An Inquiry-Oriented Approach to Span and Linear Independence: The Case of the Magic Carpet Ride Sequence

Wawro, Megan; Rasmussen, Chris; Zandieh, Michelle; Sweeney, George; Larson, Christine

doi:10.1080/10511970.2012.667516

Cited by 65 publications

(13 citation statements)

References 12 publications

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“…Our data show that this is not necessarily the case; the students in our study were able to use Computational thinking in a variety of sophisticated, productive, and reflective ways, including generating Computational justifications for claims and making strategic choices to limit the complexity of their calculations. Our work thus joins the body of literature presenting an optimistic picture of student reasoning in linear algebra (e.g., Possani et al 2010;Wawro 2014;Wawro et al 2012), and provides evidence for the utility of Computational reasoning. We also hope to have provided evidence for the analytic usefulness of including reasoning processes within the construct of Computational thinking.…”

Section: Discussionsupporting

confidence: 62%

“…This provides insight as to how students use and modify their understanding of key concepts like span and independence as they make connections between the many criteria for a matrix to be invertible. Wawro et al (2012) described the implementation of an instructional sequence in the spirit of Realistic Mathematics Education to build rich concept images for span and independence, in which vectors are identified with modes of travel through space. They showed that students continue to draw on these concrete images later as they work on more abstract tasks.…”

Section: Literature Reviewmentioning

confidence: 99%

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Students’ Use of Computational Thinking in Linear Algebra

Bagley

Rabin

2015

Int. J. Res. Undergrad. Math. Ed.

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In this work, we examine students' ways of thinking when presented with a novel linear algebra problem. Our intent was to explore how students employ and coordinate three modes of thinking, which we call computational, abstract, and geometric, following similar frameworks proposed by Hillel (2000) and Sierpinska (2000). However, the undergraduate honors linear algebra students in our study used the computational mode of thinking in a surprising variety of productive and reflective ways. This paper examines the solution strategies that the students employed to solve the problem, emphasizing their use of the computational mode of thinking.

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Section: Discussionsupporting

confidence: 62%

Section: Literature Reviewmentioning

confidence: 99%

Students’ Use of Computational Thinking in Linear Algebra

Bagley

Rabin

2015

Int. J. Res. Undergrad. Math. Ed.

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“…For instance, code G2, an interpretation of "the column vectors of A span R 3 ," refers to being able to "get to every point in the dimension." We conjecture that this interpretation of span is strongly tied to a task sequence utilized within this class that supported students' reinvention of span and linear independence or dependence by building from their intutive notions of travel (Wawro, Rasmussen, Zandieh, Sweeney, & Larson, 2012). The construction of this inventory of brief descriptions of students' conceptions given in Appendix A is a valuable contribution to what is known about how students conceptualize key ideas in linear algebra.…”

Section: Analyzing Interview Datamentioning

confidence: 84%

A Method for Using Adjacency Matrices to Analyze the Connections Students Make Within and Between Concepts: The Case of Linear Algebra

Selinski

Rasmussen

Wawro

et al. 2014

JRME

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The central goals of most introductory linear algebra courses are to develop students' proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students' interpretation of and connections between concepts. Three cases provide examples that illustrate the usefulness of this approach for comparing differences in the structure of the connections, as exhibited in what we refer to as dense, sparse, and hub adjacency matrices. We also make use of mathematical constructs from digraph theory, such as walks and being strongly connected, to indicate possible chains of connections and flexibility in making connections within and between concepts. We posit that this method is useful for characterizing student connections in other content areas and grade levels.

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“…Abraham was a junior statistics major who had completed three semesters of calculus and the discrete mathematics course. The class engaged in various RME-inspired instructional sequences focused on developing a deep understanding of key concepts such as span and linear independence (Wawro et al 2012b), linear transformations (Wawro et al 2012a), Eigen theory, and change of basis. These instructional sequences often involved engaging in a guided reinvention of key definitions or procedures.…”

Section: Methodsmentioning

confidence: 99%

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

Wawro

2015

Int. J. Res. Undergrad. Math. Ed.

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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.

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An Inquiry-Oriented Approach to Span and Linear Independence: The Case of the Magic Carpet Ride Sequence

Cited by 65 publications

References 12 publications

Students’ Use of Computational Thinking in Linear Algebra

Students’ Use of Computational Thinking in Linear Algebra

A Method for Using Adjacency Matrices to Analyze the Connections Students Make Within and Between Concepts: The Case of Linear Algebra

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

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