- Link: LAFF at edX
- Professor: Robert A. van de Geijn
- Institution: The University of Texas at Austin
- Date: Summer 2015
- Platform: edX
Linear Algebra was the first real mathematics course I decided to take for my current project. While the importance of linear algebra cannot be overstated, it is not exactly an accessible subject. I already took a Linear Algebra course in college, but I did not learn much (as usual, sadly). From my memory it was just another frustrating class that I stood in only to fulfill my degree requirements.
How It Went Wrong at College
My Linear Algebra professor at college focused the entire course on the techniques of linear algebra. These techniques included simple matrix-matrix multiplication, Gauss-Jordan elimination, projection, orthogonality, fundamental spaces, linear least-square, and eigen-blahblah. If you have some experience with the subject (and you really remember it), you should find those topics familiar. So in a way, my professor did teach everything he was supposed to teach. Except he did not.
The techniques, while indeed important to master, do not tell the whole story of Linear Algebra. It is not about applications either. I appreciated that my professor made some effort to give a recitation each week (most courses in my college did not even have recitation). He and the teaching assistants gave several presentations about the cool applications of linear algebra, such as image compression in JPEG file. Still this information did not help the students, including me, at all. The one thing that my professor forgot to teach, or deliberately avoided, is the same thing that should have made students fall in love with linear algebra: the insight.
Why LAFF Works for Me (and For You)
Techniques make up only a small part of LAFF (Linear Algebra: Foundations to Frontiers). This should not be surprising. With the ubiquity of matrix libraries as good as MATLAB, we can use a computer to apply the techniques. What we should do instead is teaching the computer to solve problems for us. In addition to computation techniques, this goal requires a deep understanding of linear algebra.
Prof. Robert van der Geijn, who teaches LAFF, insists that insight can only be gained by doing mathematics. In this class, that means understanding proofs and proving theorems. In fact, Robert make all the answer keys available at the beginning of the course. A student who just wants a certificate (like the old me in college) could just enter all the answers and get a good-looking piece of paper without even knowing what Linear Algebra is. The reason for this approach is it makes absolutely no sense to use the techniques without understanding the proofs. Proofs are not created for a specific kind of creatures called mathematician. They are important for anyone who wants to learn mathematics the right way.
Admittedly I did not dare to read even one proof in the linear algebra class in college. I even did not remember the professor mentioned any proof, or whether he included proofs in the homework and exams. (He did not!) Sadly, it was and remains the standard teaching method of mathematics nowadays. I will definitely write more about the topic of mathematics teaching in near future.
Back to the course, it was the first time I tried to make proofs by myself, without any external motives like homework or exam. It may sound extremely nerdy, but math has changed me, inside out. I’m a changed man. Suddenly I find myself interested in reading and understanding proofs. My attitude towards Linear Algebra and Math in general turns a complete 360 degrees. Now I am eager to learn more. It’s crazy, but finally it seems I have found my favorite subject.
Studying math should be like this. It should be about thinking, exploring, proving, arguing and imagining. It should be a creative process, not a mechanical one like what is taught in school.
I want to thank Prof. Robert van der Geijn, his wife, and his assistants for an excellent course. Really they taught me to love math.
(Image Source: http://ulaff.net)