In which British comedy is investigated

While my grades in Algebra 1 and 2 may lead one to believe I knew what was going on, those classes were agonizing for me. I can still vividly remember the frustration I felt because nothing seemed to make any sense and my ability to memorize steps is sub-par. To this day I find it easier to remember only the formula for the volume of a sphere and then use calculus to find the formula for surface area rather than actually memorize to formula for surface area.

Strange, I know. But it works for me.

In my last post I was thinking about what it’s like to not know something. But in addition to not knowing, I think the utter confusion one can feel in a classroom is also something important to keep in mind while teaching and lesson planning and working with students. To that end my mental index popped out the card for a skit from the British sketch comedy show That Mitchell & Webb Look called Numberwang. Yes, yes, I know it sounds like I am leading you to a dark place on the internet, but I’m not. Look:

There are other episodes along with A History of Numberwang, which I recommend watching as well. Go ahead. I’ll wait.

I asked some tweeps if they had seen the skit before, and Carl Oliver came back with this:

As that kid without conceptual understanding in algebra, this skit is pretty much exactly what it was like in class for me. Confusing, almost no stated rules I understood, and at any moment the scene might change or I might be shoved in a box for not achieving Wangernumb.

Next time I go to make lessons for others, I need to keep this skit in mind and think about how I can plan for conceptual mental grappling and not just learned memorized performance in front of a live studio audience.

That’s Numberwang.

Why I stopped putting grades on papers

I’ve been poking at this video for a while as something for Mathagogy, but it’s closer to 4 minutes than 2 and not so much about math as it is about grading. As such, I’m going to put it here and do something else for the 2 minute math project.

PCMI instigated a lot of changes in my classroom. This video is about what I believe was the biggest change and something that I think caused a lot of positive cultural shifts: I stopped putting grades on the papers I handed back.

The paper I talk about is Working Inside the Black Box: Assessment for Learning in the Classroom by Paul Black, Christine Harrison, Clare Lee, Bethan Marshall, and Dylan Wiliam.

I used SBG as my quiz structure in all classes, so students were regularly getting 2-4 question quizzes handed back (I’m not a fan of collecting homework). Prior to the article they had a number corresponding to the rubric I used and some comments. After the article they had no numbers/grades and all of them had comments (right or wrong, often in the form of a question). The change in discussion post-handing papers back was huge, but it did take time to happen and was just one part of my ongoing campaign to figure out ways to get students talking math and recognizing each other as fellow learners.

Not putting grades on things that students expect to have grades is a big shift. There’s backlash from the kids to deal with who were used to getting instant status from their grades to deal with and it takes time to change culture. Do I recommend you go and try this right now at the end of the year? Not really. Do I think you should consider this as a change for next year? Definitely. Post any questions you have in the comments and I’ll get back to you. Thanks for watching 🙂



Follow up: Emily Steinmetz over at Crazy in Math gave dropping the grades a go in favor of some student grading. Read her breakdown here.

Do you Math Forum?

The Math Forum does a lot of cool stuff. Just click the link and look at some of the tabs. If you’ve never checkout out their Problems of the Week, I highly recommend.

I bring this group up now because I’m doing an online course with them on Differentiated Math Instruction. While working through the reading, I came across this sentence:

Well-chosen problems can often replace regular exercises and assignments and accomplish multiple curricular goals more effectively.

I am very interested in the selection of problems to replace the hum-drum ‘turn to page p394, complete 1.31 odd’ where possible. Interaction, debate, making links across topics (within and outside of math!), and problems that can engage students coming from multiple backgrounds are all things I want in a classroom. Looking forward to the rest of the course.


I do what I want! (and what Anne recommends)

So Anne put up a challenge and I thought ‘why not?’

Last Saturday marked the end of a project that had been haunting my steps for a while and being free of those shackles makes the whole world seem just a little bit brighter.  This next month will be a time of significant information consumption as I work through a bunch of NCTM books (Principles to Actions, Visible Learning, 5 Practices, Fostering Algebraic Thinking), read lots of blog posts, learn how to use an iPad (hello, dark side!), prep for PCMI, and develop some PD sessions for middle grades and the NCTM HS Interactive Institute.

So for the next month I’m going to try to dedicate time every day to something small and interesting that I have read/found/am playing with.

