Ray Diagrams - templates for geometric optics

In an Honors or AP course, lenses and mirrors are covered both quantitatively and qualitatively.  This means that students must be able to draw ray diagrams to show the location and size of an image; AND they must be able to calculate image position and size using equations.  

The trick to teaching geometric optics at this level is to ignore the calculational piece.  Let the students figure out on their own how to use the calculator to solve the thin lens equation. Instead, practice drawing ray diagrams; and practice identifying the signs of each term in the relevant equations.

How I teach ray diagrams:

I only teach two principal rays for each optical instrument.  Why?  Simplicity.  Student difficulty in geometric optics does not come from the concepts themselves; the trouble is that all these lenses, mirrors, principal rays, objects, images, arrows, blend together.  

So for mirrors, I teach two rays only:  Parallel rays reflect to or from the focal point; rays toward the focal point reflect parallel to the principal axis.  For lenses, parallel rays either converge to or diverge from a focal point; rays through the center are unbent.

When the class practices drawing ray diagrams, I make them use these templates that I sketched out.  (Can you tell I flunked art class in seventh grade?)  The point is twofold: the templates force the students to use a full page for each diagram rather than cramming the diagram into a corner; and the rough sketch of the curved mirror saves the students from wasting all sorts of time using compasses or large circular objects to draw the mirror.*

* Seriously, before I used the template, a single mirror diagram could take a perfectionist student half an hour to complete, because he would obsess over the precision of the circular mirror shape. 

We identify by measurement the five relevant variables on each ray diagram: image distance, object distance, focal length, image height, and object height.  I've pre-drawn a table in the corner listing these variables in a column begging to be filled in.  The students get used to measuring di, do, and f along the principal axis, starting from the mirror or lens. 

Here, the trick is to require the students to include a sign for each entry.  What students think of as their failure to solve the thin lens equation correctly is almost never a true "math error:" rather, they generally screw up the signs of the terms in the equation.  So instead of practicing the calculation, we practice identifying signs.  

To simplify the conventions:

* converging instruments have positive focal length; diverging instruments have negative focal length.

* real images have positive image distance; virtual images have negative image distance.

* all object distances are positive [at this level].

* object and image heights are positive above the principal axis, negative below the principal axis.

I emphasize that for this exercise, a value without a sign is automatically marked wrong.  If a quantity is positive, I must see a + sign.  Otherwise, it's way too easy for a student to get most of the available credit by just putting numbers in the table; "oh, you know I meant positive, give me credit" is a disingenuous complaint when the whole point of the exercise is to force students NOT to assume that every measurement should be plugged in to the equation as positive.

It's worth practicing these diagrams for all four optical instruments until the students are bored with them.  Then optics questions on an AP or final exam will be thought of as free points.  

Reading Comprehension -- tip for teaching non-native speakers

Ever noticed how much serious reading comprehension a physics test requires?  Especially when a well-written test checks for nuanced conceptual understanding, prompts and answer choices must necessarily use complex verbiage.  

Whenever an English department gets its knickers in a twist about "reading and writing across the curriculum," I feel the words leaping to escape my vocal cavity:  "Have you ever read a physics test?  Do you realize the level of vocabulary, the verbal sophistication required to make sense of, let alone to correctly answer, a physics question?  Do you know how much writing instruction and practice must go on in my class, so that on a test students can condense broad physics ideas into a two-sentence justification, and do so in just a couple of minutes?  And on a related note, just how much are physics concepts taught or reinforced in the canonical high school English curriculum?  In studying Lord of the Flies, do you discuss how Piggy is nearsighted, thus the concave lenses in his glasses wouldn't converge parallel light rays, and thus couldn't possibly start a fire? Or do you dismiss the glaring physics error as "just a metaphor?"

[SLAP]

Burrito Girl, my wife and sidekick and English major, reminds me that at least teachers in her discipline rarely pluralize with apostrophes.  Thank you.

