THE EFFECT OF OPEN-ENDED APPROACH AND BELIEFS ABOUT MATH TOWARD STUDENTS MATHEMATICAL CONNECTION ABSTRACT

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1 Indonesian Journal of Educational Review p-issn e-issn Vol.2, No.2, November 2015, p Available online at THE EFFECT OF OPEN-ENDED APPROACH AND BELIEFS ABOUT MATH TOWARD STUDENTS MATHEMATICAL CONNECTION Sri Riyanti Faculty of Teacher Training and Education Tanjungpura University, Indonesia riyanti_mtk@yahoo.co.id ABSTRACT The purpose of this research is to determine the effect of the Open-ended approach and beliefs about mathematics toward ability connection mathematically by controlling previous knowledge of students. This research used by Quasi-experimental design with treatment by Level 2 x 2. Data were analysized by using Analysis of Covariance (Ancova). After controlling students preious knowledge, it was found that the ability of students mathematical connetions who were given with open-ended approach grouply is better than the ability of students mathematical connection who were given the open-ended approach individually, and then there was an interaction effect between the open-ended approach and students beliefs about mathematics toward mathematical connection ability. Interaction effect can be seen that the ability of students mathematical connection who were given open-ended approach in the group is better than the open-ended approach individually, especially for students in the group who have high beliefs about mathematics. Otherwise, especially for student in the group who got lower beliefs about mathematics, the ability of mathematical connection with the learning openended approach is less well than the open-ended approach individually. It can be concluded that the ability of mathematical connection is caused by the type of open-ended approach and students beliefe about mathematics. Keywords: ability of mathematical connection; open-ended approach; beliefs about mathematics; previous knowledge. In essence, the science of mathematics as a structured and systematic implies that the concepts and principles of mathematics are interrelated of each other. Mathematical connection capability is one of the critical thinking skills in mathematics that should be controlled by the student. Without mathematical connection then students should learn and remember too many concepts and mathematical procedures which are separated. For instance concepts in fractions, percentages, ratios, and linear comparison. The ability of students to connect between concepts and mathematical objects, can lead to students' understanding of concepts will be broader and deeper. This is also confirmed in the National Council of Teachers of Mathematics (2000) stated that if students can connect mathematical ideas, then they will be more in-depth understanding and more lasting. If we analyze the topics in mathematics there is no stand-alone without any connection with other topics. According to Hodgson (1996), Wilburne and Napoli (2008) that the connections between topics in mathematics can be understood if by the child if learning trains their connections ability, one of which is a meaningful learning. The connection between processes and mathematical concepts in an abstract object means that this connection occurs in the mind of the student, for example, students use their minds when connecting between symbols with representations. With math, the math connection feels more meaningful. Copyright 2015, PPs UNJ Publisher

2 Students who are able to relate mathematical ideas that will further understanding of mathematics students in because they were able to see the connection between the topics in mathematics, with another mathematics context, and with the experience of everyday life. However, the reality on the ground is not always like what we have expected. Teachers in mathematics still relies on textbooks where the learning material presentation more focused on giving examples of problems that exist in textbooks and ask the students to do it, then discuss it with students, whereas, mathematical concepts have not been properly understood by the students. Lessons like these have not been able to develop a mathematical connection ability. Students can only work on math problems based on the example given by the teacher, if the questions given are different they would have difficulty in completing. In addition, there is still lack of students' interest towards mathematics, when they are given homework assignments, they would do it in the class next day instead of doing it at home. Based on these problems required an appropriate learning strategies, through an open-ended approach which is a method that can give flexibility to the students to think actively and creatively in solving a problem, so it is useful to enhance the students' way of thinking. According to Shimada (1997) and Hudiono (2008), an open-ended approach starts from the view of how to evaluate objectively the ability of students in high-level mathematical thinking. Through these activities students are expected to address issues in many ways, so it invites intellectual potential and experience of students in the process of finding something new. Thus the learning process using the open issues can develop students' mathematical thinking that they have creatively. According to Silver (1997), Cooney (2001), Collier and Tribus (1993) that openended problems can develop a problem statement of the scope of the problem, which can give students experience in interpreting and developing ideas / different ideas to solve a problem. Students' thinking needed to decipher mathematical ideas in solving a problem so that teachers can understand deeply of students' learning styles. Learning with the open-ended approach provides maximum space for students to channel their thinking skills to solve problems according with the interests owned so that students can be able to show the corresponding relationships mathematical concepts. With open questions, the students more involved in active learning in a team, be more able to define the problem and devise a solution so that students can develop lessons that describe the real world. Before applying the open-ended approach, it needs to pay attention on students' initial knowledge which is knowledge that has been owned by the students before attending lessons will be given. This initial knowledge illustrates the readiness of students in a lesson that will be delivered by the teacher. Each student has a diverse knowledge in absorbing subject matter. Diversity prior knowledge of students will affect students mastery learning concept. According to Khan (2012) and Ruseffendi (2006), the success of students in a lesson depends on the readiness of students seen of his mental development was ready and already possessed knowledge prerequisites, so as to assist students in understanding concepts that they have learned previously. Another factor of the student who is also an effect on student achievement is the belief. There are still many students who believe that mathematics is a difficult subject and very abstract. What students believed is mostly based on the experience 56

