The Art of the Mistake in Math and Science Teaching

In retrospect, I think my first undergraduate class taught as a university professor was awful. I constructed detailed notes, which I carefully reconstructed on the whiteboard and in PowerPoint slides, which the students then dutifully transferred to their notebooks. I think they were zombies with pencils. Only the strongest students would ask questions in class, and their questions tended to reflect shallow understandings.The results of their first exam created a stunning revelation to me. The students seemed to have memorized each point I made, but they could not answer more complicated questions that required them to think independently.

I knew at the time that I had to engage the students and pass responsibility for learning on to them, but in practice I did not know how to do it. My first attempt at teaching was poor use of the traditional lecture. Don't get me wrong--some faculty are effective lecturers, and students seem to like the approach. Yet, numerous education studies have concluded that the lecture is among the least effective approaches for deep learning. The lecture is the easiest way to transfer large amounts of information into student's notebooks without engaging their brains! Sometimes this efficiency of direct transfer is needed in order to move forward to more interesting things, but when it dominates a course, it usually results in shallow learning.

Needless to say, I have revised my teaching approach. I still lecture occasionally, but I do it differently. When I simply want students to view lectures to gain first exposure to ideas, I sometimes refer them to YouTube and other courses already available online for free. Why should I teach such concepts myself when others have already done the work? Passing the burden for mundane things onto others frees up my time to focus on helping students to practice deeper applications of concepts in ways that help them better connect what they learn to the real world.

When I do lecture, I leave my notes aside, and I allow myself to make mistakes. I tell the students up front that they should not completely trust me, and that they should try to anticipate (and suggest) next steps. I know my technique is working when students comment to correct my mistakes. If I did make a mistake, I am thrilled when they report it and offer a fix. After they get acquainted with this teaching approach, when students misunderstand a concept, they often think I made a mistake, and they ask questions, so then I know they don't get it.  Many students who speak up in my classroom would never ask questions in more traditional lecture style classes, and their instructors would see little evidence of their lack of understanding until exam time. On some occasions, I work through complicated derivations in class, with an error woven into the mix. When I get to the end, if they have not discovered the error, I ask them whether the solution makes sense. If they cannot identify the error immediately, I assign them to find it, as homework.

My undergraduate course is computer programming and statistics in environmental sciences. I require the students to take a free online course in the computer language we use, then I give them datasets, which we analyze together. They use their own computers and devices in class, allowing them to play with the concepts later on their own time. I think this type of play is essential to build their creativity and their understandings of complicated concepts. In-class exercises are open ended, and we discuss and apply new techniques as they become applicable to questions that arise from looking at the data.

A large part of the course is a class project, in which the students find data online relevant to topics of interest to them. They figure out how to load the datasets themselves, and they apply analysis techniques in order to better understand the natural systems represented in the datasets.

Bottom line, I think, is that deep learning that preserves creativity requires students to actively engage with course content, with freedom to make mistakes and to experiment with their own ideas along the way. When faculty let go of a detailed script, faculty and students can engage with content, leading to real learning.