Today: Henri Picciotto’s Acceleration Posts (1/30)

I’ll start out with stating I am not a fan of separate honors classes in school and that acceleration makes me twitchy. I was accelerated along the honors math track and I developed some great rote learning skills that completely failed me in Calc 1 in college as my most excellent prof would have none of that (Thank goodness for Russ Gordon). I became a math major not because of my high school classes but rather because of an amazing 6th grade math experience and Russ Gordon’s class.

Henri lays out some discussion-worthy ideas around acceleration and honors. Tradition and parents who want sparkly transcripts are a huge challenge here. Colleges that only look at transcripts and not the actual kids are also a problem. Had a I math department, I would be interested in having some group discussions around these ideas. As it stands, I’m going to bring these to PCMI and try and get a conversation going to help further my own understanding and get others thinking about what acceleration means.

So go read Henri’s 4 posts on acceleration (I also recommend the rest of his blog). I love posts that pose questions that would shake up tradition and status quo. If we can’t defend what we have, perhaps we shouldn’t keep it? And not keeping something means working to figure out a replacement that is better, not just ‘something else’.

in which I hear from a former student

Like many teachers, I occasionally get notes or drop-ins from former students. And I don’t care how old they are; they will always be my kids. With the advent of the Facebooks, I have several that I occasionally hear from after they’ve graduated. I especially like the ones that apologize to me for not taking more math in college. Kinda adorable.

Last night I got a missive from one of my darlings worried about her math final. If she doesn’t pass it, she’ll fail the class. This is her first semester in college. The note was short, and she ended it with

Why am I so bad at the maths?! D:

I wrote back the following:

For me, maths was more about accepting a different way of thinking than anything else and that can be the hard part. We’re not brought up in a logic-based society. So much of high school math is focused on procedural thinking–if A, then do B & C, and ta-da! And that’s not really math. That’s more like advanced baking. Useful, to be sure, but not a way of thinking. Since I don’t use those types of problems as my main push in classes it’s also why the typical A-maths kids don’t like me for a few months–they’ve never really had to think for understanding before.

And that right there is the key. Maths makes sense. It’s the Queen of Science for a reason. If you think what you are doing in math does not make sense or is magical, than we need to figure out a different way for you to think about it. Luckily, there are a lot of ways to think about maths that are successful. Unluckily, finding the way that works for your brain can be grueling.
I got through high school using procedural skill (the If A, then do B stuff). I hit a brick wall in my first maths class in college because it didn’t work with that professor. I had to show him how I was making sense of the maths and since I wasn’t really making any sense I didn’t have anything to give him. There were a lot of nights spent in the math study room and I didn’t really get the thinking needed to understand, but I was moving that way. Slowly.
Sophomore year was when it was finally clicking that I needed to build my own understanding so I could stop playing the memorization game (it wasn’t possible to pass some of my classes via pure-memorization as no ones memory is that good. And I was taking Latin at the time so my memory banks were full up with vocabulary).
So yeah, maths is hard if you’re taking the memorize ever permutation of a problem and how to solve it route. Figuring out the patterns behind the maths is a challenge, but it pays a lot of dividends. If you ever want to chat about your stuff, just let me know. I have skype and I use google hangouts quite a bit for my work. Just remember: I am 3 hours ahead of you 🙂

What I did in class wasn’t enough to get her to where she needed to be to be successful in college maths. I still focus too much on the procedural at times, but every year I moved more and more away from that as I built up skills toward a teaching style I never saw in high school maths. I’m curious to see how much my classroom skills will atrophe while I am out of one or if the level of ed-research, blogs, and consulting work I do will help me hold steady. One can only hope.

I miss my kids.

in which i give a talk on the profession of teaching

IM&E hosted a thing at Berkeley October 12-14. I was working with middle grades folks and asked to give the final plenary talk entitled ‘Call to Action’. I chose to talk about the profession of teaching and how I think we get more teachers engaging with teaching as professionals. I’ve tried to type up what I said in the talk based on my copious notes, powerpoint, and memory below the cut. I know it’s not exact and I suspect my memory is editing to make me sound better, but I don’t have a video recording (thank Gauss) so it will have to do. I’ll warn you it’s longish, but I would love to hear your thoughts on professionalize and education in the comments.

Oh, and this is the tweet that spurred much of my ideas for the talk. Or rather, had me re-writing much of my ideas for the talk.

