How to teach AP students to derive equations? Annotate.

From Anna, a regular reader:

Do you have any tips you offer students for questions that ask them to derive equations?  I am finding my students get into a tissy with this type of question, and every question is so different, I can't quite put a finger on "where to start" for students...

And that - where to start - is a holy grail of physics teaching.  Fact is, the vast majority of AP Physics students are fully capable of carrying out the relevant mathematics for a college-level physics problem if they know where to start.  Sure, they might make silly algebra errors like canceling across a plus sign.  Who cares.  If we're going to successfully teach students how to derive equations as on a qualitative-quantitative translation question, those students must become comfortable figuring out where to start, and then not fretting over algebraic manipulation.

My primary word of advice for derivations is "annotate."  The phrase I use on homework, quiz, and test questions is "Annotate your derivation with a description of your approach."  Then I simply don't accept a response if there aren't clear annotations - no matter how clear, no matter how correct, a derivation without words must be redone from scratch.*

* Whereas I generally accept the math errors and move on.  It's more important to reward physics done nearly right than to use political capital to correct a math error.

Once I convince students that annotations at all are non-negotiable, it's then a long process to explain how annotating physics (like, "writing newton's second law for the top block") is different from annotating math (like, "dividing both sides by g and canceling the m").  

I really want students to tell me two things.  First, what principle of physics are they starting with?  And then, if they make an algebraic substitution, why did they do that?  

The most important annotation is that very first - where did you start?  This annotation is just as important in a numerical problem as in a derivation.  So I demand this sort of annotation even on problems that say "calculate the acceleration of the block".  

Okay, how do students learn where to start?  The good news is, AP Physics 1 doesn't include that many starting points.  Start with 

• A free body diagram and newton's second law
• A kinematics chart
• The impulse-momentum theorem / conservation of momentum
• An energy bar chart

That's about it.  Universal gravitation, circular motion, and two-body problems are just special cases of newton's second law.  Rotation questions can still be categorized exactly as above, just with torque, newton's second law for rotation, angular momentum conservation, etc.  Yeah, I'm probably missing something somewhere; yeah, often two of these approaches must be combined; but this is a really good short list for students to refer to.

Once a student decides which of these approaches to use, they should annotate each equation with where it comes from.  "Writing N2L for the block at the top of the circle using this free-body."  "Acceleration in a circle is v^2/r."  "To get the smallest speed, the normal force must be zero."  I'd love for these words to be alongside the mathematics, perhaps with circles and thought balloons or other graphical means of indicating how the words relate to the mathematics.  This process is the soul and center of the qualitative-quantitative translation: the annotation process could also be described as translating between mathematical and physical reasoning.

I try to integrate derivation into problem sets gradually throughout the year.  Students are always more comfortable at first using numbers rather than variables.  So I set them up for success with calculation problems.  But I still require an annotation process.  When I add in some full-on derivation problems without numbers, it's a straightforward step to replace their numbers with variables.  Not an easy step for most, but a straightforward step.  

By year's end, we're doing full-on QQTs from the exam.  I don't mean at all to suggest that my students think these are easy!  But from the long year of gradually developing skills, they do know how to start and generally what to do.  So they can learn effectively from the mistakes they make.

Contract Grading and the Reformation

Years ago I read a book about European history and the Protestant Reformation.  It posed the modern question: what if we found scientific evidence that your diet and exercise habits have absolutely no influence whatsoever on your long-term health prospects?  That eating a "balanced diet" definitively does NOT prevent cancer, or any other disease? That heart attacks happen quasi-randomly, whether or not you eat extra-cheese pizzas, no matter whether you jog every day or merely every gubernatorial administration?  

Americans worry incessantly about how diet and exercise affect their long-term health - including  people without any actual health problems.  They are helped along by ignorant gossip from their well-meaning friends, by journalists who carefully misinterpret scientific "studies" for maximum dramah, and by the diet-industrial complex who have a financial interest in stoking people's fears.  "Am I eating right to protect myself from cancer and heart disease?"  The answer most people come up with is a shameful "no".  

Well, the European history book said, that's how the Protestant Reformation succeeded.  Europeans in the 1400s were just as worried about how their daily actions might lead to eternal damnation as we are about how what we eat might cause disease.  They were egged on by priests who proclaimed a "The Good Place"-style system by which all actions are judged by God and his minions.  Instead of joining weight-watchers or buying books by diet gurus, people bought indulgences to save their souls.

