**Introduction**

Many studies have shown that early adolescents and many adults have difficulty with the basic concepts of fractions, rates, and proportion and with problems involving these concepts. Since students have difficulty reasoning multiplicatively and it is necessary for proportional reasoning, it is important to find ways to help students reason proportionally. This paper was conducted to construct an understanding of two grade six students’ proportional reasoning schemes namely Alice, a smart student, and Karen, an analytical student. This study was held using a constructivist theory of learning focusing on the students’ construction of proportional reasoning in appropriate problem solving tasks.

**Research Question**

How do students think about their thinking of composite unit schemes and construct or develop them into proportional reasoning with the conjecture that each child would have a unique way of accomplishing the task?

**Discussion**

Karen has not studied the method yet trying to solve proportion problem by constructing her own method. Karen uses unitizing and iterating composite unit in which she can answer a task correctly in minute. The unitizing means find the simplest rate for the two units which correspond one to another, meanwhile, iterating means determining the accurate sequence of both two units, each of rates must be corresponded, and, finally, determining the answer of the asked rate of one unit. She also was able to unitize the units in a composite and furthermore was able to deal meaningfully with composite units. In short, she was able to take a ratio as a composite unit, maintain the ratio unit of its elements, and make reasoning of what she has done.

On the other hand, Alice’s conceptualization in proportional reasoning is solely based on the unit method, a memorized procedure rather than a conceptual one. She was able to use the unit method to solve various tasks to get the answers. However, she was not able to describe her reasoning in a meaningful way, other than describing the procedures she used. She saw mathematics as utilizing a taught method in producing answers rather than making sense of the activity. She was not able to think in terms of the composite ratio unit, which explicitly conceptualizes the iteration action of the composite unit to make sense of ratio problems. She keeps using the method which has been taught to her. The researcher asked her some questions, however, and she was not able to answer correctly if the question comes to more complex problem. It was because she often made wrong calculation and had misconception of additive rule in solving proportion problem when she tried to use alternative. Even she made mistake when she find the rate for one of wrong unit. The point is she just focused on the rule or procedural way but not in conceptual orientation. I believe that her procedural orientation influenced her action in dealing meaningfully with ratio and proportion.

**Conclusion
**The conclusion of this study and also the answer of research question were we got two mental operations, unitizing and iterating which played an important role in students’ use of multiplicative thinking in proportion tasks. It is good if we start to ask some critical questions and let them to think. A student, just take one example, Karen did unitizing and iterating first before gradually using multiplication directly to correctly determine a rate of one unit based on the value of proportion given. The advantages of the method, of course, are students can construct their own knowledge, focus on the conceptual thinking, and are likely to avoid wrong calculation and technical mistake like Alice’s method. This study indicated that the unit method should not be taught to students until they have a good grasp of the unit coordination schemes. Using operation with composite units, one needed to explicitly conceptualize the iteration action of the composite ratio unit to make sense of ratio problems, to have sufficient understanding of the meaning of multiplication and division, and to have sufficiently abstracted the iteration process so one could reflect on it. Furthermore, it can be developed to more problematic problems.

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