id: 06243041
dt: j
an: 2014a.00686
au: Caglayan, Günhan
ti: Prospective mathematics teachers’ sense making of polynomial
multiplication and factorization modeled with algebra tiles.
so: J. Math. Teach. Educ. 16, No. 5, 349-378 (2013).
py: 2013
pu: Springer Netherlands, Dordrecht
la: EN
cc: H49 H39
ut: additive reasoning; algebra tiles; Cartesian product; concept-in-action;
mapping structure; models and modeling; multiplicative reasoning;
polynomial rectangle; prospective teacher education; quantitative
reasoning; relation; representation; bijections; polynomial
multiplication; polynomial factorization
ci:
li: doi:10.1007/s10857-013-9237-4
ab: Summary: 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.
rv: