Classic whirling stopper experiment, with smart phone video

In the classic old-fashioned circular motion experiment, students whirl a rubber stopper in a horizontal circle above their heads.  The stopper is tied to a string.  The string is threaded through a hollow tube - that’s what the student holds to whirl the stopper - and attached at the other end to a mass hanger. The equipment for this experiment can be procured from your storeroom  -- don’t buy the “discover centripetal force” kits  that are now down to $20 from $39.

My class's version keeps the mass on the hanger constant.  Students vary the radius r of the circular motion in each trial, measuring that and the speed v of the stopper's circular motion.   They plot a graph of v² on the vertical axis and r on the horizontal axis, and take the slope.

Experimental tips: I don't ever give handouts with procedural instructions.  For this experiment, I demonstrate the acquisition of one data point, ask for a graph of v² vs. r, and shut my mouth.  In that demonstration, though, students see rather involved and somewhat difficult technique:

• This is the one experiment all year where I insist on lab goggles.  It's easy to get poked in the eye.
• The whirler should keep an eye on the hanger.  If it's moving up or down, the stopper's speed isn't constant.
• The stopper's speed is measured by dividing the circle's circumference by the period.  A decade ago, we measured ten periods with a stopwatch - I was never happy with the precision here, because it's tough to keep constant speed for that long.  But now, we use "hudl technique" or "coach's eye" apps.  That is, we take a video of one cycle, and use frame-by-frame advancement to find the period to the nearest hundredth of a second.
• The radius of the circular motion is measured with a meterstick immediately after the video is complete.  That's a trickier measurement than you might think - students have to pay attention so that the string doesn't slip along the plastic tube.

Analysis: Since the hanger is in equilibrium and the rope is assumed to be horizontal, the tension T in the rope is equal to the weight on the hanger.  And that tension is then equal to mv² / r where m is the mass of the stopper.*

*Not the mass of the hanging thing!  When setting a net force equal to ma, the mass in question is the mass of the thing that has the acceleration.  That's the stopper here.

The goal is to use the slope of the graph to determine the stopper's mass.  In my formulaic method of teaching graph linearization, students begin by writing the relevant equation and solving for the variable on the graph's vertical axis.  Then, they write the equation for a line y = mx + b.  They circle the variables that match the y and x axis... what's left is the physical meaning of the graph's slope (or y-intercept, but that's usually zero).

We do this analysis in class, where I talk students through each step of the process above.  I check their graph, check their slope calculation, make them communicate in writing every analysis step, and then verify that they have a reasonable mass for the stopper.  Not the right mass of the stopper - they're not allowed to measure that directly with a scale!  No, I'm making sure that answers like "15 kg" or "1.5 kg" or "0.15 g" are exposed as unphysical - without even entering the lab, a person who knows what a kilogram is could reject these as possible masses of a bottle stopper.  And I'm checking that everyone is using two digits throughout.* 

*Not because we have to memorize rules of significant figures for the AP exam! No!  Because (a) two digits is the best we're going to measure pretty much anything in a first year physics laboratory, (b) the precision of this particular experiment is limited by the two-digit radius and period measurements, and (c) most importantly, insisting on only two digits throughout the year on everything (rather than ignoring excess precision, or teaching those sig fig rules that no high school student understands) is one of the ways in which I help my students develop habits of mind relating numbers to physical reality.  

Once everyone finishes, I do use a scale to show the class the stopper's mass.  I give a prize to whichever student(s) got closest.  Since we started using frame-by-frame video on smartphones, our mass values have been not just reasonable, but often correct within 10% or so.  

I've fallen back in love with this experiment now that the long road to getting the analysis right pays off with good results.  I do mean long road.  The analysis by itself might take more than an hour; data collection is 30-45 minutes for well trained students.  

Circular motion linearization: Grocery Cart

We were asked on the "Pretty Good Physics" message board to give ideas for linearization experiments with circular motion.  Here's an old-fashioned version that uses 1960s equipment.  If I remember, I'll post a second old-fashioned experiment, but with a modern twist.

Grocery cart: My colleague Wayne did this one in my first year teaching.  He cleared out the desks in the classroom, and marked with tape the biggest circle he could possibly fit.  Wayne had, somehow, acquired a standard wire grocery cart, the kind we all used to love riding in.*

* or on, much to our parents' dismay.  Hah, hah, Mom, I never fell off and** had to go to the hospital, did I.

