Teaching Multiplication of Fractions with Natural Number

Summary on:
Shanty, N.O., Hartono, Y., Putri, R.I.I, de Haan, D. 2011. Design research on mathematics education: Investigating the progress of Indonesian fifth grade students’ learning on multiplication of fractions with natural number. Indonesian Mathematical Society Journal on Mathematics Education (IndoMS. J.M.E), 2.2, 147-162.

The difficulties of students in learning fractions, according to researches, is caused by two main factors. The first one due to teaching method which emphasizes only on technical procedures. As the consequence, the students see the algorithm as meaningless series of steps. Hence, they often forget the steps or might change the procedures with the incorrect ones. The problem is also called ‘algorithmatically based mistakes’. Another factor is the concept of the fraction itself which is NOT consistent with counting principles as it is in natural numbers.

In order to deal with the problems, remodeling mathematics by developing sequence of activities which are RME based seems to be appropriate. Here, we might design learning activities by combining 5 activity-levels in learning fraction (proposed by Streefland) with the 5 tenets of RME. In addition, the 4-level emergent modelling (by Gravemeijer) was also being concerned.

A design, covering the aforementioned needs, has been implemented by Shanty toward the 5th Grade students of SDN 179 Palembang. Here, the following, describes the activity she conducted and its effects to the students.

1. Producing Fractions
The activity employed contextual problems in order to stimulate students’ informal knowledge of partitioning, as meant to be a situational activity. The problem given was about ‘locating flags and water posts on the running route’ and delivered in form of story related to the celebration of Independence Day. In the story, two people were preparing running competition from PIM (Palembang Indah Mall) (A) to Palembang District Office (B). On the way from A to B, 8 flags and 6 water posts were installed, each of them with equal distance among its kinds. The students were, then, asked to locate and label the positions of the 8 flags and the 6 water posts in the provided running map.

map
The running route from PIM (A) to District Office (B)
Picture is available on https://www.google.co.id/maps

Working with this, the students have no difficulties either in determining or labeling the required points. However, differences in labelling, that is, some students used unit fractions while the others used non-unit fractions, were found here.

2. Generating Equivalencies
In this phase, the students were hoped to recognize the relationship among fractions. To reach the goal, they were encouraged to apply number line (as a model) generated from a string of yarn (referential activity), as found in the labelled running route.

number line

At the end, she found the students were able to relate equivalent fractions, identify the relationship among fractions with the idea of multiplication of fractions.

3. Operating through Mediating Quantity
The activities done in this stage was to bridge students into the idea of fractions as operator by using the unit of length. The activity entitled ‘who runs further’ related to the ‘labelled route’ was given with a certain number of distance (say 6 km). It means that after reaching the 5th flag, a person has runned 5/8 of the total distance 6 km.

During the step, the students were slowly redirected into transforming ‘5/8 of 6’ into ‘5/8 x 6’.

4. Doing One’s Own Production
This level matches with the third characteristic of RME, namely, the use of students’ own creations and contributions. Classroom discussion (interactivity) was conducted and students’ reasonings were really encouraged. This lead to the general level of model emergent.

Two groups of anwers were identified. The first one used double number line for reasoning, and the second one apply direct multiplication. After the discussion, the students were freed up to choose which method to apply. Most of them moved into the second way.

5. On the Way to Rules for Multiplication of Fractions with Natural Numbers
This last stage guided the students to work independently using formal mathematics instead of the informal one. The activity was covered in a mini lesson including fractions as operator.

The result showed that the students have not attained the lefel of generalizing rules for multiplication of fractions with natural numbers yet. More practices were still required, in this case.

In addition to the five mentioned activities, the design also fulfil the last tenet of RME, intertwining. This by relating the concept of proportion, measuring, basic operation, and some other mathematical concepts during the lesson.

#this summary is based on my own understanding toward the article.
#for the complete article, you may visit http://eprints.unsri.ac.id/619/1/Design_Research_On_Mathematics_Education_Investigating_The_Progress_Of_Indonesian_Fifth_Grade_Students%E2%80%99_Learning_On_Multiplication_Of_Fractions_With_Natural_Numbers_-_Nenden.pdf

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