FAKULTA MATEMATIKY, FYZIKY A INFORMATIKY

 TEACHERS’ CONVICTIONS

ON MATHEMATICAL INFINITY

Bratislava 2004                                         Silvia Sbaragli

Vedný odbor:  11-17-9 Teória vyučovania matematiky

Abstract. The purpose of this research-study is to focus and investigate teachers' convinctions on mathematical infinity. Through qualitative methodologies such as the analyses of questionnaire's collected answers and the related discussion activities, several teachers' misconceptions on mathematical infinity have been pointed out, showing the consequent mental images conditioning their way of teaching. The results have proved that infinity is an unknown concept, only managed by intuition and usually reduced to an extention of the finite. These reflections have revealed that the major difficulties related to the understanding of the concept of mathematical infinity are not exclusively due to epistemological obstacles but are also strengthened and magnified by didactical obstacles deriving from the erroneous intuitive models provided by teachers to their students in the first years of education.

 

Table of Contents

Preface

Chapter 1. A basic critical historical approach to infinity

1.1   Prehistory: from 600 B.C. to 1800

1.1.1   From the Ancient Times to the Middle Ages

1.1.2   Infinity in the Renaissance

1.2   From prehistory to history of the concept of mathematical infinity

1.2.1   Bernard Bolzano (1781- 1848)

1.2.2   Richard Dedekind (1831 – 1916)

1.2.3   Georg Cantor (1845 - 1918)

1.2.4   Cantor-Dedekind Correspondence

1.2.5   Cardinality

1.2.6   The Continuum Hypothesis

1.2.7   Giuseppe Peano (1858 - 1932)

1.2.8   Cantor and the ordinals

1.2.9   Ordinals as cardinals

 

Chapter 2. International research context

2.1   The didactical contract

2.2   Images and models

2.3   Conflicts and misconceptions

2.4   The triangle: teacher, student, knowledge

2.5   Obstacles

 

Chapter 3. Primary school teachers’ convictions on mathematical infinity

3.1   The mathematical infinity and the different nature of “obstacles”

3.2   First research questions and related hypotheses

3.3   Description of theoretical framework

3.4   Description of problems

3.5   Research Hypotheses

3.6   Research Methodology

3.6.1 Teachers participating in the research and methodology

3.6.2 Questionnaire content

3.7   Description of test results, opinion exchange and verification of hypotheses outlined in 3.5

3.7.1   Description of test results and related opinion exchange

3.7.2   The idea of point

3.7.3   Potential and actual infinity

3.7.4   The need for “concreteness”

3.8   Answers to questions formulated in 3.4

3.9   Chapter conclusions

 

Chapter 4. Present and future research

4.1   The first training course on this topic

4.2   Brief description of the research carried out with primary school children in 1996

4.3   Primitive entities of geometry

4.4   The discovery of the relevance of context: the point in different contexts

4.4.1   Where the idea of point in different contexts originates from

4.4.2   Reference theoretical framework

4.4.3   A provocation

4.5   A further fundamental aspect: different representations of the point in mathematics

4.5.1   Reference theoretical framework

4.5.2   A particular case of Duval’s paradox: the primitive entities

4.5.3   Some activity proposals

4.6   The “sense of infinity”

 

Bibliography