"The Answer" vs. "The Problem"
Lesson 1 of 10
Objective: SWBAT use words to define problems for puzzles that lack words.
With two weeks to go before the winter holiday, today is to be the first day of a five-week unit about systems of linear equations and inequalities. In order to really be ready for that, however, I need to circle back to some standards about linear functions that were left undone during Unit 1. I want kids to have a better grasp of what points mean on the coordinate plane, and I want to lay all sorts of groundwork for thinking about the different representations of linear functions. I want to focus explicitly on Mathematical Practices #4, #7, and #8, which I haven't yet done. This is the real stuff of Algebra 1, and it’s time for the rubber to hit the road. So naturally, I'm thinking that I'll spend a few days going off the road entirely.
For the next two weeks, as the winter break approaches, I want to have some fun. I want to try some things I haven’t tried, and to get students excited and thinking a little differently. I want them to notice the change of unit, and that class feels different. The last two weeks before the winter holiday are one of the low points for morale and motivation among my students, so my hope is that at least they can count on having some adventurous fun in my classes. We’re going to do some math without words, we’re going to do some computer programming, we’re going to build some towers. We’re going to create foundations for all of the thinking that we need to do about linear functions, and I’m going to see how natural I can make it feel.
Also, in this and several more upcoming lessons, I’m going to steal some resources from some great math educators. The internet and my bookshelves are full of more unbelievably rich mathematics than I could ever cover in a year. Knowing what's out there, and when and how to use it is a key habit of being a successful teacher. So I'll give you a little overview of a few great resources that I've never used, and in my classroom, I’m going to see what happens. I’m going to learn with my kids, and bring them in on the experiment.
Our just-completed Statistics unit was text-heavy. Last week, students interpreted data (by writing about it), they completed a writing-prompt quiz, and on Friday, they took a multiple choice exam. Words, language, and literacy have played a major role in this class so far. So with today's wordless agenda, I'm trying to demonstrate that things are going to be different today.
As students arrive, I hand them their progress report and give them a minute or two to look at it. (Please see my lesson from earlier in the year for an overview of the mastery-based progress reports I use in this class.) This is the third progress report students have received from me, and by now, they're starting to understanding what they're looking at. When I'm done distributing progress reports, I point to the agenda and ask what everyone notices.
I take a few responses. If no one asks about the diagram in the top right corner of the agenda, I point to it to and ask if anyone can explain it. It's a calendar. The blue shaded squares represent the weeks that have already passed, the unshaded squares represent the weeks that remain in the school year, the green dot represents this week, and the orange dot represents the end of the Second Marking Period.
The first agenda item, then, is a question that is color-coded to match the calendar diagram. I hold up a progress report and ask if anyone can use words to state the question that is written here, in terms of their progress reports. As students respond, I give props to anyone who uses growth-mindset language to verbalize the question. "This progress report shows where I stand right now," someone might say, "how hard am I willing to work to get to where I want to be by January 17?"
Problems Without Words
Introduce the Activity
I point to the second agenda item, which is just a blank line, which I hope builds a little intrigue. I tell students that it's up to them to define the problem that we're going to work on today. "The problem that you'll work on today has no words," I say. "It will be up to you to put words to this problem." Then I distribute today's handout.
As I mention above in my note to teachers, I'm trying some new things today. In any school year, I find that it's very important to teach some number of lessons I've never taught before. That will be the case today and tomorrow, as my classes will investigate some of James Tanton's Math Without Words puzzles. Mr. Tanton offers a few of the puzzles on his web site, and there is an outstanding e-book for sale here.
Any of these are great place to start. I've chosen this one:
I print this graphic, by itself, as today's handout. There are two ways that my students react to receiving this: either with elation and an eagerness to figure out what's going on here, or with complete frustration that there are no instructions and that they have no idea what to do with this. But here's the thing: a puzzle like this requires a different skill set than a lot of the other work kids see in school, and the breakdown of who reacts how is different than on other days. It changes - sometimes unpredictably - the dynamic of the class. A task like this mixes up the balance. And, whether kids are thrilled or frustrated, everyone is invested.
