Freshmen: taking things one correction at a time

Test corrections have been a staple of my classes for years.  In a junior-senior course, I assign everyone to correct every answer they got wrong.  This work is done sometimes in class, sometimes as homework, sometimes in a combination of the two.  Since I'm usually either awarding points back on the test, or making the corrections worth a major grade, I generally get good effort and good understanding of the original mistakes.

I've given the same assignment to my freshmen this year.  But, this year test corrections have not been particularly effective.  Students make the same mistakes on the correction as they did on the test; or, despite repeated entreaties from me, they state claptrap like "The cart weighs 90 N because that's the only answer that makes sense." 

I'm realizing that part of the problem here is the non-immediacy of my response.  With, say, four problems to correct in a homework session, ninth graders seem much more concerned with merely getting all four done as quickly as possible than with getting the answers right.  They spew some BS, write enough that I won't call out their laziness, and then make puppydog eyes the next day when I tell them all their answers are wrong.  If they are to work on the corrections in class, freshmen sit and stare, or yap with friends, far more than my older students ever did.  Their answers are no more likely to be correct than if I assigned corrections as homework.  

I've seen it suggested that the "immediacy of evaluation" is one of the major video game features that turn on middle schoolers and teenagers.  In an adventure game, you either accomplish your quest or don't, but you find out right now, and unambiguously; then if you failed, you get another immediate chance to do it right.  Sports or shoot 'em games are the same way, just substitute "score" or "die" for "accomplish your quest."  

When it comes to video games, adults don't have to nag children about how "practice doesn't make perfect, only perfect practice makes perfect."  The video game structure ensures that only perfect practice exists.

So this week, in the context of second trimester exam review, I made my class more like a video game.

My typical "multiple choice correction sheet" is a full 8.5"x11" page, divided in half so that one correction takes up half the page.  Well, for this year's trimester review, I cut the pages in half: one small page, one correction.  One at a time.

The in-class assignment was to correct every problem missed on the recent test; then, to correct each problem missed on the last few problem sets; then, to pull from a "grab bag" of questions.  For each question they did CORRECTLY, they earned a ticket, good for extra credit and a game of skee-ball.*

* Yes, I do in fact have a skee-ball machine at my house.  Don't you?

But each person was required to show me their answers, not at the end of the period, but after each individual problem was completed.  If the answer wasn't perfect, I sent the student back to his seat to do it right.

This approach worked amazingly well.  Most got into the swing of the class quickly.  Since they were moving their bodies frequently, they did a lot less aimless staring.  Since there was a tangible, immediate way to measure their progress (the tickets), I saw a lot less yammering about non-physics topics.  

I did scan the room regularly -- when I saw a student who had been seated for more than about 5 minutes, I called him out: "What are you working on?  Why haven't I seen you?  No, you're not allowed to be 'stuck,' you've got 15 other students who can help, or you can come ask me.  You have two minutes to be up here with some sort of answer, right or wrong, capish?"

Amazingly enough, at the beginning of each class I still got some claptrap.  But this time, instead of me nagging two days after a POS problem had been submitted, I said something right away and in person: "No, 'the acceleration is left because the block is accelerating to the left' is utter nonsense.  Please read it out loud to me.  Does your sentence explain anything?  Does it help you understand why the acceleration is to the left?  Does it include a fact of physics from our fact sheet?  No?  Then please try again to justify the answer the way the class has been taught to justify answers."  

In most cases the next attempt was spot-on.  Go figure.  

First months of 9th grade conceptual physics: non-cumulative material

The diagram to the right shows a mirror.  On the diagram, draw a dotted line representing the normal to the mirror's surface.  Justify your answer.

Conceptual physics covered ray optics as the first topic of the year, back in September.  We then moved on to waves, and to circuits.  Part of the reason for this sequence was because these are easier topics than the typical kinematics and forces opening gambits.  I want freshmen to adjust to boarding school life a good bit before I hit them with the hard stuff.  But the more important reason for this sequence is that it's not sequential at all.

Kinematics and forces are self-referential.  It's important to internalize a definition of acceleration, which is used in every context imaginable.  Many force problems require kinematics to solve fully, and vice versa.  Then in whatever topic is next -- usually either energy or momentum -- it's assumed that students are comfortable with forces and motion.

