26 September 2014

Mail time: velocity-time graph that crosses the horizontal axis

A correspondent asks:

Hi Greg. I was doing some questions and I'm unsure about a certain response. I'm looking at a velocity vs time graph where the graph is linear and cuts right through the x-axis going in the neg. direction. At EXACTLY the point of intersection do we state that the acceleration is negative because the slope is negative or zero because at that instant the object has zero velocity?

My instincts want me to say negative, but last year I went with zero, sooooooo I'm not really sure.

I've made a graph of what I think you're looking at... see the picture.  You want to know, "What is the direction of the acceleration at point A?"  This is a classic question, with a corresponding classic point of student confusion.  This graph could represent, among other things, a ball thrown upward in free-fall -- it moves upward, slows, stops briefly, and speeds back up toward earth.  

Two ways I'd phrase my answer:

(1) Just because velocity is zero does not mean that acceleration is zero.  Otherwise, gravity would have to turn off just because a ball reaches the peak of its flight.

(2) Acceleration is the slope of a v-t graph.  The horizontal axis isn't special -- the slope of that line doesn't change anywhere, so the acceleration is negative everywhere, both for positive and negative and zero velocities.





24 September 2014

Can an Electric Field Be Negative?

A correspondent writes in:

I've been telling my class that the electric field cannot be negative.  1. Its direction is set and then 2. the value is set.  And since the value corresponds to the predetermined direction, it is always positive.

One of my international students from China used the vector argument.  Since electric field is a vector quantity, can't we 1. choose our direction first - independent of the field and then 2. determine the direction of the electric field and how it meshes with our predetermined direction?  Was I terribly wrong to say that E-field cannot be negative?

My response:  I say an electric field can never "be negative."  Electric field is a vector -- it has magnitude and direction.

Sometimes in a 1-dimensional problem, by convention physicists choose one direction to be positive, one negative.  For example, if south is the negative direction, then a car slowing down moving north might have a "positive" velocity and "negative" acceleration.  And the kinematics equations require algebraic use of the negative signs.  Nevertheless, the magnitude of the acceleration vector would still be 4 m/s/s*, and the direction would be south; the magnitude of the acceleration can never be - 4 m/s/s.

*  not even positive 4 m/s/s, just plain ol' 4 m/s/s

It's legitimate, though crude, to apply the same reasoning to the electric field.  Define up as positive.  Then a 200 N/C electric field that points down could be called "-200 N/C."

But there is NO REASON EVER TO DO THIS IN INTRODUCTORY PHYSICS.  EVER. 

(You can see some of my reasoning in this post: Never trust a student with a negative sign.)

Students get into trouble if they try to use F=qE, and plug in negative signs for q and E to get negative forces.  Negative forces?  What are they?  Forces also have magnitude and direction.  You can't have a -300 N force, just a 300 N force in the downward direction.*

* Again, pedants can argue that such notation can be made self-consistent.  I'm teaching introductory physics, with students who still struggle with the idea that "-300 N" doesn't mean "bad 300".  It's far more important to use notation that addresses the physical meaning of a quantity than notation that maybe, perhaps, with expertise, can be made mathematically reasonable.

So don't ever use negative signs with electric fields -- they're too easy to confuse with negative charges, which mean something completely different, and negative potentials, which are again different.  Have students state a magnitude and direction of an electric field without negative or positive signs:  "200 N/C, to the left."

15 September 2014

Giancoli's stoplight problem -- scale it down and set it up in the lab

A classic problem -- I think I first found it in the Giancoli text, but some variant is in all comprehensive problem sources -- asks for the tension in two angled ropes supporting a hanging object, given the mass of the object and the angles of the ropes.  The version I assign includes a 33 kg stoplight, a 37 degree angle, and a 53 degree angle.

Firstly, adapt the problem for AP Physics 1 rather than the traditional AP B course.  To do that, I begin the question by asking "Is it possible to calculate the tension in the left-hand rope?   If so, explain with words and without numbers how the tension could be calculated.  You need not actually do the calculations, but provide complete instructions so that another student could use them to calculate the tension."  It's okay if students choose to do the calculation first, then tell me how it's possible.  Of course, the obvious approach is to explain that we can set up two equilibrium equations (one vertical, one horizontal) with two unknowns, so the problem is solvable.

Next, I ask for the solution for the tension in the left rope.  They calculate something like 220 N.  

