Abstract

The present dissertation pursued two primary goals: first, to investigate second-grade students’ understanding of the hierarchical structure of base ten in both symbolic and non-symbolic contexts and second, to enhance their knowledge of base-ten structure. Despite the importance of structural knowledge of base ten for later mathematics learning, children do not readily acquire such knowledge in school. The present research focused on the nature of children’s understanding of the relations between base-ten units. Three studies examined second graders’ understanding of base-ten structure through their unitizing performance. Unitizing is defined as viewing one quantity in more than one way, such as interpreting a numeral using two different units (e.g., 263 as 263 ones and as 26.3 tens).
In Study 1, children completed tasks assessing both unitizing performance and principle-of-position knowledge. Results showed that children displayed strong principle-of-position knowledge but limited unitizing knowledge when interpreting numerals. Study 2 investigated unitizing performance in a non-symbolic context, namely when solving measurement division problems using manipulatives. Children demonstrated promising unitizing performance in this context. Moreover, the type of manipulative influenced their performance. Finally, Study 3 examined the effectiveness of an instructional intervention for second graders designed to improve unitizing performance with numerals. Its design was informed by the findings in Study 2 and by research on analogies in mathematics instruction. Measurement division problems served as the source of the analogy and numerals as the target. Two types of analogies were compared: explicit analogies, where cognitive supports highlighted the structural similarities between the source and target, and implicit analogies, where no such links were provided. Results showed that analogy-based instruction improved children’s unitizing performance when interpreting numerals. A visual analysis of learning trajectories suggests an advantage of providing explicit analogies in instruction, a finding that was not captured by the statistical analyses.
Together, the overall findings offer several educational implications. They emphasize the need to teach the structure of base ten more explicitly and consistently in early mathematics. They also encourage the use of pedagogical tools, such as using measurement division and analogies, to support second graders’ understanding of the hierarchical relation between base-ten units.