28 August 2014

Is projectile motion part of AP Physics 1? YES.

I got a note last night asking me to help settle an argument.  Since it's the second or third time this summer I've heard the meme, I figured I should address it publicly.

The question:  Is projectile motion part of AP Physics 1?

Answer: Yes.

The argument:  But the Physics 1 curriculum framework does not include any learning objectives with the word "projectile" in them.*  No LOs even say "two-dimensional motion" or anything similar.  Since the problems must be written directly to the learning objectives, nothing about projectiles can be asked.

* The Physics 2 framework includes one learning objective that requires students to recognize the similarity between a charge moving through a uniform electric field and a projectile.

Well, that argument takes an overly literal interpretation of the curriculum framework, I believe.   

My rebuttal:

1. Projectile motion is explicitly mentioned within "essential knowledge" 3.E.1:  "The component of the net force exerted on an object perpendicular to the direction of the displacement of the object can change the direction of the motion of the object without changing the kinetic energy of the object. This should include uniform circular motion and projectile motion."

2. Granted, the subsequent 3.E.1 learning objectives don't explicitly refer to projectile motion.  But I certainly could find a way to write a good problem involving a projectile that matches the 3.E.1 learning objectives, as well as to several others.  Problems can be written to multiple learning objectives.

3. At the AP reading, one of the points made by a College Board representative warned us about mechanistic teaching to learning objectives.  She said that in the biology redesign, a good number of teachers were flummoxed after they spent the year posting the learning objectives and teaching a course narrowly tailored to doing literally and word-for-word what the LOs said to do.  Her point:  the learning objectives are meant to be interpreted in the context of the "essential knowledge" statements, and are all interrelated.  The AP development committees are creating science exams, not the Law School Admissions Tests.  The learning objectives are intended to be flexible enough to allow for coverage of all the course material described in the framework; they are not intended to be narrow prescriptions of the precise questions that will be asked on the exam.

4. Look at the released AP Physics 1 practice exam:  questions 5 and 6 include a diagram with a projectile on it.  QED.

So teach projectiles.  Don't merely teach "here's how to plug numbers into equations to get answers to three significant figures," because that approach is just as doomed to failure in projectile problems as in other topics in AP Physics 1.  Be sure students understand the independence of vertical and horizontal motion, how a vertical acceleration alone produces a parabolic trajectory, how the velocity, acceleration, net force, kinetic energy of/on a projectile change or don't change depending on various aspects of its motion, etc.  But do teach projectiles.



22 August 2014

Required Reading: Kelly O'Shea's "Mistake Game"

Reason number 156 why I love teaching physics:  I don't care how long you or I have been doing this job, there's always something new to learn and to try.  

I looked again last night at Kelly's O'Shea's excellent "Physics! Blog!"  Kelly was the one who first articulated the holy grail of physics teaching, the ability of students to solve "goal-less problems."  While clicking some attractive links I discovered* not the holy grail, but perhaps the shroud of Turin:  Whiteboarding with the "Mistake Game."

* I also discovered that Kelly independently made a Clever Hans reference two years before I did.

I am well aware of the verb "to whiteboard," and I'm extensively tutored in its benefits.  It's not like I disagree with or disapprove of the technique; I just haven't yet found a place for it in my classes.  I've always managed to accomplish everything intended in a whiteboarding session by using pencil-and-paper.  Kelly's post has changed my mind, by introducing the critical element: the deliberate substantive mistake.

You should read Kelly's post directly for the details about her Mistake Game.  My quick summary:  Give several groups of students a problem to whiteboard... but each group is required to intentionally include one substantive mistake.*  The groups take turns presenting their solutions to the class.  Since they're charged with selling the solution, including the deliberate mistake, the students actively engage their audience.

* They're allowed to include as many unintentional mistakes as they'd like.

The mistakes are to be discovered through questioning.  Kelly says she has to teach audience members to ask substantive questions in pointing out the substantive mistakes -- not "hey, that's wrong, the slope is the wrong way" but "Which way was the car moving?" followed by "And how does a position-time graph indicate the direction of motion?"

