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.
Teach. Brian. Teach. 
I don’t carry units in calculations either.
But then I do all my calculations symbolically, and only plug in numbers at the very end, when necessary, so the question is moot. I wish I could get students to do that. So I guess I’m really “none of the above” when it comes to carrying around units.
Brian,
I’m with Chris—I want my students to do everything symbolically. But this is difficult, and a challenge for students, especially 9th graders. So I do push students to carry units through every time they write a number, especially on single line calculations. I do this because it does add a bit more burden to the “just do it with numbers” approach, that might push them over to doing things symbolically.
I guess for me, I find myself so frustrated by students not putting thought into the numbers they write that I am trying to introduce these strategies to build in habits that cause them to slow down and think about the meaning of the measurement the number represents.
Brian,
I’m with John, I make students solve the problem algebraically first, then plug in numbers at the very last step. I agree writing them over and over is ridiculous. However, writing them one time at the end I think is helpful. Most struggle to do the algebra with symbols only, but they have much better math skills at the end. Although my job isn’t to teach math directly, I think it’s a great benefit on the side for taking the course. I had one student increase his math ACT 6 or 7 points after taking my class (science ACT went way up too).
Scott
While I think writing down the units with every number is ridiculous, especially when compared to doing calculations symbolically as I would do, most of my students are also headed to some sort of chemistry course next year, and our chemistry courses, heavily emphasize stoichiometric calculations, and another small part of my justification for insisting on full numeracy with every number written, is that when students get to this in chemistry, the mathematics of what is going on will be a piece of cake, and they’ll actually be able to focus on the understanding.