[click everything to enlarge]
The last few weeks have been pretty profoundly discouraging on all three of my professional fronts. I’m sure it’s just the ebb tide of education but it’s worth mentioning that I’m specifically losing my mind over this WCYDWT thing, which is just a thing and may be much less than that, some form of digital wankery, I don’t know.
It’s like when you were a kid and you whispered the word “football” over and over again until the compound word separated and both parts seemed weird and meaningless all at once, that’s something like giving three conference sessions on the same process of curriculum design in four months. I can’t convince my fellow curriculum specialists at Google nor the teaching cohort I mentor online nor my high school colleagues of its value, which makes convincing myself of its value suddenly a real trick.
I’ll say this much for certain: if there’s value here it isn’t in the comment, “Rates! We could talk about rates!” The response to these media too often breaks down into a checklist of mathematical conversation starters and if you’re going to offer them any more than two minutes of class time then they absolutely have to be more than that. They have to earn their keep.
What do the students do with the photo? What questions will they ask? What measurements will they need? Once they’ve resolved the first question and feel like they have a stronger grip on the concept than they really do, how will you twist the scenario around to challenge them? How will you create that crisis?
Maybe it’d be better if I showed all of my cards up front, rather than this coy unveiling process which seems like a non-starter. If the exercise interests you, help me construct a narrative, an activity around these photos. Extra credit if you feel like contrasting it against Sean Sweeney’s pass at the same material some months ago.
I only know I shot this because I felt like I needed it, because the alternative is a problem involving savings accounts with different principals and different monthly deposits and none of my kids have savings accounts.
[BTW: This YouTube clip plays pretty well here.]
David PetersenNovember 20, 2009 - 3:32 am -
Well, I see that you could certainly lead a question more if you had Chris put on an old-school hockey mask or you both had paper numbers on your shirts. Another question could be led if you had an explosion going on behind you both at the left end of the photo.
David PetersenNovember 20, 2009 - 4:10 am -
The long space to the right in the first two pictures implies that we are looking for something to happen to the right of the subjects.
I would’ve questioned why you put the “origin” where you did, but in the last one, I can see that the pole there makes a nice reference point in all of them, but I didn’t notice that until the last slide here.
Along with my questions from the previous post, something to throw them off a bit might be something like Dan gets a distance-head-start, but Chris gets a time-head-start. (To lead that question with costumes, one person could wear a turtle shell and the other rabbit ears.) Or, each person begins running and there’s a bike along their pathway that they can pick up when they get to it. (or begins on the bike and gets a flat tire along the way, so he has to go on foot) I’m sure there are lots of fun extensions to the concept where different scenarios could lead to other ideas.
AlexNovember 20, 2009 - 7:25 am -
Here’s what comes to mind when I see these photos:
1) Start with a video clip of you alone running towards a destination with a clear clock in the video to open with the rather simple question of how long will it take you to get to the destination (I can’t think of a great way to frame the video to make this look like a real meaningful question). When students ask for specific positions and specific times you can throw up slides like what you have here.
2) Show another video clip this time a wider shot showing someone chasing you and ask will the chaser catch you before you reach your destination. Again more slides with positions and times. Hopefully this leads students to the concept of relative speed which is a nice example of an abstract mathematical concept but one that people still have some intuitive feel for.
3) Now switch to a ball rolling down an incline, but make sure there’s a plain background behind the incline and angle the camera so it looks like a flat surface. This time start with two slides showing the ball’s position and time and ask how long it will take the ball to get to the end (the end of the incline is painted or something).
4) After students make their predictions show the video and ask them why their predictions were incorrect. Ask them what the slides would look like if you took a picture of the ball every 0.1 seconds. Show them a sequence of slides and have them graph speed vs. time and distance vs. time.
5) Maybe you have students fit equations to the graphs and this is an activity about relating real world motion, graphs and equations (You could also go back and graph #2 and look for where your line and the chaser’s line intersect). Or maybe this leads into a discussion of derivatives and making the connection between pictures of the object at different moments to different points on a graph and how you use these points to approximate the slope.
josh g.November 20, 2009 - 8:43 am -
I have to second the explosion idea. This image is useful but it isn’t quite self-explanatory. Actually, my first thought was that this begs for video treatment, with either follow-up still shots for specific time snapshots or else coupled with a video tracker / analysis tool to extract the data straight from the source.
Barring that, I’d turn this one into a comedy and have Chris chasing Dan for stealing his twinkies, or something. But I’m a cornball that way.
For what it’s worth, as a newbie teacher desperate for all sources of student inspiration, I’ve found the WCYDWT approach to be fantastic. In my Education program, we were taught a lot about teaching through effective questioning, student-led discussions, etc. The math/science teachers in the room, without fail, would see these and think, “Great for Social Studies or English, but this doesn’t help me teach math (or science).” WCYDWT proves that statement wrong.
