| Research Questions: One way students may develop conceptual understanding is through working on strands of related mathematical tasks and thus developing and refining their understanding of the underlying mathematical concepts contained in the tasks. The purpose of this study is to illuminate this process by detailing the inherent mathematical structures in such a strand and discuss what aspects of it facilitated student learning. The research questions addressed are: (1) What mathematical structures can be uncovered by exploring/engaging with the combinatorics tasks used in the Rutgers longitudinal study? (2) In what ways are these mathematical structures revealed during students' problem-solving processes? .;Methodology: Ten tasks from the combinatorics/counting strand are selected from the Rutgers longitudinal project for this qualitative study. The data available for analysis are in the form of digitized video tapes, verified transcripts, and students' written work. The analysis focuses on decoding students' solutions into formal mathematical definitions and theorems. Concept maps are used to illustrate the overall hierarchy of the presented mathematical structures.;Findings: There are a total of sixty-three inherent mathematical structures extracted from the formal solutions of ten selected combinatorics tasks. These structures are categorized as definitions, notations, axioms, properties, formulas, and theorems. When classified with respect to the seven relevant sub-domains of mathematics, these structures pertain to: set theory, enumerative combinatorics, graph theory, sequences & sets, general algebraic system, probability theory, and geometry. The analysis suggests that the participating students uncovered many of these mathematical structures primarily in the following ways: (1) Manipulating a concrete model, (2) Listing all possible combinations, (3) Inventing different representations, (4) Seeking patterns, and (5) Making connections.;Conclusion and Suggestions: These findings support the following suggestions for practice: (1) Teachers may benefit from studying the underlying structures of a task thoroughly before assigning the task to students, (2) In determining the order of related tasks within a strand, teachers need to consider the sophistication level and the coherence of the underlying structures across tasks, (3) Using concrete models can help students to both develop and verify solutions to complex problems, and (4) Tasks whose inherent structures belong to a variety of mathematical sub-domains can help students build an increasingly interconnected view of mathematics.;Significance: This study outlined a method of extracting inherent mathematical structures from mathematical tasks. The results suggest that students have natural abilities to uncover these structures by themselves. It is hoped that this will motivate mathematics teachers to improve the way they think about using problem solving in their teaching. |
