Transcript The influence of Computer Algebra System (CAS) in teaching

The effect of Computer Algebra
System (CAS) in the development of
conceptual and procedural
knowledge
Yılmaz Aksoy, Erciyes University, TR.
[email protected]
and
Mehmet Bulut, Gazi University, TR. [email protected]
and
Şeref Mirasyedioğlu, Başkent University, TR.
[email protected]
Outline
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Introduction(Rationale of study)
A brief review of literature
Methodology
Results
Discussion
Rationale of study
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This report explores the effect of CAS in the
development of procedural and conceptual knowledge
of first year undergraduate mathematics and science
education students.
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MAPLE was used as CAS in the teaching of Calculus
concepts.
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In this study, teaching of the derivative, the
key concept of the Calculus has been studied.
As derivative is used mainly by mathematics
and science education lessons, we choose this
concept for comparing the development of
procedural and conceptual knowledge of first
year undergraduate mathematics and science
education students.
A brief review of the literature
Constructivist theory:
 According to constructivist learning theory, if an
individual construct a concept through acting an
active role while experimenting, conjecturing,
proving and applying in learning environment, this
learning can be called acquiring more than only
receiving the information. By using CAS (Computer
Algebra System) students have an active role in
mathematics classrooms.
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Conceptual knowledge
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is seen as the knowledge of the core concepts and
principles and their interrelations in a certain
domain.
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is assumed to be stored in some form of relational
representation, like schemas, semantic Networks
or hierarchies (e.g. Byrnes & Wasik, 1991).
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Because of its abstract nature and the fact that it can
be consciously accessed, it can be largely verbalized
and flexibly transformed through processes of
inference and reflection.
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It is, therefore, not bound up with specific problems
but can in principle be generalized for a variety of
problem types in a domain (e.g. Baroody, 2003).
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Procedural knowledge,
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is seen as the knowledge of operators and the conditions
under which these can be used to reach certain goals (e.g.
Byrnes & Wasik, 1991).
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Further, it allows people to solve problems quickly
and efficiently because it is to some degree
automated.
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Automatization is accomplished through practice and
allows for a quick activation and execution of
procedural knowledge, since its application, as
compared to the application of conceptual knowledge,
involves minimal conscious attention and few
cognitive resources (see Johnson, 2003, for an
overview).
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Its automated nature, however, implies that
procedural knowledge is not or only partly open to
conscious inspection and can, thus, be hardly
verbalized or transformed by higher mental
processes.
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As a consequence, it is tied to specific problem types
(e.g. Baroody, 2003).
Computer tools:
 Since the early 1980s numerous general claims have
been made about the likely benefits of using
computer tools to improve understanding of calculus
concepts
 For example, Heid (1988, p.4), commenting on a
body research conducted during the previous ten
years, states “Computing devices are natural tools for
the refocusing of the mathematics curriculum on
concepts.”
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Ellison’s study indicated that using TI-81
graphing calculators and computer software
assisted her 10 college students to mentally
construct an appropriate concept image of the
concept of derivative. However, not all the
students developed a mature concept image.
Using CAS in Calculus
 Reporting informally on a remedial teaching
program (for 22 college students) that integrated
a CAS (Maple) into a course of calculus Hillel
(1993, p.46) observed benefits to student learning:
students coming out of it had acquired different types
of insights and knowhows than the traditionallyprepared students - insights and knowhows which we
felt were closer to the essence of calculus.
Methodology
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In this study quasi-experimental research design were used.
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Sample of this study contains 49 first year students of
mathematics education and science education departments.
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The students in both groups were encountered with the
derivative concept for the first time.
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In both of the groups, students have been studied as groups of
2 or 3 students.
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The calculus potential test was administered to students in
order to determine groups were taught in a computer based
learning environment as a pretest.
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The lessons were taken in the laboratory and the
students had the opportunity to use laboratory besides
the lessons. In order to teach the derivative concept,
student cantered activities have been designed. While
designing these activities, guides were given to
students to use the MAPLE.
