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Three styles characterising mathematicians’ pedagogical perspectives on proof

Abstract

The article describes mathematicians’ pedagogical perspectives on proof in the teaching of first year university students at a mathematics department in Sweden. A conceptual frame that was used in the data analysis combines theories about proof from earlier mathematics education research with a social practice approach of Lave and Wenger. A theoretical idealised model of three different teacher styles was constructed from the data that consist of transcripts of interviews with 13 mathematicians at the department. The model gives structure to the results and can be applied and developed when comparing teaching cultures between different departments in a country as well as between different countries.

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Acknowledgement

The article was written with help of the financial support of The Bank of Sweden Tercentenary Foundation. I thank the anonymous reviewers for their valuable comments and suggestions.

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Correspondence to Kirsti Hemmi.

Appendix 1. Mathematics, basic course (30 ECTS credits)

Appendix 1. Mathematics, basic course (30 ECTS credits)

Course description:

Introductory course (7.5 ECTS credits).

Polynomial division, the factor theorem, factorisations, inequalities, absolute value, geometric sum, functions, the straight line, power functions, exponential and logarithmic functions, geometry (congruence and similarity), trigonometry, trigonometric functions. Somewhat about sets. Complex numbers.

Linear algebra (7.5 ECTS credits).

The binomial theorem, proof by induction.

Systems of linear equations, matrix algebra, determinants, vectors in 2 and 3 dimensions, linear independence, dot product, vector product, straight lines and planes, linear mappings.

Mathematical Analysis 1, (7.5 ECTS credits).

Inverse functions, cyclometric functions.

Limits, continuity, derivatives, derivation rules, derivation of elementary functions, extreme value problems, asymptotes, inequlities, integrals, relation between primitive functions and integrals, partial integration, method of substitution, integrals of certain classes of functions.

Mathematical Analysis 2, (7.5 ECTS credits).

Functions of one variable:

Applications of integrals. Differential equations (separable, linear first- and second-order equations), Taylor’s formula.

Functions of several variables: Limits, partial derivation, level curves and level surfaces, tangent plane, linear approximation, extreme value problems over compact domains, double integrals.

Teaching and learning methods:

Lectures: 8 h/wk.

Lessons in small groups: 7 h/wk.

Methods of assessment: Written examination in each of the four sub-courses of 7.5 ECTS credits.

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Hemmi, K. Three styles characterising mathematicians’ pedagogical perspectives on proof. Educ Stud Math 75, 271–291 (2010). https://doi.org/10.1007/s10649-010-9256-3

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Keywords

  • Proof
  • Mathematicians’ pedagogical perspectives
  • Transparency
  • University mathematics