A blog about physics education - student work, demonstrations, lesson ideas, and reflections on standards-based grading. Not endorsed by or affiliated with The Tatnall School
Honors Physics is wrapping up our work with the constant velocity particle model (CVPM), which is our description of how objects move with constant velocity, including several different kinds of descriptions of motion, and lots of tools to make predictions about these kinds of motions. We've now worked with five different representations of CVPM motion:
x vs t graphs
v vs t graphs
Verbal descriptions (I called them textual descriptions in the first class today, and got the expected chorus of giggles, so I gave up on that)
Algebraic models (equations)
Diagrams
We worked quite a bit today with algebraic and diagrammatic representations of motion, having used graphs a great deal for the past couple of weeks.
The best way to do it is to do it, so they jumped in and formulated their own questions about the two constant velocity buggies that I gave them; I took out one battery from the red one and replaced it with a strip of copper to make it run significantly slower. They were the equipment; the question was "what can you do with this?"
They came up with great questions - some were classics, and some were a little more out-of-the-box. All let them test their diagramming and algebraic problem-solving chops:
Cars facing each other - "where/when will they collide?"
Cars starting apart - "when/where will one car pass the other?"
Cars starting together, facing away from each other - "when will they be 10 m apart?"
Not only did they solve this one algebraically, they took their extra time and solved the problem graphically as well!
Cars starting together - "when will one car have gone twice as far as the other?"
This was very interesting; after a great deal of wrestling with their algebra, the time interval and the distance always canceled, leaving:
This is a strange result indeed - no reference to time or distance! The interpretation: if they really do start at the same spot, the question can only be answered if the faster car has twice the speed of the slower car. If it does, then any time or distance works. If not, no time or distance works!
Physics did the Timer Challenge this week - they were tasked with using an oscillator (though we hadn't used that word yet) to measure time. After 40 minutes of work or so, I cam around and put a piece of tape of each group's stopwatch, wrote a time on it, and gave them a few minutes to prepare to measure that time interval using only their oscillators.
This is great for science: jumping in, experimenting, finding out new things (not confirming things that I told them), and, ultimately, owning a task that they didn't necessarily know how to complete less than an hour ago. I mean owning, by the way: my Physics Hall of Fame records a group in 2009 with .06% error on this one. Most groups get under 1% error here.
Many things were discovered, including some of their own misconceptions: didn't that spring "slow down" when the motion got smaller? Why doesn't the time for 10 cycles change when I keep changing the pendulum's mass?
Some groups try the "easy way out" - getting the period to be a second or 2 seconds (they don't really know what period is at this point, so some pick half a cycle as their unit). It's best to make the setups resistant to this. That's pretty easy with the inertial balance and mass/spring, but requires relatively short pendula.
The first period of this went about as expected; most groups come up with some sort of proportion/cross-multiplication type of method, once they figure out that the rate of oscillation doesn't really change. Usually, that conclusion is just based on a couple of measurements or eyeballing. (Frequently, the eyes lie, and I might prompt with "if it's so obvious, then I bet you can prove it easily.)
The second period of this saw an outbreak of modeling, though: graphs were drawn, relationships were proposed, variables were written with good symbols, and constants found units and values. All in all, it was awesome.
The genesis of those graphs was genuine - I didn't require the graphs. They (I witnessed the graph-decision-moment for at least two of the groups that I saw drawing the graphs) couldn't quite decide what happened to the oscillator as its motion got smaller, and decided that graphing the time required for different numbers of cycles would tell them what was going on!
Here's one of these graphs:
They're not just making models here - they're revising them and making them ever more communicative! You just can ask for anything more on a Wednesday than that.
AP Physics brings you some investigations of Arbor Scientific's pull-back cars! These cars are pretty neat; Arbor advertises that they provide constant acceleration that's dependent upon the distance that you pull them back.
A few general observations:
They don't travel in super-straight lines. It can be difficult to get more than 2.5 meters or so of reliably straight travel
The little chrome-esque accessories tend to snap off upon collisions; that is, within 10 seconds of student use
They go a billion times (approximate measure - your mileage may vary) faster with the tops taken off, and (according to the students) also look cooler
Some insights about the cars:
Does the floor surface affect the acceleration? - Kati and Kawala
The Goal:
Our original goal was to find velocity v. time on different surfaces. Since the cars didn’t stay straight, we had to change our goal to distance v. time on different surfaces. The cars deviated too far in the end to allow the cars to go until they stopped. Our final goal was to determine the time that the car takes to travel 3m on different surfaces. We choose 3m because it was the longest part where the car stayed straight.
