This study was designed to investigate the understanding of proportion by a select group of college-bound adolescents. Gender-related differences and the influence of course experience were also studied in relation to first-order direct proportional reasoning and multiple proportional reasoning. The 901 subjects were administered the Tall-Short task as a test of first-order direct proportional reasoning. From the 474 subjects classified as successful, a random subsample of 128 subjects was administered the Projection of Shadows task. Only 22 subjects succeeded on all four subtasks of this multiple proportion task. An analysis of the transcripts of responses showed that feedback and second trials were essential for success for most (68%) of these successful subjects. In addition, this select subsample generally used a multiplicative strategy and still could not overcome the inhibiting effect of focusing on direct proportions. There were significant gender differences, in favor of male subjects, in first-order direct proportional reasoning but there were no gender differences in multiple proportional reasoning. Prior course experience in mathematics and science were each significantly related to first-order direct proportional reasoning but there was no significant relationship between either of these variables and multiple proportional reasoning.The mathematical structure of ratio and proportion problems found in secondary school science and mathematics courses includes first-order and second-order direct proportion represented by the equations, y = kx and y = kx2, respectively, first-order and second-order inverse proportion represented by the equations, yx = k and y. " = k respectively, and multiple, or joint, proportion represented by equations such as
Two group paper‐pencil batteries, the Longeot (consisting of three subtests) and three puzzles (KLR) from Science Teaching and the Development of Reasoning, were administered to 607 students from ninth and tenth grade mathematics and science classes. A subsample of 69 students was then administered three Inhelder tasks (chemicals, rods, and shadows). In general, the expected developmental trends were confirmed, with formal status being most difficult to attain on the Inhelder tasks and easiest to attain on the Longeot. The correlations between the KLR and Inhelder (0.61, p < 0.01) and the Longeot and Inhelder (0.55, p < 0.05) were moderately high. According to the method of Shayer (Note 2), it was found that each of five paper‐pencil subtests discriminates at or between the 2B (late concrete) and 3A (early formal) levels while the sixth subtest, the mealworm puzzle, was found to discriminate at the 3A level. This study indicates that either group battery may be useful in identifying transitional subjects. However, if a more stringent criterion of “formal” is needed, a “success” rate of four or five out of the six subtests may be required. Both group batteries are relatively easy to administer and score with a minimum of guidance, although the KLR scoring might need to be simplified for use by the practitioner. Sex differences found on the KLR and the Longeot are suggestive of the potential differential use of these tests by researchers investigating sex differences in achievement or aptitude.
In this second article of the 1993-94 series, a middle school teacher reflects on his work with Hispanic and Haitian students. He used what he has learned to recommend ways to help preservice teachers learn about students of other cultures.– Ed.
The creative mind balks at the ac cepted, the ordinary, the usual. Indeed, new mathematics seems to emerge as a reaction against the ordinary. If the role of the teacher is to encourage creativity by trying to guide students to think as mathematicians do, then accepted state ments must be challenged, alternate avenues explored, and comparisons made throughout the exploration.
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