I was first introduced to the idea of organizing a course around a project or a driving question in my class on student motivation and course design. It may sound like a weird combination, but it was about dealing with grade anxiety and student engagement through course design. Anyway, the idea put forward was that one way to organize a course was to have a central project or question that the course would return to over and over and that would drive exploration of topics over the semester. It provides context and motivation for the material to the students.
When I heard about project-based learning (PBL), I thought they were talking about the same thing. However, PBL apparently does not require the entire course to be structured around one project or one question, but use projects to drive whatever is the current topic. Which makes sense because I’ve been wracking my brains for single projects or questions that would encapsulate all/most of calculus or linear algebra. Oh, there are plenty of applications, but it’s very hard to have one that touches upon all of the key concepts. These courses are supposed to be toolboxes so that, when the need arises later on, the student can pick the right tool out of the box and fix the problem. So to apply PBL to such courses, projects will probably have smaller, more localized scope.
While researching possible projects for courses, I came across this site with a number of linear algebra projects. They are on a variety of applications of linear algebra, and the projects show a pretty good understanding of various linear algebra concepts. However, upon further analysis, I believe most teachers would call these research papers rather than project. See the syllabus. Note: there is also a conflation of term when it comes to the word, “research.” In graduate school, research refers to original thought, experiments, or analysis which expand the boundaries of human knowledge. In high school, research usually means read a bunch of sources and sum it up in a paper. which in graduate school would be more of a literature review. These papers are in the latter category, but at least it does require the students to understand a particular application in sufficient degree to explain them to their peers.
This bring up the question exactly what is a project. To my mind, a project starts with a question or a problem. In answering this question, the students are the ones who determine what they need to know and how to use them to address the question, though the teacher acts as a guide. The students are the ones who decide what to do and how to do it; this not a sequence of activities laid out by the teacher that the students follow. This provides room for students to explore and be creative.
I’m still trying to think of projects suitable for a college math course. For now, I’m okay with a literature review / paper presentation to break the mold of lecture/homework/midterm. Also, I’ve started compiling a list of historical/interesting math problems as material to discuss in class, as opposed to generic problems.