This approach works fine with my seniors, because they usually are in fact reasonably comfortable with forces and motion by the time I move on; and because even the slower students have enough background that they can become comfortable.  Seeing forces and motion in new contexts provides extra practice and encouragement to review previously-discussed physics.

Freshmen, though, can be absent mentally for much of our first trimester.  It's not that they don't want to do well -- just the sheer overwhelming nature of life without mom and suddenly with 400 siblings, coupled with the rate at which they're growing physically and mentally, can mean they don't remember information day to day, or even minute to minute.  At the senior level, I'm assuming a level of personal organization, daily focus, and self-driven practice that freshmen can simply not fathom.  

We just gave our first cumulative trimester exam.  Some did great; some did terrible.  My point here is, how they did this trimester doesn't matter that much to the students' overall success in the course.  When we move on to kinematics after Thanksgiving, it won't make any difference at all whether they remember whether light bends toward or away from normal.  I don't HAVE to go over the exam, I don't have to review anything; we can move on to new and different stuff, knowing that everyone can understand it in isolation from the first trimester.  Then I can sprinkle some review in over the course of the next six months in preparation for the final exam in June.

The question at the top of today's post shows one of the "justify your answer" questions on the trimester exam.  We've learned that the "normal" is "an imaginary line perpendicular to a mirror's surface," and we've extended that definition in the context of refraction across a boundary between materials.  This question requires the student to recognize that the normal is not perpendicular to the bottom of the page, but rather to the optical instrument in question; the justification just requires some statement of the definition of normal.   

So why do I ask this question on the exam?  Because it is the ONLY question I can think of that is truly cumulative with other topics we will be teaching this year... when we get to "normal forces," we'll have seen the word before; and we'll even have seen an explicit situation when the normal is at an angle to the vertical.  

Zen and the art of predicting voltage across series resistors

I have a circuit in which a 14 V battery is connected to a 15 ohm and a 25 ohm resistor in series.  What's the current through and the voltage across each resistor?

In my honors-level classes, I teach a mathematical solution using the VIR chart.  They calculate the equivalent resistance of 40 ohms; use ohm's law on the total circuit to get a current of 0.35 A; recognize that series resistors each take that same 0.35 A current; then multiply across the rows of the chart with ohm's law to get 5.3 V and 8.7 V across the resistors.

In conceptual, though, we don't use a calculator, and I want to minimize (not eliminate) calculation, anyway.  So we approach this problem slightly differently.  

Take a look at the worksheet we've used in laboratory to learn how to deal with series resistors.  Students must justify their answers to each question thoroughly.

1. Which resistor carries a larger current through it?

We start with a not-so-subtle reminder of the rule that each resistor carries the same current. 

2. Which resistor takes a larger voltage across it?

Now, I insist on an equation justification with ohm's law.  V = IR; I is the same for each because series resistors carry the same current.  By the equation, then, the larger 25 ohm resistor takes the larger voltage.

3. What is the equivalent resistance of the circuit?

Fact: the equivalent resistance of series resistors is the sum of the individual resistances.  Just add 'em up to get 40 ohms.

4. Calculate the current through the circuit.

Only now do we do a calculation.  We've learned that ohm's law applies for the total voltage and resistance in a circuit.  So, we simply say that I = V/R, with V = 14 V and R = 40 ohms.  This makes the current 14/40 amps. Yes, I allow that as the answer -- we aren't using calculators, and I don't want to mess with issues of significant figures and decimals.  Look, go ahead and haul me before the Klingon Death Tribunal for my sins.  I'm looking forward several years.  When they see circuits in a senior honors class, they'll be able to call this 400 milliamps.  For now, I'm happy that they know which values to plug in to ohm's law.

5. Estimate the voltage across and current through each resistor in the chart below.  You may use fractions for current, but not for voltage.  Answers without units earn no credit.  

Here's the Zen.  Not calculating current through each resistor -- that's the same 14/40 amps through each.  The Zen is the estimate of voltage.  I'm NOT teaching them to multiply the resistance of each resistor by the current.  Nor am I teaching them to proportionalize the voltage according to the resistance.  Nope.  I'm just saying "estimate."  

