Three experiments compared performance and transfer among children aged 83–94 months after written or manipulatives instruction on two-digit subtraction. In Experiment 1a, children learned with manipulatives or with traditional written numerals. All children then completed a written posttest. Experiment 1b investigated whether salient or perceptually attractive manipulatives affected transfer. Experiment 2 investigated whether instruction with writing would transfer to a manipulatives-based posttest. Children demonstrated performance gains when the posttest format was identical to the instructed format but failed to demonstrate transfer from the instructed format to an incongruent posttest. The results indicate that the problem in transferring from manipulatives instruction to written assessments stems from a general difficulty in using knowledge gained in one format (e.g., manipulatives) in another format (e.g., writing). Taken together, the results have important implications for research and teaching in early mathematics. Teachers should consider making specific links and alignments between written and manipulatives-based representations of the same problems. 1. Introduction The relation between symbolic and concrete representations of mathematical concepts is a recurring theme in mathematics education. On one hand, much of the value of mathematics in human thought stems from its symbolic nature. Symbolic representations of number and mathematical operations allow us to think about abstract mathematical properties and functions independent of a specific quantity or operation. For example, it is possible to know that combining two objects and one object results in three objects without specifying what the objects are. Similarly, we understand that two vehicles traveling at 50？mph move at the same speed regardless of whether both are cars or one is a boat. On the other hand, much of what we know about mathematics is grounded, at least initially, in our experiences in the world. For example, Lakoff and Nú？ez [1] suggested that even ostensibly symbolic notions, such as ordinality and logical independence, can be connected, at least metaphorically, to a physical analog (in this case, points that fall on the same line and orthogonal elements in geometry). Concrete thinking pervades numeric judgments, even among individuals who are highly numerically literate. For example, judging that requires more time than judging that , regardless of whether the problem is posed with objects or symbols [2]. Likewise, visually salient features, such as the physical
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