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Leading Students towards the Formal World of Mathematical Thinking: A Mathematician's Reflections on Teaching Eigentheory

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Title: Leading Students towards the Formal World of Mathematical Thinking: A Mathematician's Reflections on Teaching Eigentheory
Language: English
Authors: Stewart, Sepideh, Epstein, Jonathan, Troup, Jonathan
Source: International Journal of Mathematical Education in Science and Technology. 2019 50(7):1011-1023.
Availability: Taylor & Francis. Available from: Taylor & Francis, Ltd. 530 Walnut Street Suite 850, Philadelphia, PA 19106. Tel: 800-354-1420; Tel: 215-625-8900; Fax: 215-207-0050; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Page Count: 13
Publication Date: 2019
Document Type: Journal Articles
Reports - Research
Education Level: Higher Education
Postsecondary Education
Descriptors: Mathematics Instruction, Professional Personnel, Teaching Methods, Reflection, Mathematical Logic, Thinking Skills, Mathematical Concepts, College Mathematics, College Freshmen, Algebra, College Faculty, Mathematics Teachers, Undergraduate Students
DOI: 10.1080/0020739X.2019.1657598
ISSN: 0020-739X
Abstract: In this study, we analysed a mathematician's teaching journals on eigenvalues and eigenvectors in a first-year linear algebra course. The research team employed Tall's ["How humans learn to think mathematically: Exploring the three worlds of mathematics." Cambridge University Press] three-world model of embodied, symbolic and formal as a framework for understanding the mathematician and teacher's pedagogical reflections as he led the class to the formal world. In order to reach the formal world, he used a sequence of tasks that emphasized embodied and symbolic, as well as formal thinking. The analysis of the journals showed that the mathematician faced challenges in leading the class towards the formal world. The study also revealed that the mathematician strived to build a concept image, that, while perhaps mirroring his own, did not resonate with the students.
Abstractor: As Provided
Entry Date: 2019
Accession Number: EJ1228550
Database: ERIC
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  Value: <anid>AN0138667521;imt01oct.19;2019Sep19.11:06;v2.2.500</anid> <title id="AN0138667521-1">Leading students towards the formal world of mathematical thinking: a mathematician's reflections on teaching eigentheory </title> <p>In this study, we analysed a mathematician's teaching journals on eigenvalues and eigenvectors in a first-year linear algebra course. The research team employed Tall's [How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press] three-world model of embodied, symbolic and formal as a framework for understanding the mathematician and teacher's pedagogical reflections as he led the class to the formal world. In order to reach the formal world, he used a sequence of tasks that emphasized embodied and symbolic, as well as formal thinking. The analysis of the journals showed that the mathematician faced challenges in leading the class towards the formal world. The study also revealed that the mathematician strived to build a concept image, that, while perhaps mirroring his own, did not resonate with the students.</p> <p>Keywords: Concept images; IOLA; reflections; eigenvalues and eigenvectors; formal world</p> <hd id="AN0138667521-2">1. Introduction</hd> <p>How do university mathematics teachers lead and guide students towards acquiring abstract linear algebra concepts? In this study, we examined a mathematician and collaborator's linear algebra teaching journals over the period of a semester as he reflected on his teaching. The literature maintains that reflection is an essential part of teaching mathematics (e.g. Davis, [<reflink idref="bib3" id="ref1">3</reflink>]; Davis, [<reflink idref="bib3" id="ref2">3</reflink>]; Moore-Russo & Wilsey, [<reflink idref="bib7" id="ref3">7</reflink>]). According to Dewey ([<reflink idref="bib4" id="ref4">4</reflink>]), reflection is 'active, persistent, and careful consideration of any belief or form of knowledge in the light of the grounds that support it and the further conclusions to which it tends' (p. 9). Fund ([<reflink idref="bib5" id="ref5">5</reflink>]) adds that 'teachers need to develop particular skills, such as observation and reasoning, in order to reflect effectively and should have qualities such as open-mindedness and responsibility' (p. 680).