The Power of One: The Importance of Flexible Understanding of an Identity Element

Abstract

The number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of students indicated that 36/36 = 1), most students did not recognize and apply knowledge of fraction forms of one to estimate numerical magnitudes, solve arithmetic problems, and evaluate arithmetic operations. Specifically, students were less accurate in locating fraction forms of one on number lines than integer forms of the same number; they also were slower and less accurate on fraction arithmetic problems that included one as a fraction (e.g., 6/6 + 1/3) than one as an integer (e.g., 1 + 1/3); and they were less accurate evaluating statements involving fraction forms of one than the integer one (e.g., lower accuracy on true or false statements such as 5/6 × 2/2 = 5/6 than 4/9 × 1 = 4/9). Analyses of three widely used textbook series revealed almost no text linking fractions in the form n/n to the integer one. Greater emphasis on flexible understanding of fractions equivalent to one in textbooks and instruction might promote greater understanding of rational number mathematics more generally.

Keywords

• fractions
• fraction arithmetic
• mathematical flexibility
• textbook analysis
• flexible understanding of number