** Ed. note: use of "and" not "or" is deliberate here.

One student rode in the cart.  Another student pushed the cart from behind.  And a third student stood in the circle's center, pulling on one of them giant spring scales that can be read across a classroom.  The scale was connected via rope to the student in the cart.

The pushing student - chosen as someone who had rhythm - walked at a steady pace around the circle.  While the center student read the spring scale (to get the net force F), others used stopwatches to measure the time it took to go around the circle.  The crew repeated for many different speeds.

The speed of the cart v was calculated to be the circle's circumference divided by the measured time.  A graph of F vs. v showed a parabola.  To linearize, we plotted F vs. v².  The slope was the mass of the cart+rider divided by the radius of the circle.  The class compared that slope to a direct measurement of these masses via a bathroom scale.

Finally, in this K-12 school, Wayne kidnapped* a small child to ride in the cart.  The class repeated the experiment and graphed F vs. v² on the same axes. Even without any calculation of slope, it was apparent that the smaller mass in the cart had created a shallower slope.

* "Want to take a break from your class to ride in a grocery cart for a science experiment with high school students?" The child joined Wayne willingly, though I could not easily interpret their teacher's expression.

Notes from observing a poetry class - three things that DIDN'T happen.

As part of my school's faculty development program, we're asked to observe a teacher of our choice outside our department.  In 2015 I watched John Amos's 9th grade English class.  This year, I watched Ansel Sanders teach poetry to 11th graders.  I saw an enjoyable class.  But I also noted three things I didn't see: no soapboxing, no trolling, and no negging.

I'll begin with the same prologue I gave to my previous English class observation.

So many "Physics Educators," and "educators" in general, have the Soviet attitude that if only everyone did things my way, students would learn better.  I disagree.  I've always been open at my workshops and on this blog: my ideas, philosophies, and suggestions are mine alone, developed in the context of my personal strengths and weaknesses, and shaped by the ecosystems of the three schools at which I've taught.  What I do cannot work for everyone.  Yet, it's still worth sharing my thoughts, techniques, and ideas.  Not in the sense of "do these things and you will become a great physics teacher;" but rather, "here are a few ideas you may not have considered; try them, and then either throw them out or adjust them to make them your own."

So here's my extensive reaction to Ansel's class.  I will not be adopting wholesale any of his particular techniques; but I appreciate the exposure to some different ways of approaching my craft.  Many of Ansel's ideas are in the back of my brain now, ready to manifest -- consciously or subconsciously -- in my own classes.  In other words, I bought myself a few new tools.  Whether and how I use them is discourse for a future time.

What happened in Ansel’s class? Ahead of the class, Ansel sent me a packet of about five quatrains to read.  The students had been discussing the definition (or, as students and he pointed out in class, the not-definition) of poetry; they had been discussing how to read and understand poetry, and how that differed from reading and understanding prose.

In class the previous week, they had discussed Dudley Randall’s "Ballad of Birmingham" - discussed meaning, audience, how the audience’s own experiences affect the meaning, etc.  I would have loved to have seen that discussion with this particular group of students, most of whom I know and respect from other areas of school life.

In Monday’s 8:45 AM class, then, Ansel began by asking groups of students to write poetry word association on the board - basically anything they wanted to write to “get the brain juices flowing.”  This led to a review / follow-up discussion about the purpose and not-definition of poetry.

In the last 15 minutes of class, Ansel assigned groups to go deeper into poetry consumption.  We discussed three ways in which people can engage with poetry, and what each means - poetry reporter, poetry interpreter, poetry performer.  Each group was assigned to engage with a poem of their choice in each of these three roles.  By the end of class, my group had reported on the who-what-when-where of "Sadie and Maud" by Gwendolyn Brooks.  One student seemed amused to know that I had TWO great aunt Sadies, one on each side of the family.  That fact, and rareness of the names in later generations, helped us approximate the “when” of the poem.

I particularly enjoyed watching a member of my group read "Sadie and Maud" for the first time.  He did what Ansel said to do - he read it several times, even talking aloud a bit.  He reacted viscerally… “yeah, I’d rather be Sadie, you don’t have to do exactly what your parents say.”  That comment spoke to exactly what Ansel had been getting at… every person is a unique and biased audience for a poem.  This student's reaction was pure, and in the moment. He hadn’t been prejudiced by his classmates’ interpretation, or what the teacher might have said, or the compiler’s essay, or even the teacher’s expectation - Ansel wasn’t anywhere nearby, and the student was treating me like a classmate.  This was authentic.  I’d never seen that before, at least not when I knew what I was seeing.

