Learning design · Intelligent tutoring

Algebra Equation Solving Tutor

Connecting symbolic equation solving with visual explanations so learners can reason about each step.

My role: Co-design, cognitive task analysis, interface development, and tutor rules
Audience: Middle-school algebra learners
Context: Carnegie Mellon University · Collaboration with Tomohiro Nagashima

The learning problem

Learners can carry out algebraic procedures without understanding why those procedures preserve an equation’s meaning. We investigated how visual representations and self-explanation could support conceptual understanding alongside procedural knowledge.

Cognitive task analysis revealed that some learners treated the variable in each line as a different value or relied on guess-and-check. This shifted the design toward connecting each symbolic step to an interpretable visual representation.

My role and constraints

I co-designed the tutor with Tomohiro Nagashima and contributed cognitive task analysis, prototyping, learner testing, and implementation. I developed interface behavior and rules for equation solving and diagram generation.

The experience needed to support multiple equation types and meaningful solution paths. Diagrams had to help learners reason about a transformation while remaining connected to the symbolic work.

Design decisions

Investigate the reasoning behind errors

Learners reviewed erroneous worked examples, identified the error, and explained a correction. These tasks exposed misconceptions about variables, equality, and the relationship between consecutive steps.

Compare three ways to use diagrams

We prototyped a worked-example tutor, a tutor that mirrored symbolic steps as diagrams, and a tutor that asked learners to explain a future step with a diagram. Comparing these approaches helped us examine when the representation should appear and what learners should do with it.

Keep the changing representation visible

Learner testing showed that participants sometimes missed diagram transformations or did not look at the diagrams. Revisions retained transformation history and used feedback to direct attention to the relevant representation.

Model reasoning beyond one scripted sequence

We developed the future-step approach into a rule-based tutor. Rules model correct and incorrect algebraic steps, and the interface generates tape diagrams from the equation parameters.

Inside the experience

An algebra tutor shows an equation, an incorrect transformation with feedback, and the corrected solution. — The diagram-mirroring prototype links symbolic steps to visual consequences. Feedback calls attention to the unequal operation on the two sides. View full-size image

Design notes map learner-testing observations to tutor revisions. — The original iteration notes connect observations to revisions. Rows marked “not yet” record ideas under consideration, not completed features. View full-size image

The live tutor provides the interactive experience. Open it separately to inspect the equation-solving and diagram tasks at a comfortable size.

Open the algebra tutor Inspect tutor rules and code

Evaluation and learning

Early learner testing led to changes in step presentation, diagram history, and feedback. A subsequent randomized study at two schools found greater formal-strategy use and transfer, and less time per symbolic step.

The design work showed why adding a diagram is only part of the solution. The task and feedback also need to help learners attend to the diagram and connect it to the mathematical decision they are making.

Design takeaway: Visual support is most useful when the learner has a reason to use it. Research and iteration helped us decide what to represent, when to show it, and how to connect it to action.