There is a quiet crisis that happens in mathematics departments around the world. A student breezes through Calculus I, II, and III, mastering integrals, derivatives, and vector fields. They are, by all standard metrics, good at math. Then, they walk into their first upper-level proof-based course—Real Analysis or Abstract Algebra—and hit a wall.
They realize they have spent years learning to operate mathematical machinery, but they have never learned how the machine is built.
At MIT, 18.090: Introduction to Mathematical Reasoning (IMR) serves as the essential bridge over this gap. It is the course where the motto shifts from "find the answer" to "prove the answer exists." For students seeking extra quality in their mathematical education, 18.090 offers a rigorous, humbling, and ultimately empowering transformation. There is a quiet crisis that happens in
For MIT students, 18.090 is more than a prerequisite; it is an initiation. It marks the transition from being a consumer of mathematical facts to a producer of mathematical knowledge.
The "extra quality" of the Introduction to Mathematical Reasoning experience is that it doesn't just teach you math; it teaches you how to think. It strips away the comfort of plug-and-chug formulas and replaces it with the confidence that comes from constructing an ironclad argument. Accompanied by specific, actionable comments (not just a
In the end, 18.090 produces students who don't just accept the mathematical world as it is presented to them—they have the tools to question it, dissect it, and rebuild it from the ground up.
Having the resources is not enough. You must cultivate specific habits. Accompanied by specific
The Proof Linter is an in-browser, AI‑assisted tool that analyzes student-written proofs (in a structured natural language + symbolic notation) and provides line‑by‑line feedback on logical validity, clarity, and common reasoning errors — without giving away full solutions.
After submission, the tool assigns scores (1–4) in:
Accompanied by specific, actionable comments (not just a score).
“Proof Linter” – Interactive Proof Validation & Feedback Engine