Computers and Linear Algebra
Hamide Dogan-Dunlap
Mathematical Sciences
University of Texas at El Paso
Bell Hall 302
El Paso, TX, 79968
USA
hdogan@utep.edu
Abstract: This paper discusses Mathematica [1] (A Computer Algebra System (CAS)) activities
used as part of a study conducted by the author [2-5], and over all 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 on their cognitive processes. For a detailed discussion of the post-test
scores, see Dogan-Dunlap [2].
Key-Words: Linear Algebra, Visual Representations 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
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.
According to Dubinsky [12] and Harel [13],
students can achieve abstraction if flexibility
between representations of the same
concepts is established. Abstraction might
be established if concept images, defined as
all mental pictures, properties and processes
associated with the concept, and concept
definitions, defined as a form of symbols
used to specify the concept, are not
contradicting one another. One might help,
through the means of Visual representations,
students form concept images supporting
concept definitions, and as a result, establish
abstraction.
1.2 Framework
According to the constructivist view of
learning, learners construct meaning for the
subject matter [14]. That is, learners engage
in higher order cognitive skills such as
investigation and conjecture. Inquiry-based
approach is one that provides and supports
learning environments where learners
observe events, ask questions, construct
explanations, test those explanations, use
critical and logical thinking, generalize
observed patterns, and consider alternative
explanations.
Inquiry-based
learning
environments can be structured in various
forms. In one, learners are provided guided,
hands on activities and are required to arrive
at their own conclusions through
experimentation, observation, investigation
and conjecture. One may also provide
environments where learners are required to
design and carry out experiments. The
former is the learning environment provided
in the study discussed in this paper.
1.3 Technology
In the absence of advanced technologies, it
has been challenging and difficult to design
effective inquiry-based classroom activities
since 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 the quality
of concept representations, mathematics
instructors can consider addressing higher
order 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 [16]
and Wicks’ interactive approach [15] used in
teaching first year linear algebra concepts.
In order to advance students’ understanding
of abstract linear algebra concepts, the
author worked on classroom activities
supported by Mathematica to provide
inquiry-based learning environments. The
rest of the paper discusses sample
Mathematica activities and over all results
from pre-post surveys and interviews from a
study conducted in 1999.
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 research university. One of the
courses was taught traditionally, and the
other was taught in a computer laboratory
with the use of Mathematica notebooks
consisted of two- and three-dimensional
demonstrations of basic abstract linear
algebra concepts.
In both classes, the same textbook
was used. The same homework problems
were assigned, and similar quizzes were
given. Data collection included a
background questionnaire consisting of
opinion statements and a pre-test, in-class
observations, recorded interviews with
volunteers from both classes, a set of exam
and quiz questions, and a post-questionnaire.
The background questionnaire was collected
to document students’ prior knowledge in
mathematics. To investigate possible
differences due to the implementation,
students' scores on five common problems
given on exams, the final, and a quiz were
used. In an attempt to get a better insight on
students' responses on the five questions,
interviews were administered during the last
week of the fall 1999 semester.
2.1 Mathematica Notebooks
Notebooks
containing
Mathematica
commands, some of which were modified
from Wicks [15], 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 and 2 provide examples of such
notebooks. Emphases were given to twoand three-dimensional visual demonstrations
of basic vector space concepts. Each cell in
a notebook contained an example discussed
in class, and was labeled accordingly.
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 asked
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.
Solution Sets
Example 1
1.Run the following Mathematica cells, and based on the output,
provide a geometrical description for the solution set of the system of linear equations:
3+10 x=z
3+5 x=z
3-8 x=z
2. Solve the system in part 1 algebraically, and state the solution set.
3. Compare your responses given in part 1 and 2.
<<Graphics`ContourPlot3D`
p1=ContourPlot3D[ 10x-z+3 ,
{x,-2,2}, {y,0,3}, {z, 2,2}];p2=ContourPlot3D[ 5x +z+3 , {x,-2,4}, {y,0,4}, {z,-2,4},
ViewPoint->{1.300, -2.400, 2.000} ];p3=ContourPlot3D[ -8x-5z+3, {x,-2,4}, {y,0,4}, {z,-2,4},
ViewPoint->{1.300, -2.400, 2.000} ];Show[p1,p2,p3, ViewPoint->{-1.162, 4.120, 3.485}];
4
2
4
3
0
2
1
-2
0
-2
-1
0
1
Fig. 1. Mathematica Notebook on Solution sets. This shows how the notebook looks after running Mathematica
commands. Tittles are written in red, and instructions are written in blue.
