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 prepost 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
As a result of new advancements in
technologies such as digital computers and
the use of linear algebra in these
technologies [6], linear algebra classes
began to attract not only mathematics
majors, but also variety of students with
different backgrounds and majors such as
economics,
computer
science
and
meteorology. The growing heterogeneity of
linear algebra classes raised the question of
how one can modify a “first year linear
algebra curriculum” so that it can respond to
the needs of both mathematics and nonmathematics students. This resulted in a
reform movement initiated during a
calculus-reform conference in Tulane [7-9].
As a result, a linear algebra study group is
formed. In 1990, the group started working
on a list of recommendations based on the
results of the surveys and questionnaires
collected from faculty members from a
variety of colleges, universities and client
disciplines. The results of the surveys and
questionnaires indicated a high demand
from the industry and client disciplines for
the quality of concept representations,
making the first year linear algebra courses
matrix-oriented courses.
A few studies investigated the
learning difficulties occurring in linear
algebra classrooms, most of which [10-11]
reported difficulties with abstraction level of
the course material; in recognizing different
representations of the same concepts; and
the lack of logic and set theory knowledge.
One may help, through the means of Visual
representations, students form concept
images supporting concept definitions, and
as a result, establish abstraction.
1.2 Technology
In the absence of advanced technologies, it
has been challenging and difficult to design
effective inquiry-based classroom activities
because most require more time than the
time allowed for classroom meetings and
course assignments. Hence, many instructors
of mathematics have been discarding the
idea of inquiry during class time. Even take
home inquiry activities are discarded by
some of us. Now that technology is here to
shorten the time needed for effective
inquiry, increase the quantity, and enhance
mathematics instructors can consider
addressing higher order cognitive skills.
One can provide effective inquirybased learning environments through
interactive interfaces that guide students
through a process of inquiry learning. Wicks
adds “…Mathematica and Maple are two
such systems with which we can create rich
learning experience for our students.” There
has been a wide range of computer activities
such as those of the ATLAST project [12]
and Wicks’ interactive approach [13] used in
teaching first year linear algebra concepts.
In order to advance students’ understanding
of abstract linear algebra concepts, the
author worked on activities supported by
Mathematica to provide inquiry-based
learning environments. The rest of the paper
discusses overall results from pre-post
surveys and interviews conducted in 1999
and in 2003.
2 Method
A comparison method was used for the
study. Data was collected from two-fall
1999 first year linear algebra classes taught
at a mid-size U.S. research university with
higher
Anglo
engineering
student
population, and from a 2003 matrix algebra
course taught at a four year U.S. university
with higher Hispanic engineering student
population. One of the courses in 1999 was
taught traditionally, and the other was taught
in a computer laboratory with the use of
Mathematica notebooks consisted of twoand three-dimensional demonstrations of
basic abstract linear algebra concepts. The
second experimental group in 2003 was
taught in a traditional setting with the
support of web-based Mathematica activities
administered as take home assignments. The
web-based activities were similar to the
activities used in 1999.
In all the courses used in the study,
similar homework problems were assigned,
and tests were given. Data collection
included a background questionnaire
consisting of opinion statements as well as a
pre-test, in-class observations, recorded
interviews with volunteers, a set of test
questions, and a post-questionnaire.
2.1 Mathematica Based-Activities
Activities
containing
Mathematica
commands, some of which were modified
from Wicks [13], were developed by the
investigator
as
interactive,
guided
supplements to lectures. They were
primarily composed of interactive cells of
examples of basic linear algebra concepts.
Fig. 1 provides an example of such
activities. Emphases were given to two- and
three-dimensional visual demonstrations of
basic vector space concepts such as linear
independence and spanning set.
In the experimental group in 1999,
before the introduction of formal (abstract)
definitions,
related
examples
from
Mathematica cells were run in class, and
class discussions of the outcomes took
place. As more similar interactive cells with
different examples of the same concepts
were run and discussed, students were to
write their own interpretations into the
Mathematica cell that comes right after the
cells with the Mathematica commands and
the Mathematica output. Students were to
answer questions through analyzing visual
Mathematica outputs. In 2003, similar
activities were used as web-based take home
assignments. During this semester, students
were, first, introduced to formal definitions
and then assigned web-based activities.
