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Mathematical tasks designed to foster a reconceptualized view of early arithmetic

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Abstract

We discuss mathematical tasks used in a first mathematics content course for elementary teachers at our university to foster a deep conceptual understanding of early arithmetic, including basic concepts of number, number relationships and strategies, and coordinating units of different rank. Our approach is to immerse our students in a base 8 world for up to six weeks. A key aspect is that we develop base 8 vocabulary. We use base 8 analogs of instructional sequences developed in classroom teaching experiments in the elementary grades that have been proven successful to promote deep conceptual understandings of basic arithmetic and place-value numeration in young children. As a result, our students have unique opportunities to develop a reconceptualized view of early arithmetic and learn how it can be advanced.

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Notes

  1. By emphasizing early number relationships prior to working with place value we avoided the difficulty McClain (2003) describes of students attempting to develop tricks and shortcuts rather than exploring the underlying concepts of place value numeration.

  2. Muser and Berger (1988) was the textbook used in the course prior to the instructional sequences described in this paper.

  3. We do not include the typical subscript, 108, because we are operating in base 8 throughout.

  4. We acknowledge that the vocabulary we use conforms more closely to some Asian languages and does not address difficulties English speakers face with numbers from 11 through19 (Fuson & Kwon, 1992).

  5. All names used in this paper are pseudonyms.

  6. Since sums and differences to one-e are the same in base 10 and base 8, our approach does not address learning facts for sums below one-e or the corresponding differences.

  7. In the examples shown, the column for pieces of candy is labeled “candies.” We subsequently use “pieces” for purely semantic reasons. The latter clarifies that the question, “How many candies are there?” refers to the entire quantity.

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Authors and Affiliations

  1. Purdue University Calumet, 2200 169th Street, Hammond, IN, USA

    Erna Yackel, Diana Underwood & Norma Elias

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  1. Erna Yackel
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  2. Diana Underwood
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  3. Norma Elias
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Correspondence to Erna Yackel.

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Yackel, E., Underwood, D. & Elias, N. Mathematical tasks designed to foster a reconceptualized view of early arithmetic. J Math Teacher Educ 10, 351–367 (2007). https://doi.org/10.1007/s10857-007-9044-x

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