When we have a Face-Off, every group is whiteboarding the same problem. No one presents. Instead, we sit on the tables (bringing in our circle and keeping people from just doing more work in the packet and skipping the discussion) and all share our boards at the same time, then the groups talk to each other, ask questions, and try to come to a consensus (fixing their group’s board along the way as they change their minds about parts of their work).

What is the same about all of the boards? What’s different? Ask questions if you see something you don’t understand or don’t agree with. At the start of the year, they need some clear guidance about what they should do during a board meeting. After a while, they need nothing (or almost nothing) to get them going and discussing.

Keep them from presenting their work. (a) It is really, really, incredibly boring to have each group present their work for the same problem. (b) If they fix the first board, then it makes the rest seem pretty redundant (and they miss all of the cool unique things happening on any subsequent boards). (c) Everyone just whiteboarded the same problem, so they should be really familiar with it and shouldn’t need any intro to the problem, context, etc in order to be ready to think about the other groups’ work. (d) All of the above.

So instead, get them talking about differences they see among the boards. Get them to inspect the differences, decide whether both ideas can work or whether they are in conflict, and resolve the conflicts.

A really important detail here that could (but shouldn’t) be overlooked is what happens before the meeting (when they are drawing on the boards or even earlier, when they are working the problems on paper). If every board going into the meeting has a different wrong answer, most times (in my experience), the class will still end up with the correct answer by the end of the discussion. However, if every board has the same wrong answer, they’re obviously not going to get anywhere. I talk about that with them early in the year. They need to keep themselves from hearing or seeing what the other groups are doing before the meeting. You can’t really police that (nor would you really want to spend your time policing that), but it is usually pretty easy to get them to agree with the idea about how everyone going in with the wrong answer will be bad news.

Best Uses

We finish every experiment with a board meeting, but that’s not really the same as a Whiteboard Face-Off. It follows a similar format, but we’re looking for a relationship from our graphs rather than trying to agree on a solution to a problem.

I tend to use the Face-Off mode when I know we need to slow down on a set of questions because of how much depth there should be to the discussion about them. For example, I always plan to do the first set of Balanced Forces problems in my packet as Face-Offs because I expect every (or almost every) group to have a wrong idea about some part of the velocity graph that goes with each problem. When we face-off one problem at a time, they really argue out their thinking and have to immediately deal with the ideas in Newton’s First Law. The first problem takes a while, but the subsequent ones generally move more quickly.

This mode can also work well in partnership with the Mistake Game, though since I generally reserve it for problems where I expect a plethora of unintentional mistakes, students usually aren’t ready to think of a good mistake for that type of work yet. If they ask if they are supposed to be making mistakes, I say something noncommittal (“Sure”).

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*Haha, yes. So funny. Actually, one of my students this year insists on calling every meeting to order and saying, “Meeting Adjourned” when we finish.

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Other posts in the Whiteboarding series: Whiteboarding Mistake Game: A Guide | Monk Whiteboarding | Whiteboard Speed Dating

Beware The Right Thing, Said Once.

This idea is one that I’ve found I need to cycle back to again and again as I work on my teaching. It can sneak into a class via several conduits.

1. When the teacher is giving information to the class. Enough has been written on how and why clear explanations do not result in changed understandings (though they feel really good and can be really fun at the time). Here’s a link to the Veritasium video on learning physics by watching videos, just in case.
2. When the teacher is working through something with the class’s help, calling on students to fill in the gaps. Before Modeling Instruction, this activity was my most common go-to way of spending class time. I knew that I didn’t want to just tell students things, but I didn’t yet have a way of coaching them to construct their own ideas, so I basically just told them things via fill-in-the-blank “lectures”, AKA “guess what the teacher’s thinking” (at least, that’s how many students saw it). Luckily, I had some pretty amazing students who were willing to tell me what was what. One of the hugest moments in that first year of physics-teaching was when one student told me, “Listen. Ms. O’Shea. Just because one of us has happened to say the right answer to a question, that doesn’t mean that we all understand it.” Whoa. And also, thank you, Donald. Seriously. I wasn’t ready to do much better, yet, but I was starting to shape my understanding of how my classes needed to look different (and what was going wrong with them so far).
3. When students are presenting information, results, or solutions to problems (as in whiteboarding). And here’s where it starts to get even trickier. In this case, I’m not lecturing. I’m trying to guide students to develop their own ideas. The class looks a lot better. Students are doing almost all of the talking. They are using whiteboards. How can it go wrong? And yet—one of the biggest mistakes I made in my first years of Modeling Instruction was making sure students were presenting correct answers. I knew the students presenting the problem usually understood what they were doing. They weren’t just getting their information from me. It took a while to realize that the presentations weren’t doing much more for the other students than my fill-in-the-blank presentations had been doing during my first year. If the whiteboards were approved by me, then it was really me transmitting information (it was just being read by someone else). You could tell that I was the authority, not the presenters, because when students asked questions, everyone turned to look at me (even the ones up at the board). Even though The Right Thing was being said by students instead of the teacher, it was still always The Right Thing, and it was still being Said Once. Oops. The big turning point and realization moment on this front came from watching a class (that had grown bored with whiteboarding) spring to life and to action when a group went up with an incorrect board. Suddenly there was work to be done. The Wrong Things were said, and so were The Right Things. They were all said multiple times, by multiple people, and in multiple ways. Eventually, Right triumphed. Hey, now.

And even when I think I’ve understood and conquered these ideas, they are always ready to sneak back into my class under the guise of picking up the pace. In the moment, it feels like I am moving faster when we hit the correct answers and move forward, but my experience has been that these practices actually tend to slow me down.

The messier process of letting mistakes and confusions surface, of entertaining those wrong paths and ideas, of then battling through them—that all certainly makes any individual problem take more time to finish in class. But it speeds up every problem that comes after, and not just the ones that look similar to the problem at hand, either. The more my students practice working through errors, the better they are at working through their own errors and uncertainty in future, unrelated problems. They look to me less frequently for check-ins and guidance. They start pushing me away, seeing the value of holding onto their confusion for longer. None of these changes are immediate, but they start to build and gather momentum as the days, weeks, and months move forward through the year.

Some Reminders to Myself

Give a student courage to speak up when she disagrees, and normalize that willingness to question and check by letting it be a necessary part of the process of the class (checking other groups’ whiteboards during presentations). Instead of being shamed by their wrong ideas, or not even knowing that their ideas are different from what is being said in class, they are constantly listening for places where they disagree with any explanation, answer, or solution. (Those moments of, “I know I am going to disagree, but I haven’t found what I disagree with yet” that pop up during Mistake Game whiteboarding sessions are actually pretty amazing—no meekness there.)

Shake the complacent, diligently note-taking student out of just listening and writing down notes. Make sure he is thinking through and checking everything before he writes it down for himself.

Disrupt the nervous, tentative student from erasing all of her answers and replacing them with what her classmates present. If there is a large chance that the answers being presented aren’t quite right, you can’t immediately assume that yours are wrong.

Don’t let answers be the goal. Don’t let there be mysterious “secret words” that suddenly cause everyone to scream and move on to the next topic (leaving everyone, maybe including the utterer, bewildered).

When you let The Right Thing be Said Once, it reinforces every student’s idea that everyone else gets it while they simply don’t.

Actions

When you know that a group has totally nailed the understanding of a problem, make sure you assign a different group to whiteboard it. That way, you know that there will be an excellent section of question-askers to chase down the errors on the board, and you also avoid short-circuiting the discussion by having the leaders of the conversation underestimate everyone else’s confidence in the answer.

When you are up front, don’t acknowledge answers immediately. Don’t acknowledge right answers differently from wrong answers. Let there be time, redirect students to talk to each other (instead of to you), and see if you can wait out a consensus.

While you do that, be sure to go meta every once in a while and explain why you’re doing that (or you’ll drown in “I said that!”s). While you’re at it, tell them about Clever Hans. Acknowledge that it has probably been a successful strategy for them in school in the past (they will agree wholeheartedly), and also tell them why you are going to try to cut that strategy off from them. They will appreciate the big picture idea, there, even though it is making things more difficult for them than they are used to in the short run.

Any truly new idea has to come from students, be discussed by students, and be agreed upon together (or they won’t ever truly believe it, even if they can repeat it). You set the stage and put them into a situation where they can start building the idea, and you coach them on the process, but you don’t verify along the way. Any “new” information that you are presenting to them can’t really be new. It has to be something they could arrive at using old ideas. You can have them walk you there by only asking questions they have definite ways of answering (and often sitting back or getting out of the way while they work together to answer those questions before you can all move on toward the next step).

Be even more patient. Let things take time. Let them know that it is okay for things to take time.

—

I am sure that my ideas here will keep developing and evolving in the coming months and years. This, of course, is just where I am now (and how I got here).

Here’s a binder. And the handshake. [As they walk in, I'm handing out their binders (with materials, labeled dividers, spare graph paper, their names on the sides) and shaking their hands as we make the transaction. I've seen some discussion in various places online about the appropriateness of shaking students' hands on the first day of school. It seems to vary widely depending on the situation and history of the school, the teacher, the students. Since I've known most of these students for at least a year (and, in fact, was likely saying "goodnight" and turning off lights for some of them once or twice a week when they were freshmen), it makes sense for me. In a way, it's a small signal that our relationship has to be a bit more formal inside the classroom than it has been (and might continue to be) outside the classroom. We're doing some business, here. You would likely know whether that would be weird or normal at your own school and for your own situation. And also, this handshake business really has nothing to do with the constant velocity particle model, so I'll stop discussing it here.]

Hey guys, I want to show you something cool. Let’s go next door.

Motorized Cart

Come around this table in the front. Okay. Are you ready? Mentally, physically? Metaphysically? It’s going to be pretty cool. (We’re ready.) Great. So in a moment, I’m going to say “start”, and I’m going to say “stop”.  Some things might happen before I say “start”, and some things might happen after I say “stop”, but we want to focus on observing just what happens in between the “start” and “stop”. Got it? (Yep.)

Cool. [I flip the switch on the cart. ¹] Start. [The cart moves pretty slowly across the table in front of me. I watch it carefully.] Stop. [I turn off the cart. I look up at them.] Whoa. You want to see that again?

Of course. Let’s watch. [I repeat the same thing.]

Now, for this next part, let’s talk about our observations. There’s not anything in particular that I’m looking for. I just want to get your brains going. Anything that observed between the “start” and “stop”. Go for it. [A lot of responses will happen. Someone will mention the cart being turned on and off even though it happened outside the boundaries. That's fine. I don't admonish them for it (though their classmates might, depending on the group). Someone will mention that it made a noise. Someone will say, "The cart is black." Usually, too, I'll get some really helpful observations. It was going in a straight line. The speed was the same the whole time. On those, there will often be a student (or several) who observed that it didn't go in a straight line or that it didn't have a constant speed. We'll turn the cart back on to watch for that specifically. I'll ask them, "How will we know whether it is [the thing we're looking for] or not? What will we look to see? What will it look like if it isn’t?”]

(It had a constant speed.) Oh, that’s interesting. How do you know it is constant? [The conversation here varies a lot with each group. Essentially, though, we're looking for a verbal definition of constant velocity. We usually come around to something like, "it went the same distance in each chunk of time". Some variation on that. I told them I wasn't looking for anything in particular, but that was sort of a lie. I don't care about what their other observations are (unless they are inaccurate, but then we'll just challenge them). But I do care that we get to that verbal model of constant velocity.]

