Numerous 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 reasoning multiplicatively is necessary for proportional reasoning, it is important to find ways to help students reason proportionally.

What knowledge was critical for understanding ratio and proportion? The two main issues to be discussed will be the multiplicative schemes and the nature of students’ proportional reasoning.

Karen’s scheme’s of proportional reasoning
Question: to bake 6 cakes, you need 15 eggs. Using the same recipe, how many eggs do you need to bake 4 cakes?
She quickly solved it mentally and gave the answer 10 eggs.
R: can you explain?
K: in 2, 4, 6, there are three and all are times 2. Then 15 divides 3 because 1, 2, 3, got three 2 cakes – 5 eggs, 4 cakes – 10 eggs, and 6 cakes you need 15 eggs.
R: how do you know that for 2 cakes you need 5 eggs?
K: because I divided 6 cakes and 15 eggs by 3.
R: why did you divide by 3?
K: because in 2, 4, 6 there are one, two, three, so I divided by 3.
Initially, she looked at the relationship between 6 and 4 and then said in 2, 4, 6 there are three. This was an intriguing construction because “in 2, 4, 6 there are three so I divided by 3” seems to indicate that she constructed a scheme in which the composite unit consisted of three units.

Alice’s schemes of proportional reasoning
The following sections provides a report on Alice’s solutions of similar tasks.
R: Simon worked 3 hours and earned $12. How long will it take him to earn $36?
A: 9 hours (mentally)
R: Can you explain how you got it?
A: I divided 12 by 3 and got 4, 36 divided by 4 is 9 as 4 times (multiplied) 9 is 36.
R: Why did you divide 12 by 3?
A: To get 1 hour how much it is.
R: Very good
She utilized the unit strategy to solve this problem and did it very efficiently in producing the answer. I then gave her a task that I hoped would perturb her because the unit ratio does not give a whole number. I wanted to know whether she could use her method flexibly.

Interpretation of Karen’s activity
Karen’s strategies provided me with meaningful insights on multiplicative schemes in proportional reasoning. She had constructed iterable ratio units and was coordinating these units such that one ratio was distributed over the next ratio. Utilizing a similar approach in proportional reasoning, Karen 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 and maintain the ratio unit of its elements.

Interpretation of Alice’s activity
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. I believe that Alice’s procedural orientation influenced her action in dealing meaningfully with ratio and proportion.

The ability to use operation with composite units seemed to involve three essential components. First, one needs to explicitly conceptualize the iteration action of the composite ratio unit to make sense of ratio problems. Second, one needs to have sufficient understanding of the meaning of multiplication and division so that one can see their relevance in the iteration process. Third, and finally, one needs to have sufficiently abstracted the iteration process so one could reflect on it, and subsequently reconceptualize it in terms of that unit.

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