Students Perception and Intuition of Dimension within Linear Algebra Brad Taylor Virginia Polytechnic Institute and State University Btaylor8@vt.edu The purpose of this study is to investigate how students understand dimension with respect to linear algebra, and how a preconceived notion can affect this understanding. The data used in the study, collected during individual interviews, were college student responses on example generation questions asking for the possibilities of span and linear independence of column vectors in non- square matrices. The framework used when examining the responses was based off Tall s worlds of mathematics (2004) and how this can be applied to student s notion of dimension.the results suggest that students use of dimension can be captured by both the embodied world as well as the formal world, one using more real world perception while the other aligns with the linear algebra definition of dimension. Introduction University level linear algebra is a course that can be very difficult and frustrating for students. In many cases, students are learning about proofs and learning to use proofs while learning the new concepts within linear algebra. These concepts lead to many new ways of thinking that are not always easy to grasp. For instance, taking the concepts of matrices as a set of vectors, learning what linear transformations are and how they can apply to new vocabulary terms such as the span, rank, basis, null- space and many others. Within linear algebra, the definition of dimension of a vector space refers to the cardinality of the basis for that vector space. Yet there is a certain real- world aspect of dimension that can affect how students think of dimension with respect to when trying to explain a basis, vector- space or sub- space. This visual aspect of the understanding of dimension can be limiting when a student is trying to understand different concepts in linear algebra. There seems to be an obstacle of formalism (Dorier, 2000) experienced in the notion of dimension in which students are forced to leave the perceived world and use the formal definition to comprehend the material. This research paper examines how students use the word dimension in their explanations of example generating questions. Using Tall s (2004) three worlds of mathematics, students explain dimension in very different ways. The three worlds are the embodied world (looking at the perception of the world around), the symbolic world (using mathematical symbols to explain reasoning), and the formal world (using definitions and axioms as explanation). Using these three worlds, different parts of the student understanding could be categorized with respect to dimension.
Literature Review Hillel (2000) writes about the difficulties of teaching linear algebra in college. It is often the first proof- based course students will take and this alone takes a good deal of getting used to, on top of all the new definitions students will be learning to use. The author then describes the different modes of description within linear algebra. These are the algebraic view, the abstract view, and the geometric view. Hillel (2000) describes each of these views with respect to his three main factors in linear algebra which are the vectors themselves, vector operations and transformations. While Hillel doesn t actually talk about dimension, these different modes can be applied to how students think about the notion of dimension. There is a very abstract view of dimension, the understanding of it to be the cardinality of the basis. Then on the other side is a geometric understanding, for instance when thinking about the difference between two- dimensional and three- dimensional, which is the addition of a direction. Tall (2004) writes about very similar things as Hillel (2000), but he calls his differentiation, the three worlds of mathematics. These worlds can be applied to all branches of mathematics and these worlds are the embodied, symbolic, and formal worlds. When people think about the world around us and how these perceptions can be used in math, they are thinking about the embodied world. When using the numbers and symbolic representations to display our thinking, people are using the symbolic world. When using formal definition to display thinking, people are using the formal world. When applied to Linear Algebra like in Stewart (2010), we can see that the embodied view can be thought of as the graphical representation of a vector space, the symbolic world can be thought of as matrix manipulation or the algebra behind a linear transformation, and the formal view can be thought of as understanding definitions such as span, linear independence and dimension. Stewart (2010) cross- references the worlds of Tall (2004) with action process object schema (APOS) Theory and how it applies to linear algebra. She further splits up the symbolic world into an algebraic representation and a matrix representation. Stewart then shows what examples of basis and span would be in this three by four framework table. While she does not specifically talk about dimension, this paper is very useful in how the author relates Tall s (2004) worlds to linear algebra. Stewart s use of APOS is not as relevant to the present study due to the fact that these are meant to build on understanding which is not what this paper is about. Britton and Henderson (2009) talk about the difficulties of the formal nature of linear algebra. Britton states that there is an obstacle of formalism in regards to a linear algebra fog that students have when they start the course. I feel that this is evident in student understanding of dimension when trying to go from prior knowledge of perceived dimension to the formal definition. Zandieh s (2000) paper examines student understanding within the concept of derivative. While this was not the focal point of this research paper, the framework which she used was helpful in the creation of the initial table (Figure 1). Her table includes the
