## WCYDWT: Boat In The River

As you’ll recall, I take kind of a dim view towards superfluous movie soundtracks, which is to point out that the music here serves an important purpose.

Step One:

• Play the video.
• Ask the students for questions that perplex them.

DimDim doesn’t save the entire chat transcript so I can’t quote the class’ questions exactly. Several concerned speed. One student asked, “If a crowd of people is standing on the escalator, would it be faster for Dan to take the stairs?” A conciser (though less interesting) phrasing of the same: “Is Dan faster than the escalator?”

The dominant question was, “How long will it take Dan to go up the down escalator?” This was, of course, an expected outcome of the video.

Step Two:

• Ask the students to guess at the answer to our question.
• Ask the students to set an upper bound on an acceptable answer.
• Ask the students to set a lower bound on an acceptable answer.

I think our highest estimate came from Sheng Ho who gave me two minutes to walk up the down escalator.

Step Three:

• Ask the students to define the information they’ll need to solve our question.

Jason: length of escalator
abarkley: your speed, speed of escalator
alemi-thevirtuosi: how many steps on the escalator
Mark Kola: rate of movement of elevator, avg step in length, and the number of steps in the escalator
Sandra Miller: speed of the escalator, speed you run up the stairs

At this point, I asked, “Which of these is easy to measure? Which of these is difficult? Which of the difficult measurements can we decompose into easier measurements?”

For example, the length of the escalator is difficult to measure — it’s too long — but counting steps is easy and measuring the height and depth of each step is easy. My speed up the stairs is difficult to measure — do I have a speed gun? — but its components — time and distance — aren’t.

And so it goes. I wonder what effect textbook problems like this one have on our teachers’ conception of mathematical problem solving.

It’s interesting to me, also, that no one answered, “nothing,” when asked what information they needed.

Step Four:

• Give them a pile of information to use as they see fit.

Step Five:

• Give them time to work.

This is the electric classroom moment, the payoff for all your groundwork in the first four steps. Your students formulated their own question, guessed at the answer, set two bear traps for wrong answers, and discussed relevant and irrelevant information. This is useful preamble and no one is jumping into the hard work without a sense of direction, or at least a sense that we value experimentation here. No one is calling you over saying, “I have no idea where to start,” like they do when you assign “problem twenty-five on page sixty.”

This is also the moment where the DimDim experiment failed. It was maddening, feeling separated by a glass monitor from all the interesting student work out there. It was maddening, watching students try to explain their work in a constrained little chat box when it would’ve been clearer and easier to slide that work beneath a document camera, or to bring the student up to the front of the classroom to explain it. Students could have been lurking in the background but holding an amazing method or a productive error and I wouldn’t have known.

I had no idea who was finished and who needed more time. (I called us back together too early it turns out.)

Group work was impossible, also.

Step Six:

• After they compute their final answer, ask them to compare it to their error bounds from step two.
• Compare the answer to our guesses from step two. Determine who guessed closest.
• Discuss sources of error.
• Discuss follow-up questions.

Miscellaneous:

• My intent was to transform this kind of problem into something less abstract. I didn’t. It’s interesting to me how drastically you can change the skill you’re practicing depending on what information you obscure and what information you reveal. I think I sorted myself out but I’m interested if anyone knows how I should have presented this problem if I wanted to assess systems of equations.
• I should have used Google Forms for the first two steps.
• In the second draft of this problem, I added the teaser footage where I start stumbling up the down escalator. In the third draft of this problem, I added the song I was listening to in my earbuds. Both revisions resulted from test feedback from Jackie Ballarini. (Thx, JackieB!)
• A debt of gratitude to Scott Farrar, whose awesome idea I totally ripped off.
• A debt of gratitude, also, to the manager of the local multiplex, who let me run around on his escalators.
• If you attended the live session, please post a review. Did it make this WCYDWT thing clearer? Less clear? What was satisfying or unsatisfying about the experience?

2011 Mar 13: I updated the problem package to include video answering Christopher Danielson’s question, “How long if Dan rides the escalator like a normal person?”