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Then students have been studied on certain problems
which help to discover the mathematical concepts.
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Teaching the derivative concept has been designed as
two consecutive steps:
At first step; students studied on the concept of
derivative as rate of change. At this step real life
problems about rate of change have been given to
students. By solving these problems students have
discovered the concept of the derivative. Students
interpreted graphics of functions for developing
conceptual understanding of rate of change.
At second step, activities have been designed as
geometrical, numerical and symbolic (algebraic)
representations of derivative concept.
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In student activities, which were designed by researchers
before, used in computer learning environment for procedural
and conceptual learning of the derivative concept. These
activities administered with interactive worksheets prepared
with maple, animated and non-animated graphics, plotted by
maple, special applets in maple called maplet.
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By using interactive maple worksheets and animated graphics,
students have found the opportunity of numerous experiments
that provide well understanding for them.
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To provide conceptual and meaningful understanding for the
student, a maplet has been designed to see, geometrical
application of derivative as slope of the tangent line.
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At the end of the treatment, students’ understanding
of derivative was elicited through written tasks
administered to all students.
For this exam, students were given the opportunity,
but not required to use the computer to solve the
problems. These problems were considered to be
“computer neutral”. Students were presented with
tasks that assessed their conceptual understanding and
representational methods of solution of derivative.
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The open-ended written tasks used in the examination
instrument were mostly adapted from Girard (See [5])
common tasks used to assess student understanding of
derivative, in Calculus I courses and found in most textbooks
or adapted from other studies concerning student
understanding of derivative concept.
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The tasks were evaluated by a panel of mathematics
instructors (two university level) for the reasonableness of the
question for university Calculus I students. Recommendations
from the expert panel were examined and changes were made
to the instrument accordingly.
Results
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Scores for all questions were calculated by two
researchers using rubrics designed for this study. All
of the questions were open-ended also required a
written explanation.
To study the differences among students’ conceptual
and procedural knowledge we performed two
MANCOVAs using the The calculus potential test
grades as covariate, procedural and conceptual
problems’scores as dependent variables.
For comparing means of the two groups’ scores on
the questions, general linear model: Multivariate
Analysis of Covariance (MANCOVA) was used.
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Initially,we conducted an independent samples
t test having student’s pretest attainment in the
calculus potential test to examine whether
there were statistically significant differences
between the two groups.
Group Statistics
pretest
group
mathematics
fen
N
Mean
47,0227
46,4259
22
27
Std. Deviation
8,79729
8,73400
Std. Error
Mean
1,87559
1,68086
Independent Samples Test
Levene's Test for
Equality of Variances
F
pretest
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Equal variances
assumed
Equal variances
not assumed
Sig.
,001
,976
t-test for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
,237
47
,814
,59680
2,51666
-4,46607
5,65967
,237
44,890
,814
,59680
2,51856
-4,47617
5,66977
According to tables this independent-samples t-test
analysis indicates that students in mathematics group
had a mean of 47,0227 total points and the students in
science group had a mean of 46,4259 total points.So,
there is not significant difference between groups’
pre-test scores at the p>.05 level(note: p=.814).
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Then, we conducted a multiple analysis of covariance
test (MANCOVA) having student’s post-test
attainment in the questions about conceptual and
procedural knowledge as dependent variables and the
pre-test scores as covariates.
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The results showed that there were statistically
significant differences in students’ post-test
attainment between the two groups, Pillai’s
F(2,45)=16,959, p<0.05.
Conceptual questions
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Table 3 presents the results of the MANCOVA test, showing
that there were significant differences between the two groups.
Table 3: Differences between Mathematics and Science groups’ means in
conceptual knowledge
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We can conclude that the students of mathematics group
performed significantly better than the students of the science
group in conceptual knowledge questions.
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Then, we conducted a multiple analysis of covariance test
(MANCOVA) having student’s post-test attainment in the four
questions about conceptual knowledge as dependent variables
and the pre-test scores as covariates.
