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24 November 2022

Happy Thanksgiving from a different sort of family...

Although I thoroughly enjoy hunting for hand turkeys on AP exams with my friend Jen, I don't actually like turkey.  And the other two members of my household are both vegetarian.  So the main course of our Thanksgiving dinner is shown above: Shari's macaroni and cheese, based on a recipe I discovered on a sports blog.

The best part of a trimester system is giving a major exam right before Thanksgiving break.  Invariably, my students outperform what I might have predicted a month ago - because they take this major exam quite seriously, because they have matured both physically and mentally in the early part of the year, because we spend significant time in and out of class reviewing facts and basic recall items.  And I discover these predominantly positive exam results right before the national day of thanks. 

Keep the faith, folks.  I know that most of us teaching physics don't have a major exam to see direct evidence of the great progress our classes have been making.  But I know it's there.  Maybe you have to wait until a December or January exam; maybe you'll have to manufacture a few high-stakes testing situations to assure yourself that the progress you're seeking is, in fact, there.  

This is what a "summative assessment" is for.  No, it's not torture intended to shame and expose our students failings; it's like a regular season football game, a chance for our students to see where their strengths and weaknesses lie.  And, if we've been practicing well and building strength all season, we expect we'll be seeing far more strength than weakness.

The strengths are there, and will continue to get, well, stronger throughout the year.  Eventually, you'll hear your students say "remember when these kinds of problems were hard?"  Not yet.  When we get back from break, they'll say, "wow, I did way better on this exam than I thought I would!"

Right NOW, though, they're eating and watching both kinds of football, enjoying the break from rigorous academic effort.  As am I.  Happy Thanksgiving.

14 November 2022

Modified Atwood machine that *slows down*

The problem above is from 5 Steps to a 5: AP Physics 1, the "Elite Student Edition."  This situation provides an opportunity to find out how well students are doing with understanding the meaning of acceleration.

Ask: immediately after the carts are pushed or released, in which experiment is the magnitude of the carts' acceleration greater?  And, are the directions of the top cart's acceleration the same, or different, in the two experiments?

The correct answer refers to free body diagrams of both carts in each experiment.  The hanging block experiences forces from the rope and the earth; the only unbalanced force on the top block is that of the rope.*

*Yes, the top block also experiences balanced forces from earth and the surface.  Those aren't relevant to the acceleration, because they are balanced. 

Those free body diagrams are the same in both experiments!  By Newton's second law, the same unbalanced force acting on the same mass causes the same amount of acceleration.  And similarly, because the unbalanced force is in the same direction in each experiment, the accelerations have the same direction.  Done.

But what about the initial push?  All forces other than gravity** require contact.  While I was pushing the cart, the cart experienced a force from my hand.  After that - and the question explicitly is asking about AFTER the cart has been pushed - my hand isn't touching the cart, and so cannot apply a force.

** in mechanics, anyway

Okay, but isn't the cart on the surface moving to the right in Experiment B?  So its acceleration is in a different direction than in Experiment A, when the cart moves to the left.

Um, no - here is precisely the misconception this question is designed to test!  Motion is emphatically NOT the same thing as acceleration.  

When an object speeds up, its acceleration is the same as the direction of motion.  When an object slows down, its acceleration is opposite the direction of motion.  

In Experiment A, the cart is moving left and speeding up, giving it a leftward acceleration.  In Experiment B, the cart is moving right and slowing down.  Thus, its acceleration is opposite the direction of motion... so the acceleration is also leftward.

11 November 2022

Mail Time: how do you rank the forces on a block dropped onto a spring?

I was asked about a ranking task in which block 1 is dropped onto block 2, which is attached to a vertical spring.  The blocks compress the spring, and come briefly to rest in the position shown in the right-hand diagram:

We're asked to rank the (magnitude of) three forces at this instant of maximum spring compression:
  • Fa is the upward force of block 2 on block 1
  • Fb is the downward force of block 1 on block 2
  • Fc is the downward gravitational force of the earth on block 1
 Student 1's reasoning was:

We know that Fa is equal to Fb because they are a Newton's 3rd law force pair.  Because the blocks are at rest when the spring is maximally compressed, they are in equilibrium and thus forces on each block are balanced.  The forces acting on the top block are the force of block 2 upward (here called Fa), the force of the spring upward, and weight downward (here called Fc).  Setting up = down, we see that Fc has to be greater than Fa.  Final ranking: Fc > (Fa = Fb).

Which parts of this reasoning are right?  Which are wrong?