That's a summary of what happened.  

It's just as important to ask, What didn’t happen in Ansel’s class?  Three things jumped out at me.

(1) No one got on a soapbox to overemphasize the moral rectitude of their interpretation.  At least, not that I saw.  That’s part of why I would have liked to have seen the previous discussion of the "Ballad of Birmingham" in this multiethnic class.  In my college English class, too many of my classmates would have insisted on an interpretation that advanced their political agenda.  Ansel (a white guy in his 30s) mentioned his own visceral reaction to the poem from the perspective of a parent of three daughters… but in my college English class, the response would likely have been, “well, I suppose, but you don’t understand the African-American experience, so let’s stick to what the author obviously intended.”  And that would have been coming from a rich white 18 year old from Princeton, New Jersey.  Ansel emphasized taking personal meaning from each poem, even though each person in the poem’s audience brings his or her own biases, perceptions, backgrounds.  

(2) No one was intellectually dishonest - no one knowingly chose a ridiculous interpretation, then died on a hill with a smirk defending their deliberate but attention-getting stupidity.

I’ve thought deeply over my career about culture building in the classes I teach.  I’ve always focused on building a positive and “safe” class culture, where students can make mistakes and share their thoughts without fear of being put down by their classmates. Nevertheless, especially in physics, we can’t engage in relativism, we can’t think that all ideas are equal.  They’re not.  Right and wrong do exist.  In an educational environment where students have been trained that being wrong is equivalent to being a BAD BOY, it’s very difficult to strike a balance between freely sharing thoughts, and learning right answers from wrong answers.  But that’s the heart and soul of culture building.

In physics, right and wrong isn’t determined by the teacher or by peers - right and wrong is always by reference to experiment.  When a student irrationally insists that his idiosyncratic interpretation of Newton’s Second Law must be correct, I can say, “bet you $100?” and do the experiment.  It’s not mean ol’ Mr. Jacobs telling the student he’s wrong, it’s not his classmates shouting him down, it’s the universe itself.  Hard to argue with the universe.

In English, in poetry, right and wrong so often are in the eyes of the beholder.  Yes, poems can and should mean different things to different people.  But the "Ballad of Birmingham" is NOT a reference to the Odyssey; "Sadie and Maud" (published in 1962) is NOT a deliberate prequel to The Handmaid’s Tale.  

(The more likely trollish interpretations in this time and place would be something like “It’s racist against white people to assume that 'Ballad of Birmingham' might have unique personal meaning for a Black person from the South,” or “'Sadie and Maud' tells us what happens when abortion is illegal.”  Guh.  Don't know how I would've reacted.  Thank goodness I didn't have to find out.)

I know - from hearing students and faculty talk - that Woodberry students do in fact play the troll in class on occasion(s).  And I know that faculty and some students are in fact frustrated with said trolls, but don’t know what to do about them.  Engage their arguments on their merits, and you legitimize them (both the arguments and the trolls); ignore them, and they scream ever-louder for attention.  

My own approach is prevention… no one is generally shouting all-out bold bullcrap to start the year.  They test the teacher with small-scale trolling at first.  So, at the very first sign of a bad-faith comment, I shut it down hard and move on before the proto-troll can argue.  

I wonder, what has Ansel done to eliminate trolls?  Is he just lucky that this class is devoid of troll wannabees?  Has he shut down proto-trolls?  Does he have a different approach, one that I could learn from?

(3) No one put down classmates for their thoughts - not verbally, not with body language.

Just as it bothers me when students take faux-intellectual positions in defiance of reality, it’s disturbing to me to hear about classes who gang up to otherize a teacher or a peer.  I felt this as a student in my own years-ago English classes - if I said anything that peers might disagree with, or anything that my peers had to work to understand, I got withering scorn.  I know some of this was gendered, as 16-20 year olds in a co-ed setting were often as interested in jockeying for social status and sexual partners as in authentically discussing literature.