2012 Jun 16. Brian E:

By this time, the students were dying to see how close their results were to reality, so in addition to checking the answer video Dan provides for the actual time, we also used it to check their position equations.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### park_star

August 5, 2010 - 8:04 pm -

I suppose you could get worked up about not being able to make your own clips/scenarios/movies, but that’s what the internet is for.

IMO breaking down the problem like this is actually easier than teaching traditional “problem solving.” Everyone wins!

Looking forward to hearing about participants experiences.

2. #### Math Zombie

August 5, 2010 - 8:12 pm -

I LOVED this. It put me right in the shoes of your students. I had no idea ahead of time what problem I would be facing. I loved the group collaboration. Having us all work at the same time made me feel more “safe” because I could get confirmation of my own work and also get ideas from other people. It was really great how many different ideas we got doing this. Your original video pretty clearly wanted us to solve one particular question but in doing so we solved it multiple ways.

I think this particular problem jumps out at me as one of the most “useful”. It seems like you were walking through the mall, went on the escalator and thought to yourself, “I wonder how much faster the escalator makes me.” I think students can connect with this.

On a final note, DimDim seems pretty sweet. It would be really neat for an online class. I look forward to more WCYDWT problems.

3. #### Claire Thompson

August 5, 2010 - 8:28 pm -

I didn’t attend the live session, but I love how you’ve set this up here. I attempted to solve the problem and so when I watched the final clip I was *so invested* in the outcome.

4. #### Alvaro

August 5, 2010 - 8:34 pm -

My answers did not completely agree with the group when we started measuring the stairs. It was hard for me to explain my logic through typing while at the same time trying to watch the video and count the steps. Seeing different answers made me recalculate mine and re-measure a few times, which, unfortunately, allowed me to fall behind. I was so concerned about getting the correct measurements that I didn’t keep up with the class.

Using dimdim reminded me of working with friends on homework problems over instant messenger. I actually found that easier than dimdim, because it’s usually just one or two people, not 20, asking and answering questions at once. In my opinion, the chat was not the problem, but the amount of information flowing by. If I did not try to process it all at once, I might have been more efficient.

Either way, I really enjoyed the experience of solving the problem simultaneously with other people. It is much more fun than working alone. I wish I had interesting problems like these when I was learning…

5. #### Mr K

August 5, 2010 - 9:15 pm -

A) I found this one of your best yet. It transcends the WCYDWT because I don’t think you have to do much at all – it sells itself.

B)

I think I sorted myself out but I’m interested if anyone knows how I should have presented this problem if I wanted to assess systems of equations.

Ditch the timer on you climbing the down escalator. Pause after you take three steps. It’s enough of a setup to make their next question obvious, and you can replay the earlier sections to let them figure out all the data they need.

Added bonus – normalized units. The height of the escalator is immaterial, as long as it’s constant between all the runs.

6. #### David Cox

August 5, 2010 - 9:44 pm -

Using the music as a metronome…nice!

Two questions:

1. Are systems even necessary here? I approximated about 18 seconds to go up te down escalator based on the differences in time berween the stairs and escalators. (assuming the long stair is negligible as I’m not ready to wrangle this problem.) Keep in mind that it’s late, so I may be missing the obvious here.

2. What’s your endgame with the online presentation of WCYDWT? Are we talking about how to present better problems or create them? I think the paradigm has long shifted when it comes to presenting problems in such a way that it allows students to become part of the formulation process. It’s the creation of these problems that hangs me up.

It still kills me that I have Photoshop and After Effects on my school computer, bus still don’t know how to do what you just did.

7. #### Debbie

August 5, 2010 - 11:05 pm -

Have only skimmed this (am savouring it for when I have more time) but this strikes me as one of your best yet Dan!

I wonder how many of us have been thinking, “how would this work in a classroom? How would I structure this approach?” about WCYDWT. This post provides me with a format for such lessons.

Thank you.