Correct: Fa and Fb are in fact a Newton's 3rd law force pair.  The force of block 2 on block 1 is equal to the force of block 1 on block 2.

Incorrect: The blocks are not in equilibrium at the instant of maximum compression.  Equilibrium means no acceleration, so no change in speed.  Here, the blocks may be at rest momentarily, but their speed is always changing!  For the blocks to be in equilibrium, they must be at rest and not changing their speed from zero.  

Incorrect: The force of the spring does not act on block 1!  All forces (in physics 1) other than gravity require contact.  The spring is not touching block 1, and so does not apply a force to block 1.

So how to solve, then?

Draw a correct free body diagram for the top block.  This has the weight downward (called Fc), and the upward contact force of block 2 on block 1 (called Fa).  The top block has an upward acceleration: when an object speeds up, its acceleration is in the direction of motion.  This block is starting to speed up and move upward, so its acceleration is also upward.*  Acceleration is in the direction of the unbalanced force - so the forces on the top block are unbalanced upward, meaning the upward Fa is greater than the downward Fc.  This, paired with the Newton's 3rd law pair, leads to the correct answer (Fa = Fb) > Fc.

*What about right before the block came to rest?  When an object slows down, its acceleration is opposite the direction of motion.  Just before the position of maximum compression, the block was slowing down and moving downwards... also giving an upward acceleration!

25 October 2022

What does "relational learning and collaboration" mean? It means creating a team culture in the classroom.

One of the official long-term goals for my school’s faculty is to improve “relational learning and collaboration”.  Great!  But the first step in attaining this goal is to, um, define exactly what this phrase means.  And, as you may expect if you’ve read this blog for a while, I have a different view than what most would come up with.

Everything I’ve read, everything our faculty has discussed, uses the term “relationship” exclusively in the context of teacher-to-student.  One-to-one.  If the teacher builds positive relationships with each student, they’ve done a good job.

To me, that view is myopic.  Building relationships in a classroom isn’t a one-to-one process, or even one-to-many.  

It’s many-to-many.

The teacher’s role is to help build relationships among classmates, to develop a team-like atmosphere such that the whole class works together to learn a challenging subject.

What do football players or band members recall at their 30th reunion?  Probably not the details of what to do when coach calls “flood-right 213 dallas”.  Probably not the count on which horns go to the box in the last bit of the closing number.  Okay, some skills (weightlifting, preparation for performance) stick all one’s life.  But that’s not what participants remember.

What’s memorable about participating in an activity are the relationships built with teammates.  Not just winning the game/competition, but the celebration with friends afterward.  Not the hard parts of practice, but the support each player got in those difficult moments from coaches and teammates.  You recall the inside jokes, the pregame chills shared, the way the group reacted to a win - and to a loss.  Teammates - even teammates with little else in common - share an everlasting bond, because they worked together positively in pursuit of a common goal.  

Why should physics class be any different?

It’s the teacher’s job to give their class the space and independence necessary for them to build those relationships with each other.  Lab time, “come-and-show-me” exercises, test corrections, and unstructured time to solve problems together can all be culture-building activities.  But if students are always sitting and listening - no matter how communicative and empathetic the teacher’s presentation - they can’t have the small-scale unstructured interactions that build into long-term bonds.  

A good football coach is conscious of building a positive team culture.  How do you react when a teammate drops a pass they should have caught?  You love them up, making sure the dropper knows they’re still loved and supported.  What happens after the drum major screws up the very first command in the show?  No one feels worse than the drum major themself.  So, the rest of the band needs to give them love, show them they are still respected and supported.  

A good teacher must be similarly dedicated to a positive class culture.  When students make fun of classmates, it’s the teacher’s job to step in right away: "We don't do that here."  It’s on the teacher to ensure that each student feels okay in making mistakes.  Not “okay” in the sense that we’re allowed to say that an object always moves in the direction of a force; but “okay” in the sense that the whole class understands that such misconceptions are natural and persistent, and that the whole class must help each other make progress toward a correct understanding of the natural world.

That’s collaboration.  That’s “relational learning”.    

05 September 2022

An email I wrote to my AP Physics 1 class after their second problem set

We didn't do great - but we never do, on this problem set.  The two questions I'm addressing here are based on the graph below, where they were asked to calculate an instantaneous speed at t= 4 s, and where they were asked what familiar object might perform this motion.  

Please read on, 'cause there's an extra credit word buried here.