All I am looking for is an answer that fits the facts they've already stated:  the voltage across each must add to 14 V, and the 25 ohm resistor must take greater voltage.  Some students will guess 2 V and 12 V; some will guess 6 V and 8 V.  I don't care.

6.  Now, set up the circuit, and MEASURE the voltage across each resistor, and across the battery.  Record your results here.

And now the Zen must be reconciled with reality.  They make the voltage measurements, and see how the voltage is distributed:  in this case, about 5 V and 9 V.  Some students got the estimate right -- they get candy.  I praise everyone else for a "good guess;" we look to see whether their guess was high or low for the 25 ohm resistor.  

I don't teach anything, still; instead, I hand out a new sheet, with different resistors.  They fill it in again, all the way from the beginning, steps 1-6.

As you may or may not suspect, by the second or third time most students are getting pretty dang close to the right voltages.  Some folks discover the proportionality rule for themselves.  Others just recognize that "close" resistors demand "close" voltages, and "far apart" resistors demand disparate voltages.  

To me, this process is teaching good physics.  I've taught the calculations for series resistors for ages, and I've been repeatedly frustrated by students who can make calculations well but can't answer simple conceptual questions like "which takes the bigger voltage."  And as well by students who frustrate themselves because they predicted 8.25 V but only measured 8.22 V.

By the second day of filling out these sheets and measuring voltages, these freshmen are getting almost bored with the process.  That's the sign I'm looking for.  When I start to see faces saying "Gawd, not again with the voltage question..." that's when I know it's time to move on.  I give the faster guys a sheet with three, not two, resistors; then, after a multiple choice quiz, we move on to resistors in parallel.  

I'll teach parallel resistors the exact same way.

GCJ

Graph paper link, and setting up a graph for conceptual physics

Above picture from
free online graph paper

Wednesday is our first day of class.  New students arrived here on Sunday, so we're in the midst of two days of orientation and partying.*  I've spent the past week making final preparations for teaching 9th grade conceptual physics -- no wonder I haven't posted much.

*Unfortunately, the skee-ball machine is delayed, so probably won't be at my house for my advisee group's get-together tonight.  I'll have to use it for physics classes, instead.

In the first class session, we will set up a mirror and ray box.  Students will take turns using a protractor to measure angles of incidence and reflection.  Each student will make a graph of angle of incidence on the horizontal, and angle of reflection on the vertical.  The point here is to establish context for reflection and refraction, and to be sure we all know how to use a protractor before the first true laboratory exercise on refraction.

In a 12th grade course, I'd have students figure out how to scale the graph.  Dealing with a range of 0-90 degrees, when the graph paper has seven major ticks and 70 minor ticks on the horizontal axis, is a serious skill that I try to teach, and which takes significant patience and practice.  However, for 9th graders I'm happy if they can plot points accurately at all.

So I've gone to this link for free online graph paper.  This page is particularly nice because you can choose from common templates (e.g. 1 inch major with 1/10 inch minor gridlines), OR you can customize the weight of the line, number of lines per inch, etc.

For that first class session, I can bring the class to understand that we need to scale each axis from 0-90 degrees.  Making that scale will be easy as pi, because I've printed out customized paper that has nine major gridlines across the page, and five minor gridlines per major gridline.  The class can see quickly that each major line will represent 10 degrees, so each minor line will represent two degrees.  And the graph will take up the whole page.

GCJ

Conceptual physics lab: refraction without using Snell's law

We're starting the year in conceptual physics with geometric optics.  On the very first day, we'll set up a mirror and a ray box.  The class together will measure a bunch of angles of incidence and reflection, and we will together graph θr vs. θi.  We'll set up both axes from zero to ninety degrees, and show that we get a straight line.  We'll show from the graph that, say, doubling the incident angle also doubles the reflected angle.  No slopes, no calculations, just graphical experimental evidence for the law of reflection.

The next week, we'll use the ray box to shoot light into a plastic block.  We'll set up the same axes, and this time graph refracted vs. incident angle for light refracting in the block.

Now, in AP physics, we'd make a new graph of sin θi vs. sin θr; the slope of that line would be the index of refraction of the plastic.  But this year I'm not teaching AP physics.  This is ninth grade conceptual physics.  So what are we doing?  And why are we doing it?