specific concepts, crossed with the level of understanding of a function and proceeds to fill out a table per student in order to gain her results. Methods This research project is part of alarger study investigating students understanding of span and linear independence.the data for this paper draws from semi- structured individual interviews (Bernard, 1988) conducted with seven first- year, honors STEM majors. Each interview, conducted midway through the semester, lasted between 90 and 120 minutes. Each task in the interview protocol was designed to engage the participants primarily in one type of mathematical activity (Wawro&Plaxco, 2013). As students responded to each task out loud to the interviewer, and the interviewer often asked follow- up questions with the goal of understanding the student s response and way of thinking as much as possible. After watching the full videos of all the students, the question (the sixth in the interview) that I chose to analyze for this research was one designed to engage students in the mathematical activity of example generating: Create an example of a 3x5 matrix B such that the column vectors of B span R 3 and are linearly independent. If this is not possible, explain your reasoning why. Create an example of a 5x3 matrix C such that the column vectors of C span R 5 and are linearly independent. If this is not possible, explain your reasoning why. I then transcribed students responses to this question in the interviews; this allowed me to be able to look back at how the students specifically answered different aspects of the question. While doing this, I was also reading published research papers on student thinking in linear algebra as well as papers written aboutthe ability to determine student success through previously attained knowledge. Using some ideas for a table style framework from Zandieh (2000) and thinking about different modes in linear algebra from Hillel (2000), I attempted to create a table in order to look for correlations between different aspects of student thinking and different modes in linear algebra. The table in Zandieh (2000) had specific mathematical concepts across the top with different ways of thinking along the side. A table was created for each student and marks were put in the table to show how the students were thinking. The original table in this research paper had parts such as initial justification, visualization, first example generated, identity matrix, and things you would change (figure 1). Each student was then crossed with these ideas and filled out in order to create comparisons across one another. I decided this method was not givingenough concise information and I needed to look for a more narrowed down subject. I attempted to arbitrarily create different modes of thinking based off different ways I had perceived thinking about linear algebra.yet I had not clearly defined these modes and decided there was not enough research information for me to accurately create this. I found aresearch paper by Tall (2004) which talked about these
embodied worlds of mathematics and decided to try and instead apply this to the interviewees transcripts. While transcribing and labeling, the word dimension kept coming up as an interesting aspect to focus in on. When trying to label the sentences with the word dimension in it, they would be labeled differently each time, making it seem that there may be different ways to think about dimension. I decided to focus in on these passages and see what I could find on dimension. The results section goes into detail how the notion of dimension is used in the example creating question. Results All 7 of the students interviewed eventually realized that it was not possible to have the columns of a 3x5 matrix be linearly independent, as well as that fact that the columns of a 5x3 matrix cannot span R 5.Thus, the results in this paper are not based on correctness, but rather on student thinking through example generating question. The results found in Figure 1 gave a good deal of information about the overall thinking of each individual student. Yet, there were no distinct connections that I could be certain would hold true, due to the small sample size of the research. Narrowing in on one specific term could allow for a more definitive set of results. Figure 1 Name Initial Justification Visualization 1 st Example Generated Identity Matrix? Carley they would have, ya know, Did not think of 1 0 0 1 3 Yes, see 1 st too many vectors or too the 0 1 0 2 0 example few vectors, and they (in)dependence 0 0 1 0 1 would be in the wrong, with a visual Emily Josh Mark Larry So there will be 5 column vectors in R 3. So they cannot be linearly independent and span R 3 once you get past 3, if you get because 5 is above the 3. And this is not possible because uh, not possible (writes it) because we ve got like in order to be linearly independent in R3 it needs to have 3 vectors that are length 3 Those guys could be linearly independent, then when you start adding in more, you re having um, more column vectors than are in, more column vectors than n, which n equals 3 in this case. No, cause its over determined so that can t be linearly independent representation When looking at R 5 said it is not drawable Uses markers as vectors to show the linear dependence as addition of vectors Talks about a 3 dimensional space being a cloud, then trying to go to 4 cannot happen in 5x3 matrix Does not think of it visually 1 0 0 0 1 0 0 0 1 10 100 5 12 6 8 Yes, see first example. Yes, in second example. Yes, in second example. No, wrote the anti- identity What would change?. Change the fact that it has to be linearly independent.
