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When studying for a doctoral degree (PhD), candidates submit a thesis that provides a critical review of the current state of knowledge of the thesis subject as well as the student’s own contributions to the subject. The distinguishing criterion of doctoral graduate research is a significant and original contribution to knowledge.
Once accepted, the candidate presents the thesis orally. This oral exam is open to the public.
The objective of the present study was to identify the costs and benefits of directly telling students the quantitative referents for manipulatives compared to allowing them to construct meaning for the manipulatives in more open and exploratory learning environments. Sixty-five (N = 65) first graders were randomly assigned to one of three conditions that differed in the type of encoding instruction they received: direct instruction (DI), guided exploration (GE), or control. The overarching research question was: How do the ways in which children assign a quantitative referent to a target manipulative (DI vs. GE vs. control) influence their (a) learning, (b) near-transfer abilities, (c) symbolic flexibility and symbolic fluency through far-transfer tasks, and (e) problem-solving accuracy?
Results indicated that direct instruction seemed to be most beneficial for children’s learning. In terms of the learning assessment, children from the DI condition benefitted relative to children in the GE condition, in that they needed fewer items and less time before using the target manipulative in the prescribed way. Evidence suggested that children in the DI condition also outperformed their counterparts in the GE condition on a near-transfer task when looking at their initial responses, but when both initial and post-prompt responses were considered, the performance of children in the GE condition was not significantly different from the performance of children in the DI condition. In contrast, students who learned through guided exploration seemed to be more flexible in their use and interpretation of the manipulatives in the context of the far-transfer tasks than those who were told explicitly what the objects represented. The greater flexibility demonstrated by children in the GE condition also conferred an advantage on their accuracy when solving word problems with the manipulatives compared to children in the DI condition.
This study contributes to the existing literature in that it offers a nuanced view of the use of manipulatives in classroom contexts. Results suggest that teachers may wish to tailor their instructional methods to the learning objectives (e.g., learning, near transfer, far transfer) they have set for their students when using concrete representations with them.