But the Reformation preached that all the worrying wasn't relevant.  You're Saved, and going to heaven.  Or not, but if not there's nothing to be done about it anyway, God either Saved you at birth or He didn't. Don't worry about your soul.  God'll take care of it.  You take care of plowing the fields and providing for your kids.  

Now put this conversation into the context of college-bound high school students.  They worry incessantly about how their daily work habits might affect their grades and college prospects.  They're egged on by ignorant gossip among their peers, and among their parents' peers.  How many times have I talked to a 14 year old's parent, not about what activities the 14 year old might enjoy, not about what classes match their interests or their abilities, but about what will maximize the 14 year old's college resume and GPA?  How often does a senior who loathes math take yet another math course, then spend innumerous sullen hours doing the hated work for that math course, and stressing because they're getting Bs on the tests?*  When that senior has for four years gazed wistfully at the ceramics course, or beginning strings, or journalism, but parents and other trusted adults explained that they're borderline for admission to their dream college, and the only way to get off the borderline is to take some challenging "core" courses and get As.  

*And causing more stress among their peers with their contagious despondence.

How would such students feel if trusted advisors uniformly told them that college admission is truthfully a crap shoot, that they're in (or not) regardless of whether they have an A or a B in English?  That their dream college wants to see them take classes they love, not yet another year of hated French?  That the college they attend is predestined, unchangeable, and will only be revealed - not earned, revealed! - in March of the senior year?

I can't control the zeitgeist surrounding college admission.  I can, though, nail my "everyone gets an A-minus" AP physics contract to the door of the cathedral.  

Think of the weight off of students' minds once they sign their contract.  It makes no difference in their grade if they got a 3/10 or a 9/10 on last night's problem set.  All the "3/10" means is that they'll come in to redo the problem until they understand it.  The advantage to strong test performance is fewer corrections to do, not a higher grade, 'cause the grade is an A-minus no matter what.  

So no one whines that the course is too hard, or that the work is too much.  Every student knows that any time they want, they can just request a move to the general course, which will be far easier.  (But, except in rare cases, they don't request that move!)  They know that as long as they *do* the assigned work, they have the Grace of an A-minus waiting for them.  And that the A-minus will never, ever turn in to an A on the next report, no matter how much more performative work they do, or how many indulgences they attempt to purchase.

Contract grading part 4: What is the incentive for students to engage?

In my 9th grade AP Physics 1 contract grading, everyone gets an A-minus on every term report.  

The obvious question I'm asked is: If you don't assign grades based on student work, what incentive do students have to keep up, to do their assignments?  

The incentive starts with the opt-in nature of this advanced course.  It's clear that any student, at any time, could simply ask to move into the general course.  And it's clear that I would respond with a gentle "of course, I appreciate you giving AP physics a shot, I hope to see you in the AP class again in a few years!"  No shame, no pressure. 

Thus, when the work gets challenging or long or frustrating, I don't get complaints.  I mean, I have enough experience with my school's schedule and audience that I'm pretty sure I'm never making unreasonable demands in terms of out-of-class work - I do have to be careful that I'm not overplaying my hand, that I'm not dominating my students' lives at the expense of other classes or non-academic pursuits.  Yet every physics teacher has dealt with students who are angry because, for the first time in their lives, they're having a tough time understanding something academic.  This class knows they can leave the course at any time.  They also know they have the grace of an A- waiting for them regardless of performance.  So they tend to persevere, even in the face of adversity.

A poor assignment requires a redo in extra-help time.  I do put a score on each problem set; I keep track of progress on each in-class lab assignment.  In one sense, scores don't matter since everyone's grade is an A-.  What does matter is that students know I'm watching out for them.  That students have an authentic audience for their work, someone who's paying attention.  That students care enough not merely to complete practice problems and labs, but to complete them well.

I've dedicated one whiteboard to a list of students who need to redo assignments or finish labs.  I spend an enormous amount of time in the first few weeks making sure that students come in for extra help.  Again, there's no shame intended when they come in, just an opportunity to redo the problem the right way, to ask questions where they're still confused.  Generally, students are grateful for the help, happy that they figured out something that flummoxed them the first time.

The incentive, then, is to do each assignment well enough that they don't see their name on the board.  I get higher quality and more consistently completed problem sets now than I ever did when I assigned grades!  Time is more valuable as currency than grades.