Jimmy Andre w Which means that they um, there are 5 column vectors in a three dimensional space so they can t um, they may span it so they span R 3 // We know immediately that this has to be linearly dependent. Talks about a 3- dimentional subspace in R 5 Does not think about it visually Yes, see first example No, said he shouldn t With what I learned from Figure 1 in mind, coupled with the highlighting of the transcript,i found many interesting possibilities of ideas I could delve into deeper. One thing I noticed was that 5 of the 7 students mentioned the word dimension. Not only that, but the word itself was used in a variety of different ways, seeming to have some variation of meaning. I looked into how these could be split up, because when I tried to use highlighting, I found myself torn between which sections some of the passages with dimension should be placed. A few of the students used the word dimension to explain their thoughts with the perception of reality, mentioning a three dimensional space as a room or a cloud. Other times, students used dimension with respect to the formal definition such as talking about the dimension of a vector space as the number of linear independent vectors in a set. These, as well as other explanations, seemed to resemble Tall s (2004) worlds of mathematics. The first example resembled the mathematical thinking belonging to the embodied world, while the second was more in the formal world. The results section is split up into two parts; the first part details the different ways students used dimension to create and explain their examples, and the second looks at different commonalities that can be seen across the students. Examples of student thinking The five students that talked about the matrices, vector spaces, or vectors in terms of dimension did so in very different ways. One student, Carley, who talked about visualizing the 5- dimensional vector space that the column vectors of a 5x3 matrix live in, was struggling to explain why she thought you might need a square matrix to span R 5. When talking about what is needed to span she says I m calling it direction but I m not sure if it s really a direction. Um, and you can t really get that unless you have at least 5. The idea of calling each vector a direction seems to be a very visual way to think about span and how it relates to dimension. Where she struggled was in that she couldn t visualize a fifth dimension. She chose to call each vector a direction because that is what she is used to do so in three dimensions. When thinking about Tall s (2004) worlds, she seems to be trying to use the embodied world of dimension to explain something in R 5 and is restricted due to the three dimensional world we live in. When responding to the question aboutthe 5 x 3 matrix, another student, Emily, states, Pretty much that you re going to want it to span R 5 and so, um, if the set is S, then the dimension of the set is only going to be equal to three. She is explaining that you cannot span R 5 with the columns of a 5x3 matrix based off the fact the dimension of the set is three due to the three linearly
independent column vectors. She is trying to use the formal definition of dimension in this explanation although it is not mathematically correct to say the dimension of a set. Even before this she uses the definition of dimension when trying to explain the need for the columns of a square matrix to span the associated vector space and be linearly independent. She states, So if you have a set, if you wanted to span R 2 then you will need the dimension of [that set] to be equal to 2. She goes on to say that being in R 2 means you will have two vectors of dimension two, which isn t formally correct but what it seems she is trying to display is the fact that the dimension of the vector space these vectors in is would be two. Josh looked at the first problem with the 3x5 very visually, holding up markers to represent the vectors in the 3x5 matrix. He could do so, stating that the vectors of length three, meant they were in a three- dimensional space. When engaging in the second question, which dealt with a 5x3 matrix, he did not try and use markers, but instead explained that spans a 3 dimensional space inside of like inside of the 5 dimensional space. He stated that it was going to be difficult to explain graphically because he did not know what the fourth and fifth dimensions looked like. To understand the 5x3 matrix as a set of column vectors that span a three dimensional subspace inside a five dimensional vector space, Josh must have a solid understanding of the formal definition of dimension to be able to accurately compare it to the embodied view of dimension. Mark also thought of the span of the column vectors of the five by three matrix as a three- dimensional vector space being inside some five dimensional vector space. He called the three dimensional space a cloud and stated that the reason it could not span the whole vector space of R 5 was because there was no way to get to that next dimension. Mark wrote out his examples in a very number oriented way but then explained what he was doing in a very visual sense. This also shows that he has a strong enough understanding of the formal definition of dimension in order to be able to relate it to the embodied world. Finally, Jimmy seemed to have a very good understanding of the relationship between the matrix and the vector space the column vectors were elements of, and he used two separate ways to answer the two questions. In the 3x5 example, he talked about the notion of having five column vectors in a three- dimensional vector space, which would mean that there were too many vectors and the set could not be linearly independent. Then in the 5x3 example, he talked about the subspace that the three vectors created, stating that [The vectors] can only span a three dimensional subspace of R 5 just as Josh and Mark said. Comparison of dimension There were two main themes that came up in the way students mentioned dimension, and in some cases the students would use both in explaining the example matrices they