Well, I will certainly spend about five minutes asking students to predict what the θr vs. θi graph might look like.  I'd be happy if anyone recognizes that the data won't ever approach 90 degrees on the refracted angle axis.  But prediction is not really the essential issue.  My goals for this experiment are:

(1) Establish norms for data collection.  This is a simple experiment.  I showed my nine-year-old how to do it just once, and he collected data without my help.  So there's no reason we shouldn't be able to acquire data quickly, accurately, without asking a bazillion questions.  I will go around the class and hurry them along, teaching them not to second-guess data, not to be overly precise... and importantly, to collect data rather than argue with the partner.  It's also a good experiment with which to establish the rule that since no one is leaving early or doing work for another class, we might as well take as many data points as humanly possible.

(2) Practice graphing data as it is collected.  I recognize that many freshmen will struggle with a skill as simple as graphing data by hand.  So I use this straightforward experiment with axes scaled identically on the vertical and horizontal to start practicing.  Each partnership will graph data as it is acquired, one graph per group; then I'll show them where they need to collect more data in order to define the shape.  When they're done collecting, I'll give them a homework sheet, part of which will ask them to regraph their data on axes that I've prepared.  The homework is one graph per person.  This way, a partnership can split duties efficiently during lab, but nevertheless everyone will have to physically make a graph.

(3) Make a first stab at interpreting graphs.  A novice physics student might think it "stands to reason" that since the angle of incidence vs. reflection was a straight line, the angle of incidence vs. refraction should also be straight.  That's not the case.  One of the homework questions will ask whether doubling the angle of incidence doubles the angle of refraction as well... just looking at the graph shows that the angle of refraction does not quite double in this case* -- no use of Snell's Law is required at all.

*Okay, for small enough angles, doubling θi does in fact double θr as well.  If someone is actually astute enough to point out the small-large angle difference, I'll be pleased, but I'm not expecting that level of analysis.

I'm teaching ninth grade conceptual physics next year.

We used the Hewitt book
in 2012. No longer!
I recommend using no text at all.

I've spent the last few years getting my school's upper level physics classes in shape.  I now have a skeleton of problem sets, quizzes, labs, and tests that my colleagues can use to teach 11th-12th grade general physics, honors physics, and AP physics B or C.  The general physics course is based loosely on the New York Regents curriculum; the honors physics course on my version of what a future AP physics 1 test might look like.

So now it's time to tackle the only level of physics that I've never taught (cue ominous music): Ninth Grade Conceptual Physics.

(As an aside, I *have* taught ninth grade.  Once.  My first year of teaching.  "Integrated Science."  At a fluffy school.  Team teaching with an idiot who denied the results of experiment, who undermined me to students and administrators at every opportunity.  The scars still give a wee twinge on rainy days.)

The juniors and seniors I've worked with for years enter the school year as reasonably mature students; it's been my job to provide them with a challenging course to which they can apply their well-developed study skills.  Upper level physics can almost be thought of as a master class -- here's where all of the math, organization, relationships with classmates, writing, all of everything you've ever done as a student must be used in combination to conquer a difficult but manageable subject.

I am well aware that ninth grade is a different boat of gravy altogether.  I've begun talking to some of my school's best ninth grade teachers, listening to their thoughts and ideas of how they structure their course, how they develop relationships with 14 year olds, what different types of issues I can expect once I start teaching an entirely new species.  

One overriding goal over the next two or three years is to develop my own version of a ninth grade conceptual physics course, complete with a course structure, problem sets, laboratory activities, quizzes, tests, etc.  In terms of the level of physics, I want to aim at a low-arithmetic adaption of Regents-style questions, as I explain in this post.  Some of the course structure ideas that I know I'm going to implement:

Pace of the course:  I need to invert my usual approach.  With seniors, I've got to shoehorn in as much material as possible in the first half of the year.  That's when they're still motivated by their grade, that's when they are still afraid that any slackage might lead to the world ending and having to go to (gulp!) a different college than their first choice.  In the spring, I get some work out of seniors by demanding less.  They feel like I'm legitimizing their senior slide, so they actually do the minimum amount of work that I ask of them without complaint.  Thus, I'm always pushing the pace in the fall, and tapering through the spring.