8. #### Mr. H

August 6, 2010 - 12:21 am -

I must say, as someone living in Southern California, I am at a severe disadvantage for this question! I don’t think I’ve used an escalator in the last 2 years (it’s an indication of how much I shop at malls). I have no sense of how long an escalator ride should take. Give me a traffic problem! :)

For that 2-minute estimate, I got demoted to pre-algebra by Dan. :(

Probably the best thing about this activity is that students learn that reality is messy. Mathematics can be used to model reality. The precision and accuracy of our measuring instrument, namely our ability to count accurately while watching the timestamp on the video or the timecode on the vimeo player, were also tested. I must have watched those videos at least 5 times. I lost track quite a few times in my haste.

A few notes:
-Watching, pausing, and counting is not easy. Slow motion may be useful here. I thought I was pretty tech-savvy, but I guess this is a case of auto engineers aren’t always the best drivers.

-The participants today (I’m guessing) are ex/current/future math teachers or mathematically inclined people. We breezed through many parts of the activity. It seemed like we always knew what to do next. Maybe you can comment on the difference between working with a class of high school students and working with our class.

-Switching back and forth between the chat window (dimdim) and vimeo made it difficult to keep track of conversation. Too much text in too short a time, especially when we had longer answers. I like that you called on a student rep (I think it was alemi) to give an explanation so we wouldn’t drown in noise.

-It’s difficult to explore other solutions when the majority of participants already have one set way of solving. The data on the board that the class agreed upon suggests one method of solving over others.

Questions for you:
1) Is this something you do after students have learned the relevant concepts/skills and need an opportunity to apply them or do you use this as an introduction to the concepts/skills?

2) I would be interested in knowing whether the fact that you star, direct, and produce this video have an effect on student engagement? Would your students be as engaged if it were a video produced by me or by a publisher?

Possible extensions:
-Start from the top of the escalator and walk “up” but at a slower rate than the escalator is going down. How long can you remain on the elevator? How long before you reach the bottom?

-For systems of equations. Remove stairs. Leave only escalators. Walk up on both. One will be t_1(x+y) = d the other will be t_2(x-y) = d. Distance d traveled is the same. Time is obviously different. x is your rate. y is escalator’s rate. If I didn’t make any mistakes, distribute and you get the standard system of equations.

9. #### Alex

August 6, 2010 - 1:28 am -

I couldn’t think of a ‘nice’ way of doing this. But I did figure out how to do it with just the info we already had (time to walk up the stairs and the up escalator). We need speed = distance/time.

The three situations are
1. Walking up the up escalator
2. Walking up the stairs
3. Walking up the down escalator.

Time 1: ~7s (just by eyeballing it).
Time 2: ~12s
Time 3 is what we want to know.

The distance doesn’t matter (it’ll cancel out), so I’m going to pretend it’s 84m because that makes the sums easier.

Speed 1 = distance/time = 84/7 = 12
Speed 2 = distance/time = 84/12 = 7
The *elevator’s* speed is 12 – 7 = 5
Speed 3 should be 7 – 5 = 2.

Time 3 = distance/speed = 84/2 = 42 seconds.

10. #### Alex

August 6, 2010 - 1:36 am -

Addendum: turns out my answer was completely wrong. I guess that’s what you get when you eyeball your times… pausing the video it looks like it’s more like 8 seconds to walk up the up escalator.

So the escalator’s speed is 96/8 – 96/12 = 4
The speed up the down escalator is 96/12 – 4 = 4
The time up the down escalator is 96/4 = 24.

Much more acceptable – but also illustrates how just a tiny error in measurement can make a huge difference to your answer.

11. #### David L

August 6, 2010 - 3:41 am -

Sorry I missed out – I confused the date. Thanks for trying to make something interactive.

12. #### Justin Lanier

August 6, 2010 - 6:20 am -

“It’s interesting to me, also, that no one answered, ‘nothing,’ when asked what information they needed.”

Along what lines do you find this interesting? Because you think the fact that people universally asked for more information shows the engagement and real mathematical thinking that you’re looking for? Because you think the situation can be successfully analyzed without more information? Or some other reason?

