Download What is a Vector to Students? and more Exams Vector Analysis in PDF only on Docsity!
What is a Vector to Students?
Inyoung Lee Arizona State University
This study presents linear algebra studentsā vector conception found in the least-squares solution context through an IOLA (Inquiry-Oriented Linear Algebra) CTE (Classroom Teaching Experiment). Studentsā reflection writings after the CTE are the data source. Using a previously found student conception of vector in another study as a basic framing, the CTE data have been analyzed to investigate how students used the word āvectorā and what they referred to. This study offers a framework, a tool to be useful in a wide range of describing student conception of a vector emphasizing their natural way of thinking of a vector and their use of the vector.
Keywords: Vector, Linear Combination, Span, Linear Algebra, Student Thinking
Vectors are widely used in mathematical sciences. In Calculus, a vector is represented by an arrow or a directed line segment which has both magnitude and direction. Students work with functions of two or more variables incorporating vectors in that course. In linear algebra, vectors are extensively used as students learn the key concepts such as linear (in)dependence, span, basis, and vector spaces. While a significant body of studies have explored student conception of domain-specific contents, less study on student thinking of vectors has been done in linear algebra. This proposal foregrounds vectors and investigates studentsā notion of vectors to answer the research questions: How do linear algebra students think of a vector? And how do they use the vector and what do they refer to?
Literature There is little literature exploring studentsā thinking of a vector in mathematics. Hillel (2000) identified three modes of vector representation: directed line segments (geometric mode), n- tuples (algebraic mode), and elements of vector spaces (abstract mode). In the geometric mode, students interpret a vector as an arrow having the magnitude and the direction with initial and terminal points. In the algebraic mode, students perform vector operations component-wise. In the abstract mode, a vector is described as an element of a vector space, and its vector operation is performed according to the axioms of that vector space. Along the similar strand, Sierpinska (2000) defined the terms, āsynthetic-geometric thinkingā that mathematical objects are given to studentsā mind directly being seen as a shape lying in space, and āanalytic-arithmetic thinkingā and āanalytic-structural thinkingā that mathematical objects are being component-coordinatized in a given dimension. Watson et al. (2003) found that students have the imagery of āa journeyā
when adding vectors. For example, students think that the equality in = holds because it is moving from A to B and finally to C. But this may become problematic when students
interpret the commutative property for addition such as since the right side does not give a satisfying meaning to them with the journey reasoning. Kwon (2011, 2013) focused on the progress students make from embodied vector to abstract vector and provided a vector framework attending to the ontological and epistemological aspects. It incorporated Hillel (2000) and Siepinska (2000)ās modes of description and reasoning into the framework. Appova & Berezovski (2013) identified studentsā misconceptions and error patterns about vector operations. The students in their study did not distinguish between a vector and a scalar in
operations, for example, a vector minus a scalar equaling another vector or a scalar. Mikula and Heckler (2017) developed a framework to design essential skills on vector math and implemented an online training for introductory physics students. They categorized the skills required for vector operations such as addition, subtraction, dot products, and cross products using both i, j, k and arrow representations.
Theoretical Framework This study is grounded in the theoretical perspective which makes a distinction between studentsā mathematics and mathematics of students (Steffe & Thompson, 2000). Studentsā mathematics refers to the studentsā mathematical reality which we cannot access directly, whereas our interpretation of the studentsā mathematical reality is referred to as mathematics of students. Even though models constructed in this study are not representing their mathematics perfectly, it is still worth constructing the models because studentsā mathematics is indicated by observable behaviors such as the studentsā gestures, written work, and discourse occurring in their mathematical activities. In this paper, the hypothesis that various ways a student thinks about a vector are closely related to their written description of the vector and to their use of the vector in task setting, is the theoretical foundation in developing the conceptual framework and analyzing data. The conceptual framework used to analyze data in this study was developed as a tool to be useful in a wide range of describing student conception of a vector. It was initially the result of the authorās unpublished class project in which task-based clinical interviews were conducted with students who had no or little experience with linear algebra at that time. The initial framework consists of five different vector conceptions: Vector as a point, Vector as a displacement, Vector as a direction, Vector as an equivalence class, and Vector as a linear combination of other vectors. Its geometric description of the categories is shown in Figure 1. Even though a vector can be in any dimensional space, the descriptions of the categories are
illustrated in because the interview tasks mainly included 2D vectors. Vector as an arrow is set as a default because it was the initial description of vector most students made in the study.
Figure 1. Geometric description of vector conception
Using the initial framework as a basic lens, the author began coding studentsā data from a classroom teaching experiment (CTE) implementing a recently designed unit about least squares method in linear algebra. The framework was further developed by the author as the vectors in the task of the CTE include various aspects categorized under āstudent conception of a vectorā and āhow they use the vectorā in the problem context.
Vector conception categories Vector as an arrow : Students describe the tip of a vector as referring to a line segment. Vector as a point : Students use the word āpointā to describe a vector or they attend to the tip of a vector considering the tail as if it were at origin.
using the initial framework, a new set of codes emerged, the author used the updated framework to complete the coding. The author specially focused on studentsā explanation before and after the word vector when coding. When students used the symbolic notations from previous math classes to refer to a vector such as < š, š, š >, (š, š, š) or šš + šš + šš, their meaning is also analyzed. All the names used in Results are pseudonyms.
