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Research and development in the teaching and learning of advanced mathematical topics

A301 and A302 Rooms


Organizing team composition

Aims and scope

Provisional programme

Brief CVs of panelists

Call for papers

Papers and discussion documents

Team chairs:

Joanna Mamona-Downs (Greece)

University of Patras

RIO Campus 265 00 Patras

Ted Eisenberg (Israel)

Ben-Gurion University

Department of Mathematics PO Box 653 Beer-Sheva 84015

Team members:

Johann Engelbrecht (South Africa)

University of Pretoria

Dept Mathematics and Applied Mathematics Pretoria 0002

Asuman Oktaç (Mexico)


Departamento de Matematica Educativa

Ian Stevenson (United Kingdom)

King's College London

Department of Education and Professional Studies Room 1/11 Waterloo Bridge Wing Franklin-Wilkins Building Waterloo Road London

Aims and scope

Educational aspects covering students’ interaction with tertiary level mathematics are vast in number. At this level, the amount of mathematical content and the complexity of concepts spirals; techniques, lines of thinking and mental processes have to be developed either to cope with specific mathematical theory or to generate kinds of argumentation that transverses theories. Also, many general educational frameworks, models or perspectives retain significance at this level, further adding to the plurality of educational angles that can be taken. Despite (or perhaps due) to this diversity, we (the organizing team) have decided not to specialize the range of the material the Group will consider. Our stance is that any researcher working on educational issues on the ‘doing’ of University mathematics should be conversant with the whole gamut of the field, beyond particular personal interests.

In this light, we believe that the success of the group’s output will not depend on the output’s cohesion, but rather how it brings out ‘ground – breaking’ material on sundry fronts. To promote this, we plan to include a panel of experienced and open-minded researchers with the agenda to identify issues brought up by recent educational research literature that they deem to be particularly fruitful, and a main speaker to suggest directions that research should undertake in the future. The inclusive manner of the Group’s scope is reflected in the Call for Papers stated below, but we encourage authors to include components in their exposition reflecting relatively novel ideas or approaches.

Lastly, we take a slight bias towards educational output that refers to mathematical significance as well as psychological and cognitive aspects. This is partly motivated by the appearance of the term ‘Advanced Mathematical Topics’ in the title of the Group’s subject (in contrast to the more usual term ‘Advanced Mathematical Thinking’), and by our wish that the body of participants will be a mixture of educators and mathematicians with educational interests.

Provisional programme

Monday 7 July 2008 : 13:00 – 14:00

|13:00 – 13:05| Ted Eisenberg and Joanna Mamona|Opening|

|13:05 – 13:15| Johann Engelbrecht|Adding structure to the transition process to advanced mathematical activity|

|13:15 – 13:30| Pierre Arnoux and Alain Finkel |Using mental imagery processes for teaching mathematics|

|13:30 – 13:45| Anesa Hosein et al|Mathematical thinking of undergraduate using three types of software|

|13:45 – 14:00| Deonarain Brijlall and Aneshkumar Maharaj|Applying APOS theory as a theoretical framework for collaborative learning|

Wednesday 9 July 2008 : 12:30 – 14:00

|Panel discussion|

Ed Dubinsky, Derek Holton, Boris Koichu, Jean-Baptiste Lagrange|

Friday 11 July 2008 : 12:30 – 13:30

|12:30 – 12:45| Annie Selden et al|The interrelation of affect and reasoning in the proving process|

|12:45 – 13:00| Jean-Ph. Drouhard|Epistemography: How to know what students know and are supposed to know|

|13:00 – 13:15| Leigh Wood|Graduate capabilities in mathematics: putting high level technical skills into context|

|13:15 – 13:30| Jae Hee Park|Roles of teacher’s revoicing in inquiry oriented mathematics class: The case of undergraduate differential equations|

Saturday 12 July 2008 : 12:00 – 13:30

|12:00 – 12:15|Martine de Vleeschouwer|Contribution to the study of duality in Linear Algebra|

|12:15 – 12:30|Sepideh Stewart and Michael Thomas|Linear Algebra thinking: Embodied, symbolic and formal|

|12:30 – 12:45| Analia Berge|Students’ perceptions of the property of completeness of the set of real numbers|

|12:45 – 13:00|Elena Nardi et al|Beyond the ‘formalistic nonsense’: The impact of symbolisation and previously held images on students’ sense -making of formal definitions|

|13:00 – 13:15| Oleksiy Yevdokimov|Advanced problem solving activities: pro and contra in students’ terra incognita|

|13:15 – 13:30| Ted Eisenberg and Joanna Mamona|Conclusion|

Brief CVs of panelists

Ed Dubinsky

Current professional interests

Extension of Piagetian and related theories of learning to the acquisition of abstract concepts in mathematics.

Effect of computer experiences in learning abstract mathematics.

Cooperative learning in post secondary mathematics education.

Development of instructional materials for post-secondary mathematics courses using specific computer experiences based on the theoretical analyses and cooperative learning.

Calculus project – Development of a new approach to calculus based on research in learning and using computers, cooperative learning and alternatives to lecturing.