This reminds me, have any of the ideas for a central WCYDWT resource site taken off? I’ve got something I came up with last month that I need to share.
RileyNovember 20, 2009 - 9:30 am -
Oh, Dan, please know that you have inspired me immensely this year. I am personally at a point near exhaustion, but my classes have been much improved by your ideas.
I led a lesson with a similar focus six weeks ago. See my take on my (super-new) blog: http://larkolicio.us/blog/?p=42 . I’ve been working on the idea for a while, and this is not my best lesson after your example, but I think it’s there.
I think a video is a more appropriate medium than photos here. In my class, the students could navigate to different times on the video, and I never had to focus them by picking specific times or anything – they had a TON of information and had to pick some to use, which is the thing I admire most about your approach. With only photos, you tell the kids which times you think are important, what distances are important, specify units, etc.
I don’t have a totally satisfactory way to challenge kids’ assumptions in a delayed way in this lesson yet. However, in my lesson I didn’t show the kids the end of the race, and some assumptions flew that were challenged right away. That also happened in a conflict about slope, changing speeds, units of measurement, etc, but again only in the way that happens in any discussion.
Gilbert BernsteinNovember 20, 2009 - 3:30 pm -
In case no one noticed, Dan’s riding a bike in #5 and #6, and facing the oppposite direction in #7.
Dan, I think these pictures might be too small to convey anything beyond those little numbered boxes.
Kate NowakNovember 20, 2009 - 3:39 pm -
I really want to like this. They’re an improvement on the word problems we recognize under the surface. Better for helping students understand what they are dealing with and what is given and what is being asked. And I’m totally using them next time I teach Algebra 1. But if there’s a cognitive crisis lurking in there, I’m not seeing it. I don’t mean to be a pill, but I know the blowing of sunshine is not appreciated around here, so.
What would I DO? I might not bother with #1, or show it for 10 seconds. I’d leave #2 up and say very little until I heard some version of “He’s going to catch him” and hope that someone wonders when or where.
And #3, I don’t know. That would be the most important and the trickiest. I WANT to say “ask students to discuss a method for deciding where he catches him, and then share them after a few minutes” but depending on the group that might go nowhere. Failing that, and if it was a small enough group that a discussion with the whole group made sense, I would have to try to anticipate all the potential approaches, helpful and not, ahead of time, so I could keep the discussion in class making enough progress that kids didn’t lose interest.
The REALLY tricky part would be to generalize the very concrete examples to an approach that would work no matter, so that I’m teaching Algebra 1 and not 7th Grade Again. But at the same time, the concreteness is very helpful, because it keeps giving them something to grab onto. #3-8 need to be printed on handouts. That’s all I got. Someone, please do better.
Kate NowakNovember 20, 2009 - 3:47 pm -
Oh, I see the potential crisis now. In 3, you can find where they meet up on the picture provided, but in 4, it’s off the page. But at that point my kids would throw up their hands and start talking about xbox. I’d have to give it to them.
Nick HershmanNovember 20, 2009 - 5:33 pm -
First off, I’d be quick to agree with Kate and Alex’s comments. Asking when the pass occurs seems natural, or who wins the race to a specific point are useful questions. To the question of what kids do with the photo, I had to re-read your post and my first comment to think about this more concretely. What’s the acitivity? What specifically will they have to do with it?
I’d pass out the pictures in addition to projecting them, it’s nice to be able to write on this stuff.
Questions they’ll ask:
– What do you expect from me on this assignment?
– What if someone falls, speeds up, or slows down?
– What really happened?
– Am I right?
– How come there are two of the same person in the photo?
– How long is it in between exposures?
Finding the point where one person is overtaken is not going to hit the sweet spot for an eager student, for a reluctant student, it may be tricky.
I’d begin with some simple questions inviting everybody to weigh in
– What’s happening?
– Who’s going faster?
– Who’s gonna make it to the last marker (on most shots this was labeled 210) first?
Follow up with some more involved calculation questions on a handout or projected for them to complete on printouts of the photos
– How many time intervals will it take for them to reach the final marker?
– Between which two intervals do they pass?
– If this were a track, and a lap were double the distance marked, how long would it take to run x laps?
For students who breeze through, the crisis question can be stated in two ways:
– Exactly where do they meet?
– Exactly when do they meet?
For students who can do that, maybe have them extend it to a more general proof:
– Explain why if you are behind someone in a race and you are going slower than them, then you can’t catch them
– Explain how to find the exact meeting point for two racers at different points along a track at a fixed speed
General expectations that might be useful for this are:
1. Write down your process, be as clear as you can and organized, but don’t be afraid to brainstorm and think on paper as well
2. Think aloud with your neighbor. Share ideas.
3. If you can’t find an exact answer, write down an incorrect answer below and one above where you think the correct answer is.
