Selected Papers from CAME 4

Computer Algebra, Instrumentation and the Anthropological Approach

John Monaghan

University of Leeds, United Kingdom
J.D.Monaghan@education.leeds.ac.uk

This article considers research and scholarship on the use of computer algebra in mathematics education following the instrumentation and the anthropological approaches.  It outlines what these approaches are, positions them with regard to other approaches, examines tensions between the two approaches and makes suggestions for how work in this area can be extended.

 

Reflections on John Monaghan’s “Computer Algebra, Instrumentation, and the Anthropological Approach”

Glen Blume

Department of Curriculum and Instruction, The Pennsylvania State University, USA
gblume@psu.edu

Reactions to John Monaghan’s “Computer Algebra, Instrumentation and the Anthropological Approach” focus on a variety of issues related to the ergonomic approach (instrumentation) and anthropological approach to mathematical activity and practice.  These include uses of the term technique; several possibilities for integration of the two approaches; the role of the teacher, the influence of students’ actions on the teacher’s actions, and the co-construction of socio-mathematical norms in the two approaches; the extent to which affect can be accounted for in these approaches; and the applicability of the French CAS research in other settings.  Several scenarios concerning integration of the approaches are offered, several views of the role of the teacher in the two approaches are presented, and a case is made for potential emphasis on affect in both of the approaches.

 

New Technology? New Ways of Teaching – No Time Left for That!

Bärbel Barzel

Department of Mathematics, Educational Universität Freiburg, Germany
barzel@ph-freiburg. de

The title describes two challenges with which mathematics teachers nowadays have to deal concerning their classroom-arrangements: new teaching methods and integrating computers.  Many teachers are afraid that when realising both trends, curricular prescriptions can become even more difficult to achieve.  Changes in the classroom-arrangements and integrating the technology are perceived by some teachers as time-consuming and an additional burden.  In contrast to this, other teachers perceive those trends not as an impediment, but as a special opportunity to achieve their aims in terms of contents and processes.  This study investigated the question of whether the combination of both trends is an impediment or an opportunity in the frame of a research project at the University of Duisburg-Essen.  As the first step, teaching material was developed which would serve teachers as an example of a long-term-sequence relating to a compulsory topic with a combined focus on integrating CAS in an open learning-arrangement.  For this purpose, material for self-regulated learning to investigate polynomial functions integrating CAS in an open classroom-arrangement has been developed.  As the second step; this material was evaluated.  The central question of the research was to investigate to what extent this learning arrangement is suitable for simultaneously pursuing aims in both content and process.

According to the multi-faceted arrangement, a complementary research design involving the collection of both qualitative and quantitative data was chosen.  The qualitative part is an interpretive study based on video tapes.  The quantitative part is an experimental large-scale study.  The material was used in 45 classes (approximately 1200 students) from different schools in order to check whether general conclusions can be drawn.  The large-scale study also includes a post-survey and a comparative post-test. To understand the aims of the project it is necessary to grasp the ideas of the material.  Therefore, section 1 points out the main ideas of the material, section 2 explains the focus of the research project; and in section 3, early results that are organised according to teachers’ learning are presented.

 

Didactic Time, Epistemic Gain and Consistent Tool: Taking Care of Teachers’ Needs for Classroom Use of CAS; A Reaction to Barzel’s  “New Technology? New Ways of Teaching – No Time Left for That!”

Jean-Baptiste Lagrange

I.U.F.M. Reims and Didirem Université Paris VII, France
jb.lagrange@reims.iufm.fr

The relationship between teachers and Computer Algebra Systems is generally problematic.  The extensive capabilities of CAS provide opportunities for learning but also bring a new complexity that makes it difficult for teachers to take advantage of these opportunities.  Barzel’s paper contrasts with this observation: in a “Lernwerkstatt” (translated: learning workshop) project teachers were brought to accept changes in their classroom activities and management that accompanied the introduction of CAS.  The project actually took into consideration the teachers’ need to control the “didactical time” and their “praxeological” needs.  It seems that it contributed much to making CAS acceptable to teachers.  Another feature contributing to this success is the decision to limit the Lernwerkstatt project’s mathematical area to polynomials.  In other areas of mathematics, standard CAS are very problematic with regard to secondary education curricula, creating difficulties for the teacher.  That is why this reaction also introduces a discussion about the design of CAS tools for classroom use.

 

The Impact of CAS on Our Understanding of Mathematics Education
Werner Peschek

Institute for  Didactics of Mathematics – Austrian Educational Competence Centre, University of Klagenfurt, Austria
werner.peschek@uni-klu.ac.at

This paper starts with a framework of general mathematics education, which could give orientations for developments and decisions referring to the curriculum at schools of higher education.  This framework is focussing on the ability to communicate with experts and demands to reduce the expectations with regard to operative knowledge and skills and to increase the expectations with regard to basic conceptual knowledge and reflections.  At first sight, this framework seems to be rather less suitable for a sensible use of CAS in mathematics classrooms.  It will be shown, that the opposite is true: CAS can be seen as a mathematical expert and therefore assume a very important role in this framework of general mathematics education.  Finally, in this framework, the experts can be seen and used as “black boxes,” so the problem of using “black boxes” will also be discussed.

 

Interpreting and Assessing the Answers Given by the CAS Expert:  A Reaction Paper
Carolyn Kieran

Département de Mathématiques, Université du Québec à Montréal, Montréal, Canada
kieran.carolyn@uqam.ca

In this reaction to the paper presented by Werner Peschek, the author takes issue with the notion put forward that, in general mathematics education, the field of competence for operative knowledge and skills ought to be assigned primarily to the experts and, consequently, that this competence could be delegated almost completely to the electronic mathematical expert that is the CAS.  Citing research evidence from current work, the author argues that it is quite unrealistic to expect students to be able to interpret and assess the answers produced by the CAS if their general mathematical education does not include provision for developing operative knowledge and skills.  Furthermore, she argues that students themselves are not satisfied at not being able to do such interpreting.

 

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