In class, I remind everyone that any numerical solution to a physics problem is not really an "answer," but more accurately a "prediction" of the result of an experiment.  I should always be able to verify a prediction by setting up the suggested experiment.  Thing is, I don't have a 33 kg object to string up from ropes.  Why not?  Because, 33 kg is like 70 pounds.  The mass sets in my classroom don't go above about 1 kg.

And there lies the way to verify the prediction -- scale everything down by a factor of 100.  I do have 330 g to put on a hanger.  Then, I should get a tension in the left rope not of 220 N, but of 2.2 N.  

In the above picture, I've arranged the lengths of the ropes such that the angles are in the right neighborhood.  Sure enough, the left rope read a tension of 200 N, within the 0.2 to 0.3 N N tolerance I expect on a typical classroom spring scale.

Not only does this experiment reinforce the physical meaning of the problem's solution, not only do we see whether the answer is "right" or not, this experiment can help emphasize why the answer "221.43 N" is utterly ridiculous.  The scale can't read better than about 2.2 N or 2.1 N -- it can't read 2.145 N.  Only two digits mean anything (not two decimal places, two DIGITS).  Seeing a "scale reading" rather than an "answer" is the first step toward internalizing that physics is about experiments, not numbers.

08 September 2014

Where do I get AP Physics 1 multiple choice test questions? The Big Amazing Resource.

Testing isn't as simple as it used to be.  Over AP Physics B's 40+ years of existence, enough authentic multiple choice questions had been released to satisfy even the most prolific tester.  However, now that we've moved into the AP Physics 1 era, a lot of those questions are useless; even those that are in the spirit of the new exam need to be rephrased, especially to bring them down to four rather than five choices, and to minimize but not eliminate the questions that require calculation.

Of course, you can go to the College Board's official AP Physics 1 page via AP Central.  There you'll find some sample questions in a file conveniently marked "sample questions."  You can get more in the official "course and exam description.  Finally, if you've ever completed a course audit for AP Physics B or AP Physics 1, you'll be able to download the released practice exam.  Go to your account, go to "add a course," add AP Physics 1, and download the exam.

The 5 Steps to a 5: AP Physics 1 book includes a full practice exam, as well as some good questions at the end of the content chapters.  Or, try looking at the supplements to the Serway textbook's 10th edition; they've hired some seriously connected people to write sample questions for them.  And those of you who have attended my summer institutes or this past summer's open lab have a CD of materials.  Look in the "honors physics" folder, and then look at the quizzes.  Many of those daily quiz questions can be used either verbatim or with minimal revision.  

But the Big Amazing Resource for AP Physics 1 and 2 is the newest version of Matt Sckalor's AP Physics workbooks. About half a decade ago, Matt compiled every released AP Physics B multiple choice question into a single workbook, organized by topic.  Over the summer, a number of AP Physics consultants -- that is, people with intimate knowledge of the new courses -- rewrote these questions so that they meet the spirit of the new Physics 1 and 2 exams.  Now, this new and improved workbook is still not vetted by the committee.  It ain't perfect.  But it is a treasure trove for those who need to make some close-to-authentic tests.

How do I access this Amazing Resource?  The only way is to go through "Pretty Good Physics -- Secure."  Most of this blog's readers already have an account there.  If you don't, you should -- follow the link, and follow the instructions to sign up.  The process is simple but may take a few days, because it is critical that the site administrators verify that all members are honest-to-Bob physics teachers.  

Then, get into the site and search for "workbook."  Out will pop the new workbooks, all ready for you to copy and paste into your tests.

Do you know of another good resource?  Let us know in the comments.

04 September 2014

My students are only averaging 80% on the daily quizzes. What do I do?

A frequent concern this time of year for both teachers and students is grades.  The optimism of the first few days of school has faded... the students have probably gotten back a bunch of quiz and test scores, jolting them into the reality that they're gonna fail this course, and then they won't make the honor society, nor get into college, or at least the right college that would allow their parents to brag at cocktail parties.  Holy #$&*, they'd better go drop this course RIGHT NOW.


I'm sure virtually everyone reading this blog has confronted the described mentality.  One way to deal with it is reasoned logic. Explain rationally and calmly that...


Oh, yeah.  Reason and logic don't seem to be the strong points of students who get carried away with the dramahz.  So maybe we need a different approach that doesn't necessarily include a rational basis.

