Department of the Obvious: Test Corrections Work

Don't know whether you saw the December 2009 edition of The Physics Teacher. This generally excellent magazine has given me untold helpful hints, lab ideas, and physics concepts to think about in the context of teaching high school (and low undergraduate) physics. In fact, I have a co-written article being published in next month's edition, about the USAYPT, the organization that made the mistake of appointing me President. (Check us out at http://www.usaypt.org/!)

This month in TPT, Charles Henderson and Kathleen Harper explain how they use "Assessment Corrections" as a teaching tool. Great idea, obviously. What bugs me about this article is not that I think they "stole" the idea of corrections from me. Of course they didn't. In fact, I "stole" the idea from Haverford professor Lyle Roelofs -- pictured above --  who inflicted test corrections on us in Advanced Quantum Physics in 1994.  He offered half credit back on the test if we corrected out mistakes.  A classmate astutely commented, "Lyle, you know we're going to do the corrections, because without them our grades are lousy, but with them the grades are good.  So even though corrections aren't required, you're insidiously getting us to do them."  Lyle just smiled.

Anyway.  What bugged me about the Henderson and Harper article was the conceit that they were determining, through the use of a scientifically valid theory, that assessment corrections are useful, and that corrections help students learn.  The article is full of phrases like "formative assessment"  and "metacognition."  AARRGH!  Look, readers, I don't care what your "theoretical basis for assessment corrections" is, or whether you even have one.  Does anyone ever ask Roger Federer for the "theoretical basis" for his forehand?  Does anyone ask Albert Pujols for the "theoretical basis" of his swing?  No, these folks just do what works.  They're probably happy to share what they know about what works for them, but what works for them may or may not work for another professional. 

Physics may be a peer reviewed science, but physics teaching is far, far closer to art than science.   Good artists may do things in a similar manner, but they don't need peer-reviewed, buzzword-filled evidence to know they're doing something right.  All anyone -- INCLUDING ME -- can tell you about a physics teaching method is, "it worked for me, it worked for lots of other people, here's how I do it, now try it if you'd like."

Test corrections work for me.  Test corrections apparently work for Mr. Henderson and Ms. Harper, too -- you can read the article for useful examples of how other teachers have made the corrections assignment.  Corrections worked for Lyle Roelofs.  They have worked for a number of attendees at my summer institutes.  They will probably work for you.

Thermal expansion – quantitative demonstration

It’s easy to discuss thermal expansion in terms of students’ experience. They see expansion slats in sidewalks all the time. They may have noticed highway bridge expansion joints (and if they haven’t, they will probably be able to see one within 24 hours if they’re observant). Evidence for the existence of thermal expansion abounds.

However, creating measurable expansion in the classroom is a challenge. The coefficient of linear expansion for virtually all materials is on the order of 10^-5 K^-1. So even with a temperature increase of 100 K, an object will expand by about 0.1% of its length.

I can think of three experimental ways to create a reasonable thermal expansion demonstration:

1. Create an enormous temperature increase. Not practical, though, because even a 1000 K temperature change only causes a 1% length expansion.

2. Use a very, very long object, so that the 0.1% increase is big enough to see. Well, if you heat a 10 m long rod by 100 K, it will expand by about a centimeter. I don’t know about you, but I don’t have a 10 m rod handy, nor do I have room for it in my classroom, nor do I have a way of heating it relatively uniformly.  Ugh.

3. Get a measuring device that can measure itty bitty length changes.

I’ve gotten method number 3 to work in my classroom. Yesterday I used one of those metal rings with a wooden handle that come with commercial thermal expansion kits. I used a micrometer (pictured), which has can measure plus or minus 5 thousandths of a millimeter, to measure the WIDTH of the ring. You see, linear expansion happens in all directions. The width of, say, a rod will expand by the same percentage as will the length. I choose to measure the width of the ring because my micrometer can measure that easily.

The width at room temperature was 4.30 mm. I heated the ring for a minute or two in a Bunsen burner. I used an infrared non-contact thermometer – pictured to the right, available for no more than $30 – to find that the ring’s temperature rose to about 220 degrees Celsius, an increase of about 200 degrees over room temperature. Thus, an order of magnitude estimate of the ring’s expansion is that ΔL/L = (10^-5)(200) = 0.2%. Since I don’t know what my ring is made of, I can merely guess that the expansion should be in the neighborhood of a few tenths of a percent. I measured a new length of 4.33 mm, an increase of about 0.7%, which works for me.

Live PV diagrams!

It’s time to teach the ideal gas law, heat engines, and PV diagrams in AP physics. A lot of AP teachers are a bit intimidated by these topics. They’re more abstract than mechanics, and are farther divorced from our experience than, say, electricity and magnetism or waves and optics. I hope this next series of posts can help out.

