Learning Functional Relations: A Theoretical and Instructional Analysis

• Published in: Journal of experimental psychology. General 1987-06, Vol.116 (2), p.106-118
• Main Authors: Reed, Stephen K, Evans, Arthur C
• Format: Article
• Language: English
• Subjects: Biological and medical sciences; Concept Formation; Estimation; Fundamental and applied biological sciences. Psychology; Human; Olfaction. Taste; Perception; Psychology. Psychoanalysis. Psychiatry; Psychology. Psychophysiology
• ISSN: 0096-3445; 1939-2222
• DOI: 10.1037/0096-3445.116.2.106

Many scientific phenomena can be described by linear functions. In this study, we examined how well students understand the functional relations in a mixture task by asking them to estimate the concentration of an acid solution created by mixing two other acid solutions. For example, what would be the different concentrations of a 10-pt mixture created by mixing a 10% acid solution with either 9, 7, 5, 3, or 1 pt of a 70% acid solution? An examination of group means revealed fairly accurate estimates, which incorrectly suggested that students were appropriately following the integration rules of "cognitive algebra" ( Anderson, 1981 , 1983 ). However, examination of individual subject data revealed that many of the response sequences were unsystematic. We therefore studied how the estimation accuracy of individual students was related to their understanding of three principles (range, monotonicity, and linearity) that relate a concentration of a mixture to the concentration of its components. The results of three learning experiments supported the psychological validity of the proposed principles. First, significant improvements in accuracy were accompanied by significant improvements in the range, monotonicity, and linearity scores. Second, when used to predict the accuracy of individual subjects, each of these three scores correlated highly with accuracy; the multiple correlation values ranged from .87 to .95 when all three scores were used. Third, a direct statement of the principles was a more effective instructional method than the presentation of examples and a graph of the function. Furthermore, when students could use a familiar model whose principles they understood (mixing water at two different temperatures), they reached a near-perfect level of performance in the unfamiliar (acid) domain.