The effect of Computer Algebra System (CAS) in the development of

conceptual and procedural knowledge

Yılmaz Aksoy, Yılmaz Aksoy, ErciyesErciyes University, TR. University, TR. aksoyyilmaz@hotmail.com

andandMehmet Bulut, Gazi University, TR. Mehmet Bulut, Gazi University, TR. mbulut@gazi.edu.tr

andandŞeref Mirasyedioğlu, Başkent University, TR. Şeref Mirasyedioğlu, Başkent University, TR.

serefm@baskent.edu.tr

OutlineOutline

Introduction(Rationale of study)Introduction(Rationale of study) A brief review of literatureA brief review of literature MethodologyMethodology ResultsResults DiscussionDiscussion

Rationale of studyRationale of study

This report explores the effect of CAS in the

development of procedural and conceptual

knowledge of first year undergraduate mathematics

and science education students.

MAPLE was used as CAS in the teaching of Calculus

concepts.

In this study, teaching of the derivative, the In this study, teaching of the derivative, the key concept of the Calculus has been studied. key concept of the Calculus has been studied. As derivative is used mainly by mathematics As derivative is used mainly by mathematics and science education lessons, we choose this and science education lessons, we choose this concept for comparing the development of concept for comparing the development of procedural and conceptual knowledge of first procedural and conceptual knowledge of first year undergraduate mathematics and science year undergraduate mathematics and science education students.education students.

A brief review of the literatureA brief review of the literature

Constructivist theory:Constructivist theory: According to constructivist learning theory, if an According to constructivist learning theory, if an

individual construct a concept through acting an individual construct a concept through acting an active role while experimenting, conjecturing, active role while experimenting, conjecturing, proving and applying in learning environment, this proving and applying in learning environment, this learning can be called acquiring more than only learning can be called acquiring more than only receiving the information. By using CAS (Computer receiving the information. By using CAS (Computer Algebra System) students have an active role in Algebra System) students have an active role in mathematics classrooms.mathematics classrooms.

Conceptual knowledge Conceptual knowledge

is seen as the knowledge of the core concepts and is seen as the knowledge of the core concepts and

principles and their interrelations in a certain principles and their interrelations in a certain

domain. domain.

is assumed to be stored in some form of relational is assumed to be stored in some form of relational

representation, like schemas, semantic Networks or representation, like schemas, semantic Networks or

hierarchies (e.g. Byrnes & Wasik, 1991). hierarchies (e.g. Byrnes & Wasik, 1991).

Because of its abstract nature and the fact that it can Because of its abstract nature and the fact that it can

be consciously accessed, it can be largely verbalized be consciously accessed, it can be largely verbalized

and flexibly transformed through processes of and flexibly transformed through processes of

inference and reflection. inference and reflection.

It is, therefore, not bound up with specific problems It is, therefore, not bound up with specific problems

but can in principle be generalized for a variety of but can in principle be generalized for a variety of

problem types in a domain (e.g. Baroody, 2003). problem types in a domain (e.g. Baroody, 2003).

Procedural knowledgeProcedural knowledge, ,

is seen as the knowledge of operators and the conditions is seen as the knowledge of operators and the conditions

under which these can be used to reach certain goals (e.g. under which these can be used to reach certain goals (e.g.

Byrnes & Wasik, 1991). Byrnes & Wasik, 1991).

Further, it allows people to solve problems quickly Further, it allows people to solve problems quickly

and efficiently because it is to some degree and efficiently because it is to some degree

automated. automated.

Automatization is accomplished through practice and Automatization is accomplished through practice and

allows for a quick activation and execution of allows for a quick activation and execution of

procedural knowledge, since its application, as procedural knowledge, since its application, as

compared to the application of conceptual compared to the application of conceptual

knowledge, involves minimal conscious attention and knowledge, involves minimal conscious attention and

few cognitive resources (see Johnson, 2003, for an few cognitive resources (see Johnson, 2003, for an

overview). overview).

Its automated nature, however, implies that Its automated nature, however, implies that

procedural knowledge is not or only partly open to procedural knowledge is not or only partly open to

conscious inspection and can, thus, be hardly conscious inspection and can, thus, be hardly

verbalized or transformed by higher mental verbalized or transformed by higher mental

processes. processes.

As a consequence, it is tied to specific problem types As a consequence, it is tied to specific problem types

(e.g. Baroody, 2003).(e.g. Baroody, 2003).

