papers.

For further information, see http://www.s4s.org/understanding.php and http://www.achieve. org/achieve.nsf/AmericanDiplomaProject?openform.

One credit traditionally represents attendance at one scheduled 45-minute period of instruction every day throughout a semester.

For details, see http://www.cnn.com/2003/EDUCATION/09/08/early.graduation.ap/ or http://www.sptimes.com/2003/09/16/Opinion/Fast_track_to_nowhere.shtml.

See http://dailybeacon.utk.edu/article.php/12946.

For example, the University of Saskatchewan hosts The Scholarship of Teaching and Learning website that discusses the legacy of Ernest Boyer, the person who is credited with introducing the ideas of the scholarship of discovery, the scholarship of integration, the scholarship of application, and the scholarship of teaching. See http://www.usask.ca/tlc/sotl/sotl_boyer.html.

Information on PMET can be obtained from the project's home page at: http://www.maa.org/pmet/.

The classic proof that

is irrational is a ‘generic proof ’ since it immediately generalizes to the proof that

is irrational for any prime p.

In addition to espousing this view, the Lesh and Doerr book [25] takes a very broad view of modelling. As well as discussing modelling and model-eliciting activities for students, there are chapters on modelling the learner, modelling teacher development, teachers‘ models of teaching/learning, etc. Similarly, Leikin [27] has models of teachers’ interactions with their mentors, allowing construction of teacher profiles, and models of teachers' flexibility in real classroom situations.

The University of North Carolina at Chapel Hill in the US, through its School of Education, will offer an M.Ed. in (K-8) Mathematics Education. For a description of their existing M.Ed. in (K-5) Mathematics Education, with such content courses as Revisiting Real Numbers, Topics in Geometry, and Topics in Algebra and Algebraic Reasoning, see http://www.unc.edu/depts/ed/med_exp/K-5mathematics.html.

For details, see http://www.wmich.edu/cpmp/.

For further information, see the MAA-hosted SIGMAA on RUME website RUMEonline! at: http://www.rume.org/. It contains information about the organization, as well as a literature database, links to publishers, and other resources.

The ‘Moore Method’ developed out of the teaching practices of a single accomplished US mathematician, R. L. Moore, and has been continued by his students (several of whom went on to become presidents of the American Mathematical Society or the Mathematical Association of America) and their mathematical descendents. It has been remarkably successful in producing research mathematicians, but has also been used in undergraduate mathematics classes. In many versions, students are given definitions and statements of theorems or conjectures and asked to prove them or provide counterexamples. The teacher provides the structuring of the material and critiques the students' efforts, but does not lecture. While individual teachers have described their experiences [66], up to now this method has not been subjected to in-depth analysis. For more information, see the Legacy of R. L. Moore website at: http://www.discovery.utexas.edu/rlm/.

See, for example, the Research Sampler columns on MAA Online at http://www.maa.org/t_and_l/sampler/research_sampler.html. In addition, SIGMAA on RUME is currently putting together an MAA Notes volume, tentatively called Making the Connection: Research and Teaching in Undergraduate Mathematics, that will gather together a variety of expository articles on RUME.

For some thoughts on the future directions of tertiary mathematics education research, see Selden and Selden [67].