In an Inquiry-oriented math course, students are primarily engaged in mathematical activity, and math content is developed from students' activity. Researchers have identified four pillars of inquiry-based courses:
Students engage deeply with coherent and meaningful mathematical tasks.
Students collaboratively process mathematical ideas.
Instructors inquire into student thinking.
Instructors foster equity in their design and facilitation choices.
In a content course, students engage in inquiry activities in class, and the teacher uses student work to develop mathematical understanding.
Structure of a typical day
In a proof-based course, students are given a sequenced set of theorems, but are not shown proofs. They prove the theorems outside of class, and present their proofs for discussion in class.
Structure of a typical day
The websites below have extensive resources for implementing Inquiry-Based courses in college mathematics
See this page for Open Access curricula (often based on extensive research) for inquiry-based college math courses
Spencer Bagley: The four pilars of IBL and my teaching philosophy
Francis Su: Freedom through inquiry
Steven Strogatz: Teaching through inquiry: A beginner’s reflection (part 1) | part 2
David Bressoud: Reflections on a career in teaching
Dana Ernst: Soup to nuts: My approach to IBL
Robert Tabert: A turning point in my teaching
AMS blog: Active learning in mathematics: Personal reflections