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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By: Natalie Hobson, Sonoma State University

If you give calculus students graphs, they are going to draw tangent lines. As instructors we often encourage students to rely on tangent lines so heavily that discussions about rates of change become lessons about sliding lines along graphs, rather than about understanding the relationships that these graphs represent in the context of a given problem.

So how do we help students develop a deeper understanding of these relationships? Let’s consider an exercise that you might give your own students during a lesson on tangent lines and rates of change.

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[This contribution was originally posted on April 15, 2019.]

By Benjamin Braun, University of Kentucky

The phrase “mathematically mature” is frequently used by mathematics faculty to describe students who have achieved a certain combination of technical skills, habits of investigation, persistence, and conceptual understanding. This is often used both with a positive connotation (“she is very mathematically mature”) and to describe struggling students (“he just isn’t all that mathematically mature”). While the idea that students proceed through an intellectual and personal maturation over time is important, the phrase “mathematically mature” itself is both vague and imprecise. As a result, the depth of our conversations about student learning and habits is often not what it could be, and as mathematicians we can do better — we are experts at crafting careful definitions!

In this post, I argue that we should replace this colloquial phrase with more precise and careful definitions of mathematical maturity and proficiency. I also provide suggestions for how departments can use these to have more effective and productive discussions about student learning.

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