1. The instantaneous speed is given by the slope of a tangent line.  That means draw the tangent line, and take the slope.  That does NOT mean to just divide the y value by the x value!  Just because speed is in units of m/s does not mean you can divide any old m by any old s!  :-)  Look below - the black line is the tangent line at t=4 s.  The green and blue show how I've calculated the rise divided by the run.  NOT 22 m/4 s= 5.5 m/s!  But see the calculation below:

2. Yes, a cart, an airplane, and a rocket can all speed up, slow down, and turn around.  But look at the vertical axis.  This object goes 50 meters (60 m - 10 m) in 8 seconds!  My tracks are only 2 meters long.  A football field is about 100 meters long.  (For the Americans, a meter is about the same as a yard.)  So if it takes eight seconds for an airplane to go half a football field, well, that's not a very good airplane.  (Unless the airplane is taxiing, but then the speeding up and slowing down and turning around are crazy!)

I'd think a bicycle... start at the goal line and speed up, slow down as you approach midfield, and turn around.  But anything with a reasonable discussion of how long it takes to go several dozen meters is fine.

(Not a roller coaster... remember, an object in these graphs can only go two directions, toward or away from the detector.  The graph isn't a picture of the track that the coaster is on!)

3. If you would like an extra point on Wednesday's quiz, please write the word "astonishing" on the quiz.  While you may not share this word with anyone - make them read the email themselves! - you may encourage other people to read this entire email.  :-)

I'll see you Wednesday!  Keep writing facts!


23 August 2022

Mail time: a daily quiz question about average speed

What is the purpose of this question from one of the daily quizzes in your AP class? My guess is that you're highlighting a misconception.

True or false: One method of finding an object’s average speed is to find its speed during one part of its motion, add to the speed during the rest of its motion, and divide by 2.

Yup, you got it!  I'm highlighting this amazingly common misconception.  Usually the misconception appears in response to questions like "a car travels the first 50 km of its trip at a constant speed of 20 km/hr, and the second 50 km of its trip at a constant speed of 40 km/hr.  What is the average speed for the whole trip?"  The answer is NOT 30 km/hr, because more time is spent in the first 50 km than the second 50 km.  But no matter how we try, still a significant portion of the class will say "average speed?  Oh, I know how to take an average!  (20+40)/2, of course!  :-)

When I *do* pose this kind of problem to my (AP) classes, I ask it more conceptually: "a car travels the first 50 km of its trip at a constant speed of 20 km/hr, and the second 50 km of its trip at a constant speed of 40 km/hr.  Is the average speed greater than, less than, or equal to 30 km/hr?" I mean, they could do the full-on calculation of the time spent in each half of the trip, then use the definition of average speed = total distance / total time to get 27 km/hr.  But the simple conceptual approach is to notice more time was spent at 20 km/hr than 40 km/hr, so the average speed will be closer to 20 km/hr than 40 km/hr.  

(In my lowest-level conceptual physics course, the whole concept of "average speed" is out of bounds.  Let's get students understanding basic one-dimensional motion with constant speed, or with speeding up from rest / slowing down to rest.  The whole idea of "average speed" is extremely confusing, even to strong second- or third-year physics students!  So save the complexities for advanced courses.)

15 August 2022

Setting up AP questions in the lab: 2016 P1 #2, the bouncy ball experiment

A unique feature of my in-person summer institutes is the final morning of the week, when participants are asked to choose a released AP physics question to set up in the laboratory.  I think of it as "studio time", which functions similarly to my AP class's open-ended lab assignments throughout the month of April.  In studio time, we not only create these setups, we share our ideas and creations.

This is my seventh post based on experimental setups of AP free response questions!  The others:

* Block and cylinder on incline (Milo Jacobs) (2021)
Student on a raft (Stephen McAliley) (2014 practice exam multiple choice)
Which battery drains first? (Frank Prost, Tom Mellin) (2017)
Energy vs. time for a cart on rough surface (Nadia Lara, Joey Konieczny) (2015)
Bumpy Track (Zach Widbin) (2016)
Waves on a vertical string (Walter Keeley) (2016)

Today's setup and data comes from Matthew Bourbeau, a participant in my July 2022 Palo Alto High School workshop.  He set up the problem referred to by AP readers as the Black Hole: 2016 P1 #2, the bouncy ball problem.  

