\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2016a.00736}
\itemau{Caglayan, G\"unhan}
\itemti{Making sense of eigenvalue-eigenvector relationships: math majors' linear algebra -- geometry connections in a dynamic environment.}
\itemso{J. Math. Behav. 40, Part B, 131-153 (2015).}
\itemab
Summary: The present qualitative case study on mathematics majors' visualization of eigenvector-eigenvalue concepts in a dynamic environment highlights the significance of student-generated representations in a theoretical framework drawn from [{\it A. Sierpinska}, in: On the teaching of linear algebra. Dordrecht: Kluwer Academic Publishers. 209--246 (2000; ME 2016a.00721)] modes of thinking in linear algebra. Such an approach seemed to provide the research participants with mathematical freedom, which resulted in an awareness of the multiple ways that eigenvalue-eigenvector relationships could be visualized in a manner that widened students' repertoire of meta-representational competences in coordination with their preferred modes of visualization. Students' expression of visual fluency in the course of making sense of the eigenvalue problem $Au=\lambda u$ associated with a variety of matrices occurred in different, yet not necessarily hierarchical modes of visualizations that differed from matrix to matrix: (i) synthetic/analytic mode manifested in the process of detecting eigenvectors when the sought eigenvector and the matrix-applied product vector were aligned in the same/opposite directions; (ii) analytic arithmetic mode manifested in the case of singular matrices (in the determination of the zero eigenvalue) and invertible matrices with nonreal eigenvalues; (iii) analytic structural mode, though rarely occurred, manifested in making sense of the trajectory (circle, ellipse, line segment) of the matrix-applied product vector and relating trajectory behavior to matrix type. While the connection between the thinking modes and the concreteness-necessity-generalizability triad was not sharp, math majors still frequently implemented the CNG principles, which proved facilitatory tools in the evolution of students' thinking about the eigenvalue-eigenvector relationships.
\itemrv{~}
\itemcc{H65}
\itemut{undergraduate mathematics education; dynamic geometry software; visualization; representation; connection; linear algebra; eigenvectors; eigenvalues; matrices}
\itemli{doi:10.1016/j.jmathb.2015.08.003}
\end