Effectiveness of the Differential Calculus Lecture by Using Teaching Materials Based on Open Ended Approach
Oleh : Syarifah Fadillah (firstname.lastname@example.org; IKIP PGRI Pontianak, Indonesia)
Differential calculus is one of the subjects studied in the mathematics education study program. In studying differential calculus, it cannot be separated from mathematical representation. Mathematical representations are expressions of mathematical ideas presented by the students as a model or as a substitute form of situation that is used to find the solution of problems being faced by the students as a result of their mind interpretation. A problem can be represented through images, tables, graphs, words (verbal), or mathematical symbols. In calculus, students are required to have good mathematical representation ability so that they can solve problems related to mathematical concepts involving calculus. Many problems in calculus are more easily solved if they are represented in a visual form first.
Some forms of mathematical representation, such as verbal, images, numeric, algebraic symbols, tables, diagrams, and charts are inseparable parts of mathematics lesson. However, generally in learning mathematics, mathematical representations are studied or taught only as a supplement in solving mathematical problems. As essential component of learning, mathematical representation ability should be trained in the process of learning mathematics. This causes the ability of students and students’ representation is limited.
Several studies have revealed the weaknesses of student’s mathematical representation ability. The difficulties encountered among others are: the students’ difficulties in bridging representations and flexibly moving from one representation to other representations (Yerushalmy, ). Ferrini, et al  says that in studying calculus, students often feel satisfied with the different results and representations, and do not always consider that the results are inconsistent, and even contradictory. The difficulties were also encountered by the writers during teaching differential calculus at IKIP PGRI Pontianak. Most students were still having difficulties in using various forms of mathematical representation to explain mathematical ideas and to solve mathematical problems. They also were still difficult in translating between forms of mathematical representation. This condition certainly needs to be solved, considering that the students of IKIP are becoming mathematics teachers that should be able to develop the ability of representation to their students.
The selection of appropriate learning approach will support the development of the representation ability. One of alternative approaches to learning mathematics that is expected to improve the students’ mathematical representation ability is open ended approach. Sawada  says that in the open-ended approach, teachers give situation problems to the students in which solutions or answers to the problems can be obtained in various ways. The teachers or tutors then use the differences of approaches or ways used by learners to provide experience to the students in discovering or investigating something new by combining knowledge, skills, and various mathematics methods or ways which have been learned by the students previously. By various ways of solution and answer, it gives students a lot of experience in interpreting the problems and it may also generate different ideas in solving the problems (Silver, ). This will certainly open the possibility for the students to use a variety of representations to find solutions to their problems, and it can help students solve the problems creatively, so that through learning by using open ended approach, it is expected to improve students’ mathematical representation ability.
To carry out the lecture by using open ended approach, lecturers or teachers need teaching materials which are based on the approach. Therefore, this study was preceded by developing teaching materials based on open ended approach to differential calculus lecture. The teaching materials will also consider the ability to be developed, that is the mathematical representation ability. Furthermore, teaching materials that have been developed are tested in the class lecture to obtain information about the effectiveness of differential calculus lecture using the materials. In addition, the effectiveness of lecture is observed from the improvement of mathematical representation ability, students’ activities during differential calculus lecture, and students’ responses to differential calculus teaching materials based on open-ended approach which has been arranged.
Students’ Mathematical Representation Ability
Description of data on the test results of students’ mathematical representation ability after attending lectures by using differential calculus teaching materials based on open ended approach is presented in Table 1 as follows:
Table 1. Students’ Mathematical Representation Ability
From Table 1 it is seen that the initial ability of students’ mathematical representation of differential calculus is very low, with an average pretest score of 5.22 out of a maximum score of 100. After obtaining the lecture by using differential calculus teaching materials based on open ended approach, there is an increase of mathematical representation ability of students, with the posttest score of 40.03. Calculation of the average increase in the mathematical representative ability of students using normalized gain formula, it is obtained the average increase of 0.37 which is categorized as moderate.
The result illustrates that the differential calculus lecture using the open ended approach can improve the students’ mathematical representation ability, although the result has not yet been maximum because the average score is still very low at 40.03 from maximum score 100. This happens because in the open ended approach, students are given the opportunity to resolve a problem with various answers and ways/methods, so that it would appear diverse representation of the problem.
Open problems given to the students are not only oriented to get the answers or final results, but they more emphasis on how students arrive at the answers, students can develop methods, or different ways to solve the problems. It provides opportunities for students to undertake a greater elaboration, so that students can develop their mathematical thinking, as well as assisting the development of students’ creative activity in using a variety of representations in problem solving.
The results are consistent with Inprasitha’s  research, who found that by applying open ended approach, students had more opportunities to do something, to think, to play an active role, to do something original, and to draw appropriate conclusions with their own ways. The study also concluded that the learning ability of students who obtained learning with open approach was better than the conventional class.