I loved this problem when it came out* because it provides no alternative to authentic, creative approaches to lab.  It rewarded students who could look at a situation they'd never seen before, and come up with a simple experimental procedure from scratch.  It emphatically did NOT reward students whose entire laboratory experience, if any, had been following directions on a lab sheet with pre-set equipment.  (The AP laboratory problems since 2016 have been improved in terms of the simplicity of wording, and I appreciate that.)

The original problem poses a hypothesis about a bouncy ball: that its low-speed collisions are elastic, but its high-speed collisions are NOT elastic.  Students are to design an experiment that could test this hypothesis.

The elegance of this problem is that neither I, nor any test-taker, knows nor ought to know whether the hypothesis is valid or invalid!  This is true science.  Not "is this true," but "how would we know whether this is true?"

Well, it's been six years... but I've finally seen data from an experiment that tests the hypothesis.  

Matthew used the track shown above, inclined such that this lacrosse ball would roll down and hit the wall.  The motion detector at the right read the ball's speed immediately before and after the collision.  Matthew collected data for collision speeds between about 15 cm/s and 90 cm/s.

On the 2016 problem, students who knew what they were doing opted to analyze several different useful graphs.  One common approach was to graph the speed right after collision on the vertical axis, and speed right before collision on the horizontal axis.  The hypothesis predicts a graph that starts along a 45 degree slope, but then drops below 45 degrees for larger initial speeds.  

Matthew chose a less common approach (though one that I saw many, many times): use the speeds to calculate kinetic energies right before and after the collision.  Divide the KE after the collision by the KE before the collision to get an "elasticity percentage".  The hypothesis predicts this value to begin at 100% for low initial speeds, then to drop below 100% as the initial speeds increase.

Matthew's results - shown below - emphatically reject the hypothesis.
The "elasticity percentage" never is close to 100%.  In fact, the elasticity percentage for Matthew's lacrosse ball remains about 50-60% for all impact speeds he measured.  

Now, to be fair to the hypothetical student who posed the hypothetical hypothesis, the exam does state that a "new kind of toy ball" is supposed to behave this way.  A lacrosse ball is not exactly new (though I had never personally heard of the existence of lacrosse until I arrived at college in Philadelphia in 1991).  Yet, Matthew did the experiment with the materials at hand, and got clear results.  Soundly rejecting a hypothesis with experimental data is, in fact, good science.  Thanks, Matthew!

22 July 2022

Don’t let high school students revise – make them start from scratch

Greetings from the NSTA conference in Chicago.  Today’s post sprung forth after I attended a session about “chickenology” – two Ohio science teachers who are also chicken farmers use the creation of a gravity-fed chicken feeder as an engineering design project.  They bring their chickens into school to test the designs.  Fantastic.  I attended the workshop alongside Abbie Mills, Woodberry’s engineering teacher, and we got to talking about the design process in the high school classroom – how the chickenologists approached design, how Abbie does, how my wife Shari the ceramics professor does… and how I do in physics class at the smallest scale.

I’ve written already about how my test correction process became optimized when I stopped showing students their original test.  They now get only a blank version of the test, with indication of the parts for which they didn’t get full credit.  They are forced to look at every problem anew, without any indication of how they first tried to solve the problem.  Sure, some students try to reconstruct what they did originally… but they realize pretty quickly that they’re on a fool’s errand.  They just start from scratch.

Why don’t I let them see they’re original answers?  “Please can I see, so I don’t make the same mistakes again?” goes the reasonable request.  But chances are, this student won’t make the same mistake again, if they start anew with a fact or equation straight off our fact sheet, especially if they first have a conversation with a friend about the correct starting point.  And then if they do screw up again, they’re ready right now! for me to show them the conceptual error they’re making.  

But more importantly… students are unhealthily wedded to their first attempt at anything.  Having their original answer in front of them changes the exercise from “solve the problem correctly” to “see if I can explain why I was right, or at least partly right.”  Interestingly, Abbie and Shari independently have come to a similar conclusion about larger-scale design processes.  And both of them were high-level practitioners of engineering/ceramics already before they entered the classroom to teach those subjects.  They know.

Abbie holds a discussion after their first project with stick towers.  Whatever design they start with, she says, the students never, ever just tear it down and start over.  Even when their internal group discussion comes to a consensus conclusion that their design isn’t gonna work. So the post-project discussion becomes, “When did you figure out that your design wasn’t going to work?  Oh, like three class periods ago?  Why, then, did you continue to try to patch it up rather than just redesign?”  Good lesson, eh?

