Kathleen Melhuish & Kristen Lew
Texas State University

“[Functions] are completely different, which is what makes this course so challenging.” – Abstract Algebra Student

Functions are hard for students, even students in abstract algebra courses. Even if students have seen the definition and worked with examples of real functions throughout high school and college, their understanding might be stretched to a breaking point when it comes to ideas like homomorphisms on groups or rings. Fundamentally, we might know students don’t understand functions, but the extent to which they don’t understand functions goes deeper than we might think. In this blogpost, we will share some insights from our projects on key places where students’ idea of function can be detrimental to learning concepts of abstract algebra, and what we as instructors might be able to do about this.

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By: Steven Klee, Seattle University

After my day-to-day interactions with students, one of my favorite things about teaching is talking with other teachers.  There is no shortage of amazing teachers who are working hard to make their classes better and improve student learning.  Likewise, there are plenty of opportunities to find inspiration in our colleagues’ work, ranging from attending talks at conferences to simply getting coffee with coworkers to talk about how our classes are going.

A few years ago, I realized that the proportion of inspiring ideas that turned into measurable change in my classroom was essentially zero.  As I thought more about this, I realized that I was the biggest hurdle to this change.  There was a little voice in the back of my head with a constant and emphatic message: No. I can’t do that, and here are fifteen reasons why.

I know I’m not the only one who hears this voice.  Of course, the reason we have these thoughts is that they are often true.  No two people experience teaching in the same way. We have different personalities, different styles, and allow for organized chaos in different ways.  As a community, it is easy for us to despair in the challenges we face in our teaching.

Joan Baez said, “Action is the antidote to despair.” At the end of the day we are all mathematicians and we have been trained in solving problems.  To be apathetic in the face of the challenges put before us is antithetical to our training as problem solvers. And teaching, particularly teaching well, should be viewed as a problem that desperately needs to be solved.  Like many real-world problems, the problem (“What does it mean to teach well?”) is not clearly defined.  The data is messy. There is not one single correct answer.

In the rest of this post, I would like to discuss some methods for moving beyond the little voice that says “no” and changing your teaching without reinventing the proverbial wheel.  And, as with many real-world problems, I will not answer the question at hand (“What does it mean to teach well?”) and instead I will address a different question – How do I teach better?

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