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Cihampelas: An Indonesian Contextual Problem in Matrix (A part of the CASCADE-IMEI study report after First fieldwork in Bandung) by : Zulkardi
Introduction This report is about the designing and the implementation of a contextual problem in the topic of matrix. Up to now, matrix is taught in the secondary school in Indonesia in a traditional way. It means the contents deal only with numbers and operations, lack of the applications or real-life situations. In the next section, a series of problems about matrix are designed based on the realistic mathematics education theory. RME theory has five tenets or characteristics (de Lange, 1987; Treffers, 1991):(1) phenomenological exploration or the use of contexts; (2) use of models or bridging by vertical instruments; (3) the use of students own productions and constructions or students contribution; (4) the interactive character of the teaching process or interactivity; and (5) the intertwining of various learning strands. In the section 3, the reflection of these characteristics to the problems will be discussed.
Designing the contextual problems In the module matrix called 'Cihampelas' , a number of problems were designed by using Cihampelas market as the context. This problem is inspired by matrix problems in Mathematics A in the Netherlands (de Lange, 1987). All of the problems can be seen below. Cihampelas Cihampelas is a famous market in Bandung. Here, in a number of stores, people can buy various of jeans and t-shirts. One of the stores is Toko Rambo. In this store, 36 trousers Wrangler with different sizes and types are available with the following specifications: 28" (long 28 inches)= 6 30" = 12 32" = 12 34" = 6 Other types of jeans are also available with size and quantities as follows:
Levis : 6, 12, 12, and 6 Cardinal : 3, 7, 6 and 3 Esperite : 6, 6, 6 and 3 Darwin : 3, 6, 6 and 3 Problems: 1. All this information can be written down well-ordered in matrix-form. Write the matrix. 2. How many trousers fit with your sizes? Explain! 3. In a month, the number of trousers has been sold can be seen on the following matrix. Write down the number of trousers which has not been sold?
4. The average profit per pair of Wrangler jeans is Rp. 3000,-; Levis Rp. 3500,-; Cardinal Rp. 4000,-; Esperite Rp. 2500,- and Darwin Rp. 4000,-. What is the total profit on the small size? 5> Compute in the same way the profit on size 30". And in total?
Implementation of the problems Five sequence problems were given to the mathematics student teachers who have already learned topics matrix in the secondary school. However, the aims of the assessment activity using these problems not focus on whether the solution right or wrong, but on their process of mathematizing. In other word, how they tackle the contextual problems and how they use different strategies in solving the problems are the main focus of assessment. The first question is given as follows: 1. All this information can be written down well-ordered in matrix-form. Write the matrix.
In this problem student teachers are expected to engage in the Cihampelas situation in which all of the them know where the place is. Student teachers are forced to explore the situation in the problem, find and identify matrix from that problem. The students are expected to end up with two kinds of right solutions in the form of: 4 x 5 matrix or 5 x 4 matrix. However, it was only one student teacher (Sri's) has the solution 5 x 4 matrix while the others used the 4 x 5 matrix. One can see that Sri was not sure about her solution by adding the comment what the number in the column and row mean. No student teachers solved this problem as same as Sri did. Two different solutions of Sri and Dadang are shown below:
The second problem is: 2> How many trousers or jeans fit with your sizes? Explain!
It asked the students to count the number of jeans in the store that same with their size and to give the reasoning of their answer. Sri solved the problem by giving the answer 'my trouser's size 30'. Furthermore, she added all of the jeans with 30' is the size. The total of her trousers are 43 . Finally, Dadang solved the problem in a similar way, but he has different size with Sri. He totally has 24 trousers.
The second problem is the easier one from mathematical view, but it is an open-ended problem that student can have different answers and arguments. There were some nice solutions. One of the student teachers solution is ' No trousers are mine. Why? Because my size, 36, is not in the list'. The third problem related to the concept of subtraction of two matrices. The problems
as follows: 3> In a month, the number of trousers has been sold can be seen on the left matrix Write down the number of trousers which has not been sold?
From the Sri's solution to the third problem, one can see that Sri had difficulties in solving the problem comparing to the way Dadang did. Dadang used the matrix in the first problem and subtracting by the V matrix. Done. Sri, on the other hand, knowing that matrix V is not the same size as her matrix, she translated the two matrices in the first problem as well V matrix. Finally, she found the right answer as Dadang had.
The fourth problem is about the multiplication of two matrices. 4. The average profit per pair of Wrangler jeans is Rp. 3000,-; Levis Rp. 3500,-; Cardinal Rp. 4000,-; Esperite Rp. 2500,- and Darwin Rp. 4000,-. What is the total profit on the small size?
Clearly the idea of this problem, although implicitly, is that in this context the informal way of multiplying matrices becomes 'formal'. According to de Lange (1987) the informal way of multiplying matrices refer to the fact that, when adding matrices, one adds the corresponding entries, but with multiplication we do not follow this obvious rule. Regarding to the problem 4, two different strategies of multiplying matrices, informal (Sri's) and formal (Dadang's), that has the same end result can be seen below:
The last problem is also dealt with the multiplication of two matrices as an extension of the fourth problem. 5> Compute in the same way the profit on size 30". And in total?
Formally, solution of this problem is the result of the multiplication matrix V and matrix P, P is profit. (see. Dadang's solution). In this problem, Sri also used the informal strategy but had the same solution as Dadang had.
Reflections All five problems discussed earlier were designed based on the characteristics of five tenets of realistic mathematics education (Treffers, 1991): 1. phenomenological exploration or the use of contexts. In RME, the starting point of instructional experiences should be `real' to the students; allowing them to immediately become engaged in the situation. In this case, the pheomena of Cihampelas is an example source for concept formation of matrices. Here, the problems the concept of matrices, addition, subtracting and multiplication of matrices are developed. This is consistent with de Lange (1987) called conceptual mathematization, by doing the problems will force the students to explore the situation, find and identify the relevant mathematics, schematize, and visualize to discover regularities, and develop a ‘model’ resulting in a mathematical concept. 2. the use of informal strategies or models; The term model refers to situation models and mathematical models that are
developed by the students themselves. This means that the students develop models in solving problems. In this context, one can see the solutions of Sri are example of informal strategies in multiply two matrices. 3. the use of student productions and contributions In the part 3 we have already discusses two kinds of student productions. These productions can be used as a additional material for next version of the problems. 4. learning through interaction After the students solved the problems they discussed the results and compared their solution to the others. From here, they learned much the strategies of their friends. 5. Intertwined with other strands The next version of these problems will be intertwined with other strands such as graph and equations.
Conclusions In this report, how to use tenets of realistic mathematics: the use of context and the use of informal strategies or models was discussed. Five problems of matrices using context Cihampelas were developed and implemented to the student teachers in UPI Bandung. The problems were given to the student teachers at the end of the RME course as an assessment. After the student solutions were evaluated, most of the students can solve the problems. However, only a student teacher, Sri, who used informal strategy in solving the problems. This informal student contribution together with one example of the formal strategy (Dadang's) were used as illustration.
References: De Lange, J. (1987). Mathematics, Insight and Meaning. Utrecht: OW &OC. Treffers, A. (1991). Realistic mathematics education in The Netherlands 19801990. In L. Streefland (ed.), Realistic Mathematics Education in Primary School. Utrecht: CD-b Press / Freudenthal Institute, Utrecht University.