Students Ask the Questions, and the App Countdown

Inspired by Dan Meyer’s post, I shared the Apple App Countdown with my 6th 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 6th 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 6th 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)?

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Tinker, Tinker, Algebra Student

With this second post, I’d like to start writing about the sorts of things I hope to write about most – what I’m actually doing in my classroom, the recipes I’m designing and using to try to make the smartest cookies I can.  Everything I do is very much a work in progress.  I welcome all suggestions, tips, resources, and nudges in new directions!

That said, here’s what I’ve been doing for the last two days in my Algebra 2 class.

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Two and a half years ago, when I began teaching, a colleague gave me this article to read.  It’s called Habits of Mind: An Organizing Principle for Mathematics Curriculum.  It opened my mind to new ways of thinking about math education and helped to solidify and put into words (and legitimate – someone else actually does this??) the ways of thinking about math education that had already been floating around in my head.  I love the way it’s set up and the language it uses.

Many ideas from this article stuck with me, but I think the habit of mind that did most was, “Students should be tinkerers.”  Here’s what the authors have to say about mathematical tinkering:

Tinkering really is at the heart of mathematical research.  Students should develop the habit of taking ideas apart and putting them back together.  When they do this, they should want to see what happens if something is left out or if the pieces are put back in a different way.

I love incorporating tinkering into the work we do in class because it’s easy to begin, but the sky’s the limit.  It’s also a low-pressure activity – students aren’t expected to produce anything other than something interesting.  The word “tinker” also, to me, implies lots of room for student choice and creativity.  Tinkering shouldn’t be prescribed or predetermined – it’s playful and individual.  Finally, tinkering is constructive.  By tinkering, students are building their own little mathematical monsters with which they can surprise, entertain, and wow each other.  And, because they chose how to tinker, those monsters are theirs, to take ownership and be proud of.

I said I’d write about what I did in class, so I’ll stop there and start talking about the tinkering we did in my Algebra 2 class yesterday and today.

The assignment I gave my Algebra 2 students at the beginning of class was to start with the function y=x^2 and choose a single change to make to it.  Some of the examples of small changes I gave were: add or subtract a constant; multiply or divide by a constant; change the exponent on x or y; and, add or subtract terms with some power of x or y.  Then, I told them to make small changes in the change (say, add 1, then 2, then 3, then 4 to the equation) and graph each resulting function.  How does the particular change chosen affect the graph of the function?

Some background:  We just had a week off from school.  Before the break, we’d been doing a lot of work with graphing parabolas in “vertex form.”  When planning classes for this week at the end of the break, I wanted to refresh their thinking about the relationship between a relation and its graph before beginning to work on completing the square.  I really wanted to reinforce the idea that small changes in a relation will make changes in the relation’s graph that we can predict.

A few students immediately chose a single, simple change to make to the relation and started graphing, as per the assignment.  At least half of the students made a bunch of really bizarre changes and got a crazy graph.  I expected this.  Only the most conscientious student could be presented with an extremely powerful graphing program and not type in the weirdest thing he or she could think of.  While graphing insane equations wasn’t the point, I didn’t want to squelch creativity.  They really were asking, “What happens if I do this?” as they graphed their bizarre equations, which was the goal of the assignment.  And the things they were creating were quite interesting.

So, instead of redirecting them by saying, “Now, now, little Johnny, don’t do that, do the assignment,” I stopped at each student’s computer and said, “Wow!  What a crazy cool thing you made!  How did you do that?  And, what do you think will happen if you make this tiny change?”

While this strategy may have made the assignment take longer (which I have the luxury of allowing), I think it led to much more creativity and student ownership over their work.  By the end of our second class on this assignment, everyone had completed the work they were asked to do – but in their own way.  Several students wrote paragraphs describing the graphs’ transformations.  Others plotted many graphs on the same screen and labeled them – one even started using Photoshop to color-code them.  Two students challenged each other to an artistic graph-making duel.  One of those students nested almost 100 sixth-degree curves to create a wave-like figure; another graphed various versions of y=x^(2x-c), for different values of c, which made a picture that looked like fireworks.  Each student had a unique piece of work that he or she was proud of.  I’m really looking forward to the sharing session we’re going to have in class today.

I Hate Math… Class (Sometimes)

Welcome to Recipes for Pi!