My thought process when I watched the video and wanted to know how long it would take you to go up the down escalator (after a few false starts) was:

I counted how many steps you took when climbing the up escalator and got about 21.

I counted how many steps were on the escalator when it was paused and got about 28.

That meant that your speed way 3/4 you and 1/4 the motion of the escalator. I reasoned that if the escalator was working against you, your total speed would be 3/4-1/4=1/2 of your original speed. That means that your “against” trip should take about twice as long as your “with” trip, which was about 9 seconds long. So I predicted that your up-the-down-escalator trip would take about 18 seconds.

The most important thing for me to come out of this experience was observing myself on edge as I watched the answer video. I think I’ve read your WCYDWT posts before from the standpoint of a teacher, but this time I came at it like a student. And I was *so invested* in finding out what the answer was. So while I’ve always been attracted to the style of problems you’ve posed, it’s giving my students that feeling that has me really jazzed.

13. #### Alex Eckert

August 6, 2010 - 7:46 am -

You’re killin’ me smalls.

About 3 months ago I took my wife to the mall and had her videotape me walking up the up escalator, then running up the up escalator. Then walking down the down escalator, then running down the down escalator. Then, of course, running up the down escalator.

A day or two later I went into another math teacher’s classroom to discuss. I knew it had something to do with speed and how long would this take and all that, but I was looking too much into it. We didn’t come up with anything tangible, but I KNEW there was something there. You nailed it. Nice job. And great editing.

14. #### Dan Meyer

August 6, 2010 - 8:23 am -

Mr. K: Ditch the timer on you climbing the down escalator. Pause after you take three steps. It’s enough of a setup to make their next question obvious, and you can replay the earlier sections to let them figure out all the data they need.

What I’m saying is that I thought this would set up a systems of equations problem. Instead it sets up a problem where you find some rates and subtract them, then divide the rate into 1 escalator unit. (Love the normalized units, thing, BTW.)

The systems of equations problem actually requires an entirely different introductory video.

David: Are systems even necessary here?

See the first footnote under the “miscellaneous” heading.

David: What’s your endgame with the online presentation of WCYDWT? Are we talking about how to present better problems or create them? I think the paradigm has long shifted when it comes to presenting problems in such a way that it allows students to become part of the formulation process. It’s the creation of these problems that hangs me up.

Teacher One: “You should check into this WCYDWT Online thing tonight. It’ll take you thirty minutes and it’ll change how you think about math teaching.”

Teacher Two: “I’ve got thirty minutes. I’ll do it.”

Whatever format WCYDWT Online takes, it needs to pay off for that teacher to some degree in thirty minutes, not three years. Maybe that’s impossible. I know it’s impossible with my original blog format. This particular blog format seems to have paid off for several readers, though. Maybe I’ll run with that. My failures are pretty productive lately.

David: It still kills me that I have Photoshop and After Effects on my school computer, bus still don’t know how to do what you just did.

Step One: Put a camera on a tripod or on something sturdy.
Step Two: Press Record.
Step Three: Climb up stairs and escalators. Try not to cross paths visually with yourself. Stay on your own side of the screen.
Step Four: Let me know when steps one through three are done.

Thanks for the feedback, Sheng. Useful stuff.

Sheng: Is this something you do after students have learned the relevant concepts/skills and need an opportunity to apply them or do you use this as an introduction to the concepts/skills?

Both. Some skills need more direct instruction than others, but after they get a perplexing question in their heads, students are asking me for those skills, at least, rather than me delivering the skills whether they want them or not, whether there’s perplexing context or not.

Sheng: I would be interested in knowing whether the fact that you star, direct, and produce this video have an effect on student engagement? Would your students be as engaged if it were a video produced by me or by a publisher?

I’m sure it doesn’t hurt to see your teacher in the video however Graphing Stories was (reportedly) popular with a lot of students even though they didn’t know the guy climbing the stairs.