Obviously discussions aren’t generally gendered at Woodberry Forest.  [It's a boys' school.]  But our boys yearn to belong to the "Woodberry brotherhood."  The social norm of the brotherhood as I see it in many classes is that we take intellectualism seriously but not literally.  That we go through the motions, we get good grades, we have personal interests, but as a group in non-honors classes we don’t engage authentically with authority figures.  (We bro it up with teachers - that’s different.)

But Ansel’s class did engage authentically, with him and with each other.  Sure, one student was falling asleep, but he just stayed out of discussion - and the one thing he said was on-point to advance a discussion.  Pretty much everyone made a good-faith contribution.  I’m aware that this level of engagement doesn’t happen in December without some serious work on Ansel’s part in September.  Does Ansel have any techniques that he consciously uses to build this positive and open class culture?  Are there things in his personality or class structure that naturally build culture?

I see one thing that jumps out at me… Ansel is loud and enthusiastic, his eyes roam the room, fixing on student after student.  And not just his eyes - Ansel roams the room, making it seem like he’s having a one-on-one dialog with a person who asked a question, while simultaneously engaging the rest of the class with his body language.  He has, in a word, stage presence.  In two words, anyway.

It’s clear who the alpha dog is in Ansel’s classroom.  I’ve always said, freshmen are like puppies - once they know that the teacher loves them and that the teacher is unquestionably the alpha dog, they’ll do anything the teacher asks.  Juniors are big dogs… yet, Ansel has convinced them that he’s the leader of the pack, that he’s worth following.  I'm impressed.

Amy Johnson's AP Physics 1 Workbook: Use it. I will.

By now most AP Physics 1 teachers are aware of the new “Workbook”, available as a free download from a teacher's College Board course audit page.  If you’re not, get aware right away! Amy Johnson of the College Board spearheaded the production of a 350 page set of scaffolded activities.* All of them, of course, are perfectly aligned with the content, depth, and skills demanded on the AP Physics 1 exam.  They’re classroom-ready - once students have some exposure to the underlying facts and general problem solving approaches, these activities can be handed out and used without edits, even by the most persnickety AP teachers.**

* Oops, buzzword alert!  The workbook is in typical class coverage order - kinematics, then forces, etc.  The “scaffolding” means that the early units give lots of guidance, assuming less facility with the various skills we expect AP students to develop throughout the year.  For example, an early worksheet might present a justification in paragraph form but ask students to fill in the blanks; a late worksheet would just say “answer in a clear, coherent, paragraph-length response” as the exam does.

** i.e., me.

The workbook is for new or out-of-subject teachers.  We all know it’s common for people to be asked to teach AP Physics when they’re not confident in their own physics skills, let alone in their ability to choose, solve, assist students with, and evaluate practice problems. Or more bluntly: too many AP Physics teachers don’t know much about physics, and know nothing about physics teaching… yet they’re assigned to teach physics anyway.  What can these poor folks do, other than follow the poorly-written, six-decade-old* textbook?

*Yet in the 30th revised edition because the numerical values in end-of-chapter problems have been changed regularly, like diapers.

Now, they can use Amy’s workbook. A student who goes diligently through each page over the course of a year will be guided to develop a solid background in the necessary content AND skills required on the AP exam.

The workbook is also for experienced teachers.  I’ve taught physics for a quarter-century.  I’ve developed my class activities and practice problems piecemeal over that time, creating or revising one thing at a time.  

And, so has Workbook author Amy Johnson.  She’s a real physics teacher - plucked out of the trenches of the classroom, not out of the gilded halls of academia.  She’s also an experienced AP reader.  Thus, Amy has a tremendous sense of her audience.

The workbook certainly is inspired by the same best practices of physics pedagogy that underlie the AP Physics 1 exam itself.  Worksheets guide students to become fluent in multiple representations, to use verbal reasoning, to approach semi-quantitative and experimental problems.  None of the worksheets could possibly have been plucked out of the 1988 edition of Giancoli or Serway.  

Of course, there’s no shortage (anymore) of problems and activities that use best practices of physics pedagogy.  Released AP questions (on Collegeboard.com or via AP Classroom), TIPERS, The Physics Classroom, Vernier’s Pivot Interactives, and even the “Elite Student Edition” of the 5 Steps to a 5: AP Physics 1 book.  Why use the Workbook, then?

(1) It’s free.  The Physics Classroom website is free; the other items above all cost varying amounts.