Continue reading

in which i adore Peg Cagle

My interpretation of her words regarding teachers as professionals:

Don’t let anyone call you a ‘natural’. They may mean it as a compliment, but it diminishes the intellectual work that we do as professionals to become as good as we have to be to educate our children.



in which I muse on the fight

I consume a ridiculous amount of information daily. Most of it related to the things I enjoy (geekery, math, teaching, technology, art), some of it social (Facebook, twitter, tumblr), and some of it relevant to the day (news, weather). Today I came across a new book that I need to buy 20 copies of for my friends and all their kids: Heros for my Daughter.

The selling point for me was this note from the author:

As you’ll see, every hero in this book is a fighter. And as I tell my daughter in the introduction: “No matter what stage of life you’re in, when you want something—no matter how impossible it seems—you need to fight for it. When you believe in something, fight for it. And when you see injustice, fight harder than you’ve ever fought before.”

So what do our students have to fight for? And do they understand that life is just one giant battle, but it’s totally a battle worth fighting when you find those things that Light Your Fire–those things you would do in your free time when you can do anything you choose?

I can arm them for battle, but I’m not sure how to encourage them to take up their own battle cry and charge at Life. I feel all I can do is be joyous in my own life because I do know what Lights my Fire and I do know how to sound my own barbaric yawp at the world. Perhaps that is what it means to be an adult–to take part in a fight worth having.

p.s. My childhood hero was, and still is, Jane Goodall.

SBG: a reason I <3

We’re a bit over halfway through the year here and the midterms came up for 4 of my classes. Thinking back to the line of students waiting to retake skill quizzes at the end of last semester, I decided to give the kids a two-weeks-notice to get all their retakes in for the first 6 skills of the year.

This partly goes against the grain for me. I like time being a variable. I like being able to sit down with kids and have them show me that they ‘get’ the concepts and who cares if it took them a little longer. But there is also the practical me dressed in beige and pearls with a clipboard on my shoulder saying am but mortal and having students from four classes wanting to retake skills at the end of the year because they don’t know how to not do things last minute is a Bad Idea.

Yes, it is definitely my fault for not instituting a ‘only one skill a day’ rule (though I do have the ‘you may only try this skill once today’ rule).  I really need to do that next year. But then would their grades reflect their organizational ability or their mathematical knowledge?  Dog, meet Tail. Tail, Dog.

Anyhoo, in the past two weeks I’ve seen a lot of my students outside of class. I’ve had great conversations and learned so much more about what THEY know based on the skills they sunk on and the skills they flew over. On the progress reports that went out a bit ago every student had a comment with what % of skill quizzes they needed to come in and work on by today. I’ve harrassed kids in class, in the halls, and over emails. My class skill chart will have a lot more blue’s on it come Monday when I get around to updating it (blue means >90% average in the class).

But better than anything, I feel like my skill grades are more accurate now. In example: expected value (skill 17): not too good. I really need to work on that unit and come up with something better for next year because it felt like a total swing-and-miss this year and their skill scores reflect that. Writing and solving permutations (14), combinations (15), and isomorphic situations (16): awesome. Need to keep that around. Possibly should add a joint skill where kids have to write/solve a situation that demands both types of counting. Needs must ponder.

At the end of the day, I have these little slips to mark what was done. These are from the last 24 hours and represent every skill retake. It’s been a good day.


WCYDWT Response – Boat in the River

a.k.a. Where is the boat?  Find the river.
Warning: This is a 4-day lesson plan/description of how it went for my three precalc classes.  In other words, it’s a bit long.  Just letting you know.

Intro: All teachers have had horrible days.  No matter how well planned you thought you were, some days are just epic fails.  Technology can fail.  Students having bad days can lash out.  You can notice gaping holes in your Plan of Awesome™ as you are teaching it.  All of the above have happened to me.  This is not about one of those days.

I was fortunate enough to take part of Dan Meyer’s second DimDim session surrounding the book/movie Holes.  Being able to experience live his style, seeing a video, and thinking about the questions that could be ask I found myself most interested in the variety of questions the participants came up with.  So many of them could be followed to interesting mathematics and I was a bit sad to not have more time to do so.  When the Holes session was over I realized there was a whole other DimDim session I missed out on and there was a boat and a river involved.  Clearly I wasn’t checking my reader enough this summer.  I blame[thank] PCMI.  And Hiking Lots of Hiking.

