Clark, Berenson and Cavey, 2003 A comparison of Ratios and Fractions and their roles as tools in proportional reasoning
To develop shared understanding of the relationship between ratios and fractions, author did a phenomenological study to gather evidence from teachers, textbooks and students. They represent five models for this relationship and summary of evidence. They also present their model which help students' to analyze use of ratio and fractions in their solutions to proportions related problem.
ratios are in subset of fractions, i.e. all ratios are fractions. But this excludes the ratio such as Golden ratio , square root of 2 ratio of circumference with the diameter. Also the ratios are also extended to three quantities such nitrogen to potassium to phosphorous in a fertilizer which can not be described as a single fraction.
fractions are subsets of ratios. Like Van de walle says the previous sentence is true one way but not the the other way. The teachers opting this model showed understanding of fractions limited to part-whole interpretation.
Ratios and fractions are disjoint sets. Fractions represents a part of a whole where ratio is comparison of two quantities. Also some teachers said fractions are using part-whole relationship and ratios are using part-part relationship. Some textbook writers at middle school level mentioned that there are three ways of writing ratios such as 3:2, 3 is to 2 or 3/2. and hence this eliminates model 3.
Fraction and ratios are overlapping sets and have something in common. 1cup of sugar: 2 cups of flour falls under ratio, a cup of sugar/3 cups of ingredients falls in intersection of the both and ½ cup of sugar comes under fractions.
Ratios and fractions are identical sets. No textbook use this definition. But some people define fraction as 'indicated division of one number by the another.' But 12:0 is an acceptable ratio but not a fraction. This is also supported by Van de Walle (94). But still some people define ratio a:b as b restricting non-zero. This example also used by teachers pre-service to eliminate model 1 and 5.
Author is following Maher's idea that learners construct mental representations as they engage in mathematical tasks and that their thought about mathematics are unique to them as individuals. similarly conceptual development of the concept of fraction is dependent on the individual's connections between and among concepts.
The researchers presented stories and asked questions in the workshops. The conclusions are made on the basis of triad. Teachers response (oral and written), students responses (oral and written) and existing textbooks. In phase one they tried to understood the connection based on experiences and understandings of the triad mentioned above. Phase II they develop the conceptual framework for viewing the existing models and also developed their own model.
In performing the cross-multiplication procedure to solve missing value proportion problem student's path of decontextualizing ratios has no impact on producing the correct answer. Given other types of problems, however students who rely on this and other procedures often fail to reason proportionally. Students who understand ratios and fractions each in isolations may recognize ratios, write them as fractions and operate on those using memorized rules. (This is also the outcome of model 4)
Successful students seem to have used intersection model as students able to move back and fourth in ratio realm, them fraction realm and using the common realm. The movement is easier if the path for moving from one realm to other is known.