OK, I have to weigh in the units thing:
First, I don’t carry around units in calculations. I think it creates clutter and distracts me from the important thinking I need to be doing. That clutter makes it MORE difficult to diagnose mistakes I’ve made, both during and after the fact. Sure, there are a small number of mistakes I could make which would be easier to spot carrying out units, but that benefit is far outweighed by the increase in mistakes I would make because I am bothering to work out the units along the way AND by the difficulty it will be later to gander at my work at spot more meaningful mistakes.
Second, most of the physicists I know don’t carry around units their calculations either. I don’t want to pretend that making students carrying units around is some important scientific skill. Sure, thinking about quantity and how units figure into the notion of quantity is important, but carrying around units is not a stand in for understanding notions of quantity and rate. In fact, many physicists do all they can to get rid of units, so that they don’t have to carry them around in calculations. One way they do this is setting as many units as they can equal to one. Then, they go about carrying out their calculations with all units and constants all hidden in dimensionless ratios. Then, only at the end, do they introduce units back into their work. I’m not saying, we should make students do this.
I don’t know where the idea that students should carry units around came from. Does anybody know?
Last thing, I am all for teaching students to do dimensional analysis, but that’s different than carrying around units in a calculation.
There, I’m done. I said it.