3 gained during the learning of mathematics. Therefore, the experience of learning mathematics fun, diverse, and constructivists, it is important to foster positive beliefs towards mathematics, particularly in junior secondary school. To know the results of learning mathematics in an effort to improve mathematical connection with openended learning approach, students are expected to have confidence in mathematics. Mathematical belief include beliefs about the role and function of teachers, beliefs about her/his abilities in mathematics, beliefs about mathematics as a social activity, and beliefs about mathematics as a discipline (Eynde, Corte, Verschaffel, 2006; Sugiman, 2009). The expectation of this study is the result of the analysis can test / answer theories of open-ended approach, beliefs about mathematics, and mathematical connection capability, by giving open-ended aproach learning both corporately and individually that have high confidence and low confidence can improve the ability of students' mathematical connections in math. METHOD The method used in this study is a quasi-experimental design with treatment by level 2 x 2. The dependent variable is the ability of students' mathematical connections. Whereas the independent variables are (1) an open-ended approach learning, and (2) beliefs about each mathematics students who have two levels of treatment. Tabel 1. Experiment Design Mathematical Belief (B) Beliefs about mathematics are categorized as high. (B1) Beliefs about mathematics are categorized as low. (B2) Open-ended Approach Learning (A) Open-ended Approach as a group (A1) [Y]11k k = 1, 2,, n11 (A1B1) [Y]11k k = 1, 2,, n12 (A1B2) Open-ended Approach as an individual (A2) [Y]11k k = 1, 2,, n21 (A2B1) [Y]11k k = 1, 2,, n22 (A2B2) Note A1B1 A2B1 A1B2 A2B2 Y k : open-ended approach setting groups that have beliefs about mathematics are categorized as high. : open-ended approach individual settings that have beliefs about mathematics are categorized as high. : open-ended approach setting that groups have beliefs about mathematics are categorized as low. : open-ended approach individual settings that have beliefs about mathematics are categorized as low. : Score mathematical connection ability. : Respondents every cell The population was SMP in Pontianak. Meanwhile, as the study sample were students in first semester of eighth grade, each taken four classes of seventh grade in junior high school SMP Negeri 2 and 11 Pontianak randomly. The sampling 57

