Reasoning About Solutions in Linear Algebra: the Case of Abraham and the Invertible Matrix Theorem

Abstract

A rich understanding of key ideas in linear algebra is fundamental to student success in undergraduate mathematics. Many of these fundamental concepts are connected through the notion of equivalence in the Invertible Matrix Theorem (IMT). The focus of this paper is the ways in which one student, Abraham, reasoned about solutions to Ax=0 and Ax=b to draw connections between other concept statements within the IMT. Data sources were video and transcripts from whole class discussion, small group work, and individual interviews. The overarching analytical structure was influenced by a framework of genetic analysis (Saxe, Journal of the Learning Sciences, 11, 275-300, 2002), and Toulmin's Model of Argumentation (1969) was employed to analyze the structure of arguments both in isolation (microgenesis) and over time (ontogenesis). This case study, rather than focusing on student difficulties in undergraduate mathematics, serves as a compelling example of the productive and powerful reasoning that is possible as students make sense of complex mathematics. The results present an ontogenetic analysis of Abraham's use of solutions to reason about how span and linear independence of a set of vectors are related, as well microgenetic analyses of various examples of reasoning about solutions to justify connections between other concepts within the IMT.

Usos da argumentação na educação matemática

“…Layout de Toulmin para a argumentação (adaptado de Toulmin, 2006) As pesquisas na Educação Matemática sobre a temática da argumentação foram intensificadas a partir do estudo de Krummheuer (1995). Desde então, esse layout tem sido usado para analisar argumentos no Ensino Fundamental (Almeida & Malheiro, 2018), no Ensino Médio (Mariotti & Pedemonte, 2019), no Ensino Superior, na graduação (Fukawa-Connelly, 2014;Wawro, 2015), bem como na pós-graduação (Kwon et al, 2015;Metaxas et al, 2016).…”

Students’ metarepresentational competence with matrix notation and Dirac notation in quantum mechanics

“…In linear algebra research, Hillel [25] offered three modes of description (abstract, algebraic, and geometric) for the basic objects and operations in linear algebra and pointed out that "the ability to understand how vectors and transformation in one mode are differently represented, either within the same mode, or across modes is essential in coping with linear algebra" (p. 199). Wawro [26] investigated the ways in which one student reasoned about solutions to the matrix equations Ax ¼ 0 and Ax ¼ b to make and justify logical connections between a variety of concepts in linear algebra. For instance, while trying to explain why "the columns of A span R 3 " implies "Any vector b in R 3 can be written as a linear combination of the columns of A" for a 3 × 3 matrix A, the student said "I'm just rewriting it a different way to try to think about it," as he wrote the matrix equation Ax ¼ b and the corresponding vector equation.…”