SBIR Phase I: Creating Immersive Courseware for Calculus Pedagogy

The broader/commercial impact of this Small Business Innovation Research (SBIR) Phase I project is to modernize how calculus is taught and learned by addressing a growing mismatch between instructional practice and real-world problem solving. Each year, approximately 800,000 students enroll in calculus as foundational preparation for careers in science, technology, engineering, and mathematics, defined here as technical fields that rely on analytical reasoning rather than manual computation. However, traditional calculus instruction and assessment continue to emphasize low-level calculations that are rarely performed by professionals, often obscuring conceptual understanding and discouraging persistence. This project develops immersive learning exercises that allow students to make real-time problem-solving decisions while the software manages routine calculations. By shifting the focus from arithmetic accuracy to strategic reasoning, the approach enables students to practice more efficiently and build deeper understanding. Requiring continuous decision input also reduces reliance on readily available online solutions, preserving authentic learning experiences. The innovation advances technological understanding by demonstrating how interactive digital environments can better align instruction with contemporary analytical practice. The initial market includes secondary and postsecondary calculus programs seeking improved learning outcomes. A licensing-based distribution model is anticipated. By year three, the technology is projected to impact tens of thousands of students nationwide, with outcomes measured through adoption rates and demonstrated improvements in problem-solving performance. This Small Business Innovation Research (SBIR) Phase I project investigates a technically demanding approach to advancing calculus pedagogy through decision-centered digital learning environments grounded in learning engineering principles. The central technical challenge is whether expert-level calculus reasoning can be formally represented as executable decision structures that preserve student agency while enabling real-time instructional response. The research objective is to determine if immersive exercises can integrate instruction, practice, and assessment by inferring latent problem-solving states from incomplete and noisy student decision data. The proposed research constructs internally developed decision frameworks for individual calculus problems that encode multiple valid and invalid reasoning pathways under conditions of uncertainty. Students interact with these exercises by making continuous decisions in response to contextual prompts, with all actions, omissions, and timing information recorded as high-resolution behavioral traces. These traces are used to iteratively validate and refine the decision frameworks through controlled student testing. As data volume increases, machine learning methods are applied to model strategy selection, detect shifts in conceptual understanding, and adapt instructional responses at the individual level. This project will demonstrate that formally structured, data-driven decision environments can support individualized calculus learning while generating actionable diagnostic insight, establishing feasibility for a proprietary, technically scalable instructional system. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.