Certainly your students have studied the ideal gas law in chemistry. But chances are, all they did (in their minds) was plug numbers into the equation PV=nRT. A major first step in teaching this unit is to give your class a firm understanding of the physical meaning of each of these variables.

You may or may not have seen Pasco’s heat engine / gas law apparatus. It consists of a low-friction piston that attaches to a metal cylinder – see the pasco.com picture above. I use it to demonstrate the ideal gas law and PV diagrams with live data collection.

The three variables to measure are pressure, volume, and temperature of the gas in the cylinder. I attach a Vernier pressure probe to one of the ports on the front to measure, um, pressure. Temperature can be taken care of with a Vernier temperature probe inserted into the hole in the stopper on top of the metal cylinder. It’s only volume measurement that’s truly tricky.

If you can measure the height of the piston, then the volume of the gas under the piston and in the cylinder can be calculated. Pasco provides an instruction packet that suggests the use of a rotary motion sensor or smart pulley to get the piston’s position. I don’t do that, though it should work fine.

Instead, I mount a motion detector above the piston. By measuring the height of the detector above the piston’s lowest point, I can set the Logger Pro software to calculate the volume of the gas automatically from the motion detector reading. Thus, I’m collecting volume, temperature, and pressure data as many as 20 times per second.

In the picture to the right, you can see me using this apparatus to demonstrate PV diagrams at last summer’s AP Summer Institute at the University of Georgia. (Thanks to Laura Englebert, a physics teacher from the Atlanta area, for sending me the pictures. Woo-hoo!) I told Logger Pro to graph pressure on the y-axis and volume on the x-axis. Then, I slowly raised the piston, taking care not to let my hand get in the way of the motion detector. The graph showed a nicely hyperbolic curve – an isothermal process. But then I let the piston compress the gas rapidly. When gas compresses (or expands) quickly enough that there’s not enough time for heat to flow into or out of the gas, the process is adiabatic. Adiabatic compression on a PV diagram should jump to a higher isotherm, because the temperature goes up. Sure enough, while the process happens too fast to define the adiabatic curve, you can see that the graph ends up at a higher product of PV.

If you’re a bit lost in that last paragraph, don’t worry, it will make more sense once you get a chance to study the four major types of thermodynamic process that are tested on the AP exam – isothermal, adiabatic, isobaric, and isovolumetric. The 5 Steps book (now in a new and much-edited edition!) gives a good, short, readable treatment of these processes.

But note anyway that ANY portion of the ideal gas law can be tested experimentally! The linear relationship between pressure and temperature at constant volume? Plunge the metal gas cylinder into boiling water while keeping the piston from expanding. The linear relationship between volume and temperature at constant pressure? Do the same thing, but instead allow the piston to rise. And the experiment I previously described shows the inverse relationship between pressure and volume at constant temperature! Cool, eh?

Mailbag -- thermodynamics sign convention?

From Jonathan Kirby, an Atlantan:

"I have a quick question for you. We are just getting into thermodynamics, and I was wondering if I should teach "ΔU = Q-W" where W is the work done BY the system, or if I should teach "ΔU = Q+W" where W is the work done ON the system. Which way would be better (if either) for the AP Test?"

Answer:
The AP test changed to the ΔU = Q+W route in about 2002. Don't even mention the other way, unless your textbook does, in which case, good luck. :-)

(When I've used such a textbook, I've just repeated the correct definition over and over, and prayed.)

GCJ

Follow-up to multiple choice test corrections

Those of you who have attended my workshops know that, in Jacobs Physics, test corrections are one of the two most important components of the course. Sometimes, though, even the test corrections need correction.

Instead of assigning another round of “correction corrections,” I tend to just give the whole class a quiz when I find consistent misunderstandings. For example, consider the two multiple choice questions below. These were originally AAPT Physics Bowl questions, I believe…

1. A 2 kg object initially moving with a constant velocity is subjected to a force of magnitude F in the direction of motion. A graph of F as a function of time t is shown. What is the increase, if any, in the velocity of the object during the time the force is applied?

(A) 0 m/s

(B) 2.0 m/s

(C) 3.0 m/s

(D) 4.0 m/s

(E) 6.0 m/s

2. A deliveryman moves 10 cartons from the sidewalk, along a 10-meter ramp to a loading dock, which is 1.5 meters above the sidewalk. If each carton has a mass of 25 kg, what is the total work done by the deliveryman on the cartons to move them to the loading dock?

(A) 2500 J

(B) 3750 J

(C) 10 000 J

(D) 25 000 J

(E) 37 500 J

Many students showed an iffy grasp of these two questions on their test corrections. So, I posted to our class folder early last night. I noted that we would take a follow-up quiz today on these problems. I wrote the quiz to address specifically the mistakes that I had repeatedly seen on the first attempt at corrections. Here’s the quiz:

1. (a) What’s wrong with the statement “Work is done both up and to the right in order to move the boxes up the incline?”