Results As shown in Figure 2, the e vector that Gauss travels and the p vector from Home to Meetup are important pieces in answering the questions of the task sequence of least squares. While Gauss can travel around freely to meet you, your movement is restricted by the transportation vectors given in the problem context. The analysis on studentsā reflection writings revealed that their conception of a vector and their use of a vector are multifaceted. In this section, their notion of vector and their use of vectors in the Meeting Gauss task will be briefly presented focusing on the two vectors e and p.
Vector Conception- Describing the e vector that Gauss travels. Vector as an arrow : In the initial framework, this category was defined as a default, however, the author noticed that vector as an arrow conception is a way of thinking characterized when students describe the tip of a vector as referring to a line segment. For example, āGauss should meet you at the shortest point from the tip of his vector line to youā (Anton). Anton describes the path Gauss travels referring to it as an arrow. Vector as a point: Students use the word āpointā to describe a vector or they attend to the tip of a vector considering the tail as if it were at origin. Damien wrote, āGauss should meet us at the point (1.35,0.06, 2.21) ā¦the vector (Gauss travels) should be [1.35; 0.06;2.21] as that is the point he needs to reach anyways ā. Damien found the coordinates of Meetup using the GeoGebra applet and used them as the coordinates for the Gauss travel vector e. The tail of the vector e is at the Gauss location, not at the origin. Vectors and points have the same coordinates only if the vector has a tail at the origin. Another student, Annalisa noted ā I understand everything up until we had to actually find the point/vector to meet at ā. Annalisa interprets the Meetup in two ways, a point and a vector. She did not mention anything other than this quote in her writing, but still this is an indication that a vector could be identified as a point or vice versa. Vector as a segment: Students conceive a vector as a connected segment between two locations. Specifically, they use the word ādisplacementā or āmagnitudeā to describe a vector attending to the length of the vector. For example, Owen noted, āGauss will travel along (-1-a,1- b,4-c) which is the displacement vector between Gauss's old and new location .ā Owen seemed to write the Meetup as (a,b,c) and subtract it from the š vector to describe the Gauss travel vector e. His description of a vector includes two locations and a line segment connecting them. Another student, Luka wrote āWe described the entire situation as a right triangle, where the hypotenuse is the vector between the origin and Gauss , the vertical component is the vector between the plane and Gauss , and the horizontal component is the net distance travelled via hoverboard and magic carpet ā. Lukaās description illustrates that each side on the right triangle is a vector which connects the locations among Home, Gauss, and Meetup. Vector as a direction: Students use the word āvectorā referring to the span of the vector or the direction of the vector. Span of a vector refers to a line scaled either extended or shrunk along the vector. Consider the following examples, āGauss would need to move to a point that is along the line of the vectors givenā(Bianca) āThe direction he'd travel can be found with the cross product: [[i j k] [1 1 1] [6 3 8]] = 5i - 2j - 3k = [5 -2 -3] <- so either that direction or
reversed. ā (Vaki). Their description for the e vector indicates that they attend to the scaled line or the direction Gauss travels rather than the length of the vector or the vector itself.
Use of Vector- Traveling from Home (origin) to Meetup Over and up alternations: Students use vectors to get to Meetup stair-wise. Hanora noted, āSometimes it might be shorter if I moved a certain distance so he can reach me without using many modes of transportation to make zig-zags .ā and Dilon wrote ā Gauss should meet us along one of our vectors. If he wants the shortest trip he should meet. He would use V1 to get there and his distance would be very short, only 1 unit in each direction. .ā Hanora and Dilon seemed to understand the task context differently than intended, but Hanoraās description on the use of transportation vectors indicates that vectors move in over-and-up manner to get to the Meetup. Also, Dilonās explanation reveals that he recognizes šš = [ 1 , 1 , 1 ] consists of stair-wise movement 1 unit in each direction. Scaling: Students use vectors to get to Meetup by scaling. Vaki noted, āWe would get to the meeting place by using the hoverboard for 1.5 hours and the carpet for 0.5 hours .ā The hours are the amount spent on each transportation, which indicates that Vakiās vector moves in scaling manner. Iliana answered, ā I would get to the meeting point by using a combination of my transportation methods, or by using a linear combination of the given vectors .ā Also, Lurenya mentioned, ā Since Gauss is getting to the plane that is the span of the three modes of transportation , a linear combination of the three vectors will get us there as well .ā In Iliana and Lureynaās description, they would use three vectors to reach the Meetup using the span or linear combination of the transportation vectors, which indicates that they find vectors varying in length. Constructing a space: Students use vectors in generating a space such as a line or a plane. Ani noted, ā The way I am thinking about this is to locate a spot that is within the area that the 3 vectors span and that Gauss could go to. To do that, I would find the plane that the 3 vectors together span ā¦ ā and Cecil ā What I'm imagining is that if we created a "basis plane" with these two vectors then we can map that plane as well as the point Gauss is atā. There were many students who wrote the plane to be generated by the transportation vectors when they use the word linear combination and/or span.
Discussion This study, to investigate student conception of vectors and their use of vectors, began as studentsā data were analyzed with a different focus. The author noticed that the Meeting Gauss task scenario entails various ways of thinking with vectors as exploring ways to answer the guiding questions. By foregrounding vectors, student conception and its use were briefly presented using the conceptual framework developed as analyzing their data. The author finds this study important in developing instructional sequence of least squares and even linear algebra because vector is often backgrounded in studies even though student thinking of vector is multifaceted.
Acknowledgements I would like to thank the IOLA-X team members for giving me a permission to use the data collected and de-identified by them. This material is based upon work supported by the United States National Science Foundation under Grant Numbers 1914793, 1914841, and 1915156. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