Huh. So, we’ve gotten a pretty good description of this motion in words. I think a great next step in understanding this cart would be to get a more quantitative description. For that, we’d need to do an experiment. So.

What can we measure?

What can we measure about the motion of the cart? But also, what tool would we use to measure it? We don’t have a lot of time today, and this is our first experiment, so ideally we will use tools that are easily found in the lab and that we are mostly familiar with already.

There’s a space to write this down in your packet on the inside of the front page. Don’t worry about the sketch just yet. You’ll have time to do that later.

The Potential Measurements

(Distance, ruler.) Great. Has everyone used a ruler before? And we have a bunch of those, so we can definitely use that tool.

(Time, stopwatch.) Sure. Has everyone used a stopwatch before? And we have a drawer full of them right over here.

(Speed.) Cool. What tool would you use? (The ruler and stopwatch, then you just divi—) That sounds like it’s going to be more of a calculation than a measurement. What tool could we use to measure it directly? (Uh, a radar gun?) That would be so cool! I wish we had one of those. We don’t, so is it okay if I leave speed off the list for now? (Uh, okay.)

(Direction, compass.) Fair enough, but I actually don’t have compasses either. [I've had a student say they carry one in their backpack before...! If you do leave it on the list, it's easy to eliminate later because it doesn't seem to depend on anything else, and the other things don't seem to depend on it.]

(Weight, scale.) Okay, and I do have a scale. [This one doesn't always come up, but it is a pretty frequently raised idea.]

Culling the List

[So now we have something like: distance, time, direction, weight. To do the experiment, we need to figure out what relationship we are trying to find. Usually the relationship is between just two quantities, so we need to narrow down our ideas, here.]

Which of these things can we choose values for? (Definitely distance. And time. And direction. And weight.) Hold on. how would you choose the values for weight? (Oh. Actually, good point. So not weight.) Okay, so if the value for weight can’t change, it probably won’t be a dynamic part of our experiment, huh? So should I cross that one off the list? (Fair enough.)

Now. Which of the things on our list depend on the things we can choose values for? (Um, distance depends on time. Or time depends on distance.) How about direction? Does it depend on the others? (Well, it always goes in the same direction, so I guess not.) Okay, so if it doesn’t depend on these other things, it doesn’t seem so relevant for today’s experiment, right? (Yeah.) So should I cross it off the list? (Sure.)

Okay. We’re getting close, now! There is one thing I have a suggestion about.

So, I’m thinking about how in a few minutes, we’re going to split up and do experiments in different places. After, we’ll want to compare our data to each other. So I was thinking, maybe instead of just using distance, we should do a measurement that we could compare to each other. Like, if we all agreed on the same place that was zero, even if we didn’t start there. But it would be the same zero for everyone. That way we could compare where each cart was in a more meaningful way.

So instead of distance, it would be more like where you were on a number line. [Draw a number line on the board.] We would just need a common zero. An, uh, what do you call the zero on the number line in math? (Oh, um, the origin.) Yeah, a common origin.

Oh, hey! There’s an origin already taped across the room that we could use.

Wow, that’s convenient! So I guess, besides just an origin, we also need to say which was is positive and which way is negative, right? So let’s say toward physics is positive, toward chemistry (the chemistry lab is next door) is negative. For obvious reasons.

So is it fair for us to measure position instead of distance? (Sure.) It’s pretty similar. I think it still keeps the spirit of what you meant by distance, but just allows us to compare data more easily in the end.

So now we’re down to position (ruler) and time (stopwatch) for our measurements, right?

What relationship are we trying to determine?

What is our objective for this experiment? To find a relationship between what and what? (Distance and time. I mean, position and time.) Okay. So we’re finding a relationship between position and time for the motorized cart. You can write the objective down on the experiment page in the packet.

What should we do with the data?

So what are you going to measure, again? (Position and time.) Right. And so you’re going to take a bunch of data points, then where will you put them all? (In a data table?) Good idea. So what will go in the data table? (Uh, position… and time…) Okay. So let’s choose symbols for those things that will help us remember which is which. (x and y.) I think we can be even more specific.

Let’s take time, for example. What symbol would you use to represent time that would help you remember it was for time? (Seconds.) [I write (s) at the top of the column I've drawn on the board, leaving a space for the symbol in front of it.] I think seconds for the units makes sense. What about the symbol, though? (Oh, ok. t?) That’s a good idea. Now how about for position? (p?) That would make sense, but physicists have claimed that for something else already. [Moving back over toward that number line I sketched before.] So position is the spot on the number line. What symbol would you use to label the number line in math? (x?) Sure. If it were a vertical number line? (Oh, y.) So x makes sense for position to me. [Filling it in on the sample data table on the board.] How about the units? (Inches? No, centimeters?) Okay, fight! [I'd let them use whatever the decided on together, but usually centimeters wins "because it's science class" and for no other reason.]

Once you have a table full of data, what will you do with it? (Graph it?) Yeah, my favorite! [Drawing sample axes on the board.] So what will go on our axes? (Position and time?) Of course. So which one where? [Now the real fight begins. I try to coach them in the direction of saying "vertical" and "horizontal" instead of x and y for the axes. The big conflict is that they really want to put position on the horizontal because it is x, but they also really want to put time on the horizontal because it is "independent" (whatever they think that means from math classes of old). In the end, I might just end the debate by telling them "what physicists usually do" and having us agree to draw x-t graphs so that we can all compare our results more easily (after all, that's why we decided on position instead of distance).² I might throw out the line—So we'll all have a slight moment of panic when we put x on the vertical axis. But we'll get through it together.—which tends to calm them down and let us move forward.]

Off you go!

Okay, so now you know what you’re doing, right? Ready to go do it? [Very awkward pause ensues. This whole thing has moved very quickly, and the students aren't yet sure what to do. I mean, it's the first day of classes and everything. Am I actually expecting them to be ready to do an experiment? What is with this class, anyway? So I'll throw them a rope of comfort instead of just chucking them into the deep end.] Well, okay, should we try out one method together before we break into groups? [Relieved sigh of, "YES!"] Now, this is only one of many good solutions to taking data for this experiment. You absolutely do not have to use this same method. Since it’s the first day of school and your brains aren’t totally into physics mode yet, I just want to give you one example. If you have other ideas about how to take this data, please do that! I’ve seen several good ways, but I’m also always looking for new ones that are even better. [All that being said, a lot, even most, of the Honors Physics students will all do exactly what I am about to show them (down to the number of seconds between each breadcrumb). It's the first day of Honors Physics, and they.are.scared. The regular students will usually have at least one or two groups that will try their own way.]

I call this the “breadcrumb” method. I need someone who is a pro at using a stopwatch, and I need someone to grab one of those meter sticks. Just a one-meter stick, not a two-meter stick. Thanks! So basically, you’ll start the stopwatch and I will turn on the cart at the same time. Then, every, say, 2 seconds, you’ll say “now” and I will drop a breadcrumb at the new position of the cart. I’m going to put the breadcrumbs at the back of the cart because… [I mimic putting a breadcrumb in front of the cart and they can see the problem of my fingers being roadkill immediately.] We can measure the positions after that. Got it? Great. Let’s try it out. [There will sometimes be a quick discussion about whether to put the back or the front of the cart on the Start line at this point. If there isn't, they will figure that out in their groups easily enough.]

Okay, let’s try out the measuring technique. Important note! Everyone put your pencils down, now. Do not record this data. This is not your data. You will get to take your own data in just a couple of minutes. Don’t write this down! [I point to a breadcrumb.] What was the time when the cart was here? (Two seconds.) [Glance at the clock.] Two seconds after the start of time…? (No, from when we started the stopwatch.) Okay, so it’s more like a change in time, not an absolute time, right? We should come back and talk more about that later. Anyway, how about when the cart was here? [Point to the next breadcrumb.] (Four seconds after the stopwatch started.)

Where was the cart at 0 seconds, then? (Here. [Student(s) point to the Start line.]) That’s our first data point, right? So let’s measure that position. [Student who retrieved the meterstick], can you measure that position for us? [Usually, there's a discussion now about whether that position was 0 centimeters or not. The "nots" eventually win, since the Start line was not on our origin line, and the origin line is our 0 cm line. We get the student to measure from the origin line to the Start line in centimeters (and not in inches).] Is it on the positive side or the negative side of the origin? (Uh… the negative side.) Okay, so negative [measurement]. How about this breadcrumb? Can you measure the position here? Stop writing this down. [Now, almost without fail, the measurer will try to go from Start to the breadcrumb instead of from the origin to the breadcrumb, despite the discussion just seconds before. He will get corrected by his classmates, though, and we'll see that likely mistake so that they will recognize it in a few minutes when they make it again.]

Okay, I think you all have an idea now of one way to take this data, right? In a moment, not yet, let’s do groups of two or three. The carts, tape, and stopwatches are over here. The metersticks are over by the hood. Remember not to change the speed dial on the cart, even though it is incredibly tempting to do so. There might be a group in another class that isn’t finished taking data, and you don’t want to ruin their experiment.

Off you go!

[That all seemed to take quite a long time in writing, but it's not so exceptionally long in person. No more than 20 minutes, for sure, but more likely close to 10 or 15 minutes, depending on how picky the students are deciding to be in their answers on the first day of classes.]

Graphing

[The following bits are said to individual groups when appropriate as they are graphing. I don't make these as general announcements at all.] This graph is the only one for an experiment that I will ask you to plot by hand this year, so let’s do a really careful job with it. Make sure you are using a ruler to draw a single best fit line. Don’t connect the dots. Don’t just connect the first and last data points. Try to line up the ruler so that it gets as close to every data point as possible. There are more precise ways to do this work, but the eyeballed best fit line will be good enough, here, I think.

Once you have the line drawn, imagine that your data points have disappeared so that the only thing on your graph is the line. Find points on the line (not your data points) to find the slope. Use the variables on the graph (x and t) for your equation. Don’t default to x and y. There is no y-axis on the graph you’ve drawn.

Be sure to put units on the numbers in your equation, but not on the variables. That is, the slope and intercept should both have units. Go back and put units in your work to see what they should be.

Whiteboarding

When you are finished making your graph, go ahead and clean everything up. Then, grab a whiteboard and make a sketch of the shape of your graph. Sketch, not plot. Don’t make tick marks. No rulers or metersticks allowed. Just a really quick sketch, then write the equation of your line. When we are all finished, we’ll get together and have a board meeting (ha ha).

Okay, go ahead and sit on top of the tables in the circle. We will show everyone’s boards at once, and we’ll look to see what’s the same and what’s different. After a few experiments, you won’t really need me to lead this anymore because you will be able to have the conversation yourselves, but I’ll help a bit this time. In general, you want to talk to each other, not to me. So. What’s the same about everyone’s boards? [At this point, the responses can be all over the map. Eventually, you'll get to some more substantial answers, like, "they are all lines."] So everyone got a linear relationship? (Yeah.) Oh! That seems pretty significant. We were looking for the relationship between position and time for the cart, and everyone found that relationship to be linear, even though they were using different carts. Cool.

