Many research studies conclude that proportions are the most difficult problem-solving mathematical tools to master of any introductory science course. Indeed, many physical science, biology, physics, and chemistry concepts, in effect, are names given to proportional relationships. More specifically, proportional math problems can compose as much as 95% of an introductory chemistry course. Students’ ability to comprehend and effectively use proportions, therefore is a major concern of the science and math educator. Yet, little is done within the classroom to increase this proficiency.
Not only is student understanding of proportionality a concern of the science and math educator, it is a major concern of the developmental psychologist. For example, Inhelder and Piaget have studied intellectual development in relationship to students’ ability to deal with science concepts. They regard proportionality as a primary acquisition at the stage of formal operations which include subjects from 11Ð15 or 16 years. Unfortunately, there is much evidence that suggests that as much as 50% of some samples of secondary school and college-age students have failed to acquire a working understanding of proportionality.
The concept of proportions is seen as fundamental to understanding many scientific applications as well as consumer problems, advanced science and math courses, and intellectual development in general. Rates can be found in most aspects of life including cooking, navigation, physics, earth science, economics, electronics, business, and industry. Since a large percentage of adolescents are lacking this critical skill, the determination of possible ways of successfully teaching the concept is an important issue.
Sci-Math is an interdisciplinary curriculum designed to address these issues of teaching proportionality in science and math courses while using large or very small numbers. Its development and field testing were funded by the National Science Foundation. Sci-Math was cited by the U.S. Office of Education as an exemplary educational innovation worthy of national dissemination within the National Diffusion Network (N.D.N.).
Sci-Math focuses on the understanding of the concept of proportions and on the use of proportions in word problem-solving. Specifically, Sci-Math uses the rate concept and dimensional analysis used in introductory physics and chemistry courses to solve proportions (see Teaching Dimensional Analysis in the next section). This rate and dimensional analysis method has slowly moved into textbooks and has completely replaced the method of ratio-and-proportions taught exclusively in junior and senior high school mathematics textbooks.
There appears to be good reason for dimensional analysis to have replaced the ratio-and-proportion method in advanced science courses. Dimensional analysis is a simple, problem-solving, error-reducing procedure which seems to require less conceptual reasoning power to understand than does the ratio. Furthermore, it can condense mult-step problems into one orderly extended solution. However, the treatment accorded the method of dimensional analysis by too many advanced science books is confusing, too sketchy, and not logical in the approach to word problem solving.
To support Sci-Math goals, the curriculum uses hands-on activities and experiments. These experiments use simple inexpensive materials already available in schools: spoons, pennies, jars, rulers, string, etc. Proportions are of great use in everyday life as well as an important pre-algebra and physical science tool. While the Sci-Math curriculum deals with the everyday world of measuring, buying, cooking, and driving, the mathematics taught are the mathematics needed for advanced science. A good example of a Sci-Math activity is measuring, then calculating the average rate of 15 pennies to 2.2 centimeters. Using this rate, students find how many pennies would be necessary to stack in order to reach the moon some 237,000 miles away.
Developing an understanding of the Sci-Math method with its “real life” labels could help algebra students to better understand algebra and see a little more clearly its relationship to everyday life. In addition, a remedial math student, tired of writing all those labels, could easily begin to shortcut his/her work by using letters and therefore naturally begin to use algebra. The rules of adding, subtracting, multiplying, and dividing are the same for both Sci-Math units and algebra variables.
Three research studies on proportional calculations with emphasis on the rate concept, dimensional analysis, and hands-on manipulative experiments were field tested in the ninth and tenth grades. The students showed substantial improvement in proportional problem-solving skills. The studies suggest that any advanced science course is a late point at which to introduce dimensional analysis and the rate concept. It seems that the student needs to learn to understand the logic of the process using familiar experience with the concepts before applying them to the unfamiliar variables of advanced science. Hence, it is important that the rate concept and dimensional analysis be taught prior to advanced science courses. Ideally, these concepts should be taught in the seventh, eighth or ninth grade for college-bound students, in the ninth or tenth grade for non-college bound students. The research indicates both groups can significantly improve their understanding of proportions and problem-solving by using the Sci-Math techniques.
After formally adopting the Sci-Math program from N.D.N., I implemented a team-taught physical science and pre-algebra course at my school. Initially, some teaching problems developed that were particular to lower skills urban students. Despite these problems, the students demonstrated significant improvements on their science and math problem-solving skills. In my many years of teaching, I have never seen such interest in and enthusiasm for word problem-solving and science labs. For example, years later I still find students remembering with excitement how they figured out how many pennies it took to get to the moon, if they stacked one on top of each other. In addition, the algebra and advanced science teacher said that the Sci-Math skills learned in the past year are readily transferable to their courses. It is my personal experience that a firm basis in Sci-Math will also decrease avoidance of advanced science courses by students, and help science teachers who are often forced to teach the mathematics necessary for science.
Although the Sci-Math program is a powerful learning approach, the problem with the curriculum seems to be that it needs to be adapted to meet the particular needs of urban students. Research backs up my intuition. Karplus and Peterson’s 1970 research study found that while successful proportion reasoning is present in half the suburban eleventh and twelfth-grade students tested,
only one-eighth of urban students had this ability.
The teaching techniques of Sci-Math and dimensional analysis need be modified for the special needs of urban students so they better reflect a logical developmental progression with a great deal more reinforcement and more applications. This is one of the major goals of my curriculum unit.
The National Council of Teachers of Mathematics (N.C.T.M.) has stated that its first goal for the 1980’s was that problem solving must be the focus of school mathematics. According to Shirley Hill, a former president of N.C.T.M., “This means that the ultimate goal in our teaching is the ability to apply the mathematics learned.” The rate concept and dimensional analysis are two excellent tools for this purpose.
A. Teaching Dimensional Analysis
A rate states how much of one quantity per how much of another quantity (a quantity is a number and a label). It could be any rate like cost per hamburgers. Taking for example 55 miles per hour and the number of miles traveled, 495 miles, you could determine the number of hours traveled as follows:
495 miles 1 hour
———x———= 9 hours
1 55 miles
If you knew the number of hours traveled, 8 hours, you can use the reciprocal of this rate to find the number of miles covered as follows:
6 hours 55 miles
———x———= 330 miles
1 1 hour
This method extends to solving multi-step problem such as how many centimeters in 330 miles. First, let’s use the old ratio and proportion method still used in most junior and senior high school math book to solve this problem:
330 miles 1 mile
——————x =3,990 feet
x feet 5,280 feet
3,990 feet 1 foot
———=———x = 47,880 inches
x inches 12 inches
47,880 inches 1 inch
—————————x = 121,615.2 cm
x centimeters 2.54 centimeters
Now, use the short cut method of dimensional analysis to solve the problem.
330 miles 5280 ft. 12 in. 2.54 cm.
———x——x—— x———= 121,615.2 cm
1 1 mile 1 ft. 1 in.