As for a quick comparison of your pics here with Sean Sweeney’s, I’d really like Sean’s video if it were from a side viewpoint, as it is, I wouldn’t expect students to be able to generate a graph of location just watching runners run directly away from their vantage point.
My one thought stepping back from all this is that it doesn’t always need to be digital. Good questions do not require good pictures/video/digital content.
You could also try some Very Restrictive Unfair War with Dice – a game that is still not copyrighted. Give one student 15 points and another 35 The student who’s behind scores 4 each turn while the one who’s ahead scores 3. Ask the same set of questions. Iterate by giving larger numbers, decimals, fractions.
Or have students imagine a race in which each member of the class walks the a course on the perimeter of the classroom at the same time. Some go two steps at a time, some go one at a time. Have them predict when they’ll catch different members of the class, do this as a paper activity first, and then have everyone get up with their calculations and move either 1 or 2 steps at a time on your count. Have them verify their calculations at each pass.
Jason BaldusNovember 20, 2009 - 8:54 pm -
When I read Alex’s #1 about the video of one person running and asking how long they would take to reach a destination, I thought of that scene in Austin Powers when he runs over the Evil Henchman with the steamroller. It’s got that great shot from the side view where measurements could be made. Plus it cracks me up.
I think that a video is not really necessary here, but it might be helpful. Then maybe use these to give them some measurements when they ask for them. Then again, I think that being able to look at a static picture and visualize the action implied by the picture is a useful skill in math. So maybe a video would be less helpful. But not in the good kind of “less helpful” way.
As for conflict, I don’t know. Yet.
Dan MeyerNovember 21, 2009 - 2:06 pm -
Now we’re talking. Thanks for indulging my sobbing jag in the post above. I hope you all realize how much the people who read these comments will appreciate your attention to detail.
Something I should have mentioned in the post and would definitely mention in the first few seconds of class is that the strobe shots were taken one second apart.
It seems as though the commentariat is taking the “create crisis” mandate a little too literally, with hockey masks, lions, dragons, and explosions, etc. Those are good hooks and interesting motivators but the “crisis” as I understand it is best read internally.
For example, Alex creates a crisis by extending this activity to a ball rolling down an incline. (Which is more in the D=RT mold than the intersection of lines I’m looking for, but let’s suspend that objection for a moment.) Show two strobes of a ball rolling and watch as the students, confident and cocky from their successes with the running activity, quickly compute how long it will take the ball to roll down the incline only to be flummoxed by gravity and an image of the distance increasing between the strobes each second. That’s crisis.
Contra Kate, it doesn’t matter that this is a purely intellectual crisis. If I can sucker one of your easily distracted kids into leaping on the obvious-seeming-but-wrong answer and then yank it out from under her with a clear visual, I’ve just bought some time and attention.
I appreciate Nick’s attention to detail here as well as his critique of Sean’s video, which is the same as mine.
@David, I needed an origin that wasn’t Chris or Dan to motivate the fact that they both have a y-intercept. Otherwise, if Chris is the origin, we simply find the head start and divide by the difference in speed. Which isn’t a bad way to scale this down for novice audiences, I suppose.
@Gilbert, the size and resolution sucks, there’s no excusing it. Let’s treat this as a proof of concept, then, and assume that one day I’ll have the right hardware for the job. Better hardware is just a matter of budget. Better pedagogy is a much harder harder.
Kate NowakNovember 21, 2009 - 3:55 pm -
I think I’d have to see that working to believe it would work. With this particular instantiation. But, I want it to work.
And to clarify, your vision of where this is going is that the students would plot position vs time graphs and find the intersection? In my mind I was trying to force it into an rt=rt+k make substitutions and solve for t kind of direction.
Jeff PierceNovember 22, 2009 - 2:18 am -
Hmm. A passing comment from a social studies teacher…take it or leave it or throw me under the bus.
This is the first WCYDWT that I can recall leaving me positively nonplussed. The reason I always read the WCYDWTs, even though they never go into my classroom, is that I enjoy finding out how you’re going to turn an ordinary experience / easily argued about problem and scale it into all different kinds of mathematic principles.
The intellectual crises that these images lead to seem to be at their core just a repackaging of the good ole’ textbook word problem with trains leaving the station at different times. Really, who cares?
“Don’t just whinge, provide a solution.” -my last boss
A race is not static like photographs. I think that these would pack punch and hook students in if each scenario was first preceded by one second video clips from a similar vantage point, or even from a first person view.
Heck, get inside a track and put the camera inside the announcer’s box. Or spend ten minutes on youtube. Stop the tape of a 200 meter run as the athletes enter the final 50 meters. After the howls, ask the class, “Who’s going to win?”