(2) It’s aligned to the AP exam.  (TIPERS, The Physics Classroom, and Pivot are not explicitly aligned to AP.)

(3) It’s scaffolded.  (Even though the 5 Steps Elite Student questions are aligned to AP, they are not scaffolded.  5 Steps is intended as a review book.  The Workbook is intended to be used more like a textbook.)

How am I using the Workbook?  See, I already have a set of assignments and activities for each topic that are well-vetted.  That doesn’t mean I’m happy with all of them!  In particular, while my students end up doing fine with waves and sound, I know my assignments can be improved.  So, I’ll replace my in- and out-of-class waves/sound assignments with a bunch of the Workbook activities.  

Similarly, my colleague teaches an “honors physics” class that is close to AP, but doesn’t manage the entire breadth of the AP Physics 1 content.  Every year he has a few students who want to take the AP exam, but who need to learn about waves and rotation on their own.  Enter the Workbook - since my colleague has developed strong skills in these students, they will have little difficulty working through the Workbook to get themselves an appropriate background on these topics.

How are YOU using the Workbook?  I’d love to know.  Post a comment below...

Can conceptual students learn calculational methods? Yes. Look at this voltage problem.

On the conceptual physics trimester exam, I showed the circuit above, then asked, "estimate the voltage across the 10 ohm resistor."

In conceptual physics, we teach, well, conceptual approaches to all problems.  That doesn't mean we never use equations or calculations, of course; it's just that even when a problem does require mathematics, students are generally required to explain an aspect of the mathematics in words.

For example, we teach students to approach this particular circuit problem with an equation:

* The current in each resistor is the same - that's a fact that "series resistors take the same current"

* Mathematically by V = IR, then, with the same I the bigger R takes more voltage V.

* The voltage across series resistor adds to the total.  So the 10 ohm resistor takes less than 18 V, but more voltage than the 5 ohm resistor.

With this reasoning, we accept *any* answer more than 9 V and less than 18 V.

In class, the students each conduct this kind of estimation, then set up the circuit on a breadboard.  They find out experimentally that the voltages are in fact 12 V and 6 V.  After a bunch of similar predictions and experiments, my 9th graders develop for themselves an instinct - in series resistors, the bigger resistor takes more voltage, and the bigger the difference in resistances the bigger the difference in voltages.  That's not written down anywhere, 'cause it's not exactly rigorous.  But these conceptual students can estimate voltages for series resistors better than most seniors can calculate them.

Greg, when your students get to college, they'll be asked to calculate precise voltages.  I know you love conceptual physics, but the vast majority of college physics courses are calculational in nature. Aren't you doing a disservice to your students by teaching physics in a manner so different from how they’ll be asked to learn in their future?

Before I address this (legitimate!) concern on its merits, I’ll point out that it is emphatically NOT my job as a high school teacher to “prepare students for college.”  I provide for my students what they need right now… not what they should’ve had in the past, not what they may possibly need at some ill-defined point in the future. (But in Zen-like fashion, even though I adhere zealously to this philosophy, I've nevertheless received overwhelming feedback that, four years out, my students feel extraordinarily well prepared for college and college physics.  Go figure.)

I have plenty of anecdotal evidence that those who start their study of physics primarily with concepts (rather than primarily with calculation) retain more and deeper knowledge, and perform better if they continue to another physics class.  If you want them, I know you can find academic-style research studies that say the same thing.  But let's talk about just last month's trimester exam. and just this question.

Remember, I did not ever teach, model, demand, suggest, hint, or advise students about a caclulational method to finding the voltage across each resistor.  Not once.  We did sometimes ask students to calculate equivalent resistance of series resistors by adding the resistor values.  We did sometimes ask students to calculate the current through each resistor, using the fact that says "In ohm's law, use the voltage of the battery with the equivalent resistance of the circuit."  But voltages across series resistors?  Conceptual students were only taught how to make an estimate.

Nevertheless.  On this exam problem... a large majority of the class made a correct numerical calculation!

Without prompting, so many students calculated the current by using the 18 V battery with the 15 ohm equivalent resistance to get either 1.2 A (or "6/5 A" because they had no calculator).  Then, they recognized - usually wrote out in words! - that the current in the 10 ohm resistor is also 1.2 A because it's a fact that series resistors take the same current that's equal to the total current.  And finally, they used ohm's law across the 10 ohm resistor to get (10 ohms)*(1.2 A) = 12 V.