The point is, whether we even know we're doing it, physics teachers are teaching reading comprehension.  The weaker readers in our class, and the non-native speakers, are at a tremendous disadvantage in learning physics.  What can we do to promote physics understanding, and success on physics tests, for those who might be doing A-okay internalizing physics but are struggling with the words?

I had some interesting conversations with teachers at my last couple of institutes about this very subject.  Staci Babykin, of Montverde school near Orlando, in particular gave me some insights.  I teach a few students every year who arrive from Asia with book knowledge, but not practical knowledge, of English; Staci teaches far more students in this category, and her students are far less fluent than mine.  Yet she's been extraordinarily successful in developing her students' physics (and English) skills.  

Below are some of my suggestions, which freely and unabashedly borrow from discussions with Staci and others:

(1) Don't write many of your own problems from scratch.  Instead, use previously published resources from textbooks and standardized tests.  In addition to the wealth of on-level material available, an advantage to using AP, SAT II, or Regents questions is that professionals have already vetted the language for consistency, clarity, simplicity.  I know that I have a tendency to be overly colloquial in my own writing, or to use references that a non-acculturated student might not understand.  That's okay once in a while, as for a classroom demonstration where nonverbal cues are abundant.  But homework problems should prepare students for tests, which should prepare students for a cumulative exam, which should be written clearly, simply, and succinctly.*

* Unlike this blog post, for example. 

(2) Model how to communicate with mathematics.  A justification that says merely "Voltage is doubled due to Ohm's law" is not a justification at all.  I work hard with students to say something more like "By definition, the current I through series resistors is the same through each.  In Ohm's law, the R term is in the numerator, and not raised to a power.  So a doubled resistance with constant current mathematically gives a doubled voltage."

I ask native speakers to use words of explanation, because generally they can express their thoughts in words better than in equations.  The symbols of mathematics, even in basic algebraic equations like Ohm's law, are a language in themselves.  Most high school students must "translate" in their minds from words to symbols, because they are not fluent in algebra.

But non-native speakers, at least the ones I teach, are generally far more fluent in algebra than in English.  I teach them not to use words, but to use symbols:  put a line over the I to indicate it doesn't change, or just write the word "constant" and circle the I.  Then draw an up-arrow by the R and the V to show how they are related.  Or circle the R and V and write a little "x2" by each.  

By the end of the course, I would love it if my native speakers became comfortable with symbolic communication; and I would also love it if my non-native speakers began writing grammatical sentences expressing their understanding.  Just recognize that the combination of language and math skills might cause students to communicate differently.

(3) Figure out what "obvious" vocabulary needs explanation.  I do not give vocabulary quizzes.  Words that are generally new to everyone, like "adiabatic," are defined, and then learned through use in context.  Specialized vocabulary is not generally a problem for non-native speakers, because teachers and books are careful about communicating the definition and use.

It's the words that we might never think of as problems that trip up the non-native speakers.  Staci related the day she brainstormed with a group of students about all the words that, on a physics test, might as well be synonyms for "string": rope, cord, light string, cable, line, fishing line...

Or try "table":  surface, bench, desk, tabletop, hard surface, lab bench...

It is a skill, one that we can teach, for students to recognize commonly used words in physics questions; or, to guess in context what such words must mean.  

Got more ideas?  Stick 'em in the comment's.  :-)

  

Block Schedule Ideas

As I make the rounds at summer institutes, I meet folks who teach physics on all sorts of schedules.  It is a universal truth that everyone's schedule stinks, that it was designed by trained boardroom monkeys, etc.  

Thing is, there's a constant, legitimate tension between those who want lots of shorter class meetings (math, foreign language) and those who want longer chunks of time, even if that means missing a contact day or days (English, Studio Art).  And thus the problem for physics:  we kind of fall into both categories.  We need to see our students every day -- physics, like language and hard tack, must be digested in small, frequent chunks.  But, we need that weekly extended lab period, because data collection should be a slow and careful process without time pressure.