The first and most important step in damping out this particular fire is to develop a relationship with the college counselor and the director of academics -- that is, the person ultimately responsible for college placement, and the person ultimately responsible for course changes.  These folks generally do have a rationalist approach to their job, and they are used to dealing with artificial drama; what they often don't understand is the nature of a physics course.  Meet with them.  Talk to them informally as well as formally.  Bring them cases of beer.  

Explain how your course works, that it's not about memorizing facts as much as using facts in new situations.  Assure these folks that you don't intend to fail the majority of your students.  I explain that any student who diligently completes all assigned work will earn a minimum of a B- for the year, but that there may be pitfalls along the way.  John Burk likes to compare his class to a theatre production: after a week of rehearsal, the scenes are quite rough, the actors don't know their lines, and the chemistry among characters hasn't yet jelled.  So should we just give up 'cause it ain't perfect yet?  Certainly not... think of physics as year-long preparation to perform on a year-end exam.  Things are expected to be rough at the beginning.

Then work on the students.  There's vast literature -- including on this and other physics teaching blogs -- about how to develop a "growth mindset" among the class.  It's critical at the course's beginning that you NOT discuss points or grades or college plans with any student or parent.  Nevertheless, it's a reasonable expectation that students be put at ease that their 75% quiz score doesn't mean they're getting Cs and having their lives ruined.

My approach is to tell them once -- only once -- how I calculate grades.  For me, 80% is an A, 70% is a B, 60% is a C, and 50% is barely passing.  I've in the past used a "square root curve -- that works fine as well.  When we approach the first test, I show up front the AP scale, in which about 65% is a 5, 50% is a 4, and 35% is a 3.  Then I allow test corrections to earn half credit back for any points they miss.

The final, critical point to the physics teacher is: don't try too hard to make students "feel better" about being in a hard course.  

When my own offspring is upset that we -- GASP! -- ask him to mop the floor, I find that it doesn't help his mood when we sympathetically and pleasantly try to acknowledge his annoyance and assuage his irrational anger.  "Don't worry, it won't take very long" or "Hey, it's not that big a job today, the floor isn't particularly messy" provoke more tantrums from the boy than if we just go away and leave him to the task.  Similarly with your students.  The more you sympathize, the more you try to talk them through and acknowledge their irrational dramah, the more they play up said dramah and talk themselves into a negative spiral of emotion.

Just keep going with the course.  As more and more of your students start to experience success, those shrill, distraught voices will become as whispers in the wind, drowned out by the veritable thunderstorm of positive confidence.










28 August 2014

Is projectile motion part of AP Physics 1? YES.

I got a note last night asking me to help settle an argument.  Since it's the second or third time this summer I've heard the meme, I figured I should address it publicly.

The question:  Is projectile motion part of AP Physics 1?

Answer: Yes.

The argument:  But the Physics 1 curriculum framework does not include any learning objectives with the word "projectile" in them.*  No LOs even say "two-dimensional motion" or anything similar.  Since the problems must be written directly to the learning objectives, nothing about projectiles can be asked.

* The Physics 2 framework includes one learning objective that requires students to recognize the similarity between a charge moving through a uniform electric field and a projectile.

Well, that argument takes an overly literal interpretation of the curriculum framework, I believe.   

My rebuttal:

1. Projectile motion is explicitly mentioned within "essential knowledge" 3.E.1:  "The component of the net force exerted on an object perpendicular to the direction of the displacement of the object can change the direction of the motion of the object without changing the kinetic energy of the object. This should include uniform circular motion and projectile motion."

2. Granted, the subsequent 3.E.1 learning objectives don't explicitly refer to projectile motion.  But I certainly could find a way to write a good problem involving a projectile that matches the 3.E.1 learning objectives, as well as to several others.  Problems can be written to multiple learning objectives.

3. At the AP reading, one of the points made by a College Board representative warned us about mechanistic teaching to learning objectives.  She said that in the biology redesign, a good number of teachers were flummoxed after they spent the year posting the learning objectives and teaching a course narrowly tailored to doing literally and word-for-word what the LOs said to do.  Her point:  the learning objectives are meant to be interpreted in the context of the "essential knowledge" statements, and are all interrelated.  The AP development committees are creating science exams, not the Law School Admissions Tests.  The learning objectives are intended to be flexible enough to allow for coverage of all the course material described in the framework; they are not intended to be narrow prescriptions of the precise questions that will be asked on the exam.