In other words... when the class had only, exclusively been instructed on a conceptual approach, they nevertheless figured out for themselves a useful calculational approach.  And they did so in a few moments, under the pressure of an exam.  I'd say 3/4 of my class did this, and 9/10 of those who tried it did it right.

So am I doing a disservice to students by teaching them conceptual approaches?  Nope... they can learn how to calculate without difficulty.  They don't even have to be carefully taught.

Don't present work = force * distance right away. Start with energy bar charts.

I've been using annotated energy bar charts to good effect for many years now.  They force students to define a system and to identify what forms of energy are present in that system before they consider the end goal of a problem.  In this manner they serve the same purpose as free body diagrams for Newton's Second Law problems - I don't care what you're supposed to solve for, draw the free body or the energy bar chart first, and then you'll be able to solve or explain any aspect of the problem.  These devices make problem solving about physics understanding, steering the process away from a game of "let's plug some numbers in to some equations and do some algebra to get the desired answer."

But I'd always been hung up on the definition of "work."  See, students can come to an operational definition of potential energies (both spring and gravitational) and kinetic energies just through their equations.  It's simple for a first-year student to see that gravitational energy must be included in the bar chart when an object has a height above the lowest point.  "Because PE = mgh and h is not zero" is a good enough reasoning for now.  Similarly for kinetic energy: "The cart has KE because it's moving, so it has a v in KE = (1/2)mv²."  Work isn't so simple.

Work can't be understood simply from the equation "W = Fd."  Students must understand the idea of a force being parallel or antiparallel to a displacement.  The scalar nature of work.  The meaning of negative work.  That this equation only applies when a force is steady.  The difference between net work and work done by an individual force.

Until this year, I'd always spent part of a class defining work, and giving examples of when work done by various forces is positive, negative, or zero.  I showed how the net work on an object is the (scalar) sum of the work done by all forces; or, how net work done on an object is just force times distance, where "force" is the net force.   Then we did a couple of problems for practice.

And every year, I came up against the corollary to Rule Two of teaching.

• Rule Two: Your students don't listen to you.  That's okay, they don't listen to me, either.  
• Corollary: Nevertheless, students have sharp and selective recall of what you say when it provides maximum inconvenience.

I always hoped and expected that, as we moved to energy bar charts, students would adapt to using "W = Fd" as the same sort of operational definition as "PE = mgh," only with complications that we'd discussed at the outset about the meaning of work.  

Well.  My class certainly did remember W = Fd.  As an equation to be applied with brute force in any and all situations, to the exclusion of the energy bar chart.  Which stinks, 'cause the vast majority of problems which ask "how much work is done..." should be solved by finding the difference in total energy at two positions - that is, using an energy bar chart, not the equation W = Fd.

I turned the tide this year - by ELIMINATING the discussion of the definition of work.

Yeah, that's counterintuitive, isn't it... but that's how so much of teaching works.  More often than not, the less you present formally, the more and better comprehension you get.

This year, we started with energy bar charts in the simplest of situations - just with gravitational energy, kinetic energy, and work done by an external force.  Importantly, I didn't say what exactly work is.  We showed how work could be positive or negative, depending on what the bar chart requires - for example, if kinetic energy decreases with no change in gravitational energy, then negative work was done on the system in order to keep the right number of bars in the energy bar chart.  The students had access to a fact sheet which explained the direction of force and displacement for positive and negative work, but I didn't discuss or even mention this in class.

And so, every problem became an energy bar chart problem.  

Evidence, Greg?

Our trimester exam included several energy questions.  No one tried to use "W = Fd" inappropriately.

In particular, I assigned the classic problem to test this misconception.  Consider a baton: two massive objects connected by a very thin, light rod of length L.  The object of mass m is vertically above the object of mass 2m.  How much work is required to flip the baton 180 degrees so that the heavier object is on top?

Most years, the vast majority of the class uses W = Fd, getting all sorts of incorrect answers.  They make the d term equal to a part-circumference of a circle, they use all combinations of mg and 2mg and 3mg for the ill-defined "force"... ugh.

This year, no problems at all.  They made an energy bar chart.  They saw that gravitational energy went from mgL to 2mgL.  They saw that there must have been mgL of work in order to make the bar chart work.  QED.

I've never seen such success on this problem.  And all because I *didn't* teach something from the front of the room.