I've always wanted to bring the methodology of the Physics Fight into my honors physics classroom, but I never found a way that made me comfortable.  Usually a typical honors-level problem is complex, but not complex enough to provoke real discussion and conversation except on rare special occasions.  As Kelly points out, the requirement of a substantive mistake facilitates that discussion -- the presenters vie to create a mistake that is "good" enough to pass the audience's examination, while the audience members vie not just to discover but to engage in conversation about the mistake.  Imagine that... to get authentic discussion, I need to require inauthentic but authentic-seeming mistakes.  Kelly, that's practically Zen.

GCJ 

15 August 2014

Mail Time: How much detailed attention to significant figures in AP Physics 1? (Answer: NONE.)

Mo here, a retired engineer teaching AP Cal AB & BC (for 13 yrs.), and now taking over the AP Physics program. 

I can’t seem to find clear information anywhere with regard to significant figures (SF) and how they’re handled by AP Physics readers. Being an engineer, you can imagine I tend to be very strict about SF, but I don’t want to overwhelm my students unnecessarily if the College Board does not stress it.

I would greatly appreciate any advice.

Hey, Mo... this is certainly a frequently asked question.  On one hand, don't harp on students about sig figs. If they put four rather than two sig figs, that's usually not in any way an issue.  However, if they're writing down every digit on their calculator, they're missing something, and they're possibly losing credit.  

The new exam does discuss experimental uncertainty.  Rather than stress rules of significant digits, instead you might talk in context about the uncertainty in a measurement.  Two measurements of 0.19 kg and 0.20 kg are equal, because by definition "0.19" means somewhere in bewteen 0.18 and 0.20.  However, two measurements of 0.191 g and 0.202 kg are NOT equal.  

But most importantly, your question about significant figures is nearly irrelevant for the new exams.  

Take a look at the released practice exam, and see how many questions require a calculated numerical answer.  Only one part of one free response question does:  that's 1(b), which asks for a calculation of external force on a cart based on a graph of experimental data.  Now look at the rubric for that part: of the four points awarded for this calculation, not a single one is awarded for the numerical answer.  

As with everything on the new exams, rote rules about arithmetic are far subordinate to the physical meaning of predictions and experimental results.  And calculation -- whatever the number of significant figures on the answer -- is far subordinate to explanation and description.  

Hope this helps...

greg

13 August 2014

AP Physics 1: Define the system carefully when doing energy problems

For years I've taught energy conservation via an equation.  Write the energy conservation equation for the object in question, identify each term, and solve.  That gets the right answer most of the time.  However, in AP Physics 1, getting the right answer is not as important as describing clearly how to get the right answer, and why the right answer is in fact the right answer.  I need to dig deeper for this new course.

In particular, I've always been fluffy about defining the system experiencing these energy changes.  I implicitly was referring to the single object experiencing a change in position or velocity.  That's "fluffy" because potential energy is created through an interaction within a system of multiple objects; and because I'm making all sorts of assumptions about what forces doing work should be included on the left side of this equation.  The students could use this equation to get the right answer, but they could not have described in careful, correct language the energy transfer.

The NTIPERS book provided me with a guide to teaching my students to use a deeper approach.  In each energy problem presented by this book, the authors clearly define the system being analyzed, and they list the forms of energy the system could possibly possess before, during, and after the process in question.  NTIPERS suggests making a bar graph for each problem listing the amount of each energy; words and numbers are probably enough, but the bar graph is a nice approach, too.

As an example: consider 50 kg Tarzan dropping on a vine from rest, as on the 2014 AP Physics B exam problem 1.  One part asks the student to calculate the speed of Tarzan as he swings through his lowest position (call it 2 m below his starting point), neglecting air resistance.  We can approach this two equivalent ways:

1. Treating the Tarzan-Earth system:  Systems can have potential energy due to their interaction.  Systems can also have kinetic energy, which is equal to the sum of the kinetic energies of the component parts of the system.  Define the lowest position of Tarzan as the zero of gravitational energy for the system.  That means the Tarzan-Earth system begins with mgh = 1000 J of gravitational energy, and (because nothing is moving) no kinetic energy, giving 1000 J of system mechanical energy at the start.  During the process, no work is done by an object external to the Tarzan-Earth system -- the only other object exerting a force on any part of the system is the rope, and the rope is always pulling perpendicular to Tarzan's velocity, so does no work.  Since the gravitational energy is zero at the bottom point, the entire 1000 J of system mechanical energy must be converted to kinetic energy.  Now use the kinetic energy formula to calculate speed.