It’s true, though, that my students tend to find video content produced by publishers uninspiring. I chalk that less up to “unfamiliar actors” and more to “stock footage, stock music, stock photography” all of which is aimed at a professional product, but which alienates the viewer all the same.

Sheng: For systems of equations. Remove stairs. Leave only escalators. Walk up on both.

Nailed it. The only thing I’d say is that you want to end the “question” video with some teaser footage where I start to walk up the stairs, since one outcome of the problem is my speed in still water (so to speak) and we need to inspire that question, somehow. Otherwise, the students have no idea why they’re creating systems of equations.

Alex: Much more acceptable – but also illustrates how just a tiny error in measurement can make a huge difference to your answer.

Awesome. Thanks for modeling your process there, Alex.

Justin: Along what lines do you find this interesting? Because you think the fact that people universally asked for more information shows the engagement and real mathematical thinking that you’re looking for? Because you think the situation can be successfully analyzed without more information? Or some other reason?

Maybe all of the above. It’s interesting to me that our first reaction to a mediated space was to start acquiring physical dimensions, whether we needed them or not, almost like blind people feeling for the edges of a room. It’s a video. It’s 2D. It’s flat. That was irritating somehow and people wanted to resolve that irritation.

Justin, the most important thing for me to come out of this experience was observing myself on edge as I watched the answer video.

Yeah, awesome. I need to find some way to quantify this because, anecdotally, it has a huge effect when students watch the answer versus hearing it read from the back of the teacher’s edition.

Alex Eckert, great minds, etc.

15. #### Steven Peters

August 6, 2010 - 11:44 am -

I really enjoyed the video, though I immediately jumped into the details (how fast is the escalator, how fast is Dan going) without noticing the killer question of how long to go up the down escalator. Once other people asked that question, though, it sounded good.

As it started I was thinking about speed, and then Dan gave us some measurements of the stairs and some extra video so we could count them all. I was initially thinking about computing a linear velocity, but the fact that the stairs were different sizes gave me pause. He said that he used the music as a metronome so that his steps would happen at the same rate (I measured about 2.5 steps / second). Imagine two stationary staircases of the same height, one with big steps and one with small steps. If he doesn’t take the small steps two at a time, then the one with small steps should take longer. With different step sizes and a constant stepping rate, I think Dan will have a different linear velocity on each set of stairs.

Someone else mentioned counting steps, and that sounded like a good idea to me. So I counted about 32-33 steps taken on the stairs, versus 21-22 steps taken with the escalator. There’s a difference of about 11 steps between the stairs and the escalator, so my first thought was to just add that many steps to the number taken on the stairs, which would yield 43-44 steps. I multiplied this by a stepping rate and guessed 18 seconds to reach the top. I don’t think this is a mathematically valid way to do it, but it was my first guess.

I counted the steps taken during the Answer video, and it looked like 55 steps. I think the number of seconds mentioned for the stairs was 12.9 and for the escalator 8.7, while going against the escalator was around 22 seconds.

Depending on the values measured, the stepping rate is very close to 2.5 steps/second.
55/22.0 = 2.5
32/12.8 = 2.5
22/8.7 ~ 2.53

33/12.8 ~ 2.57
21/8.7 ~ 2.41

There’s some interesting aspects of measurement error and error propagation through multiplying and dividing measured values that reminds me of some exercises from physics.

So this is part of my approach, though I think it needs a final step to more accurately predict the number of steps taken on the final escalator, basically to derive the number 55 from the numbers 31-32 and 21-22. I’m still thinking about it though.

16. #### David Cox

August 6, 2010 - 1:38 pm -

I dunno, Dan, the online stuff is having more of an effect than you think. I started sending out links and such to all the math teachers in my district last year and eventually sent out your Ted talk (that one made it to out IT director as well). It has my department thinking and within a week my IT director was in my classroom to chat about ways we can encourage teachers within the district to rethink how we do stuff. The conversation was productive enough that I’m going to be out of the classroom part day and have the chance to get into others’ classrooms.

Your homerun problems have a huge WOW! factor but they may scare some folks off as they aren’t free nor are they easy. But they get people thinking.

