CONSTRUCTING THE DISK METHOD FORMULA FOR CONSTRUCTING THE DISK METHOD FORMULA FOR

THE VOLUME OBTAINED BY REVOLUTION THE VOLUME OBTAINED BY REVOLUTION A A

CURVE AROUND AN AXIS BY THE HELP OF CASCURVE AROUND AN AXIS BY THE HELP OF CAS

Muharrem AKTÜMEN* , Tolga KABACA**

* Kastamonu University, Faculty of Education, Department of Primary Mathematics Education, Kastamonu, Turkey

** Uşak University, Faculty of Arts and Science, Department of Mathematics, Uşak, Turkey

COMPUTER ALGEBRA SYSTEMCOMPUTER ALGEBRA SYSTEM

• Computer algebra systems (CAS) are originally created for the use

of applied mathematicians and engineers. Since their development

in the 1970’s and their introduction into some tertiary teaching and

learning in the 1980’s, CAS have been recognised as highly

valuable for doing mathematics and potentially valuable for teaching

and learning mathematics [1].

• Nowadays, the current systems, and most popular are: Maple,

Mathematica and Derive. And these softwares are used

progressively in mathematics education.

COMPUTER ALGEBRA SYSTEMCOMPUTER ALGEBRA SYSTEM

• Clements identified a number of main roles which a CAS can fulfil

[2].

• A mathematical laboratory or toolkit: Mathematical and allied

practitioners use CAS to help explore new ideas and new

mathematical structures. The features of CAS which are important

to such users are the ability to define new mathematical entities and

new operations on such entities. These features are essentially

those of the CAS as a mathematical programming language and an

extensible system.

COMPUTER ALGEBRA SYSTEMCOMPUTER ALGEBRA SYSTEM

• A mathematical assistant: Mathematically skilled scientists,

engineers and allied practitioners often need to carry out

computations using well established methods but of a magnitude

and/or complexity which would be too difficult, costly or time

consuming by traditional pencil and paper methods. A CAS can act

as a tireless, quick and (usually) impeccably accurate mathematical

assistant in such situations. Not only do CAS help with calculations

which would otherwise be too tedious and time consuming but their

very existence encourages users to contemplate computations

which they would otherwise have rejected as impractical.

COMPUTER ALGEBRA SYSTEMCOMPUTER ALGEBRA SYSTEM

• A mathematical expert system: Mathematically less well qualified

users can call upon CAS to carry out manipulations which they

might not feel confident to complete by hand for lack of manipulative

skill. In other circumstances such users might have consulted a

more mathematically skilled person for guidance and validation of

their mathematical activities. Thus, for such users, the CAS is acting

as a mathematical advisor or expert. Of course, it is important that

such users are aware of the possibilities which such use brings for

the introduction of inadvertent errors and they must learn to

implement routinely appropriate error checking and validation

strategies.

CAS in MATHEMATICS EDUCATIONCAS in MATHEMATICS EDUCATION

• Kutzler looks at two areas and explain the importance and

significance of technology therein. The two areas are mathematics

(intellectual) and moving/ transportation (physical) [3].

• The most elementary method of moving is walking. Walking is a

physical achievement obtained with mere muscle power. The

corresponding activity in mathematics is mental calculation (mental

arithmetic and mental algebra.) Mental calculation requires nothing

but “brain power”.

CAS in MATHEMATICS EDUCATIONCAS in MATHEMATICS EDUCATION

• Riding a bicycle is a method of moving, where we employ a mechanical

device for making more effective use of our muscle power. Compared to

walking we can move greater distances or faster. The corresponding

activity in mathematics is paper and pencil calculation. We use paper

and pencil as „external memory“ which allows us to use our brain power

more efficiently.

• Another method of moving is driving a car. The car is a device that

produces movement. The driver needs (almost) no muscle power for

driving, but needs new skills: He must be able to start the engine, to

accelerate, to steer, to brake, to stick to the traffic regulations, etc. The

corresponding activity in mathematics is calculator/ computer calculation.

The calculator or computer produces the result, while its user needs to

know how to operate it.

MATHEMATICS EDUCATION BUT HOW?MATHEMATICS EDUCATION BUT HOW?

• Many educators therefore want to follow a traditional paradigm using

sequence “Definition theorem proof corollary application” by an

approach which is more historic using the discovery chain “Problem

experiment conjecture and idea of proof ” [4].

• A CAS allows lots of experimenting by the students, thus helping to find

reasonable conjectures ([5], [6]). As a result of a great body of research, it

can be understood that using CAS is more meaningful when it was

integrated with constructivist principles in the same environment ([7], [8]).

• In a technology based constructivist learning environment, students can

find the opportunity to discover, making conjectures and construct their own

mathematical knowledge. Many researchers used a CAS like Maple,

Mathematica or Derive making students to discover by visualizing the

calculus concepts ([5], [9], [10], [11], [12])

PROBLEM SITUATIONPROBLEM SITUATION

• Calculus concepts should have been taught in a carefully designed

learning environment, because these concepts constitute a very

important base for almost all applied sciences.

• Integral, one of the fundamental themes of Calculus, has a wide

application area. In the internet site of the Turkish Language

Association, the definitions of the concept of integral “The total

consists of pieces, derivation whose function (mathematics) is

known” are given. First meaning includes finding the areas that

under curves, calculating the volume of various material things and

other application areas that is named as definite integral. This paper

focuses on that calculating the volume of various material things

through Integral. For this we used Maple as CAS.

PROBLEM SITUATIONPROBLEM SITUATION

• The concept of volume is utilized in a wide variety of applications

including the physical sciences and all of the engineering

disciplines. For example, in constructing a highway one uses survey

data to approximate the amount of earth to be moved..