Communication and conversation are critical early on.  Three weeks in, I schedule a brief (5-7 minute) meeting with every student.  I ask, how are you feeling about the course?  Do you want to remain in it?  If so, why?  (If "no", then no worries, let's put you in general physics.)  Are you getting your work done in other classes?  Is there anything you want to bring up with me?  

The point is, the contract is not an impersonal legalistic document like the Apple terms and conditions.  No, signing the contract represents a covenant.  I am dedicating myself to helping each student succeed in this college-level class; each student is promising to do the practice I ask, so that they put themselves in position for success.  

At the conclusion of each conversation, I usually provide a hard copy of the contract for the student to sign, after which I send a scan to the student's advisor.  If the student expresses minor misgivings, I ask them to give things another week to decide, and I schedule another meeting and a discussion with the advisor.

I can't shy away from difficult conversations.  Every year, a few students are overmatched by AP physics, but they valiantly keep going.  At first I just schedule extra-help to redo each problem set.  But once it's clear that their work in physics is impacting their overall academic performance - or once it's clear that they are simply not able or not willing to handle the pace or depth of this course - I need to pull the plug.  Self-perceived good students won't leave the class on their own, because they would think of themselves as "quitters."  I explain kindly that AP physics isn't telling them "no," just "not yet."  Usually, I see relief on such a student's face.  I must be willing to do what's best for each student.  And what's best is sometimes to let them gain experience in a simpler, slower-paced class.

What's best for each individual student is usually what's best for the overall team, too.  On one hand, I can't be dumping students all willy-nilly in the first weeks - that would spread despondence.  I need to stick with anyone who possibly has potential to pass the AP exam, even if their work is poor now.  Yet, a student who is leaving questions blank, a student who can't draw a free body diagram after two weeks of practice, or especially a student who passive-aggressively doesn't turn in work at all - these folks need to go.  

The rest of the class often feel embarrassed or uncomfortable around a student who clearly doesn't belong in the advanced class.  If I'm going to insist on a team atmosphere, I need to be sure that everyone on the team is capable of contributing.  

When I counsel a student into the general course, I try to be as explicit as possible about their performance.  I'll show them - and their advisor - one of their problem sets in comparison to a well-done problem set by another student.  I'll rattle off a list of unfinished assignments, and show the contract stipulation that all work must be completed.  I'll show test scores: not just 1 or 2 on an AP scale, but a raw score under 35%.  Even advisors (or parents) who occasionally start out a bit hostile can't easily argue with the evidence I present.

And, from a Machiavellian perspective, it is certainly true that I get better work from everyone else once they see that I ain't kidding about the terms of the contract.  No one is expected to be perfect, but everyone is expected to do all the work in a serious effort to get better every day.  The effort gets a bit seriouser once they see someone else was asked to leave - even if my decision to ask that person to leave was obvious to everyone.

The goal is NOT merely student compliance, though that is a first step.  My goal is that by November, I have as large a class as possible who - for most students, most days - think of physics as an enjoyable part of their academic experience.  A class who look forward to working with one another as teammates, who support each other as they learn a difficult subject.  

At the beginning, I'm working to establish norms: we do the practice problems even if we don't do them perfectly, we are kind and helpful to our classmates, we come in for extra help when asked.  Once these habits are second nature, once it sinks in that work isn't done for a shot-term grade but for long-term improvement, both the students and I look forward to our class time together.  The atmosphere becomes fun and relaxed.  The angst of being continually judged by a teacher and by peers vanishes, and is replaced by curiosity and excitement.  

Contract grading part 3: communication with parents and advisors

In my 9th grade AP Physics 1 class, everyone gets an A-minus on each term report.  The first post in this series describes *why* I chose this model, the second discusses in detail how the school administration and I worked together to develop this outlier of a grading system.

Teachers must work alongside four different constituencies, each with their own quirks: students, parents, colleagues, and administrators.  Once I developed the general idea of contract grading for AP physics and hammered out details with administrators, the next step was to communicate this approach effectively to my students' parents and faculty advisors. 

Now, I expect that the circumstances surrounding my AP Physics 1 course are nearly unique among this blog's readership: I'm teaching a 9th-grade-only section, for which the students have been pre-selected, over the summer, by a faculty committee based on admissions files.  My sense is that if our parents were straight-up given the option of general or AP physics, very few would choose AP.  This is contrary to my experience with older students, contrary to what I hear from physics teachers at other schools.   

