‘Getting Stuck’ Together: The Limits and Possibilities of Teaching Maths Online
Teaching mathematics this year has showed me that the new means to reach our students remotely requires more than an adaptation to virtual technologies. Often, it wasn’t so much the technological setup that would cause me headaches. Rather, it was the process of learning how to deal with new or other limitations of the virtual teaching and learning spaces.
One key reason for this lies in the approach I and many university mathematics teachers follow when working with our students. Developing skills in an area of mathematics is typically an interplay of Intuition (informal semantics and communication of mathematics), Proof (formal semantics), and Experiment (examples, specialisations, generalisations, and applications). This interplay is guided by the teacher, but requires students to actively engage with it. To learn successfully, students must not be “consumers only”, but actively question and challenge what is being presented. Much of the interaction between students and teachers is collaborative work on solutions to exercises and problems that builds on these three steps.
Problem solving in an actual space has a lot to do with building a rapport with students. I have noticed that it is incredibly useful for students when I “get stuck” with explaining an idea informally, or with a step of “my logic” within a proof. This mirrors how students will approach exercises and problems that are new to them: they get stuck as well. We often learn more from failing than from getting things right! More importantly, doing things wrong as a teacher helps to build the relationship with my students. It encourages them to develop a crucial scepticism, and to ask questions of me. It also helps them to develop new ideas during class.
The virtual teaching environment does not easily allow for failure or mistakes. Many of my lectures are now pre-recorded for students to “consume at their leisure”. By design, recorded lectures contain very few (if any) mistakes. They are unlikely to capture the teacher “getting stuck.” This may be because getting stuck is really something that happens organically and requires interaction with the students (“Why is what I just presented wrong?”, “How can we solve the issue based on our prior knowledge?”).
The main shortcoming of the virtual environment is that communication with students is limited to the sound and video they are prepared to share. In an actual room with students, I am able to identify those who understand what I am saying and those who are struggling by their facial expressions and body language. It is easy to receive feedback and offer clarification. Online and remotely, this is more challenging. Most students find it extremely difficult to engage in this way, because being visual and outspoken in an online classroom makes the individual student more present for everyone else. Encouragement from my side to comment and interrupt flows better in an actual room.
This limited way of communicating is effecting a further important aspect of teaching and learning: inter-student communication. In mathematics tutorials or seminars, students typically sit around tables and work on exercises in small groups. We actively encourage this, because it helps students to grasp concepts through explaining to others. Virtually, there are possibilities with breakout rooms, etc., but similar limitations exist here as well, and interaction is harder.
Teaching and learning have come a long way in the last 15 months, and I am certain that virtual teaching and learning environments are here to stay. It will be interesting to see if teachers and learners can break down further limitations of virtual learning and teaching spaces, and develop interactions comparable to those in physical spaces. This may be partly enabled by new tools and technologies. It will also require continuous learning and adaptation by teachers.
