We know from extensive research that many people—adults, students, even teachers—find the rational-number system to be very difficult.³ Introduced in early elementary school, this number system requires that students reformulate their concept of number in a major way. They must go beyond whole-number ideas, in which a number expresses a fixed quantity, to understand numbers that are expressed in relationship to other numbers. These new proportional relationships are grounded in multiplicative reasoning that is quite different from the additive reasoning that characterizes whole numbers (see Box 7-1).⁴ While some students make the transition smoothly, the majority, like Sally, become frustrated and disenchanted with mathematics.⁵ Why is this transition so problematic?

A cursory look at some typical student misunderstandings illuminates the kinds of problems students have with rational numbers. The culprit appears to be the continued use of whole-number reasoning in situations where it does not apply. When asked which number is larger, 0.059 or 0.2, a majority of middle school students assert that 0.059 is bigger, arguing that the number 59 is bigger than the number 2.⁶ Similarly, faulty whole-number reasoning causes students to maintain, for example, that the fraction 1/8 is larger than 1/6 because, as they say, “8 is a bigger number than 6.”⁷ Not surprisingly, students struggle with calculations as well. When asked to find the sum of 1/2 and 1/3, the majority of fourth and sixth graders give the answer 2/5. Even after a number of years working with fractions, some eighth graders make the same error, illustrating that they still mistakenly count the numerator and denominator as separate numbers to find a sum.⁸ Clearly whole-number reasoning is very resilient.

Decimal operations are also challenging.⁹ In a recent survey, researchers found that 68 percent of sixth graders and 51 percent of fifth and seventh graders asserted that the answer to the addition problem 4 + .3 was .7.¹⁰ This example also illustrates that students often treat decimal numbers as whole numbers and, as in this case, do not recognize that the sum they propose as a solution to the problem is smaller than one of the addends.

The introduction of rational numbers constitutes a major stumbling block in children’s mathematical development.¹¹ It marks the time when many students face the new and disheartening realization that they no longer understand what is going on in their mathematics classes.¹² This failure is a cause for concern. Rational-number concepts underpin many topics in advanced mathematics and carry significant academic consequences.¹³ Students cannot succeed in algebra if they do not understand rational numbers. But rational numbers also pervade our daily lives.¹⁴ We need to be able to understand them to follow recipes, calculate discounts and miles per gallon, exchange money, assess the most economical size of products, read maps, interpret scale drawings, prepare budgets, invest our savings, read financial statements, and examine campaign promises. Thus we need to be able to