(Because of the unusual nature of my course, I almost didn't write this post!  While I think the everyone-gets-an-A-minus approach can in fact work in many schools, my particular communication strategy is optimized for my particular 9th grade boarding school situation.  I'm not at all recommending that others do exactly as I do!  I'm sharing what I've done so that readers can adapt, or not, to their own needs.)

No, parents are not part of the decision making process.  Students don't know they've been placed into the AP course until they show up on the first day of class!  This is quite deliberate.  We don't need to add any anxieties to what is already an angst-ridden orientation period, when 14 year olds are living away from their parents - and vice-versa - for the first time.  

As soon as the first day's class is over, I send the letter below to the parents, copied to faculty advisors.  I'm trying to get word out to parents before they have that first phone call or text conversation from their kid.  My hope is the conversation goes something like:

     9th grader: I was chosen for the AP Physics class.

     Mom and Dad: I saw!  That's wonderful!  How do you feel about that?

     9th grader: We had fun doing a lab today.  I did okay, I guess.  It's cool that they chose me!

The good news is, colleagues and parents have been quite supportive so far.  The contract assuages their worries, the same way it soothes my students' grade anxieties.  

Throughout the letter, I'm trying to hit two separate emotional beats for the far-away parents: (1) Your son is special, and has been specially chosen; (2) Don't worry, we know what we're doing, we're not using your kid in some whackadoodle untested experiment.  

And those are generally the two things that our parents need and want to hear if they're going to accept that their 14 year old is taking a college-equivalent class.  

(Note that I teach at a boys school, so the gendered language below is deliberate.)

Dear Mr. and Mrs. Nti,

This is Greg Jacobs, science department chair and physics instructor.  I wanted to let you know a bit about our plans for your son this year in physics.

A committee of faculty and administrators selected a set of 9th graders to attempt the College Board’s AP Physics 1 course this year.  Woodberry has had tremendous success over the years on the AP Physics exams – in fact, my classroom is decorated with posters naming the numerous students who have earned college credit via the AP program.  The vast majority of students who take this course pass the AP exam.  Our committee carefully considered the level of challenge that each boy is likely to be able to handle, and recommended that your son attempt AP Physics 1 this year.

Below is the syllabus that your son received on the first day of class, including the course contract on the final page.  I want to assure you that I will be looking out for him!  If it turns out that the challenge of AP is too much for him, or if he’s not meeting the requirements of the course contract, I’ll counsel him into our top-rate conceptual physics course, where he will get the grounding that allows students to do extremely well in AP Physics as an upperclassman, or in the equivalent course at college.  My goal for this course is to develop a team of students who are excited about learning physics for its own sake, at a very high level.  We think your son could be an important member of that team.

If you’d like to know more about the AP Physics 1 program, please google “AP Central Physics 1”.  There you’ll see the official course overview.  

Thanks.  I’m excited to work with your son this year!

Greg

In the next post, I'll explain how I communicate with students, both on this first day and throughout the year.  Just know that these students are at first generally surprised and pleased that they have been specially chosen for this challenging course, with only minor trepidation.  And then collectively work more diligently than any of my previous first-year classes.

Contract grading part 2: How we made it happen at the school level

(The previous post explained *why* I moved to contract grading with my 9th grade AP Physics 1 class. Today I'm discussing institutionally how I worked with my colleagues and my administration to make contract grading happen.  The next posts will discuss how I communicate with parents; and then how I make this particular style of contract grading work on a day to day basis with my class.)

A few years back, a history-teaching colleague presented to the faculty and at an external conference his experience with "contract grading".  His contracts for a required 9th grade course painstakingly listed the descriptive attributes of students who get grades of A, B, and C.  He allowed the students themselves to contract for whichever level best described them.  Then, he held the students to their contract, demanding levels of in-class engagement, paper rewrites, and out-of-class effort commensurate with what the students themselves had agreed to.

My approach is much simpler.  In an AP class, there's no such thing to me as settling for B-level effort.  Students who are good fits for AP physics do all the assigned work to the best of their ability, redo whatever they bombed the first time, engage enthusiastically and diligently with laboratory exercises, and finish all test corrections.  Someone who's only partially willing to do these things shouldn't get a low grade - they shouldn't be in this advanced class to begin with.

On this colleague's model, I proposed to the dean and the headmaster that all of my AP students would contract for an A, by pledging to do each of the items in the paragraph above.  They were quite receptive to the idea, especially as this history colleague had already demonstrated that non-traditional grading approaches neither brought forth the apocalypse, nor a flood of burdensome complaints. The headmaster made the very important suggestion that all students should contract not for an A but for an A-minus, on the grounds that nobody is perfect.  It turned out that this small change was critical to the success of the approach.

