Learning Eigenvalues and Eigenvectors with Online YouTube Resources: A Journey in the Embodied, Proceptual-symbolic, and Formal Worlds of Mathematics

Figures & data

Table 1. The general rubric developed for analyzing YouTube videos.

	Embodied	Proceptual-symbolic	Formal
Action	The content creator: draws (or verbally visualizes) a mathematical object step by step without discussing the rationale behind the procedure; evaluates the geometric representation of a mathematical object(s) step by step without justifying the rationale behind the evaluation method; and highlights some properties of a mathematical object(s) by illustrating it geometrically/graphically without justification.	The content creator solves a task by following a method that involves using one or more formulas (step by step), without justifying the undertaken procedure.	The content creator: provides a formal definition for a mathematical concept; mentions some properties of a mathematical concept without justification; and points out two or more mathematical concepts are related without discussing how.
Process	The content creator: draws (or verbally visualizes) a mathematical object (step by step or might skip some steps) and explains how such a representation is created; evaluates the geometric representation of a mathematical object(s) (step by step or might skip some steps) and explains the rationale behind the evaluation method; discusses how the graphical representation of two or more mathematical objects/concepts are related; and highlights some properties of a mathematical object by illustrating it geometrically/graphically with justification.	The content creator: solves a task by following a method which involves using one or more formulas (step by step or might skip some steps) and justifies the undertaken procedure; and discusses how different symbolically presented mathematical relations and formulas are related.	The content creator: discusses how two or more mathematical concepts are related, mainly by focusing on their formal definitions; and mentions some properties of a mathematical concept with justification, mainly by focusing on their formal definitions.
Object	The content creator explicitly (or verbally) applies transformations to the geometrical representation of a mathematical object/concept.	The content creator: discusses the derivation of a mathematical relation or formula; performs transformations on symbolically presented mathematical relations and formulas to identify new properties of mathematical objects; and highlights a process as an object/a totality with justification.	The content creator uses the mathematical concept in the process of proving a mathematical theorem.
The content creator discusses the applications of the topic in the real-world and/or other disciplines.

Table 2. The rubric developed for analyzing YouTube videos related to eigen theory.

	Embodied	Proceptual-symbolic	Formal
Action	The content creator: draws eigenvectors for a given matrix; illustrates geometrically/graphically certain matrices do not have eigenvectors and eigenvalues, or only have one eigenvalue; illustrates geometrically/graphically how the value and sign of eigenvalue impact the direction and magnitude of an eigenvector; evaluates whether a vector drawn in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$ is an eigenvector for a given matrix; and calculates a matrix’s eigenvalue(s) by analyzing its eigenvector(s) drawn in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$.	The content creator: calculates eigenvalues and eigenvectors for a given matrix using $\mathrm{AX}=\mathrm{\lambda X}$; evaluates whether a vector(s) is an eigenvector of a matrix $A$ using $\mathrm{AX}=\mathrm{\lambda X}$; and determines whether a value is an eigenvalue of a matrix $A$ using $\mathrm{AX}=\mathrm{\lambda X}$.	The content creator: provides a formal definition for the eigenvalue and eigenvector (e.g., “An eigenvector of an $n\times n$ matrix $A$ is a nonzero vector $X$ such that $\mathrm{AX}=\mathrm{\lambda X}$ for some scalar $\lambda$. A scalar $\lambda$ is called an eigenvalue of $A$ if there is a nontrivial solution $X$ of $\mathrm{AX}=\mathrm{\lambda X}$; such an $X$ is called an eigenvector corresponding to $\lambda$” [16, p. 299]); highlights, without justification, that a matrix can have multiple eigenvalues but not more than the number of its rows/columns; highlights, without justification, that a matrix in $\mathbb{R}$ can have no eigenvalues; and points out eigen theory is related to linear transformation, determinants, changes of basis, and linear systems of equations without discussing how they are related.
Process	The content creator: considers the given matrix in a task about eigen theory as a linear transformation (on basis vectors), draws the corresponding eigenvectors in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$, and provides justifications for both the output and the undertaken method; justifies geometrically/graphically why certain linear transformations (e.g., rotation by 90°) do not have any eigenvectors in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$; justifies geometrically/graphically why certain transformations (e.g., $\left[\begin{array}{cc}3& 0\\ 0& 3\end{array}\right]$) with a single eigenvalue could have multiple forms of eigenvectors; justifies geometrically/graphically how the value and sign of an eigenvalue impact the direction and magnitude of the corresponding eigenvector(s); and justifies geometrically/graphically how certain known linear transformations impact the eigenvalues (e.g., when the linear transformation is a rotation, the eigenvalue is 1).	The content creator justifies the method undertaken for calculating eigenvalues and eigenvectors, drawing on several procepts such as determinant and linear systems of equations.	The content creator: justifies (mainly by the definition of eigenvectors and eigenvalues) why a matrix can have multiple eigenvalues but not more than the number of its rows/columns and/or why in $\mathbb{R}$, a matrix can have no eigenvalues; discusses the relationship between eigenspace and linear span; and discusses the relationship between eigen theory and linear transformation, determinants, changes of basis, and systems of linear equations.
Object	The content creator discusses that eigenvectors “can be picked up mentally” [23, p. 953] and performs linear transformations on their graphical representations to explore whether they can span the full space. Here the content creator probably illustrates geometrically/graphically what can happen if eigenvectors do not span the full space.	The content creator: performs operations on $\mathrm{AX}=\mathrm{\lambda X}$ when solving linear algebra problems/theorems (e.g., multiplying both sides of $\mathrm{AX}=\mathrm{\lambda X}$ by $A$ in the process of proving $A^{k}X={\lambda }^{k}X$); and highlights, with justification, that the resulting object in each side of $\mathrm{AX}=\mathrm{\lambda X}$ is a vector [30].	The content creator uses eigen theory in proving mathematical theorems (e.g., the diagonalization theorem).
The content creator discusses the applications of eigen theory in various fields, such as in differential equations and electrical engineering (for example, see Section 5.7 in [16]).

Table 3. A summary of the findings across the two cases.

	Embodied		Proceptual-symbolic		Formal
	Case 1	Case 2	Case 1	Case 2	Case 1	Case 2
Action	9 (21.9%)	0	13 (31.7%)	6 (46.1%)	13 (31.7%)	4 (30.8%)
Process	4 (9.8%)	0	2 (4.9%)	0	0	0
Object	0	0	0	2 (15.4%)	0	0
The application of the topic in the real-world and other disciplines			Case 1		Case 2
			0		1 (7.7%)

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