Major Results of Undergraduate Linear Algebra

I have grown steadily unhappier with my linear algebra course. So this term, I am going to rewrite it in a radical way. I would like to make the experience a bit closer to my (successful) IBL Euclidean Geometry course. In EG, we use Euclid’s Elements as our “extant literature,” and then I have an IBL task sequence designed to go with it.

To make this project work, I need to write something analogous for Linear Algebra. For lack of a better title, I am calling this Elements of Linear Algebra. I would like to emulate the structure of Euclid’s work, too, where each book is just enough material to get to an important goal result. Then I can design tasks & homework for the students to do so that they work around the text and extend it. For context, here are the big results at the end of the first four books of Euclid’s Elements:

  1. The Pythagorean Theorem and its converse
  2. The equi-decomposability of rectilineal planar figures with squares
  3. If x lies outside a circle O, and a line through x meets O at points b and c, then the line xa is tangent to O if and only if (xa)^2 = (xb)(xc).
  4. the construction of a regular pentagon

Those are four beautiful theorems.

How should I choose the Beautiful Theorems for an undergraduate linear algebra course?

For context: (1) my course is is pre-rigorous, we don’t write “proofs.” Instead, we argue from examples to generality. (2) this course has been entirely stuck in Euclidean space, but it is not quite a matrix algebra course. (3) I have taught out of Strang for several years.

I want your feedback! Here is my current list of potential big results for the to explore. Each of these would be a chapter of my text.

  1. Descriptions of lines and planes in the plane and in 3-space using both standard forms and parametric forms.
  2. How to solve a system of equations, including how to decide if it is solvable, how big the solution set is, and a reasonably efficient algorithm for writing down the solution set.
  3. the invertible matrix theorem
  4. the fundamental theorem of linear algebra (borrowed from Strang — the four subspaces and how they fit together)
  5. How to find a ‘best approximate solution’ when there is not a given solution
  6. The spectral theorem for diagonalization of symmetric matrices

I would love to include the singular value decomposition, but I can barely get to #6 right now.

Note that these are just the big results! Each chapter would have other material in it to help us understand and contextualize the big result. For example, the invertible matrix theorem will necessitate determinants coming up in that chapter.

Anyway… What do you think?


An Important Part of Success

So far, this term seems to be off to a good start. I realize that one thing I am doing much better this term than last is really important. If I write it down, maybe I will always remember.

The attitude I present in the classroom is crucial to having a good first few weeks. In particular, this means projecting an overwhelmingly patient, cheery, positive, and helpful face, with as much energy as I can muster at any given moment.

This is especially important for my largish Math in Decision Making courses (55 students each). But it is also helping set a better tone for my linear algebra class.

Last term things were not so easy for me personally. It bled into my work. I did not consistently get across a positive attitude about students, their learning, and my involvement. This term, things are going better. I just have to keep it up for 13 more weeks.

Grabbing Attention with Student Inquiry

I have been pretty lazy the last two days, but I have been thinking about the upcoming semester “as a background process.” Right now, my main concern is that I will be teaching at least one section of Math in Decision Making, our liberal arts/quantitative reasoning course. The classes tend to be large (50-60 students), many of the students are either mathmatically bruised or have small motivation for the class.

Usually, I combat this by picking topics to study that I find interesting to let my enthusiasm shine through. Also, I try to sprinkle in a few high energy activity days. That energy comes from me, though. It doesn’t always infect the students. This term, I will try to get a better hook into the students by supplementing our intro days by asking them to ask questions about the subject the activity introduces. I don’t have a better plan, yet. It can be hard to get students to generate meaningful questions when confronted with new material. So, I will spend some time this week thinking about how to support them through this activity.

Tomorrow, Spring 2016 begins in earnest. I have a dangerously large to-do list forming for the week, but I refuse to worry about it until 9am.

Back to Writing

I haven’t written much in the last year. This is not particularly a problem—this blog is mostly an outlet for thoughts in progress.

But the lack of posting here is a symptom of my distance from deep thinking about my work in the last year. I have plenty of excuses, even some good ones. But I did find it useful to write here at one point. I found it even more useful when people decided to read and comment. I am still amazed that some of you have done this. Thank you.