We ended up with the contract that you can read here.  

In practice, now, a small committee including the academic dean and the head of admissions select students who they think can handle the AP Physics 1 course as freshmen.  I've asked them to cast a wide net!  That is, they don't just rank by standardized test scores - which they can't anymore anyway since our school's admission process went test-optional.  These folks are all quite familiar with the incoming class.  They make their best guess at choosing a team.

Then, on the first day of class, I explain to the students (orally, and in writing on the course syllabus) that they have been selected for the AP physics section.  In three weeks, I tell them, I'll have an individual meeting with each student.  If by then they've lived up to the terms of the contract so far, and if I judge that they are likely to be successful in the college-level course, I will offer them the contract to sign - at which point they may choose to sign, or to switch into the general physics course.  I remind them that I have chosen them to be part of this team, so that I am invested in their success.  That I will be honest with them if I think they can't handle the course, so that they can just concentrate on each assignment and leave the long-term planning to me.

But what about their GPA?  

Here's what the contract says about grades:

Your marking period report will indicate an (unweighted) grade of A- each term during the year.

After AP score reports are released in July, your transcript will be adjusted according to the scale below.  This will increase the overall GPA for all those who earn 3 or above – which, historically, has been virtually everyone.

However, if your in-class performance is better than your AP score equivalent, your transcript will reflect the in-class performance.

AP Score     Transcript Grade

5             A and honors bump

4             A- and honors bump 

3             B+ and honors bump

1 or 2             B

Note the out that I've left myself and my students: someone who's done well all year won't be penalized for having one bad day on May 9.  That said, the AP exam is pretty darned consistent.  It's rare that someone significantly underperforms what I've seen from them all year.  In practice, my goal is to eventually counsel out anyone whose in-class performance is below the 3 level, so that *everyone* will earn a weighted B+ or better.  Those students who aren't getting 3s on most practice exams are invariably better off building skills in our very strong conceptual physics class, and then returning to the AP course junior or senior year.  

Here's where the headmaster's genius suggestion solved problems I hadn't anticipated.  A student who truly is trying to game the GPA system in the short term is better off with an earned A in general physics than the automatic A- in the AP class!  And a student willing to take a class below their intellectual level for the purpose of earning a higher grade is someone I don't need in AP physics, any more than the football team needs someone who only joined to impress potential sexual partners.  The students who choose to stay - which, so far, has been all of them - understand that they're taking a small GPA hit for now, but in return are freed from the angst of worrying incessantly about whether they're perfect.  

When someone inevitably asks in class about grades or GPAs, my response is gentle, but clear about two points:  (1) Read the contract for details, and (2) If GPA will have any influence on your decision to remain in the college-level section, you don't belong here;  I'm happy to help you switch into the general class, where you're likely to earn a natural A.  They get the message very, very quickly - such that the SECOND student who tries to ask about grades is hurriedly and firmly shushed by classmates.

The next post will describe how I communicate all of the above to parents and advisors.  Then I'll get to discussing the nitty-gritty of how the class works on a daily basis.

Contract grading in AP Physics 1 part 1: Why I do it.

The most important qualification for success in AP Physics 1 is the want-to barrier.  I mean, yes, there's a level of intellectual horsepower required.  Yet, the borderline students who willingly opt in to the challenge of a college level course tend to do well in the long term, while borderline students who are pushed by parents or counselors - or by the transactional potential of a higher GPA - to take AP physics tend to perform poorly.

Much more importantly, the borderline students who opt in for the right reasons don't bring their classmates or the class culture down.  And such students don't bring *me* down.  Rather, they make me happy.

A number of years ago, I decided I need to limit my work with advanced students to those who truly want to work with me.  I'm sick of fighting with smart students who use my class to game the college admission system.  I'm done with students who have no interest in physics, just in getting a weighted A on their GPA.   I have no patience left for those whose goal is to simultaneously maximize their honors grade while minimizing the engagement necessary to earn that grade.  

My first step was to focus my work on the youngest students.  Our boarding school 9th graders have already made the personal decision to seek out a challenging environment.  They could all have remained at their local public or independent high school.  Already these 14 year olds have shown themselves to be not entirely risk-averse.  They see their teachers as kindly parental figures whose goal is to know, challenge, and care for them.  They *don't* yet see teachers primarily as mean, demanding authoritarians who are obstacles to their success.