With freshmen, I recognize that the fall is NOT the time to push hard in physics.  Adjusting to high school, and in my case to boarding school, is a difficult process for an adolescent.  Sure, a few students are ready for serious academics from day one -- these folks will be siphoned into Honors Physics within a few weeks.  Most need a gentle introduction to high school.  Then, in February or so (just as the seniors start to slack), freshmen are ready to move fast.

Sequence of coverage:  It's been argued that a physics class can seem friendlier by starting with more straightforward topics like ray optics.  At the AP level, I completely disagree -- the last thing I want to do is to give the immediate impression that memorizing facts alone will lead to physics success.  I want to start tough and get easier.

But in ninth grade, we will start with ray optics.  Refraction, total internal reflection, lenses, and mirrors all can be taught well diagrammatically and conceptually, with absolutely no mathematics.  But, I can use Snell's Law and the thin lens equation as an "application" for the honors course -- students who *can* handle quantitative predictions with these equations can be moved out, while the remaining students learned some serious physics without feeling bowled over by mathematics.  

I haven't decided on a precise sequence of coverage, but I do know that we want to gradually add arithmetic and basic algebra as the year progresses.  By the end we will certainly have covered the "Big Three" skills of reasoning with equations, interpreting graphs, and understanding the meaning of numbers.  It's just that we'll get to these skills gradually, after we start with a topic that allows straightforward conceptual prediction and straightforward experimental verification.

Types of test questions: I talked extensively to Bruce Oldaker, who at one time was in charge of helping the physics department at West Point streamline their testing at all levels.  He put into words a point about test construction that I have always done by feel:

First consider the portion of a test that is essentially recall, asking students to state facts or solve simple problems in situations they've seen before.  Then consider the portion of the test that asks students to synthesize multiple concepts, to extend problem solving techniques to new situations.  By sorting test items into bins of "recall" and "synthesis," it's possible to control the perceived difficulty of the test while adjusting the rigor of the evaluation of the students' physics knowledge and ability.

Now, as a long-time AP teacher, I have always advocated (though I didn't say it this way) keeping the "recall" and "synthesis" portion of a test consistent throughout the year, and in similar proportion to what students will see on the cumulative national exam.    Sure, that makes the first test of the year seem difficult; but soon enough students are old pros at physics tests involving considerable synthesis.  The shock of a test that doesn't just present homework problems with the numbers changed is going to happen sometime in the year; so, I say, deal with the shock right away when the class has plenty of time to recover.

Freshmen, though, need to build up to "synthesis" testing gradually.  Bruce and some of my colleagues point out that the same students who can't do anything but spit back facts at the beginning of the year will often develop their reasoning skills so that they can handle difficult questions by June.  Freshmen are growing that much physically and intellectually.  So, Bruce suggests that early tests be as much as 80%-90% recall... and that by the end of the year the recall percentage can be reduced to 40%-60%.  Without wasting too much time on meta-analysis, I'm going to be conscious of starting simple, and adding complexity to my tests throughout the year.

Got any ideas?  I'd love to hear 'em.  I've got an enormous amount of work to do to develop the freshman course to my liking.  It will take several years, and it will take plenty of failed attempts, too.  Maybe in a couple of summers I can hold a "Conceptual Physics Summer Institute" where we all get together to talk about teaching freshmen...

Update Feb 2021: I did hold that Conceptual Physics Summer Institute last year - it was well received, see the reviews in the comment section of the linked post!  I will be doing another CPSI in 2021, likely the weekend of July 24-25 and July 31-Aug 1. I'll post details soon.  

Framing appropriate questions for conceptual physics

I've been working a bit with my colleagues on our 9th grade conceptual physics course.  We teach a rigorous physics (not "physical science") class to all 9th graders.  It's a difficult proposition to aim the material at the correct level.  Many students have not taken algebra, and those who have are certainly not fluent in algebra skills.*  We want to minimize arithmetic, and concentrate on conceptual skills.

* Those top-15% students who are fluent in algebra are broken out into a section of Honors Physics I.

Nevertheless, we want to teach serious physics, not merely a set of facts to be learned or situations to be memorized.  We still teach physical reasoning from equations, for example... but in the sense of "mass doesn't change, speed doubles, so by ½mv² , kinetic energy quadruples."  This is some of the same fundamental understanding expected from AP-level students, but at a slower pace, with fewer equations, and without a calculator necessary.

