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Personal Research Journal 2013 TOPIC “ MULTIPLICATION OF A FRACTION WITH FRACTION” Ronal Rifandi Entry Level

: 5th Grade of Primary Students

Goal

: To develop a local instructional theory and construct models to support students’ understanding of multiplication of fraction

LITERATURE REVIEW Article 1. Keijzer, R. (2003). Teaching formal mathematics in primary education. Fraction learning as mathematising process. CD-Beta Press, Center for Science and Mathematics Education.

The point of view in research about mathematic education Keijzer (2003) argued about learning mathematics as “mathematising”. This is in line with the point of view which stated that mathematising is “watching the world from a mathematical perspective to thus make it more mathematical” (Freudenthal, 1968, as cited in Keijzer, 2003). Treffers (1987, as cited in Keijzer, 2003) stated that in teaching mathematics teacher need to consider the initial knowledge of students and relate it with the realistic contexts as a base for the learning activity. The realistic contexts which consist of meaningful problems give a chance to students to build their understanding of the mathematics (Greeno, Collin & Resnick, 1996, as cited in Keijzer, 2003). Furthermore, as in RME approach, the mathematisation of the meaningful problems became a tool for students to construct the formal notions about the concept (Van den Heuvel-Panhuizen, 1996, as cited in Keijzer, 2003). The “mathematisation” instead of just transfer the mathematics to students Keijzer (2003) proposed that “When discussing this mathematising process, we actually discuss the process of modeling, symbolizing, generalizing, formalizing, and abstracting”. These kind of activities reflects the journey of the students in reaching the formal and abstract structures of the mathematical concepts. They feels by their own every part of the activities which led to the meaningful learning. Moreover, instead of just transfer the knowledge from teacher to students, Freudenthal (1991, as cited in Keijzer, 2003) suggested that mathematics teaching and learning should be a process of “reinvention” by the students. The role of the teacher is to support Ronal Rifandi- 16 August 2013

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Personal Research Journal 2013 students in this learning process. We hope that they don't only do the mathematics but they understand the meaning behind the process. NOTES: This article gives me insight about meaningful learning. How we support children in constructing their understanding about the concept of multiplication of fractions. The idea of “mathematising” within the five stages stated in the article could be a consideration for my educational design although I will not come up until the abstract structures. I also pointed out about starting a lesson by providing a rich realistic context in order to support students in working at their informal level and then continue to a higher level of understanding.

Article 2. van Galen, F., Figueiredo, N., Gravemeijer, K., van Herpen, E. J. T. T., & Keijzer, R. (2008). Fractions, Percentages, Decimals and Proportions: A Learning-teaching Trajectory for Grade 4, 5 and 6. Rotterdam: Sense Publishers. Chapter 4. Core insights into fractions Van Galen et.al (2008) stated that in primary school mathematics, fractions are still an important topic to be taught because it often relates in our daily life even when the fraction itself is not explicitly appear. Moreover, they stated that the understandings of fractions become a base or a starting point to learn about proportion, decimal numbers and percentages (van Galen, et al, 2008). They also argued that in general there are two situations which involve fractions; sharing and measuring. The “sharing” activity produce fraction if we distribute a number of things into a number of people in which the these numbers are not the same. Meanwhile, in a measuring activity fraction emerge as an effect of the need of a smaller measurement unit (Van Galen et. al., 2008) Moreover, when dealing with fraction, it always relates to part-whole relationship and a fair sharing activity. These activities concern about the reasoning not about the empirical procedure and bar and number line model can be used as the model for reasoning (van Galen et. al., 2008). In the case of multiplication of fractions, initially most students are influenced by their notion about the multiplication of the whole number that is seen the multiplication as a repeated addition. However, not all cases in the multiplication of fractions can be seen at that point of view. Moreover, also based on their prior knowledge that multiplication produce a larger result, but it is not applied on the multiplication of fractions which produce a smaller number. Ronal Rifandi- 16 August 2013

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Personal Research Journal 2013 To overcome these kind of problems, van Galen et.al (2008) proposed that we can give meaning to a multiplication of fractions by assuring that students also can see multiplication as a factor not only as repeated addition. They stated that the notion about the proportion is more necessary instead of just repeated addition (van Galen et. al., 2008) Furthermore, in the operation of fraction, Van Galen et. al. (2008) argued about two routes that should be followed; “reasoned operation” and “number relationships”. Lastly, another challenging stage that teacher should be aware of is in promoting the relation of “part of” to “times” is a difficult problem (van Galen et. al., 2008).

NOTES: From this article I get the information about the importance of learning fraction of students that is as a basic foundation for the topic of proportion, decimal number and percentage. This article also explains about two activities that produce fraction; fair sharing and measurements. Moreover, the part-whole relation becomes an important part of the Regarding the multiplication of fractions, this article stated the fraction can play a role as a factor not only seen as repeated addition.