After actually watching the video and reading the post I realized there was no boat or river.  Thanks to Jason Buell I was able to watch a screen capture of the whole thing and it set my brain ticking about having the audience propose the questions from the video.  Dan had a virtual session full of teachers do it and most came up with “how long will it take Dan to get up the down escalator?”  The question seemed obvious when you stopped the video as Dan’s a few seconds up the down escalator.  Obvious to me as a trained educator.  Apparently it wasn’t as obvious to a live session of high schoolers.

Day 01

The first day of school.  I have three Precalc classes back to back at the end of the day.  After the regular hullos from myself and my Teacher Candidate (TC, think student teacher but better trained and using a co-teaching model from the start of the year), we introduced them to Dan and then had them watch the video twice—stopping when Dan’s about 4sec up the down escalator.  The first time they were instructed to watch and think about what type of questions they could ask.  The second time we asked they jot down observations about the video to share with their table group.  My classroom has eight 4-seat groups that we put a quarter sheet of paper on and asked that in 5 minutes each group write down the question they would ask and the information they would need to answer the question.  This is my version of having people type their question and all hit enter at the same time.

During the discussion time we walked the room and checked in with groups.  There were a few who admitted being baffled at what they were supposed to do so we clarified.  There were groups who said they couldn’t think of a question so we pressed them on what they observed and when they couldn’t explain the why we suggested that maybe they did have a question.  One group wrote down the name of the song and the artist under their question and was pleased with Dan’s taste in music.  There was definitely some non-math chatting going on throughout this but the directions got them talking with their groups and sharing ideas, which would help to set class norms for the year.  I’ve found that if I start them off discussing (even if it’s not 100% maths discussions) it is much easier to foster discussions for the rest of the year.  I want my room to be a place they expect to have discourse and debates in and a room where they feel safe doing so.  That is my Big Goal for the year.

After we collected the slips, I shuffled them while explaining that we would take a look at the questions as a class and see what similarity and differences the groups had.  The following is a sample of what some of the quarter slips said from the three classes.  Almost all of the questions were repeated in some form and the classes were fairly homogeneous with their questions.

-“If he stood on the escalator, would the escalator still be faster than the stairs?”
-“Was there a significant difference between how long it took him to go up the steps versus going down them?”
-“Why does he put headphones in?”
-“How fast was he going?”
-“How many beats per second are in the music?”
-“Was the time it took to reach the top different from the time it took to reach the bottom of the escalator?  On the stairs?”
-“What speed is Dan going up the escalator?”
-“What was the speed of the escalator?”
-“Was his speed the same on all trials?”
-“How far did he walk?”
-“Where is the boat?  Find the river.”

The speed question was most common followed by distance questions.  Exactly one group out of 24 asked “How much time does it take to go up the down escalator?”  One.  It was not the result I was expecting.

The last question listed above was seconded by a lot of students and gave me a heads-up that they pay attention to file titles.  I evaded by nonchalantly saying that there is a collection of math problems referred to as “boat in the river” problems and that the one in the video is just one version while silently praying none of them Google it and ruin my punch line.  The rest of class was spent discussing the music and I was surprised at how few students caught on that Dan was walking to the beat after two showings.  Two of the classes asked to see the video again and I obliged (love being able to do that).  The classes agreed that he started at the same time from the top and bottom, but even though consensus had been reached that he was walking to a beat, a good portion of the classes didn’t think that meant Dan would end his stair walking at the same time.  Some students proposed that you go faster going down the stairs than up so the Dan walking down the stairs would reach the bottom a bit before the Dan going up.  Counter arguments were made about the beat of the music and then class ended so we shelved the discussion until the next day.

After school my TC and I sat down with the questions each class came up with and decided that since the big question we expected barely happened, we would have the classes answer their own questions in a logical order.  We would start with figuring out the distance, move onto speed of Dan, then find the speed of the escalator, and then finish with how long it takes Dan to go up the down escalator.  Before any of those questions could be answered we came up with a field trip for the next day to try and clarify the walking to the beat questions.

Day 02

I enjoy online videos a lot.  I’m not a YouTube person as I find it cluttered with too much stuff I’m not interested it.  I prescribe to Video Sift as it leans geeky and clips from QI are regularly tossed up and I ❤ Stephen Fry.  While thinking about the escalator problem and walking to a rhythm, I was reminded of a clip from a few weeks ago billed as “Synchronized Walking Competition” (SFW, but the comments/ads may not be).

