On the conceptual physics trimester exam, I showed the circuit above, then asked, "estimate the voltage across the 10 ohm resistor."
In conceptual physics, we teach, well, conceptual approaches to all problems. That doesn't mean we never use equations or calculations, of course; it's just that even when a problem does require mathematics, students are generally required to explain an aspect of the mathematics in words.
For example, we teach students to approach this particular circuit problem with an equation:
* The current in each resistor is the same - that's a fact that "series resistors take the same current"
* Mathematically by V = IR, then, with the same I the bigger R takes more voltage V.
* The voltage across series resistor adds to the total. So the 10 ohm resistor takes less than 18 V, but more voltage than the 5 ohm resistor.
With this reasoning, we accept *any* answer more than 9 V and less than 18 V.
In class, the students each conduct this kind of estimation, then set up the circuit on a breadboard. They find out experimentally that the voltages are in fact 12 V and 6 V. After a bunch of similar predictions and experiments, my 9th graders develop for themselves an instinct - in series resistors, the bigger resistor takes more voltage, and the bigger the difference in resistances the bigger the difference in voltages. That's not written down anywhere, 'cause it's not exactly rigorous. But these conceptual students can estimate voltages for series resistors better than most seniors can calculate them.
Greg, when your students get to college, they'll be asked to calculate precise voltages. I know you love conceptual physics, but the vast majority of college physics courses are calculational in nature. Aren't you doing a disservice to your students by teaching physics in a manner so different from how they’ll be asked to learn in their future?
Before I address this (legitimate!) concern on its merits, I’ll point out that it is emphatically NOT my job as a high school teacher to “prepare students for college.” I provide for my students what they need right now… not what they should’ve had in the past, not what they may possibly need at some ill-defined point in the future. (But in Zen-like fashion, even though I adhere zealously to this philosophy, I've nevertheless received overwhelming feedback that, four years out, my students feel extraordinarily well prepared for college and college physics. Go figure.)
I have plenty of anecdotal evidence that those who start their study of physics primarily with concepts (rather than primarily with calculation) retain more and deeper knowledge, and perform better if they continue to another physics class. If you want them, I know you can find academic-style research studies that say the same thing. But let's talk about just last month's trimester exam. and just this question.
Remember, I did not ever teach, model, demand, suggest, hint, or advise students about a caclulational method to finding the voltage across each resistor. Not once. We did sometimes ask students to calculate equivalent resistance of series resistors by adding the resistor values. We did sometimes ask students to calculate the current through each resistor, using the fact that says "In ohm's law, use the voltage of the battery with the equivalent resistance of the circuit." But voltages across series resistors? Conceptual students were only taught how to make an estimate.
Nevertheless. On this exam problem... a large majority of the class made a correct numerical calculation!
Without prompting, so many students calculated the current by using the 18 V battery with the 15 ohm equivalent resistance to get either 1.2 A (or "6/5 A" because they had no calculator). Then, they recognized - usually wrote out in words! - that the current in the 10 ohm resistor is also 1.2 A because it's a fact that series resistors take the same current that's equal to the total current. And finally, they used ohm's law across the 10 ohm resistor to get (10 ohms)*(1.2 A) = 12 V.
In other words... when the class had only, exclusively been instructed on a conceptual approach, they nevertheless figured out for themselves a useful calculational approach. And they did so in a few moments, under the pressure of an exam. I'd say 3/4 of my class did this, and 9/10 of those who tried it did it right.
So am I doing a disservice to students by teaching them conceptual approaches? Nope... they can learn how to calculate without difficulty. They don't even have to be carefully taught.