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.