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Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

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Abstract

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

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Notes

  1. The notion of mathematical knowledge for teaching corresponds to, and represents an effort to further refine, Shulman’s (1986) notions of subject matter knowledge and pedagogical content knowledge. The correspondence between mathematical knowledge for teaching and Shulman’s notions are the focus of a paper in preparation (Ball, Thames, & Phelps, in preparation).

  2. This term is analogous to the term situations for justification coined by Cobb, Wood, Yackel, and McNeal (1992) to describe classroom moments whose issue is the legitimacy of mathematical activity, the accountability “as it occurs during the negotiation of mathematical meanings and practices” (p. 576). Since proof pertains to every mathematical activity, the situations for proving are not associated with any particular content area (algebra, geometry, etc.).

  3. This is not to say, however, that strategies like systematic enumeration cannot be used in the proof of statements that involve infinitely many cases. Consider for example the task: ‘Prove that if x is odd, then x 2−1 is divisible by 8.’ One way to prove this statement is to first use systematic enumeration to identify all four odd residue classes mod 8 and then develop a general argument to verify that the statement holds for each of the identified classes. Furthermore, we draw attention to the limits of the feasibility of using systematic enumeration, even when a finite number of cases are involved. Consider for example the proving task: ‘How many towers can you create that are 15 blocks tall using only yellow and red blocks? Prove your answer.’ It is unreasonable (from a practical point of view) for someone to solve this task by enumerating systematically all 32,768 cases.

  4. Analysis reported elsewhere (Ball & Bass, 2000b) offers some insight into the relationship between proving tasks that involve a single case and the proving activity that can be provoked by these tasks. This analysis focused on an episode where the students in a third-grade class were debating over the claim of a student, Sean, that the number 6 was ‘even and odd.’ The proving activity provoked by the classroom community’s efforts to verify or refute Sean’s claim brought to the fore issues of mathematical language, especially in the domain of definitions. Specifically, several students tried to refute Sean’s claim by using the classroom community’s definitions of even and odd numbers: the number 6 fulfilled the definition of even numbers but not that of odd numbers. Yet, as it turned out, Sean had in mind a special family of even numbers—those that have an odd number of groups of two (≡ 2 mod 4)—but he lacked the mathematical language to appropriately describe them. Thus, Sean’s claim could be verified, if one defined (as Sean intended) ‘even-and-odd numbers’ as the numbers that belong to this special family of even numbers.

  5. By reductio ad absurdum we mean the method of proof that proceeds by stating a proposition and then showing that it results in a (logical) contradiction, thus demonstrating the proposition to be false. In this sense, we use reductio ad absurdum as a method of refuting propositions. Also, we use proof by contradiction, which is closely related to reductio ad absurdum, as the method of verifying the truth of a given proposition by the supposition that it is false and the subsequent drawing of a conclusion that contradicts a proposition that is proved or assumed to be true.

  6. In order to distinguish between the second author’s two roles in this article—as the teacher in the episodes and as a researcher who examines the teacher’s practice—we use third person when we refer to the teacher and first person when we comment on the teacher’s practice. The teacher’s self-reflections on her practice (whether published in research reports or unpublished in her journal) are presented in third person.

  7. The (implied) domain of discourse in these definitions is the set of non-negative integers.

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Acknowledgments

We wish to thank Terry Wood, Steve Galovich (†), Gabriel Stylianides, anonymous reviewers, and the participants of the fourth Nuffield Seminar on Mathematical Knowledge in Teaching (Cambridge, U.K., January 2008) for useful comments on earlier versions of the article.

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  1. University of Oxford, 15 Norham Gardens, Oxford, OX2 6PY, UK

    Andreas J. Stylianides

  2. University of Michigan, Ann Arbor, MI, USA

    Deborah L. Ball

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Correspondence to Andreas J. Stylianides.

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Stylianides, A.J., Ball, D.L. Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving. J Math Teacher Educ 11, 307–332 (2008). https://doi.org/10.1007/s10857-008-9077-9

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