Problem is, it's tough to find questions at this appropriate level.  Hewitt's conceptual physics text is a great source, of course, but I'm talking about finding a huge bank of questions that will allow you to write numerous quizzes, tests, and exams.  For college-level physics, the AP program provides more questions than you'll ever need.  At the general but quantitative level, the New York Regents exam is the way to go.  I have not yet found a good non-quantitative, published source of questions that are ready to copy-and-paste into your tests.

Now, the Regents exam includes occasional qualitative questions.  These can be used nearly verbatim in conceptual physics.  Most of the Regents questions include arithmetic or algebra, though, often emphasizing the mathematics through the phrase, "show all work, including the equation and substitution with units."  I have no complaints about this quantitative approach; in fact, I train my junior-level general class to handle Regents-style questions.  I just know from our department's experience that, for freshmen, "substitution with units" presents a considerable barrier to physics understanding.

Try turning a quantitative Regents question into a no-calculator conceptual physics question.  For example, from the January 2006 exam:

The speed of a wagon increases from 2.5 m/s to 9.0 m/s in 3.0 s as it accelerates uniformly down a hill.  What is the magnitude of the acceleration of the wagon during this 3.0-second interval?

(1) 0.83 m/s²     (2) 2.2 m/s²      (3) 3.0 m/s²       (4) 3.8 m/s²

Four different ideas occur:

Ask about the acceleration's direction instead of its magnitude.  Freshmen can learn the fundamental fact that speeding up means acceleration and velocity are in the same direction, while slowing down means acceleration and velocity are in opposite directions.  I'd write...

A wagon travels down a hill.  The wagon's speed increases from 2.5 m/s to 9.0 m/s in 3.0 s.  What is the direction of the wagon's acceleration?

(A) up the hill       (B) down the hill   

(C) straight down     (D) straight up

Ask for a straightforward calculation of the acceleration.  Even though I'm making the problems accessible without a calculator, I'm not ignoring quantitave reasoning entirely.  It *is* important that a student recognize that acceleration depends on the change in an object's velocity, not on the velocity itself.  So, I'd write...

The speed of a wagon increases from 9 m/s to 12 m/s in 3 s as it accelerates uniformly down a hill. What is the magnitude of the acceleration of the wagon during this 3.0-second interval?

(1) 1 m/s²     (2) 3 m/s²      (3) 3.5 m/s²       (4) 4 m/s²

Trying to just divide any old velocity by 3 s leads to an incorrect answer.  The the correct answer can be determined at a glance, even by a mathematically inept ninth grader.  (Math teachers will cheer now, because we're forcing students not to grab a calculator to manipulate (12-9)/3.)

Ask for a comparison to familiar values.  The only acceleration that our students probably have a feel for is g.  So, ask...

The speed of a wagon increases from 2.5 m/s to 9.0 m/s in 3.0 s as it accelerates uniformly down a hill.  Is the magnitude of this wagon's acceleration...

(A) greater than Earth's free-fall acceleration

(B) less than Earth's free-fall acceleration

(C) equal to Earth's free-fall acceleration

Such a question does NOT necessarily require a direct calculation of the wagon's acceleration.  If the student thinks in Hewitt-ese, then speeding up in free-fall means gaining 10 m/s of speed every second of fall.  This wagon accelerated for 3 s, and gained nowhere near 30 m/s of speed, giving (B) as the only possible answer.

Ask about the physical meaning of numbers.  Even without calculators, our students should develop a feel for the physical reality represented by numerical answers.  Speeds in m/s can be estimated in mph by multiplying by 2 and adding a bit.  But I'm not asking anything truly quantitative here:

The speed of an object increases from 2.5 m/s to 9.0 m/s in 3.0 s as it accelerates uniformly.  Which of the following objects could NOT reasonably perform this motion?

(A) A car on an interstate  

(B) An airplane during takeoff  

(C) A lab cart on a track in the classroom

(D) A bicyclist going down a hill

There you have it -- FOUR different conceptual physics multiple choice items inspired by a single Regents question.  And any one of these questions can be expanded into an open-response test item,  or assigned for homework, by adding the phrase, "justify your answer."