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The results showed that there were statistically significant
differences in students’ post-test attainment between the two
groups, Pillai’s F(4,43)=7,625, p<0.05.
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On the other hand, in question 4 there was not significant
differences in students’ post-test attainment between the two
groups, p>0.05.
Procedural questions
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Table 4 presents the results of the MANCOVA test, showing
that there were significant differences between the two groups.
Table 4: Differences between Mathematics and Science groups’ means
in procedural knowledge
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We can conclude that the students of mathematics group
performed significantly better than the students of the science
group in procedural knowledge questions.
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Then, we conducted a multiple analysis of covariance test
(MANCOVA) having student’s post-test attainment in the three
questions about procedural knowledge as dependent variables
and the pre-test scores as covariates.
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The results showed that there were statistically significant
differences in students’ post-test attainment between the two
groups, Pillai’s F(3,44)=5,219, p<0.05.
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On the other hand, in question 5 there was a significant
differences in students’ post-test attainment between the two
groups, p>0.05.
Discussion
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Most of the students in mathematics group
answered this question correctly. Students in
science group can do the computations but
they couldn’t show the equality.
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Students in the mathematics group showed
better understanding of the concept of the
derivative (such as the meaning of the
derivative) than the science group and there
was also a significant difference on procedural
skills.
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Specifically, they were able to express ideas in
their own words and their conceptualizations
were broader, clearer, more flexible and more
detailed than students in the control group.
These results can be interpreted as evidence
that students can understand calculus concepts
showing that it was possible to reorganize the
order in which calculus is taught to students, to
focus on concepts prior to teaching procedures.
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The students reported feeling that the computer
relieved them of some of the manipulative aspects of
calculus work, that it gave them confidence on which
they based their reasoning, and it helped them focus
on more global aspects of problem solving.
During the instruction the students were involved in
discussing ideas and were required to make sense of
calculus related language, including terminology and
symbols.
References
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[1] Artigue, M. Analysis. 1991. In Tall, D. (Ed.),
Advanced Mathematical Thinking. Kluwer Academic
Pub. pages 167-198.
[2] Baroody, A. J. (2003). The development of
adaptiveexpertise and flexibility: The integration of
conceptualand procedural knowledge. In A. J.
Baroody & A.Dowker (Eds.), The development of
arithmetic concepts and skills: Constructing adaptive
expertise (pp. 1-33).Mahwah, NJ: Erlbaum.
[3] Byrnes, J. P., & Wasik, B. A. (1991). Role of
conceptualknowledge in mathematical procedural
learning. Developmental Psychology, 27(5), 777-786.
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[4] Cooley, L. A. Evaluating the effects on
conceptual understanding and achievement of
enhancing an introductory calculus course with a
computer algebra system. 1995. (New York
University). Dissertation Abstracts International 56:
3869.
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[5] Ellison, M. J. The effect of computer and
calculator graphics on students’ ability tomentally
construct calculus concepts. 1993. (Volumes I and II).
(University of Minnesota).Dissertation Abstracts
International, 54/11 4020.
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[6] Fey, J. T. Technology and mathematics education:
A survey of recent developments and important
problems.1989. Educational Studies in Mathematics,
20, pages 237-272.
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[7] Girard, N.R. Students’ representational
approaches to solving calculus problems:
Examining the role of graphic calculators. 2002.
(University of Pittsburgh)
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[8] Heid, M. K. Resequencing skills and concepts in
applied calculus using the computer as atool. 1988.
Journal for Research in Mathematics Education,
19(1), pages 3-25.
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[9] Tall, D.& West, B. Graphic insight into calculus
and differential equations. 1986. In A.G. Howson &
J. P. Kahane (Eds.), The influence of computers and
information on mathematicsand its teaching.
Cambridge: Cambridge University Press. pages 107119.
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[10] White, P. Is calculus in trouble? 1990. Australian
Senior Mathematics Journal, 4(2), pages 105-110.
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Contact:
Mehmet Bulut
Gazi University
Faculty of Gazi Education
ANKARA-TURKEY
[email protected]