Elementary students' use of relationships and physical models to understand order and equivalence of rational numbers
This study examined the interaction between the use of physical models and children’s understanding of fractions as demonstrated through their ability to compare and order fractions. Although physical models are recommended to help children developing an understanding of fraction concepts, there are multiple ideas about how to use the materials in classroom instruction and the results concerning the effectiveness of physical models have been mixed. Post, Behr, and Lesh (1986) suggested students must develop a “quantitative notion of rational number” (p. 40) which was directly connected to their ability to compare and order fractions. Smith (1995) identified four perspectives (Parts, Components, Reference Points, and Transform) to categorize general approaches for solving order and equivalence problems, which provided a framework for this study. Thirteen students from a multi-level third, fourth, and fifth grade class participated in this study. The teacher’s mathematics instruction was organized around problem solving and discussion of solutions. Daily classroom observations were videotaped over a three month period. All thirteen students participated in individual clinical interviews prior to and after the unit; eight students participated in interviews midway through the unit. All interviews were videotaped and summarized. The analysis of the data identified the relationships students attended to when comparing and ordering fractions. These relationships were grouped into eight perspectives (Limited, Pieces, Part-Whole, Unit Fraction, Within-Fraction, Between-Fraction, Equivalence, and Transform) extending Smith’s (1995) work. Many of these perspectives were connected to developing a quantitative notion of fractions and were influenced by the use of physical models. Physical models were used for more than just finding answers. Pre-partitioned area and linear models helped students learn equivalent relationships; however, some students acted as though the pieces were unrelated to the whole or used materials without thinking about relationships. Relationships were extremely important for comparing and ordering fractions in the part-whole perspective. Students who were able to identify, use, and extend relationships had a stronger understanding of fractions and could move between perspectives to solve problems efficiently and effectively.