2025 Theses Doctoral

Comparing Students’ Reasoning With Function Composition Using a Dynamic Graphical Representation to Traditional Representations

Chen, Yuxi

As one of the central topics in mathematics education, “function” has received massive attention from scholars. Function composition, on the other hand, as one of the primary operations on functions that has multiple implications in advanced mathematics, such as the chain rule, is still underexplored. In particular, the function composition concept in school curricula is still approached procedurally with substitutions of algebraic equations or discrete points, and research about students’ understanding of function composition has also mainly focused on algebraic representation. Very limited research has examined students’ reasoning and understanding of graphically depicted function composition. Furthermore, there is a dearth of literature about the impact that dynamic representations can bring when students reason about function composition tasks, while their affordances and limitations have been investigated for some other function topics.

This study addressed this gap by exploring how students reason about function composition through different representations with and without a three-dimensional dynamic graphical representation, with particular attention to its affordances and constraints on students’ reasoning and understanding.Through a qualitative multiple case study methodology, fifteen graduate mathematics education students participated in individual task-based interviews that contained three rounds of function composition problems. In the first round, participants sketched composite functions given static graphs of parent functions. In the second round, they worked with algebraic equations to sketch composite functions. In the third round, participants had access to a dynamic graphical representation demonstrating particular function composition problems alongside static graphs. Data collection included video and audio recordings of participants’ processes and their written work. The data analysis examined students’ reasoning performance and patterns through: (a) distinct types of reasoning (graphical, algebraic, and discrete point reasoning); (b) levels of understanding as characterized by APOS theory; and (c) integrated versus fragmented reasoning approaches particularly in reasoning for domain and range.

The findings reveal that without a dynamic graphical representation, students employed longer reasoning processes with the need for diverse types of reasoning. The reasoning pattern of alternating between graphical and discrete point approaches proved effective, while solely relying on algebraic or discrete reasoning displayed limited success. In comparison, with the dynamic graphical representation, students demonstrated higher reasoning efficiency with shorter reasoning processes, predominantly using graphical approaches. Their reasoning became more integrated, dynamic, and coherent. The results also indicated that without a dynamic graphical representation, students presented more Action-level understanding of function composition, accompanied by their fragmented reasoning approaches and reliance on explicit equations or discrete points.

In contrast, with the dynamic graphical representation, the Process-level understanding moderately increased to be more than the Action-level understanding. The use of the dynamic graphical representation particularly enhanced students’ reasoning about domain, range and continuity. While the dynamic graphical representation prompted graphical reasoning and reduced reliance on algebraic equations and discrete points, some students struggled with the three-dimensional model or unfamiliar dynamic presentations, which also illustrated the constraints of this tool.