By Leann Ferguson

A highlight of teaching for me is teaching others how to teach. I regularly teach a “Teaching Practicum'' class, during which the “student teachers” teach three lessons to a live class of Calc 1, Calc 2, or Calc 3 students, and then do an extensive reflection on their preparation and presentation of the lesson. Their self-reflections mention feelings of nervousness, satisfaction, disappointment, surprise, frustration, and realizations of just how much work, effort, thought, and emotion goes into teaching. They had their eyes opened by the glimpse behind the scenes of teaching they experienced first-hand as they started to realize, understand, and internalize the effort required to be an effective instructor. In short, these lessons were ripe with intentional and unintentional learning opportunities for both the student teachers and the students.

I wanted to share some of the takeaways from my observations of these lessons. I hope you find at least one comment useful and/or find something that sparks some productive thought towards developing your own teaching craft.

Write down anything and everything you want your students to walk away from class with.  During a typical lesson, only about 40% of what happens during class time is written down; the rest is verbal. If all a student does is write down what you wrote down, then they will miss 60% of the lesson. That's not good for studying days later when many other things have filled the student’s head. 

For our freshmen, copying down everything the instructor writes (and nothing more) is what they think taking (good) notes means. As we teach them otherwise and show them how to take notes, we need to bridge the gap and write anything and everything down that we want them to walk away with.

Think about the entire process of seat/board work. This process starts with choosing problems:  the problems we do on the board to demonstrate and explain the content, the problems we ask the students to do to discover or practice the content, and the problems we ask the students to do to review previous (and related) content. Once we have the problems in mind, we must decide if the students work individually or as a group, at their seats or at a board. Most importantly, we must decide how they get feedback on their work.

My go-to is having the students work at the board in groups of 3 or 4, and then one of the students (preferably not the one that wrote on the board) explain the group’s solution. In addition to the benefits of students explaining something to someone else, I can publicly demonstrate and reinforce the communication ideal of a written solution. Other logistics that we must consider are what we do after the students finish seat/board work, how to challenge those that blow through the content and need more, and what we do when (not if) we run out of time to allow the students to complete seat/board work. 

With an eye toward what the students will take away from the lesson, plan for what your board(s) will look like (especially what you will erase and not erase) and whether you want to show the use of a computational or graphical program (like R, MATLAB, or CalcPlot3D). In my classroom, 3 of the 4 walls are covered with whiteboards (the 4th wall is a chalkboard).  I use the left “side-board” to write out the objectives for the lesson, as well as the review problem that the students complete immediately upon entering the classroom. I write a student’s name next to the “side-board problem” and that student completes the problem and explains it to the rest of the class. I purposefully pick a problem from the previous lesson that has components I will use throughout the current lesson. This means the students’ work and any commentary or work I added will remain on the sideboard throughout the lesson so I can constantly refer to it as needed.

Think about how you ask questions of the students … and give the students time to respond.  How do you ask them if they understand or are following the lesson? In my experience, asking the students to raise their hands if they are completely lost or solely asking if everyone understands or if they “got it” has utterly failed to elicit useful and productive responses. I must prefer something along the lines of "I see some confused looks. What can I clarify or explain differently?" 

Soliciting student responses using questions like, "What questions can I answer for you?" asks students to process what you’re asking about, to formulate their response, and then to gather the courage to verbalize their response. All of this takes time. And more often than not, we don’t give the students the time they need … give them at least five seconds. Rather than awkwardly staring at them while waiting for a response, I recommend taking a sip of water.

Plan for the use of technology and/or visual aids. A surprising amount of class time can be "lost" to issues with technology (such as projecting a document, website, or computational/graphical program) or to setting up visual aids. Having these prepped and ready the moment you want them will maintain the flow/rhythm of your conversation and avoid unnecessary breaks in the learning process. This might mean providing scaffolded copies of R or MATLAB code or a weblink to a CalcPlot3D visualization before or after class as appropriate. If you want the students to use the computational/graphical program in real time, you will need to allow time for the students to open the program and then type — I often have a student at the keyboard instead of me to demonstrate or walk through the computation or visual.

How do you connect with the different perspectives and different interests in the classroom? One student may really understand a physics example, another student a space example, and still another student a weather example. What choice do you make for the motivation of a topic? For an example problem?

This is also an excellent place to incorporate names, cultures, or histories that really help the students “see” themselves in the math content. For example, when I needed to label some problems in a Number Theory class, I chose to use the last names of contemporary Number Theorists like Watkins, Radziwill, Soundararajan, Erdős, and Zhu. For another example, I made up a scenario that was loosely based on the life of Dr. Euphemia Haynes (the first African American woman to earn a PhD in mathematics) as the context for a word problem.

And finally, take some time to think back over your student "career" and identify a really good instructor and a not-so-good instructor. Why do you remember them? What made them good or not so good? When you teach or interact with students, are there things from the good instructor that you can lean towards and incorporate into your interactions? What about things you should avoid doing? 

Whatever you incorporate into your interactions (from this trip down memory lane, from a list of Classroom Strategies, or from what you are observing real-time in your classes), please go with what works with your personality and your strengths. We quickly become a "not-so-good" teacher if we try to cram ourselves into a mold that isn't us. Your lessons and interactions will be much more genuine and effective if you let your personality shine through and be yourself.

Author’s Note:  The views expressed in this article are those of the author and do not necessarily reflect the official policy or position of the United States Air Force Academy, the Air Force, the Department of Defense, or the U.S. Government.

Lieutenant Colonel Ferguson is an Assistant Professor with the Department of Mathematical Sciences at the U.S. Air Force Academy.  After graduating from Colorado State University, she commissioned into the U.S. Air Force and has served in various technical and training assignments across several commands. In 2012, she received her PhD from Indiana University, Bloomington for her work in Curriculum & Instruction, specifically Mathematics Education at the post-secondary level.