How'd You Do It?
We tested the car on three different surfaces: ground, table and carpet. In order to keep the car going straight, we set three meter track by using rulers as an acceptable travelling distance. The gap between the tracks was 12cm and the distance between the “pulling back” place and the start point was 36cm so that the car’s engine was well-prepared to go forward. We set up these steps as precise as we could every time we changed surfaces. We first pulled back the car to let it start on the ground and measured how long the car took to pass through the three meter track. We then did the same thing on table and carpet and recorded 8 data sets for each surface. Time was the only variable here. We figured out the relationship between the time and different surfaces from the car’s motion, and the following data and graph showed the results.
What Happened?
From our data the ground and the table’s time for the
car to go 3m is close enough for it not to be negligible. The carpet was
significantly affected the motion of the car. The data table shows the ground
was more consistent than the other materials.
How does the acceleration vary with pull-back distance? - Alex, Alex, and Brandon
The Goal:
We wanted to determine
whether two times the pull on the car will give us two times the acceleration
of the car.
How'd You Do It?
In order to figure
whether two times the pull would equal two times the acceleration, we did a
simple set-up using a motion detector at a 1 meter distance from the starting
line of the car to determine what the acceleration would be. We began with pulling the car back to a
distance of 10 cm and recording the acceleration. Next, we pulled the car back to 20 cm doubling the initial
distance, and record data from that test.
We also recorded the acceleration of the car with multiple other pull
back distances. 15cm, 30cm, 35cm,
40cm, 50cm, 60cm, 70cm, 80cm, and 100cm.
What Happened?
On the right is a picture of the 60 cm pullback test. Though the graph does not obviously
represent any function, we decided to approximate the slope of the line to find
the average acceleration.
Once the pull back distance
reached 70cm the car started to click.
From the data and the graph below we see that this is the maximum amount
of pullback distance and anything above has no effect.
This is the graph of the data we collected. We wanted to find a function to predict
how fast a car would accelerate for how far we pulled it back so we found this
function.
/ .275 (x) ^ .503 for
0 <
x < 70 cm
Acceleration (x drawback) =|
\ 2.15
for 70
< x
*Where x is the draw back in centimeters*
We
discovered that two times the pull back does not give two times the
acceleration. Up to 35cm pull back
twice the pull back actually gives approximately root(2) times more acceleration. We would are able to see that this is
true because of the x^.503 in our best-fit equation for the graph. We discovered another cool thing in
that once we reached 70 centimeters the car would not accelerate any faster.
John Burk had the awesome idea for the Physics Teacher Camp this summer - a professional development opportunity without any lame PowerPoints or a thousand talks that you passively listen to. He brought a dozen or so physics teachers together at St. Andrews in DE for four days of... whatever we wanted to do.
The balance between freedom to pursue what we wanted and the availability of collaborators made it the most productive and fun thing you've ever called professional development!
I made a few WCYDWT? videos, but I haven't had the time to polish and post them yet (soon, I promise!). Apart from that, I got a bunch of great ideas for lab practica and general pedagogy. The amount that you can grow by being in prolonged contact with other excited physics teachers with no other tasks to perform can't be overestimated!
John and I did, however, write an article about it in for the October edition of The Physics Teacher. The permalink is here, though you'll need a subscription to TPT to view it.
I've been starting the honors course with the traditional kinematics -> forces sequence for a few years now, after a few variations. I do displace 2D kinematics until after we've done all of the force stuff, but before we do energy. That gives them a great chance to really own (I guess that they say 'pwn' these days) this stuff; it's a great moment about "wow, remember when we thought this was so hard?!". The physics curriculum, on the other hand, is dramatically different.
Anyway, I try to get the honors kids into a few great habits very quickly, like using units (not just writing them down, but using them! More on that here) and doing algebra with no numbers until the last line. That last bit takes some doing, because they certainly have some ingrained tendencies and, while it's not actually any different, it takes a couple of weeks for them to realize that it's just the same old algebra.
It can be a bit tricky to do that well in the first part of motion (I bit the bullet and starting framing it as CVPM this year :) ), because it really just isn't that complex. A lot of symbolic abstraction here is, for the most part, a little forced. There are ways to build precursors to it, but full 'no numbers until the end' (NNTE) application isn't super-well applied here, at least as a beginning.
