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Cramer, Kathleen; Monson, Debra; Whitney, Stephanie; Leavitt, Seth; Wyberg, Terry. (2010, February). Dividing Fractions and Problem Solving. Mathematics Teaching in the Middle School v15 n6 p338-346 Feb 2010.


Dividing Fractions and Problem Solving

Kathleen Cramer, Debra Monson, Stephanie B. Whitney,
Seth Leavitt and Terry Wyberg
NOTE --- At this time, NCTM is granting permission for this site to publish just the abstract, summary and references for this article.


See how a class of sixth graders used concrete and pictorial models to build meaning for arithmetic operations with fractions.


The part-whole model for fractions helps students understand the use of rational numbers (English and Halford 1995). Although it is important for middle school students to expand their interpretation of fractions to other constructs (operator, ratio, quotient, and measurement), the part-whole model is a crucial gateway to making sense of rational numbers (Lamon 2007). These sixth-grade students’ extended use of concrete and pictorial models provided them with a deep understanding of the part-whole model that in turn supported their construction of a procedure for dividing fractions. This initial fraction interpretation supported students’ understanding of a more complex idea: how to divide fractions.

Students’ understanding of the part-whole model went beyond naming and representing pictures and concrete models using fraction symbols. Our students’ understanding of the part-whole construct was strongly tied to the idea of “flexibility of unit.” Early in their exposure to naming and representing fractions, students were asked to reflect on the unit. The idea that a certain piece of a fraction circle could be named in multiple ways depending on the unit is a fundamental idea tied to the part-whole model. Our students have many experiences reinforcing this idea; this is not the case in traditional curricula.

Our students’ understanding of fraction division is closely tied to their actions with pictures. Helping students make the connection be- tween symbolic procedures and the actions they perform with concrete or pictorial models can enhance their understanding of a procedure (English and Halford 1995). In this case, meaning for the symbolic procedure is developed from students’ initial actions with pictorial models. There is a strong connection between actions on the pictures and steps to the common denominator algorithm. Students constructed the common-denominator strategy for fraction division by solving a measurement story with pictures, with some guidance from the teacher. Instructional activities linked their pictorial strategy to an abstract one.

We should not underestimate what students are capable of doing in mathematics. As we have seen, this group of sixth graders were given the opportunity to develop a rich under- standing of the part-whole model for fractions. They were able to construct for themselves a meaningful interpretation for fraction division on which future symbolic work can be built.

In future lessons, students should be given story problems involving partitive division. These types of problems and experiences can be used to help students develop meaning for the invert-and-multiply algorithm. A more complete understanding of fraction division would involve working with measurement and partitive contexts as well as constructing more than one symbolic procedure.


English, Lyn D., and Graeme S. Halford. Mathematics Education: Models and Processes. Mahwah, NJ: Lawrence Erlbaum Associates, 1995.

Lamon, Susan. "Rational Numbers and Proportional Reasoning: Toward a Theoretical Framework for Research." In Handbook of Research on Mathemat- ics Teaching and Learning, edited by Douglas Grouws, pp. 629−67. Reston, VA: National Council of Teachers of Mathematics, 2007.

Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Pearson Education, 2007.


Kathleen A. Cramer and Terry Wyberg are colleagues at the University of Minnesota. Their research interests focus on teaching and learning rational numbers.

Debra S. Monson and Stephanie B. Whitney are doctoral candidates in mathematics education at the University of Minnesota working on the Rational Number Project.

Seth Leavitt is a teacher at Field Middle School in Minneapolis. He is interested in improving mathematics instruction in kindergarten through high school classrooms.