After giving, approximately, one week for
the activities, students were to have in-class
discussions on their responses to the
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
similar to the one in fig. 1 were used to
discuss linear independence, and its
connection to the concepts of span, spanning
sets, and bases. The activities were mainly
used to help students gain deeper
understanding of the formal (abstract)
definition of linear independence stated on
the textbook by Larson and Edwards [14] as:
“ A set of vectors S={v1, v2,,...,vk} in a vector
space V is called linearly independent if the
vector equation c1 v1+c2 v2+...+ck vk=0 has
only the trivial solution, c1=0, c2=0,...,ck=0.
If there are also nontrivial solutions, then S
is called linearly dependent.”
3 Results
3.1 Student opinion
In both 1999 and 2003, for the majority, the
groups’
opinions
on
pre-survey
questionnaire were the same. Students’
opinions for the statements on students’
feelings toward mathematics and the use of
technology in the mathematics classroom
did not show significant difference. Fig.2
summarizes the percentages of students
agreeing on some of the pre-questionnaire
statements. Approximately, the same
percentages of experimental (41% in 1999
and 51% in 2003) and traditional groups
(45%) indicated 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.”
percentages
pre-post surveys from 1999 and 2003
90
80
70
60
50
40
30
20
10
0
Trad
Exp 1999
Exp 2003
pre-Favorite
Subject
postEnjoyed
pre-Use of post-Use of
Tech
Tech
Enhances Enhances
post-Tech.
Appropriate
opinion statements
Fig. 2.
Percentages of students’ responses to pre-post survey questions.
Opinion statements similar to the
pre-questionnaire statements were also
included in the post-questionnaire in order to
document possible changes on students’
feelings, attitude and motivation toward
mathematics and instructional technology.
The questionnaire was administered in class
during the last week of each semester. Fig.2
provides percentages of students who agreed
on some of the post-questionnaire
statements.
The majority (74% in 1999 and 76%
in 2003) of the experimental groups agreed
with the post-questionnaire statement
“Technology we used is appropriate for this
course.” This may imply that the use of
Mathematica activities may have caused the
majority of the experimental group students
feel positive about the role of technology in
teaching and learning. This is also supported
by the high percentages (see fig.2) of those
in the experimental groups (70% in 1999
and 72% in 2003) agreeing on the postquestionnaire statement, “Computer assisted
instructions, such as MATHEMATICA,
MathCad, DRIVE, can enhance learning of
the material covered in this class.” Notice
should also be given to the high percentage
(79%) of traditional group students agreeing
with the statement even though they have
not experienced the use of technology in
their course. One possible explanation is that
the difficulty level in the traditional course
may have been higher than those in the
experimental groups. As a result, students
may have turned to technology as a remedy
for
the
learning
difficulties
they
encountered.
The post-questionnaire statement, “I
have enjoyed the class,” was used to
document students’ motivation. The seventy
percent (82% in 2003) of the experimental
groups and 50 percent of the traditional
group agreed with the statement (see fig. 2).
Here, notice should be given to the large
difference between the percentages of
students in tradition and experimental
groups who expressed that they have
enjoyed the class. This result, contrary to the
lower percentage (50 percent) of the number
of students in the traditional group, indicates
that the majority of the experimental groups
seem to have enjoyed the class, and as a
result, they may have been highly motivated
to participate in class activities.
On a post-questionnaire statement
seeking students’ opinion on the difficulty
level of some of the linear algebra concepts,
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 visualduring 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
Technology can be used to introduce
abstract linear algebra concepts via quality
visual representations in a shorter time
frame. This study used Mathematica to
introduce basic abstract linear algebra
concepts through mainly visual inquirybased activities. Without the powerful
computations, and quality and quantity
visual representations Mathematica offered,
the inquiry activities used in the study would
have been difficult to implement during the
time frame allowed.
Overall
the
cognitive
and
pedagogical benefits learners in this study
gained from the inquiry-based visual
activities provide evidence that technology
and computers can have powerful roles on
enhancing learning environments, and
maximizing the understanding of abstract
mathematics concepts.
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ICTCM, Chicago, November 2003.
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