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Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles

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Abstract

This study is about prospective secondary mathematics teachers’ understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations—referent preserving versus referent transforming compositions—acting on these quantities. Although multiplicative structures can be modeled by additive structures, they have their own characteristics inherent in their nature. I situate my analysis within a framework of unit coordination with different levels of units supported by a theory of quantitative reasoning and theorems-in-action. Data consist of videotaped qualitative interviews during which prospective mathematics teachers were asked problems on multiplication and factorization of polynomial expressions in x and y. I generated a thematic analysis by undertaking a retrospective analysis, using constant comparison methodology. There was a pattern which showed itself in all my findings. Two student–teachers constantly relied on an additive interpretation of the context, whereas three others were able to distinguish between and when to rely on an additive or a multiplicative interpretation of the context. My results indicate that the identification and coordination of the representational quantities and their units at different categories (multiplicative, additive, pseudo-multiplicative) are critical aspects of quantitative reasoning and need to be emphasized in the teaching–learning process. Moreover, representational Cartesian products-in-action at two different levels, indicators of multiplicative thinking, were available to two research participants only.

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Notes

  1. Examples:

    • A 4 by 2 rectangle—made of 8 irreducible units of 1—conceptualized as the unitizing of the even number 8

    • A 2x + y + 3 by x + 1 rectangle—made of 2 irreducible units of x 2, 5 irreducible units of x, 3 irreducible units of 1, 1 irreducible unit of y, 1 irreducible unit of xy—conceptualized as the unitizing of the polynomial expression.

  2. Areal is an adjective meaning of or pertaining to area.

  3. Respectively as [1], [x], [y], [x 2], [y 2], [xy].

  4. Respectively as (1, 1), (1, x) or (x, 1) (1, y) or (y, 1), (x, x), (y, y), (x, y), or (y, x).

  5. It took John quite some time to realize that it was possible to express the area of a same-color-box as a product of two (combined) linear quantities.

  6. With relational notation, this can be expressed as \( (2x + y,x + 2y + 1) \equiv [[x^{2} ,x^{2} ],[xy,xy,xy,xy,xy],[y^{2} ,y^{2} ],[x,x],[y]] \).

  7. With relational notation, this can be expressed as \( (2x + y,x + 2y + 1) \equiv [(2x,x),(y,x),(2x,2y),(y,2y),(2x,1),(y,1)] \).

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Acknowledgments

I would like to thank the five prospective teachers for being part of this research study, and the reviewers and the editors for their very helpful comments and suggestions.

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    Günhan Caglayan

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Caglayan, G. Prospective mathematics teachers’ sense making of polynomial multiplication and factorization modeled with algebra tiles. J Math Teacher Educ 16, 349–378 (2013). https://doi.org/10.1007/s10857-013-9237-4

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