Today's work is as pure an exercise in Mathematical Practice #1 as I can imagine. I refer to this standard all the time, because I do think it must and does happen all the time, but here there is no ambiguity. There is something to solve, but we have to make sense of it and figure out what the problem is, then we have to persevere - probably through some frustration - to solve it.
Getting Started: Five Minutes of Sense-Making
I start by asking the question that I ask my toddler every time he sees something new: "What do you see?" I tell students that they may discuss what they see with colleagues, or they may work alone, but I am going to wait five minutes before I take any questions. This five-minute requirement is really hard for me to uphold, because I want to respond to the brilliant things I hear and I want to assist kids who are foundering. But it's so important for me to slow down and just observe for a little while. Sense making is temporal, and that's a difficult point for many 14-year-old algebra students to understand. Taking time to make some sense here is a step in the right direction.
Using Student Questions to Define the Words We Need
After at least five minutes (more if the kids and I can handle it), I say that I'll take questions. Before I answer any of them, I try to get four or five to write on the board. Usually, this first round of questions will only scratch the surface, like these. Once they're written on the board, I answer all questions of the kind, "What or how are we supposed to do, find, or figure this out?" by saying that that's exactly what we're trying to figure out.
A lot of students are fixated only on "finding the answer." The question is, "the answer to what?" I write the "The Answer" on the board and point to these words. I say, "I could tell you a number, and say that it's the answer, but what would that tell you? Would you know what it meant or where it was coming from? Before we can find an answer, we need to figure out what the question is."
In the questions that I've written on the board, I can count on finding at least one word that requires some explanation from students. If it's "triangles," for example, I turn it around and ask where students see triangles. This in turn leads students to use other words, which then need defining. "In order to begin to discuss this problem," I say, "we're going to need to start defining some of these words, so that we can talk about it." To prompt students just a bit further, I ask what the numbers (on the right side of the page) mean. In order to answer this question, we need to figure out what we're going to call each figure (they're "groups" in the case of this class and this class), and what to call a set of figures ("families" and "classes" here). It's fun to watch kids agree on this, and the rather tame words we settled on in these examples do little to indicate the fun we had coming up with them.
Once we have these words, we can use them to talk about what we see on the page. I try to say as little as possible here, but I indicate that someone has said something useful by writing it on the board. If students are engaged, I set them to it, and work continues in small groups. If they need my help, I'll gently guide a class discussion until I see that a majority of kids are ready to go.
Each class gets to a different place, so it's hard for me to say exactly what should happen by the end of class. Generally, students are sketching diagrams and working through the problem, as they understand it, when it comes time to close up shop for the day.
We watch this video:
After watching, I ask if everyone thinks we should get in on this, and I take questions from students. Tomorrow, I'll go just a little more in depth about what Wednesday's class will entail, but this video is certainly enough to get kids excited.
Today's exit ticket is an index card. The task, as I'd originally planned it, was for each student to write their questions about today's "Math Without Words" puzzle. As it turns out, my exact use of this task depends on how the lesson goes.
In some lessons, it might be best to distribute the index cards during the lesson, have each student write their questions, collect them, and then read these questions aloud, midway through class. This is a nice strategy for getting all kids involved, surfacing student-generated vocabulary, or keeping kids engaged if it they seem ready to give up.
If that happens, I'll return the cards and ask them to write another question or two on the other side of the card. I'll improvise the prompt according to how class went today. I might ask students to write the question in their own words by using vocabulary that we defined as a class. If we already have a well-define problem, I might ask them to write extension questions. I might just ask them what questions they still have. Whatever happens, I'm looking to see what kinds of questions they're asking - this will help me plan for another round of this sort of problem solving tomorrow.