Most U.S. citizens measure their weight in pounds and their height in feet and inches. Some U.S. citizens can easily convert their weight to kilograms and their height to metersespecially if they know that a meter is 39.37 inches and a kilogram is 2.205 pounds. Becoming able to make such conversions is one of the most important first steps toward understanding science. 
Mathematics is the fourth level
of abstraction...
and the beginning of science.










Usually, the first topic that really stumps
in school is fractions. Some find fractions easy, even "obvious."
Others have to work hard. Some never quite get the hang of fractions.
In college, science and math courses assume
facility with fractionsthat is, with ratios and proportions. During
the past few decades, college science teachers have become aware that many
of their students cannot use ratios, cannot convert between different units,
do not understand simple proportions, such as Ohm's Law or Newton's Second
Law of Motion, However, research done in the past few decades has shown that most students can usefully understand those mathematical abstractions. The route to understanding lies in approaches to teaching and learning that have been rather rare until recently: "The process of science cannot be learned by reading, listening, memorizing, or problemsolving. Effective learning [useful understanding] requires active mental engagement." (From note to students in "Physics by Inquiry," a product of the Physics Education Group, PEG, at the University of Washington.) Active mental engagement means solving puzzles, resolving contradictions, "figuring things out for oneself." So, here's a puzzle: Many health brochures on obesity give a "formula" for calculating BMI. You divide your weight in pounds by your height in inches and then divide by your height again. The number you get is further multiplied by a number they give you. Caluclate that number.All you need to know is that a meter is 39.37 inches and a kilogram is 2.205 pounds. You need to understand the use of ratios and proportions. You need to "see" a certain mathlevel abstraction. Can you convert firkins to pottles? 