</p> <p>The instructor often emphasized that his goal was to reach the eigentheory section of the course, which motivated the focus of this paper. While studies suggest that undergraduate students have trouble learning the eigentheory concept, they also have shown several methods whereby this difficulty can be alleviated (e.g. Thomas & Stewart, [<reflink idref="bib18" id="ref6">18</reflink>]; Caglayan, [<reflink idref="bib1" id="ref7">1</reflink>]; Gol Tabaghi & Sinclair, [<reflink idref="bib6" id="ref8">6</reflink>]; Salgado & Trigueros, [<reflink idref="bib8" id="ref9">8</reflink>]). Thomas and Stewart ([<reflink idref="bib18" id="ref10">18</reflink>]) showed that students appeared confident with symbolic procedures (i.e. calculating the characteristic polynomial), but not embodied ones (i.e. linking diagrams and eigenvector properties). While the students referred to 'being stretched', or 'unchanging direction', when describing eigenvectors, it appeared very few actually implemented this definition in context (i.e. trouble determining whether is an eigenvector given that is an eigenvector). They also noted that despite their relative strength in symbolic manipulation, they did not as a whole properly symbolically manipulate the equation into the equivalent equation mainly due to the students apparent inability to properly think about the matrix in this context. Gol Tabaghi and Sinclair ([<reflink idref="bib6" id="ref11">6</reflink>]), as well as Caglayan ([<reflink idref="bib1" id="ref12">1</reflink>]), reported that usage of dynamic geometric environments (DGEs) appeared to help students learn the eigenvectors and eigenvalues concepts, perhaps by encouraging related embodied thought as Thomas and Stewart ([<reflink idref="bib18" id="ref13">18</reflink>]) suggested. Gol Tabaghi and Sinclair utilized a theory of instrumental genesis to talk about dragging modalities, and additionally leveraged the embodied cognition lens 'to analyse visual and kinesthetic understandings'. Due to the DGE's tendency to encourage students to reason in an embodied way which tended to include motion-based conceptions of eigenvectors and eigenvalues, Gol Tabaghi and Sinclair expanded Sierpinska's synthetic-geometric mode of thinking to <emph>dynamic-synthetic-geometric</emph>. They further reported that students narrowed in on a synthetic-geometric way of thinking about eigenvalues and eigenvectors through various types of dragging objects in the DGEs, including dragging an object in an experimental way (<emph>wandering</emph>), dragging in some purposeful way (<emph>guided</emph>), dragging to preserve a certain property (<emph>dummy-locus</emph>) and dragging along a line <emph>(line</emph>). They reported that the usage of <emph>Sketchpad</emph> made eigenvalues more apparently dilation factors than the corresponding symbolic process, and that students as a whole integrated the analytic-arithmetic mode of thinking with the synthetic-geometric mode. They describe two emergent features of DGEs as 'independence from the coordinate system', and 'both ways of thinking are based on properties of objects not calculations'. Caglayan ([<reflink idref="bib1" id="ref14">1</reflink>]) utilized a DGE in an unstructured way to show that mathematics majors successfully conceptualized eigenvectors and eigenvalues with a DGE even in an unstructured implementation. Salgado and Trigueros ([<reflink idref="bib8" id="ref15">8</reflink>]) claim that via a model designed to capture student interest, the students constructed an object conception of eigenvalues and eigenvectors the previous literature described as 'almost impossible to achieve'. In general, all of these studies make the case that students should be encouraged to coordinate multiple representations of eigenvectors (e.g. corresponding the geometric diagrams with the algebraic symbolic), and it seems that strategic use of models, DGEs and embodied representations are effective tools for reaching this goal.</p> <p>In this study, Tall's ([<reflink idref="bib16" id="ref16">16</reflink>]) three-world model of embodied, symbolic and formal mathematical thinking was used as a framework for understanding the mathematician and teacher's pedagogical reflections.</p> <hd id="AN0138667521-3">2. Theoretical framework</hd> <p>Building on Tall and Vinner's ([<reflink idref="bib17" id="ref17">17</reflink>]) notions of <emph>concept images</emph> and <emph>concept definitions</emph>, Vinner ([<reflink idref="bib19" id="ref18">19</reflink>], p. 69) believed,</p> <p>We assume that to acquire a concept means to form a concept image for it. To know by heart a concept definition does not guarantee understanding of the concept. To understand, so we believe, means to have a concept image.</p> <p>Developing these two notions further, Tall's ([<reflink idref="bib15" id="ref19">15</reflink>], [<reflink idref="bib16" id="ref20">16</reflink>]) three worlds of mathematical thinking (embodied, symbolic and formal) endeavours to lay out the individual mathematics learning journey, a path, from childhood to a research mathematician. According to Tall ([<reflink idref="bib15" id="ref21">15</reflink>]), the embodied world is based on 'our operation as biological creatures, with gestures that convey meaning, perception of objects that recognize properties and patterns ... and other forms of figures and diagrams' (p. 22). In other words, the various ways of thinking in the embodied world can also be characterized as giving body to an abstract idea. The symbolic world is the world of practicing sequences of actions which can be achieved effortlessly and accurately. In Tall's ([<reflink idref="bib15" id="ref22">15</reflink>], p. 22) views</p> <p>The world of operational symbolism involves practicing sequences of actions until we can perform them accurately with little conscious effort. It develops beyond the learning of procedures to carry out a given process (such as counting) to the concept created by that process (such as number).