The effects of CAS in the development of problem

solving abilities

Mehmet Bulut, Gazi University, TR. Mehmet Bulut, Gazi University, TR. mbulut@gazi.edu.trmbulut@gazi.edu.trandand

Yılmaz Aksoy, Yılmaz Aksoy, ErciyesErciyes University, TR. University, TR. aksoyyilmaz@hotmail.comaksoyyilmaz@hotmail.comandand

Şeref Mirasyedioğlu, Başkent University, TR. Şeref Mirasyedioğlu, Başkent University, TR. serefm@baskent.edu.trserefm@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 problem solving abilities 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 and science education lessons, we choose this concept for comparing the development of problem solving abilities first year undergraduate mathematics and science 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.

PROBLEM SOLVINGPROBLEM SOLVING

Problem solving is considered central to Problem solving is considered central to school mathematics. NCTM (2000) school mathematics. NCTM (2000) states,Instructional programs should enable all states,Instructional programs should enable all students to build new mathematical knowledge students to build new mathematical knowledge through problem solving; solve problems that through problem solving; solve problems that arise in mathematics and in other arise in mathematics and in other contexts;apply and adapt a variety of contexts;apply and adapt a variety of appropriate strategies to solve problems; and appropriate strategies to solve problems; and monitor and reflect on the process of monitor and reflect on the process of mathematical problem solving. [p. 52]mathematical problem solving. [p. 52]

Similarly, Kilpatrick et al. (2001, p. 420) explained, Similarly, Kilpatrick et al. (2001, p. 420) explained, Studies in almost every domain of mathematics have Studies in almost every domain of mathematics have demonstrated that problem solving provides an demonstrated that problem solving provides an important context in which students can learn about important context in which students can learn about number and other mathematical topics. Problem-number and other mathematical topics. Problem-solving ability is enhanced when students have solving ability is enhanced when students have opportunities to solve problems themselves and to see opportunities to solve problems themselves and to see problems being solved. problems being solved.

Further, problem solving can provide the site for Further, problem solving can provide the site for learning new concepts and for practicing learned learning new concepts and for practicing learned skills.Thus, problem solving is important as a way of skills.Thus, problem solving is important as a way of doing, learning and teaching mathematics. doing, learning and teaching mathematics.

If problem solving should be taught to If problem solving should be taught to students, then it should be taught to preservice students, then it should be taught to preservice teachers who are likely to not have been taught teachers who are likely to not have been taught it in an explicit way. If it is to form a basis of it in an explicit way. If it is to form a basis of teaching mathematics, then preservice teachers teaching mathematics, then preservice teachers should understand it from a pedagogical should understand it from a pedagogical perspective. perspective.

Problem solving In mathematics, it is considered that problem

solving to be the ability to use mathematical tools for solving real world technical problems. Characteristic of problem solving are the three steps shown below

Today, problem solving is treated at school only half-heartedly. The main emphasis is put on the second step, calculation, and its execution with paper and pencil.

Let’s look at optimization problems. Typically we treat them in a manner so that we can do, may be, three optimization problems in an hour using 80% of the time for (hand) calculations so that only 20% remain for modeling and translating. Hence, most problem solving exercises turn into exercises for practicing the required calculation skills. And this we call “problem solving training”?!

Since translations from the real world into mathematics and vice versa rarely are taught explicitly, it is understandable that a majority of students don’t develop this ability. Hence, they are afraid of exercises requiring such translations. With the (extensive) use of powerful technology such as CAS for the calculation step, we can dedicate a lot more time to teach the choice of models and how to translate problems and results.

We may be able to treat ten or more optimization problems in an hour spending 80% of the time on modeling and translation and only 20% on calculations. This would be the proper “problem solving training”![1]

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.”

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.

There is a commonly held belief that problems There is a commonly held belief that problems associated with algebraic manipulations can easily be associated with algebraic manipulations can easily be overcome by using the symbolic algebra facility of CAS overcome by using the symbolic algebra facility of CAS (Bennett, 1995; Day, 1993; Heid, 1988; Tall, 1996). Bennet (Bennett, 1995; Day, 1993; Heid, 1988; Tall, 1996). Bennet (1995) and his teaching colleagues monitored a one-term (1995) and his teaching colleagues monitored a one-term general education calculus course with several classes of general education calculus course with several classes of undergraduate students of varying abilities. The college undergraduate students of varying abilities. The college teachers informally evaluated the course and they found that teachers informally evaluated the course and they found that their students were able to tackle problems in a variety of their students were able to tackle problems in a variety of different ways. Bennet found that he spent less class time different ways. Bennet found that he spent less class time dealing with algebraic problems and, in consequence, he dealing with algebraic problems and, in consequence, he was able to spend more time discussing calculus concepts was able to spend more time discussing calculus concepts with the students. He strongly recommends use of CAS with the students. He strongly recommends use of CAS (Derive) to minimize the detrimental effects of students’ poor (Derive) to minimize the detrimental effects of students’ poor algebraic backgrounds.algebraic backgrounds.

Frid (1992), in a wider study of comparative Frid (1992), in a wider study of comparative methods of instruction for college calculus, methods of instruction for college calculus, interviewed students and observed that interviewed students and observed that those from the class that had used Tall’s those from the class that had used Tall’s approach were better able to explain the approach were better able to explain the underlying notion of local linearity using the underlying notion of local linearity using the notion of local straightness.notion of local straightness.