• In this paper it is found out that a CAS can have an effect on a

discovery application in an integral calculus course.

METHODOLOGYMETHODOLOGY

• In this study, a semi-structured interview was carried out. In this

interview, it is tried to construct the disk method formula.

Levels of constructing the disk method formula in this study:

• Entrance to the concept: evaluate the volume of an Egypt pyramid.

• Evaluate the volume of a cone (By Maple worksheet)

• Designing their own rings and evaluating its price (By Maplet).

• The interview has been presented as a dialog between teacher and

students.

PRECONCEPTPRECONCEPT

• Following steps, which are used to construct the integral giving the volume

of a three dimensional object, are assumed to be known by students;

PRECONCEPTPRECONCEPT

• Following steps, which are used to construct the integral giving the volume

of a three dimensional object, are assumed to be known by students;

PRESENTATION OF FINDINGSPRESENTATION OF FINDINGS

• First, a problem was given to students;

• Problem: Find a general formula for the volume of a square based Egypt

pyramid by using the integral concept.

• Key question: How can you divide into regular cross-section areas the

pyramid and how can you define these cross-section areas by the term of

the cutting variable?

• Teacher: Consider a pyramid. It is obtained a cross-section area by cutting

the pyramid, perpendicular to its height.

PRESENTATION OF FINDINGSPRESENTATION OF FINDINGS

PRESENTATION OF FINDINGSPRESENTATION OF FINDINGS

PRESENTATION OF FINDINGSPRESENTATION OF FINDINGS

At this point, students studied on the maple worksheet about volume of a Cone, before calculating its volume with paper and pencil method.

MAPLE WORKSHEETMAPLE WORKSHEET

MAPLET: DESIGN YOUR RINGMAPLET: DESIGN YOUR RING

STUDENTS’ RINGSTUDENTS’ RING

• f(x)=1

• g(x)=1.5

• Interval:

[-0.3,0.3]

• Volume:

• 2.3561

• Price: 1178 YTL

• f(x)=x^2+0.9

• g(x)=x^2+0.6

• Interval:

[-0.3,0.3]

• Volume:

• .8821

• 441 YTL

STUDENTS’ RINGSTUDENTS’ RING

• f(x)=-ln(x)+1.6

• g(x)=ln(x)-0.6

• Interval:

[3.1,3.3]

• Volume:

• 0.07915

• 39,57 YTL

• f(x)=x^2+.8

• g(x)=cos(x)+.2

• Interval:

[-0.2,0.6]

• Volume:

• 1.39456

• 697,28 YTL

STUDENTS’ RINGSTUDENTS’ RING

• f(x)=.4+tan(x)

• g(x)=1.8-tan(x)

• Interval:

[3.6,3.9]

• Volume:

• 0.46635

• 233,17 YTL

• f(x)=sin(x)+1.4

• g(x)=cos(x)+0.2

• Interval:

[-1.2,-1]

• Volume:

• 0.1054

• 52,7 YTL

STUDENTS’ RINGSTUDENTS’ RING

• f(x)=cos(2*x)

• g(x)=cos(2*x)+.2

• Interval:

[-0.3,0.3]

• Volume:

• 0.7849

• 392,45 YTL

• f(x)=sin(2*x)

• g(x)=cos(2*x)+.2

• Interval:

[0.3,0.5]

• Volume:

• 0.199601

• 99,8 YTL

0,1 br = 50 YTL2

You can download this maplet file to below web-site:http://w3.gazi.edu.tr/web/aktumen/diskmethod/disk.htm

1. Pierce. R., Stacey, K. (2002). Monitoring Effective Use of Computer Algebra Systems. In B. Barton, K.C. Irwin, M. Pfannkuck & M. O. J. Thomas (Eds.), Mathematics Education in the South Pacific (Proceedings of the 25th Annual Conference of the Mathematics Education Research Group of Australasia, 575-582.

2. Clements, R. (1999). Essential Mathematical Concepts Needed by User of Computer Algebra. Teaching Mathematics and its Applications, Volume 18. No 4.

3. Kutzler, B. (2000). The Algebraic Calculator as a Pedagogical Tool for Teaching Mathematics, The International Journal of Computer Algebra in Mathematics Education, (7) 1, 5 – 24.

4. Cnop, I., New Insight in Mathematics by Live CAS Documents, Albuquerque Conference, 2001.

5. Zehavi, N., Exploring the relationship between reflective thinking and execution in solving problems with a Computer Algebra System, International Workshop on research in Secondary and Tertiary Mathematics Education, 7-11 July, 2006.

ReferencesReferences

6. Dubinsky, E., Schwingendorf, K., Calculus, Concepts, Computers and Cooperative Learning (C4L), The Purdue Calculus Reform Project, 2004.

7. Leinbach C., Pountney, D.C. and Etchells, T., Appropriate Use of a CAS in the Teaching and Learning of Mathematics, International Journal of Mathematical Education in Science and Technology, Vol.33, No.1, 2002, pp.1-14.

8. Malabar, I. and Pountney, D.C, When is it appropriate to use of a Computer Algebra System (CAS)?, Proceedings of ICTMT4 Playmouth, 9-13 August 1999.

9. Chundang, U., Using CAS for the visualization of some basic concepts in calculus of several variables, TCM Conference, Japan, 1998.

10.Galindo, E., Visualization and Students’ Performance in Technology based Calculus, 17th PME-NA, Columbus, OH, October 21-24, 1995.

11.Cunningham, S., Some strategies for using visualization in mathematics teaching, ZDM, 3, 1994, pp 83-85.

12.Kabaca, T., Constructing the Limit Concept by using a Computer Algebra System [CAS], International Conference of Teaching Mathematics (ICTM), June 30 – July 5, 2006.

ReferencesReferences

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