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A Hypothetical Learning Trajectory for Eigen Theory
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Michelle Zandieh
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Arizona State University
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\textit{zandieh@asu.edu}\textit{}
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Chris Rasmussen 
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San Diego State University
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\textit{chrisraz@sciences.sdsu.edu}
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Christine Larson 
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Indiana University \&
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San Diego State University
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\textit{larson.christy@gmail.com}
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An understanding of eigen theory can provide students with powerful ways of analyzing and understanding systemic-level problems in many areas of mathematics, engineering, and sciences.  
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Most mathematics, engineering, and physics majors will encounter eigen theory at least twice in their undergraduate career: in linear algebra and in differential equations.  Prior research documents the many struggles that students face as they attempt to bridge their informal and intuitive ways of thinking with the formalization of concepts in linear algebra (Dorier, Robert, Robinet and Rogalski, 2000; Carlson, 1993). Contemporary theories of learning and advances in instructional design theory, however, offer fresh ideas for addressing these well-documented problems. 
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The purpose of this paper is to report on one such research-based approach to improve the learning and teaching of linear algebra. In particular, this paper will articulate a hypothetical learning trajectory (HLT) for the development of eigen theory. This HLT will be grounded in analysis of data collected from a semester long teaching experiment in linear algebra. As such, the HLT we describe will be both retrospective and prospective. It will be retrospective in the sense that the HLT is informed not only by the literature, but also by our ongoing work with learners. It is prospective in the sense that what we learned from working with students informs revisions and changes to our HLT. This, in turn, will be the basis for our next classroom teaching experiment.
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We define a HLT to be a storyline about teaching and learning that occurs over an extended period of time (cf, Simon, 1995).  The storyline includes four aspects, all of which are reflexively related and revisable: (1) Learning goals about student reasoning, (2) a storyline of how students' mathematical learning experience will evolve, (3) the role of the teacher in the storyline, and (4) a sequence of instructional tasks that students will engage in. In our view, a HLT can be a useful tool for researchers and instructional designers interested in studying the evolution of student reasoning in classroom settings.  
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Our instructional design efforts are informed in large part by the theory of Realistic Mathematics Education (RME), with particular emphasis on the heuristics of guided reinvention and emergent models (Gravemeijer, 1999). The heuristic of guided reinvention suggests means by which teachers and instructional designers can promote students' ability to develop the intended mathematics for themselves. The emphasis of guided reinvention is on the character of the learning process, rather than on inventing as such. The heuristic of emergent models can be thought of in terms of a global transition in which students and the teacher develop a \textit{model-of} their informal activity which gradually develops into a \textit{model-for} more formal mathematical reasoning. This global transition is a process by which a new mathematical reality emerges, grounded in informal and situation-specific activity (Zandieh \& Rasmussen, 2008). 
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For example, in our teaching experiment we found that students could essentially reinvent the determinant as a way of measuring the area of the image of the unit square under multiplication by an arbitrary 2x2 matrix.  More importantly, and related to the notion of emergent models, the relationship between the column vectors of a matrix and the determinant of this matrix has the potential to become a powerful reasoning tool. Figure 1 gives one student's work on a task that asked them to find an expression for the area of the image of the unit square when multiplied by the matrix 
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Figure 1. A student shows how he found the area.
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After finding this area, students were asked to make predictions about 
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Figure 1. A student shows how he found the area
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After finding this area, students were then asked to make predictions about the area of the image of a unit square when multiplied by a 2x2 matrix whose column vectors were linearly dependent.  This helped students to develop a visual intuition for the relationship between the determinant of a matrix and the linear (in)dependence of the column vectors that make up the matrix.  Nearly a month after this introduction to determinants, one student, who we will call Karl, explained his thinking about how this idea connects to eigen theory:
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When you look at the, uh, vectors, what does the determinant give us?  It gives us the area between any two given vectors.  And if, if our determinant equals zero, that basically means that the vectors that we're solving for have no area in between.  So therefore they lie along the same line.  
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As he spoke, Karl held his hands in a v-shape, presumably emulating two vectors pointing out from the origin.  When he made reference to the determinant being zero, he made a motion of flattening his hands together to indicate that the two vectors now lie along the same line.
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\hfill{}This type of reasoning has inspired us to reframe the development of the eigen unit. In particular, we conjecture that it might be more intuitive for students to first think about the process of finding eigenvectors, as opposed to eigenvalues. This ``eigenvector first'' approach goes along with the goal to find those vectors whose image lies along the same line as the original vector -- and these vectors can be found by forcing the determinant to be zero. Such an eigenvector first approach has also been documented to be more conceptually accessible to student in differential equations (Rasmussen \& Blumenfeld, 2007). Our full paper will detail the four components (Learning goals about student reasoning, a storyline of how students' mathematical learning experience will evolve, the role of the teacher in the storyline, and a sequence of instructional tasks) for our new HLT for this innovative ``eigenvector first'' approach.
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\textbf{References}
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Carlson, D.  (1993).  Teaching linear algebra: Must the fog always roll in\textit{?  The College Mathematical Journal}, 24(1), 29-40.
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Dorier, J.L., Robers, A., Robinet, J., \& Rogalski, M.  (2000).  The obstacles of formalism in linear algebra.  In J.L. Dorier (Ed.), \textit{On the teaching of linear algebra} (pp. 85-124).  Dordrecht: Kluwer.
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Gravemeijer, K.  (1999).  How emergent models may foster the constitution of formal mathematics.  \textit{Mathematical Thinking and Learning}, 1, 155-177.
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Rasmussen, C., \& Blumenfeld, H. (2007). Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions. Journal of Mathematical Behavior, 26, 195-210.
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Simon, M.A. (1995).  Reconstruction mathematics pedagogy from a constructivist perspective.  \textit{Journal for Research in Mathematics Education, 26,} 114-145.
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Zandieh, M. \& Rasmussen, C. (2008). A case study of defining: creating a new mathematical reality. Manuscript submitted for publication.
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