4 technique using cluster sampling technique that was taken a sample of 72 students from each school were randomly. Instruments questionnaire beliefs about mathematics receipts Likert with scale reliability coefficient 0,896.Instrumen mathematical connection ability test in the form of essay test with reliabilitas0.798 coefficient. While the initial knowledge of mathematics test instruments such as double selection with reliability coefficient of Data analysis techniques used are: a descriptive analysis and hypothesis testing using Anacova by applying univariate GLM procedure of SPSS version 17.0 refers to the opinion of I Gusti Ngurah Agung (2006: 63-81). RESULTS Descriptive research results are presented as follows. Table 2. Description Score Mathematical Ability Student Connections Group N Minimum Maximum Standard Average Score Score Deviation A ,889 4,010 A ,361 4,022 B ,042 5,059 B ,208 3,860 A1B ,694 3,345 A1B ,083 4,010 A2B ,389 3,643 A2B ,333 3,397 Based on the description of the average ability of students explain that the mathematical connection of the open-ended approach in groups (A 1 ) is better than the open-ended approach individually (A 2 ). So also be seen from the standard deviation showed more homogeneous A 1 from A 2. On average mathematical ability to connect students with high confidence (B 1 ) is better than the ability of students' mathematical connection that have low confidence (B 2 ), but the standard deviation shows that the more homogeneous B 2 from B 1. On average ability students' mathematical connection to the cell A 1 B 1 is better than A 1 B 2, A 2 B 1 and A 2 B 2. Neither seen from the standard deviation indicates that the more homogeneous A 1 B 1 from A 1 B 2, A 2 B 1 and A 2 B 2. Further analysis of hypothesis testing using analysis techniques kovariat used GLM Univariate procedure with significant level in this study were taken by 5%. Table 3. Tests of Between-Subjects Effects for ANCOVA test. Source Type III Sum of Squares Df Mean Square F Sig. Partial Eta Squared Corrected Model a Intercept X A

5 B A * B Error Total Corrected Total a. R Squared =.743 (Adjusted R Squared =.735) Based on Table 3, shows that there are differences in mathematical connection ability by learning the open-ended approach in groups and open-ended approach individually after controlling for initial knowledge of mathematics students. The results obtained indicate that the F count = and Sig. = Because Sig. = less than α = 0.05, H 0 is rejected. Further analysis shows that there is interaction effect between open-ended and approach beliefs about mathematics students to the mathematical connection ability after controlling for initial knowledge of mathematics students. In Table 3, shows that F count = and Sig. = Because Sig. = less than α = 0.05, H 0 is rejected. Table 4. Mean Parameter Estimation Y Between All Levels Factor A Factor B After the For Each Control X 95% Confidence Interval Std. Partial Eta Parameter B T Sig. Error Squared Lower Bound Upper Bound Intercept X [B=1.00] [B=2.00] 0 a [A=1.00] * [B=1.00] [A=1.00] * [B=2.00] [A=2.00] * [B=1.00] 0 a [A=2.00] * [B=2.00] 0 a a. This parameter is set to zero because it is redundant. For students groups who have beliefs about higher mathematics, there are differences in students' mathematical connection ability by learning the open-ended approach in groups and open-ended approach individually after controlling for initial knowledge of mathematics is shown students. Things that count = 5846 and Sig. = Because Sig. = less than α = 0:05 H 0 is rejected. As for the group of students who have math confidence is low, there are differences in students' mathematical connection ability by learning the open-ended approach in groups and open-ended approach individually after controlling for initial knowledge of mathematics students. shown that t = and Sig. = Because Sig. = less than α = 0:05 then H 0 is rejected. Results of this analysis are shown in Table 4. 59

6 DISCUSSION Based on the research that has been done that the application of open-ended approach in varies and belief of mathematics are able to improve student's thinking skills, specifically the ability of mathematical connections. The results of the first hypothesis test shows that learning by using open-ended approach in group are indeed superior compared to the open-ended approach individually. In this study, learning by using the open-ended approach in groups can improve student s mathematical connections in order to develop their thinking skills to associate one concept to another, the concepts of other disciplines and the concepts of daily life in the process of learning of mathematics. According to the research by Alhadad (2010) and Sari, Kurniawati, Pramesti (2013) that the application of learning treatment with open-ended approach in the groups are able to improve thinking skills of students, so that students can work together to discuss issues given. Next, based on research by In Prasitha (2006) it was shows that learning by using open-ended approach proves that the students become proficient in observation, better know how to work together in a cooperative and more reliable in asking "why?" and "how?". Results of this study are supported by Sawada (1997) that learning with an open-ended approach, the teacher gives a student a problem situation in which a solution or answer to the problem can be obtained in various ways. Teachers then use the differences in approach or method used by students to provide experience to students in finding or investigate something new by combining the knowledge, skills and methods in ways math student has learned before. Steps of learning towards a group of students who were given open-ended approach individually is similar to the open-ended approach in groups. The learning with an open-ended approach is more focused on the individual student learning is done individually and guided by the teacher if students has difficulty im learning. According Hamalik (2011) that the student-centered teaching is a learning process based on the needs and interests of students so that the studentcentered teaching is designed to provide a flexible learning system based on the student s life and learning styles. Teachers are not act as the central but only as a supporter. The open-ended approach that individually given to students are still not optimal for learning process due to the approach hasn t reveal the significant result in when the students are given a problem, explore issues, record student responses, the discussion of student responses, and to summarize what have they learned. In this case, contradicts to what is revealed by Silberman (2006) when students learn in their own way, they develop the ability to focus and reflects. Also, working in their own way give students the opportunity to personally accountable for what they have learned. Lessons are given open-ended approach provide less individual basis for enhancing the impact of student s mathematical connection capabilities. Most likely due to the students are not involve to cooperate with peers so that students are unable to express their opinions in class during the discussions when they are ought to present the results of their work. The second hypothesis results shows that there is an interaction between the effect of the open-ended approach and beliefs of maths to mathematical connection ability after controlling for the initial knowledge of maths in students. Therefore, openended approach and student s beliefs about maths influence to determine the ability 60