So what’s different about everyone’s boards? [Hopefully they will talk to each other a bit about how the slopes are different, the signs of the slopes are different, and the intercepts are different. Then we can dig into what each of those differences represents. Asking, "Who had the fastest cart? Who had the slowest cart?" can help a lot. Eventually, we get to the idea of the slope telling both the speed and the direction of the motion. When we're there, we wish for a word that meant both parts of that info, and we get velocity.] Sweet. It seems like we’ve defined this relationship pretty well at this point. Let’s try using it in new problems!

Next Steps

Next, we might have a really brief discussion that pulls together the pieces of our model (the verbal piece from our observations, the features of the evenly-spaced “breadcrumb” motion map on the table, the position-vs-time graph features). Maybe 2 or 3 minutes, there. Then on to working some problems in the packet. I let them encounter the velocity-vs-time graphs on their own. They get confused and nervous for a small time, but they can figure it out pretty easily (especially if they are using the collaborative work correctly and arguing at their tables). Within a day or two, we’re Mistake Gaming the problems with whiteboards.

And in general, we are also wrapping up the unit as quickly as possible. We’ll come back to everything in this model when we build the constant acceleration model in a couple of weeks. We want to move into forces as quickly as possible and start explaining, not just describing, with our nascent physics powers.

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1. The cart in my photo is the crazy-expensive PASCO version. There is a cheaper (but less straight-line-going, I hear) cart that has been oft-discussed on Modeling Instruction listservs.

2. I might also just let each group decide on their own. I might do a hybrid, making a suggestion of what we usually do, but giving them the freedom to make their own choices. In general, I haven’t found that giving them the choice here is worth the time unless it’s a group that I am pretty sure is going to be able to deal with the graphs being different (and therefore the slopes and intercepts meaning different things) during the discussion really efficiently and easily. (A group this year decided to go rogue on the graph, but quickly realized that you could compare it to the others if you rotated the board, then pretended to look at it through the back of the not-transparent board—not every class is going there on day 1 though!) Even though it’s the first day of class, knowing my students pretty well before they get here (small boarding school + working on freshman dorm) can help me make that call in the moment. It just varies from class to class.

Observe an interesting phenomenon

Bring your packet and pencil. Let’s go next door and look at something cool. [Okay, actually, we are already next door looking at something cool because we just finished pre-lab-ing the Fg experiment. Anyway—] Springs are awesome, right? [Now I'm stretching the spring and they are itching to pick it up and start mashing it together.]

I think you know what I’m going to ask next.

What can we measure?

So, what can we measure?

Spring force. How? What tool can we use to measure it? (The scale thing from before.) The spring scale? (Yeah, that.) Alright. [It doesn't bother them how circular this measurement idea is going to be. Oh, well.] What else?

How much the spring stretches. Okay, so the change in length of the spring. (Yeah.) What can we use to measure it? (A ruler…) Great. [At this point, there might be some confusion on their part between focusing in on what we can measure about the thing we literally just observed and asking different kinds of unguided questions (ex. "How much we can stretch the spring before we break it." — that might be interesting, but it is not the same as identifying what we can measure about the current phenomenon). In any case, they are usually pretty easily guided back to the process of defining the experiment.] Okay, so we’re deciding to measure how much the length of the spring changes, right? Not the length of the spring itself.

What’s our objective? (Find the relationship between spring force and spring stretch.) So you know what to do now? [Nodding.] After you make all the measurements (of spring force and spring stretch), what will you do with the data? (Make a graph.) Perfect!

Hey, before you go—what symbol are you going to use for spring force? (Fs) How about for spring stretch? (Uh… s?) That’s sort of a terrible variable, right? It kind of looks like a 5, or like seconds, or something. Maybe ∆x or ∆L (change in length) would work. I usually use “d” because I know that eventually, one day, you’re going to end up squaring that thing, and it gets kind of annoying to keep writing the parentheses you need because of the ∆. (What does “d” stand for?) Um—”da spring stretch”? Not perfect, I know.

[There are some quirks in taking data for this experiment. Some students will do something like taping down the spring scale so that it stays in place while stretching the spring with their hands. In that case, the "fixed" end of the spring is not actually fixed (as the force changes, the spring scale gets longer, so the "fixed" end of the spring moves, too, making the measurements that assume spring end location incorrect. The easiest way to take the data is to hold the spring in place (with a hand or something else) and to pull the other end of the spring with the scale itself. I don't tell them any of that before they start, but go around and guide them a bit as they take data. I want to let them have and create their own ideas about how to take the data. And they need to see that their idea really is a problem in order for anything I say to be meaningful for them.]

Graphing the data

[When they start taking this data, they have already used LinReg once (in the Fg experiment just a few minutes ago), so they are already pros.]

[They quickly get their data input and start examining their graph.]

What is the intercept on your graph? (Uh, 0.02.) 0.02 what? (Uh, cm. No wait, Newtons.) Is it 0 N? (No, it says 0.02 N.) Oh. I think it says 0 N, though. (???) Look at the range for the intercept. It includes 0 N, right? (Ok, yes.) So you can’t say it isn’t 0 N. (Well, ok.) Let’s change tactics—should it be 0 N? (Um.) What would it mean to have an intercept of 0 N on this graph? (Well, if the spring isn’t stretched, then there isn’t any force. Okay, yes. It should go through (0,0).) And your graph says it does, or at least says that you can’t say it doesn’t. So I think you don’t need to write down an intercept for your equation.

Now that you’ve got the graph all set, just make a quick sketch of it in your notes and write the equation of the line using meaningful symbols and units. When you’re ready, go ahead and make a quick whiteboard of your graphs and equations for both experiments.

Board Meeting

When you’re finished with the experiment, go ahead and make a whiteboard of your graph and your equation. Make sure you use the symbols from your graph instead of “x” and “y”. Make sure you put units on your slope. [Note: The board below has the graphs from both experiments at once (which I alluded to in the intro to the post). We talk about them one at a time during the meeting, so I won't discuss the first graph here.]

Alright! Let’s have a board meeting! Do you remember what we do during these meetings after experiments? Remember that we’re looking to talk about what is the same about our graphs and what is different about our graphs so that we can flesh out the relationship we were hoping to find.

So. What’s the same? What’s different? (They’re all linear.) Everyone had a linear graph? Actually, wait. Did everyone have a linear graph that also went through (0,0)? (Yeah.) So in that case, it’s an even more specific thing. Everyone found the relationship to be directly proportional. What about the slopes? [Are they all the same, like in our weight and mass experiment? They quickly point out that they were using different springs, of course, and so it makes sense that we all have different slopes.]

So what does it mean to have a steeper slope on your graph, then? (Ummm… oh! It means it’s not as stretchy of a spring.) Cool. So that slope tells you something about how stretchy the spring is.

Post-Mortem

You were looking for a relationship between spring force and spring stretch. Was there a relationship? (Yes.) What relationship did you find? (It was directly proportional.)

Cool. So—

And so—

What did that constant tell you? (That’s the slope. It’s telling you about the stretchiness.) Nice. So let’s call this constant the “spring constant”. How about just putting a quick subscript on that constant? So we can write the whole relationship as—

Remember that spot on the front of the packet where we made a table of the forces? There’s a column that we ignored before for equations, and you could store that one there.

—

Other posts about the Balanced Force Particle Model:
Building the Balanced Force Particle Model
Common Types of Forces
Force Vector Addition Diagrams
Balanced Forces before Constant Acceleration
Empirical Force Laws: Gravitational Force Experiment
Building Newton’s 3rd Law (Next in the series—coming relatively soon!)

Observe an interesting phenomenon

Bring your packet and pencil. Let’s go next door and look at something cool.

Are you ready? Here we go. [I dramatically put the mass hanger onto the spring scale. Then a mass onto the hanger. Then another. Even though it's only the third week of school, the students are ready to ham it up and react as though this phenomenon is the most utterly cool thing they've seen in at least several weeks.] I know, right? So—

What can we measure?

We’re going to do this just like the first experiment. We’ll follow that same sort of procedure of setting things up. Actually, we’ll always use that same sort of procedure, and soon enough you won’t even need me to walk you through it.

Anyway, what could we measure about this really cool thing? [The first suggestion is pretty much always to measure the weight.] What do you mean by weight? (Gravity.) Gravitational force? (Yeah.) Okay, great. What tool could we use to measure it? (The scale thing.) Oh. Huh.

What does this scale measure? [A lot of answers, but they usually settle in on it measuring the spring force. They want to say right away, too, that it's the same as the weight.] Okay, could everyone sketch a quick FBD for the hanger, then? [They do.]

Is it balanced or unbalanced? (It’s balanced.) How do you know? [A lot of answers. "It's not moving." But I'm holding out, if possible, for "it's at a constant velocity."] Great, so how does the spring force compare to the gravitational force? (They’re the same.) So we can use the spring scale to measure the weight (Fg) as long as we keep the hanger at a constant velocity.

What else can we measure? [Sometimes mass is suggested quickly, sometimes not. Sometimes the students haven't caught onto the whole scheme here, and they start saying things like, "how far you can stretch it until..." which is asking a different question of the situation, not saying what you could measure about what you've already seen. They can be redirected pretty easily. If they don't get to mass quickly, sort of idly looking at the masses and putting them on the table in front of me gets them to see the writing on the masses and be prompted to it, usually.] What’s the difference between mass and weight? [Interesting ideas from various places, here.] Which changes when you go to the moon? Your mass or your weight? (Weight.) Okay, so they’re definitely different. How can we measure the mass? (Just use the numbers on the things?) That works! [Using a balance to get the values also works. I just go with the path they elect each time.]

Okay, so what relationship can we find? The relationship between….? (Weight and mass.) Great. You can write that down as the objective.

So you know what to do now? [Nodding.] After you make all the measurements (of weight and mass), what will you do with the data? (Make a graph.) Perfect!

Graphing—now with more computers!

Instead of graphing this by hand, we’re going to start using a computer program called LinReg to do that work for us. Using LinReg makes graphing easier (no more hand-drawn graphs), and it also forces us to think about the range of every measurement. For example, when you’re reading this spring scale and want to say what the force is, you can say it’s definitely between what and what? [The student nearest me will look at the spring scale and say something like, "Uh, 0.5 N and 0.7 N."] Great. So when you’re entering data into LinReg, it will want you to write that as 0.6 N for the measurement and 0.1 N for the uncertainty (half the range). Then when you graph it, it will put range bars on every data point to help you see how big the data point is. (Better (more precise) equipment would result in smaller data points). It’s also going to make up 99 extra experiments by using random points within the range of each of your measurements, and it will use that to figure out a range for the slope and intercept of your graph. Pretty cool, right?

You should be able to find it on all of the computers, and it’s pretty self-explanatory, but I will also walk around and make sure you’re figuring out how to use it. You can open it right away and take data right into LinReg. Just make sure you save your file every once in a while.

So everyone knows what they’re going to do now? (Yep.) Right. Ready, set, go!

[Eventually the groups start telling me they are finished, and I walk around to each one and check in on their graphs. I ask them to change the window so we can see (0 g, 0 N). I ask whether the data looks linear. I ask them to do the linear fit, then drag through to see the fictitious teams.]

What is the intercept? Does the graph go through (0 g, 0 N)? [A variety of answers. Many say, "no" because the intercept value is not given as 0 N.] What about the range? See the range it gives for the intercept? Is 0 N inside that range? [Oh! Yeah, it is.] So you can’t say that the intercept is not 0 N. But more importantly, should the intercept be 0 N? [No. Or, wait, yes.] Why? [Because if you don't put any masses on the scale, it shouldn't have any weight.] Okay, so we feel pretty good about that intercept, then. So you don’t need an intercept for your equation.

