This post is inspired by an article by Karen Marrongelle and Chris Rasmussen , in which they discuss the false dichotomy between all lecture and all student discovery as the two exclusive teaching strategies available for mathematics teachers. I’ve noticed that many discussions among postsecondary mathematics teachers lead to a debate of the merits of these two classroom teaching strategies, with the result that interesting teaching practices are left undiscussed. Below I describe three key teaching practices that I’ve learned about and used over the past several years that fit between and beyond these extremes. I’ve observed that when I use these practices, students are generally more engaged in the course, e.g. attending office hours, asking questions in class, forming study groups, etc. Though they appear simple, using these practices successfully has required perseverance and effort on my part, and a willingness to regularly revise their implementation.
Use student questions as discussion prompts
When a student asks a question during class, I’ve found that it often reflects an underlying misunderstanding held by a sizeable subset of the class. While it is common for student questions to be answered by the instructor, it can be helpful to provide students with a few minutes of class time to come up with an answer on their own in small groups. While this isn’t an appropriate response for all student questions, such as situations where a negative sign in a computation is overlooked, I’ve found that using student questions as discussion prompts is typically more effective than my answering questions directly. One of the best aspects of this technique is that it either completely resolves the question or else prepares students to seriously think about the explanation that I provide after the discussion time. I’ve found that students are much more attentive listening to my answers if I’ve given them a couple minutes to focus on the problem themselves before I start talking.
For example, when I was teaching a calculus class recently, a student asked a question about computing a limit that required multiplying the function being considered by cos(x)/cos(x). I took an informal poll to see how many students were confused by this problem, and over half of the class was stuck. Instead of telling them to multiply by the appropriate quotient of cosines, I had the students talk with each other for two minutes to share their ideas. Each of the groups had at least one student who knew what to do, and because of this my role in the classroom was changed from being an instructor to being a guide, leading the students to successful peer learning.
A frequent question about this technique (and the next) is how to balance allocating class time for students to collaborate with covering content. In small courses where I am the only instructor, I can rearrange the course schedule as needed, so this isn’t an issue. When I teach large-lecture calculus courses, the online homework, lecture schedule, and examination dates are determined by the course coordinator, so I don’t have much flexibility to rearrange content. In this setting, I generally choose to cover fewer examples in more depth, whereas when I first started teaching I chose to cover many examples with less discussion.
Collect multiple student ideas for approaching a problem
A related technique I’ve used is to gather suggestions from students on how to start examples. My goal in this is not to have the students simply take the lead when solving problems, but instead to explicitly discuss as many entry points into each example as possible. My typical approach to this is to write the problem on the board/screen and record a list next to it with ideas that students suggest for how to begin. Regardless of whether or not a valid starting strategy is provided, I continue drawing ideas out of the students until I have five or six items on the list. Once this list is on the board, then I’m in a position to discuss each of these ideas in turn, identifying the ideas that I know will fail, the ideas that I know will work, and any new ideas that I hadn’t thought about before. I’d rather address these in public than in multiple one-on-one discussions during office hours, in the hope that this will guide students to develop better overall strategies for investigating problems.
There are three key aspects of this technique that I’ve learned about the hard way while teaching. First, it is important that I truly value every suggestion given by students. Even if there is an idea for starting a problem that seems rather ridiculous, I consider it carefully and make sure that I provide respectful critical commentary. If students feel that their mistakes are being mocked in the slightest, then they won’t continue sharing them and the entire technique stops working. Second, it is good to frequently remind students of my motivation for going through the process of collecting and critiquing ideas, including mistakes and errors. I want students to have in mind that the purpose of this is both to think critically about errors and to determine the correct answer to a problem. Third, I respond to each student suggestion by thanking them for sharing their idea. I appreciate any student who is willing to risk sharing a misguided idea in public, and I want to make sure students know this.
Assign critical essays regarding readings from course texts
When my students complete reading assignments regarding the mathematics they are studying, one outcome is that they learn more and are more engaged in class. However, like many other teachers, I’ve found it difficult to motivate students to regularly read their course texts. The most effective tool I have to address this is to assign short essays regarding reading assignments. Typically these are two to three pages long, and require students to critically analyze/review a section of their text. I instruct students to think of their essays as being similar to a book or movie review, where they have to highlight both the strengths and weaknesses of the reading, and justify their critical commentary.
I give an assignment of this type several times per semester, and try to have the essays focus on the most important topics in each course. The general result of this assignment is that engaged students who are already reading are more focused and retain more from the texts, while disengaged students who ordinarily would rarely open the book are forced to at least complete a reasonable skim through the material while writing their essay. In general, I’ve found that the result of assigning critical essays is that all students get more from the readings than they otherwise would have. To keep the grading of these essays consistent, I use a grading rubric for mathematical writing that I have developed . An interesting side effect of assigning these essays is that it provides me a window into how the students are thinking about the ideas under consideration, which allows me to be more responsive regarding specific issues that students are struggling with.
 Marrongelle, Karen and Rasmussen, Chris. Meeting New Teaching Challenges: Teaching Strategies that Mediate between All Lecture and All Student Discovery. Making the Connection: Research and Teaching in Undergraduate Mathematics Education. Carlson, Marilyn P. and Rasmussen, Chris, eds. MAA Notes #73, 2008. pp 167-177.
 Braun, Benjamin. Personal, Expository, Critical, and Creative: Using Writing in Mathematics Courses. To appear in PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies. Special Issue on Writing and Editing in the Mathematics Curriculum: Part I, 2014. http://dx.doi.org/10.1080/10511970.2013.843626