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.

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…

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.

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.

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.

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

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.

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.

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!!!!

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.

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.

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!

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?

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.

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.

So the world wants to know…what is the bean/house painting problem?

Trade secret. Got it.

WOW. You are changing my teaching. I read a ton of your blog this summer, and as it’s my first year with the capability to show video in my room, I figured I’d jump right in. Just finished showing my first class this problem, and “electrifying” is exactly the right word. I’ve never seen students so excited. Amazing. Thanks.