</p> <p>Finally, Tall defines thinking in the formal world as that which 'builds from lists of axioms expressed formally through sequences of theorems proved deductively with the intention of building a coherent formal knowledge structure' (p. 22). Tall's ([<reflink idref="bib16" id="ref23">16</reflink>]) framework considers a wide range of learners</p> <p>from those who struggle with mathematics to the gifted, taking account of the possible developments of those who require practical mathematics in their every day lives, theoretical mathematics in a wide range of applications, or formal mathematics at the frontiers of pure mathematical research. (p. 406)</p> <p>Using Tall's model of mathematical thinking, Stewart, Thompson, and Brady ([<reflink idref="bib11" id="ref24">11</reflink>]) investigated a mathematician's (and co-author) movements between the worlds while teaching algebraic topology. The instructor reported that students experienced the most difficulty in moving from the embodied world into the formal world. Believing the struggle would stimulate mathematical growth in his students, this instructor 'refused to give students proofs that were pre-packaged. More specifically, he wanted to provide students with intuitions and pictures that would help them understand the conceptual nature of the proof and ultimately lead them to it' (p. 2262). In a similar study, examining reflections of a mathematician and collaborator (Stewart & Schmidt, [<reflink idref="bib10" id="ref25">10</reflink>]) teaching abstract algebra, the instructor made a path through the worlds and helped students to walk through it. In his views 'Mathematics drives the class, I don't even think about pedagogy in some sense' (p. 47). In a different study, Stewart ([<reflink idref="bib9" id="ref26">9</reflink>]) created a set of linear algebra tasks designed to help students move between the three worlds. In another study, Stewart, Troup, and Plaxco ([<reflink idref="bib12" id="ref27">12</reflink>], [<reflink idref="bib13" id="ref28">13</reflink>]) examined a mathematics educator's (and co-author) decision-making moments and found that even though at times he anticipated students' difficulties with linear algebra concepts, his moves between the worlds left some students confused and initially were not as effective.</p> <p>These studies are generating some noteworthy results and indicate that movements between the worlds are a rich research topic worthy of more investigation.</p> <p>The research questions guiding the current study were: (a) Which routes did the mathematician take to move the linear algebra students to the formal world? (b) What were some of the challenges he experienced?</p> <hd id="AN0138667521-4">3. Method</hd> <p>This qualitative narrative study (Creswell, [<reflink idref="bib2" id="ref29">2</reflink>]) examined the linear algebra instructor's mathematical thought processes while teaching a first-course in linear algebra. This study took place over the course of a semester at a large research university. The analysis focused on a mathematician's reflections, as recorded through journal entries, over a five-day period, while implementing tasks from the Inquiry-Oriented Linear Algebra (IOLA) curriculum (Wawro, Zandieh, Rasmussen, & Andrews-Larson, [<reflink idref="bib20" id="ref30">20</reflink>]). While the instructor valued the visual aspects of the IOLA curriculum, this study is not intended to evaluate its efficacy. The research team consisted of a mathematician specializing in differential geometry (the instructor, postdoctoral fellow and co-author), two mathematics educators and an undergraduate research assistant student. Throughout the semester, the instructor recorded his observations on how his class reacted to a variety of teaching styles and ideas. He additionally met with the research team once a week throughout the semester and the following summer to discuss these experiences and reflections. This allowed the researchers to triangulate data via member checking with the instructor directly and additionally afforded him ample time to share a wide variety of teaching experiences, as well as his reasoning and thought processes while making these decisions. To collect additional data on the instructor's teaching from the student's perspective, the research team administered several surveys, and conducted a student interview. The research team converted the instructor's journal and the worksheet results into Excel spreadsheets to expedite coding and sorting the data to search for themes after coding. In keeping with a narrative study, the research team performed a retrospective analysis of the journal (Creswell, [<reflink idref="bib2" id="ref31">2</reflink>]) by iteratively coding the data. The team started with a combination of categories developed from the previous study (Stewart et al., [<reflink idref="bib12" id="ref32">12</reflink>]) and an open coding (Strauss & Corbin, [<reflink idref="bib14" id="ref33">14</reflink>]) scheme to allow for the possibility of discovering new categories unique to this study. The main themes for this study were: Teaching, Students, Class Activities, Math (instructor's math, students' math), Reflection and Tall's worlds. By instructor's math we mean, the math he was doing and talking about, and by students' math, we mean his reflections on students' mathematical abilities, and conversations on math in class. For the purpose of this paper, we will only present the analysis from the instructor's teaching journals.