Yet, precisely because these barely-teenagers have just taken an enormous leap outside what was comfortable for them, they often don't want to compound leap upon leap.  Our freshmen congregate to the popular activities that their peers say are cool - mostly football and soccer in the fall, rarely theater or the outdoors program.  When we tried giving the freshmen the option to choose to join an AP physics section a few weeks after arrival, many qualified candidates stayed away.  They (and their parents) wanted a "solid start" to their boarding school career.  They were still gaining their footing in discovering who they were socially and athletically - they didn't want to risk "failing" academically.

So, we've gone to a contract grading system in the 9th grade AP Physics 1 class, in which all students in the course get an A- on each term report, no matter what.  After a year of just seven students opting in to AP Physics 1, I've had class sizes of 15, 20, and 20 - out of only 90 or so total 9th grade students at the school.  The students are happy, they're enthusiastic, they're fun to be around.  They're learning physics well enough to pass the AP exam (13/15 passing in 2020, 18/20 passing in 2021, probably similar in 2022.)

I'm sure you have two major questions:

(1) How did I and my school make this happen?  In a school that emphatically publishes grades six times per year, how do I get away with my class being such an outlier?

(2) Without term grades as a motivator, what techniques do I use to keep students invested and engaged for the long haul of a school year?

I'll address each of these questions in the next posts.  

Describing a laboratory procedure: speed of a cart at the bottom of a ramp

 Samar, who teaches in Maryland, called my attention to a question in the AP Physics 1 Workbook:  

In order to perform an experiment, two students need to determine the velocity of a cart just as it reaches the bottom of a ramp.  In a few short sentences, describe an experimental setup that they could use to determine the [instantaneous] velocity of the cart at the bottom of the ramp.

This is posed as an extra suggestion to discuss with students, and so doesn't have a solution in the teacher version.  Which of course - experimental physics is a creative endeavor, where right and wrong certainly exist, but where numerous correct approaches are available.  

Nevertheless, it's worth me giving a few examples of how I'd suggest answering the question such that (a) the procedure is correct, (b) the procedure is clearly communicated, and (c) the procedure is described in "just a few short sentences" rather than in a multi-page lab report full of vacuousness.

I've graded more AP experimental questions than anyone else on the planet*, so please trust me when I say that you shouldn't accept any response longer than about 80 words.  Seriously - no matter how thorough, no matter how accurate, a long response is no good!  For one thing, the student just used all sorts of time writing all these words here, when that time could have been more productively used elsewhere - on other problems in an exam, or perhaps at home playing with the family dog.  It's not possible to earn extra credit, or a "plus one!" on an AP exam.  Just answer the question, then stop.  If you "lose" a point for not saying something important, well, the extra five sentences you wrote at the end aren't gonna help.

* I'm probably not kidding. Guinness Book, please contact me!

So, how would I answer?  Here are four ideas off the top of my head.  I'm sure others will chime in with other thoughts!  

1. Put a dual-beam photogate above the cart at the position where the cart leaves the track.  Tape a thin slice of an index card to the top of the cart, such that the card breaks the photogate beam.  Then drop the cart down the ramp, and the photogate will read the instantaneous speed at the bottom.

2. Place a meterstick horizontally at the bottom of the track.  Record the cart's movement on video.  Pause the video in consecutive frames when the cart is at the bottom of the track.  The distance the cart traveled between frames - read on the meterstick - divided by the time for each frame (known from the video camera) is the instantaneous speed.

3. Place a motion detector behind the cart.  Have the detector create a velocity-time graph for the cart's motion down the ramp.  The maximum reading on the vertical axis is the cart's instantaneous speed at the bottom.

4. Use a smartcart that can create a velocity-time graph of the cart's motion down the ramp.  The maximum reading on the vertical axis is the cart's instantaneous speed at the bottom.

Bean Dad and physics pedagogy: they're not the same at all, but I can understand why people have this impression.

I truly hope you missed the brief internet celebrity of "Bean Dad." Brief, tragic summary: a less-than-empathetic parent tried to teach his hungry six-year-old daughter to use a manual can opener by denying her food until she, without help, figured out how to open a can of beans.