Yes, good meta-lesson, one that I’m glad is being taught.  Yet, I want to focus on the physics lesson at hand, not the meta-lesson.  I want to skip the whole three-day process of using hope and duct tape to finish, and instead force the student to begin anew.  For the long-term goals of my class, I want students to develop confidence in the correctness of their solutions so that they can scaffold their understanding to more and more complicated physics problems.  So I never ask students to revise anything at all.

Understand that this penchant for sticking with the first approach to a school project until death do us part is not unique to physics or engineering!  Shari faces the exact same issue with art students.  They are quite resistant to improving a project that needs more work – they want to be finished.  But if she takes the first attempt away and makes them start over, they produce an improved piece every time.  Same thing goes, she thinks, for her English teaching days: she was so, so frustrated that students “revised” an essay only by making small cosmetic changes in response to direct input from the teacher.  She thinks that had she taken the first draft completely away, made general comments, and demanded a new essay from scratch, her students’ writing would have been much improved.

As always, I’m transparent with my pedagogy.  As students bring me responses to check – lab predictions, test corrections, in-class problem solving exercises, anything – I either stamp them off as correct, or ask students to redo from scratch on a blank sheet.  The first time I place a student’s incorrect response in the garbage can, I see horrified faces: omg, that’s so harsh!  They see quickly that my calm, smiling, polite explanation of why, pedagogically, they need to start from scratch is in no way harsh.  They see that other students meet the same fate, such that my objection to their wrong physics is in no way personal.  And over the weeks they see how quickly they’re improving their physics understanding… to the point that the students themselves begin to trash partially correct responses without any intervention from me.  

19 July 2022

Lab always takes longer than you think...

I just finished this year's Conceptual Physics Summer Institute.  It's always fun to share laboratory ideas, especially the "come and show me" exercises that form the ground state of my courses.  

At the institute, participating teachers did two of these activities - one with motion graphs, one with circuits.  The question I was asked throughout the weekend was, "how long do you spend on these labs in your class?"  And the answer is, for activities that took only minutes for each individual teacher, I generally spend a full week of class.  

Folks were a bit incredulous, I think.  That's a long time for what seem to be simple "worksheets".  Why do they take so long?  I've got two reasons, among others that people can suggest in the comments:

(1) I'm not just expecting students to get answers and experimental results.  For the prediction portion of the lab exercise, I'm expecting thorough justifications in exactly the format and style I've modeled for them.  No shortcuts are allowed - I only allow them to do the experiment once I've checked their prediction for that thoroughness.  

For the experimental portion of the exercise, I expect clear data that stands up to scrutiny.  At least one second of motion detector data that matches the predicted graph.  Properly obtained speeds, masses, or times that match prediction within 20%.  (For most exercises other than resistors-in-series, if they don't match prediction, they redo either the prediction or the experiment.  We don't just say "human error" and move on.  Grrr.)

(2) Teachers - myself included! - oft underestimate the discrepancy between those tasks which we can do on automatic, and those which the students can NOT.  See, even teachers who are unfamiliar with the "come and show me" methodology are familiar with the ideas that underlie the facts and equations used for prediction.  Students are not.  No, not even if you spend 10-20 minutes going over the facts and equations with the students, showing them exactly how they apply to the exact situations they'll be dealing with.  Your class is starting from square one, and the only way for them to advance to [square>1] is through slow engagement.  They must find the right fact - even if sometimes they find the wrong fact and have to try again.  They must use the equation correctly - even if they not only use the equation incorrectly, but they also are using the wrong equation to begin with.

Oh, then consider the data collection process itself!  Making the simple measurements I ask for with motion detectors, photogates, smartcarts, voltmeters, etc. should be a two minute process - 90 seconds to set up, 30 seconds to acquire the data.  I even demonstrate this two minute data collection process for the class.  Yet.  Figuring out the simplest of ideas - like, how do I quadruple the distance this cart travels along a track? - takes forever for an inexperienced student.  Not because they're dumb, not because they're a lazy slacker, but because they are utterly ignorant, ignorant of not just lab equipment uses but of life in general.*  

* My wife found this out the hard way in her ceramics class.  The class had made cylindrical mugs from molds.  She gave everyone string and rulers, and asked them to measure the circumference of the mug so that they could cut a wraparound decoration to the right length.  She was astounded that very few students could even figure out how to make this measurement at all, and that even those who made the measurement got it way wrong.  

I'm in no way intending a negative rant about the danged kids these days!  I'm just making an observation about the reality of our students' practical skills upon entry to our class.  