For instance, the Mathematica
notebook shown in figure 1 was given to
students before discussing types of solution
sets for linear systems. During the class,
instructor referring to the example as the
first example of the day had students run the
Mathematica cells, and discuss, based on the
graphical representation of the system, the
solution type for the system. Next, students
were asked to experiment on their own
examples of linear systems, discuss the
solution sets of their systems, and as a result,
make conjecture on all solution possibilities
for linear systems. Once students made
reasonable progress with the guidance of the
instructor, they were to find, algebraically,
solution sets for the same linear systems,
and to compare the algebraic results with the
graphical outcomes. The purpose of the
comparison was to get students to make
Linearly Independent (Dependent) Sets
Example 2


Run the commands given in the following Mathematica cells.
Based on the outcome of the commands, answer the following questions (write your response in a new cell
that comes right after the cells with Mathematica Commands and Output):
1. State the solution for the vector equation a i + b l + c j = 0 where i is the vector in green, l is the vector in
blue, and j is the vector in red.
2. Is the set {i, l, j} linearly independent? Explain your reasoning.




Enter your own vectors from R3 in to the first cell with vectors i, j, l and k.
Repeat Steps 1 and 2.
Now, for the set of vectors you have used in step 3, discuss whether the set
{i, l, j, k} is linearly independent or not.
Discuss Span of the same set {i, l, j, k}.
Fig. 2. Mathematica notebook addressing linearly independent (dependent) vectors, span and spanning set. This
shows how the notebook looks after running Mathematica commands. Tittles are written in red, and instructions are
written in blue. Some of the Mathematica commands used in this notebook are modified from Wicks [15].
connections between the algebraic and
geometric approaches, and to have an
appreciation for various representations of
the concept.
Mathematica notebooks similar to
the one in fig. 2 were used to discuss linear
independence, and its connection to the
concepts of span, spanning sets, and bases.
These 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
[17]
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
For the majority, the two 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.3 summarizes the percentages
of students agreeing on some of the prequestionnaire statements. Approximately,
the same percentage of experimental (41%)
and traditional group (45%) indicated that
they
agreed
with
the
statement,
“Mathematics is my favorite subject,” and
40% of the traditional and 50% of the
experimental group agreed with the
statement, “Use of software, such as
Mathematica, MathCad, or Derive, enhances
learning of college algebra.”
pre-post questionnaire
90
80
70
%
60
50
Trad
40
Exper
30
20
10
0
prepost-Enjoyed
Mathematics
is my favorite
subject
pre-use of
tech
enhances
post-use of
tech
enhances
Fig. 3. Percentages of students’ responses to pre-post questionnaires.
Opinion statements similar to the prequestionnaire statements were included in
the post-questionnaire in order to document
possible changes on students’ feelings, and
motivation towards mathematics and
instructional technology. The questionnaire
was administered in class during the last
week of the semester. Fig.3 provides
percentages of students who agreed on some
of the post-questionnaire statements.
The majority (74%) of the experimental
group agreed with the post-questionnaire
statement “Technology we used is
appropriate for this course.” This may imply
that the use of Mathematica notebooks in
the classroom made majority of the
experimental group students feel positive
about the role of technology in teaching and
learning. This also seems to be supported by
the high percentage (see fig.3) of those in
the experimental group agreeing on the postquestionnaire statement, “Computer assisted
instructions, such as MATHEMATICA,
MathCad, DRIVE, can enhance learning of
the material covered in this class.”