The day started with telling the students we needed to get some better visuals of walking to a beat going and that we would watch a video and then take a little field trip to a staircase.  Students were highly unimpressed with the start of the video especially as the only intro I gave was asking them to check out how the gentlemen were all walking to the same rhythm and had trained to walk the same sized step.  At about 1:50 into the clip they perked up and a few even applauded.  I paused when they do the ‘synchronized sitting’ bit to keep class moving.

Next we laid out the plan for the field trip.  We would run 3 trials with two students per trial where one would walk up the staircase and the other would walk down stepping to the beat of Daft Punk’s Around the World.  My school is two stories and we are lucky enough to have a staircase wide enough that students could line the sides for a good view and still have enough space for the two walkers.  Music volume was a bit of an issue, but we made it work.  Next year I will have students snap to the beat to help everyone hear but not disturb other classes.  I demoed what it would look like and then had ‘volunteers’ line up at the top and bottom (in one class we had more than enough offer to help, in another no one offered so ‘volunteers’ were selected).  With an 8-count call out to make sure they would both start at the same time, the classes got a nice visual of what it means to walk to a beat and, outside of a few missed steps, every trial ended with the two students finishing at virtually the same time.  When that didn’t happen, students were quick to point out “He started late!” or “She took an extra step!”  As a bonus, I was able to corral a passing teacher in one class and a passing administrator in another class as ‘volunteers’ which the students found hilarious.

The field trip took a good 15 minutes out of class, but helped get all students on board with what the beat in the video means in terms of Dan’s walking pace.  It’s also amusing to listen to students’ whispered excitement about taking a field trip in a math class and how they’ve never done such a thing before.  I’ve found that by doing unusual things at the start of the year I get the students used to expecting me to be a bit odd and thus have more capital to try odd things without them resisting.

Back in the classroom we put all the question slips from the previous day that dealt with speed under the document camera (mmm, student ownership) and announced that we would be answering their questions and that Dan’s speed on the stair and on the escalator would be the first.  Next we reviewed the video and asked if they wanted to focus on Dan going up or going down?  Thankfully, multiple students pointed out with backup reasoning that it shouldn’t mapper which left me feeling positive about the field trip.  They choose to focus on the ‘up’ portion of the video because you can actually see Dan’s feet when he hits the top.

Using the same quarter sheet system as on Day 1, we gave them a few minutes to discuss with their group what information they needed to calculate Dan’s speed and then collected.  The responses fell into 4 categories:

-number of steps
-length of a single stair
-start and end times
-beats per minute

That last one is something we’d decided to save as a post-activity problem so we focused on the first three.  Enough groups saw the extra-long landing in the middle of the stair that it was brought up before we showed the pictures of the steps.  There were some great discussions here about accuracy and if it was okay to round off to the nearest inch or half inch.  I enjoyed seeing who wanted it ‘perfect’ and who was okay with the rounding.  One of the classes had multiple students asking for the more accurate read on the tape measurement so they could work with those numbers.  Counting the number of steps was harder than it should have been due to a dying projector bulb but they were able to do it.  At this point class was at an end so we assigned calculating the distance Dan walks up the stairs (not the escalator) as homework.

When students had left we planned out Day 3 with hopes of getting to the final question by the end of class.  Ah, hopes.

Day 03

What we thought would be a 5-10 minute check of student work turned into an awesome conversation about distance, accuracy, and sharing of student ideas.  We asked the groups to share their calculations and be ready in 5 minutes to provide their answer to the question “what distance did Dan walk up the stairs?”  We called each table and wrote them all down on the board.  We marked matching answers with matching symbols and asked the groups with the largest and smallest numbers to share their thinking.

Answered ranged from 200 inches (whoops, missed a button on the calculator) to slightly over 700 inches.  Most landed around 530 inches.  Two groups in two different classes added all the heights and all the lengths.  Many used Pythagorean Theorem which caused some students to facepalm that they hadn’t.  A few broke the stair into three pieces and added two diagonals and one horizontal together.  Even had a student volunteer to come to the board to draw a picture.  She was rewarded with whispers of “wow, nice” and head nods from other students followed by class agreement that that was probably the most accurate calculation on the board.  Other students pointed out that her calculation is within a foot of their Pythagorean calculation that ignores the landing and just takes total horizontal and total vertical in one triangle.  I made note that a bit more work led to a slightly more accurate number and that sometimes it’s a judgment call when to do more work and when to say good enough.