Well, true to the spirit of Twitter, it took no time at all for the no-context sweeping generalizations to be pronounced.  I tend to ignore ridiculous Twitter debates that don't involve football, but an author whom I greatly respect jumped in with a barb that hurt: 

"The Bean Dad approach is STEM pedagogy in a nutshell," @jonnysun said.  (He's since deleted the tweet, I think - I found a reference to it, but I cannot find the tweet itself.)

I know other teachers heard similar not-so-flattering remarks about how science is taught.  Folks got defensive about their methodology.  And that way madness lies.

I mean, any good science teacher teaches by inquiry, by modeling, by discovery, or by whatever buzzword means "don't just talk about science, do science".  And none of these buzzword approaches, done correctly, bears any serious resemblance to Bean Dad.

Yet, before we turn our shoulders in a huff... please consider why so many intelligent people think that STEM pedagogy is like Bean Dad.  Fact is, this is a general perception of our craft.  Why?  That's an uncomfortable question.

I know that my very own students have this perception early in every school year. And in my first few years of teaching, I didn't know how to help students and parents and colleagues understand the difference between Bean Dad and "I can't help you with a blank page, I need to see your serious written attempt."  Insisting that students engage authentically with the material rather than demand that I solve their problems for them means that I will always, forever, deal with the charge that I "refuse to help."

And, well... a lot of our peers try to teach via inquiry or the like, but don't really understand what they're doing.  They don't lecture, but in good faith they don't know what guidance to give, just that they're not supposed to lecture.  Or in not-so-good faith they don't care what guidance to give.  Or they assume that since they figured things out on their own, so should their students.  These folks are, in fact, the school version of Bean Dad.  

We're deluding ourselves if we don't acknowledge the existence - maybe even prevalence! - of Bean Dad science teachers.  Their well-poisoning means that everyone else has to work ten times as hard to establish a positive class culture that gives appropriate guidance, but also allows students appropriate freedom to make mistakes.

We can't avoid complaints.  Yet, we can help students, parents, and colleagues understand our methods.  We can be transparent about our pedagogy.  We can de-emphasize the value of right answers and over-emphasize the value of correct approaches.  We can publicly prioritize progress over performance, long-term goals over short-term goals.  We can stand up for colleagues who share our values.  

And we should, must, keep on going in the face of pressure each fall.  When alumni of your course are the ones shutting down the complaints that "Mr. Lipshutz doesn't help us learn," then you know you've done well.

An ill-posed problem involving the work-energy theorem

A source of physics questions showed a diagram like the one to the right.  A person can exert a 150 N force parallel to the frictionless ramp.  How much work does this person do on the 20 kg box to bring it from the bottom to the top of the ramp?

Approach 1:  Define the system as the box-only.  Ignore the work done by the earth, because the problem asks only for the work done by the person on the box.  Because the force of the person is parallel to the ramp, the work done is just force times distance - (150 N)(5 m) = 750 J.

Approach 2:  Define the system as the box-earth.  The work done by the person should equal the potential energy gain of the box-earth system.  That's mgh, where h is the 2.5 m vertical displacement (i.e. 5 m times sine of 30 degrees).  The work done is thus (20 kg)(10 N/kg)(2.5 m) = 500 J.

Which is correct?  

Neither.  But the problem is ill-posed.

The easiest way to see the ill-posediness is to try to draw an energy bar chart.  Let's use the box-earth system.

The gravitational potential energy bars are easy - from zero at the bottom of the ramp to something at the top.  

On an energy bar chart, conservation of energy is written by equating the number of bars on the left plus the bars of work done by an external force with the number of bars on the right - I describe the process in this video as "bars plus bars equal bars."  Right away, you can see why the problem is ill-defined!  What's going on with the kinetic energy?

We could assume constant speed.  Then the bar chart shows that we need the same amount of work done by the person as the gravitational energy at the top of the ramp.  That's a totally reasonable assumption!  This gives the work done by the person as 500 J, as in approach 2.  

But then what of approach 1?  The 150 N force applied over 5 m does give 750 J of work done!  But that would cause the box to have 250 J of kinetic energy at the top of the ramp!  Is that okay?  Well, sure, but wouldn't the box fly off the top of the ramp, then?!  I suppose, the problem said the person "can" exert 150 N of force, so perhaps the person is applying less force than they "can"?!?  But then I'm sounding like a lawyer, so by definition I'm wrong!!!

Creating multiple exclamation points in a blog post means the problem is ill posed.  Now, "ill-posed" doesn't mean "One student personally can't figure out the answer."  Were such a problem to accidentally show up on an in-class or standardized exam, I'd expect a student to explain the issue in writing - not to come to the front of the room to argue with the proctor. 

