An Excessive Fondness for Fractions - by Bill Marsh, Ph.D.

An Excessive Fondness for Fractions

Bill Marsh, Ph.D.
October 16, 2008

“The importance of fractions to mathematics cannot be overstated. ... For middle and high school students, real numbers are mostly taken for granted ... The point is that fractions are an essential intermediary step between whole number and real numbers. ...It is impossible to overstate the importance of fractions. Numbers and geometry are at the heart of mathematics, and fractions are required for both. You can’t do mathematics without an understanding of fractions and their operations.” -- Mathematician W. Stephen Wilson in “Review of Mathematical Soundness”, 2008.

“[D]oes not the difficulty begin with fractions? Should we have such a notion of these numbers if we did not previously know a matter which we conceive of as infinitely divisible – i.e., as a continuum?” -- French mathematician and philosopher Henri Poincare in Science and Hypothesis, 1903.

“The device beyond praise that visualizes magnitudes and at the same time the natural numbers articulating them is the number line, where initially only the natural numbers are individualized and named.” -- German/Dutch mathematician and educator Hans Freudenthal in The Didactical Phenomenology of Mathematical Structures, 1983.

“For students of all ages, definitions of basic mathematical concepts have to be framed with care: not too formal, not too informal. The New Math movement gave rigorous definitions starting with whole numbers, i.e., 5 is the collection of all sets that contain 5 elements, that were out of touch with children’s and parents’ mathematical experience. The common sense notion of a whole number as a counting number, to count how many items are in a collection, provides an adequate informal definition in the early grades. Later, in anticipation of rational and real numbers, whole numbers can be identified with appropriate points on the number line.” -- Mathematician Alan Tucker in “Preparing for Fractions”, 2006 PCMI Workshop.

In his review for the Washington State Board of Education, Professor Wilson says "The importance of fractions to mathematics cannot be overstated." He then proceeds to do what he says is impossible.

He says "The point is that fractions are an essential intermediary step between whole number and real numbers." Some fractions, yes, but not all of them. All you need is a dense subset of the reals. Two such suggest themselves: decimals, whose denominators are powers of ten, and the fractions used by carpenters and computers, those with denominators a power of two.

Following the excellent advice on definitions given by Professor Tucker, first or second graders can define, think of, and use "measuring numbers" as "good names for dots on number lines." To see a six-year-old explain in less than six minutes a "naming trick" that gives a notation system for the non-negative real numbers, go to http://www.youtube.com/watch?v=d90wWqYBMOQ

Professor Wilson says: "Numbers and geometry are at the heart of mathematics, and fractions are required for both." Fractions are necessary in geometry, but we all know that they are not sufficient. Neither the circumference of the unit circle nor the diagonal of the unit square has a length expressible as a fraction. For geometry and measuring, fractions are too complicated and not good enough. There are both too many of them and too few.

Fifty years after reading Science and Hypothesis what I most remember is the model of non-Euclidean geometry inside Euclidean, which shows that we can deny an axiom. I think that the answers that Poincare expected to the questions above means that we can deny the widely held assumption that fractions are what should come next after the counting numbers as children learn arithmetic. In k-4 arithmetic, we can be natural, get real, not rationalize, and not be negative.