THE ROLE OF THE COMPUTER IN THE TEACHİNG AND
LEARNİNG LİNEAR ALGEBRA
Asst.Prof. Sinan Aydın
Yüzüncü Yıl University, Faculty of Education
Department of Primary School Education, Van/Turkey.
Sinanaydin1704@yahoo.com
ABSRACT
The availability of computers has forced mathematicians to rethink the way they are teaching mathematics. When an operation can be
performed quickly and satisfactorily by a computer, an instructor has to ask `What is it that a student really needs to learn in a course and
How?’. One of the essences of teaching is to help students learn the material that they need.
In this study we discuss The Role of the Computer in the Teaching and Learning Linear Algebra. Also it deals with some strategies which were
used and the computer projects that were developed for linear algebra courses. 27 students in 2-A class (experiment group) and 28 students
in 2-B class (control group) in Mathematics education of Primary Department in the Education Faculty of Yüzüncü Yıl University, which is in the
2nd Term of 2006-2007 Academic Years, were assigned as the sample of the study. The resulting improvement in the students learning has
been remarkable.
Keywords: Linear Algebra, Computer Projects, Teaching Strategies.
INTRODUCTION
The innovations in the technology create the necessity to
re-examine our approach to teaching the subject in linear
algebra. One of the basic essences of teaching is to help
students learn the subjects that they need. The
availability of computers has forced mathematicians to
rethink the way they are teaching mathematics. If a
mathematical operation can be done quickly and
satisfactorily by a computer, an instructor has to ask
`What is it that students really need to learn in a course
and How?’.
A simple reality that every teacher of this course can see
easily is that linear algebra is a very useful in the modern
scientific world. Also another reality for this course is that
many applications of linear algebra don’t happen without
a computer. We know that some mathematical software
programs such as Matlab, Mathematica, Drive and Maple
can do many computations in the basic linear algebra
course. In this point, it is very important to make an
effective integration of computing into the linear algebra
classroom. Some researchers studying on linear algebra
are working on how to incorporate computing effectively,
how we will teach and how the students will learn
(Carlson, Johnson, Lay, Porter, 1993; 42 ; Dorier, 2002;
877 ; Uhlig, 2003; 148 ). One widespread idea is that
mathematic students tend to be less computer user than
some other majors as physics, engineering and
architecture. When a mathematic instructor integrates
computing into the courses, Students will have more
confidence and more useful computer experience. So an
answer to the question above mentioned is that students
need to understand well the main mathematical concepts
ant to be able to apply them in different situations via
computer.
Many linear algebra instructors agree with the idea that
students of this course need to take first the basic
concepts and the theorems, than to master in the basic
hand calculations and finally, after these steps, to use a
computer to do arithmetic (Sierpinska, Trgalova, Hilel,
Dreyfus, 1999; 125). If it is done so, students can concentrate
much on the conceptual ideas instead of trying to get the
arithmetic right in the solution. Also when a linear algebra
instructor uses a computer, he can do some calculations that
are not otherwise possible in class. For instance, he can ask
questions, in talking about an advance calculation, to the class
how to their approach the problem. He is able to do exactly by
computer what his students tell him to do. By help of this
procedure, students will be willingly, and are motivated to
participate in next steps of the course.
Using computer in linear algebra makes it possible to ask
theoretical questions which are arithmetically too complicated
to do with pencil and paper in class. There is a similar situation
in this question that which vectors are in a subspace? And how
we understand two subspaces are the same subspace. For
example, in vector space theory, every subspace of a vector
space has infinitely many bases. It is possible for many linear
algebra students to know that every subspace has a special
and predetermined basis. When the students find basis of the
row space of a matrix by computer, they can meet some
different basis for the subspace. This computation brings out
the idea that there can be more than one basis for a subspace.
Some linear algebra researchers say that some pure
mathematicians can worry that using computers in a
mathematic course will turn students into unthinking way
(Axler, 1995; 141; Harel, 2000, 182; Rogalski, 1996; 215). This
approach could be true in some conditions but it is easy to
avoid in linear algebra; in my opinion, computer can be used to
motivate learning the concepts and the theory of linear algebra.
Also in this course, the theory plays a basic role in
computations. For instance, in solution of linear equations
systems, we know the famous theorem that every solution of a
linear equations system can be written as the sum of a
particular solution of the system. This theorem can be seen as
to be difficult by students to understand without computer.
Matlab gives us an approach to understand the formal theory
and a procedure to find the general solution of the linear
equations system.
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This article deals with the two special strategies that
were used and some computer projects that were
developed for linear algebra.