</p> <hd id="AN0138667521-5">4. Results</hd> <p>In this section, we will analyse the instructor's journals on five class periods of an introductory linear algebra course during which the fundamentals of eigentheory were presented. The class met three times each week for a period of 50 min. The classes were structured around a sequence of four tasks designed by the IOLA project (Wawro et al., [<reflink idref="bib20" id="ref34">20</reflink>]). The tasks use the ideas of 'stretch direction' and 'stretch factor' of a linear transformation to develop the formal notions of eigenvector and eigenvalue. Several of the requisite concepts, such as bases, coordinates and matrix representations of linear transformations, were covered earlier in the term so that the IOLA sequence could be used. In analysing his 5-day teaching segments, we will examine the instructor's (a) movements between Tall's ([<reflink idref="bib16" id="ref35">16</reflink>]) worlds, (b) pedagogical decision-making moments and (c) reflections on self and students. In addition, we provide diagrams that illustrate how the instructor understood each class period to be situated among the three worlds and the amount of emphasis on each world. The circles labelled with E represent embodied thinking, S symbolic and F formal, while the relative sizes reflect the given emphasis.</p> <hd id="AN0138667521-6">4.1. IOLA task 1: emphasizing on the embodied world and including some symbolic computation</hd> <p>The first IOLA task (see Figure 1) describes a linear transformation geometrically, in terms of 'stretch directions' and 'stretch factors', and presents three questions related to it. This task is primarily situated in the Tall's ([<reflink idref="bib16" id="ref36">16</reflink>]) embodied and symbolic worlds. By withholding any matrix representation of the transformation, the task was meant to force students to interpret the action of the transformation on vectors via the embodied world. Ideally, this will build intuition and facility. The instructor very quickly noted that students were having difficulty with embodied thinking and decided to take a more active role in guiding them through the task on the board. His next intention was to move students to a more symbolic representation of an idea of stretching, which he wrote as a 'mathematical one'. The instructor mentioned in his journals that the students struggled again.</p> <p>We needed to iron out the common misunderstandings: for every linear transformation the zero vectors gets sent to the zero vector, points are identified with vectors, etc. Then we needed to understand what stretching means. After one or two attempts and a geometric description, I asked for a mathematical one. Although no one could articulate it precisely, at least one student had the right idea: scalar multiplication.</p> <p>In question 2, the instructor symbolically computed the images of vectors under the transformation, and had a feeling that students were able to follow. However, their understanding faltered when the instructor presented a vector not in one of the stretch directions. 'So in question 2, we converted the two vectors into linear combinations of vectors in the stretching direction. Then used the linearity of the transformation to find their images. I'm not sure if this made sense to them'. In question 3, students did not give much feedback. The instructor gave a handout – the preview of the next task – and hoped that ' ... perhaps the motivated student will see the connection of how to use it and then be more prepared for the next task'.</p> <p>PHOTO (COLOR): Figure 1. IOLA task 1.</p> <p>This task was designed to activate students' embodied world thinking and help them to think visually about the concepts. By calculating the precise locations of the images of vectors and computing a matrix representation of the linear transformation, this task also bridged the embodied into the symbolic world.</p> <p></p> <hd id="AN0138667521-7">4.2. IOLA task 2: keeping the embodied ideas fresh, emphasizing on the symbolic world, introd...</hd> <p>The second IOLA task continued to build the concept image in much the same way as the first, but instead of a figure 'Z', there is a collection of discrete points (see Figure 2). Moreover, both the standard coordinate grid (referred to as the 'black' coordinates), and the one determined by the eigenvectors (referred to as 'blue' coordinates) are overlaid on the collection of points. At the start of the task, the instructor perceived that the students were not engaging with the tasks in a meaningful way. He remarked on having 'difficulty getting the students to be active participants'. As a result, he 'decided to do the worksheet together', meaning that he would guide the class by doing the various parts at the board. He conjectured that, 'part of the reason that the worksheet took so long was because most students don't have a facility with coordinate vectors'.