Answer to the Research Questions

As in the first chapter of this research, there were two research questions.

The first research question will be answered by relating the analysis of the pre-test of second cycle as described in Subsection 4.4. to students‟ strategies in solving problems in the teaching experiment phase as described in Subsection 5.4.2.

Meanwhile, in order to answer the second research question, the researcher also referred to Subsection 4.4.2 until 4.4.3, but the researcher more focused on what kind of support given to extend students‟ understanding of each learning phase.

5.1.1 Answer to the First Research Question

The initial knowledge of students more or less affected their learning process in the teaching experiment. Students who were familiar with formal algorithms seemed to have a tendency to use the algorithms in solving problems.

Once they know that the question is about multiplication of fraction, they will use

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the algorithms of multiplication of fraction to solve it. One of the cases can be seen in Subsection 4.4.2.2.2.

Further, since the fifth graders involved in this research already learned about multiplication of a fraction by a fraction, then they changed the whole number to be fraction so that they could use the rules of multiplication of a fraction by a fraction. The problems could appear then when the students made mistakes in converting the whole number or when they did not convert the whole number but directly multiplied the whole number with numerator and denominator (see Subsection 4.4.2.2.2)

Moreover, the knowledge of fraction division also gave influence in their strategy. The students, who associate „sharing‟ with „division‟, directly used the algorithms of fraction division when they were faced to activity of sharing some pieces of cake to some people (Subsection 4.4.2.2.1).

5.1.2 Answer to the Second Research Question

As stated in the beginning of Subsection 5.1, in the following the researcher will answer the second research question by focusing on support given in each phase of learning process.

1) Fractions are related to division and multiplication

Since the students are familiar with fair sharing context, then the researcher could use the context as starting point to learn about multiplication of fraction with whole number. The idea that fractions are related to division came when the students were faced to the context, for instance when they were asked to share one cake to a number of people. As the result, they will use unit fraction as

. Regarding to sharing more than one cakes, the students also tend to relate it to division as

. However, as the student already proficient enough regarding to the idea that fraction are related to division, then they do not need much support.

Since the students already learned about a fraction times a fraction, then the researcher could use the knowledge to convey the idea of the relation between fraction and multiplication. By asking the students to determine fraction got if some part of cake divided to some people, the student will come fraction of fraction, which is actually multiplication of a fraction by a fraction.

One of the issues found in fair sharing activity was about the whole of fraction. Even though the students already knew about formal algorithm of multiplication of a fraction by a fraction, they need more support to notate fraction of fraction (see Subsection 4.4.2.2)

Further, asking students to find fractions of something can be used to elicit the idea that fractions are related to multiplication. As in the case of Ara and friends (Subsection 4.4.2.2), one of the supports could be given was by posing such question that provoke students to give reason about their answer. Since they already learned about equivalent fraction, then relating the problem to the pre-knowledge could help them to find the answer without directly used algorithm of multiplication of fraction by whole number. Then, once they recognize the word

„of‟ means multiplication, they will use the algorithms.

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2) Repeated addition of fractions as multiplication of whole number by fraction Since the idea of multiplication as repeated addition already emerged when learning about multiplication of whole numbers, then the students did not need much support to relate it to multiplication of whole number by fraction, as stated by Schwartz and Riedesel (1994) in Subsection 2.2. However, the use of rice to make lontong in the beginning of the meeting seems could draw students‟

enthusiasm to learn.

3) Inverse property of unit fractions

The researcher hope to elicit the idea of unit fractions by using colouring activity, where the students have to determine the number of yarn colour needed to make one meter of colourful knitting. Since the idea of this activity more or less similar to preparing number of menus activity, then the students also did not need much support in solving the problems. One small support might be given for students in this activity was by giving guidance (see vignette in Subsection 4.4.2.4) to make mathematical conclusion, such as „when a unit fraction multiplied by its denominator, then the result will be one‟.

4) Commutative property of multiplication of fraction with whole number The students seems already knew that multiplication of whole number by fraction ( ) had the same result as multiplication of fraction by whole number ( ). The support needed was to guide students to understand how to find and in some ribbons (Subsection 4.4.2.5).

5.2 Local Instruction Theory for Extending the Meaning of Multiplication