Cover

Title

Copyright

Contents

Foreword

Report of the Strasbourg Meeting

THE INFLUENCE OF COMPUTERS AND INFORMATICS ON MATHEMATICS AND ITS TEACHING

PART I

THE EFFECT ON MATHEMATICS

1.1 Introduction

1.2 New and revived areas of mathematical research

1.3 Proof

1.4 Experimentation in Mathematics

Simulation

Exploratory Data Analysis

1.5 Iterative methods

1.6 Algorithms

1.7 Symbolic Manipulation Systems

1.8 Computers and Mathematical Communication

1.9 The intellectual, economic and social dangers

PART II

THE IMPACT OF COMPUTERS AND COMPUTER SCIENCE ON THE MATHEMATICS CURRICULUM

2.1 The Common Mathematical Needs of Students in Mathematics, Science and Engineering

(a) Preparation for University Mathematics

(b) The University Mathematics Curriculum

2.2 A Discussion of Particular Curriculum Areas on whichComputers and Informatics have an Impact

(a) Discrete Mathematics Courses

A Discrete Mathematics Syllabus

(b) Calculus in the Computer Age

I The Role and Relevance of Calculus

II. The Content of Calculus Courses

III. Computers for Learning and Teaching Calculus

(c) Logic for Mathematicians and Computer Scientists

2.3 Exploration and Discovery in Mathematics

PART III

COMPUTERS AS AN AID FOR TEACHING AND LEARNING MATHEMATICS

Introduction

3.1 A changing view of mathematics

3.2 Computers change the relation between teacher and student

(a) The mathematical activity of the student

(b) The role of the teacher

3.3 Some particular uses of the computer in the classroom

(a) Graphic possibilities

(b) Self-evaluation and individualised instruction

(c) Assessment and Recording

(d) Student errors

3.4 Student responses to work with computers

3.5 The provision of software

3.6 Cultural, social and economic factors

3.7 Conclusion

APPENDICES

A. Symbolic Mathematical Systems

B. Exploratory Data Analysis

REFERENCES

List of Participants

Supporting Papers

Contents List

General

Mathematics education and curricula

Some experiences

Symbolic systems and algebra

Stochastics

Discrete and continuous mathematics

Visualizations

Mathematics and the Computer Revolution

1 A HISTORICAL PERSPECTIVE

2 MATHEMATICS AND THEORETICAL COMPUTER SCIENCE

3 COMPUTERS AS AN AID TO MATHEMATICAL RESEARCH

4 THE INTELLECTUAL DANGERS

5 ECONOMIC DANGERS

6 EDUCATIONAL DANGERS

7 CONCLUSION

Living with a New Mathematical Species

New Mathematics for a New Age

The Core of the Curriculum

Computer Literacy

Computer Science

Computers in the Classroom

Let Us Teach Guessing

Metaphors for Mathematics

REFERENCES

Checking Mathematics with the Aid of a Computer

REFERENCES

On the Mathematical Basis of Computer Science

1 AROUND THE NOTION OF COMPUTATION

2 AROUND THE NOTION OF ALGORITHM

3 AROUND LOGIC: SYNTAX AND SEMANTICS

CONCLUSION

REFERENCES

The Mathematics of Computer Algebra

INTRODUCTION

ELEMENTARY CALCULATIONS

ADVANCED METHODS

FACTORISATION

INTEGRATION

PHYSICAL APPLICATIONS

CONCLUSION

REFERENCES

Mathematical Education in the Computer Age

1 INTRODUCTION

2 THE INFLUENCE OF COMPUTERS ON MATHEMATICAL EDUCATION

3 A NEW WAY OF TEACHING MATHEMATICS

3.1 Method of teaching

3.2 Feasibility of exploratory mathematics

4 CONCLUSIONS

REFERENCES

A Fundamental Course in Higher Mathematics Incorporating Discrete and Continuous Themes

THE CURRENT MATHEMATICS CURRICULUM: TRADITIONS AND CONCERNS

RESPONSES TO THE CHALLENGE OF DISCRETE MATHEMATICS

A FIRST-YEAR CURRICULUM INCORPORATING DISCRETE AND CONTINUOUS THEMES

A. Functions

1. Numbers and Relations

2. Functions and Operations

3. Models

B. Behavior of discrete functions

1. Sequences: Iteration and Recursion

2. Difference Operators

3. Summation

4. Landau Notation (0,o) and Limits of Sequences

C. Behavior of continuous functions

1. Limit Heuristics

2. First Derivative

3. Differentiation Rules

4. Monotone Functions and Local Extrema

5. Second Derivative

6. Extreme Values

7. Related Rates

D. Estimation and error

1. Mean Value Theorem

2. Solution of Equations

3. Interpolation

4. Approximation

E. Integration

1. Introduction

2. The Definite Integral

3. The Indefinite Integral

4. The Fundamental Theorem of Calculus

5. Evaluation of Integrals: Analytic Techniques

6. Evaluation of Integrals: Numerical Techniques

7. Applications of Integration: Aggregation

8. Applications of Integration: Modeling

IMPLICATIONS FOR THE CURRICULUM

REFERENCES

Graphic Insight into Calculus and Differential Equations

DIFFERENTIATION

NON-DIFFERENTIABLE FUNCTIONS

AREA CALCULATIONS

FIRST ORDER DIFFERENTIAL EQUATIONS

QUALITATIVE ANALYSIS OF DIFFERENTIAL EQUATIONS

NEWTONfS LAWS

SYSTEMS OF DIFFERENTIAL EQUATIONS

SECOND ORDER DIFFERENTIAL EQUATIONS

CHANGES IN LEARNING STYLE

IS PROGRAMMING ESSENTIAL ?

CONCLUDING REMARKS

REFERENCES

Calculus and the Computer. The Interplay of Discrete Numerical Methods and Calculus in the Education of Users of Mathematics: Considerations and Experiences

1 NEW POSSIBILITIES

1.1 New possibilities for the user

1.2 New possibilities in the teaching-learning process

2 THE INTERPLAY DISCRETE - CONTINUOUS

2.1 General considerations

(a) Insufficiency of continuous analysis for obtaining concrete numerical results

(b) Most concrete models of analysis have a discrete basis

(c) The transition from models to concrete numerical results in general, cannot be accomplished without continuous analysis

2.2 The context of dynamical systems

3 EXPERIENCES

REFERENCES

Computer-based Symbolic Mathematics for Discovery

INTRODUCTION

1 A Curricular Project

2 Some Specific Examples

2.1 Curve Sketching

2.2.1 Coefficient Patterns

2.2.2 Coefficient Growth

2.2.3 Computing Time

2.3 Other Considerations

3 Some Discovery Projects Using Computer Algebra

3.1 Elementary Algebra:

3.2 Matrices and Determinants;

3.3 Summation

3.4 Generating Functions and Power Series:

3.5 Integration and Differentiation

3.6 Non-linear equations

References

Computer-aware Curricula: Ideas and Realization

1 INTRODUCTION

2 CHANGES IN MATHEMATICS

3 CURRICULA

REFERENCES