Algebra Project – Development of a new approach to abstract algebra based on research in learning and using computers, cooperative learning and alternatives to lecturing.

Building a community of researchers in undergraduate mathematics education.

Derek Holton 

Professor of Mathematics

University of Otago, Dunedin, New Zealand

My main interests in mathematics education are bright students and how to help them have a deeper understanding of mathematics. This has led me to look at ideas around problem solving and what mathematics is. In recent years I have become interested in matters relating to tertiary maths education and I am currently spending a great deal of time looking at recruitment and retention issues in undergraduate maths courses (with the ICME 11 Survey Team). In mathematics I have worked with permutation patterns and with cycles and matchings in graphs.

Boris Koichu

Senior Lecturer, Mathematics Education

Technion – Israel Institute of Technology

Haifa, Israel

My research interests focus on problem solving, with special reference to the issue of transfer of learning, promoting mathematical giftedness and teaching and learning of university mathematics. In relation to advanced mathematical topics, these interests are reflected in the following studies. One study with gifted high school students has explored their heuristic behaviors while looking for new theorems in spatial Euclidian geometry, difference equations and calculus. An on-going research with undergraduate students is aimed at characterizing their conceptual understanding in linear algebra. I am also engaged in a study on incorporating advanced methods of evaluation in mathematics curriculum at pre-university level.

Jean-Baptiste Lagrange

Team of Didactics of Didactics (University of Paris 7) and IUFM (University of Reims)

My involvement in research and development in the teaching and learning of advanced mathematical topics is at upper secondary level before university. In France, at this level, students have to consolidate their algebraic proficiencies in order to tackle pre-calculus.

The curriculum recommends non formal approaches of calculus concepts, but also that students should be introduced to abstraction and demonstration. It is then not easy to think of the role of algebraic techniques with regard to conceptualization. Rehearsing “rote” techniques certainly does not help, but it is important that students understand the equivalence of expressions and the benefit of algebraic transformations. They should also be able to perform basic transformations without too much difficulty in order to handle problems with inventiveness, intelligence and rigour.

My research group is developing and experimenting a Dynamic Geometry and Computer Algebra tool (Casyopée). This tool can be described as a symbolic calculator of functions and it is also designed to help students deal mathematically with problems of geometrical dependencies (for instance the area of a figure against a length of a segment). We are currently experimenting on a series of lessons with this tool at 11th grade. We prepared specific tasks for helping students to develop algebraic techniques and reflect on these. We expect to learn about the role these development and reflection can play in students’ conceptualisation of advanced notions.

Call for papers

We solicit both theoretical and empirical papers to be presented at the sessions for TSG 17. Any paper that contributes to at least one of the following mathematical educational issues will be considered:

Cognitive studies that examine students’ mental processes in the doing of mathematics (such as in proof, in problem solving and in definition making). Such studies must explain their relevance to ‘advanced thinking’.

Cognitive and pedagogical studies at the tertiary level concerning specific mathematical concepts and theories.

Studies on how an expert’s reflection on the nature of specific mathematical topics leads to the identification of some crucial thinking processes that are imperative for students to master. Such studies preferably would include suggestions about suitable channels of student support in attaining these processes.

(iv) Studies that investigate affect, attitude, and / or socio-epistemological aspects of mathematics education as they relate to Advanced Mathematical Thinking. Also studies on specific approaches towards general teaching at this level are encouraged.

(iv) Studies that examine the role that digital technologies play either in enhancing the teaching and learning of advanced mathematical topics or in how students can mediate unfamiliar mathematical issues on their own.

Researchers that are interested in presenting a paper, should send an abstract of between 500 and 1000 words. The abstracts submitted will be reviewed by the organizing team in order to select those individuals who will make presentations. The abstract must be in English. Please send your abstract by January 20, 2008 through e-mail to both e-mail addresses: and

Authors will be notified by February 15, 2008 concerning the status of acceptance. Authors whose abstracts have been accepted must complete a full paper by June 1, 2008 to be placed on the official web-site of the ICME-11. Depending on the response, it is hoped that the papers contributed may be published in a special issue of a journal or as an edited book.

Papers and discussion documents

Arnoux and Finkel (Full paper) (321.00 KB)

Berge (Full paper) (74.00 KB)

Engelbrecht (Full paper) (99.00 KB)

Hosein, Aczel, Clow and Richardson (Full paper) (112.00 KB)

Selden, McKee and Selden (Full paper) (79.00 KB)

Stewart and Thomas (Full paper) (2.00 MB)

Vleeschouwer (Full paper) (211.00 KB)

Wood (Full paper) (73.00 KB)

Yevdokimov (Full paper) (295.00 KB)

Nardi, Biza and Iannone (Full paper) (235.00 KB)

Mamona - Downs and Downs (Abstract) (29.00 KB)

Oh Nam Kwon et al (Abstract) (31.00 KB)

Holton (Panellist summary) (111.00 KB)

Koichu (Panellist summary) (20.00 KB)

Lagrange (Panellist summary) (24.00 KB)

Dubinsky (Panellist summary) (24.00 KB)

Drouhard (Full paper) (94.00 KB)

Brijlall and Maharaj (231.00 KB)