I was a little surprised by how many students had trouble with the following homework question, which was intended to be an easy point for them:

Estimate C2(touching) − C2(not touching), the capacitance of a finger touch on the packing-tape and foil sensor, by estimating the area of your finger that comes in contact with the tape, and assume that the tape is 2mil tape (0.002” thick) made of polypropylene (look up the dielectric constant of polypropylene on line). Warning: an inch is not a meter, and the area of your finger tip touching a plate is not a square meter—watch your units in your calculations!

Remember that capacitance can be computed with the formula

where is the dielectric constant, 8.854187817E-12 F/m is the permittivity of free space, A is the area, and d is the distance between the plates.

The problem is part of their preparation for making a capacitance touch sensor in lab—estimating about how much capacitance they are trying to sense.

There is a fairly wide range of different correct answers to this question, depending on how large an area is estimated for a finger touch. I considered any area from 0.5 (cm)^{2} to 4 (cm)^{2} reasonable, and might have accepted numbers outside that range with written justification from the students. Some students have no notion of area, apparently, trying to use something like the length of their finger times the thickness of the tape for A.

People did not have trouble looking up the relative dielectric constant of polypropylene (about 2.2)—it might have helped that I mentioned that plastics were generally around 2.2 when we discussed capacitors a week or so ago.

What people had trouble with was the arithmetic with units, a subject that is supposed to have been covered repeatedly since pre-algebra in 7th grade. Students wanted to give me area in meters or cm (not square meters), or thought that inches, cm, and m could all be mixed in the same formula without any conversions. Many students didn’t bother writing down the units in their formula, and just used raw numbers—this was a good way to forget to do the conversions into consistent units. This despite the warning in the question to watch out for units!

A lot of students thought that 1 (cm)^{2} was 0.01 m^{2}, rather than 1E-4 m^{2}. Others made conversion errors from inches to meters (getting the thickness of the tape wrong by factors of 10 to 1000).

A number of students either left units entirely off their answer (no credit) or had the units way off (some students reported capacitances in the farad range, rather than a few tens of picofarads).

A couple of students forgot what the floating-point notation 8.854187817E-12 meant, even though we had covered that earlier in the quarter, and they could easily have looked up the constant on the web to figure out the meaning if they forgot. I wish high-school teachers would cover this standard way of writing numbers, as most engineering and science faculty assume students already know how to read floating-point notation.

Many students left their answers in “scientific” notation (numbers like 3.3 10^{-11} F) instead of using more readable engineering notation (33pF). I didn’t take off anything for that, if the answer was correct, but I think that many students need a lot more practice with metric prefixes, so that they get in the habit of using them.

On the plus side, it seems that about a third of the class did get this question right, so there is some hope that students helping each other will spread the understanding to more students. (Unfortunately, the collaborations that are naturally forming seem to be good students together and clueless students together, which doesn’t help the bottom half of the class much.)

Thanks for sharing this experience (I’m sure it isn’t the first time you’ve had students with large problems of “knowing where they are” because of not visualizing or estimating or checking, etc.

In the old slide rule days — where the slipstick gave only roughly three decimal places of accuracy at best and no order of magnitude, one had to do pretty good estimating in one’s head in order to nail down where things were. I think most of us from that era found the mental estimations hugely helpful in just “getting into where the relationships were” for a deeper sense of what was going on.

I’ve seen several relatively recent books that try to provide heuristics for mental estimations e.g. “Street Fighting Mathematics” (I think that’s the title), and my favorite book in the last few years “Cell Biology By The Numbers” a truly marvelous book of “what’s going on” and “ways to think about it”.

Comment by alanone1 — 2017 February 5 @ 12:58 |

That units and reasonableness of answers will be expected are things engineering and science students must be taught some first time. Unfortunately, it often doesn’t happen before serious, practical courses like yours. Keep up the good work!

Comment by chaikens — 2017 February 5 @ 14:45 |

But this is college! How did they get admitted to the engineering program without some fluency? — or are these not engineering or science majors?

Comment by alanone1 — 2017 February 5 @ 14:48 |

These are all engineering majors. But if we only admitted those who already had fluency, we’d have to reject 95% of our applicants, instead of only 71%. Furthermore, students are admitted to UCSC independent of what major they say they want to do, so criteria relevant only to engineering and science students wouldn’t be used for admission anyway.

What I haven’t checked (because I’m afraid of the answer), is whether there is any correlation between how far along the students are in the major and their basic engineering competence with things like algebra and using units. (The class is a mix of sophomores through seniors this year, though I’d really like to have mainly freshmen and sophomores.)