Most frequently, this sort of question asks the minimum amount of work the person needs to do to bring the box to the top of the ramp without specifying the force applied by the person.  Then the assumption of constant speed is not just reasonable but required to find the minimum work.  It's the additional specification of the force of the person on the box that causes the issue.

That doesn't mean this scenario couldn't be used for a well-posed problem!   "Explain why this problem is unsolvable without knowing how the box changes speed on the ramp" would be an excellent question.  Or, remove the "frictionless" statement, specify that the block is at rest at the bottom and the top, and ask for a justification of whether the ramp is frictionless or not.

Describing motion from position-time graphs

 

The position-time graph to the right represents the motion of a cart on a track.  The motion detector points to the left.  Explain how to reproduce this graph.

When describing motion, students only need to address two elements: 

a. Which way is the cart moving?

b. Is the cart speeding up or slowing down?

Yet, students trying to answer this question will tie themselves in knots addressing how the cart is "accelerating", trying to use words like displacement, velocity, vector, negative, positive, etc.

Don't even give your class the string with which to tie knots.  Get them in the habit of addressing each of these two questions only; and get them starting every response with a fact of physics.

For the initial foray into position-time graphs, only three facts are relevant: 

1. A position-time slope like a front slash / means the object is moving away from the detector.
2. A position-time slope like a back slash \ means the object is moving toward the detector.
3. To determine how fast an object is moving, look at the steepness of the position-time graph.

a. Which way is the cart moving?  Do NOT accept an answer that begins with "The cart is moving right."  I don't care what else this response says, it is wrong on its face.  A response must begin with a fact of physics.*

*"But that's not fair, Greg! A student who says "the cart is moving west because the slope is negative" is right!"  See, I'm not concerned with whether this particular answer is right or wrong - I'm concerned with my students developing a long-term deep understanding of position-time graphs, such that they can handle any question on a high-stakes exam as easily as Serena Williams handles a shoulder-high volley at the net.  Imaging Serena as a wee lass asking someone to hit her volleys for practice... and that someone kept lobbing her.  "Please hit me volleys."  "But I won the point!" says her suddenly FORMER practice partner.

"A position-time slope like a back slash \ means the object is moving toward the detector.  This graph is always a back slash, so the cart moves opposite the way the detector is pointing - the cart moves RIGHT."

b. Is the cart speeding up or slowing down?  Similarly, do not accept any answer that doesn't start with a fact of physics - especially do not accept an answer that references acceleration.  Yes, it is technically possible for an experienced physicist with a deep understanding of mathematical physics to recognize that the concave-down graph means the second time derivative is negative, and the negative slope means negative velocity, and to connect that to the magnitude of the velocity vector getting larger.  Aarrgh!  No!  No introductory physics student thinks this way!*

* And the vanishingly rare unicorn who can in fact think this way should have no trouble whatsoever using the much simpler facts above to reason through this graph.  So make the unicorn do so.

"To determine how fast an object is moving, look at the steepness of the position-time graph.  This graph is always getting steeper, so the cart always speeds up."

c. Now go reproduce the graph with a motion detector and cart on a track.  Amazingly, even after answering these two questions correctly, about 20% of the class will still set up the situation incorrectly.  They'll claim they need a curved track.  They'll have the cart moving away from the detector because "the graph is sloped down", even though they just wrote clearly on their own paper that the cart moves toward the detector!

Let them mess up.  

When the students have trouble reproducing this graph, they're confronting their personal misconceptions.  Even if a friend just shows them what to do, they'll see for themselves, "oh, the cart had to move toward the detector and speed up!  Now I get it!"  Or, you can ask them to read back to you what they wrote: "Which way is the cart moving, again?  And is the cart speeding up or slowing down?  So did you set up the cart moving toward the detector and speeding up?"

I don't like having students use their hands or their bodies to reproduce motion.  You'll end up in arguments about whether the graph does or doesn't look like it's supposed to, because it's really tough to keep a hand or a whole body continually speeding up for a second or two.  No, use a cart on a track!  This can be done with a fan cart on a flat track, or with a free-wheeling cart on a slanted track.  Insist on seeing one full second of motion - then you won't see just the 0.1 s when the student pushed the cart to get it started on the incorrect motion, and you won't have to argue about why that's wrong.  It's pretty much impossible to do this wrong and get a good-looking graph that's at least one second long.