Quantizing and Visualizing the Influence of Scaffolding Moves on Mathematical Modeling Competencies

• In Event: Innovative Mixed Methods Research Practices in Mathematics Education Contexts

Abstract

Mathematical modeling requires learners to construct mathematical representations of real-world systems, a process involving multiple cognitive phases from understanding scenarios to validating results. While research documents the challenges learners face during modeling, less is known about which instructional supports effectively promote modeling competencies. This study addressed two objectives: identifying which scaffolding moves promote or suppress specific mathematical modeling competencies, and developing visualization methods to communicate relationships between qualitative instructional moves and quantitative competency outcomes.
The study draws on Blum and Leiß's Mathematical Modeling Cycle (MMC), which delineates six phases of model construction: understanding, simplifying/structuring, mathematizing, working mathematically, interpreting, and validating. We coupled this with Stender and Kaiser's Scaffolding Moves for Modeling Framework (SMMF), which classifies cognitive supports for modeling. Our approach conceptualized scaffolding as contingent support—assistance tailored to learners' current reasoning within the modeling process—rather than predetermined interventions. This contingency requirement entailed analyzing scaffolding effectiveness through clinical interactions rather than experimental manipulation.
We implemented a conversion mixed-methods design, transforming qualitative interaction data into quantitative form for statistical analysis. Data comprised 51 video-recorded clinical interviews with 25 undergraduate STEM majors working on differential equations modeling tasks involving six open-ended scenarios including disease transmission and predator-prey dynamics. Two analysts independently coded videos using protocol coding procedures with the MMC for modeling engagement and the SMMF for scaffolding moves, generating 5,596 modeling engagement segments and 3342 scaffolding move instances. Reliability was established through percentage overlap measures for temporal coding and consensus resolution of disagreements. We quantized coded segments by recording frequencies and temporal locations of each code. Cluster analysis examined the underlying structure of scaffolding moves based on their references to modeling cycle stages, while logistic regression estimated move-competency relationships, calculating adjusted log odds to account for baseline occurrence rates of each competency.
Cluster analysis identified three distinct scaffolding categories: moves supporting quantitative reasoning about scenarios, moves facilitating mathematical representation, and moves promoting reflection and revision. Statistical analysis revealed that most scaffolding moves showed no significant effect on competency uptake beyond baseline expectations. However, specific moves demonstrated clear impacts: moves attending to units and noting errors promoted validating behaviors, while moves focused on mathematical procedures suppressed simplifying/structuring and interpreting. Importantly, the same scaffolding move category could promote or suppress identical competencies depending on which modeling stage it targeted, indicating that contextual application determines effectiveness.
This work addresses methodological challenges in mixed-methods education research by developing visualization techniques that preserve qualitative nuance while supporting quantitative analysis. Four novel displays—competency roses, ternary graphs, caterpillar plots, and azulejos plots—communicate complex relationships between instructional moves and modeling engagement. These visualizations demonstrate how to meaningfully integrate qualitative and quantitative data beyond simple triangulation, moving the field from descriptive characterizations of teacher-student interactions toward predictive models of instructional effectiveness. The approach offers methodological advances applicable to other educational contexts requiring fine-grained analysis of teaching-learning interactions. The findings also contribute substantive knowledge about scaffolding effectiveness, revealing that instructional impact depends on contingent application rather than move type alone.

Authors

• Jennifer Czocher, Texas State University

• Alexander White, Texas State University-San Marcos

• Sindura Kularajan, Utah State University

• Elizabeth Anne Roan, Texas State University

• Andrew Baas, Texas State University