Toward the end of the previous century, a good number of physics-clueless administrators were trying to sandwich a year-long AP physics course into a single semester.  While it is true that a full year of 45-minute classes is the time equivalent of a half year of 90-minute classes, anyone with a pea for a brain and a few days of experience in a physics classroom can explain that time-equivalent and actually-learning-stuff-equivalent are not the same thing.  The College Board even experimented with offering a January AP exam to those who were forced to take AP physics from September to January only, with disastrous results.*

* Try proposing that the football team condense their practices... instead of two hours a night, four nights a week in the fall, just take the two weeks at the end of the previous school year after AP exams.  The team can practice all day, every day, to get ready for the season.  Then in the fall, we cancel football practice so as to teach classes all the way to 6:00 PM.   And the team will be ready to show up and win on Friday nights.  Right?

When I first became a consultant for AP, my mentor warned me to budget a good half hour per workshop to discuss schedule complaints, especially block schedule complaints.  I've tried to turn complaint time into a positive discussion, by collecting thoughts and ideas from participants about adjusting to whatever necessary schedule.  

In my own mind, only two schedule approaches are truly untenable for AP Physics: (1) Single-semester AP Physics B as a first-year course, and (2) Any sort of sandwiching both Physics C Mechanics and Physics C E&M into a single, first-year course.

Everything else can be finagled.  Generally, wherever possible, design an approach such that students see the same material multiple times throughout the year.  Four weeks on a topic in September is much less effective than two weeks in September, with the other two weeks of practice scattered throughout the rest of the year.

* Teaching 45-60 minutes daily, but without an extended lab period?  No problem.  After the first experiment, take care of data collection in class, even over multiple class days.  But do most of the analysis out of class as homework.

* Teaching 90 minutes per day on alternate days?  That's fine.  You'll have to do some tricks to maintain attention for 90 minutes, since not even physicists can sit still for 90 minutes of physics lecture.  The point is, the students' brains will have the necessary time to process what they've learned, and to develop the "muscle memory" they need.  This is one of the better block schedule approaches.

* What if you get AP Physics B for just the spring semester on a block schedule, but everyone had physics last year?  That's not a problem.  Be sure that the prerequisite physics course is rigorous, preferably at the AP level but just much slower than AP.  (Here's a post discussing just such a course.)  Then don't do explicit "review," but start the year with something new; review previous topics in context throughout the year.  Just be sure that no one can get into the AP course without the prerequisite.

* Here's a common scenario:  "Honors Physics" in the fall on a 90-minute daily block, then AP Physics B in the spring on a 90-minute daily block.  I would approach this the same way as the previous situation.  Teach the honors course at the AP level, but slowly.  Don't just do the mechanics portion of AP -- actually branch out and cover many topics.  Then the students who just take the first half of the course will be well served with a broad-based introductory course, and the AP students will have a wide foundation to build on.  Again, be absolutely firm about requiring the prerequisite.

* What about teaching AP Physics C - mechanics as a first-year course?  Many folks do this successfully, because Physics C - mechanics is really a semester-long course in college.  My own approach would be to spend the first half of the year teaching at the algebra-based physics B level.  Then, start the course over from the beginning in January, overlaying calculus approaches onto the previously-learned concepts.  

I have no doubt that other teachers have more and better ideas.  Post them below!  But please, no kvetching.  You're stuck with your schedule.  Share your ideas of dealing successfully with difficult situations; ask how others would approach your own schedule.  But there's no point in telling the world how awful your administration is.*

* Mainly because the world probably already knows.  :-)

Goldfish

I just finished out a four-day physics institute at Dunwoody High School near Atlanta.  I love running these institutes because I meet fun and interesting physics teachers,* and I always learn something new to bring back to my class for next year.

*Is there any other kind?

This year I have been showing off Wayne Mullins' electrostatics demonstration in water, in which I attach 20 V AC across two metal blocks in a shallow container of water.  We use a multimeter to map the voltage at different spots within the water.  The advantages of the water over the more traditional "conductive paper" for this experiment are twofold:  (1) You can feel the voltage viscerally by sticking two fingers of the same hand in the water -- the fingers tingle more with bigger voltage across them -- and (2) Tap water is much cheaper than conductive paper.

