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Elementary students’ early development of embedding and disembedding is complex and paves the way for later STEM learning. The purpose of this study was to clarify the factors that support students’ embedding (i.e., overlapping shapes to form a new shape) and disembedding (i.e., identifying discrete shapes within another shape) through the use of filled shapes as opposed to shape frames. We recruited 26 Grade 1 students (~6–7 years old) and 23 Grade 3 students (~8–9 years old), asked them to work on two layered puzzle designs from the Color Code puzzle game, and interviewed them about their thinking processes. The first graders had higher success rates at fixing and embedding the tiles correctly, and students at both grade levels improved on the three-tile design when encountering it a second time about two months later. The four-tile design was more difficult, but students improved if they could identify a correct sub-structure of the design. Successful students used a combination of pictorial shape strategies and schematic location strategies, systematically testing tiles and checking how they could be embedded. The results suggest that helping students focus on sub-structures can promote their effective embedding. 

This study investigated 37 third graders’ explanations of subtraction worked examples shown in number sentence or visual form (ten frame or number line) and their justifications for which visual and numerical worked examples corresponded to the same subtraction strategy. Results showed that third graders gave more detailed explanations in number sentence form than in visual form; whereas, they had higher accuracy in matching number sentences to visuals than vice versa. When matching, they were more likely to reason sufficiently when identifying processes represented in the worked examples as opposed to reasoning about the order of the numbers. When using worked examples, teachers should make use of visuals to help students focus on how the visuals represent the operations.