By Steven Klee, Contributing Editor, Seattle University
When I first started incorporating active learning in the classroom, I struggled with getting my students to buy into being active. I made worksheets, put the students in groups, and excitedly set them off to discover and play with mathematical ideas. Despite this, many students were inclined to sit silently in a group of four and work on the problems on their own.
But really, who can blame them? First, this propensity towards solitude can be explained by basic human nature: specifically, the fear of being wrong. We don’t want to be wrong. At least, we don’t want to be wrong in front of other people. From that perspective, working alone is safe and comfortable. We should view our job as teachers as one of helping our students overcome this basic human inclination, as opposed to viewing it as a failure or shortcoming on their part.
Beyond this, the desire to work alone can be attributed to culture and expectations. Many students’ formative educational years have been spent sitting silently in desks passively absorbing lectures. If they feel this is what is expected of a math class, then it is natural for them to continue to sit silently, even if the environment is meant to be collaborative. Of course, it is not my intention to imply that this is an issue that is entirely the students’ fault – maybe my questions weren’t sufficiently open-ended, maybe I wasn’t doing a good enough job at “selling it,” maybe the students just like working alone, maybe, maybe, maybe… The list goes on.
I tried some of my standard tricks to foster communication among the students. I would prepare impassioned pep talks about the benefits of working with your peers. This technique flopped for obvious reasons – no one wants to listen to what they are told is good for them. Otherwise, cigarette companies and fast food restaurants would go bankrupt and I would be much more diligent about flossing. I’d try to lighten the mood, saying “this isn’t a library, you’re welcome to talk to one another.” I’d give a difficult problem and leave the room to get a drink of water, forcing the students to rely on one another. These strategies helped, but never served to create the classroom of my dreams – one where students discuss math problems at such a frenzied pace that time ceases to exist; one that causes passersby to wonder whether we are having a math class or developing some bizarre scientific improv comedy troupe.
Over time, I continued to reflect on my own teaching and sought advice from more experienced practitioners of active learning. As a result, I have developed a few strategies that have been effective in my classrooms. One of the most effective strategies for me has come from eliminating those pesky desks that keep getting in the way of my students’ learning.
Desks? Where we’re going we don’t need desks!
One day after watching Back to the Future for the umpteenth time, Doc Brown gave me a spark of inspiration: “Roads? Where we’re going we don’t need roads!” But now replace “roads” with “desks.” Maybe the desks were the problem.
In the spring of 2014 I was teaching a Graph Theory class in my favorite classroom on campus. It has chalkboards on three of the walls, and the fourth wall is a bank of windows that looks out onto our campus quad, which is very pretty with a fountain and trees. It’s sort of a mathematician’s paradise. The most important aspect of this room is the board space – there’s enough room for 24 people to work on the board at the same time. I decided to test this desk-free learning idea: instead of sitting in their desks, what if my students spent most of their class time at the board? This had a noticeable impact on the quality of conversations and engagement in the classroom. I’ve found success in implementing this strategy in different ways in different classes. It has led to more dynamic exam review sessions for lower-level calculus and linear algebra classes, deeper learning in my introduction to proofs classes, and hotbeds of mathematical ideas in my graph theory classes. I’d like to offer some tips and tricks on how this can be done in a variety of settings.
I typically prepare a worksheet with problems for each class that are meant to guide the students through a certain topic or idea. If necessary, I start class with a short lecture introducing some new ideas or definitions and then set the students to work on the problems I have prepared, telling them that I want them to get up and work at the board in groups with their peers.
From a practical teaching perspective, there are a number of benefits to doing this. Because I can see what everyone is doing from just about anywhere (as long as the room is convex), I can easily assess each group’s progress from a vantage point in the center of the room and quickly determine which group needs the most immediate attention. If everyone seems to be making a common mistake, I can pull the group back together and add clarification or facilitate a large group discussion. Similarly, if one of the groups has something interesting to share – for example, an interesting example/counterexample or a solution that is different from what everyone else has done – it is easy to bring the whole class to their work area and let them share their ideas.
This practice can be particularly effective in a graph theory class where dynamic problem solving is so important. For example, solving problems about planar graphs on paper can be frustrating if you have to keep redrawing the same graph until you find a nice planar drawing. On the board, it is much easier to just erase the problematic edges without having start from scratch. In a different activity, I give the students pseudocode for an algorithm (for example, Dijkstra’s Algorithm), without telling them what the algorithm does. They decipher the pseudocode and run through the algorithm on an example graph to try to figure out what the algorithm does. After that, we discuss what the algorithm does and prove that it works as a group. This is easier once the students already have a solid intuitive understanding of what the algorithm does because it separates the difficulty of such proofs into more manageable pieces: first, understand what the algorithm does, and second understand techniques for proving an algorithm works.
Overcoming solitary work
Most importantly, being at the board helps students overcome the fear of being wrong. Because work on the board is inherently temporary, students don’t have the same reservations about writing down some ideas and sharing their thoughts, even if they are incomplete and especially if they may be wrong. I call this the “Bob Ross approach to mathematics.” On The Joy of Painting, Bob Ross taught us “We don’t make mistakes. We just make happy little accidents.” Overcoming their initial fear of making mistakes helps students get to the point where real learning happens.
Unlike working on a piece of paper, which carries a natural expectation that you will start writing at the top and finish writing at the bottom, work at the chalkboard can flow more organically. Students come to see that solutions often take serpentine paths through different parts of the problem and various examples until you figure out how to put all the pieces together. Then, once they find a solution, they can write out a clean version of the solution in their notebooks. In many cases, they just take pictures on their phones and transcribe their notes later.
Having students stand at the board also makes for a more social environment that naturally fosters collaboration and does seem to create a more active classroom. Standing helps everyone be more engaged, more physically active and, as a result, more mentally active. In a course evaluation, one student commented “Using the chalkboards in class was a great way to get our blood flowing and keep focused during class.” When you are standing, it is more reasonable to expect that you should be talking with the people around you.
As a result of spending more time collaborating with their peers, students come to see they are not alone in their confusion or struggles. They learn to ask questions, which can be as simple as “I didn’t get what you just said, can you say it again?” When I was in grad school I started asking that question, perhaps to the extent that people got tired of hearing it. It has been hugely beneficial to me and my students. When working together, students see different approaches and different ways of thinking about problems. As a result of having to answer questions posed by their peers, they reflect more deeply on the way they approach problems.
Heuristically, this also seems to help the students develop better problem solving skills. They realize that in order to solve a problem, it helps to write something, anything, to get your brain wrapped around the problem. Students learn to explore small examples, reflecting on their observations, and thinking about how to generalize those observations. By the end of the quarter, students who had initial reservations saying “do we have to work at the board?” have changed their attitude, with some saying “do we get to work at the board today?”
What if I’m not teaching graph theory?
The tips I’ve discussed here are applicable beyond a course in Graph Theory and can be used beyond classes at a small liberal arts school with small classes. Having students work together to solve practice problems at the board without their notes or books can be valuable in helping them prepare for an exam. Teaching Assistants can implement similar practices in recitation sections with smaller groups of students, even if they are part of a large lecture course. At Seattle University, we don’t have a graduate program, but we have undergraduate Learning Assistants who facilitate study groups for our lower-level math courses. We train our LAs to facilitate group work in this way so that students are actively engaged during their study groups.
In mathematics, we often pride ourselves on the fact that our research can be done wherever there’s a chalkboard. We should strive to include this in the way we help our students learn!