One of the hoped-for outcomes of a physics (or ceramics) class is that students have gained some kinesthetic experience with how the world works.  Few school courses outside art or science require students to work with their hands.  So don't expect your class to be able to do simple tasks at all quickly.  Everything in lab always, always takes longer than it should.  And that's okay.  It's time well spent.

04 July 2022

Teaching stories 1 - Martha's group work

“Okay, boys and girls, get out your notebooks and copy down today’s notes,” Martha pontificated. Oy, her singsong voice sounds worse than fingernails on a blackboard, Alex thought. Wait. My dad used to use that phrase… but I’ve never been in a classroom with actual chalk or a blackboard. Maybe “sounds worse than dry erase markers smell” is the modern version?  Martha’s sharp tone snapped Alex to attention.

“Right, Mr. Alexander?”

“Please?” said Alex.” 

“Please show the class a velocity vector, like I was saying?”

“Gotcha,” Alex replied, as Martha gave him the Stare of Death.  Alex drew an arrow up and to the right, labeling the angle from the horizontal as 30 degrees.

Martha took over Alex’s position at the front of the room as the big-dog takes over little-dog’s spot in line for doggie dinner.  “Now, I draw the horizontal component like this.  In math class we’d use the kohssin to figure out the horizontal component, but here we’ll just measure with a ruler.”

“You mean cosine, Mrs. F?” said the freshman Will Jefferson.  Martha switched the Eye of Sauron from her colleague to her student in the front row.  “No, Will, I mean kohssin.  See: C-O-S-I-N.” Martha wrote the letters on the board. “Kohssin.”

“Mrs. F!  That’s pronounced “cosine!” the freshman Will Jefferson said.  “It’s a natural consequence of projecting a vector quantity along the line of a second vector using the “cross product,’ aka the ‘scalar product.’  The cross product of vectors A and B is represented by AB cosine theta, where theta is the smallest angle measure between the two vectors.”

While Will’s lungs audibly refilled to continue the lecture - Will had asthma, every intake or outtake of breath was audible to the surrounding congressional district - Martha frowned, folded her arms, and tapped her feet.  Will wasn’t taking the hint.  “And so,” he said, “---”

“Mr. Jefferson, you are, of course, 100 percent correct.”  Twenty-one sets of eyes snapped to Alex.  “Although, you’ve conflated the technical term ‘cross product’ with ‘dot product.’  That’s an easy enough mistake to make, no worries.  Thing is, Mr. Jefferson, right now I’m happy for our class just to understand how to draw and measure a vector component with a ruler.  No trigonometry necessary.”  Alex put on his biggest smile.  “If you, or anyone else, would like to stick around after school one day this week, I’d be happy to show you some advanced vector operations and how they apply to physics beyond this course.  

“Right now, though, y’all take a look at my diagram here.  This is all we need for today.  Watch what I do so you can do it on the homework assignment.” 

As Alex turned to draw on the board, he registered a gaggle of open mouths among the students.  Except for the freshman Will Jefferson.  His eyes were wide, but his mouth was closed.  The rest of the students seemed stunned.  Alex heard a whisper somewhere in the back: “Woah! Will shut up!  How’d that happen?


“Will sure is smart,” Martha mentioned to Alex conversationally.  All students had filed out, and Alex was sprawled in his desk chair.  “Yup,” Alex replied.  He was tired.

“He’s too smart, I think,” said Martha.

“How is he ‘too smart?’ I mean, isn’t that our job to help students get smarter?”

“I mean, he’s way ahead of the class, and that’s not good for him or for overall class morale. And he needs to learn when to speak, and when to keep quiet. Let’s put Will in a group tomorrow to help some other students who aren’t as smart as he is.”  

Alex frowned. Martha’s tone sounded ominous, as if a soundtrack were playing a scary string crescendo over her words.  But Alex didn’t have the mental energy to discuss further.


“Good morning, boys and girls!” Alex cringed, outwardly as well as inwardly.  He couldn’t help it.  Why does Martha have to address the class that way?  Why not “folks,” or “everyone,” or even “class?”  Calling 14 year olds “boys and girls” is guaranteed to breed resentment; it preemptively destroys relationships. 

“You have today’s lab sheet on your desk.  We are measuring the circumference of circular things with a string, and the diameter with a meter stick.  Please graph circumference on the vertical axis and diameter on the horizontal.  Your groups are listed on the screen.  I’ll give a score for the accuracy of your data, for your ability to work together as a group, and for the analysis.  One graph per group.  Okay…. Go!”

