## Prospective Mathematics Teachers’ Sense Making of Polynomial Multiplication and Factorization Modeled with Algebra Tiles

Source: *Journal of Mathematics Teacher Education, Vol. 16, Iss. 5*, p. 349-378, October 2013

This study examines 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.

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.

The author generated a thematic analysis by undertaking a retrospective analysis, using constant comparison methodology.

There was a pattern which showed itself in all the 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.

The 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.