European Research in Mathematics Education I.I + I.II
Proceedings of the First Conference
of the
European Society for Research in Mathematics Education, Vol. I + II

Editor: Inge Schwank, 1999
Publishing House: Forschungsinstitut fuer Mathematikdidaktik, Osnabrueck.

 

Internet-Versions	Vol. I:	ISBN 3-925386-50-5 pdf-file, 2.706 kB, V1.01
	Vol. II:	ISBN 3-925386-51-3 pdf-file, 1.321 kB, V1.0
Paper-Versions	Vol. I:	ISBN: 3-925386-53-X how to order
	Vol. II:	ISBN: 3-925386-54-8 how to order
Overview including direct link to complete single contributions in pdf-format	Vol. I:	Table of contents I html-file Abstracts I html-file
	Vol. II:	Table of contents II html-file Abstracts II html-file
Contents - Sketch	Vol. I:	Contents, Preface, Introduction, Keynote Addresses, Group 1 "Nature of Mathematics", Group 2 "New Technologies", Group 3 "Teaching Practices", Group 4 "Social Interactions", Keywords, Author Index
	Vol. II:	Contents, Group 5 "Cognition", Group 6 "School Algebra", Group 7 "Methodologies", Keywords, Author Index

Abstracts Vol. II
Group 5, Group 6, Group 7

MATHEMATICAL THINKING AND LEARNING AS COGNITIVE PROCESSES
Inge Schwank, Emanuila Gelfman, Elena Nardi

The work was prepared in such a way that - after the reviewing process - all accepted papers were distributed to the prospected group members. In the spirit of CERME the group leaders decided that during the sessions the accepted papers would not be orally presented one by one. For the purpose of the stimulation of a goal-orientated, in depth discussion six general themes had been identified and two members of the group were asked to give a general introduction referring to the papers fitting each theme and to current research developments. The themes and the introduction presenters are as follows:

1. The Nature of Cognitive Structures - Introduction and Overview
   Emanuila Gelfman & Inge Schwank
   
   
2. Individual Styles of Cognition
   Sara Hershkovitz & Marina Kholodnaya
   
   
3. Cognition and Emotion/Motivation
   Elena Nardi & Rosetta Zan
   
   
4. Cognition and Language Pierre
   Luigi Ferrari & Pearla Nesher
   
   
5. Cognition and New Technologies
   Tatiana Oleinik & Elmar Cohors-Fresenborg
   
   
6. Mathematical Thinking in Modern Conditions – Situated Cognition
   Jarmila Novotna & Elena Nardi

(pdf-file, 84 skB)

COOPERATIVE PRINCIPLES AND LINGUISTIC OBSTACLES IN ADVANCED MATHEMATICS LEARNING
Pier Luigi Ferrari

This paper is concerned with the role of language in advanced mathematical thinking. It is argued that some behaviors may arise from the application to mathematical language of some conventions of ordinary language. Grice’s Cooperative Principle (CP) is introduced in order to explain some episodes that are not easily accounted for in terms of semantics only. Some examples of (undue) application of CP to mathematical language are given. It is argued that the application of CP to mathematical language in problem solving is closely linked to the poor use of mathematical knowledge and, more generally, to the attitudes and behaviors that Vinner (1997) names ‘pseudo-analytical’.

(pdf-file, 77 kB)

THE ROLE OF WAYS OF INFORMATION CODING IN STUDENTS' INTELLECTUAL DEVELOPMENT
Emanuila Gelfman, Marina Kholodnaya

Ways of information coding are subjective means with the help of which the surrounding world is reproduced in individual experience. Mastering mathematical concepts presupposes solution of two didactic tasks: firstly, including in the process of teaching three ways of information coding: verbal, visual and sensual - sensory with consideration of certain requirements to introduction of each of them; secondly, organisation of self-transference in the system of these three ways of information coding. Such activity in the process of work with concepts should lead to success of individual intellectual behaviour.

(pdf-file, 108 kB)

ON TEACHING STUDENTS TO WORK WITH TEXT BOOKS
Emanuila Gelfman, Raissa Lozinskaia, Galina Remezova

Ability of using different ways of work with a text-book is one of the most important intellectual characteristics of student’s personality. It may be formed by a teacher and by a student himself by means of specially designed tasks.