Now that you’ve got the graph all set, just make a quick sketch of it in your notes and write the equation of the line using meaningful symbols and units.

Board Meeting

When you’re finished with the experiment, go ahead and make a whiteboard of your graph and your equation. Make sure you use the symbols from your graph instead of “x” and “y”. Make sure you put units on your slope. [Note: The board below has the graphs from both experiments at once (which I alluded to in the intro to the post). We talk about them one at a time during the meeting, so I won't discuss the second graph here.]

Alright! Let’s have a board meeting!

Do you remember what we do during these meetings after experiments? Remember that we’re looking to talk about what is the same about our graphs and what is different about our graphs so that we can flesh out the relationship we were hoping to find.

So. What’s the same? What’s different? (They’re all linear.) Everyone had a linear graph? Actually, wait. Did everyone have a linear graph that also went through (0,0)? (Yeah.) So in that case, it’s an even more specific thing. Everyone found the relationship to be directly proportional.

What about the slopes? (They’re all 0.01 N/g.) That’s so weird! You all had about the same slope? (Yep.) Huh. But you all used different masses and different spring scales, right? (Yeah.) There must have been something in common, still, about what you used. [At this point, I usually shift over to the spring force discussion so they can see that they all had different slopes for their graphs in that experiment. They quickly point out that they were using different springs, of course, and we shift back into thinking about what must have been the same about our materials in the first experiment. Eventually, they start talking about how gravity was the same and we get to the idea that we all used the same planet for the interaction.]

That’s so neat. So, hang on. What if we all went to the moon and did the experiment again? Would you all get the same slopes as each other again? (Yes!) And would it be the same slope as you got this time? (No!) Would it be steeper or less steep on the moon? [Some thinking takes place. Then they come to the right conclusion pretty quickly, having talked their way through what the slope must mean.] Alright. Can I try to sum up what you’ve all just said? (Uh-huh.) Great; you can put the boards back, then.

Post-Mortem

You were looking for a relationship between weight and mass. What relationship did you find? (It was directly proportional.) Do you know that symbol, for directly proportional, from math? (No.) It’s like this—

No, it’s not a fish. It’s the directly proportional symbol. Anyway, I think you do have this next idea from math, though. We can replace that sign with “equals, times a constant”. I mean, the directly proportional things are basically equal to each other, but with some scaling factor. (Okay, right.)

And since we all found the same slope, it seems like this proportionality constant is some particular number (at least while we’re on Earth). Are you ready to have a name for it? (Sure.) So we give it a special symbol (g) and we call it gravitational field. We can’t call it gravity, because Fg is gravity, so we have to be careful about how we refer to it. We could call it “g” or “little g” or “gravitational constant” or “gravitational field”—just not “gravity”. (Okay.) If we had more precise equipment, we could have gotten smaller data points (that is, the boxes that represent our points would have been smaller because the ranges would have been smaller), so we would have had a smaller range for that slope and a more precise value for it. Do you want the accepted value for g near the surface of the Earth? (Yes.)

I think that’s in the range for our slopes, right? (Yes!) Nice. So using our new symbol, here’s the way that people usually write that weight and mass relationship—

Remember that spot on the front of the packet where we made a table of the forces? There’s a column that we ignored before for equations, and you could store that one there. You can write down the value for g somewhere around there, too. You’ll probably need to refer to it for the next week or so, but you’ll just know those facts pretty soon because you’ll use them a lot.

—

Other posts about the Balanced Force Particle Model:

Building the Balanced Force Particle Model

Common Types of Forces

Force Vector Addition Diagrams

Balanced Forces before Constant Acceleration

Empirical Force Laws: Spring Force Experiment (Part 2 of this post. I will update this one with the link when it’s ready!)

Diagram Attributes

I will run through the typical way I introduce the diagram in class in just a minute. First, here’s the teacher preview of some key energy pie chart features.

Before doing anything else, list the objects in your system. The diagram will only make sense in the context of your chosen system, so you need to be explicit about that. Teacher note: if you want students to pick up the habit of defining a system first, you need to be very careful to always do it that way yourself.

The size of the pie matters. The size of the pie determines the total amount of energy stored in the system. So, if the pie becomes larger, there is now more energy stored in the system. If the pie becomes smaller, there is now less energy stored in the system.

The size of a slice doesn’t matter on its own. Whether the slice has grown, shrunk, or stayed the same between snapshots is what matters. Students don’t have energy equations yet. They know what each flavor of energy depends on (per the flavors of energy discussion immediately before this activity), so they should have ideas like, for example, kinetic energy will be bigger if the object is moving faster. They don’t have a way of reasoning, yet, about whether the kinetic energy should take up half of the pie or a quarter of the pie at any particular snapshot. We’re looking to show how the energy storage changes, not what it necessarily is. Energy is boring. Change in energy is interesting.

For the situations we’re going to encounter, represent, and analyze in this class, we would be surprised to see energy being stored as Etherm in one snapshot, then stored as Kinetic or Gravitational energy in a subsequent snapshot. For our purposes, when there is a ∆Etherm, the energy is essentially “stuck” that way. We’ll never get to use that slice of the pie again. That slice of the pie is dead to us. It might get larger, but it will never get smaller. Since it’s a dead part of the pie, instead of writing ∆Etherm over and over again, we can just shade it in. We’ll all know what that means.

Introduction of diagram

Right. Ready for a new representation to add to our arsenal? When’s the last time you drew a pie chart? Third grade? (Students: Yep, that sounds about right. Back with the bar charts.) Okay, but these pie charts are going to be a little more sophisticated than what you were drawing back in the day.

Just inside the packet there’s a page with some energy pie chart practice. We’re going to do the first one together. Actually, we’re going to do the first one together twice. After that, I think you’ll get the idea, and I’ll let you argue the rest of them out without me.

So the problem is a ball being dropped to the ground, right? I think it looks like this? [I'm drawing the diagram on the board, but having done this problem at least 15 or 20 times by now, I don't have the packet in my hands.] And it has a sort of motion map next to it, right? The arrows are supposed to be showing how fast the ball is moving at that snapshot. The dot means it isn’t moving.

The first thing we need to do is define what we’re saying is in our system. Let’s put all the objects that store energy into our system.  We could put the entire universe in our system, but that seems a little overkill. Can we narrow it down to just the relevant objects for this problem? (Likely answers: The ball. The ground. The air.) I think you’re missing something. Something really big. (Oh, the Earth.) Okay. Does the ground store energy? (Yes. Wait, it doesn’t hit the ground. So no?) So we probably don’t need the ground. It won’t change anything to include it, if it doesn’t store energy, but we don’t really need it. Why does the air store energy? (Air resistance.) Okay, so question—from how high above the ground is this ball being dropped? (It doesn’t say.) Let’s do it twice, then. Let’s do it the first time with it being dropped in this room. Then we can come back to it a second time and consider it being dropped from the Sears Tower. (I don’t think it’s called that anymore.) Okay, the second time we’ll do it being dropped from the Sears Tower in 1992, when I know for sure it was still called the Sears Tower. When we are dropping it in this room, what do you think about the air resistance? Will it be significant? (Probably not.) So we don’t need the air in our system right now, but we will need it on the second time through.

Okay. We’ve got our system now! Ball and Earth. There are three snapshots shown in the picture, so we’ll draw three pies. We put all of the objects that would store energy into our system, so the total energy should stay the same. In the future, we could have the total energy changing, but while we are practicing this diagram for the first time, let’s try to keep it so the total energy stays the same. So we don’t have to keep track of too many new things all at once. So if the total energy is the same, then the size of the pie should always be the same.

Everyone’s favorite—drawing circles! We need to draw three circles that are all about the same size. [I start drawing my totally inadequate circles on the board. The flavors of energy chart is still up on the board next to the circles, for easy reference for the next part.]

Oh! Big warning! Do NOT write this next part down yet! Trust me, you’ll just get really angry with me if you write it down now. You’ll be able to write it down at the end.

Alright! Finally! Now, it’s time to play the Energy Pie Chart Game! Who wants to go first? [Choose an arbitrary kid who is excited to play the "game" even though they don't know what it is yet. Let's call the first kid Brandon.]

Great! Brandon, what’s your favorite flavor of energy, besides thermal? [You want to do thermal energy last. Trust me. You'll see in a minute.] (Uh, gravitational interaction energy, I guess?) Super! In the first snapshot, is there any energy being stored as Ug? (Yes.) [I section off a corner of the pie and label it Ug on the board.]

Okay, second snapshot. Does the amount of energy stored as Ug get bigger, smaller, or stay the same? (It’s smaller.) [I section off a smaller chunk of the pie for Ug in the second pie.]

Final snapshot. Does the amount of energy stored as Ug get bigger, smaller, or stay the same? (Smaller.) [I start sectioning off a still smaller slice for the third pie. If the class (or Brandon himself) hasn't steered us off toward needing a y = 0 m line yet, it usually happens at this point as some think there should be no Ug in the third pie, but others think there should. At whatever point seems reasonable, there's a discussion of needing to agree where Ug will be zero and the drawing of a y = 0 m line on the picture. Most classes decide that the ball is above the ground, that they want the y = 0 m line on the ground, and that there is therefore still some Ug left at the end. It really doesn't matter what they decide as long as it is consistent and makes sense to them. It's probably not a good idea to put the y = 0 m line above the object because, while a negative amount of Ug is fine, it's really tough to draw a negative slice of a pie (and hurts your brain a bit to even think about).]

Thank you for playing! who wants to play next? Zoe? Great! What’s your favorite flavor of energy (not thermal)? (Okay, spring energy.) Fantastic! In the first pie, is there any energy stored as spring interaction energy? (No.) Second pie: does the amount of energy stored as Us get bigger, smaller, or stay the same? (There’s still none.) Third pie: does the amount of energy stored as Us get bigger, smaller, or stay the same? (None again.) Excellent, thank you for playing!

Next! Kirstin. What’s your favorite flavor of energy (not thermal)? (I guess it must be kinetic energy, then.) A fine choice. Okay, first pie. Is there any energy stored as kinetic energy? (Well, it’s not moving, so no.)

Okay, second pie. Does the amount of energy stored as kinetic energy increase, decrease, or stay the same? (It increases.) [I add a section of Kinetic energy to the second pie. It doesn't matter how much. Students might get concerned about how big the kinetic slice should be compared to the Ug slice, but they have no way of making that comparison yet, so it really doesn't matter. It just matters that the K slice has gotten larger and the Ug slice has gotten smaller since the first pie.]

Third pie. Does the amount of energy stored as kinetic increase, decrease, or stay the same? (Increase.) [I draw a larger slice of K in the final pie.]

Thank you so much for playing! Next! Victor. What’s your favorite flavor of energy? (That would be change in thermal energy.) Aces.

Is there any interaction that would cause energy to be stored as thermal energy between the first pie and the first pie? (No. Wait, what?) Remember that we’re always talking about ∆Etherm, not Etherm itself, so we will always have to keep comparing the current snapshot back to the first snapshot. (Aha.) So, could there ever be ∆Etherm in the first pie? (No.) Sweet.