4. Look at the released AP Physics 1 practice exam:  questions 5 and 6 include a diagram with a projectile on it.  QED.

So teach projectiles.  Don't merely teach "here's how to plug numbers into equations to get answers to three significant figures," because that approach is just as doomed to failure in projectile problems as in other topics in AP Physics 1.  Be sure students understand the independence of vertical and horizontal motion, how a vertical acceleration alone produces a parabolic trajectory, how the velocity, acceleration, net force, kinetic energy of/on a projectile change or don't change depending on various aspects of its motion, etc.  But do teach projectiles.



22 August 2014

Required Reading: Kelly O'Shea's "Mistake Game"

Reason number 156 why I love teaching physics:  I don't care how long you or I have been doing this job, there's always something new to learn and to try.  

I looked again last night at Kelly's O'Shea's excellent "Physics! Blog!"  Kelly was the one who first articulated the holy grail of physics teaching, the ability of students to solve "goal-less problems."  While clicking some attractive links I discovered* not the holy grail, but perhaps the shroud of Turin:  Whiteboarding with the "Mistake Game."

* I also discovered that Kelly independently made a Clever Hans reference two years before I did.

I am well aware of the verb "to whiteboard," and I'm extensively tutored in its benefits.  It's not like I disagree with or disapprove of the technique; I just haven't yet found a place for it in my classes.  I've always managed to accomplish everything intended in a whiteboarding session by using pencil-and-paper.  Kelly's post has changed my mind, by introducing the critical element: the deliberate substantive mistake.

You should read Kelly's post directly for the details about her Mistake Game.  My quick summary:  Give several groups of students a problem to whiteboard... but each group is required to intentionally include one substantive mistake.*  The groups take turns presenting their solutions to the class.  Since they're charged with selling the solution, including the deliberate mistake, the students actively engage their audience.

* They're allowed to include as many unintentional mistakes as they'd like.

The mistakes are to be discovered through questioning.  Kelly says she has to teach audience members to ask substantive questions in pointing out the substantive mistakes -- not "hey, that's wrong, the slope is the wrong way" but "Which way was the car moving?" followed by "And how does a position-time graph indicate the direction of motion?"

I've always wanted to bring the methodology of the Physics Fight into my honors physics classroom, but I never found a way that made me comfortable.  Usually a typical honors-level problem is complex, but not complex enough to provoke real discussion and conversation except on rare special occasions.  As Kelly points out, the requirement of a substantive mistake facilitates that discussion -- the presenters vie to create a mistake that is "good" enough to pass the audience's examination, while the audience members vie not just to discover but to engage in conversation about the mistake.  Imagine that... to get authentic discussion, I need to require inauthentic but authentic-seeming mistakes.  Kelly, that's practically Zen.

GCJ 

15 August 2014

Mail Time: How much detailed attention to significant figures in AP Physics 1? (Answer: NONE.)

Mo here, a retired engineer teaching AP Cal AB & BC (for 13 yrs.), and now taking over the AP Physics program. 

I can’t seem to find clear information anywhere with regard to significant figures (SF) and how they’re handled by AP Physics readers. Being an engineer, you can imagine I tend to be very strict about SF, but I don’t want to overwhelm my students unnecessarily if the College Board does not stress it.

I would greatly appreciate any advice.

Hey, Mo... this is certainly a frequently asked question.  On one hand, don't harp on students about sig figs. If they put four rather than two sig figs, that's usually not in any way an issue.  However, if they're writing down every digit on their calculator, they're missing something, and they're possibly losing credit.  

The new exam does discuss experimental uncertainty.  Rather than stress rules of significant digits, instead you might talk in context about the uncertainty in a measurement.  Two measurements of 0.19 kg and 0.20 kg are equal, because by definition "0.19" means somewhere in bewteen 0.18 and 0.20.  However, two measurements of 0.191 g and 0.202 kg are NOT equal.  

But most importantly, your question about significant figures is nearly irrelevant for the new exams.  

Take a look at the released practice exam, and see how many questions require a calculated numerical answer.  Only one part of one free response question does:  that's 1(b), which asks for a calculation of external force on a cart based on a graph of experimental data.  Now look at the rubric for that part: of the four points awarded for this calculation, not a single one is awarded for the numerical answer.  

