The seed for this special issue was planted when the co-editors were invited to be the co-chairs of the Topic Study Group (TSG) on “New trends and developments in tertiary mathematics education” at the 10th International Congress for Mathematics Education (ICME-10) in July 2004. The work of the TSG was completed with considerable help from the members of the TSG organizing team – Meira Hockman, Karen King and Alexei Sossinsky – along with the many people who made written and verbal contributions. The articles in this special issue have been developed from selected presentations at the TSG at ICME-10.
This special issue starts with the paper by Annie Selden, who puts forward a provocative title for her paper: “New Developments and Trends? Or, More of the Same?” Selden identifies a variety of topics of interest in undergraduate mathematics education, points to a subset of these topics that is addressed in this collection and foreshadows many of the subsequent articles, situating them in a broader context of tertiary mathematics education. This volume addresses several overlapping themes related to the theory and practice of mathematics education: issues of transition as related to both curriculum and pedagogy, teacher education, research in undergraduate mathematics education, technology and its influence of curriculum, pedagogy and research. And we note that many papers have common ideas and threads.
The issue of challenges in transition to undergraduate and advanced mathematics is of significant concern to the mathematics education community. Lovric points to a trend that school graduates are less prepared in dealing with university level mathematics using the result of the recent imposed change in the province of Ontario, Canada. Luk provides a personal account of transition, describing the challenges he faced both as a student and as teacher of undergraduate mathematics in Hong Kong. Hockman presents a concern of “watering down” courses in order to comply with the need to accommodate a larger amount of students and lack of support from administration in South Africa.
These papers raise a universal concern – the concern of deterioration. This goes hand in hand with what Selden refers as “two contradictory trends”: the advocacy for school graduates who are better prepared mathematically for both university and the work place, and the seeming desire by legislatures and administrative bodies to reduce levels of certification.
Teacher Education is another important focus of this issue. After all, a significant part of tertiary mathematics education is the education of future teachers of mathematics. In relation to teacher education, the considered issues pertain to curriculum, pedagogy and research. Martinez Luaces presents a case for the use of modelling in the curriculum for mathematics teachers and provides several examples of modelling activities that seemed to have a positive impact on the participants who engaged in these activities. The papers of Zazkis and Liljedahl present reports on research conducted with a population of pre-service teachers. Zazkis investigated the ways in which students perceive irrational and prime numbers and pointed to common features of these two sets considering how these numbers can or cannot be represented in a standard algebraic notation. Liljedahl's investigated the impact of successful mathematical discovery – referred to as an Aha! experience – on the beliefs and attitudes of pre-service elementary teachers. Leikin's research considers both undergraduate and graduate mathematics education students and acknowledged the similarities in the interaction between a teacher and a student and between a mentor (supervisor of student-teachers) and a mentee. Leikin presents an example of connection between theory and practice of mathematics education, specifically, how a model of interaction developed by analyzing the work of in-service teachers can be used with preservice teachers in order to raise the quality of their discourse about teaching.
There is an ongoing effort to develop curriculum and pedagogy to address better the needs of all students as well as of specific populations of students. As a snapshot of this effort, Safuanov presents a view on pedagogical development implementing a “genetic” approach to teaching, that is, pedagogy that recognizes historical, logical and socio-cultural development of the subject matter. Further, acknowledging that “understanding” is the ultimate goal of teaching “mathematics”, Kannemeyer provides an innovative instrument attempting to identify what such understanding entails in the context of a Calculus course.
Technology is the theme that intertwines with all the areas of mathematics education. As a snapshot from research that investigates the influence of technology on learning, Gurevich, Gorev and Barabash discuss the impact of the usage of various computerized tools on students’ achievement in plane geometry and in analytic geometry.
In the closing remarks we wanted to look into the future. However, rather than foretell the future – a task that would be impossible considering the rapid changes in technology – Holton in his concluding paper “Tertiary mathematics education for 2024” presents a wish for the future. This wish includes emphasis on the “creative” side of mathematics, rather than on its “created” side, that is, emphasis on the activity of doing mathematics rather than the focus on the artifacts of such activity of others. Technology appears to be one of the means to this end. The paper further includes an emphasis on research in mathematics education that will help understand better the learning process and in turn influence pedagogy. “That mathematics is seen to be something that is to be enjoyed and not feared” – is one of the aspirations put forward by Holton. We hope it is shared by the readers.