In a comment on her post Student Thinking About Abstracting, Mylène says
What frustrates me and disorients my students is that those justifications are never discussed, and even the fact that this is a model is omitted. To further “simplify” (obscure) the situation, most discussions of the matter don’t distinguish between two ideas: “the model has a change in behavior at 0.7V,” vs. “they physical system has a change in behavior at 0.7V.” Finally, the chapter starts with the most abstracted model (1st diode approximation) and ends with the less abstracted (3rd diode approximation).
On getting students to understand models: I agree that this is a huge problem. I’ve been trying various techniques and can’t claim to have found a silver bullet.
One thing I tried in class yesterday (disguised as a gnuplot tutorial) was to build up a model a little at a time to match measured data. I was trying to build an equivalent-circuit model for a loudspeaker, so I started by gathering data (rms voltage measurements across the loudspeaker and across a series resistor at different frequencies) and plotting magnitude of impedance vs. frequency from the data, then building the model a component at a time. Before doing the modeling, we had spent some time looking at the behavior of building-block circuits (R+C, R||C, R+L, R||L, C||L, C||L||R) using gnuplot, so I could ask them things like “how can we model the impedance increasing with frequency above about 1kHz?” We could then immediately modify the model and plot the results. Once things were close, we could use gnuplot’s “fit” command to tweak the parameters.
We didn’t start with “loudspeakers are …”, though we did start with one of the specs—that this was an 8Ω loudspeaker—for our first model. I didn’t even point out to the students that the frequency of main resonance peak is given as a spec on the data sheet. The data sheet gives it at 191Hz, while our measured data show 148Hz (more than 22% off, while factory tolerances for the resonant frequency are usually ±15%). They also give the voice coil inductance as 0.44mH, while our model gets 35µH, a factor of 12.6 difference! And they give the Qes of the resonance peak as 3.52, while our model of the R||L||C for the peak has .
Maybe the inductance difference can be explained by the standard measurement for the voice-coil inductance being made at 1kHz for the Theile-Small parameters, while I fitted for a wider frequency range and added an extra 112µH inductor in parallel with a 32Ω resistor to bump up the impedance around 10kHz. Or maybe my fitting is a really bogus way to get the inductance, since I’m only looking at the amplitude and not the phase of the signal, and non-linear resistance could throw things off. Or maybe the Parts-Express people mis-measured or had a typo—I have no idea what measurements they made to get the parameters they report, or maybe these loudspeakers were so cheap because they didn’t meet the specs, though they are certainly good enough for our lab.
I think that one could do the same sort of model-building with diodes (the part whose models Mylène’s students were confusing with reality): start by measuring the I-vs-V characteristics. The setup I used to get a lot of data points with the Arduino for characterizing the FET in an electret mic might be a good one for them to use, though the unipolar ADC in the Arduino might be more challenging for characterizing diodes. Then try fitting different curve families to the data. Forget about physics for explaining how the diodes work, but concentrate on finding simple models that fit the data. For example, the FET models we used for the mic are not quite the standard ones, since there is a clear slope in the saturation region, and it doesn’t match the channel-length modulation model—but it can be fit with some simple curves.
Of course, I gave up on some modeling before even having the students collect data themselves—the power FETs they are using are incredibly messy, having threshold voltages that shift a lot as the transistors warm up and having an undocumented negative dynamic resistance region when diode-connected.
So it is important that their attempts to build models be of phenomena that are relatively easy to model, but they should build and fit the models (with some guidance) rather than just be handed them. I made the mistake of handing them models to fit for the electret mic lab and for the electrode lab. They not only didn’t understand the models, but they didn’t understand how to do the fitting.
I’m planning next year to do the model-building/gnuplot tutorial much earlier in the quarter, before they do the electrode labs, so that they can build the electrode models with some understanding. I’ll need to rearrange some other material, to do inductors much sooner, if I plan to use the loudspeaker data again. I may want to rearrange the labs a lot next year, since all of my first three labs involved model fitting, and the students weren’t ready for it. It may be better to move the sampling lab (which is currently lab 6) into the beginning, so that students can learn to use the Arduino in a simpler lab. As currently written, though, that lab calls for designing a high-pass filter for DC level shifting and a low-pass filter for removing aliasing, neither of which are suitable for a first-week lab in circuits.
Scheduling the labs and the classes is difficult. Fitting in all the topics they need before each lab is a tricky jigsaw problem, particularly when I discover them having problems with topics that I assumed they knew or could pick up quickly. Sigh, some stuff in the first week or two of lab is probably going to have to be “magic” as they’ve learned so little in physics classes that I can’t count on them having any useful lab or modeling skills when they come into the class. I just have to decide which things I’m willing to give them, rather than having them do for themselves.
Currently, I’m leaning toward having every lab have a design component, and to have them build models for important concepts, but I’m willing to give them a model for thermistor behavior that they just have to fit the parameters for. The design in the first two labs this year is very light (selecting a resistor value), but the measuring and model fitting is pretty heavy. The electrode lab has no design currently, but a lot of measuring and model fitting. I think I underestimated the relative difficulty of model fitting and design for these students, and may need to move the model fitting later in the quarter. I don’t think I can start with RC filters in the first week though, as they need voltage dividers, complex numbers, sinusoids, and complex impedance—probably at least 4 classes worth of material. Maybe by week three, though.