“Dammit!”  said Lindsay, sotto voce but not so sotto that her friends couldn’t hear her voce.  “Why do *I* have Will in my group?”  The students surrounding Lindsay laughed.  Most of them, anyway.  

Will hung his head.  But then he fixed a smile, sat next to Chris and Wilson, and grabbed the lab sheet.  “Hi, Christopher, Hello Wilson.  Here, I have four strings so that we can divide the measuring work.  Here is the graph paper; let’s each mark our data points on the graph ourselves, does that sound good?”

Chris and Wilson grunted noncommittally.  They were both eyeing Lindsay surreptitiously as she flounced toward their table.  “Ugh.  So you’ve already given us our orders, right, Will?  You’re so smart and diligent, aren’t you.”  Eye roll.

“Lindsay, I made a suggestion about optimizing our efforts so that we can be done quickly and accurately.  You don’t have to agree with me, of course - do you have an alternative suggestion about the division of labor in this enterprise?”  Lindsay said nothing, just sat still and defiant.  

Chris grabbed a string and started measuring the circumference of the wall clock above their desk.  Will worked on a table leg.  Wilson started to get up to find something circular. 

Lindsay looked around… no one was paying any attention to her.  

She fixed her face in a determined look, adjusted her blouse, and leaned across the table toward Wilson.  

“Wilson, your headphones have circular ear attachments.  How about we measure those?”  Wilson turned around and stared… he could see directly down Lindsay’s blouse - not exactly into her cleavage, because what was there had already been cleaved and was now hanging straight down like an udder.  Wilson’s mouth was open.  He carefully removed his headphones from his neck, while keeping his eyes stock still.  He passed the headphones ever so slowly to Lindsay, who kept the show open for a long moment.  She sat down and busied herself with her measurement, now pointedly ignoring Wilson.

The Freshman Will Jefferson rose from the table leg.  “6.2 centimeters circumference and 2.0 centimeters diameter, though that diameter is plus or minus a number of millimeters because I can’t put the ruler through the center of the table leg, ha!”  Wilson and Lindsay stared.  Will graphed his data point, and started measuring around the large circular table itself.  

“Ach!  Will, what are you doing close to me?  Stay away!”  Alex turned his head at the shrill exclamation.  Lindsay had backed up as if Will were a skunk.  In truth, contrary to Will’s classmates’ expectations, Will was both cleancut and, well, clean.  Alex knew what it was like to be thought of as the smelly kid.  While Alex himself had been quite a clean, not-smelly 9th grader, he had in fact been a greasy and odiferous 7th grader.  Alex’s classmates hadn’t adjusted to that changing nasal reality for an entire presidential administration.

“Lindsay, I just measured the smallest circumference our group is likely to find; now it is time to measure the largest available circumference.  I’ll only be a moment!”  

“Harumph!” Lindsay said.  When no one noticed her Harumph - except perhaps Wilson, who was still standing still with his mouth open - Lindsay looked around, and made a beeline for a different group.  

Chris, Wilson, and Will settled into the experiment.  Chris put a point on the graph.  After a wistful look at Lindsay’s receding tuckus, Wilson acquired data from his headphones.  Will looked at the graph.  “Okay, if we each get a couple more points in between the table leg and the table, we’ll be good!”  The others were silent, but seemed happy to follow Will’s lead.

Martha strode up to the table.  “This graph is good so far.  Where is your fourth group member?”  

“Mrs. F, our fourth group member has abandoned us, but we are making significant progress in acquiring linear data!” said The Freshman Will Jefferson.  Martha gave the three boys her most condescending frown.

“Well, that doesn’t seem like an appropriately functioning group!  The boys taking over all the tasks and ostracizing the girl!  That is unacceptable.  LINDSAY!  PLEASE COME BACK!”  

Lindsay aimed one last giggle at her friends, then walked meekly back to the table where Martha waited for her.  “Yes, Mrs. F?”

“Lindsay, I’m sorry you felt you needed to leave.  These boys will be inclusive from now on.  Right?  Right?

Chris and Will stared daggers at Martha.  Will looked angrier by the moment.  He started to say something, but Chris walked in front of him.  “Yes, Mrs. F,” Chris said.  “We understand.”  He looked back at Will, mouthing “it’s okay” with what he hoped was a gentle face.  

“Sure,” said Wilson, “we are very happy to have Lindsay with us!” He looked at Lindsay.  Looked at her chest.  Looked back up at her eyes.  Lindsay smiled evilly.