The results are also in line with Dewanto’s  research on learning with problem-based learning approach (BBM) which concluded that through non-routine problems, including an open issue, could improve students’ mathematical representation ability.
The average of students’ activities during the differential calculus lecture using teaching materials based on open ended approach for all activities is 61.5%. Students’ activities during the differential calculus lecture using teaching materials based on open ended approach are quite active, it is seen from the average activity of active student at 61,5%. This is in line with the Nohda’s  opinion who says that learning by using open ended approach can help the development of creative activities of students. Through this approach, each student has the freedom to solve problems according to their ability and interest, students with higher abilities can perform a variety of mathematics activities, and students with lower abilities can still enjoy mathematics activities according to their own abilities.
The responses of students to differential calculus teaching materials based on open ended approach overall are quite good. The overall results of processing the data on student responses to the differential calculus teaching materials based on open ended approach quite good. By using the materials, it makes the students more motivated and enjoy learning, the students feel more easier to understand the materials and make them think more critically. The students also feel quite satisfied with the content, quality of writing, and images of teaching materials. The lowest average score on the aspects of belief is some students still feel that the information in the teaching materials is too much so that it is difficult for them to take important ideas and to remember them. Most students also states that the tasks in the teaching materials are too much and too hard, however, learning by using the teaching materials makes them confident in learning and in solving the tests.
Based on the data analysis, generally it can be concluded that the differential calculus lecture by using teaching materials based on open ended approach is effective. In detail, it can be summed up as follows: (1) the average increase of students’ mathematical representation ability using normalized gain formula of 0.37 is categorized moderate; (2) the average of students’ activities during the differential calculus lecture using teaching materials based on open ended approach is 61.5%; and (3) the average score of students’ responses to teaching materials of differential calculus based on open ended approach of 3.33 is categorized quite good.
 NCTM, Principles and Standards for School Mathematics. Drive, Reston, VA: The NCTM, 2000.
 Yerushalmy, M, “Designing Representations: Reasoning about Functions of Two Variables,”. Journal for Research in Mathematics Education, vol. 27, pp. 431-466, 1997.
 Ferrini, et al, An Overview of the Calculus Curriculum Reform Effort: Issues for Learning, Teaching, and Curriculum Development. The American Mathematical Monthly, 98(7), 627-635, form http:// ortal.acm.org/citation.cfm?id=115400. downloaded on 28 April 2008.
 A. Sfard, Operational Origins of Mathematical Objects and the Quandary of Reification-The Case of Function. in E. Dubinsky & G. Harel (Eds). The Concept of Function: Aspects of Epistemology and Pedagogy. USA: Mathematical Association of America, 1992.
 T. Zachariades, et al, The Difficulties and Reasoning of Undergraduate Mathematics Students in the Identification of Functions. University of Athens, 2002
 Y.Y. Hong, et al. “Understanding Linear Algebraic Equations via Super-calculator Representations,” in T.Nakahara & M. Koyama (Eds). Proceedings of the 24th Annual Conference of the International Group for the Psychology of Mathematics Education, pp. 57-64, 2000.
 T. Sawada, Developing Lesson Plans on Shimada, S. and Becker, J.P. (Ed). The Open Ended Approach. A New Proposal for Teaching Mathematics. Reston: VA NCTM, 1997.
 E. A. Silver, Fostering Creativity through Instruction Rich in Mathematical Problem Solving and Problem Posing. form http://www.fizkarlsruhe.de/fiz/ publications/ zdm/2dm97343.pdf. downloaded on 19 May 2008.
 S. Thiagarajan, et al, Instructional Development for Training Teachers of Exceptional Children. A Source Book. Bloomington: Center of Innovation on Teaching the Handicapped. Minneapolis: Indian University, 1974.
 R. Hake, Analyzing Change/Gain Scores, form http://www.physics. indiana.edu/-sdi/AnalyzingChange-Gain.pdf. downloaded on 7 February 2009.
 J.E. Kemp, et al, Designing Effective Instruction. New York: Macmillan College Publishing Inc, 1994.
 E. Suherman, et al, Strategi Pembelajaran Matematika Kontemporer. Bandung: Jurusan Pendidikan Matematika FPMIPA UPI, 2003.
 P.D. Eggen and Kauchak, Strategies for Teachers : Teaching Content and Thinking Skills. New Jersey : Prentice Hall, 1988.
 M. Inprashita, Open-ended Approach and Teacher Education. from http://www.criced.tsukuba.ac.jp /math/apec2006/progress_report/Symposium/Imprasithaa.pdf. downloaded on 15 May 2008.
 S. Dewanto, Meningkatkan Kemampuan Multipel Representasi Mahasiswa melalui Problem-based Learning, Dissertation on SPS UPI. Not published, 2008.
 N. Nohda, A Study Of “Open-Approach” Method In School Mathematics Teaching – Focusing On Mathematical Problem Solving Activities, from http://www. nku.edu/~sheffield/nohda.html. downloaded on 31 March 2008.