One of the most common misconceptions I see while working with students in secondary school is the notion of an inverse. The idea isn’t too complicated, but the reason that I see students making mistakes with it is because they are in the process of learning about functions and it becomes a cognitive burden to think about these abstract processes such as inverses and other transformations. However, I firmly believe that giving students the right idea of how these different concepts fit together will help them navigate their classes with ease.
Back when I was in CÉGEP, I spent my summer months working as a gardener. It was a radical change from my usual routine of doing mathematics and physics to planting flowers and cleaning flower beds in the heat of the summer. It wasn’t bad by any means, but it was quite different from what one would normally expect myself to do. In essence, it was a job of convenience. I didn’t hate it, but I did look forward to doing something else. In fact, I remember telling my coworkers for multiple summers that I was working as a gardener only until I could finally work in my own domain of interest.
If you were talking to a friend about the drive you made this morning, you might tell them something along the lines of, “I drove from my place to hear in about an hour, going east for the entire time.” Your friend, who knows where you live and has a rough idea of the route to get to where you both are, understands implicitly that you don’t really mean that you went and drove in one direction only for the entire hour. After all, you first had to exit the property of wherever you were parked, then you probably had to navigate through the neighbourhood of where you live, before finally getting onto a highway or a direct route to your destination. It’s only at that point that you actually began going east. Before then, you may have driven in all directions, for varying amounts of time.