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DNR perspective on mathematics curriculum and instruction, Part I: focus on proving

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Abstract

This is the first in a series of two papers whose goal is to contribute to the debate on a pair of questions: (1) What is the mathematics that we should teach in school? (2) How should we teach it? This paper addresses the first question, and the second paper, to appear in the next issue of ZDM, addresses the second question. The two questions are addressed from a particular theoretical framework, called DNR-based instruction in mathematics. The discussions in the current paper are instantiated mainly in proof-related contexts. The paper offers a definition of mathematics as a union of two categories of knowledge: ways of understanding and ways of thinking. The latter are generalizations of the notions, proof and proof scheme, respectively. The paper also discusses cognitive-epistemological and curricular implications of this definition, focusing mainly on the inevitable production of narrow or faulty mathematical knowledge and the asymmetry in educators’ attention to ways of understanding and ways of thinking.

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Notes

  1. Throughout the two papers, a term with a special DNR meaning is italicized until after it is first defined.

  2. The adjective “eventual” is to acknowledge that deductive reasoning is developmental. The nature and level of rigor of deduction taught must carefully aligned with the current conceptual development of the students.

  3. There are different reasons for using the terms way of understanding and way of thinking, rather than the terms, subject matter and conceptual tool (see Harel, 2008).

  4. In DNR, mental act is a primary term (see Harel, 2008).

  5. In most contexts throughout the two papers, a reference to “individual” is applicable to “community; hence, it is not necessary from now on to mention both.

  6. The question concerning the extent of the empirical observations of someone’s proofs that determine her or his proof scheme is a methodological question and will not be addressed in this series of papers.

  7. Institutionalized ways of understanding are those the mathematics community at large accepts as correct and useful in solving mathematical and scientific problems. A subject matter of particular field may be viewed as a structure of institutionalized ways of understanding.

  8. A company produces quilts made out of small congruent squares, where the squares on the main diagonals of the quilt are black and the rest are white. The quilts are square; that is, the length and width of the quilt are equal. The cost of a quilt is: for material, $1 for a black square and $0.50 for a white square; for labor, $0.25 per square. To order a quilt, one must specify the number of black squares, the number of white squares, or the total number of squares. April, Bonnie, and Chad ordered three identical quilts, and each filled out an order form like the one below. April entered the number of black squares in the Black Cell. Bonnie and Chad entered the same number as April’s, but accidentally Bonnie entered her number in the Whites Cell and Chad in the Total Cell. April was charged $139.25. How much were Bonnie and Chad charged?

    Number of black squares

    Number of white squares

    Total of squares

     

     

     

  9. Pseudonym.

  10. For the exact characteristics of these and other induction problems, see Harel (2001).

  11. Given the definition of mathematics presented earlier, the term mathematical knowledge refers collectively, here and elsewhere in the two papers, to desirable and institutionalized ways of understanding and ways of thinking.

  12. The issue of teacher knowledge will be discussed in more detail in Paper II. Also in this paper, teacher’s knowledge base will be defined in terms of DNR concepts.

  13. I thank the anonymous reviewer who recommended to list these examples of critical issues in which the two fundamental questions must be addressed.

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Authors and Affiliations

  1. University of California, San Diego, CA, USA

    Guershon Harel

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  1. Guershon Harel
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Correspondence to Guershon Harel.

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I wish to acknowledge the helpful comments from Evan Fuller, Nitsa Movshovitz-Hadar, and anonymous reviewers. Preparation of this paper was supported in part by National Science Foundation Grant No. REC-0310128. Any opinions or conclusions expressed are those of the author and do not represent an official position of NSF.

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Harel, G. DNR perspective on mathematics curriculum and instruction, Part I: focus on proving. ZDM Mathematics Education 40, 487–500 (2008). https://doi.org/10.1007/s11858-008-0104-1

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