Okay, second pie. Has there been any interaction that would cause energy to be stored as thermal energy between the first pie and the second pie? (Well, no.) How about between the second pie and the third pie? (Does it hit the ground? Oh, it doesn’t. So, no.) Alright! Thanks for playing!

So that’s it, right? We’re finished? [Look at the board and see the weird-looking pies with blank slices.] Okay, there’s clearly something weird going on. When we started, we didn’t know how much of the pie each slice would take, so we didn’t end up using the whole pie. Is there anything outside the system that would take energy out or put energy in? (No, we put everything that mattered in the system.) Okay, so the total energy must stay the same. So we just need to redistribute the slices so that we take up the whole pie each time.

And now you see why I told you that you probably didn’t want to write this down while we did it, right? [At which point, the kid who didn't listen to that is grumbling while reaching for an eraser, but also can only really blame himself.]

Okay, we just need to make sure that we keep the idea of whether the slices get bigger or smaller. The first pie has only Ug, so that must be the entire pie. The second pie has Ug and K. We don’t know whether more energy is stored as Ug or as K yet, but we know that Ug is smaller than it was before. [Usually, they want to make it half and half. Even though they wouldn't agree with that in a couple of days, it really doesn't matter here, so I go with what they want.] Final pie, we know Ug has to be even smaller and the rest has to be K. [They usually want a very little slice because the ball is just barely above the ground (which is usually their y = 0 m line).]

That looks a lot better, huh?

And—we’re finished! Okay, should we try that one again?

One more time

[The second time through is much faster. They have a second space to write the new version of the problem, but I usually just modify what was on the board—that makes more sense for the class conversation, anyway. The new version of the problem is to drop the ball from much, much, much higher (the Sears Tower in 1992) so that we have to take the effects of air into account.

We quickly decide that the main thing that will be different is that we now have an interaction between snapshots that causes energy to be stored as thermal. That is, there is now some ∆Etherm. Let's join the conversation back at that point, when they've decided what is different is that there is now some ∆Etherm.]

Okay, let’s give Victor another shot at playing the pie chart game.

Victor, is there any interaction between the first pie and the first pie that would cause some energy to be stored as thermal? (Yes. No, wait, no. Not in the first pie.) Can you ever have a change in Etherm in the first pie? (No, that doesn’t make sense.)

Okay, is there any interaction between the first pie and the second pie that would cause some energy to be stored as thermal? (Yes. Air resistance.) Why does that cause a change in Etherm? (The ball is hitting the air.) [Reference back to the matter model and look at what happens to the particles when you hit the matter models together. Faster random motion. ∆Etherm.] Great, so we need to take a slice of the pie back and make it ∆Etherm. Do you think we should take it out of Ug or out of K? [Good, brief discussion usually happens here. We settle in on taking it out of K because with air resistance, it wouldn't be moving as fast as it would have without air resistance. I section off a part of the K slice, but I don't label it yet. I pause.]

Remember earlier when we were first talking about thermal energy? We said that we never see a box start suddenly sliding across a table while everything gets colder. In fact, once the energy ends up stored as thermal, it’s usually kind of stuck there. At least, in most of the simple examples we’re thinking about right now, that’s true. Right? It’s tough to get the energy that is stored as Etherm to be stored as another flavor of energy. Basically, if any slice of the pie represents ∆Etherm, that part of the pie is going to stay that way for the rest of the problem. It sort of becomes dead to us. So knowing that, it will be pretty easy to just shade in any part of the pie that represents ∆Etherm. We’ll all know that it means it’s the dead part of the pie. [So now I shade in the ∆Etherm slice instead of labeling it.]

Last one. Third pie. Is there any interaction that happens between the second and third pie that would cause even more energy to be stored as thermal energy? (Yes, it keeps hitting the air.) So is the amount of ∆Etherm bigger, smaller, or the same as the second pie? (It’s bigger.) Could it be smaller? (No, it has to stay ∆Etherm.) Right—there might be some cases where that happens, but not in the situations with simple objects that we’re looking at right now. [We decide to slice out the kinetic energy again, and I shade in the last slice. On the whiteboard, I just draw diagonal lines to "shade" the area instead of using all of that ink, but they tend to color it in when they are using pencil.]

That’s great, everyone! Do you get the general idea so far with pie charts? (Yep.) Wonderful. There are a bunch more problems on this page and the next one to get some pie chart practice and a deeper understanding of what’s going on with energy storage. My recommendation would be to work in groups and to do the problems on the whiteboards at your tables first so that you can try out your division of slices and erase to make revisions without getting frustrated about having to erase so much on the paper. You can do whatever works for you, though. Let’s work on these for a bit, then we’ll whiteboard.

Okay, you’re ready for me to stop talking now, right? [At this point, most of them probably aren't listening to me anymore anyway. They're grabbing markers or fighting out the next problem with their groups, already.]

I should note that I teach using Modeling Instruction (MI) and that some of my model names differ slightly from those in the canon materials. The Model Building page on this blog collects my posts about my paradigm experiments, the graphical representations we use, and anything else related specifically to how I implement MI. It also tries to give some disambiguation on the names of the models. I am hoping that link will be helpful in the case of confusion since I am not duplicating that information here. The comments, of course, are also available for questions/thoughts/complaints/etc.

Honors Physics

This class is about a 70/30 (%) mix of high school sophomores and juniors. The majority of the students are in Honors Algebra II/Trig, which is the same (essentially) as regular Precalculus (here). This list is pretty similar to last year’s, but a little sharper and more focused in some areas.

In the past few years, the semester exam (in January) has included everything through (including) energy transfer. The full 10 units are usually finished by around the start of spring break. The fourth quarter this year will include a culminating project/essay/exhibition/cool new thing (descriptive post coming, but probably not until the spring) and (hopefully) the Why is the Sky Blue? storyline (some Light Particle Model, Mechanical Wave Model, sound, etc—with more or less breadth depending on the time remaining) since it’s a crowd pleaser (and great way to end the year).

Vectors
V.1  A  I treat vectors and scalars differently and distinguish between the two.
V.2  A  I can graphically add and subtract vectors.
V.3  A  I can break a vector into components.
V.4  B  I can use a graphical vector construction to calculate 2-D kinematics quantities.

Constant Velocity Particle Model (CVPM)
CVPM.1  A  I can draw and interpret diagrams to represent the motion of an object moving with a constant velocity.
CVPM.2  B  I differentiate between position, distance, and displacement.
CVPM.3  B  I can solve problems using the constant velocity particle model.

Balanced Forces Particle Model (BFPM)
BFPM.1  A  I draw properly labeled free body diagrams that show all forces acting on an object.
BFPM.2  A  When given one force, I can describe its N3L force pair.
BFPM.3  A  I relate balanced/unbalanced forces to an object’s constant/changing motion.
BFPM.4  B  I can use N1L to quantitatively determine the forces acting on an object moving at a constant velocity.
BFPM.5  B  I can draw a force vector addition diagram for an object experiencing no net force.

Constant Acceleration Particle Model (CAPM)
CAPM.1  A  I can draw and interpret diagrams to represent the motion of an object moving with a changing velocity.
CAPM.2  B  I can describe the motion of an object in words using the velocity-vs-time graph.
CAPM.3  B  I can solve problems using kinematics concepts.

Unbalanced Forces Particle Model (UBFPM)
UBFPM.1  A  I use multiple diagrams and graphs to represent objects moving at a changing velocity.
UBFPM.2  A  My FBDs look qualitatively accurate (balanced or unbalanced in the correct directions, relative sizes of forces).
UBFPM.3  B  I can solve problems using Newton’s 2nd Law.
UBFPM.4  B  I can draw a force vector addition diagram for an object experiencing a net force.

Momentum Transfer Model (MTM)
MTM.1  A  I can draw and analyze momentum bar charts for 1-D interactions (IF or IFF charts).
MTM.2  A  I treat momentum as a vector quantity.
MTM.3  B  I can explain a situation in words using momentum concepts.
MTM.4  B  I can use the conservation of momentum to solve 2-D problems.
MTM.5  B  I can use the relationship between the force applied to an object (or system) and the time duration of the force to calculate the impulse delivered to that object (or system)

Projectile Motion Particle Model (PMPM)
PMPM.1  A  I accurately represent a projectile in multiple ways (graphs, diagrams, etc).
PMPM.2  B  I can solve problems involving objects experiencing projectile motion.

Energy Transfer Model (ETM)
ETM.1  A  I can use words, diagrams, pie charts, and bar graphs (LOLs) to represent the way the flavor and total amount of energy in a system changes (or doesn’t change).
ETM.2  A  I identify when the total energy of a system is changing or not changing, and I can identify the reason for the change.
ETM.3  B  I identify thermal energy as the random motion of the tiny particles of a substance.
ETM.4  B  I can use the conservation of energy to solve problems, starting from my fundamental principle.
ETM.5  B  I can use the relationship between the force applied to an object (or system) and the displacement of the object to calculate the work done on that object (or system).

Oscillating Particle Model (OPM)
OPM.1  B  I can draw/interpret motion, force, and energy graphs for an oscillating particle.
OPM.2  B  I identify simple harmonic motion and relate it to a linear restoring force.

Central Force Particle Model (CFPM)
CFPM.1  A  I can calculate the magnitude and direction of the acceleration for a particle experiencing uniform circular motion (UCM).
CFPM.2  B  I can use Newton’s 2nd Law to solve problems for a particle experiencing UCM.
CFPM.3  B  I can use the Universal Law of Gravitation to solve problems.
CFPM.4  B  I can use the conservation of energy to solve problems involving a significant change in the distance between an object and a planet.

Momentum Transfer and Energy Transfer, Part II (MTET)
MTET.1  A  I can qualitatively represent the energy stored before and after any collision.
MTET.2  B  I can determine whether or not a collision was elastic by analyzing the motion information.
MTET.3  B  I can solve a problem by employing two fundamental principles.

Physics!

This class is almost entirely juniors (this year and last year there has been only one senior in the class). Most students are also in the regular Precalculus class, though there is usually a good range (in the past it has ranged from Algebra 2 to Calculus BC in the same section).

I am trying out a new sequence this year. I’m not ready to write more details about it yet, but will certainly write more once I’ve tried it with my students. We will cycle back to energy and momentum transfer between unbalanced forces and projectile motion, and we will start testing on the italicized objectives from those units after that time.

Energy Transfer Model (ETM)
ETM.1  A  I can use words, diagrams, pie charts, and bar graphs (LOLs) to represent the way the flavor and total amount of energy in a system changes (or doesn’t change).
ETM.2  A  I identify when the total energy of a system is changing or not changing, and I can identify the reason for the change.
ETM.3  B  I identify thermal energy as the random motion of the tiny particles of a substance.
ETM.4  B  I can use the conservation of energy to solve problems, starting from my fundamental principle.
ETM.5  B  I can use the relationship between the force applied to an object (or system) and the displacement of the object to calculate the work done on that object (or system).

Constant Velocity Particle Model (CVPM)
CVPM.1  A  I can draw and interpret diagrams to represent the motion of an object moving with a constant velocity.
CVPM.2  B  I differentiate between position, distance, and displacement.
CVPM.3  B  I can solve problems using the constant velocity particle model.