17. #### Mark

August 6, 2010 - 3:18 pm -

I really appreciated the exercise. It served three purposes for me.

1. My dad taught math for 33 years. I learned a lot about how he approached problems during that time. I wonder what he would do with technology and teaching today. I think that even though the technology is out there, it is utilized so little.

2. My daughter is taking summer school to advance a grade in math. After comparing the coursework with this one example, I hope more teachers follow suit.

3. My biggest value from this exercise is applying the principles to developers. I will be utilizing many of these techniques to expand innovation, creativity, and a general passion for solving problems. Well done!!!!

18. #### William

August 7, 2010 - 5:16 pm -

Hey Dan,

Little late with the comment, but I found the dynamic interesting in that I found myself holding back for fear of making a mistake and looking silly. Especially given the fact that the problem “seemed” easy so I kept thinking there must be a catch.

This is so different from the way kids are taught that I think you have to build the trust necessary to really make this work. In high school we’re working against 8-10 years of indoctrination about the way things are supposed to be done.

Wm

19. #### Jason Buell

August 7, 2010 - 9:59 pm -

Thanks for doing this Dan.

Those of you downloading the video, I apologize for the occasional chop. i basically had to drop out and just watch because when I’d do anything my computer would freak out.

As for the session itself, it was definitely a step up from just reading the blog.

I was there mainly to get a feel for the flow of the whole thing rather than solving the problem. As such, I might have a slightly different perspective than others. My comments are specifically directed towards the session as PD, not the problem itself.

I’m also going to try to approach this as someone who hasn’t been reading your blog for years.

I think, if I were brand new, I would have preferred a quicker run through of the start to finish process with probably more time for you to talk about why you set up certain things in specific ways. For example, the iterations that the video went through before you landed at that version. The advantages of using the escalator vs some other setup. Showing the river boat textbook problem and talking about the differences between the two. What is the thought process you went through even before filming?

So my main recommendation: It was 20ish teachers/math people who are regular readers and know your stuff. It was clear (at least it appeared to be clear) your motivation for setting up things in the way you did. I know why you ask for lower/upper bounds. I get the specific advantages of the format vs live vs textbook. I got the tap at the beginning, the music, the advantage of an escalator vs say skateboarding kids or a moving walkway, etc. I don’t think a non-regular reader would get all that. The advantage of having you live would be a more interactive Q and A for why each section is how it is. Without that, I could just watch the screencast.

20. #### Stacy

August 8, 2010 - 3:46 pm -

Finally getting around to watching this whole thing and WOW! The thing you did that I would not have thought of was the video and picture measurements. If I could ever have dreamed of this problem and the setup on my own, I probably would have taken the measurements myself, written them down, and simply told students when asked. The way you did it is much better. Now they’re just one step from having been there with measuring tapes themselves. Please give us a great and rigorous geometry problem next. :-)

21. #### Dan Meyer

September 16, 2010 - 5:51 pm -

a) A propos of Mythagon’s post – were you trying to walk to the beat?

That was the point of the music, which would otherwise be the sort of affectation I try to avoid. I think I’m off by a few frames on the down escalator, though, which isn’t to say that I missed the beat, just that I aligned it poorly in post.

I’d be real curious how this runs with your preservice teachers. I’m particularly interested in how many watch the video and wonder to themselves how long it will take me to go up the down escalator.

22. #### Darcy

September 18, 2010 - 3:10 pm -

So when I approached this problem my first thought was to think what is the equivalent distance in steps and then divide it by your walking speed since it seems to be fairly consistent. I counted that for every 32 steps you went up the escalator would bring you 21 steps down so equivalently this is the same as taking between 60 and 66 steps up the escalator. Your walking speed I found to be between 2.27 and 2.57 steps/sec. This gives the answer a range between 23.24 and 29.07. There was a lot of guesswork involved with this. It was a neat problem and I am using it for a reflection piece in my curriculum class! I hope you don’t mind.