(pdf-file, 46 kB)

PROJECTS AND MATHEMATICAL PUZZLES - A TOOL FOR DEVELOPMENT OF MATHEMATICAL THINKING
Marie Kubinova, Jarmila Novotna, Graham H Littler

The teaching of mathematics in the Czech Republic has traditionally been of an ‘instructive’ nature. This teaching strategy has alienated most students in mathematics because they have been expected to use the skills they have learnt without understanding the underlying concepts. They have not been able to appreciate the usefulness of mathematics nor get enjoyment from the subject. From our own teaching experience, by using projects and mathematical puzzles we have found that our students have gained the necessary understanding, enjoyed their work and developed other important attributes such as the ability to conjecture, to work systematically and to communicate. In this paper we set out the benefits of using this practical approach to the teaching of mathematics together with an analysis of the processes necessary to set up such teaching strategies in schools etc.

(pdf-file, 66 kB)

THREE-TERM COMPARISONS
Pearla Nesher, Sara Hershkovitz, Jarmila Novotna

Based on previous work by Novotna et al., 12 three-term multiple comparisons problems were analyzed and field tested with teachers (in Israel) and students (in the Czech Republic). The variables in the study were: The compared and the reference in problems, the use of the verbal expression “more than” and “less than”, and the underlying schemes. First analysis and the preliminary results are now presented here.

(pdf-file, 65 kB)

TEACHER STUDENTS' RESEARCHES ON COGNITIVE ACTIVITY WITH TECHNOLOGIE
Tatyana Oleinik

This paper represents the results of special courses given to undergraduate teacher students of «mathematics-computer science» speciality. A general idea that integrated these courses, consist in the training of future teachers for understanding the possibilities and limits of the use of technologies (systems like DERIVE and Cabri-geometre) for support of pedagogical control of learners’ cognitive activity. It is very important to develop teacher students’ abilities for their researches on why and how they should use technologies particularly like microworlds and computer algebra systems. It gave an opportunity to encourage teacher students’ interest in teaching algorithmic and half-algorithmic problems as well as heuristic problems and therefore heuristic methods of operations that intrinsic of creative activities.

(pdf-file, 57 kB)

ON PREDICATIVE VERSUS FUNCTIONAL COGNITIVE STRUCTURES
Inge Schwank

Predicative thinking is thinking in terms of relations and judgments; functional thinking is thinking in terms of available actions and achievable effects. Depending on the way of thinking the orientation in the world, the type of sources for getting insight are not the same. E.g. it should be visible in different eye movements. In addition to our qualitative experiments, recently we started to run a study based on EEG-methods while students were solving logical pattern fitting tasks. The EEG complexity during predicative thinking decreased in comparison to functional thinking and mental relaxation, with this reduction being most pronounced over the parietal and right cortex. A reduction in dimensional complexity during functional thinking which was concentrated over the left central cortex, although significant, was less clear.

(pdf-file, 199 kB)

WINNING BELIEFS IN MATHEMATICAL PROBLEM SOLVING
Rosetta Zan, Paola Poli

The aim of this study is to define the relationship between the ability in solving mathematical problems and the beliefs about mathematical problem solving. We compare the beliefs of children with or without difficulties in solving mathematical problems, by using a questionnaire. The results show that good solvers and poor solvers have significantly different concepts of a mathematical problem: some beliefs appear to be winning in that they are able to activate the correct utilization of knowledge.

(pdf-file, 51 kB)

Synthesis of the Activities of WG-6 at CERME 1
Paolo Boero

Pre-conference preparation was coordinated by C. Bergsten (Sweden), P. Boero (Italy) and J. Gascon (Spain). Ten contribution proposals were received, eight of which were accepted, distributed to all participants before the conference and discussed during the conference. Coordination of Working Group activities was ensured by C. Bergsten and P. Boero. From ten to twelve participants took part in the WG sessions; nine of them were present at all sessions. Countries represented within the WG were France, Germany, Hungary, Italy, Spain, Russia and Sweden.

(pdf-file, 37 kB)

ON THE CONSTRUCTION AND INTERPRETATION OF SYMBOLIC EXPRESSIONS
Luciana Bazzini

Recent research studies have pointed out the crucial role of constructing and interpreting letters in algebra. Many difficulties emerge because of the incapability to relate the algebraic code to the semantics of the natural language. A teaching experiment, carried out with 16 year old students, attending the second year of the Gymnasium (a humanistically oriented High School) is described here. This experiment was aimed at analysing the cognitive behaviour of the students when facing learning situations dealing with a productive use of symbols and their understanding.

(pdf-file, 66 kB)

FROM SENSE TO SYMBOL SENSE
Christer Bergsten

In this theoretical paper figurative aspects of algebraic symbolism are discussed and related to the theory of image schemata, opening up for one way of understanding the development of symbol sense in mathematics. The ideas of mathematical forms (referring to spatial characteristics of mathematical formulas), and form operations, are at the core of this analysis.