Momentum Transfer Model (MTM)
MTM.1  A  I can calculate the momentum of an object (or system) with direction and proper units.
MTM.2  B  I can draw and analyze momentum bar charts for 1-D interactions (IF or IFF charts).
MTM.3  B  I treat momentum as a vector quantity.
MTM.4  B  I can explain a situation in words using momentum concepts.
MTM.5  B  I can use the relationship between the force applied to an object (or system) and the time duration of the force to calculate the impulse delivered to that object (or system)

Balanced Forces Particle Model (BFPM)
BFPM.1  A  I draw properly labeled free body diagrams that show all forces acting on an object.
BFPM.2  A  When given one force, I can describe its N3L force pair.
BFPM.3  B  I relate balanced/unbalanced forces to an object’s constant/changing motion.
BFPM.4  B  I can use N1L to quantitatively determine the forces acting on an object moving at a constant velocity.
BFPM.5  B  I can draw a force vector addition diagram for an object experiencing no net force.

Constant Acceleration Particle Model (CAPM)
CAPM.1  A  I can draw and interpret diagrams to represent the motion of an object moving with a changing velocity.
CAPM.2  B  I can describe the motion of an object in words using the velocity-vs-time graph.
CAPM.3  B  I can solve problems using kinematics concepts.

Unbalanced Forces Particle Model (UBFPM)
UBFPM.1  A  I use multiple diagrams and graphs to represent objects moving at a changing velocity.
UBFPM.2  B  My FBDs look qualitatively accurate (balanced or unbalanced in the correct directions, relative sizes of forces).
UBFPM.3  B  I can solve problems using Newton’s 2nd Law (Fnet = ma).
UBFPM.4  B  I can draw a force vector addition diagram for an object experiencing a net force.

Projectile Motion Particle Model (PMPM)
PMPM.1  A  I accurately represent a projectile in multiple ways (graphs, diagrams, etc).
PMPM.2  B  I can solve problems involving objects experiencing projectile motion.

Central Force Particle Model (CFPM)
CFPM.1  A  I can calculate the magnitude and direction of the acceleration for a particle experiencing uniform circular motion (UCM).
CFPM.2  B  I can use Newton’s 2nd Law to solve problems for a particle experiencing UCM.
CFPM.3  B  I can use the Universal Law of Gravitation to solve problems.
CFPM.4  B  I can use the conservation of energy to solve problems involving a significant change in the distance between an object and a planet.

—

Last year’s post: Honors Physics 2012 Objectives

I am curating a set of 12 goal-less prompts to be handed out at the start of the year. All of the prompts are based on students’ lives at St. Andrew’s School (SAS). They are going to spend the year working on incrementally better analyses of objects and situations that interact with them every day. This class just got relevant. Or, more relevant. Physics was already pretty pervasive in their lives. You know.

The Plan

Assign in some way (random? let them pick? something in between?) a problem to each student. That problem belongs to them for the year.

All students get to work on all problems, but the student who has ownership of each particular situation will be the point person for taking any needed measurements and for (hopefully) getting a little obsessed with what is going on there.

Every two or three weeks, spend a class day (a double period?) devoted to working on the analyses.

As they move through the course and acquire more ways of representing, explaining, and predicting their world, they will see more aspects of their situation that they can model, and they will see pieces of their earlier work that can be enhanced.

At the end of the year, the point person for each problem will put together some sort of presentation (probably a poster) that communicates the work that they’ve done this year. I think (and hope) they will also feel pretty legit about their work and understanding in physics class this year with something that significant in their pockets.

The Prompts

Some of these prompts are going to need more explanation than others since they are all tied to the particulars of SAS life. The actual text of the prompts that I will give to the students will be only the short part in bold type. The rest is an explanation here to give more context for non-SAS readers.

I am also still short about 2 or 3 prompts. I have a couple more weeks to figure it out, and I think some good ones will come up quickly when I get to chat with faculty friends after meetings in the near future.

Salt shaker game: A popular lunchtime game while waiting for announcements to start, the salt shaker game involves sliding the container across the table and trying to get it as close to the edge as possible without it falling off. 1 point for getting within a finger-length of the edge, 2 points for getting within a knuckle-length of the edge, 3 points for hanging off the edge (unless the other person is able to successfully flip the shaker and have it land correctly).

Jumping off the floating dock: Down past the front lawn, you can swim from the T-Dock to the floating dock (while being properly supervised, of course). As long as you have fewer than 5 friends with you, and as long as you aren’t diving, you can then jump off of the floating dock. Bam! Great physics problem.

Campus elevators: Enough said?

Soccer: This sport is played in the fall (both boys and girls teams) and in the winter (SAISL—St. Andrew’s Indoor Soccer League), so it should provide plenty of opportunities for analysis. I thought I would choose a sport, broadly, for one prompt, and soccer seemed like a good one for various reasons. Of course, we’ll see!

Riding a skateboard down the gully (while wearing a helmet!!): The gully leads down from where I live near the junior girls’ dorms to the crew dock. It is dangerously traversed by many a freshman boy on a skateboard. Just how dangerous is it? I think we might find out this year. By physics, not by accident. I hope! I’m not sure if the picture does the hill justice, but I couldn’t find a quick way to get a better perspective shot of it.

SAS school bus trip with Lonnie: Everyone’s favorite SAS bus driver (seriously, the girls lacrosse team even has a play named after him), Lonnie’s trips are always an adventure. In the right hands, I think this could end up as an amazing project. I also wish I had a photo of Lonnie to put here.

Squash cannon: First, a note from the squash/tennis coach in response to an email inquiry from me (with the context stock photo added by me—thank you Google Images!)—

We have two squash cannons and a tennis ball machine. The squash cannons we use a lot, and we had a lot of trouble with them last year–problems with the drum cracking and some with the motor or something, but when they worked they were great. The tennis ball machine isn’t is as good but it can still be useful; the main issue is we only have one plug out there on the courts, so you have to use court 5 to use it. The squash one allows you to set and vary the pace, height, & timing of the ball, and the tennis one you can adjust the topspin and underspin as well as the elevation.

I was looking for a repeatable projectile launcher that students encounter “in the wild” of the SAS experience, and the squash cannon seems to be a good fit.

Sledding down the hill in front of Amos: Assuming we get some snow around here eventually, there will most certainly be some sledding down the hill. The window in my classroom points directly at the action, and there is a lot of fun physics to model. Note: do not sled down the hill (a) on top of other students, (b) while standing on a sled, or (c) while on a bicycle (even if it worked the first few times). No broken collarbones in the name of a physics project, please! Also note: the photo proved more challenging to take than expected. Sorry.

The Prowler: I’ve seen students pushing this up the gully before, though apparently it has moved to its new home in the field house. A note from the trainer in response to my inquiry about it—

It’s called a “Prowler”. It’s a push sled with multiple height handles and a spot to add weights. I changed the “skis” on the Prowler so they only will work indoors now on the track. I think it’s a great idea to use them for class!

Apparently it has a notice on it that says “may cause nausea”, and according to this video that a quick youtube search produced, professional MMA fighters apparently train with it. I think this one will be great!

Baseball pitching machine: I am not as sure about this one, since I am not yet 100% sure that it will be possible to take data before the spring (I’m working on that). If it is, though, I think students would find it compelling.

Mattress collision: Apparently some students like to organize matches where everyone grabs his mattresses, runs toward each other (2 at a time), and collides. Ah, boys’ dorm. I’m not 100% sure about this one yet either, but if it works out, it could be a good analysis.

Next Steps

As I said before, I need a few more good situations for my collection to be ready for the start of the year. I’m especially hoping for another one that involves a good (and safe) collision. I can think of some SAS events that would be really fun to analyze, but only happen at a particular point in the year, so they aren’t going to work for this project.

At the end of the year, assuming this project is a success and that I would want to do it again, I plan to ask students on course evaluations which topics were the best, which should be chucked, and what should have been on the list, but wasn’t.

I’m sure there will be a new post about this project in May/June with some results.

What do you think, Internet physics friends?

—

* Note: Josh’s students regularly play against my students in sports. Whenever I am watching the two teams play, I’m always thinking something like, “If they were to start talking about IF charts or PMPM** or what objectives they need to test right now, both teams would totally get it! Ha!”***

** IF charts are momentum bar charts. PMPM is Projectile Motion Particle Model (pretty relevant to most sports).

*** It’s no surprise I’m just that nerdy, right? I write a blog about teaching physics…

“It’s against the law to think in the doldrums!”

So there you are. Stuck in description-land. The juicy parts of physics (explain and predict) are far away and out of reach. You can easily spend ages and ages there. The students start to think that physics is kinematics, and have no idea the beauty and power that awaits them. It drags out longer and longer. You’re obsessing over motion maps or equations, distance versus displacement, but it’s really not even clear why we’re learning about constant acceleration when we have no idea why it’s special and deserves its own consideration and unit. We’re itching to use the word force, or the idea of momentum, but those ideas are out of reach. We have no explaining principles yet, and no way to make physical meaning of things. We’re just describing, describing, describing our lives away.

BFPM is the natural motivation for CAPM

After the paradigm investigation in my balanced forces unit, we start off with a truly excellent 5-page activity that I got from Matt Greenwolfe. It’s 5 versions of the same problem with just slight changes to the situation each time. It essentially involves a box that you first push along a rough floor, then let go of while it is already moving. Students draw system schemas, FBDs, and velocity-vs-time graphs for each stage and altered scenario. We whiteboard them as board meetings (every group whiteboards the same problem, then we circle up and argue it out until we come to a consensus).

Drawing the v-t graphs really pushes students to come to terms with the rule they’ve just invented (Newton’s 1st Law). When the forces are balanced, their v-t graph must look like the graphs they were just drawing in the constant velocity unit. When the forces are unbalanced, the graph must look different from that (have a non-zero slope). We decide pretty quickly to use straight-line segments for our graphs (instead of curvy segments) so that we’re comparing the simplest representations. Even though they haven’t studied constant acceleration graphs yet (nor do they even have the physics definition for the word acceleration, so they’re not using the word), it is pretty obvious how to draw a graph with an increasing or decreasing speed. Through the slightly changed scenarios, they also start to put together an idea of how “close” the line will be to being constant velocity (zero slope) or how “far” from constant velocity (very steep) based on how “close” or “far” the forces are from being balanced.

We’re starting to develop a qualitative feel for how the unbalanced-ness of forces is related to the slope on a velocity-vs-time graph. That velocity-vs-time graph slope seems to have a meaning that would be worth spending some time investigating soon, huh?

Later in the BFPM packet, we get to the problem that originally gave me the idea that BFPM just makes sense as a lead-in to CAPM. The football player problem (also via Matt Greenwolfe) is just a killer. Here are the packet pages.

It starts out innocently enough, with a really straightforward (and by page 12 in the packet, usually rather simple for the students) balanced force problem. Check, check, check, and they are off and running past page 12 in a hurry. Then comes page 13, and (in the whiteboarding at least), we screech to a halt.

We start playing variations on the football player’s theme. Part c (what if he pushes with more force, but Ff doesn’t change?) isn’t too tough. He’s going to speed up, right? Parts e and f are similarly straight-forward. Sure, I can trick a few students on those parts (we’re still just learning about balanced forces, and it takes quite a while to really internalize N1L). Part d (what should the player do to go from a constant 2 m/s to a constant 3 m/s?), though, consistently leads to one of the best whiteboarding discussions of the year.

While they are first encountering the problem on paper, most groups will come to the same conclusion (just push with 3/2 as much constant force, duh), and they’ll get there quickly and without much discussion. A couple of groups will be really bothered by it, as they have two ideas that both seem like they must be true. On the one hand, “push with 150 N” seems like it has to be the answer. On the other hand, don’t the forces have to be balanced for the sled to move at a constant velocity?

