Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.
Learning fractions by making patterns – An Ethnomathematics based approach Sathya SANKARAN, Harini SAMPATH and Jayanthi SIVASWAMY Center for IT in Education, International Institute of Information Technology, Hyderabad, India
[email protected] Abstract: Mathematics is one of the difficult subjects children encounter. This is attributed to the fact that mathematics is taught as an abstract set of symbols and rules. Many innovative approaches have been tried to clear this misconception and make children see the real applications of mathematics. One such approach is using ethnomathematics – the mathematics present in the cultural forms of an ethnic group. In this work, we explore the effectiveness of this approach to teach a difficult mathematics concept – fractions. Fractions have been chosen due to its complexity and the inherent difficulties they pose to children. A tool was developed to teach fractions by engaging the child in two activities – making a bead necklace and tiling an area. The evaluation results indicate that such an approach is very effective in teaching the concept. Keywords: Patterns, fractions, ethnomathematics.
Introduction Mathematics is often viewed by school children as a set of rigid rules. In many cases students have a good procedural knowledge to perform the operations but lack the corresponding conceptual understanding. This motivates the use of alternate methods of teaching so that students appreciate the subject more and relate it to real life. One such method is ethnomathematics. Ethnomathematics [1] is the study of mathematics that is present in various cultures around the world. Researchers have discovered that there are many mathematical concepts that go into the cultural forms of an ethnic group such as its architecture, arts, crafts and jewels. The use of ethnomathematics to teach mathematics in schools has been motivated by the factors that students from different backgrounds should take pride in the achievements of their people and they must realize that mathematics arises out of real needs and interests of human beings [2]. Hence, a multicultural and interdisciplinary approach to mathematics education can be pedagogically beneficial. The idea of using Ethnomathematics to teach mathematical concepts is being successfully carried out in Africa as a part of the Africa meets Africa project [3]. In this paper, we explore the effectiveness of using ethnomathematics to teach fractions. To this end, we built a software tool that teaches fractions using examples from Indian culture. Our target audience is children who are beginning to learn the concept of fractions.
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Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.
1. Fractions Fractions are considered to be the most complex mathematical concept encountered by a child in the primary school years [4]. The children are familiar with whole numbers that are countable and the concepts of addition and subtraction at this age. However, fraction is a composite entity with a numerator and denominator and is more related to the concept of division. This requires the children to make a qualitative leap in understanding [5] and so makes the concept of fractions difficult for them to assimilate. Fractions can be considered as a set of five interrelated constructs – part-whole, ratio, operator, quotient and measure. One could start teaching fractions based on any of these constructs. But it has been shown that students perform best when they are taught fractions using the part whole subconstruct [6] where fractions occur when a larger quantity is divided into parts. The limitation in using this subconstruct would be the fact one can teach only proper fractions where the numerator is lesser than the denominator. However, once students grasp this fundamental idea of what fractions are, we can expand it to teach more complex ideas from other constructs. Our tool uses the part-whole subconstruct to introduce the concept of fractions. Before we present our work, we briefly review the software tools that are available for teaching fractions. Joyce [7] is a PDA based multi-player game system that aims at teaching addition and subtraction of fractions with equal and unequal denominators and reduction of fractions. Shemesh [8] aims at teaching the concept of fraction equivalence. Internet-based Fraction-learning Integrated Learning Environment (IFILE) [9] aims at teaching the addition/ subtraction of fractions with unlike denominators. All these tools basically aim to teach how to perform operations on a fraction. In contrast, our tool aims at teaching the basic concept of fractions itself. 2. EthnoFracto – A tool to teach the concept of fractions In our tool, children are given activities in which they make some jewel or pattern well known in Indian culture conforming to certain simple rules. Then they are explained the mathematical concept behind the patterns they have made. This approach should appeal to their aesthetic sense, let them relate to the mathematics in the real world and resonate with the constructivist philosophy of learning. Another design feature in our tool is the possibility of multiple correct solutions for any given problem. This is included in the tool to enhance creativity in students and make them appreciate that real world problems do not have a single unique solution. While the culture-based approach to teach mathematics is an effective one, there are some practical issues when incorporating the idea in the classroom. First is the lack of structure in the process. Supervision and correction is required during the process. Since multiple solutions (patterns) are possible, validating every pattern and making explicit the underlying mathematical concept becomes a demanding task for the teacher. Next is the recurring resource requirement. A set of materials needs to be provided to every child to make the pattern. The teacher may have to create new set of materials every time the activity is to be repeated. This justifies the creation of a software tool which facilitates the students to make patterns, automatically validate them and explain the mathematics behind the pattern. There are two different activities in our tool. In the first activity, the children make a bead necklace and in the second, they tile an area.