(pdf-file, 68 kB)

THE ROLE OF ALGEBRAIZATION IN THE STUDY OF A MATHEMATICAL ORGANIZATION
Pilar Bolea, Marianna Bosch, Josep Gascón

Algebra is considered here, not as a particular mathematical organization (such as arithmetic or geometry, for instance) but as a process, the process of algebraization, that can affect either a whole mathematical organization or, as is nowadays the case in secondary education, some aspects of it. Depending on the case examined, a more or less algebraized mathematical work results: the algebraization process can in fact be considered as the modelling of a whole initial mathematical work in order to deepen our knowledge of it. Algebraization is thus, in this sense, a specifically didactic activity, relative to the study of mathematics. Institutional restrictions on the algebraization process constitute a paradigmatic example of the fact that mathematics (the object of study) and didactics (the process of study) are inseparable.

(pdf-file, 67 kB)

SIMULATIN OF DRAWING MACHINES ON CABRI-II AND ITS DUAL ALGEBRAIC SYMBOLISATION: DESCARTES' MACHINE & ALGEBRAIC INEQUALITY
Veronica Hoyos, Bernard Capponi, B. Geneves

This paper focus on the connection between geometrical constructions and algebraic descriptions derived of simulate some mathematical machines in CABRI-II learning environment. We present a scenario experimented with pupils in a first degree of a high school (eleventh grade). The action’s motive is modelling a Descartes’ machine as a CABRI-II diagram. The starting hypothesis leads on the fact that such situations promote connections between geometrical properties and their dual algebraic symbolisation. The animation of CABRI-II diagrams provides the visualization of geometrical aspects, and calculus on geometrical objects provides the algebraic symbolisation. This inquire into the meanings emerged of this mathematical experience can be completed by asking the pupils how to refute (using the geometrical tools and objets experienced) a frequent statement about the usual algebraic inequality at this academic level.

(pdf-file, 77 kB)

THE INTERWEAVING OF ARITHMETIC AND ALGEBRA: SOME QUESTIONS ABOUT SYNTACTIC AND STRUCTURAL ASPECTS AND THEIR TEACHING AND LEARNING
Nicolina A. Malara, Rosa Iaderosa

After a brief introduction and a few general considerations on syntactic difficulties in the interweaving of arithmetic and algebra, we analyse the conflict between additive and multiplicative notation in arithmetic-algebraic realm and present the first results of some activities carried out according to our hypothesis of research with the aim of overcoming such conflict and promoting the semantic control of complex writings. We conclude with some reflections concerning the choice and difficulties of the didactic activities in middle school on these topics.

(pdf-file, 77 kB)

SYNTACTICAL AND SEMANTIC ASPECTS IN SOLVING EQUATIONS: A STUDY WITH 14 YEAR OLD PUPILS
Maria Reggiani

The capacity to write and to solve equations is a crucial point in the construction of algebraic thinking, since it involves both the mastery of the formal rules of algebraic language and calculation, as well as the correct interpretation of the meaning of the symbols used. Our research study is mainly concentrated on examining the possibility of improving the capacity of students to solve first grade equations during their first year of high school, by establishing a dialectical relationship between the application of properties and their semantic control. In this report we would like to present certain observations concerned with the problem exposed in relation to equations without solutions or with an infinite number of solutions.

(pdf-file, 67 kB)

FORMING ALGEBRA UNDERSTANDING IN MPI-PROJECT
Sam Rososhek

Research in how to form Algebra understanding has been done in frames of MPI-project for many years. It is based on H.Weyl-I.Shafarevich conception about Algebra as the collection of coordinatizing quantities systems. There are three main informative lines in MPI-project systematic course of Algebra-functional, algebraic structures and mathematical modelling. Two last of them are discussed here.

(pdf-file, 60 kB)

SOME TOOLS TO COMPARE STUDENTS' PERFORMANCE AND INTERPRET THEIR DIFFICULTIES IN ALGEBRAIC TASKS
Éva Szeredi, Judit Török

The aim of this paper is to show how some research tools coming from different disciplines and theoretical frameworks were used to explore students’ difficulties in a rather complex algebra task and the differences between two different groups of students, in order to produce interpretative hypotheses about emerging phenomena.

