Computers and Linear Algebra

Hamide Dogan-Dunlap Mathematical Sciences University of Texas at El Paso Bell Hall 302 El Paso, TX, 79968 USA

Abstract: This paper introduces a Mathematica [1] (A Computer Algebra System (CAS)) activity used as part of a longitudinal study conducted by the author [2-5] in 1999 and 2003, and overall results from pre-post surveys and interviews. The paper will attempt to show the positive effect of the use of technology on students’ motivation and attitude, and on their cognitive processes. For a detailed discussion of the post-test scores from 1999, see Dogan-Dunlap [2, 5].

Key-Words: Linear Algebra, Technology and Abstraction.

1 Introduction 1.1 Linear Algebra making the first year linear algebra courses As a result of new advancements in matrix-oriented courses. technologies such as digital computers and A few studies investigated the the use of linear algebra in these learning difficulties occurring in linear technologies [6], linear algebra classes algebra classrooms, most of which [10-11] began to attract not only mathematics reported difficulties with abstraction level of majors, but also variety of students with the course material; in recognizing different different backgrounds and majors such as representations of the same concepts; and economics, computer science and the lack of logic and set theory knowledge. meteorology. The growing heterogeneity of One may help, through the means of Visual linear algebra classes raised the question of representations, students form concept how one can modify a “first year linear images supporting concept definitions, and algebra curriculum” so that it can respond to as a result, establish abstraction. the needs of both mathematics and non- mathematics students. This resulted in a 1.2 Technology reform movement initiated during a In the absence of advanced technologies, it calculus-reform conference in Tulane [7-9]. has been challenging and difficult to design As a result, a linear algebra study group is effective inquiry-based classroom activities formed. In 1990, the group started working because most require more time than the on a list of recommendations based on the time allowed for classroom meetings and results of the surveys and questionnaires course assignments. Hence, many collected from faculty members from a instructors of mathematics have been variety of colleges, universities and client discarding the idea of inquiry during class disciplines. The results of the surveys and time. Even take home inquiry activities are questionnaires indicated a high demand discarded by some of us. Now that from the industry and client disciplines for technology is here to shorten the time needed for effective inquiry, increase the quantity, and enhance the quality of concept representations, mathematics instructors can consider addressing higher order cognitive skills. similar homework problems were assigned, One can provide effective inquiry- and tests were given. Data collection based learning environments through included a background questionnaire interactive interfaces that guide students consisting of opinion statements as well as a through a process of inquiry learning. Wicks pre-test, in-class observations, recorded adds “…Mathematica and Maple are two interviews with volunteers, a set of test such systems with which we can create rich questions, and a post-questionnaire. learning experience for our students.” There has been a wide range of computer activities 2.1 Mathematica Based-Activities such as those of the ATLAST project [12] Activities containing Mathematica and Wicks’ interactive approach [13] used in commands, some of which were modified teaching first year linear algebra concepts. from Wicks [13], were developed by the In order to advance students’ understanding investigator as interactive, guided of abstract linear algebra concepts, the supplements to lectures. They were author worked on activities supported by primarily composed of interactive cells of Mathematica to provide inquiry-based examples of basic linear algebra concepts. learning environments. The rest of the paper Fig. 1 provides an example of such discusses overall results from pre-post activities. Emphases were given to two- and surveys and interviews conducted in 1999 three-dimensional visual demonstrations of and in 2003. basic vector space concepts such as linear independence and spanning set. 2 Method In the experimental group in 1999, before the introduction of formal (abstract) A comparison method was used for the definitions, related examples from study. Data was collected from two-fall Mathematica cells were run in class, and 1999 first year linear algebra classes taught class discussions of the outcomes took at a mid-size U.S. research university with place. As more similar interactive cells with higher Anglo engineering student different examples of the same concepts population, and from a 2003 matrix algebra were run and discussed, students were to course taught at a four year U.S. university write their own interpretations into the with higher Hispanic engineering student Mathematica cell that comes right after the population. One of the courses in 1999 was cells with the Mathematica commands and taught traditionally, and the other was taught the Mathematica output. Students were to in a computer laboratory with the use of answer questions through analyzing visual Mathematica notebooks consisted of two- and three-dimensional demonstrations of Mathematica outputs. In 2003, similar basic abstract linear algebra concepts. The activities were used as web-based take home second experimental group in 2003 was assignments. During this semester, students taught in a traditional setting with the were, first, introduced to formal definitions support of web-based Mathematica activities and then assigned web-based activities. administered as take home assignments. The After giving, approximately, one week for web-based activities were similar to the the activities, students were to have in-class activities used in 1999. discussions on their responses to the In all the courses used in the study, questions from the activities.

