10 Minutes of Thoughts on My SBG Linear Algebra Class

I’ve been meaning to write a post about my standards-based Linear Algebra course for months, but the hectic schedule of the semester has kept me away from this task until now. Today was my last “content” day of Linear Algebra — we have two more classes remaining, one for a test day and another for a re-assessment day. This seemed like a good time for me to take ten minutes to gather some thoughts about how the semester went.

Standards List for Linear Algebrahttps://www.overleaf.com/read/kycvnvzdvksw  (Availablle on Overleaf, which is awesome and I can’t recommend enough)

What Went Well: We ended up having 20 standards this semester. This is a little more than one per week (our semester has 16 instruction weeks). Overall, I think this was a good number of standards to have, and I’m happy with how they turned out. I tried to group them again by “Big Questions” to have a reference frame of what it is we’re trying to do in the course. Oddly, we tackled “Big Question 5” last (on inner product spaces), but I kept it numbered like that because of the textbook we are using. My basic idea was to come up with a Big Question for each chapter. For some stuff, this worked well (e.g., eigen-everything) but for other stuff we didn’t cover a whole lot (e.g., determinants).

I think I’m doing a better job of the sales-pitch aspect of a standards-based course. Many of my students expressed to me at various times that they really appreciated the ability to improve on past performance and that they were under less stress than in a traditional class. In a recent class meeting, a student wasn’t happy with the performance on the last quiz, and exclaimed, “Oh, thank goodness we have an exam on this soon!!!” [I asked the student for permission to share this quote.] I think this is one of the best things about my SBG courses — students really want to take an exam just to show what they know, whether that means showing mastery of current material, or showing mastery of material they struggled with earlier in the course.

My SBG approach definitely has some pros and also some cons, but the way it has shaped my interactions with students has always been a huge positive. Even with the sticky details that need to be cleaned up from this semester, I can’t imagine going back to a traditional grading scheme.

Room for Improvement: This semester was a little odd because we lost several days because of weather. Tropical Storm Hermine hit us, and we lost almost a week because of Hurricane Matthew. The re-shuffling of the academic calendar created a speed-bump that I never really recovered from. I hope next semester our calendar runs much more smoothly.

In particular, I am wondering about how I can improve in three areas. First, I want to expose my students to more applications of the material we are learning. I felt rushed all semester (related to shuffling of course calendar, maybe?) and so I didn’t ever feel like I had time to fit in cool applications, or videos on where people use this stuff “in the real world,” etc. A colleague teaching the same course required students to do group projects on applications of linear algebra & I believe the students presented them to the class at the end of the semester. This seems like a great idea, but I’m always nervous about assigning group projects because I remember how much I hated doing them as a student. It’s something I should consider more.

Second, all of my course standards are weighted equally. This has served me well in Calculus II and in other courses. But in Linear Algebra it became a little tricky, because part of what I was aiming to do was to have my students attempt to write proofs of mathematical statements. (The only mathematical background required for entry into my course is Calculus I, and that is for “mathematical maturity” as opposed to content reasons.) So some of my students were concurrently taking our “Introduction to Proofs” course, but others weren’t taking this course and won’t need it for their major. In general, my idea was to ask them to prove elementary results they had already seen in class. The problem I encountered is that a “write a proof” standard is really tough. How do I let them have multiple attempts? Is it okay if they end up never being able to prove stuff about, say, matrix inverses, but they can prove stuff about, say, subspaces of a vector space?

One idea I’ve had is to have the students keep a “Proof Portfolio” and grade it as either “complete” or “not” at the end of the semester. I’m sure there’s some specs-based approach I could implement for this, but I haven’t worked out what it would look like yet.

Third, trying to put together all my course materials on the fly is hard. All of the time, I was working on: Plans for class, writing exams, writing quiz questions, writing reassessment questions, putting together online homework, meeting with students for several hours a week outside of class, updating the list of standards regularly… I would admonish my summer-month self that I should do more of this “in my free time” before the term begins so I’m not under such a time crunch during the semester. But I am not great at this because I like building a course as it goes, as I see how the students are responding, as I see how the pace of the course unfolds, etc. Having to get all this done ahead of time would probably help me out a lot, but it’s tough to do. Thankfully some of my stuff from this semester can be re-used when I teach Linear Algebra next semester.

My ten minutes are done so I have to move on to the next task on my queue! I hope to add more later.

Standards-based Linear Algebra

This semester I’m teaching our introductory linear algebra course. As I did for Calculus II, I’ve implemented a standards-based assessment system. I’ve taken our course content and split it into “standards”, or little pieces of mathematics that I want my students to master. These standards are grouped together by what I call “Big Questions”. Here is what we’ve covered so far this semester:

  • Big Question #1: What are the tools for solving systems of linear equations?
    • 1.1: I can solve systems of linear equations using row operations. I can use Gaussian elimination with back-substitution to solve systems of linear equations. I can use Gauss-Jordan elimination to solve systems of linear equations.
    • 1.2: I can characterize the solutions to systems of linear equations using appropriate notation and vocabulary.
    • 1.3: I can use matrix inverses to solve systems of linear equations.
    • 1.4: I can find and use an LU-factorization of a matrix to solve a system of linear equations.
  • Big Question #2: What is the fundamental structure of the algebra of matrices?
    • 2.1: I can perform algebraic operations with matrices, including addition, subtraction, scalar multiplication, and matrix multiplication. I can compute the transpose of matrices.
    • 2.2: I can find the inverse of matrices using Gaussian elimination. I can find the inverse of matrices using a product of elementary matrices.
    • 2.3: I can demonstrate theoretical connections about properties in the algebra of matrices.
  • Big Question #3: How can we characterize invertible matrices?
    • 3.1: I can find determinants using cofactor expansion. I can find determinants using row or column operations.
    • 3.2: I can demonstrate theoretical connections between statements equivalent to “the matrix A is invertible.”
    • 3.3: I can demonstrate theoretical connections between matrix equations, vector equations, and systems of linear equations, and their properties and solutions.
  • Big Question #4: What are vector spaces & how can we describe them?
    • 4.1: I can prove whether an algebraic structure is a vector space (or not) using the vector space axioms. I can prove whether or not a subset W of a vector space V forms a subspace. I can determine and characterize subspaces of $\mathbb{R}^n$.
    • 4.2: I can write a proof showing whether a subset of vectors from a vector space forms a spanning set for the vector space (or not). I can write a proof to show whether a subset of vectors from a vector space is linearly independent (or not). I can determine whether a set of vectors forms a basis for a vector space. I can find the dimension of a vector space.
    • 4.3: I can find a basis for the row space, the column space, or the null space of a matrix. I can determine the rank and nullity of a matrix. Given a consistent system Ax=b, I can describe the general solution in the form x=xp+xh
    • 4.4: I can demonstrate knowledge of the theory of vector spaces by proving elementary results and theorems.

The remaining Big Questions are:

  • Big Question #5: What are inner product spaces and how can we describe them?
  • Big Question #6: What kinds of functions map one vector space into another while preserving vector space operations?
  • Big Question #7: What are eigenvalues and why are they useful?

Our first exam was last week, so today has been Re-Assessment Central in my office. I’ll hand back our exams tomorrow and I’m hoping to talk with my students more about standards-based grading and how they can improve their standing in the course.