23. #### Ben Blum-Smith

October 4, 2010 - 7:27 am -

Hey Dan, so

a) With the 12 preservice teachers, the natural current was strongly toward “how long to go up the down escalator?” When we asked for questions, we got about four suggestions, of which that was 1 (the other interesting one to me was “how fast is the escalator going?” – I forget the other two). I asked, with due self-consciousness about the morbidness, “If you were going to die immediately after class, which of these questions would you be most upset about not getting answered?” Something like 10 of the 12 of them voted for the intended question.

b) About 1/3 of them caught the walking to the beat thing. I didn’t point it out till the post-mortem. I can’t say why more folks didn’t catch this, but why I myself didn’t catch it I can speculate: I was strongly inclined toward boat-in-the-river logic when I approached the problem, a fundamental assumption of which is that the boat’s speed relative to the water is a constant. This assumption is false in the video (though not disastrously false) because since the escalator steps are bigger than the stairs, and you’re taking them at the same steps/sec rate due to the music, your speed on a still escalator would be greater than your speed on the stairs. My brain actually tried to preserve this assumption, at least approximately, by making me think that you were going up the stairs at a higher steps/sec rate than the escalator. I mean I literally thought that’s what it looked like. So I couldn’t bring myself to believe you were stepping to the beat until you told me so.

c) While I’m with you on the general principle of affectation-avoidance, I think you might be underestimating some significant value, beyond the walking-to-the-beat thing, that the music is adding to the video: i) it just definitely makes it cooler. You inevitably boogie a little in your heart as you watch. This could have been a distraction, but it’s not, at all, because ii) it participates in the storytelling. It’s so driving, it totally suggests “something’s going on here, what is it?” Like both a race and a mystery. To me, anyway. More importantly, the moment the clock starts running is the moment the beat drops! This totally adds to the like click-of-the-stopwatch, gunshot-to-start-the-race quality of that moment. As above, I think the natural current toward the desired question is already very strong, but if a person wanted to make it even stronger (cf Mythagon’s post), one way I could imagine doing that is to stop the music after the four copies of you do their thing, and then have the beat drop again when you start to head up the down escalator.

d) While when I first looked at it it seemed to me that this problem is a middle school-thru-algebra I problem, now I think this problem could actually also be done productively in a calculus class! When we did it, 3 of the four groups came up with what appeared to be mathematically more-or-less sound methods (the more-or-less b/c 2 of them made the incorrect boat-in-the-river assumption I described in (b)) producing answers that were off by 50% or more, while 1 group came up with a mathematically naive method that produced the only reasonably close answer. (13 sec on stairs, 8 sec on escalator; 13 – 8 = 5: going with the escalator took away 5 seconds, so going against the escalator should add 5 – 13 + 5 = 18 sec) Why were the more mathematically sophisticated answers so off while the wrong-ish one ended up being right-ish? Because all the sophisticated answers were functions of the measurements that had very high derivatives, so small errors in the measurements (and perhaps the inaccuracy due to the false boat-in-the-river assumption) led to big inaccuracies in the answers. Meanwhile, the unsophisticated method is a linear function trying to capture a nonlinear situation, but I suspect (haven’t checked) that it is actually a linearization of the correct function, so no big shock that it produces a pretty good answer. How mathematically rich! I fully didn’t expect all this.

24. #### Dan Meyer

October 4, 2010 - 4:50 pm -

Thanks for recapping this one, Ben.

With the 12 preservice teachers, the natural current was strongly toward “how long to go up the down escalator?” When we asked for questions, we got about four suggestions, of which that was one.

I’m curious if you controlled for group bias. Something I enjoy about the WCYDWT live sessions is that we all hit “enter” at the same time on our questions, which mitigates against a person’s tendency to follow or subvert a crowd.

I think you might be underestimating some significant value, beyond the walking-to-the-beat thing, that the music is adding to the video: i) it just definitely makes it cooler.

Agreed, but I don’t want my students’ engagement with math contingent on a peppy soundtrack. This strikes nine out of ten of my readers as overly picky and I agree it’s not the biggest deal, but everything counts in the long game. If I had to do it again, I’d probably just dub in a metronome.