Fig. 1. Mathematica web-based activity, from 2003, addressing linearly independent (dependent) vectors, span and spanning set. Some of the Mathematica commands used in this activity are modified from Wicks [13].

Mathematica web-based activities 3.1 Student opinion similar to the one in fig. 1 were used to In both 1999 and 2003, for the majority, the discuss linear independence, and its groups’ opinions on pre-survey connection to the concepts of span, spanning questionnaire were the same. sets, and bases. The activities were mainly Students’ opinions for the statements used to help students gain deeper on students’ feelings toward understanding of the formal (abstract) mathematics and the use of definition of linear independence stated on technology in the mathematics the textbook by Larson and Edwards [14] as: classroom did not show significant difference. Fig.2 summarizes the “ A set of vectors S={v1, v2,,...,vk} in a vector percentages of students agreeing on space V is called linearly independent if some of the pre-questionnaire the vector equation c1 v1+c2 v2+...+ck vk=0 statements. Approximately, the same has only the trivial solution, c1=0, percentages of experimental (41% in c2=0,...,ck=0. If there are also nontrivial 1999 and 51% in 2003) and solutions, then S is called linearly traditional groups (45%) indicated dependent.” that they agreed with the statement, “Mathematics is my favorite subject,” and 40% of the traditional and 50% (both in 1999 and 2003) of the experimental groups agreed with the statement, “Use of software, such as Mathematica, MathCad, or Derive, enhances learning of college algebra.” 3 Results

pre-post surveys from 1999 and 2003

90 80 70

s 60 e Trad g

a 50 t

n Exp 1999

e 40 c

r Exp 2003 e 30 p 20 10 0 pre-Favorite post- pre-Use of post-Use of post-Tech. Subject Enjoyed Tech Tech Appropriate Enhances Enhances opinion statements

Fig. 2. Percentages of students’ responses to pre-post survey questions.