(pdf-file, 74 kB)

Research Paradigms and Methodologies and their Relationship to Questions in Mathematics Education
Milan Hejny, Christine Shiu, Juan Diaz Godino, Herman Maier

The seven papers which comprised the starting point for our discussion and are published above seemed to fall naturally into three types: two dealt with global theories of mathematics education (Rouchier, Godino & Batanero), three were empirical studies which addressed specific questions in mathematics education (Bagni, D’Amore & Maier, Stein) and the remaining two presented accounts of elaborated methodologies which have been devised and developed for specific purposes in research into mathematics education (Marí, Stehlíková).

(pdf-file, 50 kB)

THE ROLE OF THE HISTORY OF MATHEMATICS IN MATHEMATICS EDUCATION: REFLECTIONS AND EXAMPLES
Giorgio T. Bagni

The effectiveness of the use of history of mathematics in mathematics education is worthy of careful research. In this paper the introduction of the group concept to an experimental sample of students aged 16-18 years by an historical example drawn from Bombelli’s Algebra (1572) is described. A second sample of students was given a parallel introduction through a Cayley table. Both groups were asked the same test questions and their responses examined and compared.

(pdf-file, 124 kB)

THE MEANINGS OF MATHEMATICAL OBJECTS AS ANALYSIS UNITS FOR DIDACTIC OF MATHEMATICES
Juan Diaz Godino, Carmen Batanero

In this report we argue that the notion of meaning, adapted to the specific nature of mathematics communication, may serve to identify analysis units for mathematical teaching and learning processes. We present a theory of meaning for mathematical objects, based on the notion of semiotic function, where we distinguish several kinds of meanings: notational, extensional, intensional, elementary, systemic, personal and institutional. Finally, we exemplify the theoretical model by analysing some semiotic processes which take place in the study of numbers.

(pdf-file, 79 kB)

DIDACTICAL ANALYSIS: A NON EMPIRICAL QUALITATIVE METHOD FOR RESEARCH IN MATHEMATICS EDUCATION
José Luis González Marí

The methods used for the research in Mathematical Sciences Education are common for Psychology, Pedagogy and other related fields. But, to be honest, many of these approaches lead to too punctual results, not very important and with few possibilities to modify substantially the educational practice. The reason could lies in the inadequacy of such methods to cover the complexity of the field, in which many factors are operating in a non isolated way and interconnected to each other through relationships which need to be identified and analysed previously in a global framework under the mathematical knowledge as a common factor. To carry out this previous task, about which we must decide later the usage of the most suitable methods, we have been using a non empirical integrating research procedure, called Didactical Analysis, built up in the conjunction of meta-analysis and qualitative approaches. This paper exposes the principles, the conceptual framework and the techniques that make up this still being studied methodology.

(pdf-file, 94 kB)

INVESTIGATING TEACHERS' WORK WITH PUPILS' TEXTUAL EIGENPRODUCTIONS
Bruno D’Amore, Hermann Maier

More and more researchers in mathematics education recommend mathematical writing by pupils as an important activity besides their verbal contribution to classroom communication. A written text in which pupils express their own mathematical ideas in their own language, i. e. by use of words and formulations which are within their individual active language competence and performance, will be called here, according to Selter (1994) a ‘textual eigenproduction’ (tep). After a brief characterisation of teps and discussing their didactical functions, we present the design and the results of case studies with 16 teachers from Italy and Germany, on how far they are prepared to and experienced in work with teps in their mathematics classroom.

(pdf-file, 94 kB)

RECENT TRENDS IN THE THEORY OF DIDACTICAL SITUATIONS
André Rouchier

During the past thirty years the theory of didactical situations developed by Guy Brousseau and other people was at the centre of the identifications of the main characteristics of didactical systems as far as they are set up to teach mathematics. Following the great trends of the theory in its historical and notional development, from the first modelisation of the mathematization process to the identification of the crucial role of institutionalization, we focus our attention on the more recent questions calling for evolutions as new fields for research: regulation of didactical systems, organization of sets of problems for teaching, relationship between teaching devices and concepts of the theory in various contextual environments such as teachers training and the use of so called new technologies of information.

(pdf-file, 60 kB)

THE RESEARCH METHODS EMPLOYED BY CONTRIBUTORS TO THE PRAGUE DIDACTICS OF MATHEMATICS
Nada Stehlíková

The article focuses on the methodology of research within the Prague Seminars of Didactics of Mathematics. Its main characteristics will be identified: the phenomenon of formalism as a research question, models of a cognitive net and their use for diagnosis, re-education and prevention as the main aims, introspection as a research method. Finally, a method of implementing research results will be presented.

(pdf-file, 72 kB)

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