But... I don't really have to do that yet. Sure, some kids will get a nice 'Not Proficient' on the first assessment involving the algebra standard, because they'll be missing that core skill:
Algebra (A)
Core Skills
Apply percentages appropriately and accurately
Use no numbers in algebraic solutions
Proficiency Indicators
Recognize the need for and properly apply the quadratic formula
Be fluent in algebraic operations
Use ratios accurately in problems requiring comparison of the same expression
Advanced Indicators
Use the calculator’s solver to solve an intractable equation
That's where the SBG magic comes in: it doesn't really matter! I'm most concerned with them nailing what we've done so far, so the feedback that comes along with that grade (lots of check marks, with that solitary little x by the NNTE skill) will speak to the idea that they're doing well, but just need to work on that one skill. They'll have plenty of time to work on that, for sure. The grade is much less important than the feedback, because the grade changes.
Some kids will be fine with it at this point - they'll certainly have been exposed to it, and could do it, in principle. More power to 'em.
For many, it's less an issue of ability to do it than disbelief that I'm serious about them doing it. In that case, this'll serve as a much more specific and tangible reminder that I am serious about it.
At the end of all of this, I can assess a bit earlier and more often than with traditional grading, because I don't need to worry about the ethical implications of giving an assessment that I'm not sure that everyone can nail (well, that's not exactly what I mean, but the "average knowledge bar" for the assessment can be lowered). It's just that I wouldn't want to give a big one (lots of standards, lots of time, some standards not assessed again for a long time, etc.) without lots of time, but that's why the big summative tests are at the ends of the model units. The smaller ones leading up to it are just for the benefit of the students; I wouldn't even record a grade, if I didn't want to have as much data as I could. All I need to do is educate the kids about why we're having the assessment and what they should do after it's over (hint: read the feedback, apply the feedback, maybe look at the grade next week); after all, taking a test is better test prep than cramming! (Can anyone find the reference to that paper? I couldn't, after a cursory Google)
Assessments aren't for me, they're for you.
PS: Why am I so super-sold on NNTE? Other than the fact that you need to be able to do it to work at the next level, it means that you don't have to re-derive the whole thing when you change a given quantity, allows you to actually solve any problem that involves a system of equations (two unknowns, one of which isn't the desired quantity, is enormously difficult for 99% of kids that haven't mastered NNTE), gives you a chance to check your work by checking the units of the answer (without having to carry around units in your algebra), lets you learn about function dependencies (look: the coefficient of friction's in the denominator, so increasing it will decrease the stopping distance) and lets you use those dependencies to check your work a different way (should the stopping distance decrease with increased coefficient of friction? Yes, it should - check.), and lets you evaluate (and again use to check your answer) special cases of the relationship (what happens to the acceleration of the cart if the angle of the ramp becomes 90 degrees? Well, sin(90) = 1, so the expression reduces to g - that's exactly what it should do!). Also, that was the longest sentence e v e r.
There are lots of things to try to get students to do with units. High on the list tend to be:
Write them down when recording data. This one can be an epic fail: ever take a bunch of data in the lab, go home to try to analyze it, and have no idea if you measured A or mA?
Write them down on answers. Certainly, you won't go on a run with me if I tell you that we're going "30." I could be a crazy ultramarathoner; who knows, if I don't give you a unit there?
Write them down in the work leading up to that answer.
Now this last one is where the rubber meets the road. The previous reasons are pretty straightforward, but this is a reason both subtle and deep. There are a variety of advantages that keeping track of these units during the process give you:
You don't *really* know what the units on your answer are if you haven't checked the cancellation of the units in the work leading up to it - you're just assuming that you won't end up with ftlb/weeksecond or some such hogwash
Checking those units is just another chance for you to find your mistakes. If your units aren't just silly (ftlb/week, for a problem about power), but instead wrong (ftlb/weeksecond for the same problem), then it's not worth your time getting out your calculator - it's wrong. The converse isn't true (units working does not imply a correct answer), but this can catch quite a few mistakes.
Paying attention to the units, and the answer's size in general, gives you a real leg up in terms of catching other kinds of mistakes; if you get 700 million meters per second for the speed of the car (or anything!), that's a pretty good hint that you did something wrong.
A lot of these issues about carrying units around everywhere are geared towards mistakes. If you're in a point-crazy lust for a correct answer, then there's real psychological incentive not to check your units! Any sort of check that you do on your answer could reveal that it's wrong (*gasp*)! This isn't a terribly rational motivator, because it gets in the way of you getting more correct answers, but for a kid afraid of math/science, it can be real.