Computer tools:Computer tools: Since the early 1980s numerous general claims have Since the early 1980s numerous general claims have

been made about the likely benefits of using been made about the likely benefits of using computer tools to improve understanding of calculus computer tools to improve understanding of calculus concepts concepts

For example, Heid (For example, Heid (19881988, p.4), commenting on a , p.4), commenting on a body research conducted during the previous ten body research conducted during the previous ten years, states “Computing devices are natural tools for years, states “Computing devices are natural tools for the refocusing of the mathematics curriculum on the refocusing of the mathematics curriculum on concepts.”concepts.”

Ellison’s study indicated that using TI-81 Ellison’s study indicated that using TI-81 graphing calculators and computer software graphing calculators and computer software assisted her 10 college students to mentally assisted her 10 college students to mentally construct an appropriate concept image of the construct an appropriate concept image of the concept of derivative. However, not all the concept of derivative. However, not all the students developed a mature concept image. students developed a mature concept image.

Using CAS in CalculusUsing CAS in Calculus Reporting informally on a remedial teaching Reporting informally on a remedial teaching

program (for 22 college students) that integrated program (for 22 college students) that integrated a CAS (Maple) into a course of calculus Hillel a CAS (Maple) into a course of calculus Hillel ((19931993, p.46) observed benefits to student learning: , p.46) observed benefits to student learning: students coming out of it had acquired different types students coming out of it had acquired different types of insights and knowhows than the traditionally- of insights and knowhows than the traditionally- prepared students - insights and knowhows which we prepared students - insights and knowhows which we felt were closer to the essence of calculus. felt were closer to the essence of calculus.

MethodologyMethodology In this study In this study quasi-quasi-experimental research design were used. experimental research design were used.

Sample of this study contains 4Sample of this study contains 499 first year students of first year students of mathematics educationmathematics education and and science education science education departmentdepartments.s.

The students in both groups were encountered with the derivative concept for the first time.

In both of the groups, students have been studied as groups of 2 or 3 students.

The calculus potential test was administered to students in The calculus potential test was administered to students in order to determine groups were taught in a computer based order to determine groups were taught in a computer based learning environmentlearning environment as a pretest. as a pretest.

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.

Then students have been studied on certain problems Then students have been studied on certain problems which help to discover the mathematical concepts. which help to discover the mathematical concepts.

Teaching the derivative concept has been designed as Teaching the derivative concept has been designed as two consecutive steps: two consecutive steps:

At first step; students studied on the concept of At first step; students studied on the concept of derivative as rate of change. At this step real life derivative as rate of change. At this step real life problems about rate of change have been given to problems about rate of change have been given to students. By solving these problems students have students. By solving these problems students have discovered the concept of the derivative. Students discovered the concept of the derivative. Students interpreted graphics of functions for developing interpreted graphics of functions for developing conceptual understanding of rate of change. conceptual understanding of rate of change.

At second step, activities have been designed as At second step, activities have been designed as geometrical, numerical and symbolic (algebraic) geometrical, numerical and symbolic (algebraic) representations of derivative concept.representations of derivative concept.

In In studentstudent activities, which were designed by researchers activities, which were designed by researchers before, used in computer learning environment for procedural before, used in computer learning environment for procedural and conceptual learning of the derivative concept. These and conceptual learning of the derivative concept. These activities administered with interactive worksheets prepared activities administered with interactive worksheets prepared with maple, animated and non-animated graphics, plotted by with maple, animated and non-animated graphics, plotted by maple, special applets in maple called maplet. maple, special applets in maple called maplet.

By using interactive maple worksheets and animated graphics, By using interactive maple worksheets and animated graphics, students have found the opportunity of numerous experiments students have found the opportunity of numerous experiments that provide well understanding for them. that provide well understanding for them.

To provide conceptual and meaningful understanding for the To provide conceptual and meaningful understanding for the student, a maplet has been designed to see, geometrical student, a maplet has been designed to see, geometrical application of derivative as slope of the tangent line. application of derivative as slope of the tangent line.

At the end of the treatment, students’ understanding At the end of the treatment, students’ understanding of derivative was elicited through written tasks of derivative was elicited through written tasks administered to all students. administered to all students.

For this exam, students were given the opportunity, For this exam, students were given the opportunity, but not required to use the computer to solve the but not required to use the computer to solve the problems. These problems were considered to be problems. These problems were considered to be “computer neutral”. Students were presented with “computer neutral”. Students were presented with tasks that assessed their conceptual understanding and tasks that assessed their conceptual understanding and representational methods of solution of derivative.representational methods of solution of derivative.