Opinion statements similar to the not experienced the use of technology in pre-questionnaire statements were also their course. One possible explanation is that included in the post-questionnaire in order the difficulty level in the traditional course to document possible changes on students’ may have been higher than those in the feelings, attitude and motivation toward experimental groups. As a result, students mathematics and instructional technology. may have turned to technology as a remedy The questionnaire was administered in class for the learning difficulties they during the last week of each semester. Fig.2 encountered. provides percentages of students who agreed The post-questionnaire statement, “I on some of the post-questionnaire have enjoyed the class,” was used to statements. document students’ motivation. The seventy The majority (74% in 1999 and 76% percent (82% in 2003) of the experimental in 2003) of the experimental groups agreed groups and 50 percent of the traditional with the post-questionnaire statement group agreed with the statement (see fig. 2). “Technology we used is appropriate for this Here, notice should be given to the large course.” This may imply that the use of difference between the percentages of Mathematica activities may have caused the students in tradition and experimental majority of the experimental group students groups who expressed that they have feel positive about the role of technology in enjoyed the class. This result, contrary to the teaching and learning. This is also supported lower percentage (50 percent) of the number by the high percentages (see fig.2) of those of students in the traditional group, indicates in the experimental groups (70% in 1999 that the majority of the experimental groups and 72% in 2003) agreeing on the post- seem to have enjoyed the class, and as a questionnaire statement, “Computer assisted result, they may have been highly motivated instructions, such as MATHEMATICA, to participate in class activities. MathCad, DRIVE, can enhance learning of On a post-questionnaire statement the material covered in this class.” Notice seeking students’ opinion on the difficulty should also be given to the high percentage level of some of the linear algebra concepts, (79%) of traditional group students agreeing with the statement even though they have forty three percent of the traditional and Mathematica activities supporting learning thirty four percent (13% in 2003) of the of abstract linear algebra concepts. experimental groups indicated that they found the learning of vector space concepts 3.2 Students’ Cognitive Processes very difficult. Some students (4%) in the Interviews from 1999 revealed that some traditional group, none in the experimental students (majority of the traditional groups, stated that matrices and system of students) perceived linearly independent linear equations (8% in 1999 traditional vectors as those with different angles in group, and 8% in 2003 experimental group) between, and some perceived linearly were very difficult to learn. Thirty nine independent vectors as those that are percent of the traditional group, and 34 perpendicular to each other. For instance, percent (36% in 2003) of the experimental according to a student from the traditional groups thought that the learning of linear group, a set of vectors is linearly transformations was very difficult. Note that independent if the vectors in the set do not the percentage difference for the most have the same angle between themselves concepts is favoring the experimental and x-axis. His cognitive processes used to groups. This may imply that many students construct meaning for the concept can be in the experimental groups went through the detected in the following statement he made process of learning at relative ease. One may during the interviews: attribute this to the use of visual “ Okay, I am (pause) I come to apply the same thing. There is no vector in the set that can be produced by adding any of the other two vectors in the set but okay that can be produced by linear combination of any other vectors in the set so I would, if there is like n vectors in the set. I would draw whole bunch of them none of them would be on the same, have the same angle between themselves and x-axis like that so look like that, there will be no vectors that are just shorter versions of each other. ” The student seemed to have been understanding of geometrical meaning of struggling to fit the formal definition linear independence. Here is a student in the (possibly memorized) into his/her graphical experimental group describing his/her understanding of linear independence. One conceptualization: can see that the student had an incomplete “…Well umm, you had (pause) three vectors, and they are all you know coming, passing through zero then umm that you know that they definitely have the trivial solution but you could also may be see that umm if this you know vector was multiplied by something that would bring it this way, and the other was multiplied by something that might bring it umm you this way by a certain amount then you could see that, that this vector could be a result of ...see it looks like ohh, you were just to add these two together but send them in the opposite direction (pause) then you would get opposite of that vector, and then you would get it to be zero. That would say that it is not linearly independent...” This student’s understanding of the and power of visual Mathematica activities concept seemed to have been more visual- during the process of the conceptualization oriented. His/her argument resembles of linear combination and linear arguments made during the Mathematica independence. activities and class discussions. Again one can see, in the student’s response, the role 4 Conclusion Undergraduate Mathematics, Technology can be used to introduce College Mathematics Journal, abstract linear algebra concepts via quality Vol. 24 (1993). visual representations in a shorter time [7] D. Carlson, Teaching Linear frame. This study used Mathematica to Algebra: Must the Fog Always introduce basic abstract linear algebra Roll in? Resources for Teaching concepts through mainly visual inquiry- Linear Algebra, MAA notes, based activities. Without the powerful Volume 42, Pages 39-51 (1997). computations, and quality and quantity [8] D. Carlson, The Linear Algebra visual representations Mathematica offered, Curriculum Study Group the inquiry activities used in the study would Recommendations for the first have been difficult to implement during the Course in Linear Algebra, The time frame allowed. College Mathematics Journal, Overall the cognitive and Vol.24, No.1 (1993). pedagogical benefits learners in this study [9] G. Harel, The Linear Algebra gained from the inquiry-based visual Curriculum Study Group activities provide evidence that technology Recommendations: Moving and computers can have powerful roles on Beyond Concept Definition, enhancing learning environments, and Resources for Teaching Linear maximizing the understanding of abstract Algebra, MAA notes, Volume 42, mathematics concepts. pp. 107-126 (1997). [10] M. A. Dias, M. Artigue and E. Didirem, Articulation Problems References Between Different Systems of [1] Wolfram Inc. http://wolfram.com/ Symbolic Representations in [2] H. Dogan-Dunlap. “Visual Instruction Linear Algebra, University of abstract concepts for non-major Paris, ED411135, V2 (PME), students,” the International Journal July (1995). of Engineering Education (IJEE). In [11]. J. Hillel and A. Sierpinska, On One press. Persistent Mistake in Linear [3] H. Dogan-Dunlap, 2003. “Technology- Algebra, PME 18th Proceedings, Supported Inquiry Based Learning Vol. III (1994). in Collegiate Mathematics,” The [12]. S. Leon, E. Herman and R. proceedings of the 16th annual Faulkenberry, ATLAST ICTCM, Chicago, November 2003. Computer Exercises For Linear [4] H. Dogan, 2001a. “A comparison study Algebra, Upper Saddle River, Between a Traditional and NJ: Prentice Hall (1996). ExperimentalProgram,”Proceedings [13]. J. R. Wicks, Linear Algebra; An of the ISCA 10th international Interactive Laboratory Conference on Intelligent Systems, Approach with Mathematica, June 13-15, 2001, Arlington, Addison-Wesley Publishing Virginia. Company, Inc. Reading, [5] H.Dogan, 2001b. “A comparison study Massachusetts (1996). Between a Traditional and [14]. R. Larson and B. Edwards, ExperimentalProgram.”Unpublishe Elementary Linear Algebra, ddissertation. University of Third Edition, D.C. Heath and Oklahoma, Norman. Company, Lexington, [6]. A. Tucker, The growing Importance Massachusetts (1996). of Linear Algebra in