The whiteboarding discussion gets all tangled up in those thoughts. Eventually, they will draw good velocity graphs or motion maps that show what must be happening, then they draw FBDs that must match the sections on their motion graphs. Then they come to an answer.

But! What force does the player have to exert while the sled is in the “speeding up” part of the graph? 150 N? More than 150N? Less than 150 N? Several arguments and graphs later, they get to an idea that anything more than 100 N will make the player speed up, but the higher the force (above 100 N), the quicker he’s going to get to 3 m/s, so the steeper the slope on the graph. And bam! Motivation for building the constant acceleration model is now complete. Moreover, students enter the exploration of non-constant velocity-time graphs with really good ideas about the meaning of them, having already practiced drawing graphs for objects that speed up and slow down, and with an idea of how they’re going to be able to use them moving forward. When I used to do CAPM immediately following CVPM, they never had any of those on their side.

By the way, Matt’s materials are about to be moved to a new Internet home, which is why I haven’t linked to them above. I will update this post with a link to those materials once they have settled themselves in to their new abode.

CAPM is better when you understand forces

Forces creep into the conversation throughout the unit. From the beginning with the constant acceleration paradigm experiment (which I wrote about in my model-building post for the constant acceleration model), to drawing diagrams during the “mini-goalless problems” in the problem solving section, to pattern-finding while drawing kinematics graphs.

Here are the Stacks of Kinematics Curves pages from last year’s packet.

We work through these graphs and whiteboard them in two rounds (about a page at a time with the different problems split among different groups). This was a great activity even back when we did it before forces, but it becomes even more powerful with the addition of that last piece (are the forces balanced or unbalanced (and if unbalanced, in which direction) during the time shown on the graph?). Balanced or unbalanced is pretty simple for them by this point (constant velocity? balanced! not constant velocity? not balanced!), but the direction the forces are unbalanced is a tricky thing to decipher. Different students have different ways of coming to an idea about it, but the most common way to think about it that the forces are unbalanced in the same direction as a speeding up object and in the opposite direction of a slowing down one.

The really cool part, though, comes during the whiteboarding. As the examples start to accumulate, and as we settle into agreement about the correct sets of graphs, at least one student in each class usually starts to see a pattern: the sign of the acceleration always seems to be the same as the sign for the unbalanced forces. We keep checking each new example to see whether it fits the pattern. I don’t push this idea on them at all, and I’d leave it alone if no one at all started bringing it up, but it’s really very cool. Now they have an idea about the steepness of the velocity-time graph being related to how unbalanced the forces are, plus the sign of that slope being related to the direction that the forces are unbalanced. You can just feel the energy rising toward a really big idea, can’t you?

After a recent Twitter discussion with Joe Kremer, I’m also mulling over the idea of actually adding an Fnet vs time graph to the stack instead of just asking whether the forces are balanced or unbalanced (+/-). That could be great priming for the momentum transfer unit down the line, and it would be more visual than what they currently do. It would also be easier to show the situations where the forces are unbalanced, but not always in the same direction.

In any case, with the qualitative beginnings of a model for how unbalanced forces change the velocity of an object, the investigation of a quantitative relationship (the paradigm experiment in the next model, unbalanced forces) a very natural next step. The stage has been set to start building UBFPM.

BFPM before CAPM

Of course, there’s not One Right Way to learn physics. This post is just a way for me to share a small change that has made a huge difference for my classes, hoping that it might help others, too, or instigate good discussion/questioning about sequencing. It has certainly (and thankfully) kept me out of the kinematics doldrums for the past couple of years. I think it’s a bit faster for my classes, overall, too, since the constant acceleration unit isn’t the brick wall for my students that it used to be in the old order, but I just have a feeling about that, nothing so concrete.

So that’s my sell. Get to explaining and predicting more quickly—remember that any CVPM concept that is a little fuzzy will be practiced in BFPM and examined in more detail in CAPM, so you don’t need to spend much time with constant velocity at all before you’re ready to play with interactions.

I also want to point out some potential pitfalls of using this type of whiteboarding, give some tips on how it has worked best for me, and talk about some of the benefits. I’d better start, though, with a description of what I mean by the “Mistake Game”.

What is the Mistake Game?

In a moment, I’m going to describe the first day of whiteboarding in my classes using the same sort of style that I use in my model-building posts. I think that will give a better picture of how it looks in my classroom than my trying to describe it.

Before that, I should talk a little bit about what happens before we start whiteboarding.

The students have been working “individually-together” in groups of two or three on a few problems (not necessarily an entire worksheet) from our packet. We don’t divide up time by the worksheet numbers; rather, we just keep working continuously through, with different students always at slightly different places, then pause when we’ve all done at least a certain chunk (usually about 5 problems/parts of problems) of work.

While they were working, I was moving around the room, doing some coaching, but also trying to look sort of unavailable and not like I’m looking directly at their work. That is especially important at the start of the year, before the students “get” the whole groove of the class. The answer-driven, nervous students will seize me if they think I’m available or even just looking at them. They are too eager to check every step of every solution, to scan my face for some tiny signal that they are doing everything correctly, and to ask me questions that I really want them to answer themselves.

Later in the year, I won’t worry so much. They know that they are going to be whiteboarding the problems, so they know they will soon have all of the answers available to them. They know that the best way to learn the concepts is the wrestle their way through the problems, so they actually don’t even want me to help. They are more likely to shoo me away, or tell me to stop looking at their work, than to play Clever Hans.

In September, though, they aren’t there, yet. They don’t come in that way. They haven’t been allowed to make mistakes before. Not in the way that they’re required to make them in physics class. Right now, it’s scary for them. If a group ensnares me, it will take a long time to get myself detached from that table without making them feel abandoned or frustrated (though they’ll probably feel both anyway, since I won’t answer the questions the way they want me to do). Trying to answer their questions without answering my questions involves a lot of side-stepping, responding with a different question, and explaining of why I’m not being as helpful as they’re used to teachers being. In the meantime, I’ve missed out on hearing the discussions, arguments, and resolutions at the other tables, so I’ve lost the pulse of the class for several minutes, and I need to get reoriented.

Eventually, I know that all of the tables have done the first 5 or 6 bits of work. It’s time for our first ever whiteboarding session. I’m psyched; I love whiteboarding!

The First Day

Hey everyone, can we pause for a couple of minutes? We’re about to do something for the first time that we’ll be doing really frequently this year. It’s called “whiteboarding.” Let me tell you a little bit about how to do it, then we’ll try it out right away. In a moment, not yet, each group will need a whiteboard and a marker. Each table will write a solution to one of the problems on their board. When you’re finished writing it, you’ll take your board and put it up backwards on the ledge of the big board up front.

When all of the boards are up, we’ll know we’re ready to start. Each group will get the chance to come up and present their solution to everyone.

Oh wait, I forgot an important part! There’s a little wrinkle in the solution-writing part of whiteboarding. When we whiteboard in this class, we almost always play the Mistake Game. So the wrinkle is that you have to include at least one intentional mistake in your work. You can also include as many unintentional mistakes as you’d like.

For your intentional mistake, you should be trying to make a really good mistake. That means it should be something that you think your classmates might actually do. A lot of the time, a good way to come up with a quality mistake is to use something you actually did wrong when you were first solving the problem.

Let’s try to avoid Where’s Waldo? types of mistakes. So your intentional mistake should NOT be something like labeling the axes backwards, changing a number in an arbitrary manner, or spelling your names wrong. You might do some of those things unintentionally, but for the Mistake Game, you’re goal is to include a really quality, conceptual mistake that will lead the class to a good discussion when you present your board.

Okay, let’s get working on the boards. Would you two do #1? This table #2? #3? 4? And then you three take #5? Great. And don’t worry; you’re going to get better at making mistakes. [Now it takes a few minutes for them to grab boards and markers and to write up their problems. I basically walk around a bit and encourage them along. I don't comment on whether their answers are correct or not, since we're about to have that discussion. It doesn't take them long (in the year) to stop asking those sorts of questions because they know that they will have the correct answer by the end of the discussion.]

Are we all ready? Excellent. Now I need to tell you what your job is during the boards that you’re not presenting. For everyone not presenting, your job is to ask questions of the group up front. If you think there is something they need to change about their board, you need to ask them questions to get them to change it.

Asking questions is probably (definitely) the hardest part of this whole process. Okay. So examples of bad questions: “Isn’t that part wrong?” or “Don’t you think you should change it to be like such-and-such?” Okay, any statement that you’ve basically just put a question mark on the end of is not going to be a very good question. Good questions usually try to lead people to seeing an inconsistency or contradiction in their work. And you’ll get better at asking questions, too. It will take time.

Now, at the beginning of the year, a lot of times students have an impulse to look at me while they are presenting or while they are asking questions after the presentation. I’ll help facilitate the conversation a lot today, but one of our goals it to get to the point where you don’t really need me to talk during whiteboarding.

Okay, before you get worried that you’ll never know what the “right answer” is in this class, let me tell you a couple of other things. First, I’ve read about how these kinds of group discussions work—even if everyone in the room has the wrong answer at the beginning, as long as you don’t all have the same wrong answer, you’ll end up with the correct one through the discussion. So that’s one of the reasons why I tried to get you to keep from checking with all of the other tables while you were working—I wanted to make sure that we preserve diversity in any of the wrong answers. And second, I’ll obviously let you go up with a wrong answer on your board, but I won’t let you sit back down with one. If you ever finish your discussions with a wrong answer, I’ll chime in with a question or two at the end to redirect the conversation. What I’ve found, though, from my past classes, is that I actually very rarely have to do that. You guys tend to do a really good job with these discussions.

One two last comments before we get started. Remember to be nice to each other. So none of the… “That’s wrong and you’re stupid!” type comments, right? And also remember that you never know whether a mistake on the board was intentional or not, so be gentle. Finally, for the people presenting—you shouldn’t act and draw it out when you are asked a good question that addresses an error on your board. Even though you put the intentional mistake in your work, you don’t have to act like you’re totally and hopelessly confused about the mistake. You can acknowledge a good discussion and make the change pretty quickly when it happens. But also don’t jump to change the board after a question that just hints at how to fix the mistake. It’s sort of a balance. We’ll figure that out along the way, too.

Alright! First problem, here we go! Go ahead and do the dramatic reveal of your board guys, and then walk us through your work. Everyone else, please give your respectful attention to the front of the room. [All of that took a while to write, but doesn't take more than a few minutes to actually say before getting right into the real work of whiteboarding.]

Why it rocks

After trying it myself, I completely recommend starting the year with the Mistake Game as the primary way of whiteboarding problems. It was a good way to teach the students how to have effective whiteboarding discussions. Asking questions to uncover the intentional mistakes sets up an artificial, but comfortable, way of showing students how to have a productive conversation that leads to an increase in understanding. At the start of the year, when the problems are relatively easy for the students, it also helps isolate those difficult discussion skills from confusion about problem solving.

The following comparisons are based on my observations of students in my classes before and after I started using the Mistake Game.

Without mistakes: Student presentations often consisted of them standing silently next to their board, occasionally pointing. It was awkward, and they didn’t know what to say because they assumed that their work was self-explanatory and that everyone already had the same work.
With mistakes: There is an obvious need for explaining their work, since they assume it will be different from everyone else’s. Moreover, the way to explain their work now seems clear—they must walk their peers through their thinking so that their mistake will be highlighted.

