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In order to resolve ambiguities remaining after our earlier investigations,112 we have continued our investigation of students' ability to apply the ratio concept in a simple task of measurement and prediction. We can now report significant progress toward clarifying the relationships among students' responses to the task. It appears that many of the subjects who do not use proportional reasoning have access to several alternate procedures; they make use of one suggested by some particular aspect of the task presentation. This personal preference, we believe, reflects the individual's cognitive style3 rather than a developmental level in the Piagetian sense.4 These conclusions were derived from the use of a new form of the ratio task administered to 616 students in grades 4 through 9. We shall call the new procedure Form B, to distinguish it from the original version to be called Form A. Method of PresentationFor all groups studied, Form B was administered by one of the authors. The subjects were given answer pages as described previously (Fig.

In an earlier report, elementary and secondary school students5 ability to apply the ratio concept in a simple problem of measurement was described [1 ]. Subjects were asked to report a measurement, predict a result they could not measure, and explain their predictions. It was found that students5 responses could be grouped into six categories, ranging from guessing to proportional reasoning. The authors speculated about a developmental sequence of these responses but were unable to come to any definite conclusion. We have now repeated the administration of the task to 155 sixth, eighth, and eleventh grade suburban students who had been tested two years ago as fourth, sixth and ninth graders. In addition to the participants in this longitudinal study, comparison data will be reported for 141 eighth grade students who had not participated in the earlier testing.These fourth graders are using polar coordinates to describe the location of a flag on their playground. One boy is determining the direction with the aid of a sighting device on the ground, while the other ''s about to pace off the distance to the flag from the starting point. Scaling their measurements when they draw a map of the flag position will involve them with ratios.

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