</p> <p>PHOTO (COLOR): Figure 2. IOLA task 2.</p> <p>The instructor made the decision before the class started to present the definition of eigenvalue and eigenvector after the first two tasks. His rationale was that introducing them half way through gives some resolution to the first two tasks, while also providing a framework within which the last two tasks can be situated. In the journal, he explained that, the two class periods exploring the connection between coordinates and linear transformations would be sufficient as 'a segue to define eigenvalues and eigenvectors'.</p> <p>Despite recognizing the importance of everyday thought modes for developing concept images, the instructor still views the definition as the most important element in the concept image. Not only does he choose to present it after only two class periods, but he also expresses frustration at not arriving at the definition sooner. 'Finally, I was able to define eigenvalue and eigenvector.' In fact, he makes the decision to cut short the discussion of Task 2, Part 3 in order to present the definition. He remarked, 'Problem 3 was useful and I wish I had more time to go through it.'</p> <p>By representing vectors with respect to different bases, the students are situated in the symbolic manipulation. Yet the visual representation of the vectors on different grids and the definitions of eigenvalue and eigenvector connect this to activity to both the embodied and sybmolic worlds.</p> <p></p> <hd id="AN0138667521-8">4.3. Blending all three worlds of mathematical thinking</hd> <p>The instructor made the decision to use Day 3 not for the next IOLA task, but instead to synthesize the various embodied, symbolic and formal aspects of eigentheory that the students have so far encountered. To do so, he used exclusively a lecture teaching style. First, he showed how the black and blue coordinate matrix representations of the transformation from those tasks are related by conjugation by the change of coordinate matrix. Next, starting with the standard coordinate representation of the linear transformation, he used GeoGebra to demonstrate visually the effect to the linear transformation on vectors in the unit circle, and in particular how it exactly stretches some, but not all, directions. At this point, he reiterated the eigenvalue and eigenvector definitions, and derived the standard way of computing them from the characteristic polynomial and finding the nullspace of A – λI. From here, he presented a series of examples including the transformation from the IOLA tasks, an eigenspace with more than one dimension, and the differentiation operator acting on function spaces.</p> <p>The instructor did not make any remarks on how the students respond to the lecture. Instead, his journal entry was a rather clinical report of the content from the lecture, mainly the instructor's math and no mention on students' math. From this one could infer that the instructor was engrossed in conveying his own concept image, and how he experiences the mathematical concepts of eigenvalues and eigenvectors.</p> <p></p> <hd id="AN0138667521-9">4.4. IOLA task 3: computations in the symbolic world together with some formal theory</hd> <p>On Day 4, the instructor returned to the IOLA sequence with task 3. This task is the most like standard textbook exercises for eigentheory. For three distinct two-by-two matrices, the students are asked to (<reflink idref="bib1" id="ref37">1</reflink>) find the stretch factors given the stretch directions (see Figure 3), (<reflink idref="bib2" id="ref38">2</reflink>) find the stretch directions given the stretch factors and (<reflink idref="bib3" id="ref39">3</reflink>) find both the stretch factors and directions. After observing their work for the first part, the instructor noted that even though 'They had a WebWork assignment due the same day that was mostly about computing eigenvalues and eigenvectors', he 'was surprised to see how many were unsure where to start'. The WebWork assignment he mentioned contained only column vectors and matrices, while the IOLA task describes stretch directions. Hence, the instructor interpreted this as a lack of synthesis between the ideas of 'direction' and 'column vector'. This motivated the instructor's pedagogical decision to use the blackboard to guide the class through the task, reinforcing certain connections in the image concept. First, he 'decided to go slowly through some fundamental concepts that might be getting in the way of using the eigentheory'. Among the fundamental concepts that the instructor covered were the embodied-symbolic connection between nonzero vectors and 'directions' in the plane. Next, he reiterated how shapes in the plane can be thought of as collections of vectors. 