All of this is predicated on the idea that finding mistakes is good, because it shows you where the work is needed. That doesn't come automatically; it requires a lot of re-education of your kids. We want to find their mistakes, because it shows us where there's a chance for improvement. That's why the homework is so hard! :)
If we have the proper incentives in place (no immutable points for initial quizzes, labs, HW, etc.) and a focus on a better goal (real understanding - knowing that it won't be complete right away, but that finding the issues will result in a better knowledge down the road), then maybe kids can buy into really using units, and not just write them because I told them to.
The first model that we make sued to center around the dots on the floor of my lab. With the renovation, I got a new floor, complete with much less 'spotty' tiles, so that kind of went out the window. Instead, I went to filling my board with dots, but that wasn't great either.
This year, I took advantage of our great art studio and superstar Mrs. Silverman, and made myself an art.
"One art, please!"
I took one of those large rolls of white paper, some brushes, paint, and water, and made a drip painting (think Pollock, but dots without many streaks, and not as good).
This was the subject of our first model. It's great because:
The number of dots isn't necessarily knowable (because of the super tiny ones), so we must come up with some sort of predictive and approximate description
The number of dots isn't necessarily a single number, because there are some streaks there - what counts as a dot?
The most common approach involves sampling a portion of the area and using proportional reasoning; this is right in their wheelhouse, so we can concentrate on the other demands of creating their first algebraic model
The uncertainty comes in two distinct varieties: random (from the selection of the sample - can be reduced by sampling more areas or bigger areas) and systematic (I left few dots near the edges of the paper, and the left side has a higher density than the right side)
Here are a few of the first whiteboards of the term!
We'll need to work on good symbol choice, but fun's a good place to start :)
Today the physics classes played the mistake game for the first time. We've been talking about making predictive models, and they've read (refreshed, hopefully) about several different algebraic models that will come up frequently: proportional, linear, quadratic, power (>1), power (<1), log, exponential growth, and exponential decay.
This morning, I prepared each group's whiteboard with 6 graphs, and gave them the task of identifying the algebraic model or models that would correspond with the graphical models given. They also had the task of making a (hopefully, just the one!) mistake, and of justifying that mistake in the same way that they justify the others to the class. Per Kelly O'Shea's awesome suggestion, I framed the discussion by having them ask questions of each other, rather than "hey, that one's wrong!" They got in the hang of that pretty well, and I tried to get in the hang of adding not much to the discussion.
It's easy to let presentations be valuable only for the presenters, because we tend to let the kids off of the hook as far as the burden of real probing questions goes. This did a great job of keeping everyone on their toes, and will hopefully set the tone for the oh so many productive whiteboard discussions that we'll have this year!
Here's a good representative. Can you find the mistake?
I hadn't originally intended for this to be a during-the-year-only blog, but it turns out that that's what we've had. Things are just different when the kids are here (in a good way!).
We ran the much-vaunted Marshmallow Challenge for the first time. Actually, most of the kids did it last year in class, but without some of the unpacking of ideas that we did this year. Many folks have blogged about this, but here are my impressions and my spin on it:
The TED talk is pretty essential. It raises some great questions in the kids, and really helps us get a discussion started about grading, trying, and the value of failure
Much of the talk is about teamwork, which is a good lesson, but which isn't really my goal for the exercise. Hopefully, a little of that will splash on them, too
The idea about failure leading to success is huge. This is a great springboard to talking about why older students are more averse to failure than little kids, which leads smack dab into SBG
The coupling of high stakes (ahem: tests) and novice skills (shouldn't they be novices at everything they study, since it's the first time?) really motivates SBG as a better representative of what they know than traditional grading
I kept track of the number of collapses that each team had and the time at which they first engaged the marshmallow. The correlation's a bit spotty, but I might be able to add some graphs later. It certainly plays out as advertised (most of the time).
It's super-important to add a heaping helping of "it's the feedback that matters!" in here, lest kids just here "infinite chances!". This hopefully spills over into their traditionally graded classes, too; even though the feedback won't help them change the past grade, it'll certainly help their future ones, if they let it. Failure is very helpful, but only if students learn from it - reassessment by itself is not a silver bullet.
Don't get to specific with how your flavor of SBG is implemented yet - maybe wait until the first assessment, but certainly roll it out over a period of days. Get the big ideas in first.
My attempt at a bit of a Prezi to introduce those big ideas (sometimes the views at each point in the path aren't optimal, but it's heavily dependent on the settings of the machine you're viewing it on, so it's not really fixable on my end) is here. It certainly needs to be narrated (like any good presentation, it's not all on the slides. That's not to say that I claim it to be a good presentation. :) )
Some pictures of structures (I didn't get all of the pics that I'd like, because I didn't check the camera battery - boo!):
