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Classroom engagement towards using definitions for developing mathematical objects: the case of function

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Abstract

For mathematicians, definitions are the ultimate tool for reaching agreement about the nature and properties of mathematical objects. As research in school mathematics has revealed, however, mathematics learners are often reluctant, perhaps even unable, to help themselves with definitions while categorizing mathematical objects. In the research from which we take the data presented in this article, we have been following a group of prospective mathematics teachers studying functions. In the course of learning, the students gradually accepted the definition as the ultimate criterion to identify examples of function. And yet, this use was hindered by the difficulty the students experienced while trying to understand the logical structure of the definition. Our close analysis has shown that the determiners “for every” and “unique” constituted the main source of the difficulty. We propose that a brief introduction to logic and, in particular, to parsing complex mathematical sentences, may be a useful addition to mathematics curriculum.

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Notes

  1. Note that the participants in this study are prospective elementary-school mathematics teachers. As such, they have studied functions during their middle-school and high-school learning. We therefore followed those students' first steps in their first year in college, as second-time learners.

  2. The word narrative often seems to be misleading as it relates to a certain “timeline” whereas definitions, axioms and theorems often seem timeless. And yet, there are stories that are timeless. E.g., “trees are plants” is a factual timeless narrative. Mathematical definitions are narrative as they are developed by means of communication in the community of mathematicians, and then are communicated time and again in the classroom community.

  3. Fischbein considered the dual nature of the geometrical figure: formal and figural. Theoretically, a geometric figure is an entity totally controlled by its definition and is therefore an abstract concept. In practice, however, what counts when people think about such figures are their figural visualizable properties, such as shape, size, and even color (Fischbein, 1993).

  4. The course was taught in Hebrew. For the purpose of this paper, parts of it were translated to English by the authors, maintaining classroom register.

  5. We use the word vignette to refer to a part of the lesson that is around a certain theme. In this case—about specific ideas related to using the word function in class.

  6. Although the formal definition of the word function is different, this is the definition used in various middle school textbooks.

  7. Note that turn 7 belongs to both parts. Often, the boundaries between segments of discourse are fuzzy.

  8. The students were referring to the numbers that were written in the diagram on the board (Fig. 2). We chose to write the symbols (1 and (−1)) instead of the related words (one and minus one), as this is what they saw on the board.

  9. While speaking, Jane pointed to the board. It is not clear which was the point that she pointed to.

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Acknowledgments

This study is supported by the Israel Science Foundation, no. 446/10.

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Authors and Affiliations

  1. School of Education, Tel-Aviv University, Givataim, Israel

    Michal Tabach

  2. Levinsky College of Education, Tel-Aviv, Israel

    Talli Nachlieli

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  1. Michal Tabach
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Correspondence to Michal Tabach.

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Tabach, M., Nachlieli, T. Classroom engagement towards using definitions for developing mathematical objects: the case of function. Educ Stud Math 90, 163–187 (2015). https://doi.org/10.1007/s10649-015-9624-0

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