Inspired by Dan Meyer’s post, I shared the Apple App Countdown with my 6^{th} graders today in class. I’d never done any of Dan’s “What Can You Do With This?” problems in a class before, but I like the idea and I’d been meaning to try it. I went for it this time because it struck me as a particularly good “What Can You Do With This?” problem. Two things that feel really important to me in a good “What Can You Do With This?” problem are: the situation being naturally curiosity-inspiring, and the data being right there in front of you, in the world you typically inhabit, albeit needing to be found. The Apple App Countdown is an existing thing that really does motivate a question that can be solved mathematically. The page screams at you, “How can I predict when the 25 billionth download will occur so that I can immediately download an app???” regardless of whether or not you actually want the prize (and my 6^{th} graders did want that prize). The only information hunting-ground you need is your computer screen. Great recipe for mathematical fun!

I decided to not show any of the videos posted on Dan’s blog and, instead, just set a computer at the front of the room with the ticker going, because the problem in its natural habitat, with no framing, was so compelling already. I felt that the students would be able to figure out what data they needed to collect and how to collect it without much prompting from me, so I didn’t feel the need to emphasize a clock or a change over time.

So here’s what happened: I set my laptop where everyone could see it and asked the kids to inspect the screen for about 30 seconds. Then I asked, “What is this?” and “What does it make you wonder?” I got a range of responses for the second question, such as, “I wonder how Apple made this.” Kids expressed interest in trying to download the 25 billionth app, but didn’t automatically treat it as something they could calculate. Their immediate response was to think, Well, I should check this site periodically over the next several days and when it gets close, start downloading apps like crazy. I had to probe them to have them come around to thinking that this was something they could predict, with some calculations.

This surprised me. They are, generally speaking, a very mathematically inquisitive bunch. It doesn’t take much to get them engaged with any mathematical situation. I’ve also worked hard with them to develop investigatory skills and techniques, but mostly with purely mathematical situations. And, when I thought about it more, I realized that these investigations mostly started with questions I’ve posed. We always continue investigations with student-generated questions – I strongly encourage these – but I almost always start the investigation off with a question of my own.

Once that idea that this situation was solvable as a math problem was in the air, however, they were completely on-board with doing the work. To get them going, I asked what piece of information they would need to get started predicting when they should try downloading the 25 billionth app. The overwhelming response was, how fast is the ticker moving? And, when asked how what data they would need to calculate the speed of the ticker, they recognized that they would need to time the progression of the numbers.

They spent the rest of the class taking measurements, comparing measurements with each other, and making calculations. By the end of class, most of the students had calculated an amount of time – a REALLY long one. I let them stew on this for a while… until someone realized that they’d calculated how long it would take the app store to reach 25 billion downloads from zero downloads. Not from 24 billion and a lot of change. At this point, class ended, so I sent them with the problem as a homework assignment.

From doing this problem with my 6^{th} graders, I realized how while I make a great effort to encourage my students to pursue their own mathematical questions, I rarely begin our mathematical activities with questions, problems, and situations generated by my students. It’s always me who says, “Let’s do this cool problem about ________,” and then opens the floor to students’ ideas. This seems like an imbalance to me. If I claim that my students’ mathematical interests are worthy of class investigation, then their ideas and questions should motivate our work, not just continue it. I believe (and hope) that they know that if they bring in a problem or situation, we will investigate it as a class. But it isn’t a “thing” in our class. I would love to find a way to encourage my students to bring mathematical questions – including “What Can You Do With This?” questions, as well as more purely mathematical problems – to our class for us to do on a regular basis.

Has anyone done problem-collecting with their students? Any suggestions for good ways to encourage and implement such a thing, with middle or high schoolers?

P.S. – I noticed (as did several of my colleagues, who did this problem in their classes as well – and we didn’t coordinate this) that the App Countdown page isn’t consistent when you open it at different times. Open several tabs of it, and you’ll see that the countdown is off by possibly hundreds of millions of downloads from screen to screen. Also, if you let it run open on your computer for a while, the number of downloads will be drastically off from the number of downloads when you open a fresh tab. (I draw this conclusion from a single data-point – I let it run on my computer overnight last night.) This made me wonder… does the ticker actually increase linearly, in some way imposed by Apple, instead of according to the actual downloads? And, if so, is the rate of increase the same each time you open the page, or does it change according to the actual rate close to the time you opened the page (or some other criteria)?

Nice post. I also did this with 10th graders and 8th graders, and I had similar responses. They were much more interested in talking about all of the ways in which a linear model was going to be terrible for this, rather than execute the computations to get a baseline estimate.

What you said about always initiating the question really resonated with me. I’ve been trying to get around this all year long, especially with my algebra 1 students. Often, I just show up to class, stand at the front for a while, wait for them to quiet down, and then ask, “OK, what now?”

Unfortunately, they almost never have a great idea for how to move forward. This truly bothers me. I am still wrestling with the notion that a teacher is the disseminator of knowledge, and that a large part of my role is setting “the course.” I really want to change this dynamic, but MAN it is hard.

I keep imagining meeting a new batch of students and asking them, “so how does this work – this whole math class thing? What are we supposed to do?” These things never go as expected, however.