7 of student's mathematical connection that require them to think critically when they are given such challenging questions. It means that the interaction effect will have good results and optimal if it s done on the effects of each treatment level adapted to the conditions and circumstances between the open-ended approach and beliefs of maths. Learning of maths will work well if the student s beliefs of maths gave a positive influence on learning by using the open-ended approach. Which is similar to Goldin (2002) opinion that a person s mathematical beliefs are formed from their attitude towards maths and its continuous conviction will form the mathematical value in that person. Because the factor of attitude is the one that determine the success or failure of studying maths. How can student s confidence on maths became an essential issue?. According to the research by Sugiman (2009), almost half of the students which have a very good level of confidence in maths, this is a potential of a class that can greatly support the learning process. Even some students who achieve a score of 95% mathematical with confidence. A Group of students who have high beliefs of maths are have faith on that everything that has done will definitely gives maximum results or success. Therefore, students who have a high confidence of mathematics learning with an open-ended approach as the group that will be easier to cooperate each other to achieve an agreement to solve challenging problems, having sense of responsibility and respect for other s opinions. In contrast, a group of students who have low confidence of maths will have fear, anxiety issue and feeling insecure. Therefore, students who have beliefs about mathematics learning with low given the open-ended approach by individually will motivated them to learn and gives a high curiosity towards learning maths because they realize of their weaknesses. Thus, in this study, there are significant interactions between the open-ended approach and beliefs maths towards mathematical connection ability after controlling the initial knowledge of maths in students. Since that there is an interaction between the open-ended approach and beliefs of maths to mathematical connection ability, then results will support the third hypothesis that students who have beliefs of maths in sub-group of high confidence has differences in student s mathematical connection ability by learning the openended approach in groups and open-ended approach individually after they controlling the initial knowledge of maths in students. Results of the average corrected calculation shows that the students who have belief of maths in sub-group of high confidence, the student's mathematical connection ability by learning with an open-ended approach in the group had an average of better than the students with mathematical connection ability by learning the open-ended approach individually which has an average of This results are consistent with the opinion cited by Kaunang Woolfolk (2012) that the most powerful belief is the one that influence motivation in school is the belief of his ability, and has the characteristics of stable properties and the uncontrollable individual has characteristics that unchanged. On the other hand, the ability are not stable and controllable-repertoire of skills and knowledge that always expanding. With hard work, study, or training, knowledge can be improved. Students who have positive beliefs of maths, of course will be well prepared to improve its ability, optimistic, rational, responsible in resolving a problem. Those 61