'I think it's always worth repeating that a vector "lies in a shape or object" if the tail sits at the origin and tip sits at a point in the shape'. Also, he showed the class how finding the stretch factor (given the stretch direction) is equivalent to solving a linear system with one unknown and usually more than one equation. With these fundamental notions in place, he proceeded with the work of completing the task. As on Day 3, there was no mention of students' math in his journals.</p> <p>Graph: Figure 3. IOLA task 3, question 1.</p> <p>The instructor laments not showing how the linear system that must be solved to obtain the stretch factor will be inconsistent if it is set up with a non-stretch direction. 'What I should have done in addition, is to point out that when you choose a vector not in one of the eigenspaces, then solving for a stretch factor will lead to an inconsistent system.' Later, he regrets not connecting the formalism of solving linear systems to finding eigenvectors. 'But perhaps I should have gone through the derivation of the nullspace of a matrix, rather than appealing to their experience with WebWork calculations.'</p> <p>In this lecture, he equally stressed the symbolic and formal worlds, since, in using the concepts of eigenvalue and eigenvector, the students must first recall the formal definitions, and then move to the symbolic world to produce answer.</p> <p></p> <hd id="AN0138667521-10">4.5. IOLA task 4: emphasizing the formal and stressing the need for computation</hd> <p>The fourth IOLA task (see Figure 4) aimed to introduce students to a subtlety, thus far hidden, of eigentheory: multiplicity. The entire task involved a single linear transformation of <emph>R</emph><sups>3</sups>, presented as a matrix. As in the previous task, the first two parts involved finding either a stretch direction or a stretch factor, given the other. In particular, it is found that a certain stretch factor has two stretch directions. That is, the corresponding eigenspace is two-dimensional. The third and final part poses a rather provocative question: given that 2 and 3 are stretch factors and the former has two distinct stretch directions, could there be additional stretch factors? At the heart of this question is the observation that eigenvectors for distinct eigenvalues must be linear independent. A counting argument then shows that we already have a basis of eigenvectors and hence there can be no other eigenvalues.</p> <p>Graph: Figure 4. IOLA task 4.</p> <p>The instructor appears eager for the class to spend time with this last part. He makes the pedagogical decision to go 'through [the first two parts] together on the board. 'My hope was that this would put everyone on the same page to try the third part.' Once the students have had an opportunity to think about the third part, he observes:</p> <p>Every students' work that I saw was the same. To decide if there was another eigenvalue or stretch direction they all computed the characteristic polynomial to see if there was another root. I anticipated this, so I then presented a solution that crucially uses the fact that all three eigenvectors form a basis for R<sups>3</sups>. I did not get very much feedback from the class on whether they were internalizing this.</p> <p>Although there are multiple ways to approach the third part, the students all reached for the most symbolic, computable solution. They found the characteristic polynomial in order to find all the eigenvalues. He expected this and presented a contrasting formal solution. In this way, he hoped to show the students alternatives to the symbolic world, and perhaps build a connection between the concept of basis and eigentheory. The final piece of eigentheory was diagonalization. After presenting an example with insufficiently many stretch directions, the instructor was in a position to explain diagonalization and when it can be done.</p> <p>In this day, a higher emphasis was given to the formal world thinking. By asking the students to reason about the existence of eigenvalues, they were pushed to reason formally. However, to verify or test formal reasoning, they needed to use techniques from the symbolic world.</p> <p></p> <hd id="AN0138667521-11">5. Discussion and concluding remarks</hd> <p>Eigentheory is among the most sophisticated mathematical concepts the students have encountered, and the analysis of the five days revealed that the mathematician's emphasis on the three worlds of mathematical thinking gradually changed from more embodied and visual to more formal and abstract. This is in line with Tall's ([<reflink idref="bib16" id="ref40">16</reflink>]) view that, 'As mathematical ideas progress into more sophisticated levels, the balance between embodiment and symbolism changes' (p. 408).