Summarizing: 1000 J gravitational energy before --> no work during --> 1000 J kinetic energy after.

2. Treating Tarzan as a single object:  Objects can have kinetic energy, but not potential energy.  So Tarzan begins with no kinetic energy.  During the process, two forces act on Tarzan: the force of the string, and the force of the earth.  The string force does no work on Tarzan because it is always perpendicular to Tarzan's velocity.  The earth does work on Tarzan equal to Tarzan's weight multiplied by the distance Tarzan travels parallel to the weight, or 1000 J.  This is positive work, because the force of gravity is parallel, not antiparallel, to the downward component of Tarzan's displacement.  At the bottom, then, we know 1000 J of work has been done on Tarzan; this must change Tarzan's kinetic energy by 1000 J, since we're not treating Tarzan as a system with internal structure.  Now use the kinetic formula to calculate speed.

Summarizing: 0 J total energy before --> 1000 J work done by gravity during -->  1000 J kinetic energy after.

This may sound like semantics to you.  It certainly does to me.  But I'm telling you, the new exam will ask students to explain energy conversion and conservation with direct reference to an appropriate object or system.*  We must teach the students to describe energy conversion in the kind of language I've included above.

* Look at learning objective 5.B.1.2 in the curriculum framework, for example.

Furthermore, the students are expected to be able to understand when the model of an object or system fails.*  What does that mean?  We calculated 6.3 m/s for Tarzan at the bottom.  A follow-up question might suggest that Tarzan was actually moving 5.0 m/s at the bottom, and experiments verify that air resistance has no measurable effect on Tarzan.  The question may ask for a resolution of this discrepancy.

* That's LO 5.B.2.1 for those of you with an education doctorate.

A reasonable explanation is that treating Tarzan as an object isn't reasonable.  He could be spinning; the "missing" mechanical energy is accounted for by his rotational kinetic energy.  Similar questions could be posed in which an object seems to have less energy than predicted by a point-object model, but some internal structure in motion accounts for the total mechanical energy budget.


09 August 2014

Trading and Grading Homework Problems In Class: The Context of Your Feedback Matters.

Years ago I described how I grade nightly homework assignments quickly.  Point of emphasis: I don't write on the students' papers.  No one reads my comments carefully.  I just put a number on the paper, record that number, and give it back the next day.

I've also suggested that it's not effective to "go over" a homework problem in detail in class.  Johnny might have asked you to go over the problem, but he'll tune out once he finds where he thinks he lost points; he won't learn a lasting lesson.  And everyone else in the class has tuned out, because they either got the problem right to begin with, or they just don't care right now.

The purpose of grading homework, for me, is NOT to give detailed feedback.  It's to be sure that students are keeping up with the assignments, that they are practicing their problem presentation, that they are engaging appropriately with the material.  If I don't grade homework, the homework doesn't get done properly.

It's reasonable to ask, then: If I'm not writing more than a number on a homework problem, and I'm not discussing the problem in class, how should a student who wants detailed feedback get it?

Timing matters.  Your students have a bobzillion things going on in their lives.  Like it or not, physics just isn't a constant priority.  In order for feedback to be meaningful and useful, it has to be given at a time when the student is ready to receive it.  

One effective time for feedback would be as the student is struggling with the homework problem, immediately after he's completed the problem.  Thing is, you're usually not present at just the right moment; and, if your student is actually doing the homework problem in your presence, he's going to want the feedback* before the time is right.  

* And you're likely to give him the feedback too early, too.  Students must engage with the problem.  It's a good thing for them to get stuck.  One of the more important physics teaching skills is to find a way to politely yet firmly deny assistance to a working student until the time is right.

One way to provide appropriately-timed help is to set up part of class such that the time will be right for feedback.  Have the class solve a problem while you sit in the front of the room, encouraging students to see you after they've completed each step.  Just the fact that there are twenty other students needing your attention will usually convince students to keep working until they're well and truly stuck.  Then the faster workers can experimentally verify the answer to the problem while you help the slower folks.

But as for homework, students need to learn to rely on their classmates.  Whether the students are working in small physics parties at night, or whether they just send questions to each other via text, email, facebook, etc, they are helping each other become un-stuck.  That's the whole point of collaboration -- not only so everyone can make appropriate progress on tough problems, but so they develop the experience of teaching the problems to someone else.

Context Matters.  So Set Up A Useful Context For Feedback.

