First Day

Yesterday was the first day of classes, and things went fairly well. The problems with the classroom computer had been straightened out, and I was able to use both Mathematica and Geometer's Sketchpad in my lectures.

I don't think I did a very good job teaching DiffEq this summer, and I am determined to use the lessons learned this semester. The first day of class is important, and establishing some sort of rapport with the students is key. I think I did a decent job of that; the students were, at least intermittently, responsive to the questions I tossed at them, and most of the responses were more-or-less correct. We'll see; I'll be giving quizzes every Friday (except for midterm days and the last day of class), and that should give me a clue as to how well they're following.

Getting ready for the geometry class, I couldn't find the textbook - it turned out I'd left it at home after last Fall Semester - but I was able to put together some GSP slides. When I got to class, I briefly borrowed the text from one of the students and reminded myself of the topics that needed to be hit. Things seemed to go well, but we'll see.

Today is the first session of Linear Algebra I. The beginning is mostly a recap of systems of simultaneous linear equations, stuff they saw in College Algebra; making it interesting is a bit of a challenge. Again, we'll see what happens.

:cracks knuckles:

Second Day

Linear algebra is, for me, an interesting course to teach. One of my fundamental maxims of mathematics is this: "Algebra yields information. Geometry yields understanding." (I sometimes call it "Stoutfellow's First Law".) Linear algebra is a hybrid field, which can be viewed algebraically or geometrically, and I always push my students to be prepared to view it either way.

Today, we discussed systems of linear equations and the method of Gaussian elimination. GE is a procedure for changing a system of equations without changing the solutions, pushing it towards a form where the solutions are easy to compute. To illustrate how it works, I wrote down a pair of equations, say, x + y = 3, -x + 4y = 5. (Those aren't the equations I used, but they'll do.) Then I drew coordinate axes and the lines corresponding to those equations: two lines, intersecting in a single point, and that point's coordinates are the unique solution to the equations. Then I began working the transitions.

"The first operation of GE is to swap the equations - put them in a different order. What does that do to the geometry?"
"Nothing."
"Right - the lines get relabeled, but nothing else changes."
"The second operation is to multiply one of the equations, say equation two, by a nonzero constant. What happens to the geometry?"
(Some fumbling, a student talking to himself): "Nothing."
"Right; if you multiply an equation by a nonzero constant, its solutions don't change."
"The third operation is to add a multiple of one equation to another." (I do an example, and draw the new line.) "Line two rotates around the intersection point. If I add different multiples of equation one, line two rotates to different places. In particular, making the right choice, I can rotate it to be horizontal. What does that mean algebraically?"
(Hem, haw.) "The variable x no longer appears in that equation."
"Yes!" (I then point out the connection between this and the algebraic concept of "row echelon form". With many gestures, I try to display what happens if you're working with equations in three variables instead of two.)

Algebra yields information. Geometry yields understanding. I hope I can drive that through to my students.