Without mistakes: When a group presents a board with a wrong answer, the atmosphere in the classroom gets increasingly uncomfortable. Often, the error is pointed out very quietly, and an embarrassed group member quickly edits the board, hoping to cut off any discussion.
With mistakes: Errors are a normal, expected part of every presentation. In the case of an unintentional mistake, the presenting group always has the option to act as though it were intentional. Remarkably, most students feel no need to “cover up” their unintended errors. Since mistakes are expected, they feel okay about both making one (or two or three) and having a discussion about it.

Without mistakes: If there are no immediate comments or questions, the presenting group often tries to put their board away (below the big whiteboard) and go back to their seats as quickly as possible. The teacher often has to keep the group up front long enough for anyone who has a different answer on their paper to formulate a question.
With mistakes: After many presentations, there is a comfortable period of silence while everyone scrutinizes the board in front of them. Sometimes there are immediate questions, but sometimes it takes time to find and digest the mistake(s). Since everyone knows there is at least one thing wrong with the board, there is a need for that quiet reflection. The presenting group does not make haste for their seats because their mistake hasn’t yet been resolved (some groups do still try to run at the first opportunity, but that opportunity comes later).

Without mistakes: Students vied to be assigned a problem they knew they had done correctly. If they weren’t already sure about an answer, they try to check with as many people as possible, including the teacher, before putting their board on the ledge.
With mistakes: When groups are assigned a problem that they aren’t sure they’ve correctly solved, they don’t worry. They write as much as they can, knowing that by the end of their presentation, they’ll know how to solve the entire problem (with the help of their classmates). There is no right-answer anxiety from any of the groups while they are writing on their boards.

Without mistakes: Especially in the regular classes, little thought is put into conventions and format when writing on whiteboards. No students ever correct mistakes in symbol use, axis labels, etc. The teacher is left to ask these picky procedural questions at the end of every presentation, much the annoyance of everyone (teacher included).
With mistakes: Students are delightfully picky about each other’s work. They have permission to be respectfully picky without having to be a jerk (and also without leaving the teacher to be the jerk). Students think about symbol use, labels for graphs, formatting for algebra, etc when writing their whiteboards. Good habits are reinforced.

A secret of the mistake game—it’s actually the best test corrections ever. Over the course of the year, as the students get better at making worthwhile mistakes, something wonderful starts to happen. Students start to work out their mistakes from old tests by, I guess, role playing them in front of their peers. They present a problem using a mistake they’ve made before, but this time the joke is on the mistake. This time, the student knew it was a mistake all along. Even better—especially near the end of the year, students express pride in a particularly good mistake. When that happens, it is almost always about an old mistake that they are now showing mastery over. That’s not to say that students aren’t thinking of great conceptual mistakes that are original for them; rather, students are just exceptionally proud of exorcising an old demon.

That’s all great, but how do I win the game?

Okay, you’ve got me. It’s not really a game. At least, it’s not a competitive one. There’s no winning. No awards. No badges. No rankings.

I don’t build it up as a game, though, and I haven’t yet had any problems with students wanting to “win” it.

Building it up as a “game” is probably one of the potential stumbling blocks of using the activity in class. That leads me to the next portion of the post—

Possible pitfalls

Over the past year or so, it’s been really fun to hear from other teachers about how they’ve used this whiteboarding strategy in their own classes. Among the situations I’ve seen in my own classes, as well as stories from other teachers, I’ve recognized some ways that the activity can get stuck in a more inefficient/ineffective rut.

Here are the ones I’ve heard about or seen so far. I’ll add to the list as I hear about or experience others.

Where’s Waldo? mistakes

Misspelled names (no, seriously), arbitrarily changing a digit in the solution (often without that change even affecting the rest of the work, though it should have), switching the labels on the axes of a graph—I see these the most near the start of the year. I’ve also heard a lot about these. Sometimes they seem to be generated by a misunderstanding of the game. If students think that the goal is to hide a mistake in their board, Where’s Waldo? is clearly the way to go. It’s hard to argue with a classic!

Of course, the goal of the Mistake Game isn’t to hide a mistake. Everyone already knows that your board has a mistake. The goal is to generate good discussions by working through errors in reasoning and understanding that are likely to besiege a few students in the class at one point or another.

The other main source of Where’s Waldo? types of mistakes is probably weak metacognition skills (or a weak understanding of the problem—or a combination of those two). It takes a while to develop the ability to think about how you might go wrong while working a problem, and although students have been making mistakes for years, they haven’t often been asked to make them on purpose. So they probably need more guidance, to have good examples pointed out when they arise, and time to practice.

“Our mistake is… that we didn’t make a mistake!”

Some of the best, most productive, and funniest results of playing the Mistake Game come from situations where students either (a) think that they wrote a board with no mistakes (yes, everyone think’s they’re hilarious for thinking of that mistake at the beginning of the year) or (b) think their board still has an error when it is actually correct (that is, they thought the correct solution was actually a mistaken one). In the first case, the “no mistake” board almost always contains the most unintentional mistakes of that batch. In the second case, rather than running back to their seats, the presenting group tends to stand around for a while at the front, even though no one has any more questions.

Both of these possibilities could lead to uncomfortable situations, but again, with the mistake norming that is being done on a daily basis through the whiteboarding routine, the presenters are usually good-natured about it. I guess this one is actually more of an anti-pitfall, unless the “we didn’t make a mistake…!” becomes a routine. If there is a student or group who is really intent on not making (intentional) mistakes (I haven’t had this happen, but I could imagine such a student), one option might be playing Mark‘s Mistake Game spinoff: the Mystery Mistake Game. It is exactly the same, except that each group may choose (without telling the other groups) whether or not to include an intentional mistake.

Whiteboarding with mistakes takes forever

This is a complain that I haven’t really seen myself. I haven’t timed it, but adding mistakes hasn’t seemed to make my whiteboarding process any longer than it was before. Of course, whiteboarding at all is going to take much longer than lecturing or “going over” answers to problems. If you make the jump from no whiteboarding to whiteboarding with mistakes, it will probably take a lot more class time than originally planned.

I suspect that much of the other taking “forever” aspects come from the other items on this list, so the solution is probably to attack those problems.

If the time problem seems to happen while students are writing the whiteboards (rather than during the discussions), having a timer somewhere visible might help students become more aware of the time they are spending. I also find the “put your boards up backwards” part of my process to be helpful here; a slow group will see that the other groups are visibly ready, even if they are chatting, because the boards have been accumulating at the front. That can help keep some groups on track when they were wandering, as they start to realize that the class is waiting for them.

In my regular classes, we tend to whiteboard every problem that we do (though we don’t always do every single problem in the packet). In my honors classes, the students tend to get pretty good at knowing which problems they think are worth whiteboarding. They usually want to skip the discussions for problems that they already feel confident about, and that can save some time, too. Of course, I just have to be careful to make sure that I always agree with their assessment of problems that are “skippable” during whiteboarding since they don’t always know when a problem has some extra subtlety that will make for a good argument in presentations.

Presenters who love playing “dumb”

I saw a lot of this one when I was first trying out this activity and hadn’t yet completely understood what would work best. It can be a little tricky at the start of the year to calibrate the “acting” required in presenting boards with mistakes. Basically, when they talk through their written work, they should include the mistake fluidly with everything else. Once the presentation shifts to answering questions, they should stop acting as though they totally believe their mistake and instead fix it when prompted by questions. They shouldn’t, though, fix the mistake in response to really poor or irrelevant questions. They can engage the asker a little. The problem comes when they take that engagement too far, or when they respond with mock-confusion to even decent or good questions.

So. Part of the fix here is just a little teacher intervention when it happens (one of the reasons why I talk more during whiteboarding in the first few weeks than I typically do during the rest of the year). The other fix is making sure that the students really understand that they aren’t supposed to be “hiding” their mistakes. The jig is up; we all know you have a mistake.

“I got 5 for the velocity.”

Ignoring the naked numbers in that statement, the real problem is two-fold: (a) it is not a question and (b) it is about the student’s work, not the presented work. I usually intervene immediately when “questions” like this arise, whenever they happen. “Ask about their work, not your work.” If the students have enough time to be quiet and thoughtful after a presentation (instead of rushing to ask questions or move forward to the next problem), then the problem is usually not that they don’t have time to compare the work on the board to their own. In that case, these questions are probably symptomatic of a larger problem: the students talk sequentially, not to each other. That is, they wait patiently to make their own statements (usually not listening to anyone else—except maybe the teacher) instead of responding to classmates. They often make the statement to the teacher. They are stuck in an individualized view of school and are used to learning next to, not with, other students.

I am still working to get my regular students up to the level of discussion that I’d like during whiteboarding. When I’ve had success at changing this mindset/behavior, it’s been due to short talks with the class about what great physics classes do (subject of a future post, hopefully). I basically start a class one day with a quick 2 or 3 minute chat about how I think they are a really decent class, and about how I want to share something I’ve noticed about what really great physics classes do. Then I tell them about talking to each other, making each statement a response to what has been said by someone else. I’ve seen really huge changes happen with just those kinds of short talks in an honors class this past year, and I’m hoping to try to apply it more in my regular classes this fall. I think the key is to remember that they don’t know how to have those kinds of discussions in physics class. So patience and some explicit, well-timed pointers seem to help them develop those skills.

Asking questions is tough!

This potential pitfall is such a big one, the headline deserved a bigger font.

Asking good questions of the presenters during whiteboarding is one of the most challenging things I ask students to do all year. And I start asking them to do it around the second or third day of school. Every student will have trouble coming with questions that highlight contradictions and that aren’t just statements with a raised voice at the end.

What I’ve found to be most helpful is pausing the class, acknowledging how difficult it is to formulate good questions, asking for a reworded question, and occasionally rewording a question myself to help show them how to do it (and especially to interpret the meaning if I know what they are trying to ask, but the presenters are at a loss about how to understand/answer the question as posed). Critically important to that whole process is allowing silence and thinking to happen without trying to rush to the quickest solution. If a question is badly phrased, I try to let the same student ask again instead of letting someone else just jump in. Without letting them reword it, that student won’t move forward in his skill, even though the class might resolve the problem by having another student ask a great question. I want all of the students to keep wanting to participate, and I want all of them to get good at asking questions.

Another question-asking difficulty—some students only ever want to ask questions of me, never of the presenters. The Mistake Game itself does help refocus those students, but some persistent ones will still exist. If they turn to me before or while asking, I just point back up to the front of the room (I always sit/stand at the side or back during presentations). If they say that they have a question for me, I direct them back to the presenters anyway. They will usually ask the presenters if I insist, thinking (I guess) that I will answer their question once I hear it and realize it is actually for me. The presenters almost always do a really nice job answering the generalization question in those cases, and I’m always glad that I gave the students up front that opportunity when I hear their responses.

Final note about asking questions: A lot of this blog post, and especially this section, was prompted by reading this post about confidence in reasoning (and having a discussion in the comments). More good thinking about setting up discussions among students is contained over there.

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Links to the old posts along my mistake whiteboarding journey: The Mistake Game | Whiteboarding with Mistakes

More about challenges for whiteboarding: Understanding the Pressures Against Whiteboard Meetings