# The relationship between proportional thinking and achievement of selected science and mathematics concepts at the knowledge, comprehension, and application levels

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Introduction Two important considerations for improving student achievement are the sequence of the curriculum and the developmental readiness of the learner. Gagne has developed an instructional strategy for improving the sequence of curriculum. Piaget has identified various stages of cognitive development and has described the logical thinking skills characteristic of each stage. The task analysis strategy of Gagne and the developmental theory of Piaget should be combined to enhance student learning. This can be accomplished if logical thinking skills necessary for learning specific concepts can be determined. Purpose The purpose of this study was to determine the relationship between proportional reasoning and students' ability to demonstrate knowledge, comprehension, and application of simple machines, structure of matter, and equivalent fractions. If a relationship does exist, then a student's achievement of knowledge, comprehension, and application of the selected concepts should be a function of his level of proportional reasoning. Procedure A sample of 136 subjects was randomly selected from a population of 444 white, middle class students who were taking ninth-grade physical science as a required class. Equal numbers of boys and girls were selected. These students had studied the selected science concepts as part of their regular science curriculum. Piagetian tasks were administered to assess students' proportional reasoning skills and to group them into four levels of proportional reasoning: high and low quantitative proportional reasoners, and high and low qualitative proportional reasoners. A paper/pencil test was used to assess students' achievement of knowledge and understanding (comprehension and application) of the selected concepts. Discriminant function analyses were applied to the data to determine if a relationship existed between proportional reasoning and achievement of the selected concepts at the knowledge, comprehension, and application levels. Findings In general, a positive relationship was found to exist between proportional reasoning and achievement of the selected science and mathematics concepts. Quantitative proportional reasoners (formal operational) achieved significantly greater knowledge of the science concepts and significantly greater understanding of equivalent fractions concepts than qualitative proportional reasoners (concrete operational). Specifically, high quantitative proportional reasoners achieved significantly (p < .01) greater knowledge and application of simple machines, knowledge, and comprehension of structure of matter and application of equivalent fractions than high and low qualitative proportional reasoners. Low quantitative proportional reasoners achieved significantly (p < .01) greater knowledge of simple machines and structure of matter than low qualitative proportional reasoners. Low quantitative proportional reasoners achieved significantly less application (p < .01) of simple machines and comprehension (p < .05) of structure of matter than low qualitative proportional reasoners. And low quantitative proportional reasoners achieved significantly greater application of equivalent fractions than high qualitative proportional reasoners (p < .05) and low qualitative proportional reasoners (p < .01). Conclusions The findings of this study generally support the hypotheses investigated: achievement of selected science and mathematics concepts is related to proportional thinking, and quantitative proportional reasoners achieve significantly greater knowledge, comprehension, and application of the selected concepts than qualitative proportional reasoners. Analysis of the means suggests a particular interaction pattern between proportional reasoning and level of achievement (Bloom's taxonomy). It suggests that when achievement is low, proportional reasoning interacts at the knowledge level. When achievement is high, proportional reasoning interacts at the application level or higher. Recommendations Since a general relationship was found between proportional reasoning and achievement, it is recommended that experimental studies be conducted to investigate a cause and effect relationship between proportional reasoning and achievement. Implications Because most high-school-age students would not be expected to possess quantitative proportional reasoning, teachers instructing students in these concepts would probably be unwise to teach for mastery at the application level. By employing a sound instructional program, they could probably teach for mastery at the knowledge and comprehension levels and minimal achievement at the application level. The identified relationship between proportional reasoning and achievement of the selected concepts permits the task analysis strategies of Gagne and the developmental theory of Piaget to be combined and utilized to enhance student achievement of these concepts.