</p> <p>While the instructor's decision to use the IOLA tasks shows that he values the embodied and symbolic worlds as part of the concept image, the instructor's goal of reaching the formal world was extremely important to him. For example, on Day 5, the instructor tries to speed through what he considers 'rote' so that the class can get to something more formal that generates connections between concepts. In fact, his decision to present the definitions of eigenvalue and eigenvector at precisely the midpoint of the unit reflects the significance they hold for him. They represent, for the instructor, a single idea which unites the various notions from all three world that the students have been exposed to. A mathematical understanding of eigentheory (to him) involved primarily the definitions, but also how those definitions manifested themselves in the embodied and symbolic worlds. For example, he was able to think of a definition of eigenvector in symbols as described by the equation <bold>,</bold> in the embodied world as a picture of an image vector collinear with its preimage, and additionally various properties related to eigenvalues and eigenvectors in the formal world. The instructor believed the more connections between eigentheory and other linear algebraic concepts that he can convey to the students, the more robust their concept image. This belief is supported by the literature, which claims that students learn linear algebra more completely when they can connect eigentheory concepts across different modes of thinking (Thomas & Stewart, [<reflink idref="bib18" id="ref41">18</reflink>]; Caglayan, [<reflink idref="bib1" id="ref42">1</reflink>]; Gol Tabaghi & Sinclair, [<reflink idref="bib6" id="ref43">6</reflink>]; Salgado & Trigueros, [<reflink idref="bib8" id="ref44">8</reflink>]).</p> <p>The mathematician and teacher in this study has negotiated the mathematical journey himself, and knows the path well and values all three worlds of mathematical thinking. Nevertheless, he would still gravitate more towards the formal world as the most important part of a mathematical concept. In Tall's ([<reflink idref="bib16" id="ref45">16</reflink>]) view 'formal mathematics is more powerful than the mathematics of embodiment and symbolism, which are constrained by the context in which the mathematics is used' (p. 152).</p> <p>The instructor's objective was a mathematical treatment of eigentheory, hence, he used IOLA to present a web of connections surrounding the formal definitions. Throughout the course, the instructor tried to follow the objectives of IOLA materials designed for each task. His intention was to have the students work in small groups to complete each task first, and then come together as a class to discuss solutions. However, in many occasions when he noticed that progress among the students was much slower than anticipated, he often reverted to a more standard lecture format. While encouraging participation, he would lead the class through the tasks at the blackboard. In Tall's view, 'Over the longer term, embodied strategies may give insightful meaning at various stages of development, but as the mathematics becomes more complicated, symbolic strategies offer greater power and precision' (Tall, [<reflink idref="bib16" id="ref46">16</reflink>], p. 407).</p> <p>One may speculate that the instructor underestimated the time necessary for establishing new connections between mathematical ideas. What appears 'rote' and part of his 'everyday' mode of thinking is completely foreign to the typical undergraduate linear algebra student. Hence, the connections between the formal definitions and surrounding concepts that appeared so strong to the instructor were quite tenuous with the students. Tall ([<reflink idref="bib16" id="ref47">16</reflink>]) asserts that,</p> <p>working at a particular level- either in a community of mathematicians or as an individual learner operating with one number system prior to shifting to an extensional blend- the individual may feel comfortable at the current level yet the shift to another level may be a stimulating challenge for some while being problematic for others. (p. 410)</p> <p>The research team valued the instructor's reflections preserved in his teaching journals and found his dual role as a teacher as well as a researcher indispensable to this analysis. Although we discussed some reasons for the instructor's pedagogical decisions wherever possible, there are instances where such reasons are not as clear, even to the instructor. This may be due to the fact that he could not recall the precise circumstances of the decision.</p> <p>Our study suggests exploring ways of motivating students to achieve a more holistic understanding of linear algebra concepts across the three worlds. The research team is in the process of analysing the data from students' surveys and the interview. The results of this investigation will help the team to further explore Tall's framework, suggest ways to improve the instruction of linear algebra and most importantly, advance students' understanding of the abstract linear algebra concepts.</p> <hd id="AN0138667521-12">Acknowledgements</hd> <p>We would like to thank David McKnight, for his tremendous contributions to this project.</p> <hd id="AN0138667521-13">Disclosure statement</hd> <p>No potential conflict of interest was reported by the authors.</p> <ref id="AN0138667521-14"> <title> References </title> <blist> <bibl id="bib1" idref="ref7" type="bt">1</bibl> <bibtext> Caglayan, G. (2015). Making sense of eigenvalue–eigenvector relationships: Math majors' linear algebra – Geometry connections in a dynamic environment. Journal of Mathematical Behavior, 40, 131 – 153. doi: 10.1016/j.jmathb.2015.08.003</bibtext> </blist> <blist> <bibl id="bib2" idref="ref29" type="bt">2</bibl> <bibtext> Creswell, J. W. (2013). Qualitative inquiry and research design: Choosing among five approaches (3rd ed.). Thousand Oaks, CA: SAGE.