Mathematical modeling (MM) is considered a fundamental skill for students in STEM disciplines. Practicing the MM skill is often the most effective when students can engage in group discussion and collaborative problem-solving. However, due to unevenly distributed teachers and educational resources needed to monitor such group activities, students do not always receive equal opportunities for this practice. Excitingly, large language models (LLMs) have recently demonstrated strong capability in both modeling mathematical problems and simulating characters with different traits and properties. Drawing inspiration from the advancement of LLMs, in this work, we present MATHVC, the very first LLM-powered virtual classroom containing multiple LLM-simulated student characters, with whom a human student can practice their MM skill. To encourage each LLM character's behaviors to be aligned with their specified math-relevant properties (termed "characteristics alignment") and the overall conversational procedure to be close to an authentic student MM discussion (termed "conversational procedural alignment"), we proposed three innovations: integrating MM domain knowledge into the simulation, defining a symbolic schema as the ground for character simulation, and designing a meta planner at the platform level to drive the conversational procedure. Through experiments and ablation studies, we confirmed the effectiveness of our simulation approach and showed the promise for MATHVC to benefit real-life students in the future.

Learning how to create, test, and revise models is a central skill in scientific reasoning. We argue that qualitative modeling provides an appropriate level of representation for helping middle-school students learn to become modelers. We describe Vmodel, a system we have created that uses visual representations and that enables middle-school students to create qualitative models. Software coaches use simple analyses of model structure plus qualitative simulation to provide feedback and explanations. This system has been used in several studies in Chicago public school classrooms, using curricula developed in collaboration with teachers.

Learning how to create, test, and revise models is a central skill in scientific reasoning. We argue that qualitative modeling provides an appropriate level of representation for helping middle-school students learn to become modelers. We describe Vmodel, a system we have created that uses visual representations and that enables middle-school students to create qualitative models. We discuss the design of the visual representation language, how Vmodel works, and evidence from school studies that indicate it is successful in helping students.

Learning how to create, test, and revise models is a central skill in scientific reasoning. We argue that qualitative modeling provides an appropriate level of representation for helping middle-school students learn to become modelers. We describe Vmodel, a system we have created that uses visual representations and that enables middle-school students to create qualitative models. Software coaches use simple analyses of model structure plus qualitative simulation to provide feedback and explanations. This system has been used in several studies in Chicago public school classrooms, using curricula developed in collaboration with teachers. We discuss the design of the visual representation language, how Vmodel works, and evidence from school studies that indicate it is successful in helping students.