We are learning about linear transformations in linear algebra. These are just functions from one vector space to another that preserve the structure of the vector space. I had students do a team exercise to determine if a number of transformations, such as derivatives, integrals, translation of a vector by a constant vector, multiplication by a matrix are linear transformations.
I get that the linear algebra is new and confusing to students, but I was amazed (I probably shouldn’t have been) by how many found the function definitions to be difficult. After all, we learn about functions in high school. It’s just the linear algebra that’s new and confusing, right? Not quite, as it turns out.
I think this is a case where new knowledge confounds old knowledge. Because we are learning about something new, students lose their ability to keep their mathematical wits about them and apply their old knowledge to the situation. They start to believe that something funky must be going on, and then they try to use the new stuff all over the place. It doesn’t work.
Between my first section and the second, I reinforced the basic function concepts, and it might have helped a little bit. It didn’t help a lot. I know many students got it. I know some didn’t. My hope is that those who have been keeping up with the material might have had a learning curve, but then they learned from their teammates what was going on (or me, when I solved the exercise), and now they know what to do.
I hope I can reinforce the idea that you can go a long way in life if you can keep your mathematical wits about you. Uncommon sense. Apply your uncommon sense. Others call it common sense, but I don’t think it is!