Here are some conversations that come up every year.
1. Zero Current
Student: “I tried to measure current, but I couldn’t get a reading.”
Me: “So the display was blank?”
Student: “No, it just didn’t show anything.”
(Note: Display showed 0.00)
2. Zero Resistance
Student: “We can’t solve this problem, because an insulator has no resistance.”
Me: “So it has zero ohms?”
Student: “No, it’s too high to measure.”
3. Zero Resistance, In a Different Way
Student: “In this circuit, X = 10, but we write R = 0 because the real ohms are unknown.”
(Note: The real ohms are not unknown. The students made capacitors out of household materials last week, so they have previously explored that the plates have approx. 0 and the dielectric is considered open)
4. Zero Resistance Yet Another Way
Student: “I wrote zero ohms in my table for the resistance of the battery since there’s no way to measure it.”
What I Wonder
- Are students thinking about zero as indicator that means “error” or “you’re using the measuring tool wrong?” A bathroom scale might show zero if you weren’t standing on it. A gas gauge shows zero when the car isn’t running.
- When students say “it has none” like in example 2, what is it that there is none of? They might mean “it has no known value”, which might be true, as a opposed to “it has no resistance.”
- Is this related to a need for more concreteness? For example, would it help if we looked up the actual resistance of common types of insulation, or measured it with a megger? That way we’d have a number to refer to.
- #3 really stumps me. Is this a way of using “unknown” because they’re thinking of the dielectric as an insulator that is considered “open”, so that #3 is just a special case of #2? Or is it unknown because the plates are considered to have 0 resistance and the dielectric is considered open, so we “don’t know” the resistance because it’s both at the same time? The particular student who said that one finds it especially hard to express his reasoning and so couldn’t elaborate when I tried to find out where he was coming from.
- Why does this come up so often for resistance, and sometimes for current, but I can’t think of a single example for voltage? I suspect it’s because both resistance and current feel concrete and like real phenomena that they could visualize, so they’re more able to experiment with its meaning. I think they’re avoiding voltage altogether (first off, it’s about energy, which is weird in the first place, and then it’s a difference of energies, which makes it even less concrete because it’s not really the amount of anything — just the difference between two amounts, and then on top of that we never get to find out what the actual energies are, only the difference between them — which makes it even more abstract and hard to think about).
- Since this comes up over and over about measurement, is it related to seeing the meter as an opaque, incomprehensible device that might just lie to you sometimes? If so, this might be a kind of intellectual humility, acknowledging that they don’t fully understand how the meter works. That’s still frustrating to me though, because we spend time at the beginning of the year exploring how the meter works — so they actually do have the information to explain what inside the meter could show a 0A reading. Maybe those initial explanations about meters aren’t concrete enough — perhaps we should build one. Sometimes students assume explanations are metaphors when actually they’re literal causes.
- Is it related to treating automated devices in general as “too complicated for normal people to understand”? If that what I’m reading into the situation, it explains why I have weirdly disproportionate irritation and frustration — I’m angry about this as a social phenomenon of elitism and disempowerment, and I assess the success of my teaching partly on the degree to which I succeed in subverting it… both of which are obviously not my students’ fault.
One possibility is that they’re actually proposing an idea similar to the database meaning of “null” — something like unknown, or undefined, or “we haven’t checked yet.”
I keep suspecting that this is about a need for more symbols. Do we need a symbol for “we don’t know”? It should definitely not be phi, and not the null symbol — it needs to look really different from zero. Question mark maybe?
If students are not used to school-world tasks where the best answer is “that’s not known yet” or “that’s not measurable with our equipment”, they may be in the habit of filling in the blank. If that’s the case, having a place-holder symbol might help.
This year, I’ve really started emphasizing the idea that zero, in a measurement, really means “too low to measure”. I’ve also experimented with guiding them to decipher the precision of their meters by asking them to record “0.00 mA” as “< 5uA”, or whatever is appropriate for their particular meter. It helps them extend their conceptual fluency with rounding (since I am basically asking them to “unround”); it helps us talk about resolution, and it can help in our conversation about accuracy and error bars. Similarly, “open” really means “resistance is too high to measure” (or relatedly, too high to matter) — so we find out what their particular meter can measure and record it as “>X MOhms”.
The downfall there is they start to want to use those numbers for something. They have many ways of thinking about the “unequal” signs and one of them is to simply make up a number that corresponds to their idea of “significantly bigger”. For example, when solving a problem, if they’re curious about whether electrons are actually flowing through air, they may use Ohm’s law and plug in 2.5 MOhms for the resistance of air. At first I rolled with it, because it was part of a relevant, significant, and causal line of thinking. The trouble was that I then didn’t know how to respond when they started assuming that 2.5MOhms was the actual resistance of air (any amount of air, incidentally…), and my suggestion that air might also be 2.0001 MOhms was met with resistance. (Sorry, couldn’t resist). (Ok, I’ll stop…)
I’m afraid that this is making it hard for them to troubleshoot. Zero current, in particular, is an extremely informative number — it means the circuit is open somewhere. That piece of information can solve your problem, if you trust that your meter is telling you a true and useful thing. But if you throw away that piece of information as nonsense, it both reduces your confidence in your measurements, and prevents you from solving the problem.
Some Responses I Have Used
“Yes, your meter is showing 0.00 because there is 0.00 A of current flowing through it.”
“Don’t discriminate against zero — it isn’t nothing, it’s something important. You’ll hurt its feelings!”
Not helpful, I admit! If inquiry-based learning means that “students inquire into the discipline while I inquire into their thinking”*, neither of those is happening here.
Some Ideas For Next Year
- Everyone takes apart their meter and measures the current, voltage, and resistance of things like the current-sense resistor, the fuse, the leads…
- Insist on more consistent use of “less than 5 uA” or “greater than 2MOhms” so that we can practise reasoning with inequalities
- “Is it possible that there is actually 0 current flowing? Why or why not?”
- Other ideas?
*I stole this definition of inquiry-based learning from Brian Frank, on a blog post that I have never found again… point me to the link, someone!
Note for future exploration:
Before I started teaching, I was so embedded in my own understanding that I couldn’t empathize with the experience of not knowing zero. How could anyone not think that way? I knew that Europe had no concept of zero until the 13th century, that the gigantic and wealthy Roman Empire, and many others before that, had risen and fallen, handling navigation through their enormous territories and accounting for their staggering tribute income, somehow without it. But I couldn’t relate.
But now that I teach, I notice that my students internalize the concept of zero only partially. I think about why and how. I remember that hundreds of generation of their forebears, some of the most brilliant minds in history, didn’t internalize it either.