Research Findings

The findings from both studies give valuable insights into how students tackle algebraic problems and how teachers can receive improved support from data-oriented tools.

Findings from Study 1: Mechanisms of Mathematical Performance

1. Much of the students' performance was influenced by the perceived design of the problems. The uniform spacing of the problem pairs and the use of color as an initial cue prompted differences in the types of strategies they used, as well as their efficiency and accuracy.

2. Logged behaviors permitted us to identify productive strategies as well as unproductive strategies; our analysis also demonstrated that students who paused to reflect, reset, and/or reconsidered their original approach to a problem tended to have stronger and deeper learning outcomes.

3. Frequent pauses and purposeful resets are not indications of failure; rather, they are often indicators of conceptual engagement and productive struggles, components of deep mathematical learning.

Findings from Study 2: Analyzing Real-Time Strategic Thinking with AI

1. Through the use of MathFlowLens (MFL), we learned that students who explored potentially dead-end paths had higher conceptual understanding and procedural flexibility than students who followed only optimal paths. 

2. Students who followed only optimal paths often lacked variability in terms of strategic decision-making, suggesting they had a more superficial or way to thinking of strategies for problem-solving.

3. By rebuilding the entire problem-solving pathway using graph-based classification methods, we were able to bring to light pathways, including pauses, detours, and false starts, that standard correct/incorrect measures have failed to account for.

Cross-study findings: Implications for instruction

• Process data > correctness: By analyzing students' strategies from moment to moment, this provides richer and more usable data than correctness at the end of a problem.

• Mathematical flexibility can be fostered: Students benefit from looking around in environments that do not penalize them for exploring. Supporting students while they are flexibly, creatively engaged with something leads to better procedural and conceptual learning.

• Dashboards make teachers more responsive: Teachers who had access to the MathFlowLens, together with the integrated dashboard, demonstrated improved noticing, were able to address misconceptions immediately during the lesson, and provided more personalized feedback. Teachers even expressed that the dashboard provided them with a deeper understanding of the thinking students had done, not just whether or not they were correct.