Anyway. I am going to try regular reflective writing again.

Here I go.

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An Approach to Specifications Grading: Guest Post by John Ross

I have been involved in a lot of discussions about assessment strategies lately. There is a bit of a swell of young faculty who are rethinking their assessment strategies carefully. For some, this is a first serious step to rethinking their jobs as educators, and for others it is further step into the details of how to be effective.

Today we have a guest post by John Ross of Southwestern University. I met John at the Legacy of R.L. Moore meeting this summer, so I already know he is interested in effective teaching methods. This past weekend he mentioned lightly on twitter that he is using a new assessment setup. I wanted to hear the details, so I invited him to write about it. I am very pleased that he accepted my challenge.

My Version of Specs-Based Grading

by John Ross, Southwestern University
This semester I am running my calculus class using a specifications-based grading system. The decision to do this was made after discovering Robert Talbert’s blog and reading the many informative things he had to say about specs grading. If you’re unfamiliar with this style of grading, I’d recommend starting there (

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Change of assessment: SBG & Specs down

In the last few years, I have experimented with different types of assessment strategies. In particular, I used something that would be recognizable as Standards Based Grading, and something that would be recognizable as Specifications Grading.

But this term, for at least one course, I am abandoning both. My best course is Euclidean Geometry. Standards based Grading didn’t work because the learning goals are big, process-oriented things. This made it seem like Specs Grading would be a decent fit, especially because a typical assignment (“find a proof of this theorem, present it to the class, and write a paper about it”) is a complicated, professional activity. In the end, the quality might vary, but you either did it well enough, or you didn’t. But the explicit statements like “do seven of these to earn an A” broke my spring class. Students rushed to their desired grade and then stopped. So, I won’t do either of them in the canonical way for this course again.

I have learned a lot from trying SBG and Specs. I think that a knowledgeable person can still see evidence of each in my work. I now have a clear statement of what the (previously nebulous) learning goals are (SBG). And I have much better language to describe what acceptable work looks like (Specs).

But I won’t use explicit promises about grade conversions anymore. Instead, I have described what typical achievement looks like for each grade in looser terms.

If you want to peek, go here.

Eugenia Cheng and Knowledge, Belief, and Understanding

Recently I finished How to Bake π by Eugenia Cheng. It is nice to pick up a bit of mathematics IMG_20150816_173003129_HDRpopularization every now and again and look for new ways of explaining the essential nature of our subject with the uninitiated. Since Cheng is a category theorist, and I don’t know any category theory, this gave me some hope for a new perspective or two. These hopes were reinforced by the fact that she has chosen to use making dessert as the analogy to describe ideas!

The book is enjoyable, and Cheng does an admirable job of trying to explain what mathematics is and how one does it. Of course, she then moves on to focus on what the essential nature of category theory is (her answer: “the mathematics of mathematics”). I found a few nuggets of new analogies and viewpoints to share with my students who are struggling to write their first proofs and find their first conjectures.

The part I most enjoyed was in the last few pages. Cheng describes three different viewpoints on truth:

  1. Knowledge
  2. Belief
  3. Understanding

I really like that these are clearly differentiated. Most importantly, she describes the process of mathematical communication really well, and with a useful diagram. (I suppose that a useful diagram is to be expected from a category theorist.) Here is the diagram.


I think what is truly great here is that the picture is a narrow ravine (she points out that it is difficult and dangerous to jump across). And her accompanying description of mathematical communication is as a process by which you (1) somehow pack your understanding into a proof, and then (2) share that rigorous proof. This plays up the role of a axiomatic proof as the kind of thing which can be done without (or maybe just with a lot less) ambiguity, while recognizing that it might obscure the understanding a little bit. But it also highlights that there is a job for the reader, which is really an inverse to the job of the writer. The reader must (2′) confirm receipt of and agreement with the logical argument, but then (1′) unpack it to construct their own understanding of the ideas.

This is a model I can share with my new proof-writers to help them do their jobs better.