Examining Students’ Procedural and Conceptual Understanding of Eigenvectors and Eigenvalues in the Context of Inquiry-Oriented Instruction

Abstract

This study examines students’ reasoning about eigenvalues and eigenvector s as evidenced by their written responses to two open-ended response questions. This analysis draws on data taken from 126 students whose instructors received a set of supports to implement a particular inquiry-oriented instructional approach and 129 comparable students whose instructors did not use this instructional approach. In this chapter, we offer examples of student responses that provide insight into students’ reasoning and summarize broad trends observed in our quantitative analysis. In general, students in both groups performed better on the procedurally oriented question than on the conceptually oriented question. The group of students whose instructors received support to implement the inquiry-oriented approach outperformed the other group of students on the conceptually oriented question and performed equally well on the procedurally oriented question.

Appendix: Grading Scheme for Assigning Points to Open-Ended Response Questions 8 and 9b

Q #	Points awarded and criteria
8	3 points: Method 1: Full points were awarded to students who reasoned about the determinant to arrive at the correct conclusion without making computational or conceptual errors. Examples of this kind of reasoning are shown below.     (i) $\det \left( {A - \lambda I} \right) = 0$ implies $\left( {\lambda - 1} \right)\left( {\lambda - 4} \right) = 0$ implies $\lambda = 1$ or $\lambda = 4$ implies $\lambda = 2$ is not an eigenvalue for the matrix $A$.     (ii) $\det \left( {A - 2I} \right) = - 2 \ne 0$ implies $\lambda = 2$ is not an eigenvalue for the matrix $A$     (iii) $\det \left( {A - \lambda I} \right) = \left| {\begin{array}{*{20}c} {3 - \lambda } & 2 \\ 1 & {2 - \lambda } \\ \end{array} } \right| = \left( {3 - \lambda } \right)\left( {2 - \lambda } \right) - 2 = \lambda^{2} - 5\lambda + 4$. Substituting 2 in the characteristic equation gives $4 - 10 + 4 = - 2$ implies $\lambda = 2$ is not an eigenvalue for the matrix $A$. Method 2: Full points were awarded to students who reasoned about $A - \lambda I$ without using the determinant to arrive at the correct conclusion without making any computational or conceptual errors. Examples are shown below.     (i) $\left( {A - 2I} \right)\left[ {\begin{array}{*{20}c} x \\ y \\ \end{array} } \right] = 0$ implies $x = 0$ and $y = 0$ which is the trivial solution, so $\lambda = 2$ is not an eigenvector for the matrix $A$.     (ii) $\left( {A - 2I} \right) \cong \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & { - 2} \\ \end{array} } \right]$, and the column vectors of this matrix are not linearly dependent, so $\lambda = 2$ is not an eigenvalue .     (iii) $rref\left( {A - 2I} \right)$ does not have a free variable, so $\lambda = 2$ is not an eigenvalue .     (iv) The first column of $\left( {A - 2I} \right)$ is not a scalar multiple of the second column so $\lambda = 2$ is not an eigenvalue so $\lambda = 2$ is not an eigenvalue .
	2 points: Two points were awarded to students to students who take a conceptually correct approach (either by reasoning about the determinant or by reasoning about $A - \lambda I$ without using the determinant) but either     • made a computational error (e.g. factoring the characteristic polynomial incorrectly) or     • did not offer a clear conclusion about whether 2 is an eigenvalue or not, or     • arrived at the correct conclusion without a full explanation of why
	1 point: One point was awarded to students whose response included some evidence of conceptual understanding, but who made a conceptual error (which might be accompanied by a computational error).
	0 points: No points were awarded to students who left the page blank, or whose response: (i) gave no evidence of conceptual understanding, or (ii) said something like “I don’t know.” Example of responses we considered to include no evidence of conceptual understanding are “Yes, because A = PDP −1” and “I say it is… because… there are 2’s in the problem.”
9b	3 points: Three points were awarded to students whose response appropriately coordinated with the eigen-concept, referenced (either by directly naming or by explicitly referring to their work shown in 9a) all three correct vectors, and provided a correct rationale for this selection.
	2 points: Two points were awarded to students whose response provided at least two correct explanations (e.g. $Mx = \lambda x$ is written and student writes that “an eigenvector tells you the direction of stretching”) but did not identify and explicitly describe what happens to all three correct vectors.
	1 point: One point was awarded to students who either     (i) Provided one correct explanation (e.g. by either writing “$Mx = \lambda x$” or “an eigenvector tells you the direction of stretching”) and explicitly connected this explanation to at most one correctly selected vector     (ii) Suggested components of $M$ that would transform x into one of the given choices, such as $M = I$, $- I$, $or\; 0$.
	0 point: No points were awarded to responses that do not coordinate with the eigen-concept, do not suggest components of $M$ that would transform x into one of the given choices, says I don’t know, or leaves the page blank. An example of student response to question 9 which was awarded 0 point was “all are the same size.”