A Rational Approach to Fractions

Abstract

Are there substantial intellectual questions worth consideration in remedial arithmetic? Could the source of our students' errors be genuine intellectual doubt, or is the problem essentially one of student deficiencies in trivially obvious areas? I believe that within arithmetic lie substantial concepts. Further the concepts are complex enough to make plausible a variety of interpretations. Many of the interpretations are appropriate, many misleading, and many, mainly those initially in the minds of students, are incorrect. The confusion from which springs such arithmetic horrors as + 2 = 1 has an obvious rational basis. It has been my experience that many arithmetic errors have some logical basis. To deal with these errors, I believe it is necessary to challenge the ideas upon which they are based. For many reasons, this is a formidable task for mathematics teachers. In light of the kind of experience that characterizes the personal mathematical development of most math teachers, it is clear where there could arise innumerable problems in remedial teaching. Many teachers of mathematics enter the profession due to a high level of ability and the personal excitement derived in the study of the discipline. In many cases intellectual success with mathematics spurs teachers to levels far beyond even their own initial expectations. They often build (though never completely) both a macroscopic view of the subject, and a strong ability to produce rigorous results. The remedial student's experience is perhaps the antithesis of what has been described for his teacher. Due to his often repeated experience of failure, most people would agree that the remedial student needs understanding, support, and concern. In addition, I contend, he needs organization, mental discipline and a sense of a developing understanding. But perhaps at the apex of all of these needs is the student's need for ongoing intellectual challenge (motivation). He must feel that what he is learning is of value. Thus the question naturally arises, is there intellectual challenge in arithmetic? Posed differently, is arithmetic worth knowing? Calculators and technology not withstanding; yes is the answer to both questions. Rather than attempting to make a case for the conceptual content of all of arithmetic, let us look specifically at fractions.