TWO STRATEGİES IN USİNG COMPUTER
a. Using software programs to explore new linear
algebra concepts
A computer classroom with individual stations needs to
be available. One of the main mathematical software
programs, as matlab, can be used as a resource. Instead
of giving examples that satisfy a definition or checking
that they satisfy a certain list of properties about a
concept, instructor lead students through a series of
exercises to work with examples and explore their
properties. This approach is the central point of using
computer in linear algebra. For instance, noncommutativity of the product of matrices becomes more
real and clearer for students if they produce examples by
themselves in computer. Of course, a computer is not
necessary to do this arithmetic but using computer
makes it possible to do it in class without spending much
time.
b. Using computer projects to motivate students
The abstractness of the linear algebra concepts allows
students to model a wide variety of concrete problems.
To show this abstractness, Instructors give individual
projects to them. Also every student should have a
chance to choose an applied problem in his/her area of
interest. Since many of the current linear algebra
textbooks emphasize applications to different fields, there
are many source of problems(Bretscher, 1997; Cullen,
1996; Larson & Edwards, 1996; Lay, 1997; Moore &
Yaqub, 1992). Students will make an exploratory work,
look for required patterns and make conjectures during
these applications.
ON A SPECİAL PROJECT
What follow is part of the project on linear
transformations. The parts consist of a problem that has
been divided into several sections. Students discover
pieces of linear algebra information than use it to arrive
at conclusion. Also they use computer and a software
program in necessary steps.
METHOD
The study took place during the 2nd Term of 2006-2007
Academic Years in Mathematics education of Primary
Department in the Education Faculty of Yüzüncü Yıl University.
27 students in 2-A class (experiment group) and 28 students in
2-B class (control group) were assigned as the sample of the
study. While no computers were available in the classroom of
the control group, and the topics followed the traditional order
found in most linear algebra textbooks, the experiment group
was received computer assisted teaching. The classes of the
experiment group were hold in a computer laboratory room.
Exploratory work with linear algebra computer programs and a
few projects constituted during the semester. Only the final
exam was used for the evaluation of the study and there were
no additional writing requirements. The data obtained was
analyzed with the help of descriptive statistics.
FINDINGS
The following table compares the grades obtained by the
students in the pre-test exam of the groups.
Table 1: Summary of the pre-test results with respect to the
groups.
Group
N
St.Dev. df
t
Sig.

Experiment
27
14,51
1,37
Control
28
12,67
1,10
1.
2.
3.
4.
5.
Apply L to the vectors in the standard basis of
R5. Write the resulting vectors as columns and
use them in the given order to from a matrix M
( we call M the matrix associated to L with
respect to the standard bases in the domain
and in the range of L).
Check that to find the image of any vector v,
we obtain the same result if we use the
definition of L or if we multiply the matrix M by
the column v = ( x1, x2, x3, x4, x5 )t.
Find the nullspace (or kernel) of L.
Find the dimension of the image of L.
Check that
dim (R5) = dim (kerL) + dim (Im
L).
1,04
0,3
According to the tablo1, there was no difference in the level of
the groups when they started the education program at the
probability value of 0,5. So we can say that the experiment
group and the control group were the same level in linear
algebra course at the beginning of the term.
The following table compares the grades obtained by the
students in the post-test (the final) exam of the groups.
Table 2: Summary of the post-test (the final) exam results with
respect to the groups.
Group
N
St.Dev. df
t
Sig.

Experiment
27
69,92
13,03
Control
28
58,75
16,02
Let L: R5 → R4 be defined by
L( x1, x2, x3, x4, x5 ) = (x1- x5, 2x1- x2, x1+ x3, x4 ).
53
53
2,57
0,01
As shown at the table2, the difference, between the experiment
group and the control group, is significant (p<0,05). According
to this finding, the improvement in learning of the experiment
group is due to the using of the teaching strategies described in
this article.
RESULTS
Linear algebra provides a vital arena where students can see
the interaction of mathematics, and machine computation. The
integration of computation and theoretical mathematics is so
natural in linear algebra that students can use their experience
with linear algebra as a starting point for seeking similar
integration in other mathematical areas. So linear algebra
deserves a central place in the curriculum of mathematic
program. Students need to learn how to integrate a theoretical
and computational understanding of mathematics.
Computer-based instruction may lower the quality of learning
linear algebra if too much emphasis is placed on individual
work with computer. An improvement in learning can be due to
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the combination of the teaching strategies. Incorporating
computer into teaching linear algebra works best when it
is done with strategies that promote critical thinking and
increase communication between students and teachers.
When the algorithms of linear algebra are properly
understood, students spend less time on computations
and more time discussing problems, theorems and
proofs.
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