(b) What is the direction of the force necessary to carry one box up the incline at constant speed? Justify your answer. Your justification should include a free body diagram.

2. (a) Explain why the average force during the time interval t = 1 s to t = 5 s is NOT 1.0 N.

(b) How do you get impulse from this graph WITHOUT trying to find an average force?

Going over a test

I know it's happened to you and it's frustrated you.  You give back a test, you discuss one of the more frequently missed questions, hoping for a teachable moment.  But half the class is rooting through the rest of the test, sitting back with a vacant expression, or simply absent mentally.  What to do?

One option, which I've discussed before, is to allow corrections for half credit.  Then there's no need for you to take too much class time to go over the test -- it's the students' job to figure out what they missed, and to convince you they understand now.  A related idea is to announce a "fundamentals quiz" over commonly missed concepts from the test.  Either way, the students are forced to think about the test beyond just "what did I get?"

Of course, test corrections are time- and manpower-intensive. You have to give time in or out of class to get the corrections done, you have to grade them as thoroughly as you would a test.  I only do corrections in my AP class -- I find the general class moves slowly enough that those who missed important points will pick them up soon. 

So how do I go over a test in general physics?  Well, keeping my comments brief and to the point helps.  But the key little trick is to HOLD THE TESTS IN MY HAND while I go over them. 

Here I'm playing with the students' minds.  They desperately want their tests back, but only so they can see the grade.  Once they see that grade, their mind is done for a while, and they don't want to think about physics.  So I use the grade as a carrot.  I dangle the papers with the grades on them right in front of the class.  Not obviously or obnoxiously, of course, but they are never sure when I'm going to shut up and hand out the tests.  And, they're nervous about what they did right or wrong.

So they listen.  And ask questions.  They want to hear what I say, so they can figure out whether they were right or wrong.  The same discussion AFTER I give the tests back would be fruitless.

How do I know this technique works?  Well, I don't for sure.  But I do note that folks occasionally note to their friends whether they did or didn't make the mistakes that I discussed... so they must have paid some attention.

GCJ

Introduction to vectors in general physics

Folks are often surprised that, in AP physics, my "vectors unit" consists of one problem, in which we show that a rope at an angle is equivalent to two ropes, one vertical and one horizontal.  We use sines and cosines to calculate the vertical and horizontal components of the rope's tension.  That's it.  And that's enough -- my class developes the skills necessary to break all kinds of vectors into components, and to add vector quantities, just as well now as when I used to do a week-long unit on vector math.

In general physics, I do a bit more mathematical prep work as we move into forces at angles.  But not much.

By now, my general physics students can deal with Newton's second law, as long all forces are either horizontal or vertical.  For example, we do problems with cars slamming on the brakes, and slowing due to friction.  Next, I want them to deal with forces at angles, like the lawnmower that's pushed along its handle. 

We start the process of breaking vectors into components with displacement vectors, not force vectors.  Why?  Because students are familiar with the concept of cardinal directions and shortcuts:  going 3 miles north and 4 miles west is the same thing as going 5 miles northwest, as long as you choose the angle correctly.  Everyone gets that.  If I can show them the mathematics in the context of displacement, then they can transfer those mathematics to force vectors.

Take a look at the assignment below.  Note that Will Collier is a real student, whom I taught in 2001-02.  He really does run a transport unit in Iraq, or at least he did last year.  The rest of the exercise is merely convenient science fiction.  I've done similar exercises before using maps of New York City, Berlin, and South Africa -- pick something of interest to one of your students!

Instructions for vector assignment:

Imagine that a new type of pilotless aircraft is developed. The plane is reliable and safe, yet inefficient; due to the vagaries of its design, this plane can fly ONLY along the cardinal directions: north, south, east, or west.

Will Collier, Woodberry class of 2002, runs a military transport unit. We will imagine that he has commandeered a large number of these aircraft to deliver goods throughout Iraq from a base in Baghdad. Your job as Lt. Collier’s unit navigator is to give directions to other Iraqi cities such that these special aircraft can fly there.

So, for each of the ten destinations you choose for Lt. Collier’s supplies, draw a displacement vector to the town.

Then, for each displacement vector, on unlined paper, using half a page per town, do the following:

• sketch the displacement vector and its components on x and y axes.

• state the magnitude and direction of the displacement vector.

• calculate the x and y components of the displacement vector, showing all work carefully as instructed in class

• write out, in words: "To fly to [whatever town], fly 200 km north, then 300 km west." Of course, fill in the correct distances and directions.

Follow these instructions (which are exactly what I said in class) carefully. I expect thorough, neat work. This should not take an exceptionally long time; work quickly but carefully. You should find this assignment to be the easiest all year. If you do not, come see me ASAP!