Cognitive Development in School Context • Fractions and Manipulatives

Students come to the learning of the ratio concept with well-developed ideas about whole numbers. These ideas can interfere with understanding fractions; students often interpret fraction symbols like 1/3 as two separate whole numbers, 1 and 3. For students to overcome these interpretations, they need opportunities to reinterpret fractions. Working with physical materials like manipulatives can help in this regard. When they work with hands on materials, students take some fairly automatic actions, like separating pieces, collecting pieces into sets, and partitioning those sets. These automatic actions turn into the mathematical actions of counting, grouping, and collecting. During these actions, children develop new interpretations of the physical array they have changed.  We have found that the process of active reinterpretation benefits learning, both in terms of immediate changes in performance and in terms of applications and learning in new settings. In particular, children develop three-fold interpretations in which a piece can be a unit, a member of a group, and a member of a whole. This multiple interpretation appears to be critical to the development of the ratio concept, at least in the context of parts and wholes.

To help children develop multiple interpretations and notice the important features of ratios, we have created a new manipulative called an Annaboard, shown in the figure. The board juxtaposes different quantities that help children notice different relations among the quantities (e.g., twice as big). The children invent notational systems to characterize the differences between the quantities. The tandem processes of perceptual noticing and symbolic invention prepares students to understand the significance of a conventional notation when it becomes available.