The open-ended written tasks used in the examination The open-ended written tasks used in the examination instrument were mostly adapted from instrument were mostly adapted from Girard (See [5]) Girard (See [5]) common tasks used to assess student understanding of common tasks used to assess student understanding of derivative, in Calculus I courses and found in most textbooks derivative, in Calculus I courses and found in most textbooks or adapted from other studies concerning student or adapted from other studies concerning student understanding of derivative concept.understanding of derivative concept.

The tasks were evaluated by a panel of mathematics The tasks were evaluated by a panel of mathematics instructors (two university level) for the reasonableness of the instructors (two university level) for the reasonableness of the question for university Calculus I students. Recommendations question for university Calculus I students. Recommendations from the expert panel were examined and changes were made from the expert panel were examined and changes were made to the instrument accordingly.to the instrument accordingly.

Results 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.

Initially,we conducted an independent samples t test having student’s pretest attainment in the he calculus potential test calculus potential test to examine whether there were statistically significant differences between the two groups.

Group Statistics

22 47,0227 8,79729 1,87559

27 46,4259 8,73400 1,68086

groupmathematics

fen

pretestN Mean Std. Deviation

Std. ErrorMean

Independent Samples Test

,001 ,976 ,237 47 ,814 ,59680 2,51666 -4,46607 5,65967

,237 44,890 ,814 ,59680 2,51856 -4,47617 5,66977

Equal variancesassumed

Equal variancesnot assumed

pretestF Sig.

Levene's Test forEquality of Variances

t df Sig. (2-tailed)Mean

DifferenceStd. ErrorDifference Lower Upper

95% ConfidenceInterval of the

Difference

t-test for Equality of Means

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).

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.

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.

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

We can conclude that the students of mathematics group

performed significantly better than the students of the science

group in conceptual knowledge questions.

Conceptual questionsConceptual questions

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.

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.

On the other hand, in question 4 there was not On the other hand, in question 4 there was not significant differences in students’ post-test attainment between the two groups, p>0.05.

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

We can conclude that the students of mathematics group performed significantly better than the students of the science group in procedural knowledge questions.

Procedural questionsProcedural questions

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.

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.

On the other hand, in question 5 there was a On the other hand, in question 5 there was a significant differences in students’ post-test attainment between the two groups, p>0.05.

DiscussionDiscussion

Most of the students in mathematics group Most of the students in mathematics group answered this question correctly. Students in answered this question correctly. Students in science group can do the computations but science group can do the computations but they couldn’t show the equality.they couldn’t show the equality.

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.

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.

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 [1] Artigue, M. Analysis. 1991. In Tall, D. (Ed.), Advanced

Mathematical Thinking. Kluwer Academic Pub. pages 167-198.

[2] [2] Baroody, A. J. (2003). The development of Baroody, A. J. (2003). The development of adaptiveexpertise and flexibility: The integration of adaptiveexpertise and flexibility: The integration of conceptualand procedural knowledge. In A. J. Baroody & conceptualand procedural knowledge. In A. J. Baroody & A.Dowker (Eds.), A.Dowker (Eds.), The development of arithmetic conceptsThe development of arithmetic concepts and skills: Constructing adaptive expertise (pp. 1-and skills: Constructing adaptive expertise (pp. 1-33).33).Mahwah, NJ: Erlbaum.Mahwah, NJ: Erlbaum.

[3] [3] Byrnes, J. P., & Wasik, B. A. (1991). Role of Byrnes, J. P., & Wasik, B. A. (1991). Role of conceptualknowledge in mathematical procedural learning.conceptualknowledge in mathematical procedural learning. Developmental Psychology, 27(5), 777-786.Developmental Psychology, 27(5), 777-786.

[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.

[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.

[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.

[7] Girard, N.R. Students’ representational approaches to solving calculus problems:

Examining the role of graphic calculators. 2002. (University of Pittsburgh)

[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.

[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 107-119.

[10] White, P. Is calculus in trouble? 1990. Australian Senior Mathematics Journal, 4(2), pages 105-110.

Köszönöm szépen!

Contact:

Mehmet Bulut

Gazi University

Faculty of Gazi Education

ANKARA-TURKEY

mbulut@gazi.edu.tr