Imagine you've talked to a student explicitly about how Newton's third law force pairs can't act on the same object.  Perhaps you said so in class, and perhaps you also helped this student get un-stuck on a homework problem where he was trying to have the force pairs "cancel" each other to equilibrium.  And then, on the test two weeks later, he made the same mistake again.  Aarrgh!

Think about this student's response when you helped him.  In class, he listened attentively, maybe even wrote down some words; on the homework, he nodded, said "okay," and corrected his mistake.  Remember, everything we do in physics class is fleeting.  We like to think that students are motivated by their grade or their love of the subject to remember everything we say and everything they're asked to do.  But really, once the problem set is done, so is the student.  Even if the timing of your feedback to this student was right, the context wasn't.  

I like to create multiple contexts, each of which has some level of import.  For example, I give a quiz every day.  On the quiz, I might ask directly: "Can Newton's Third Law Force Pairs act on the same object?"  or, "The earth pulls on an astronaut, and an astronaut pulls on the earth.  What is the acceleration of the earth-astronaut system?"*  Even though the student had to use this fact on the previous night's homework, he might not have really internalized what he was doing -- after all, students are answer-focused, no matter how hard we try to change them.  This quiz provides the same question in another context.  When we go over the quiz, now the student hears the answer, and hopefully will make the connection to the homework problem.

* Expecting the answer "It can't be determined," knowing that too many folks will say "zero."

And finally, I've taken to having students grade homework problems to a rubric once every week or two.  While generic "going over" a problem is ineffective, describing the solution in the context of awarding points to a friend's paper is extraordinarily effective.  Without my prodding, students often write detailed explanations on their classmates' papers.  After the in-class grading exercise, students have done the problem, seen my solution, applied my solution to grade a friend's problem, thought about what score they themselves might have earned, then seen their friend's detailed commentary.  Now THAT'S context.

31 July 2014

Horse Games -- Don't Play 'em.

Take a look at this cart.  It's moving to the right as it comes to a stop.  What is the direction, if any, of its acceleration?  Mr. Van Houten, what do you think?

"Um... right.  NO!  LEFT!"

Okay, which one?  Right or left?

"Left."

Why?

"Oh, so I guess it's right.  Right?"

Mr. Van Houten... don't play horse games with me, man!  Let's figure this out the right way so you don't have to guess.


In the early 1900s, word spread around Germany that a special horse named "Clever Hans" could solve simple arithmetic problems.  He would answer by stamping his foot the correct number of times. Amazing!  Even skeptics couldn't figure out how Clever Hans kept getting the answers right.  They made Hans's trainer go away when they asked the questions -- but the horse still stamped the correct answers, so no one was cheating.  Either the skeptics had to acknowledge a horse with mathematical ability superior to the average politician, or else something unknown was afoot.

Hans had essentially unfettered success until he was asked questions by people he couldn't see; then he failed.  Hans also couldn't answer questions asked by people who didn't already know the answer.  Hmm.

The logical conclusion was that the horse was extraordinarily attuned to peoples' body language.  Those who asked Hans questions would tense up as Hans approached the correct answer... then they would sag with some emotion -- relief, for most, or resignation, for skeptics -- when Hans would make the final hoof tap.

For many physics questions, your students could have an easier time getting the right answer than even Clever Hans would.  The choices are often limited to "right," "left," or "neither."  Or perhaps "greater," "less," or "equal."  In any case, students who equate physics understanding with merely choosing the right answer -- or students who just want the teacher to shut up and stop talking to them -- can read the teacher's body language to nudge them in the correction direction.  

Now, I may not be a good poker player with cards, but I'm a pretty good physics teaching poker player.  No matter my student's answer to an in-class question, I keep my face flat, and ask "why."  We establish early on that trying to read my face does no good.  I look the same for a wrong answer as a right answer, waiting for the explanation.  It's impossible to feign understanding.

When, inevitably, a student enters into the dialogue described above, I tell him -- and the whole class, if possible -- the story of Clever Hans.  "Don't play horse games with me.  Physics is not about getting the right answer by any means necessary.  It's about explaining the natural world using the facts and equations we've learned.  It's about communicating clear predictions about experiments, then verifying those predictions with equipment in the lab.  It's about refining those predictions when they turn out to be incorrect.