Dallas Turner, of Rockford, Illinois, noted that he had seen this very demonstration decades ago.  A college professor he knew had set up the electrodes in a slightly deeper tank than I use, and with dechlorinated water.  

Then the professor dumped in a bunch of little goldfish.

Aha!  The goldfish will align themselves along the lines of equipotential!  Just as you can spread your fingers as far as you want without a tingle as long as you spread them perpendicular to the electric field, so the goldfish can sit comfortably in the water as long as there's no potential difference between head and tail.  

In response to the obvious question from the animal lovers in our group, neither Dallas nor anyone thought that the goldfish were or could be harmed by this demonstration.  For humanitarian reasons,* when I do this demo I will be sure that the fish have a tank and caretaker** awaiting them after they perform for my class and for science.  We'll be sure to videotape the event.

*Defined as "avoiding the wrath of my wife, Burrito Girl"

**my nine-year-old son, who will name each individual goldfish

Use unlined paper for problem sets in honors / AP physics

I do a lot of posting about what has worked for my class.  Today I post about an idea I had that didn't work like I expected.

About a year ago, I redesigned my problem sets for honors students.  For over a decade, I had simply assigned end-of-chapter-style problems via email, and required students to answer each on a fresh page of unlined paper.  I changed the assignments so that each problem was typed out on the front and back of a page, AP free response style, with space to answer under each part.  These new problem sets looked like worksheets, not end-of-chapter problems.

The theory was, I wanted to get better attention to the meaning of the answers, and specifically to each step in the problem solving process.  By asking the question in stages, each of which couldn't be ignored, I thought I'd set students up for success, minimizing the helpless feeling of "I don't know where to start."  Furthermore, I figured I'd be able to grade more quickly and accurately, because I could turn directly to the part of a question that I most wanted to see answered.  No more hunting through a poorly-presented page of work!

In general, my "theory" was proven correct.  Grading was quicker, I got fewer students trying to hide the fact that their work was incomplete, I got full sentence answers to direct questions like "justify the reasonability of your answer to (c) by comparing it to speeds with which you are familiar."

It was the unintended consequences that proved dastardly.  Primarily, the perception of the homework as a worksheet to be filled out damaged the collaborative culture of the course.  

Students would check their answers to part (d) with each other.  But if they found that their answers were different because of an issue earlier on in part (b), they would not take the time to deconstruct part (b).  Previously, when the problem had been presented on a blank canvas, everyone saw the problem as a whole (even if the original text on a different page used parts (a) (b) (c) ) -- and so they discussed their whole solutions. 

I asked a lot of "place a check by one of the following, and justify your answer."  Students would verify with each other that they had checked the correct box, but would not discuss their justifications (figuring, I guess, that they had "collaborated" by looking to see that the right box was marked).  Without the checkbox and the space for answers, I used to get too-long essays as justifications; but the students communicated with me in writing, and with each other both in writing and in face-to-face discussions.  That's what I wanted. 

Yes, grading was quicker and less intellectually taxing,  because I could quickly find the student's response to each part of the question.  The unintended consequence was that collaboration became less intellectually involved, too, because the students never had to read each other's work.  When using a blank page, I often saw students going line-by-line through each others' solutions, trying to find mistakes or common ground.  (And I saw their presentation get better as not only I but also their peers criticized sloppy and haphazard solutions.)  When I dictated presentation style through my worksheets, student-to-student discussions became nasty, brutish, and short.

Note that I've been very happy with worksheet-style assignments in regular physics, and I'm going to use that style in conceptual next year.  In those classes, though, most students aren't ready right away for substantial multi-step, creative problem solving.  

I had wanted to move to the AP worksheet-style homework assignments for years, and only recently found the time to write the worksheets.  Turns out, that effort was essentially wasted.  Next time I teach honors or AP, I will be back to requiring a full page of unlined paper for each problem.

Three principles for "technology in the classroom"

Computers -- or, now, iPads -- in the classroom for every student sounds like a great idea, and can be a successful grant pitch.  But beware how they're used...