Comment by gasstationwithoutpumps — 2017 February 5 @ 18:43 |

The correlation between fluency and how far they are along would be interesting (and even important) to know….

Comment by alanone1 — 2017 February 6 @ 03:48 |

Excellent. I teach dimensional analysis to my 5th grade math students, and now I feel justified :-). You could suggest your students refer to https://www.khanacademy.org/math/algebra/units-in-modeling/intro-to-dimensional-analysis/v/dimensional-analysis-units-algebraically as I think it’s unreasonable for you to have to cover it in class.

Comment by whatisron — 2017 February 5 @ 21:11 |

I agree that 5th–7th grade is a good time to introduce dimensional analysis. It is unfortunate that some juniors and seniors in engineering majors seem still unfamiliar with the concept.

Comment by gasstationwithoutpumps — 2017 February 5 @ 21:56 |

What is more interesting is how many teachers do not use units in their teaching. I see it all the time in 5 – 8 teachers. They themselves are uncomfortable with the basics. The math/science background for K 8 teachers can be very limited. In high school students can get through a science program and see very little of labeling and converting units. When I teach Algebra II to high school sophomore I do a whole unit (about 2 weeks) on area and volume. It is amazing how many cannot see the square and cubic relationships. I think some just do not have the cognitive capability yet. And as for units it is like pulling teeth to get them to figure them out. Alan’s comment about the slide rule era is so true. I actually bring slide rules in (I have about 50. I collect them. Weird math hobby.) and have the kids use them just so they have to exercise their estimation skills. Works for some, not others.

Comment by gflint — 2017 February 6 @ 07:57 |

Odd that you correctly prefer pF (as I do) over 10^{-11} F, but then use the 1960s line printer version E-12 for the permitivity. Do you also accept e-11? That is common on the CS side of the world, leading to such abominations as writing e^{-11} instead, thinking it actually means 10^{-11}. Ever see that?

I like that my current physics text uses standard SI prefixes all the way out to p and f and T and E in relevant problems. EHz is still rather unusual usage for light frequencies, but THz is now commonplace.

Your problem with dimensional ignorance is heartening to me. It is good to know that CC students are the equal of yours. It is like shooting fish in a barrel to give cm^2 in a problem until they have faced it more than a few times, but they forget the correct approach very quickly. (I’ll find out on a future exam that has a similar problem.) I strongly recommend a followup on preparation, but I already know the answer. Have they had chemistry at UCSC? If so, the odds are that they spent several weeks on converting anything and everything. Find someone teaching it and ask. Our freshman chem class does, and many universities use the same book/curriculum that my CC does. Our chem faculty coordinate with those at a nearby university. The problem is that they both actively forget it and compartmentalize it as “not engineering” because it is chemistry.

Try this if you discover that conversion of cm^2 and cm^3 and ft^3 is actually on their exams: Ask the ones who missed it if they had chemistry, and if they covered this topic. If they say “no”, feign horror and ask them who they had for chem, then followup. You’ll probably discover that it was on the first test they passed, but they don’t actually learn much (in the sense of knowing it for more than the next exam) in prerequisite classes.

Comment by CCPhysicist — 2017 February 15 @ 18:08 |

I accept, but don’t like, 1e-11. It is important that students understand floating-point notation, because they will see it a lot. I’ve had GRAD students who thought that 4.5e3 meant 4.5 e^3, rather than 4.5 10^3. Engineering students generally can’t keep track of orders of magnitude any more (haven’t since slide rules became obsolete), so I don’t object to them converting back and forth between 10^k notation and the prefixes, but I want all answers to be in the range 1≤n<1000 with the proper prefix. (I accept numbers in the range 0.1≤n<10000, but for numbers <1, I get upset if there is no leading zero.)

Most of these students have had 6 or more chemistry classes. That doesn't mean that they can convert pg of something in 50µl into mM without someone writing the formula down for them.

I'm sure many of them have had unit conversions in the past—they just haven't internalized the notion that units matter—perhaps because they have always gotten credit for writing down numbers without units.

Comment by gasstationwithoutpumps — 2017 February 15 @ 18:55 |

It might be interesting to visit the chemistry department and compare notes. Maybe they think units only matter in chemistry and not other classes, or maybe they don’t have to put down units because the other exams are multiple choice. But our students appear to choose to forget everything until forced to remember or relearn, and yours could be the same.

Comment by CCPhysicist — 2017 February 16 @ 18:57 |