"But reading my face to get the right answer so I'll shut up?  That's not physics.  That's not even possible, as you just discovered.  So don't be Clever Hans.  Just answer the question as best you can, and if you're wrong, that's okay, you'll learn something."





23 July 2014

Dashboard Monk -- where to find the "spring popper" toy


Remember the 2009 AP Physics B exam problem 1?  It described an experiment in which a "spring in a

small jumping toy" was compressed, allowing the toy to propel itself to a measured height.  Varying the mass of the toy caused the measured height to change.  By appropriate graph linearization, the spring's force constant could be determined.

But where to get such a toy?  Thanks to Drew and Robbie in my Tampa APSI for showing me.  Try googling "Dashboard Monk," or click right on the link.  I've seen these toys with Easter Bunnies, pastel eggs, and other themes.  But I've never found them other than serendipitously in dollar stores.  Now you (and I) know how to order these online.

Once you have a class set of these $5 toys, you can do the experiment posed in the 2009 exam.  To vary the mass, stick some putty on the top of the toy; or, wrap some solder around the base.  How do you measure the height to which the toy rises?  Well, that was the last part of the 2009 problem!  Many ways are possible.  I think I will pose just this question one day to a class, saying "determine the maximum height of this toy with nothing other than a stopwatch."  

When you actually do the full experiment, you could certainly use additional equipment to determine the maximum height.  Procedures suggested by students on the AP exam itself frequently referred to video analysis -- put a meter stick in the background and go frame-by-frame near the top.  A fellow reader joked that a student could put his forehead above the toy and see if he got bopped, moving his head progressively higher until the toy merely brushed his eyebrows.*

* No idea whether that's an apocryphal or an authentic answer.

One of the best parts of physics teaching is going to toy stores with the science department credit card in hand.  Thanks Robby and Drew for showing me where to shop.

18 July 2014

Open Lab 2014 -- last call

Folks, the 2014 Open Lab begins a week from Sunday.  On July 27, at least ten physics teachers will descend upon Woodberry Forest School to discuss whatever is on your mind.  I've got plans to show and develop some in-class lab exercises, work with my new Vernier and Pasco rotational equipment, take an after-lunch trip to the hardware store to see if we can build something cheap and useful... mainly I'm looking forward to hanging out with interesting physics teachers in my classroom and around town.

If you're planning on coming, I need to know right away -- I'm leaving town for an AP Institute shortly, so I'm making nearly-final preparations.  I won't turn you away even if you show up last minute, of course.  However, you might not have a seat at our Sunday night dinner, a nametag, or a seat on the minibus.

Remember, the Open Lab itself is free; but you need to arrange your lodging (I recommend the Holiday Inn Express in Orange), and you need to be prepared to pay for a couple of meals while you're here.  We start officially at 4:00* on Sunday, and we're done officially at noon on Tuesday.  I can also be available a few hours before and after these official times.  

* 4:00 P.M, not A.M..  I'm crazy and dedicated, but not THAT crazy and dedicated.

If you can't make it in 2014, I'll likely run something similar again in 2015 if things go well this year and if there's interest.  Just let me know...

GCJ

14 July 2014

Direct Measurement Videos

Youtube has been a staple of the physics classroom for a while now.  There's no shortage of interesting videos to analyze, some of which are specifically created for science purposes.  My favorite of the produced-for-science infotainment genre is probably the Smarter Every Day series.  Take a look at the episode about how cats land on their feet.  It's got amazing footage, an entertaining host, high quality production, solid physics, and CATS!  I wouldn't use this video as a teaching tool, though.  My in-class demonstrations and labs are never designed purely as a show.  Everything I set up in class has a quantitative predictive element.  My students will certainly enjoy Smarter Every Day, but as a supplementary extra-fun part of learning physics, not as something integral to the course.

So can a video series possibly be integral to an introductory physics course?  Yes.

Take a look at Direct Measurement Videos, a series produced by Peter Bohacek, Matthew Vonk and several other Wisconsin physics teachers.*  Each video in their library shows live footage of an experiment.  Post-production work provides enough information to make, well, direct measurements.  For example, the frame number and frame rate are displayed prominently.  Where appropriate, a length scale is superimposed on the action.  Multiple camera angles are shown when useful.  

* The full credits on the site list Peter Bohacek, Matthew Vonk, Ellen Iverson, and Karin Kirk.  I mention Matthew in particular because he was my table leader at the AP Physics reading.  Most of the videos seem to be credited to Peter.