</bibtext> </blist> <blist> <bibl id="bib3" idref="ref1" type="bt">3</bibl> <bibtext> Davis, E. (2006). Characterizing productive reflection among preservice elementary teachers: Seeing what matters. Teaching and Teacher Education, 22, 281 – 301. doi: 10.1016/j.tate.2005.11.005</bibtext> </blist> <blist> <bibl id="bib4" idref="ref4" type="bt">4</bibl> <bibtext> Dewey, J. (1933). How we think: A restatement of the relation of reflective thinking to the educative process. Boston, MA : D. C. Heath & Company.</bibtext> </blist> <blist> <bibl id="bib5" idref="ref5" type="bt">5</bibl> <bibtext> Fund, Z. (2010). Effects of communities of reflecting peers on student-teacher development e including in-depth case studies. Teachers and Teaching: Theory and Practice, 16, 679 – 701. doi: 10.1080/13540602.2010.517686</bibtext> </blist> <blist> <bibl id="bib6" idref="ref8" type="bt">6</bibl> <bibtext> Gol Tabaghi, S., & Sinclair, N. (2013). Using dynamic geometry software to explore eigenvectors: The emergence of dynamic-synthetic-geometric thinking. Technology, Knowledge and Learning, 18 (3), 149 – 164. doi: 10.1007/s10758-013-9206-0</bibtext> </blist> <blist> <bibl id="bib7" idref="ref3" type="bt">7</bibl> <bibtext> Moore-Russo, D., & Wilsey, J. (2014). Delving into the meaning of productive reflection: A study of future teachers' reflections on representations of teaching. 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International Journal of Mathematics Education in Science and Technology, 48 (1), 40 – 49. doi: 10.1080/0020739X.2017.1360527</bibtext> </blist> <blist> <bibtext> Stewart, S., Thompson, C., & Brady, N. (2017). Navigating through the mathematical world: Uncovering a geometer's thought processes through his handouts and teaching journals. In T. Dooley & G. Gueudet (Eds.), Proceedings of the tenth congress of the European society for research in mathematics education. (pp. 2258 – 2265). Dublin : DCU Institute of Education and ERME.</bibtext> </blist> <blist> <bibtext> Stewart, S., Troup, J., & Plaxco, D. (2018). Teaching linear algebra: Modeling one instructor's decisions to move between the worlds of mathematical thinking. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education. (pp. 1014 – 1022). San Diego, CA : The Special Interest Group of the Mathematics Association of America (SIGMAA) for Research in Undergraduate Mathematics Education.</bibtext> </blist> <blist> <bibtext> Stewart, S., Troup, J., & Plaxco, D. (in press). Reflection on teaching linear algebra: Examining one instructor's movements between the three worlds of mathematical thinking. The International Journal on Mathematics Education.</bibtext> </blist> <blist> <bibtext> Strauss, A. L., & Corbin, J. (1998). Basics of qualitative research: Grounded theory procedures and techniques (2nd ed.). Newbury Park, CA : Sage.</bibtext> </blist> <blist> <bibtext> Tall, D. O. (2010). Perceptions operations and proof in undergraduate mathematics. Community for Undergraduate Learning in the Mathematical Sciences (CULMS) Newsletter, 2, 21 – 28.</bibtext> </blist> <blist> <bibtext> Tall, D. O. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. 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This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.</bibtext> </blist> </ref> <aug> <p>By Sepideh Stewart; Jonathan Epstein and Jonathan Troup</p> <p>Reported by Author; Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib18" firstref="ref6"></nolink> <nolink nlid="nl2" bibid="bib16" firstref="ref16"></nolink> <nolink nlid="nl3" bibid="bib17" firstref="ref17"></nolink> <nolink nlid="nl4" bibid="bib19" firstref="ref18"></nolink> <nolink nlid="nl5" bibid="bib15" firstref="ref19"></nolink> <nolink nlid="nl6" bibid="bib11" firstref="ref24"></nolink> <nolink nlid="nl7" bibid="bib10" firstref="ref25"></nolink> <nolink nlid="nl8" bibid="bib12" firstref="ref27"></nolink> <nolink nlid="nl9" bibid="bib13" firstref="ref28"></nolink> <nolink nlid="nl10" bibid="bib20" firstref="ref30"></nolink> <nolink nlid="nl11" bibid="bib14" firstref="ref33"></nolink>
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  Data: In this study, we analysed a mathematician's teaching journals on eigenvalues and eigenvectors in a first-year linear algebra course. The research team employed Tall's ["How humans learn to think mathematically: Exploring the three worlds of mathematics." Cambridge University Press] three-world model of embodied, symbolic and formal as a framework for understanding the mathematician and teacher's pedagogical reflections as he led the class to the formal world. In order to reach the formal world, he used a sequence of tasks that emphasized embodied and symbolic, as well as formal thinking. The analysis of the journals showed that the mathematician faced challenges in leading the class towards the formal world. The study also revealed that the mathematician strived to build a concept image, that, while perhaps mirroring his own, did not resonate with the students.
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