Notice also the high percentage (79%) of
traditional group students agreeing with the
statement even though they have not
experienced the use of technology in the
course. One possible explanation is that the
difficulty level in the traditional course may
have been higher than those in the
experimental group. 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 in both classes. 70
percent of the experimental group and 50
percent of the traditional group agreed with
the statement (see figure 3). Here, notice
should be given to the large difference
between the percentages of students in both
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 group
may have enjoyed the class, and as a result,
may have been highly motivated to
participate in class activities.
On a post-questionnaire statement
seeking students’ opinion on the difficulty
level of concepts, forty three percent of the
traditional and thirty four percent of the
experimental group indicated that they
found the learning of vector space concepts
very difficult. Some students in the
traditional group, none in the experimental
group, stated that matrices (four percent)
and system of linear equations (eight
percent) were very difficult to learn.
Thirty nine percent of the traditional group,
and 34 percent of the experimental group
thought that learning linear transformations
was very difficult. Note that the percentage
difference for each of the concepts (topics)
is favoring the experimental group. This
may imply that many students in the
experimental group went through the
process of learning at relative ease. One may
attribute this to the use of visual
Mathematica activities supporting learning
of abstract linear algebra concepts.
3.2 Visual Representations
Students’ Cognitive Processes
and
Since the technology in this study was
mainly used to introduce various visualrepresentations of abstract linear algebra
concepts, the author also looked at the
possible differences on the way students
from both groups conceptualized the
abstract
concepts
such
as
linear
independence. That is, the possible effect of
the visual representations on cognition was
investigated.
To examine students’ ability to
answer visual-based problems, the following
question was given on the postquestionnaire:
The correct response for this
question was options C and D. Option C has
two vectors with an acute angle in between,
and option D has two perpendicular vectors.
All the other options had three or more
vectors with angles varying from acute to
perpendicular. Table 1 shows percentage of
students who chose options C or D as the set
of linearly independent vectors.
Table 1. Students’ Responses on the post-questionnaire problem.
Options
C
D
Traditional (25)
56%
64%
Experimental (21)
67%
76%
Table 1indicates that the experimental
group might have been slightly better at
answering visual-oriented problems than the
traditional group. 56 percent of the
traditional group and 67 percent of the
experimental group indicated C as the set of
vectors that are linearly independent. There
was 64 percent in the traditional and 76
percent in the experimental group who
circled the vectors in part D. Some students
circled other options as well as linearly
independent vectors. In both groups,
responses that included all options (A
through F) were around 20 percent within
the difference of plus and minus two.
Interestingly, both sections had higher
percentage of students who chose option D
than option C. The analysis of selected
interviews shed some light on what bases
the students made their decision. Interviews
revealed that some students perceived
linearly independent vectors as those with
different angles in between, and some
perceived linearly independent vectors as
those that are perpendicular to each other.
For instance, according to a student from the
traditional group, a set of vectors is linearly
independent if the vectors in the set do not
have the same angle between themselves
and the x-axis. His cognitive processes used
to construct meaning for the concept can be
detected in the following statement he made
during the interviews:
“ 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 seems to be struggling to
fit the formal definition (memorized) into
his/her graphical understanding of linear
independence. One can see that the student
has incomplete understanding of geometrical
meaning of linear independence.
Here is a student in the
experimental group describing his/her
conceptualization:
“ Umm to represent them, three coming from the same point, I don’t think so...Because one of
them will always be able to be represented by the, the sum of, of you know scalars times the
other two......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 concept
seems to be more visual-oriented. His/her
argument resembles arguments made during
Mathematica
activities
and
class
discussions.
The interview results and the
difference of 12 percent between the two
groups who circled options C and D on the
post-questionnaire problem discussed earlier
does seem to indicate that the experimental
group might have had stronger conceptual
understanding of abstract concepts than the
traditional group. Here, it should be noted
that significant differences were found on
the conceptual post-test questions favoring
the experimental group. A detailed
description of the results of the statistical
analysis of post-test questions can be found
in Dogan-Dunlap[2].
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 inquiry-based activities.
Without the powerful computations, and
quality and quantity visual representations
Mathematica offers, the inquiry activities
used in the study would have been difficult
to implement during a 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
have powerful roles in enhancing learning
environments,
understanding
concepts.
and
maximizing
the
of abstract mathematics
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