Personal story:  A few years ago, our amazingly awesome IT guy asked if I’d like to try out a tablet computer.  He wanted feedback about whether he should move the faculty or classroom computers from standard laptops to tablets.  I said, sure – I thought the tablet would be especially useful in drawing diagrams when I designed physics problems, and to organize the handwritten notes I often take at conferences or events.*

*Turns out I was wrong.

I trundled out the tablet for the first time at the AP Physics reading.  All the table leaders were gathered together to present and discuss the rubrics we had written.  I took notes on the tablet for 90 minutes.  Nevertheless, I felt the call of Evil Homer throughout… I wonder how Tiger is doing in the US open?  What’s the score of the World Cup match?  What time do the Reds play tonight? 

Look.  I’m an old man, a professional, and I was thoroughly invested in the meeting at hand.  Yet I almost had to slap myself several times when I started to check sports scores.  Imagine if instead I were a horny teenager* trolling for unwholesome trysts via facebook. 

* Ed. Note: remove redundancy 

Now, I’m not disputing that there are times when a class set of computers, ipads, whatever can be useful.  Most obviously, laboratory exercises with computerized data collection are fun and pedagogically effective.   But even when everyone involved has the best of intentions, the internet-enabled computer is a distraction.  Connectivity reduces productivity in a classroom-style setting.  

How, then, do you make useful use of technology?  I recommend three basic principles.

1. Be sure your unsexy, fundamental equipment is rock solid FIRST, before you go fancy.  Do you have a reliable xerox machine, scanner, networked printer, LCD projector, teacher's desktop or laptop workstation, and computerized data collection hardware and software (such as a labpro or passport)?  If not, what the heck is your school doing mucking around with iPads?  Get the basics taken care of before playing with anything else.

2. Don't go out of your way to design activities for your computers / iPads.  Most student-centered laboratory work in a first-year physics course should be done with 1950s-era equipment: masses, pulleys, stopwatches, resistors, lenses, etc.  You can and should use computerized data collection regularly for in-class demonstrations, of course.  I'm suggesting, though, that the computer can too easily become a "magic box" rather than a tool.  Be sure your students can graph by hand, and have done so bazillions of times, before you allow them to use excel.  The front-end instruction time teaching every student how to use, say, a vernier current probe, plus the frustration involved when they screw up, isn't worth it -- use a simple ammeter instead.

So when would you use a computer (or iPad)?  When the computer can do something that can't be done easily otherwise.  For example, give everyone a set of motion graphs, and ask them to duplicate them using a motion detector and a cart.  Use the iPad magnetometer app (or a Vernier magnetic field probe) to measure the magnetic field near a current carrying wire; a compass can work, but it takes a lot of math to get to the magnetic field value.  Use a free app like freqgen  (review forthcoming) as a continuous digital frequency generator.  I do astronomy simulation labs and investigations using "Starry Night," inexpensive computer astronomy software.  Certainly you can find numerous uses for the computer or iPad.  Just be sure that you're not using technology for its own sake, or to please an administrative request.

3. Be active during the activity.  I'm not suggesting that you be a warden, scanning the crowd for inappropriate websites.  Nor am I suggesting you go all Miami Dophins by imposing ridiculous punishments for misuse.*  Just walk around the class, and show authentic interest in their activity.  If you're on your feet, helping with software issues, getting involved in everyone's work, the class will be unlikely to screw around.  And if anyone does, you can deal with it immediately, quietly, and firmly.

* The Miami Dolphins, an American professional football team, gave each player on the team an iPad with their playbook included as an interactive application.  Great idea -- normally pro football playbooks are phonebook-thick.  Problem is, some idiot in the organization decided to threaten the players with a $10,000 fine for "inappropriate" use of the iPad, apparently including youtube, social media, or (presumably) porn.  Never mind that a professional football player can afford to buy his own dang CASE of iPads so he can do what he wants with them.  The threat of a fine was no more than bigshot coaches showing their players who's boss.