As with everything on the new exams, rote rules about arithmetic are far subordinate to the physical meaning of predictions and experimental results.  And calculation -- whatever the number of significant figures on the answer -- is far subordinate to explanation and description.  

Hope this helps...

greg

13 August 2014

AP Physics 1: Define the system carefully when doing energy problems

For years I've taught energy conservation via an equation.  Write the energy conservation equation for the object in question, identify each term, and solve.  That gets the right answer most of the time.  However, in AP Physics 1, getting the right answer is not as important as describing clearly how to get the right answer, and why the right answer is in fact the right answer.  I need to dig deeper for this new course.

In particular, I've always been fluffy about defining the system experiencing these energy changes.  I implicitly was referring to the single object experiencing a change in position or velocity.  That's "fluffy" because potential energy is created through an interaction within a system of multiple objects; and because I'm making all sorts of assumptions about what forces doing work should be included on the left side of this equation.  The students could use this equation to get the right answer, but they could not have described in careful, correct language the energy transfer.

The NTIPERS book provided me with a guide to teaching my students to use a deeper approach.  In each energy problem presented by this book, the authors clearly define the system being analyzed, and they list the forms of energy the system could possibly possess before, during, and after the process in question.  NTIPERS suggests making a bar graph for each problem listing the amount of each energy; words and numbers are probably enough, but the bar graph is a nice approach, too.

As an example: consider 50 kg Tarzan dropping on a vine from rest, as on the 2014 AP Physics B exam problem 1.  One part asks the student to calculate the speed of Tarzan as he swings through his lowest position (call it 2 m below his starting point), neglecting air resistance.  We can approach this two equivalent ways:

1. Treating the Tarzan-Earth system:  Systems can have potential energy due to their interaction.  Systems can also have kinetic energy, which is equal to the sum of the kinetic energies of the component parts of the system.  Define the lowest position of Tarzan as the zero of gravitational energy for the system.  That means the Tarzan-Earth system begins with mgh = 1000 J of gravitational energy, and (because nothing is moving) no kinetic energy, giving 1000 J of system mechanical energy at the start.  During the process, no work is done by an object external to the Tarzan-Earth system -- the only other object exerting a force on any part of the system is the rope, and the rope is always pulling perpendicular to Tarzan's velocity, so does no work.  Since the gravitational energy is zero at the bottom point, the entire 1000 J of system mechanical energy must be converted to kinetic energy.  Now use the kinetic energy formula to calculate speed.

Summarizing: 1000 J gravitational energy before --> no work during --> 1000 J kinetic energy after.

2. Treating Tarzan as a single object:  Objects can have kinetic energy, but not potential energy.  So Tarzan begins with no kinetic energy.  During the process, two forces act on Tarzan: the force of the string, and the force of the earth.  The string force does no work on Tarzan because it is always perpendicular to Tarzan's velocity.  The earth does work on Tarzan equal to Tarzan's weight multiplied by the distance Tarzan travels parallel to the weight, or 1000 J.  This is positive work, because the force of gravity is parallel, not antiparallel, to the downward component of Tarzan's displacement.  At the bottom, then, we know 1000 J of work has been done on Tarzan; this must change Tarzan's kinetic energy by 1000 J, since we're not treating Tarzan as a system with internal structure.  Now use the kinetic formula to calculate speed.

Summarizing: 0 J total energy before --> 1000 J work done by gravity during -->  1000 J kinetic energy after.

This may sound like semantics to you.  It certainly does to me.  But I'm telling you, the new exam will ask students to explain energy conversion and conservation with direct reference to an appropriate object or system.*  We must teach the students to describe energy conversion in the kind of language I've included above.

* Look at learning objective 5.B.1.2 in the curriculum framework, for example.

Furthermore, the students are expected to be able to understand when the model of an object or system fails.*  What does that mean?  We calculated 6.3 m/s for Tarzan at the bottom.  A follow-up question might suggest that Tarzan was actually moving 5.0 m/s at the bottom, and experiments verify that air resistance has no measurable effect on Tarzan.  The question may ask for a resolution of this discrepancy.

* That's LO 5.B.2.1 for those of you with an education doctorate.

A reasonable explanation is that treating Tarzan as an object isn't reasonable.  He could be spinning; the "missing" mechanical energy is accounted for by his rotational kinetic energy.  Similar questions could be posed in which an object seems to have less energy than predicted by a point-object model, but some internal structure in motion accounts for the total mechanical energy budget.


