

5 Using
Manipulative Models to Kathleen Cramer Apryl Henry 

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THE Rational Number Project (RNP) has reported on several longterm teaching experiments concerning the teaching and learning of fractions among fourth and fifthgrade students (Bezuk and Cramer 1989; Post et al. 1985). A curriculum created for these teaching experiments and then revised on the basis of what was learned reflects the following four beliefs: (1) children's learning about fractions can be optimized through active involvement with multiple concrete models, (2) most children need to use concrete models over extended periods of time to develop mental images needed to think conceptually about fractions, (3) children benefit from opportunities to talk to one another and with their teacher about fraction ideas as they construct their own understandings of fraction as a number, and (4) teaching materials for fractions should focus on the development of conceptual knowledge prior to formal work with symbols and algorithms (Cramer et al. 1997). A decade of research on the teaching and learning of fractions among fourth and fifth graders has shown us that of the four pedagogical beliefs listed above, the second is the most important. In order to develop fraction sense, most children need extended periods of time with physical models such as fraction circles, Cuisenaire rods, paper folding, and chips. These models allow students to develop mental images for fractions, and these mental images enable students to understand about fraction size. Students can use their understanding of fraction size to operate on fractions in a meaningful way. The multiple models mentioned above were used in RNP teaching experiments. The fraction circle model used in combination with the RNP activities (see fig. 5.1)was the most powerful of the models. During interviews, students consistently referred to fraction circles as the model that helped them order fractions and estimate the reasonableness of fraction operations.


Fig. 5.1. Fraction circles 



CONCLUSION Often you hear teachers argue that there is not enough time to use manipulative materials. Even when manipulatives are used, teachers often make the transition to symbols too soon. The RNP students discussed in this article used manipulative models virtually every day during five weeks of instruction. The predominant model used was fraction circles. The samples of students' thinking presented here show the benefits of using manipulative models for five instructional weeks. RNP students developed number sense for fractions. In general, they had an understanding of fraction size evidenced by the type of ordering strategies they comfortably used. They were able to estimate reasonable answers to fraction addition problems. They were also able to verbalize their thinking. Students using a traditional program did not develop number sense. Developing an understanding of fraction size and estimating a reasonable answer to fraction operation problems are appropriate goals for elementary schoolaged children. Much of the symbolic manipulation of fraction symbols done in fourth and fifth grades can be adequately addressed in the middle grades. Students will be more successful if teachers in elementary school invest their time building meaning for fractions using concrete models and emphasizing concepts, informal ordering strategies, and estimation. REFERENCES Bezuk, Nadine, and Kathleen Cramer. "Teaching about Fractions: What, When, and How?" In New Directions for Elementary School Mathematics, 1989 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Paul R. Trafton, pp. 15667. Reston, Va.: NCTM, 1989. Cramer, Kathleen, Merlyn J. Behr, Richard Lesh, and Thomas Post. The Rational Number Project Fraction Lessons: Level 1 and Level 2. Dubuque, Iowa: Kendall/Hunt Publishing Co., 1997. Cramer, Kathleen, Thomas R. Post, and Robert del Mas. "Initial Fraction Learning of Fourth and Fifth Graders Using Commercial Curricula or the Rational Project Curriculum." Journal for Research in Mathematics Education, in press. Post, Thomas R., Ipke Wachsmuth, Richard Lesh, and Merlyn J. Behr. "0rder and Equivalence of Rational Numbers: A Cognitive Analysis." Journal for Research in Mathematics Education 16 (January 1985): 1836. 