generated. One way the students used dimension in their explanations was to try and relate it to their perceptions of dimension in the real world. This seems very similar to the examples of the embodied world from Tall (2004). While there was no real symbolic world for dimension, I think it could be said that an understanding of the linear dependence/independence and number of column vectors can give the dimension. This would be a symbolic way of finding dimension, even though there isn t really a symbolic way to represent it.student understanding in the formal world appears when students talked about the dimension as the number of linear independent vectors in a set. For instance, saying that the columns of a 5x3 matrix can span a three dimensional subspace shows a students knowledge of dimension in the formal world. Based on the research done from these interviews, four of the students had a strong understanding of dimension in both the formal and embodied world. Carley did not seem to have as strong of an understanding of the formal world, making it more difficult to solve part be of the question which had a vector space of R 5. Once she understood the first part, she immediately resorted to saying it was difficult to visualize the five dimensions. She looked back at the first question, implying that the 5x3 must not be possible because the 3x5 was not possible. She did not seem to have a clear understanding of why the column vectors did not span R 5, other than just stating I mean I feel like it just makes sense that you should have at least the same number of vectors as this number [n in R n ]. In comparison, Josh first picked up markers to explain how a 3x5 matrix could not represent a linearly independent set of vectors, yet as soon as he saw the next question was in R 5 he went straight into writing his example matrices showing that the three vectors could not span all of R 5. His reasoning, along with Jimmy and Mark, were all similar, along the lines that they could span some three dimensional subspace inside of R 5 which represents a five dimensional vector space. Each of these three students had knowledge enough about the matrices and what it meant for the dimensions of the vector spaces, that they could give algebraic representations of the matrices and give a very embodied description of why each matrix would not work for the given example. The students were able to get past the restrictions of the embodied world due to their understanding of dimension in the formal world. Conclusion While there has not been much research done on student understanding of dimension and its usefulness in linear algebra, it seems that based off the research, it is a very important tool in student understanding of vector spaces and their relation to a matrix as a set of vectors. The role of dimension in this example generating question came not in the initial writing of the matrix, but the explaining of why the matrix could not fulfill the question requirements. The students with a strong formal understanding of dimension in the realm of linear algebra were able to relate it to the embodied view of dimension in order to explain span and linear independence in laymen s terms.
When teaching linear algebra, teachers will have a tough time trying to force students out of their perception of dimension in the embodied world. The notion of dimension is know and heard long before it is taught it a mathematics classroom. You hear about movies being 3- D, people talk about time being the fourth dimension and understand the three dimensional space we live in. Then, this idea of dimension is further exhibited when students learn about the Cartesian plane in algebra 1 (Vitsas & Koleza, 2000). This allows students to see functions on the x- y plane, and when switching to vectors, it is easy to see why students want to create them starting at the origin and going to the points listed in the column vector. This is, for obvious reasons, restricted by three dimensions, leaving a gap for how students first perceive vectors, and how they are being taught in linear algebra. Based on the small sample of interviewees in the research, it seems that students who have been able to bridge the gap of dimension, are able to understand many other aspects of linear algebra in a much more conceptual manner than those who struggle with relating to the formal definition. References Bernard R. H. (1988). Research methods in cultural anthropology. London: Sage. Britton, H. & Henderson, J. (2009). Linear algebra revisited: An attempt to understand students conceptual difficulties. International Journal of Mathematical Education in Science and Technology, 40(7), 963-974. M. Bogomolny, (2007) Raising students understanding: linear algebra, in: J.H. Woo, H.C. Lew, K.S. Park, D.Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, vol. 4, PME, Seoul, 201 208. J- L. Dorier, A. Robert, J. Robinet, and M. Rogalski. (2000) The obstacle of formalism in linear algebra, On the Teaching of Linear Algebra. 85 124 Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.- L.Dorier (Ed.), On the teaching of linear algebra 191-207 Stewart, S., & Thomas, M. O. (2009).Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematical Education in Science and Technology, 41(2), 173-188. Tall,D.O. (2004) Building theories: The three worlds of mathematics, Learn. Math. 24(1), 29 32. Vitsas, T. &Koleza, E. (2000). Student s misconceptions on the concept of dimension, Proceedings of the 2nd Mediterranean Conference on Mathematics Education. 108 119 Nicosia: Cyprus Mathematic Society- Cyprus Pedagogical Institute.
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