09 August 2014

Trading and Grading Homework Problems In Class: The Context of Your Feedback Matters.

Years ago I described how I grade nightly homework assignments quickly.  Point of emphasis: I don't write on the students' papers.  No one reads my comments carefully.  I just put a number on the paper, record that number, and give it back the next day.

I've also suggested that it's not effective to "go over" a homework problem in detail in class.  Johnny might have asked you to go over the problem, but he'll tune out once he finds where he thinks he lost points; he won't learn a lasting lesson.  And everyone else in the class has tuned out, because they either got the problem right to begin with, or they just don't care right now.

The purpose of grading homework, for me, is NOT to give detailed feedback.  It's to be sure that students are keeping up with the assignments, that they are practicing their problem presentation, that they are engaging appropriately with the material.  If I don't grade homework, the homework doesn't get done properly.

It's reasonable to ask, then: If I'm not writing more than a number on a homework problem, and I'm not discussing the problem in class, how should a student who wants detailed feedback get it?

Timing matters.  Your students have a bobzillion things going on in their lives.  Like it or not, physics just isn't a constant priority.  In order for feedback to be meaningful and useful, it has to be given at a time when the student is ready to receive it.  

One effective time for feedback would be as the student is struggling with the homework problem, immediately after he's completed the problem.  Thing is, you're usually not present at just the right moment; and, if your student is actually doing the homework problem in your presence, he's going to want the feedback* before the time is right.  

* And you're likely to give him the feedback too early, too.  Students must engage with the problem.  It's a good thing for them to get stuck.  One of the more important physics teaching skills is to find a way to politely yet firmly deny assistance to a working student until the time is right.

One way to provide appropriately-timed help is to set up part of class such that the time will be right for feedback.  Have the class solve a problem while you sit in the front of the room, encouraging students to see you after they've completed each step.  Just the fact that there are twenty other students needing your attention will usually convince students to keep working until they're well and truly stuck.  Then the faster workers can experimentally verify the answer to the problem while you help the slower folks.

But as for homework, students need to learn to rely on their classmates.  Whether the students are working in small physics parties at night, or whether they just send questions to each other via text, email, facebook, etc, they are helping each other become un-stuck.  That's the whole point of collaboration -- not only so everyone can make appropriate progress on tough problems, but so they develop the experience of teaching the problems to someone else.

Context Matters.  So Set Up A Useful Context For Feedback.

Imagine you've talked to a student explicitly about how Newton's third law force pairs can't act on the same object.  Perhaps you said so in class, and perhaps you also helped this student get un-stuck on a homework problem where he was trying to have the force pairs "cancel" each other to equilibrium.  And then, on the test two weeks later, he made the same mistake again.  Aarrgh!

Think about this student's response when you helped him.  In class, he listened attentively, maybe even wrote down some words; on the homework, he nodded, said "okay," and corrected his mistake.  Remember, everything we do in physics class is fleeting.  We like to think that students are motivated by their grade or their love of the subject to remember everything we say and everything they're asked to do.  But really, once the problem set is done, so is the student.  Even if the timing of your feedback to this student was right, the context wasn't.  

I like to create multiple contexts, each of which has some level of import.  For example, I give a quiz every day.  On the quiz, I might ask directly: "Can Newton's Third Law Force Pairs act on the same object?"  or, "The earth pulls on an astronaut, and an astronaut pulls on the earth.  What is the acceleration of the earth-astronaut system?"*  Even though the student had to use this fact on the previous night's homework, he might not have really internalized what he was doing -- after all, students are answer-focused, no matter how hard we try to change them.  This quiz provides the same question in another context.  When we go over the quiz, now the student hears the answer, and hopefully will make the connection to the homework problem.

* Expecting the answer "It can't be determined," knowing that too many folks will say "zero."

And finally, I've taken to having students grade homework problems to a rubric once every week or two.  While generic "going over" a problem is ineffective, describing the solution in the context of awarding points to a friend's paper is extraordinarily effective.  Without my prodding, students often write detailed explanations on their classmates' papers.  After the in-class grading exercise, students have done the problem, seen my solution, applied my solution to grade a friend's problem, thought about what score they themselves might have earned, then seen their friend's detailed commentary.  Now THAT'S context.