TopoMath: Rapidly Learning How to Construct Mathematical Models of Static Systems

• In Event: 63.025 - Supporting Modeling Epistemologies in the Science Classroom

Abstract

Objectives. The goal is to develop an instructional treatment that teaches students to construct complex algebraic models of static systems. The key scaffolding is provided by an intelligent tutoring system (TopoMath) that uses a novel node-link representation instead of equations.

Theoretical Framing. Having students construct mathematical models of natural and engineered systems is thought to be more effective than exploring models built by others, but teaching students how to construct models takes time away from learning the science. In earlier work, our Dragoon tutoring system reduced the time required to learn how to construct models by scaffolding and fading a node-link model notation, an explicit problem solving strategy, explicit schemas, feedback and the linguistic complexity of the system descriptions. Dragoon students’ biology learning gains were higher than the baseline students’ gains, and yet both treatments took the same amount of time (Authors, 2016). However, applying Dragoon to teach electronic circuits to technicians revealed that its notation was confusing when applied to static systems, such as a battery-resistor circuit. Analysis of errors in a university modelling class confirmed that the static parts of systems were often not modelled properly (Authors, 2017). Dragoon uses a node-link notation similar to those of Model-It, Stella, Simulink and other dynamic system modeling tools. These notations are founded on the idea of numbers flowing along links, but mathematical models of static systems are founded on the idea of satisfying constraints. The TopoMath system replaces the traditional notation with a novel node-link representation based on constraint satisfaction.

Methods/Data. A TopoMath curriculum is being developed and evaluated with students learning high school algebra. We are using a design-based research method, wherein the instructors, students and researchers all work together to both enact and improve the instruction. Data include notes taken during and immediately after each class meeting, and log data from TopoMath.

Results and Scholarly Significance. The curriculum has been enacted twice with students who failed to complete a College Algebra class and were enrolled in a remedial class. Our observations so far include: (1) Existing schema-based instruction assumes that one schema application suffices to solve the whole problem. That assumption doesn’t scale. When applied multiple times per problem, two abstract schemas suffice for most problems. (2) Student can learn the two abstract schemas by analogy to concrete ones. (3) The means-end strategy, which is supposed to dominate novice model construction, is not what our students do nor is it easy to teach them to do it. Instead, they notice immediately a few obvious schema applications, then guess to find others. (4) When the system description is bulleted sentences such that each sentence matches one schema, students can construct impressively complicated models. This powerful scaffolding may be novel.

Authors

• Kurt VanLehn, Arizona State University

• Chandrani Banerjee

• Sachin Grover

• Fabio Augusto Milner, Arizona State University

• Jon Wetzel