From Linear to Inverse Square: Students’ Difficulties with Mathematical Relationships in Virtual Science Investigations

• In Event: Identifying and Harnessing Students’ Math Errors for Intervention and Instruction

Abstract

Objectives
We examined the difficulties students had while building best-fit mathematical models of data within a virtual science investigation (Authors, 2013; 2023a). We also examined whether embedded real-time scaffolding remediated students’ understanding of these models. To do so, we analyzed students’ fine-grained performance and interaction logs gathered while they completed the investigations.

Perspectives
The Next Generation Science Standards (NGSS, 2013) expects students to develop mathematical modeling competencies (Practices 2 and 5), which can, in turn, deepen their understanding of scientific phenomena and prepare them for future STEM opportunities (National Research Council, 2012). Here, we focus on competencies related to selecting appropriate mathematical functions for best-fitting curves of data, a common difficulty for students (Casey, 2015). For example, when an inverse square (1/x) mathematical model predicts scientific phenomena (e.g., Coulomb’s Law), students struggle to transfer their understanding of the algebraic equation to the graphical representation (Moynihan et al., 2019). [SYSTEM] assesses students on these competencies and aims to help develop them. [SYSTEM] uses knowledge-engineered and data-mined algorithms to assess students’ competencies on fine-grained sub-practices. It uses the algorithms’ outputs to deploy automated scaffolding via a pedagogical agent. These are designed to support students on specific difficulties, such as choosing mathematical functions that do not fit the shape of the data or having poor fitting mathematical models (Authors, 2023b).

Methods
Participants were 59 high school students (2 classes), taught by two teachers in two U.S. states. Students completed three [SYSTEM] investigations on electromagnetism (NGSS DCI PS2.B). Each investigation contained four auto-scored and scaffolded stages (Hypothesizing, Collecting Data, Plotting Data, and Building Models; Table 1). Two investigations contained phenomena best modeled by a linear relationship, and one by an inverse square relationship (Table 2).

Results
When analyzing logs of students’ performance, we found that they performed at a high-level overall in the two math stages during investigations with a linear graphical relationship (Tables 3 and 4). However, when the underlying scientific phenomenon was an inverse square mathematical function, 23 students (39%) struggled, requiring the scaffolding.
We analyzed the interaction logs of these 23 students further and found that they constructed several mathematical models, between 2 and 15, while attempting to find one with an appropriate functional form and fit. All 23 first selected “inverse” (1/x) to describe the relationship of the data rather than “inverse square” (1/x2), indicating that students had trouble disambiguating between the two similar looking models. Students required different levels of scaffolding support, ranging from general to a bottom out hint (Figure 1). After receiving scaffolding support, all 23 students were able to correctly identify the inverse square relationship.

Contribution
When applying mathematical modeling competencies to analyze scientific phenomena, students may struggle to identify nonlinear mathematical relationships. Theoretically informed learning environments with embedded real-time scaffolding can help students overcome these challenges. By reacting to students’ specific difficulties, operationalized at a fine grain size, a learning environment can provide the timely feedback students need so they do not fall behind and are set up for future STEM success.
 

Authors

• Ellie R. Segan, Rutgers University

• Amy E. Adair Midyett, Rutgers University

• Janice D Gobert, Rutgers University

• Michael Sao Pedro, Apprendis