A Unit Factor Method for Solving Proportion Problems.

In the academic marketplace, scien tists are among the best customers for the mathematician's product, and all in dications are that the customers aren't happy. Visit any gathering of science teach ers?the faculty room of the junior high school, the science preparation room in the high school, the kaffeeklatsch in a college science department, or any professional meeting?such as the National Science Teachers Association or the American As sociation for the Advancement of Sci ence?and you are likely to hear a lament concerning the poor preparation of stu dents in mathematics. Criticism ranges from the students' inability to do simple computations to their total lack of under standing of basic algebra. More than likely someone in the group will call attention to the evils promulgated by the new math ematics, a convenient scapegoat for the col lective frustrations of science teachers who are unable to teach science as they would like because students are unable to do the mathematics they need. Actually, science teachers have always complained about the mathematical incom petence of their students. Thus the problem is not likely to be related to new mathe matics, nor is it likely to be a result of poor teaching. But it may very well be related to the cognitive development of students. That is to say, students at a certain stage of intellectual development may not profit from an explanation that is crystal clear to the teacher because they don't think like the teacher. We believe it is time for mathematics teachers to consider an alternative widely practiced in the science class but almost never taught in the mathematics classroom, namely, solving proportion problems using the unit factor approach. The importance of proportional logic in science can be seen from a cursory examination of concepts and principles that are taught. In physics one encounters the concepts of density (the ratio of mass to volume), velocity (the ratio of distance to time), acceleration (the ratio of velocity to time), pressure (the ratio of force to area), and the laws that govern the behavior of gases, where pressure, volume, and temperature of a gas are related by either direct or inverse proportions; all are obvious examples of the application of pro portional logic. From chemistry we could add stoichiometric calculations, which in volve proportions between reactants and products in a chemical reaction; concentra tion terms, such as molarity (the ratio of moles of solute to liters of solution); and equilibrium constants (ratios of the concen tration of products to the concentration of reactants in a chemical system at equilibrium). Obviously, these concepts and principles are only a few of the many topics in science that depend on proportional logic. The dif ficulty of proportions is widely recognized by both science and mathematics teachers. In junior high school, mathematics teachers observe the difficulty students have with such problems as "If eggs cost 60 cents a dozen, how much would five eggs cost?" When students are asked to convert twelve inches to the equivalent in centimeters, many of them are unable to use the equiva lence, 1 inch = 2.54 centimeters, to get the correct answer. If the problem is to convert centimeters to inches, the task proves even