

The Role of
Representations
Kathleen A. Cramer, Terry Wyberg and Seth Leavitt 

NOTE  At this time, NCTM is granting permission for this site to publish just the abstract, summary and references for this article.  
Summary The concrete models we choose to help students build meaning for fractions and operations are important. In our work, we have determined that the fraction circle model supports students understanding of the partwhole model for fractions and provides students with mental representations that enable them to judge the relative size of fractions. Students are able to estimate answers to fraction addition and subtraction tasks and judge reasonableness of answers to fraction operation tasks because theyhave strong mental representations for these numbers and are able to manipulate them mentally. In our latest teaching experiment, we developed lessons to help students build meaning for finding exact answers to fraction addition and subtraction problems using common denominators. We found that the fraction circles vividly demonstrate the need for finding common denominators when adding and subtracting fractions and that the fraction circles show the steps to exchanging given fractions with equivalent ones with common denominators. We will continue to study how fraction circles and other models contribute to students mastery of this difficult procedure. REFERENCES Cramer, Kathleen, and Apryl Henry. “Using
Manipulative Models to Build Number Sense for Addition and Fractions.”
In Making Sense of Fractions, Ratios, and Proportions, 2002 Yearbook of
the National Council of Teachers of Mathematics (NCTM), edited by Bonnie
Litwiller and George Bright, pp. 41–48. Reston, VA: NCTM, 2002. Cramer, Kathleen, Thomas R. Post, and
Robert C. delMas. “Initial Fraction Learning by Fourth and FifthGrade
Students: A Comparison of the Effects of Using Commercial Curricula with
the Effects of Using the Rational Number Project Curriculum.” Journal
for Research in Mathematics Education 33 (March 2002): 111–44.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics. Reston, VA: 2000.
Post, Thomas R., and Kathleen Cramer, “Children’s Strategies in Ordering Rational Numbers.” Arithmetic Teacher 35 (October 1987): 33–35.
Authors Kathleen A. Cramer and Terry Wyberg are colleagues at the University of Minnesota. Their research interests focus on the teaching and learning of rational numbers. Seth Leavitt is a teacher at Field Middle School in Minneapolis. He is interested in improving mathematics instruction in kindergarten through high school classrooms. 