… all the sophisticated answers were functions of the measurements that had very high derivatives …

Can you help me out with this?

Pretty obviously, I need to find one of those moving sidewalks things and run it again.

25. #### Marcie

October 8, 2010 - 7:02 am -

Thanks for a great lesson! I teach physics and have been able to adapt some of your things (this lesson in particular).

Thanks for inspiring us edcuators to put in the extra bit to make sure we have the kids thinking critically!

26. #### Murf

October 24, 2010 - 1:31 am -

27. #### Murf

October 24, 2010 - 1:49 am -

PS: I dig the beat in this video (and understand its purpose). A few students and I can’t get it out of our heads. Did you make the music in Garageband (or equivalent) specifically with this video in mind, or did you get lucky and find something you could use?

28. #### Dan Meyer

October 24, 2010 - 7:06 am -

Hey thanks for the feedback, Murf. In general, I’m not too particular about credit. I think your concern about students linking back to the answer is warranted. You can feel free to toss my name down with no link, if that’d make things easier.

The audio track is the instrumentals from Drake’s track “Over”, which probably isn’t something you’d want to play in class.

29. #### Tao Wang

February 9, 2011 - 12:34 pm -

I’m a little late to the party, but I just wanted to thank you for the video. Yes, it’s not perfectly relevant to systems, but it did get my students thinking about what your speed is on the escalator — which is a big step in parsing those boat in the river problems.

30. #### Max

March 4, 2011 - 6:50 am -

What would the question video need to do to make this a problem about rational functions?

I know there’s a rational function problem involved in the question of planes flying on windy days, the outbound trip being into the wind and the inbound trip being with the wind.

Oh! What about if you asked if the total time for running up and down the escalator was the same as or different from the total time running up and down the stairs. What would make that question pop out?

Max

31. #### Brian

March 21, 2011 - 2:49 am -

OK. So maybe no one will look at this because this post is from so long ago. And maybe someone already said this. I tried to scroll through the comments to see if anyone else noticed this, but it is like forty some comments. But has any body thought about HOW similar this is to the painting the house problem that Dan turned into putting beans in the cup problem?!?!!

Going up the down escalator is like having someone paint the house while someone else removes it! Or someone filling the cup with beans as someone is taking away beans.

This also helps to connect what Alex said about the distance not mattering. The size of the house or the size of cup doesn’t matter. It’s all about the times, because the amount (distance, area, or volume) is fixed.

Makes me think about new questions about the Bean question: How long will it take them fill if someone starts being a jerk and begins taking beans out at the rate they had been filling? What if they decide to be a jerk after the cup is half way filled? Depending on who decided to be a jerk, would the cup get filled or cup get emptied?

Sorry if this is a repeat of other people’s insight. But I was excited, and had to share.

32. #### Dan Meyer

March 22, 2011 - 4:59 am -

Good eye, Brian. I don’t think the bean counting problem is a matter of public record here but, yeah, both problems require a reconception of speed. Escalators per second in one case. Cups per second in the other. (Or percent per second, if you prefer.) If you can get students to that reconception — rather than giving them a formula that’s good for a single use-case — they can handle all kinds of variants and extensions.

33. #### Dan Meyer

March 23, 2011 - 3:56 pm -

You’ve seen the house painting problem: “if I paint a house in four hours and you paint a house in six hours how long will it take us to paint the house together?” As for [WCYDWT] Bean Counting, if I posted all my curricula then I wouldn’t have much to talk about in presentations and workshops, I suppose.

34. #### Christopher Danielson

March 23, 2011 - 6:19 pm -

A friend and I have had many arguments about these house painting problems. She maintains that they are an application of the work equation. I maintain that the work equation is ridiculous and only good for these absurd problems; that instead when students bump against these problems (unavoidable in our Math Center curriculum, I’m afraid), we should encourage them to use whatever strategies make sense to them.

But if they don’t use the work equation on these problems, what is the point of having them in there? she asks.