Using the CAS should also help students Using the CAS should also help students overcome algebraic difficulties.overcome algebraic difficulties.

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 before, used in computer learning environment for problem problem solvingsolving of the derivative concept. These activities of the derivative concept. These activities administered with interactive worksheets prepared with maple, administered with interactive worksheets prepared with maple, animated and non-animated graphics, plotted by maple, animated and non-animated graphics, plotted by maple, special applets in maple called maplet. 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 common tasks used to assess student understanding of derivative, in Calculus I courses and found in most of derivative, in Calculus I courses and found in most textbooks or adapted from other studies concerning textbooks or adapted from other studies concerning student understanding of derivative concept.student 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 instructors (two university level) for the validity of validity of qquestionuestions.s. Recommendations from the expert panel Recommendations from the expert panel were examined and changes were made to the were examined and changes were made to the instrument accordingly.instrument accordingly.

Results

Means of problem solving questions were evaluated and compared for all students.

Scores for all problems 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 ANCOVA using the calculus potential test grades as covariate, procedural and conceptual problems’scores as dependent variables.

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).

The statistically non-significant difference between the mathematics and the science groups as measured by the pretest scores revealed that both the groups appeared to be equivalent in their mathematical knowledge prior to this study. However, in a quasi experimental design, the possibilities of pre-existing differences in mathematical abilities between the groups can not be completely ruled out. The statistical procedure of the analysis of covariance takes into account any such pre-existing differences between the groups while comparing their means. So for further analysis of the data, the analysis of covariance was used with the pretest scores used as the covariate.

To address the study’s null hypothesis, student participants in both mathematics and science groups were tested on their problem solving ability at the posttest after the treatment completed. The mathematics group mean score was 19,3636 with a standard deviation of 4,4886, and the science group mean score was 15,6667 with a standard deviation of 5,2404 (Table1).

Table 1: Descriptive Statistics for Students Performance by Group on the Posttest Table 1: Descriptive Statistics for Students Performance by Group on the Posttest ExaminationExamination

An ANCOVA comparison of this measure revealed significant difference in problem solving ability between the groups following the treatement (p=.013) at an a priori determines alpha level of .05 (Table 2). Equality of variances was assured with a Levene’s Test (α=.163)

Table2: Comparative Analysis of Students Performance by Group as Measured by the Posttest Exam with Pretest Measure as a Covariate

we can conclude that the students of mathematics group performed significantly better than the students of the science group in problem solving questions.

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.

Problem Solving QuestionsProblem Solving Questions

1.1.An eagle’s height is changing related to An eagle’s height is changing related to time. In order to find the eagle’s height to its time. In order to find the eagle’s height to its nest nest H (t)H (t) function where function where H H represents the represents the height and height and t t represent the time, is used.5 represent the time, is used.5 seconds after taking off from its nest an seconds after taking off from its nest an eagle’s height to its nest is 100 meter. Its rate eagle’s height to its nest is 100 meter. Its rate of change in height is 3 meter per second. of change in height is 3 meter per second. What is the value of ?What is the value of ?

Correct Correct answer answer sample of sample of studentsstudents

Wrong Wrong answer answer sample of sample of studentsstudents

2. A farmer has a 1000 meter hedge. He wants 2. A farmer has a 1000 meter hedge. He wants to hedge his terrain whose one side is a wall. to hedge his terrain whose one side is a wall. As he will not use hedge for the wall, find the As he will not use hedge for the wall, find the dimensions of the terrain in reactangular form dimensions of the terrain in reactangular form which has the greatest area. Calculate the which has the greatest area. Calculate the maximum area of the terrain. maximum area of the terrain.

Correct Correct answer answer sample of sample of studentsstudents

Wrong Wrong answer answer sample of sample of studentsstudents

3.A gas company wants to furnish 3.A gas company wants to furnish pipes from the point B in the pipes from the point B in the island to the point A at the coast as island to the point A at the coast as shown in the figure. The distance shown in the figure. The distance of the point B to the coast of the point B to the coast perpendicularly is 1,6 km. The perpendicularly is 1,6 km. The distance from this point to A is 3,2 distance from this point to A is 3,2 km. The coast of furnishing the km. The coast of furnishing the pipes at the coast is 27 YTL and pipes at the coast is 27 YTL and the coast of furnishing the pipes the coast of furnishing the pipes under sea is 143 YTL. In order to under sea is 143 YTL. In order to furnish the pipes with minimum furnish the pipes with minimum coast find the value of the x. coast find the value of the x.

Correct Correct answer answer sample of sample of studentsstudents

Wrong Wrong answer answer sample of sample of studentsstudents

References

[1] Artigue, M. Analysis. 1991. In Tall, D. (Ed.), Advanced Mathematical Thinking. Kluwer Academic Pub. pages 167-198.

[2] 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.

[3] 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.

[4] Fey, J. T. Technology and mathematics education: A survey of recent developments and important problems.1989. Educational Studies in Mathematics, 20, pages 237-272.

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

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

[6] 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.

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Contact:Contact:

Mehmet BulutMehmet Bulut

Gazi UniversityGazi University

Faculty of Gazi EducationFaculty of Gazi Education

ANKARA-TURKEYANKARA-TURKEY

mbulut@gazi.edu.trmbulut@gazi.edu.tr