Hierarchical Planning and Hierarchical Representation for Algebraic Expressions
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Human cognition relies on hierarchical organization to represent and handle complex symbolic information across domains such as language, music, and motor behaviour. Mathematical expressions are similarly organized according to hierarchical operator structure, yet little is known about how this structure is represented and used. Specifically, it remains unclear whether hierarchical organization reflects a stable cognitive representation of symbolic structure or a planning strategy that reduces working-memory load.
The present study investigated both hierarchical representation and hierarchical planning in the copying of algebraic expressions. Here, hierarchical representation concerns how operator-defined structure is maintained in working memory, whereas hierarchical planning concerns how that structure is used to organize copying into meaningful units rather than writing symbol by symbol. The study examined whether these forms of hierarchical organization are reflected in behavioural performance and how they differ between children and adults.
Twenty-five adults and twenty-seven sixth-grade children completed two complementary copying tasks. In the Hidden Copying Task, the expression remained on screen but was occluded, so that participants could consult it as often as they wished but could not look and write at the same time; this task was designed to capture copying strategy and planning. In the Delayed Copying Task, the expression was shown briefly and then removed, so that the complete expression had to be maintained in memory before writing; this task was designed to probe memory representation. Within each task, we systematically manipulated operator structure—that is, the arrangement of multiplication and addition operators—while holding symbol identity and visual complexity constant. Performance was assessed using copying accuracy, error patterns, and inter-token timing.
The results showed different expressions of sensitivity to operator structure across tasks and age groups. In the Delayed Copying Task, children’s copying accuracy differed between the two critical operator structures, whereas adults showed little corresponding difference, providing evidence of hierarchy-congruent representation in children. In the Hidden Copying Task, adults’ copying accuracy differed between the two critical operator structures, while children did not show a comparable structure-dependent difference, providing evidence of hierarchy-congruent planning in adults. In both tasks, the inter-token gap profiles showed descriptive structure-related differences, but the critical Position × Condition interactions did not reach statistical significance. In the Hidden Copying Task, the adult timing-error correlation was significant in one critical structure but not the other; however, the direct difference between these correlations was not statistically significant.
Together, these findings suggest that operator structure can influence both the representation maintained in working memory and the planning strategy used during copying. The different patterns across age groups indicate that hierarchy-congruent representation and planning are not expressed identically in children and adults. Future research should clarify how these components develop and whether making operator-defined grouping explicit can support the learning and writing of mathematical expressions.