Or put on that Jason mask. Seriously.
Maria DroujkovaNovember 22, 2009 - 7:10 am -
I think of “What Can You Do With This” as a framework for educators or probably education researchers. It’s been helpful to me personally. For example, the other day I saw my math club members’ cars in the drive up the hill, but they would not enter. I found the lot of them outside, extremely happy in a big PILE OF LEAVES.
I asked myself, “WCYDWT?” and sat in the pile for a while, thinking it over. Then we devoted the whole club to “counting all the leaves in the pile” – well, estimating, really. Kids got to 100, 000 very meaningfully, and worked through a couple of important snags in the process, like counting “Ten thousand, two thousand, three thousand…” (the typical place value mistake). It was a very good math experience for six year olds, as evaluated by kids, parents, and myself.
I largely blame WCYDWT for this success. I could probably write up a journal article about it, with formal references to relevant research frameworks. However, it’s your baby, so maybe you should do that.
John S.November 22, 2009 - 7:44 am -
I found this series of photos very confusing. A lot of that confusion I think was due the fact that the wide angle view obscures the detail of the runners into near obscurity. Sean Sweeney’s video suffered from the problem of the increasing difficultly of judging the position of the runners as they get farther from the finish. These photos suffer from lack of detail caused by the wide angle view.
I was really confused by the apparent time transport that occurs between photos 3 and 4. (as well as between other adjacent photographs). The fact that photos 1 through 3 are obviously a set led me to think that the photos were all related to the same race. With the help of the comments above, I now think that you didn’t intend for the photographs to be a series but to have each stand alone as an individual problem.
The information that the strobe shots were taken one second apart definitely re-sparked my interest and led me
to consider each photo as two overlaid snapshots from a separate race from which to have students make predictions as to the entire course of the race. My confusion now subsides but my interest also does. Now each problem presented becomes a pair constant rate problems presented visually.
I liked the problem better when I was totally confused.
Here’s my suggestion on how to “be less helpful”.
1) Drop the strobes.
2) Start with a wideangle of the entire course, the position of the runners and a time reference. Ask the students, who is going to win?
3) Add closeups of sections of the race course with time references. Have lots of of these and show the students only what they ask for. Give an index of the available closeups to the students and challenge the students to predict the winner by viewing the fewest number of close ups as possible.
4) Make the first problem linear.
4a) Make the second problem linear.
4…) linear again
5) Crisis – introduce a non-linear problem (have a runner stop to tie his shoe or trip and fall.)
Optional lesson. Graphing stories. Hand out a random set of snapshots to different groups of the students and have them graph the course of the race and then discuss why different groups got the same or different results.
Optional race: Have a runner race a basket ball shot. Linear versus quadratic. Do you graph horizontal versus
vertical or position versus time?
Kathy SierraNovember 23, 2009 - 10:36 am -
Dan, Dan, Dan. (youngster.) Changing the world takes *time*. Be patient and keep pushing. One day you’ll be whining about the opposite problem… that the ideas you helped champion are so commonly accepted nobody can remember there was ever another way. It’s just the way it’s done. You’ll be all emo over someone saying — with respect to WCYDWT — “duh”. :)
But it won’t happen unless you and your cohorts keep at it.
monika hardyNovember 23, 2009 - 5:13 pm -
i must be the oldest one here – or the one with the worst eyes. i need bigger. :)
i love what you’re doing dan. and more importantly – my kids do.
the frankness and authenticity of this post in particular….thank you for that as well.
josh g.November 26, 2009 - 11:21 am -
Hey, I get intellectual crisis. I just like explosions. =)
Knowing that the snapshots are 1s apart makes a big difference; the lack of a clear time reference was one reason why I felt that video would be better. I totally agree that having printouts of the photos to write on is a plus, so maybe the ideal I-have-infinite-time-to-prepare-lessons version would use both.
I’m too sleep deprived to imagine exactly how this would all play out. Is there a specific intellectual crisis that these media uncover in students?
curmudgeonNovember 26, 2009 - 1:21 pm -
When you and a hobbit are being chased by a hungry dragon, you don’t need to outrun the dragon. You only need to outrun the hobbit.
Dan MeyerNovember 27, 2009 - 10:18 am -
@josh g., the part, I think, that excites me most in what’s a pretty average multimedia package is the moment where we toss up photo #1 and ask who is faster, Chris or Dan?
That’s one of the essential ingredients for the intellectual crisis: a question that’s impossible to answer, one on which everyone can speculate regardless of mathematical ability, one that illustrates a need for more information.
josh g.November 27, 2009 - 10:32 am -
Thanks, that makes sense. I think I was too used to your usual WCYDWT formula of only showing the first image. Good lesson for me there – focus on the opener and find the good stuff there first.