Type of mathematical proof: personal preference or adaptive teaching behavior?

Abstract

In our study, 32 German and Swiss 8th/9th-grade classes of lower-secondary school worked with their teacher on the same proving problem. The sample belongs to the Swiss-German study “Quality of Instruction, Learning Behavior and Mathematical Understanding”. Our data analyses relate to the teachers’ approaches to generating a specific form of evidence with their classes when dealing with a particular elementary number-theory problem. We address the question of how the different strategies can be characterized as manifestations of a certain approach to proving and try to clarify in which way the observed approach can be interpreted as adaptive teaching behavior. For this purpose, we searched for possible correlations between three main strategies or types of generating a specific form of evidence (experimental, operative, formal-deductive approach) on the one hand and (a) the teachers’ beliefs and personal characteristics and (b) the students’ prior knowledge of algebra and mathematics in general on the other hand. As our analyses show, three main approaches to proving occurred but not in equal proportions: there is a predominance of the approach that entails the highest extent of formalization and abstraction. Nevertheless, an operative way of proving is widespread too. On the whole, the findings indicate that one particular approach to proving can be interpreted as a personal preference of a specific group of teachers and, at the same time, with respect to the students’ mathematical skills as a manifestation of adaptive teaching behavior.

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Fig. 1
Fig. 2

Notes

  1. 1.

    De Villiers (1990) describes the following five functions of proofs: verification, explanation, systematization, discovery, and communication.

  2. 2.

    In the education systems of Germany and Switzerland, lower-secondary education (grades 7 to 9) is organized in a streamed model. Depending on their academic achievements, students are assigned to one of three different school types (or tracks) that differ in their demands.

  3. 3.

    The survey of the teachers confirmed this assumption: the teachers reported that they often (four-point scale: 1: never, 2: rarely, 3: often, 4: very often) base the planning of their mathematics lessons on the learning-related preconditions of the students in the class (M = 3.38; SD = 0.61; N = 32) and on the results of the previous lesson (M = 3.56; SD = 0.50; N = 32).

  4. 4.

    In one class, the students first solved the problem in small-group work and thereafter presented their solutions in a whole-class discussion. In all other classes, the predominant pattern consisted in classroom discourse in which the teacher and the students jointly dealt with the problem.

  5. 5.

    ‘Prototypical mathematics teachers’ are teachers who teach (more or less) exclusively mathematics and (nearly) no other subjects while ‘all-rounders’ are teachers who teach a range of different subjects in parallel (e.g., mathematics, biology, chemistry, physics, and computer science).

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Acknowledgements

We thank the Swiss National Science Foundation (SNSF) for supporting the project.

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Correspondence to Esther Brunner.

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Brunner, E., Reusser, K. Type of mathematical proof: personal preference or adaptive teaching behavior?. ZDM Mathematics Education 51, 747–758 (2019). https://doi.org/10.1007/s11858-019-01026-y

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Keywords

  • Mathematical proving
  • Mathematics education
  • Lower-secondary level
  • Teacher beliefs
  • Generation of evidence

Mathematics Subject Classification

  • 03F03
  • 97B20
  • 97B50
  • 97C70
  • 97D40