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<p>A core task in&#xa0;linear algebra is determining whether a set of vectors can span a particular space. To better understand how students approach this problem, we analyzed 196 student responses to an exam question asking whether the columns of a matrix span ℝ<sup>3</sup>. The responses were gathered from 14 instructors across six semesters at various U.S. tertiary institutions. Using Balacheff’s cK¢ model of conceptions, we categorized the distinct justifications (as control structures) evident in the responses. We found that students employed a wider range of control structures than those in their textbooks and that the correctness of these control structures varied. Specifically, students’ strategies and correctness differed by textbook format (HTML vs. PDF) and testing environment (in-person vs. online). Our correctness coding not only aligned with instructor grading but also provided a more nuanced description of student justifications. The proposed process of classification could be used with other core tasks offering instructors detailed and nuanced information about students’ responses.</p>

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Conceptions of span in linear algebra through analysis of student exam responses

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月20日

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  • Kenny Deng,
  • Francisco Pizarro,
  • Vilma Mesa,
  • Thomas Judson

摘要

A core task in linear algebra is determining whether a set of vectors can span a particular space. To better understand how students approach this problem, we analyzed 196 student responses to an exam question asking whether the columns of a matrix span ℝ3. The responses were gathered from 14 instructors across six semesters at various U.S. tertiary institutions. Using Balacheff’s cK¢ model of conceptions, we categorized the distinct justifications (as control structures) evident in the responses. We found that students employed a wider range of control structures than those in their textbooks and that the correctness of these control structures varied. Specifically, students’ strategies and correctness differed by textbook format (HTML vs. PDF) and testing environment (in-person vs. online). Our correctness coding not only aligned with instructor grading but also provided a more nuanced description of student justifications. The proposed process of classification could be used with other core tasks offering instructors detailed and nuanced information about students’ responses.