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Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.
2.1 Activity I –Whole is a discrete Set – Making Jewels In this activity, children make a bead necklace locally known as 'mala'. It is a simple to make, yet very common and popular jewel in India. It is made by stringing together colored beads or seeds. The tool presents the children with a string of given length. A tool box is provided with different color beads. In this activity ‘the whole’ is the set of all beads in the mala and ‘the parts’ are the beads of particular color. The task for the child is to use these beads and construct a mala. Some simple rules are laid out to structure the activity. Some sample rules are- a green bead should always be followed by a yellow bead, and there should be at least one green bead in the mala etc. The activity is designed such that there are multiple ways to string the beads while adhering to a given set of rules. Once the child completes the task, the software checks if the rules are violated, and if so, the location of error is indicated and the child is asked to repeat the activity. On completion of the activity, the child is asked to count the total number of beads in the mala. Then they are asked to count the number of beads of each color. Then the child is asked to write down the notation - Number of beads in a particular color / Total number of beads (see Figure 1). An explanation that the concept of fraction occurs when we divide a larger quantity (in this case the set of all beads in the mala) into parts (the beads of a particular color) is given. The final correct answer is displayed as a pie chart (as shown in Figure 1) to reinforce the part-whole concept.
. Figure 1: Screenshot of the tool showing a completed mala making activity
Figure 2: Patterns in 2D space: South Indian Kolam (left) and Traditional Indian flooring (right)
2.2 Activity II –Whole is a continuous area – Tiling The second activity is based on making patterns in 2-D space. 2D Patterns contain lots of mathematical ideas and are abundant in cultures around the world. A few of examples are South Indian Kolams which employ graph theory ideas [11] and traditional floor patterns that exhibit tessellations. The level of mathematics that can be taught using such 2D patterns is therefore wide-ranging. In the tiling activity, the task for the child is to fill an area with tiles of different colours and same size. Here, ‘the whole’ is in the form of a continuous area where as it was a discrete set in the earlier activity. The tool presents the children with a grid and a tool box of tiles as seen in Figure 3a. As in the previous case, a set of rules are provided to structure the activity. On successful completion, the children are asked to count the number of tiles of each color. Then the child is asked to write down the notation Number of tiles of particular colour / Total number of tiles used. The children are then explained that they are partitioning an entire area into different colors and the area occupied by each colour is a part and hence fraction of the whole area. Sample screenshot of a completed tiling is shown in Figure 3b. 343
Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.
Figure 3a: Snapshot of tool with grid and toolbox of tiles. 3b: A completed pattern. 3. Evaluation Two experiments were conducted to study the effectiveness of teaching fractions using pattern making. In the first experiment, children made a mala using beads (both virtually and physically) and in the second, the children tiled an area.
3.1 Experiment I Methodology – A pre test was conducted and twenty children from third grade were chosen to participate in the study. The pre test consisted of 9 questions drawn from a teacher’s manual [10]. These students had not been introduced to fractions formally through their curriculum. The children were randomly assigned into two groups of size 9 (as only 18 children participated in the experiment). One group of children made mala with real string and beads. Written instructions and explanations were given to this group on paper. The experimenter worked with each child and pointed out any violation of rules. The second group performed the activity using the tool we had developed. The content of instructions and explanations were same in both cases. After both the groups finished their activity, they were given a post test and the improvement in performance was assessed. Pre test (Mean Correct and 3.1.1 Wrong Answers for 20 3.1.2 3.1.3 students) Wrong 3.1.4Correct 1 7
Post test (Number of Students who provided correct and wrong answers and nature of errors made) Correct Answer Wrong Answer Concept Notation Real Beads 6 1 2 IT Tool 7 0 2
Table 1: Results of the experiment. Mean scores in pre test (left) and student performance in post test (right).
