# DIP: Graphical Model Construction by System Decomposition: Increasing the Utility of Algebra Story Problem Solving

9/1/16-8/31/19

PIs: Kurt VanLehn, Jon Wetzel, Fabio Augusto Milner
Arizona State University
Award Details

This project studies a new genre of learning technology that may remove a notorious bottleneck in STEM education: mathematical model construction. These days, computers can solve complex mathematical problems, but humans must still define the problem for the computer, which is called constructing a model of a system. Many students can learn procedural skills, such as solving a quadratic equation, but constructing a model frustrates them because there is no procedure. This effectively stops their progress in math and blocks their entry to STEM professions. That may be why model construction is one of the few practices that appears in both math (CCSSM) and science (NGSS) standards. The key innovation for solving these problems is a new genre of learning technology based on two ideas. First, it emphasizes decomposing the given system description into subsystems. Second, although the final model is a set of algebraic equations, the model is first constructed as a node-link graph that shows which quantities are connected to which relationships. This notation is called TopoMath.

TopoMath builds on prior success with the Dragoon intelligent tutoring system, and represents a revision of that system to support a novel graphical representation to allow learners to recognize distinct problem-solving schemata. Stealth assessment using Bayesian Knowledge Tracing will allow feedback on student submitted models in response to word problems in modelling. When a model is represented as a TopoMath graph, it can usually be drawn such that distinct subsystems correspond to distinct subgraphs. This makes it easier for students to understand the relationship between the model and the system that is represents. Moreover, when constructing a model by decomposing a system into subsystems, blank areas in the TopoMath graph suggest which subsystems still need to be modeled. Students can learn model construction schemas by comparing and generalizing systems that have visually similar TopoMath models so that when constructing a model by decomposing a system into subsystems, if a schema matches a subsystem, then a whole section of the model can be filled in without further decomposition. These are just a few of the synergies of combining system decomposition and TopoMath’s graphical representation of mathematical models. This project will explore sequences of TopoMath learning activities with the goal of bringing students to model construction mastery with just 20 hours of instruction. The instruction will be developed in the context of remedial college math classes that are equivalent to high school algebra 2 classes. The instruction will include individual, small group and whole class activities using the TopoMath technology. Qualitative analysis of verbal protocols will be undertaken using the Knowledge-Learning-Instruction framework both for system evaluation, and to better understand the processes of learning in model construction that are supported by the system’s representations and scaffolds for modeling.