We moved onto the escalator distance calculations and had a bit of a struggle counting the number of steps.  Watching my computer screen I’m pretty sure there are 29.5 steps, but one class went with 31, the second went with 32, and the third said 33.  Again, the projector resolution was not up to the task.  Students quickly pulled out Pythag and consensus was reached in each class for the distance Dan travels on the escalator.

Figuring out the times was something we knew would be a bit of an estimate.  To try and get better accuracy, we split the class and had half look at the escalator and half look at the stairs.  I asked how we should figure out the time.

Class: Take the average of our numbers!
Me: Which average?
Class: The average.
Me [confused voice]: Which average?
Class [getting annoyed]: THE average!
Me [confused voice]: Which one?
Class [convinced I have hearing damage]: the AVERAGE!
Me: Which. Average?
Student: The Mean!
Class [groans]: The Mean!
Me [sweetly]: Just checking.

Each group averaged their scores and my TC and I snagged their averages and put them on the board.  And then class time was up so we sent them home with an assignment of calculating Dan’s speed on the stair and the escalator.

After school we poured over the differences between the data each class was working with and pounded our heads trying to smooth the connection between the escalator speeds and vectors that we wanted to make.

Day 04

Timing was thrown off by needing to get books checked out at the start of class.  The process cost us over 15 minutes, but gave me time to chat with students in line and learn a bit more about my classes.  Returning to class, we got to work summing up the speed calculations.  The average between the classes put Dan’s speed up the stairs at around 40 in/sec and Dan’s speed up the escalator around 60 in/sec.  When doing a quick start-of-class check in with the groups a student asked me if the math was supposed to be hard as he’d been working with s = d/t since middle school.  I agreed that there are probably kids in elementary school that could calculate speed, but I wondered if they could explain the ‘why’.  I reminded him of all the ‘why’ questions that had been posed the past three days and he agreed that no, his middle school self probably would not be able to justify as well as his current self.  I also told him that part of the project was to get everyone used to talking with others in their group, sharing their ideas and calculations with the class, and helping me learn more about the personality of the individuals in the class and the class as a unit.  He seemed appreciative of the plan and then asked if they would ever get to see the end of the video.  I promised he would see it by the end of class.  I heard a few muttered prayers of thanks from eavesdropping students in nearby groups.  I added:

Me: “It’s a guy walking up the down escalator.  Why do you want to see the end so badly?”
Students: “We just want to know what happens!”

They want an ending.  They want closure.  We haven’t even proposed guesses for how long it’s going to take to climb that escalator, but they want Dan to reach the top and to see him do so with their own two eyes.  I let them know we have two more questions to answer before they get their wish.

On the front board is a summary of student calculations/estimates for distance, time, and speed for both the stair and the escalator.  I throw down the first gauntlet while holding up the quarter slip one of them originally wrote it on: how fast is the escalator moving?  Some students have total idk looks on their faces and others are muttering that it’s easy.  We let them know they have about 5 minutes to work at their tables to find a solution and that they need to be ready to explain their answer as we will call on a few groups.  I’ll note that since there are two of us in the room we’re able to give the groups good attention.  We don’t offer a lot of help here—they have the information they need, they just have to think through the ‘why’ in a logical manner.  Several of the groups I hit up first have a number, but are fumbling with the why.  I push at getting them to say what they know and ask questions about where the numbers we have already calculated come from.  I ask about our field trip.  I have them remind me why he had headphones on.  I sneak away when they start debating the wording of their argument.

When we have the groups share out (closer to 10 minutes later than 5) I am looking for the key point that Dan is walking the same pace on the stair and the escalator.  Questioning techniques is definitely a weak spot of mine (as NBCT proved to me last year) so my brain is flying here.  In the first Precalc class of the day after a few groups use some great language about pace and how if you remove the ‘Dan’ element from the ‘Dan’ + ‘Escalator’ speed you should have just the ‘escalator’, a student puts forth that she doesn’t think you can just subtract the numbers since the escalator steps are larger than the stair steps.  Half of me freaking out about the monkey wrench she just threw into my head as I never thought about that question and I have a sneaking suspicion she’s right and the other half is cheering on her thoughts because it’s a great point to bring up for clarification.

Keeping the straightest face I can, I repeat that the step-height is different and then ask the class if that’s going to affect our current calculation.  The class splits here between yes, no, and not quite understanding the problem.  We let them talk it through for a few minutes.  I interject a reminder to think about how we calculated speed.  It’s noted that speed is total distance travelled over total time.  The class concludes that the difference in step height has already been taken care of when we calculated the distance travelled.  At the time I was with them, but there was nagging doubt.