Results - The pre test results revealed that most students did not have a proper understanding of the concept of fractions. In the post test, however, we found that their performance had significantly improved. The group that used real beads and the group that used IT tool showed almost equal progress. We found that the children had some trouble with the notation of fractions. The results are given in Table 1. After the activity we asked the students how they understood the concept of fractions. They responded that “a fraction is the number of objects of a particular type /Total number of objects”. 344
Kong, S.C., Ogata, H., Arnseth, H.C., Chan, C.K.K., Hirashima, T., Klett, F., Lee, J.H.M., Liu, C.C., Looi, C.K., Milrad, M., Mitrovic, A., Nakabayashi, K., Wong, S.L., Yang, S.J.H. (eds.) (2009). Proceedings of the 17th International Conference on Computers in Education [CDROM] . Hong Kong: Asia-Pacific Society for Computers in Education.
3.2 Experiment 2 The aim here is to assess the effectiveness of teaching fractions via the tiling concept which involves pattern making in 2-D space. This is only a preliminary study and a controlled experiment with a larger number of participants is planned. Methodology - This study included three students who had completed the 2nd grade. All the three children were administered a pre test. Then they were asked to complete the tiling activity. Once they completed the activity, they were given a post test. Results - We found that the performance of the children improved significantly in the post test and they were able to understand the concept of fractions. 4. Discussion and Conclusions In this paper, we argued that ethnomathematics could be used to teach difficult mathematics concepts effectively to students. Our results show that the concept of teaching fractions via pattern making is effective and that activity in virtual or physical space is a key enhancer in learning. One can view this as a proof of concept of the broader goal of teaching difficult mathematical concepts using ethnomathematics. The advantages of this approach are - teaching mathematics using ideas familiar to students helps in improving conceptual understanding and software tool such as ours lets the student construct their own knowledge. It is well known that this constructivist approach to learning, where students learn from their own experience is very effective. In future, we plan to expand the tool to teach improper fractions and other difficult concepts. References [1] D'Ambrosio, U. (1985). Ethnomathematics and its Place in the History and Pedagogy of Mathematics. In For the Learning of Mathematics, 5(1) 44-48 [2] Zaslavsky, C. (1994). Africa Counts and Ethnomathematics. In For the Learning of Mathematics, 14(2). [3] Africa meets Africa: Making a Living through the Mathematics of Zulu Design, http://www.africameetsafrica.co.za/, Accessed on 01, May 2009 [4] Kamii, C. & Clark, F.B. (1995). Equivalent fractions: Their difficulty and educational implications, Journal of Mathematical Behavior 14, 365–378. [5] Lamon,S.J.(1999).Teaching fractions and ratios for understanding. NJ:Lawrence Erlbaum. [6] Charalambos Y. Charalambous & Demetra Pitta-Pantazi (2007). Drawing on a theoretical model to study students’ understandings of fractions, Educational Studies in Mathematics , Vol. 64: 293–316 [7] Kuang-Cheng Feng, Ben Chang, Chih-Hung Lai & Tak-Wai Chan (2005). Joyce: A Multi-Player Game on One-on-one Digital Classroom Environment for Practicing Fractions, In Proceedings of the Fifth IEEE International Conference on Advanced Learning Technologies, 543-544 [8] Ilana Arnon, Pearla Nesher & Renata Nirenburg (2001). Where do fractions encounter their equivalents? Can this encounter take place in elementary-school?, International Journal of Computers for Mathematical Learning, Vol. 6: 167–214. [9] Siu Cheung Kong, Lam For Kwok (2002). Modeling the Process of Learning Common Fraction Operations: Designing an Internet-Based Integrated Learning Environment for Knowledge Construction, Proceedings of the International Conference on Computers in Education, 767-775. [10] Subramaniam, K. (2005). Maths for Every Child: Teachers' book to accompany text-cum-workbooks of the Homi Bhabha Curriculum in Mathematics for class 3, Homi Bhabha Centre for Science Education, Mumbai. [11] G. Siromoney & R. Siromoney (1987). Rosenfeld's cycle grammars and kolam, Graph grammars and their application to Computer Science, Lecture Notes in Computer Science, 291, pp 564-578, Springer-Verlag
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