UNDERSTANDING THE CONCEPTS OF PROPORTION AND RATIO CONSTRUCTED BY TWO GRADE SIX STUDENTS
There are many students having problems with the basic concepts of fractions, rates and proportion and with problems involving these concepts. Students have difficulty reasoning multiplicatively that uses in proportion problems. They often use additive reasoning in solving a problems where multiplicative reasoning is required. This research tried to find ways to help students reason proportionally. They have a problem to identify a litany of task variables that affect problem difficulty. It is important to analyze mathematical structures and children’s solution processes in developmental precursors to the knowledge needed to function competently in a domain.
MULTIPLICATIVE SCHEMES IN PROPORTIONAL REASONING
To determine what knowledge was critical for understanding ratio and proportion is the goal of this analysis. There are two main issues to be discussed about multiplicative schemes and the nature of students’ proportional reasoning. There are three researches studying about multiplicative structure. First, Vergnaud (1983) researched about building multiplicative structure as consisting in simple and more complex situation of proportion. The analysis of students work and teaching situation are his approach. Second, Confrey (1995) doing approach to multiplicative structure is based on the idea of splitting where it is an action of duplicating and constructing simultaneous splits of an original. Steffe (1988) has argued that the key to students’ meaningful dealings with multiplication is its ability to iterate abstract composite units.
The differences between Confrey’s and Steffe’s approaches are that Confrey observes multiplicative actions as independent of addition ideas, in other side, Steffe’s approach considers early multiplicative acts as making natural use of counting based mental structures. There are two significant related changes –in what the numbers are and in what the numbers are about- when children move from additive to multiplicative. Researcher believe that multiplicative reasoning is an entry point to the world of ratio and proportion. He would elaborate how students’ thinking of composite unit schemes can develop into proportional reasoning.
The research question:
How students in grade six understanding the concept of proportion and ratio and how they construct some methods to solve many different task?
This study was conducted using a constructivist theory of learning that focuses on a child’s construction of proportion reasoning. The researcher monitors each student how he/she think and use the concept of ratio and proportion. He set up to explore how two students construct proportional reasoning and use a unique way of accomplishing some tasks.
The methodology adopted for this study was clinical interviews that have two main advantages. First, allowing to interventions where students were encouraged to elaborate on their statement and judgments (Opper, 1975). Second, this approach to gathering data provides for a continual interaction between inference and observation (Cobb, 1986b). The use of interviewing as a successful tool of research must be accompanied by appropriate learning tasks. A set of learning tasks developed by the researcher and adapted from the literature was used to analyze the ways learners construct proportional reasoning knowledge from the problem solving activity.
The researcher did observation in the two beginning grade six girls (11 years) Alice and Karen. They were in different class but a common mathematics teacher. The teacher said that Alice was a very bright student and A students. In the other side, Karen was a hard working student and in the second top class of the school for sixth graders.
Content knowledge of Students
Schoolteacher said that Alice had had been taught the ratio type of questions in a section on the topic of money though no terminology of ratio or proportionality was used. From Alice’s work sheet, all the tasks used the unit method, finding the rate for one and multiplying to get the rate of money. Alice had studied this method in class while had not as her class was slower than Alice’s class.
ANALYSIS AND RESULTS
Interpretation of Karen’s activity
Karen used strategy with meaningful insights on multiplicative schemes in proportional reasoning. She constructed iterable ratio units to find the answer by coordinating these units such that one ratio was distributed over the next ratio. She was able to unitize the units in a composite and was able to deal meaningfully with composite units. She was able to take a ratio as a composite unit and maintain the ratio unit of its elements. The key foundation in Karen’s meaningful dealing with proportional reasoning was the ability to iterate composite units. Karen used relationship as a countable unit to find the answer a ratio unit method. There are two advantage that method, first to avoid any fractional or decimal computation, and secondly, to become a powerful method for solving all missing value tasks within the contexts that made sense to Karen.
Researcher believed that Karen’s method was a scheme according to Glasersfeld (1980) that consisted of three part: the experiential situation, the child’s specific activity or procedures, and the result that students expect. Karen was able to articulate the goals of her actions and give mathematical meaning to the procedures or reasoning she undertook and internalize her actions. She used her fingers on several occasions to re-present pattern. Finally, She was able to generalize her actions across similar proportionality task.
Interpretation of Alice’s activity Alice’s conceptualization in proportional reasoning is based on the unit method, a memorized procedure rather than a conceptual one. She used the unit method to get the answers and solve various problems. She could not describe her reasoning in a meaningful way and the procedure she used. She was not able to make sense of activity and think in terms of the composite ratio unit. Researcher believed that Alice’s procedural orientation influenced her action in dealing meaningfully with ratio and proportion.
Different levels of multiplication reasoning can be seen in the strategies utilized by these two students- Karen and Alice. Karen had constructed multiplication schemes in solving proportional reasoning while Alice used procedural method than conceptual one. Alice tried to memorize a procedure so that she could not construct and explain whether her strategies were or were not viable. Her thinking was based on additive reasoning rather than multiplicative one in which each sequences of counting acts into composite units. Karen unitized the composite unit to find a ratio unit and then iterated it to its referent point. She coordinated two number sequences and was able to preserve the relationship in the iteration. She was able to deal meaningfully with composite units. In the other hand, Alice was not able to think in terms of composite of ratio units and make decision about which unit strategy to use. Although she was successful to use that method to solve some tasks, she was not able to use it to solve a problem requiring qualitative proportional reasoning.
Researcher believed that there are varying degrees of sophistication between Karen’s and Alice’ solutions to the problems based on whether they formed composite ratio units and reinterpreted the problems in terms of those units. Karen’s method of iterating composite units seems to avoid the additive strategy that Alice used. The iteration scheme is based on a simple action scheme that can underlie the solution to proportionality. Unit ratio strategy (find the rate for one and multiplying to get the rate for many) may not help students develop multiplicative reasoning. Interpreting a situation of that unit may help to understand the nature of composite units. The unit method-a meaningful understanding-becomes essentially instrumental for the children to development of insight (Freudenthal, 1979) in multiplicative reasoning.
CONCLUTION One of the goals of this study was to investigate the ways in which the students began to develop proportional thinking. There are three essential components to use operation with composite units. First, one needs to explicitly conceptualize the iteration action of the composite ratio unit to make a sense of ratio problems. Second, one needs to have sufficient understanding of the meaning of multiplication and division. Third, one need to have sufficiently abstracted the iteration process. The unit method should not be taught to students until having a good grasp of the unit coordination schemes. Further research is need to elaborate how students’ composite uit schemes of multiplicative structure can develop into proportional thinking.