Pondering the question now, if the escalator was frozen and Dan walked up it and the stair at the same pace, he would hit the top of the escalator first since he’s been travelling slightly further per beat.  If you run the math, Dan is going about 14.8 in/step[beat] on the stairs and 17.4 in/step[beat] on the escalator.  The quick and dirty subtraction rests on the assumption that the steps are the same size.  The difference isn’t huge, but it’s there and something I’m going to have to account for next week.  On the bright side, if my excel work is correct it explains why the students have an overestimate.  I’m hitting myself upside the head for not considering this beforehand and being ready to help students think about the problem or propose it in any class that didn’t consider it.  I owe that student some cookies.

We moved onto the last question: “How much time does it take to go up the down escalator?”  There are only about 20 minutes left in class at this point and ending with showing the video answer would be sweet and is my goal.  Students set to work and this is our chance to probe the thinking of some of the quieter students.  As I check in with groups, when one student offers up a number I ask a different student at the table for the why.  The Quiet Ones always seem shocked that I spoke to them, but every time I did this they were able to answer fully, if not very loudly.  We fielded a few out-of-the-box ideas (some really want to use the beat/sec but I asked them to skip that idea for now as I wanted to follow up next week looking at that).  In general, the groups took Dan’s stair speed (40ish in/sec), subtracted the escalator speed (20ish in/sec, it had been concluded earlier that the escalator moves the same speed up and down), and then divided the distance by the speed to get a time (around 24 sec).

Given the timing, my TC and I were able to chat with each group but we didn’t have a chance to have them share out and show the video solution so we chose the latter.  Before starting the video we put on the board the guess the students came up with and I told the class to say bye to Dan (which they did).  We also asked if they thought their time would be perfect (“No!”) and why (“We rounded!”).

We started the video from the beginning and when Dan started up the down escalator every set of eyes in the class was glued to the screen.  At first there were mutters for him to go faster as they seemed to think he would go over 24 seconds.  When it became clear he would finish before 24 seconds they called for him to slow down.  When one class realized they were within two seconds, they cheered.  They.  Cheered.  There were high-fives.  Students were congratulating their group for being awesome.  The bell rung.  Great first week everyone.  Have a good weekend.

Bonus Round

My school is a PLTW school and a film crew from South Korea was in the school filming for a documentary on different education practices in the US, Japan, and elsewhere.  They have this entire lesson on film and I am trying to see if they will let me have a copy because that would be awesome.  The camera guy went right up close and it made the kids both nervous and amused.  I am curious to know what they thought of the classroom.

End Notes

No worksheets were used during this process.  Where we ended up is where I hoped, but it took longer than I expected and didn’t take the path I expected (really, only one group is curious how long it’ll take him to get up the down escalator?  Really?).  A worksheet would have locked us into certain expectations and I didn’t feel it necessary.  I plan on collecting the scratch work students did (if they have it), but I didn’t demand any records be kept.  Lastly, it was interesting to walk around and see how different students organized their work—the carefully organized with full sentences versus the vague sketched out numbers in random places on a page.

Will I do this again?  Yes.  I had fun discussing the problems with students and I felt it had a lot of entry points for students of all levels.  Everyone has been on (or at least seen) an escalator.  Speed = distance/time is straightforward.  Pythagorean Theorem is comfortable.

Next Steps

Though we know it’s not a perfect fit, we’ll be using the ideas students discussed surrounding adding and subtracting the speeds to intro vectors and we’ll even have them doing a “boat in the river” problem to kick things off.  Vectors will lead into triangles and angles which will lead into Trigonometry.  Trig is the main goal, but vectors are nice because (a) a lot of these kids are in physics or will take it in the next year or two and, (b) historically students suck at notation and vector notation will help prime them to pay attention to details (I hope).


If you have any ideas/comments/questions/quandries, please feel free to post them below.  I have some pondering to do about this lesson, but I need some distance from it first and commentary will be super-helpful to look back on.

Big thanks to Dan Meyer for the awesome video.  Thanks to everyone who was in the DimDim session that I’ve watched more times than is probably healthy.  Thanks to Jason for the screencast.  And thanks to my TC for being awesome and supporting my sometimes crazy ideas and being way better versed in vectors than I am.