“Good,” said Martha.  I have taken off 10% from your lab group for this temporary breaking up.  Let’s not allow that to happen again, okay?”  

Chris still had his back to Martha, and was watching Will.  As Will drew his audible squeaky breath, it seemed as if he were a dragon preparing to level a city block.  Chris gently trod on Will’s toes.  “Not now, later,” he mouthed, practically pleading with his eyes.  Will relaxed marginally; his exhalation was a damp squib.  

Martha moved on to harangue another group.  Chris closed his eyes in relief and sat down.  Will’s anger had been turned from medium-high to simmer.  He simply stared at Lindsay.  “Well, I guess we’d better try to get the other 90% of the points,” Will said, keeping his eyes locked with Lindsay.  Finally he broke away and started measuring the knob on the classroom thermostat, on the wall just behind Wilson.   

Wilson and Lindsay smiled at each other, but mischievously, not kindly.  Wilson measured his other headphone.  He put a data point on the graph.  And another data point.  He looked back up at Lindsay, and moved aside so that she could watch the show.

And sure enough, as Will put his next data point on the graph, he gasped.  “What’s this?  Who measured a 10 centimeter diameter and a 60 centimeter circumference?  And this other point, too - that’s not possible!”  Chris shrugged.  Wilson just looked back and forth from Will to Lindsay, alternating between “who, me?” body language and a shit-eating grin.  

“Are you sure of these, Wilson?  Can I see what you measured?  Because these data points don’t make sense!”  

Wilson didn’t say anything.  He just leaned over with his pencil, and put another point on the graph - randomly.  Will’s face contorted.  “You can’t do that!  This isn’t science, this is unethical in the extreme!”  Wilson kept smiling through Will’s increasingly desperate lecture - Lindsay was looking at him with wide eyes, and Wilson felt all tingly from her attention.

When Lindsay and Wilson laughed with each other, while not even looking at or acknowledging Will, Will left.  He started at the wall for a moment, composing himself; and setting himself the most neutral face he could muster, walked to the front of the room.  Alex was helping a group draw a best-fit line.  Martha was sitting at her desk.

Will stated his case to Martha.  “Mrs. F, my partners are not doing what they were assigned to do.  They are falsifying data and making our results inaccurate.” 

Martha looked up from her computer screen.  “Will, in the working world, you will have to learn to cooperate and get along with all types of people.  I’m sorry you’re frustrated with your partners, but you simply must learn to work with others.”

“Ah,” Will said.”  “This is like the working world?  Excellent.  So you are going to fire your useless employees who aren’t doing their jobs properly, right?  You’re not going to finish projects successfully to make the quarterly earnings totals with clowns like them on your staff!”

“Will, that’s enough - we don’t talk like that about our classmates.”

“Are you going to fire them, then?”

“No, Will.  Learn to work together.”

Wheezing intake of breath.  Back to the carefully neutral face.  “Then I’m sending my resume to a competitor, I’m giving my two weeks’ notice, and we are going to drive you out of business.  Furthermore, I will be filing a whistleblower ethics complaint against your company.”  Will gave Martha his gentlest smile.

“Will, I’m assigning two demerits for disrespect to a teacher.  I need you to go take a break, get some water, and come back when you can manage your out-of-control temper.”

“Temper?” Will said, placid as the Lake.  “No, Old Woman F, no temper.  Just returning what’s due with interest.”  He turned his back and walked slowly back to his chair.

Now Martha was visibly angry.  She stood up to continue the argument, but there was Alex. “You know, Martha, he was totally calm, no temper.  He wasn’t even arguably disrespectful to you until after you gave him demerits for disrespect.”  Alex spoke very quietly and gently.  But Martha snapped. 

“He called me ‘Old Woman F!”

“Well, you called them “boys and girls.”

“That’s accurate!”

“Accurate and demeaning.  Exactly like ‘Old Woman F.”

“Goddammit,” Martha hissed.  “It’s your job to back up your teaching partner, not to make excuses for impertinent kids!  Don’t ever do that to me again!”  

Alex didn’t respond - he just walked away to the middle of the classroom, where he saw Bill Clark beckoning to the freshman Will Jefferson.  “Will, come join our group!  We’d be happy to have you over here!”  Will gratefully headed toward Bill’s table before Martha arrived to talk to Will’s former group.  Alex didn’t hear what was said there, but he saw smiling and laughing from Martha as well as from the students.  It was hard to tell who was more uncomfortable - the 13 year old, or the 23 year old.