Article 3 Streefland, L. (Ed.). (1991). Fractions in realistic mathematics education: A paradigm of developmental research (Vol. 8). Springer. Chapter 4. The course in Theory and Practice According to Streefland (1991), students in primary school can use the division situations as a tool for constructing the understanding in learning fractions. He proposed five stages that can be used in the learning activities, as follows: 1. Serving up and distributing (producing fractions and their operational relations) 2. Seating arrangements and distributing (intertwining with ratio and generating equivalences). 3. Operating through a mediating quantity (the four main operations). 4. Doing one’s own productions at a symbolic level. 5. On the way to rules for the operations with fractions. (Streefland, 1991) At the first stage, the focus is to introduce the concrete contexts to the children and allow them to produce fraction by using the estimation and distribution based on this context (Streefland, Ronal Rifandi- 16 August 2013

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Personal Research Journal 2013 1991). Streefland (1991) also argued that students can use their notion about repeated halving as a reference for estimating in doing the fair sharing activities. In the second stage, Streefland (1991) stated that “Seating arrangements, illustrated by an appropriate symbol and diagram, are eminently suited for producing equivalency situations”. This activity will help students to build their idea about generating equivalency within the context. In the third stage, the fraction is seen as an operator not only as in a part-whole relationship. Several representations can help children to deal with this operation such as strips and bars. Regarding the multiplication students might have a confusion about the term “times” which is symbolized by “x” and the new term “of” on fractions which is also symbolized by “x”. (Streefland, 1991). We need to assure that students understand that we can use the symbol “x” to represent the term “of” when we talk about the amount less than 1 and clarify the meaning behind the symbols. In the fourth stage, Streefland (1991) stated that “Attention is paid here to taking fractions apart and putting them together in order to acquire skill in producing equivalent fractions and to sharpen one’s own concept of the operation”. However, we cannot assume that students can do this by themselves, we need to support them by providing the learning situation that will come up with several “though-model situations” (Streefland, 1991). Lastly, on the way to arithmetic rules for fractions, we donot intend to push students to gain the formal structure immediately. Streefland (1991) argued that we support student to work on their own informal way. NOTES: This article gives me an insight about teaching fractions to students. By taking the five stages mentioned above as a consideration, I can draw a sort of learning activity regarding multiplication of fractions especially on multiplication of fractions and fraction.

Article 4. Streefland, L. (1993). Fractions: A realistic approach. Rational numbers: An integration of research, 289-325. Streefland started the discussion at the beginning of the paper by posing several mathematical sentences consisting the operation of fractions. And he used a bar of chocolate which contain six parts as a model. One of the mathematical sentences is ½ x 1/3 = 1/6. For this problem to get the result, the partitioning is done step by step. “Of the two parts representing one third of the bar, one half of one part must be taken, which means ½ x 1/3 = 1/6” (Streefland, 1993). Ronal Rifandi- 16 August 2013

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Personal Research Journal 2013 Streefland (1993) stated that by using a “mediating representation” such as the bar consisting six parts, the main operation of fractions, included the multiplication, can be done. However the most important thing is whether the students understand the meaning of the process or not. Based on his study, Streefland (1993) proposed two activities that can help students get the notion about fraction that is “fair sharing” and “splitting up the group of sharers into subgroups”. In the activity of fair sharing students usually tend to use such reference points for their estimation instead of using the one that is given by others (Streefland, 1982, as cited in Streefland, 1993). Streefland (1993) gave an example of dividing 3 pizzas among 4 students. The solution that might come up by the students as follows:

(Streefland, 1993) We support students to see fraction as an operator when dealing with a measure, weight, or price to what is being distributed (Streefland, 1993). Streefland (1993) gave context about dividing 18 pizzas on a table where 24 people are seated. He used the activity of table arrangement which represent the “splitting up the group of sharers into subgroups”. The diagram below gives the illustration of the table arrangements.

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The diagram can be extended and modified become more than two tables and many conditions can be explored. One of the questions arise from the activity was “someone is sitting at table 4 – 5. What is the table where she would be served only half that amount?”(Streefland, 1993).

(Streefland, 1993). That represents the process of multiplication of fraction with another fraction and fraction play role as the operator. Streefland (1993) also proposed about the influence of the notion about natural number in students initial knowledge. He called it with N-distractors. Students might also use the property or the rule of natural number operations when they deal with fraction operation, and it is not correct. Ronal Rifandi- 16 August 2013

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Personal Research Journal 2013 NOTES: We can use mediating representation in the learning process especially when students work on their informal level. By using a model, students can elaborated and explore the problem to get the solutions. Moreover, the notion about fair sharing is important as a base for students in learning about fractions. Teachers also need to think about how to help students to omit the Ndistractors.

