Motivation

Characterizations of algebra as foundational for success in advanced mathematics and STM are commonplace (Chen, 2013). Even though students are often engaged by algebraic notation in an experimental way, many students struggle with the notation, the conceptual understanding of notation, and the flexibility to turn to different strategies that may apply to an unknown problem (NMAP, 2008; Rittle-Johnson et al., 2015). Educational technologies have emerged for algebra and can support algebra learning on a larger scale, but most only reveal the final answers a student has generated and pay little to no attention to the processes a student unfolds to derive those answers.

Why Strategy Matters

Mathematical proficiency encompasses not only correct answers but also strategic flexibility, or the ability to choose and apply multiple approaches to problem solving in varied contexts (Baroody, 2003; Threlfall, 2009). Environments that include gamified learning exist for a variety of reasons and are less directed than conventional instructional strategies—for example Graspable Math and From Here to There (FH2T) could be considered more exploratory environments as compared to more conventional instruction. Research suggests students learn better when provided the opportunity to explore multiple solution paths, rather than being directed toward one "correct" solution path (Clark et al., 2016; Wouters et al., 2017).

There is still substantial work to be done in the area of mathematics assessment, and so far these assessment platforms only document the final answer as well as the learning journey through the processes a student goes through to derive the answer. This limits our ability to identify productive struggle, misconceptions, or creative thinking. Finding or constructing definitions for productive struggle, misconceptions, or creative thinking are all necessary components of deeper mathematical learning.

The Need for Process-Level Data

Recent research (e.g., Chan et al., 2022; Pradhan et al., 2024) has demonstrated that student behaviours such as:

• Extended pauses prior to a solution

• Frequent reattempts/resets

• Non-linear, exploratory paths

The above are tied to enhanced conceptual understanding and increased procedural flexibility.

Unfortunately, very few systems have the capability to collect and think with this fine-grained, process-level data. Most systems collect correctness-based metrics that fail to deal in how students think and learn.

Creating a New Pathway with MathFlowLens

To help support this need for process-level data we created MathFlowLens (MFL)—a new tool that represents problem-solving pathways using graph-based algorithms (e.g., Dijkstra’s, A*) classifying their pathways, as:

1. Optimal

2. Sub-Optimal

3. Incomplete

4. Dead-end

We visualize these classifications using interactive network graphs that make student thinking visible and manipulable for both researchers and educators.

By emphasizing the variation in student strategies or pathways, rather than their outcomes, we can use MathFlowLens to:

• Fall back into how students approach algebraic problems

• Connect strategies/behaviours with learning gain

• Support the development of teacher-facing dashboards to provide targeted, real-time feedback

This work is motivated by the idea that understanding learning is key to creating adaptive, equitable, and engaging learning environments, not just understanding the outcome.