While the production quality is solid, Direct Measurement Videos are emphatically NOT edited for infotainment purposes.  You won't see a narrator.  Many of the clips would seem humdrum to the non-physicist: a car braking on an ice rink, a marble colliding with a wooden block, a doll rotating on a turntable.  The excitement of these clips is that they bring to life the end-of-chapter problems that we've been assigning for decades.  

Aside:  I got myself in trouble at a consultant meeting when I cheeked a non-physics-teacher presenter.  His brief speech was full of enthusiasm but empty of substance -- I wanted to get back to talking physics teaching.  The guy kept going on about how we as consultants should emphasize real-life physics.  So I raised my hand and asked him, "Could you please give us an example of physics that is NOT 'real-life' physics?"  The reaction was as if I had cited Adam Smith at a 1980 Moscow State University economics department meeting.

Of course all physics is "real-life."  The central tenet of my own teaching has been to highlight the experimental nature of physics problem solving, to the extent that I refer not as much to the "answer" to a problem as to the "prediction" made by the problem.  I set up quantitative demonstrations that have little wow-factor, but which verify the prediction made by in-class example problems or homework problems.  

Direct Measurement Videos have taken that philosophy to a whole new level.  In class, I'm limited to the equipment I own, the space in my classroom, and the tools (such as Vernier's live computer data collection) that allow for immediate analysis.  DMVs have no such limitation.  They can show a roller coaster at 200 frames per second -- I have no nearby roller coaster.  They can show the slow-motion, frame-by-frame results of a dart sticking to a cart -- while I can do that experiment, I analyze with Vernier motion sensors only, and my students can't repeatedly rewind the live action to see the moments before and after collision.

My intention is to use a Direct Measurement Video about once a week as a homework problem in a different sort of "flipped class."  Traditionally I've assigned a textbook-style problem for homework, and then we've set up the physical situation in class.  Now, though, I can have the students work through a textbook-style problem in class to make a prediction, then assign the data analysis off of the video as homework.  Or, rather than assign a problem which gives all relevant input data, I can link a video and say "determine the coefficient of friction between the doll and the surface."  The students have to do more than just make a calculation; they have to figure out what data must be acquired to make that calculation, and then they have to figure out how to acquire said data from the video.  We can get away from the idea of physics problem solving as putting given numbers together in the right equations; we can get away from the idea of laboratory as a separate, distinct, disconnected portion of a physics class, instead integrating predictive and experimental physics on an everyday basis.  Wow.  Thanks, Peter, Matthew, et al.


08 July 2014

5 Steps AP Physics 1 is out...

The 5 Steps to a 5: AP Physics 1 book is out.  You can buy here via amazon.  I've completely rewritten the text from previous Physics B version because, well, the AP Physics 1 exam is completely different.  Last week in Barnes and Noble, I couldn't find any other prep books for the new exam.  And I don't trust the competing companies' books, anyway.*

* I suspect they're all superior to Barry Panas's "1 Step to a 1," which offers a money-back guarantee.

I really like the approach to content in this year's book.  Instead of textbook-style exposition, example problems, and practice problems, each of the content chapters intertwines exposition and examples.  In the spirit of the new exams, the "examples" don't pose a particular problem to solve -- getting hung up on the exact correct answer rather than the process and explanation will be a particularly harmful bugaboo on this new style of exam.  Rather, the examples pose situations which are fertile ground for all sorts of kinds of problems.  Then, I suggest what kinds of questions could be spawned from each situation, and I show how to answer each of these questions.

The book includes a variety of practice questions in each content chapter, and a complete practice exam written directly to the AP Physics 1 learning objectives and science practices.  And already I've found the first major error... that's inevitable, of course, in a first edition.  Even though I worked through everything multiple times, the first time I read the published version I found the stupidicism.  In the forces chapter, I ask a question about a block pulled along the ground by a string angled up 30 degrees above the horizontal.  Great... except that in the solution I state that the normal force on the block is equal to its weight.  Duh.  Since the string force has an upward vertical component, the normal force on the block must be LESS than the block's weight.  Grrr.  I'll have to rewrite that one for the second edition.

Any other comments, ideas, complaints, or suggestions can go in the comments, or can be sent to me via email.  Enjoy the book