Article 5. Mack, N. K. (1998). Building a Foundation for Understanding the Multiplication of Fractions. Teaching Children Mathematics, 5(1), 34-38. The goal of the paper The goal of the author is to support students in understanding of taking a part of a part of a whole and make it meaningful in students’ mind. The students in this paper already learned about addition and subtraction of fractions and they already have the notions regarding these two operations. Important issues to consider Mack stated that it is important to build students’ informal knowledge before teach them about the formal form of the multiplication of fractions. Mack (1998) posed three questions related to understanding the multiplication of fractions; What mathematics do I want the students to learn? What informal knowledge do the students already have? And How might I help the students build on their informal knowledge to lay a foundation for their understanding? Students need to know about two things when they learn the multiplication of fractions, those are equal-sized parts and equal representation which each fraction has. They also should be able to do a fair sharing process. In the understanding of equal-sized parts, the students have to know that the size of the part is based on the size of the unit. Based on the previous notion about multiplication, students only know that multiplication can be seen as the repeated addition. However, when they deal with the multiplication involving fractions, this notion should be extended. The repeated addition nation (as in whole number) cannot apply in all cases of the multiplication of fractions. For example, the problem of onefourth of one-half of a whole cookie. In that case, the students need to understand about taking a part of a part of a whole. Ronal Rifandi- 16 August 2013

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Personal Research Journal 2013 Initially, most students see the operation of it as division instead of multiplication. Although in informal way they can perform the process of taking a part of a part of a whole but they still cannot relate it to the notion of multiplication. For instance, an example from Mack’s article, in taking ¼ of ½ of a whole, the students modeled the situation by using a circle or rectangle. The students known that the ½ part is based on the unit and the ¼ is based on the 1/2. So the ½ become another “whole”. And the result, 1/8, that they get by using the circle or rectangle is equivalent to one-fourth of one-half of a whole cookie (Mack, 1998). How to help students to build the informal knowledge ? Mack (1998) stated that deepen students’ understanding about fair sharing and the equal-sized parts are important because it is needed when we deal with the operation of fractions. Furthermore, teacher should support students with understanding of finding of a fraction of a whole number amount. Finally, the teacher helps students to build their informal knowledge on understanding the process and meaning of taking a fraction of a part of a whole. There are several forms of case involving multiplication of a fraction and fraction, start with the simple to the complex one (Mack, 1998). 1. 2. 3. 4.

The denominator of the first fraction was the same of the numerator of the 2nd fraction. The denominator of the first fraction is the multiple of the numerator of the 2nd fraction The denominator of the first fraction is a factor of the numerator of the 2nd fraction. The GCD of the denominator of the first fraction and the numerator of the 2nd fraction is 1.

The students are encouraged to draw their informal knowledge related to equal-sized parts and the meaning of portioning a quantity into fractional amounts (Mack, 1998). NOTES: Building the informal knowledge about the multiplication of fractions is important before providing the formal notion. In the case of multiplication of a fraction and fraction, students should understand the process of taking a part of a part of a whole situation.

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Personal Research Journal 2013 My 2nd Presentation, 15 August 2013

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Personal Research Journal 2013 Notes and reflections after presentation 15 August 2013.

To begin the discussion after I presented about my topic, I posed a question about the content of my design. Should I start from the beginning, or directly go to the informal level of multiplication of a fraction with fraction? Andrea said that it could be, but depend on the situations, I should assure whether the previous knowledge of the students already enough to start a new lesson about this topic or not. Maarten said that in the past, learning multiplication of fractions perhaps not as same as what our traditional teacher did. It is not a standard that the multiplication was taught before “addition of fractions�. Look at the history of learning multiplication of fractions. Moreover, he said that building an understanding about the topic of multiplication should start from the activity on a concrete level. Nenden asked about the initial knowledge of the students. It should be clear before teacher conduct the next lesson about multiplication of a fraction with fraction. Mieke suggested the use of an appropriate real context to begin the learning activity. The story relating to an area which then can be represented in an area model could help students build their understanding about multiplication of a fraction with fraction. Overall, based on this discussion, I need to define what is the preliminary knowledge that students should have before starting the lesson about multiplication of a fraction with fraction. I should check it to the participant by conducting an activity about it. Furthermore, regarding the multiplication of a fraction with fraction, I need to design a sequence of activities start from a concrete level, such as providing a story as a context, then support students in the emergent of models. Help them to represent the situation in a model so that they can use it as a tool for thinking. The steps that are proposed by Streefland and Keijzer could be as consideration for me on making the educational design about this topic.

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