8 students may help themselves to appreciate school subjects and develop their own confidence. Consistent with the results of research by Tarmizi Suthar (2010) that there is a significant relationship between beliefs of maths and confidence in the ability of the achievement mathematics. In contrast, the fourth hypothesis results shows that students who have beliefs of maths in lower sub-group have differences in mathematical connection ability by open-ended approach in groups and open-ended approach individually after controlling the initial knowledge of maths in students, The mathematical connection ability with an open-ended approach in the group has an average at , while the students with mathematical connection ability with an open-ended approach individually has an average of Based on the calculation results show that students who have beliefs of maths in the lower sub-group, their ability of mathematical connection to a group of open-ended basis is less well than the open-ended approach individually after controlling for the influence of the initial knowledge of mathematics in students. The Results are different with the opinion of Greer, Verschaffel, and De Corte (2002) which state that the beliefs and negative attitudes unable to help students to appreciate maths and can t help them to develop confidence of their abilities. This is due to many factors that shape student s mathematical beliefs that attitude, interest and motivation to learn. Students have negative beliefs due to the attitude that unconfident doing mathematical tasks and less interested in learning. They tend to avoid, unmotivated or lazy in following maths. As the process to form confidence in students towards maths requires a relatively long time. Confidence and attitudes are formed bit by bit which is the result of the interaction of students with maths. Therefore, students who have low confidence that taught by an open-ended approach to individually will lead them to be motivated to learn, curiosity and learning independently high in the study of maths than those who have low confidence that taught with an open-ended approach in groups which more depending on his team than themselves. As expressed in the research Sugiman (2009) that students who have excellent confidence in maths are very supportive in the learning process in their class. Based on the above, the overall findings of this research are the mathematical connection ability is influenced by the type of open-ended approach and the level of students' beliefs about mathematics. To that end, learning mathematics with an openended approach can be used as a reference of teachers in teaching students with attention beliefs about mathematics. For teachers of mathematics, learning with an open-ended approach and students who have beliefs about mathematics can significantly affect the results of studying mathematics. Therefore, teachers should make an open-ended approach and students who have beliefs about mathematics as a learning strategy that builds thinking skills of students to participate actively in the implementation of learning mathematics. The implication, efforts to improve the ability of mathematical connections every teacher should prepare everything related to the learning process. Teachers only act as mentors or facilitators of student learning rather than as assessors. Therefore, teachers must prepare themselves if there is problems faced by students during the learning process takes place, with the characteristics of different students. On learning activities, the teacher as the facilitator can lead students to learn actively participate in classroom learning is done either individually or in groups. 62

9 For mathematics teachers that provide an open-ended approach learning to the students as a group, it can improve their learning outcomes, especially in their mathematical connections. As learning with an open-ended approach in the group is more suitable in small groups so that they can discuss with friends in one team discuss the material given by the teacher. Through this learning process, students can actively participate in group discussions so that those who have high ability are able to help others who have low ability. Thus, a fun learning environment are created, because students' begins by introducing students to the open issues. Students will have the freedom to express their opinions in according to their knowledge. In this case, learning activities can improve student s thinking skills to develop an idea or concept ideas so that they can relate one concept to another. During the learning process there are internal and external factors that are difficult to control so that the researchers found some limitations in this study in terms of the assessment instruments, mathematics learning outcomes is only focused on the ability of mathematical connection with the coverage of material covering SPLDV and the Pythagorean Theorem. So the assessment is limited to the realm of cognitive aspects, but to the realm of psychomotor and attitude are not done in this assessment. In addition, researchers can not strictly control the possibility of communication between the subjects of the two schools were given with different treatments. CONCLUSION Based on the results of this study concluded that overall mathematical connection capabilities are influenced by the type of open-ended approach and the level of student's beliefs of maths. The student s ability of mathematical connection by learning to approach a group of open-ended basis is better than the open-ended approach individually after controlling the initial knowledge of maths. Then there is an effect of the interaction between the open-ended approach and student s belief of maths to the mathematical connection ability after controlling the initial knowledge of maths in students. By this interaction, the ability of mathematical connection with open-ended approach in groups is better than the open-ended approach to individual who have high confidence in the sub-group math after controlling the initial knowledge of maths in students. In contrast, student's mathematical connection ability by learning to approach a group of open-ended basis is less well than the open-ended approach individual who has the confidence of mathematics students in lower sub-group after controlling the initial knowledge of maths in students. REFERENCES Agung, I Gusti Ngurah. (2006). Statistika Penerapan Model Rerata-Sel Multivariat dan Model Ekonometri dengan SPSS. Jakarta: Yayasan SAD Satria Bhakti. Alhadad, Syarifah Fadillah. (2010). Meningkatkan Kemampuan Representasi Multipel Matematis, Pemecahan Masalah Matematis, dan Self Esteem Siswa 63

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