I’d like to introduce myself to writing about teaching math by giving some history about me and math while explaining part of why I’m starting to write this.  This blog was in part born out of a mixture of enthusiasm and anger about the ways that the world learns and shares math, triggered and brought to a head by an event in my own math education.  For me, writing is the best way to process my feelings about and reactions to events.  I was motivated to share my writing, and to begin a blog, when one of my students asked me about the event in question.  So, read on and enjoy!

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I was recently reminded of how much I dislike math class – and how I learn math best.

Last semester, I registered to take a graduate-level math class.  I hadn’t taken a math class since college, and had begun to miss being on the learning – rather than teaching and learning-for-teaching end – of the classroom.  I figured that taking the class would be difficult – both because it was a graduate-level class and because I’d be taking it while teaching – but was motivated to face the challenge because, well, I like doing math.

Before about 5 ½ years ago, I hated math.  That’s surprising to those who’ve gotten to know me best in the past 5 ½ years, but the fact that I love math now is a total shocker to those who know me from before.  Throughout elementary, middle, and high school, I was one of those students who often baffle math teachers most – the extremely conscientious kid who does all of the homework, stays after class to ask questions, and aces tests, but is convinced she’s terrible at math.  I always felt as though while I could eventually do anything with enough memorization, I never really understood what was going on.  If you’d asked me then what it meant to do math, I would have said, “Math is learning complicated, nonsensical procedures and algorithms, and using them again and again and again until you can do it with your eyes closed (or at least without your notes).”

But when I got to college, something changed.  Here’s what I think happened:

  1. I stopped solving problems by memorizing everything the teacher said to do, because the problems weren’t like that.  Most of the problems weren’t like the examples – you had to build your own methods from the tools you were given, not copy a procedure.
  2. I stopped worrying about the answer and started paying attention to the process, because the problems weren’t about the “answers.”  The problems we were doing were proofs – and when you’re doing a proof, you already have the answer.  Math wasn’t working the machine properly, but building the machine you needed.
  3. I started working with other people.  My classmates and I figured out how to do the problems together, and the professors came to our informal homework meetings to talk to us, too.  Math was social, cooperative, a conversation among people – students and teacher – learning together.

As I started really thinking about math and bringing my tastes to my mathematical work, I began to see how many ways there are to interact with math.  I also began to see how very beautiful the work of finding, simplifying, proving, and changing patterns can be.

And when I began teaching math at Saint Ann’s, I grew to love math even more.  I had to really think about interesting things to do with math, and do them – mostly for the first time, because these were precisely the activities I avoided in elementary and high school.  My peers’ love of doing math was infectious, and their patience with and appreciation of my pace of thinking were blessings.  I got to try to build classes where, whether by conscious or sub-conscious effort on my part, my students learned math the way I learn math best – by talking with each other, figuring things out for themselves, and certainly not just listening to the teacher.

I probably thought that taking a graduate math class was a good idea because the images of math class I had in my head were of math classes at Haverford and the classes I try to run. And the feelings I had after one week in this math class were probably so very negative for those reasons as well.

After about a half an hour into doing the homework assignment, I could feel my old math anxiety returning.  Here was math class at its near-worst for me – math as competition, math as listening to a professor spout information, math as a race to accumulate facts.  I was working alone.  My professor quoted from a textbook for two hours and did nothing to create an environment in which a bunch of people really interested in math (himself included) could work and learn together.  There were a TON of problems for homework, because we’d covered an entire chapter – itself covering all of introductory group theory – in two hours.

And I began to feel math anxiety all over again.

This made me angry.  Math anxiety??  Me??  I like math!  I know that, in working on math with others, I can contribute ideas, ways of thinking, and ways of structuring thoughts that are valuable.  I also know that I can solve and find the resources to help me solve problems that I come across and find interesting.

There are many, many ways to learn and participate in math (and any subject of study, for that matter).  One of those ways is listening to a professor in lecture and working through chapters-worth of math problems every week – and maybe that accelerated, private, listening-based way of learning is great for some people.  But it certainly isn’t good for me.  I can learn more modern algebra by working through a good book on my own, at my own pace, by sharing interesting problems with friends and peers, and, most importantly, teaching it.

So, I dropped the class.  And I know, better than I did before, how I learn math and how I want to teach it.