Making Sense of Problems and Persevering in Solving Them
“The time has come,” the Walrus said,
“To talk of many things:
Of shoes—and ships—and sealing-wax—
Of cabbages—and kings—
And why the sea is boiling hot—
And whether pigs have wings.”
One of my favorite books since I was very young has been Lewis Carroll’s Through the Looking-Glass and What Alice Found There. So when my eleventh grade English teacher assigned us the task of memorizing a famous poem, I chose The Walrus and the Carpenter. After enthusiastically reciting all eighteen verses, I looked across the room at my classmates’ bland expressions, wondering why no one else seemed to enjoy the poem as much as I did.
Several years later, I was standing in front of my first classroom, teaching one of my favorite subjects: math. I saw the same bland expressions as twenty-seven fourth graders worked their way through the four steps of “problem-solving” as mindlessly as they had recited the Pledge of Allegiance that morning. I didn’t get it. Solving problems was fun for me; why wasn’t it fun for them?
I realize now that the fun part for me has two components: the puzzle, and the accomplishment. The puzzle is about comprehending something that at first is incomprehensible. The accomplishment comes from working through that puzzle to completion, whether on my own or with some assistance. This is the essence of Math Practice #1: making sense of a problem (the puzzle) and persevering in solving it (the accomplishment of seeing it through).
One summer, several years ago, my family spent a week in the Outer Banks. The beaches are wide and windy, and my kids loved flying kites. This, of course, meant the inevitable tangled string. It was usually a fairly routine matter to loosen the right loop, undo the knot, and send the kids on their way so I could get back to my book.
Until one day when my son came to me with Cobble’s Knot.
Jerry Spinelli, in his wonderful book Maniac Magee, tells the story of a legendary knot of string which had hung outside of Cobble’s Corner, the local pizza shop, for over a year. The owner offered free pizza for a year to anyone who could untie the knot. Many had tried. None had succeeded.
“To the ordinary person, Cobble’s Knot was about as friendly as a nest of yellow jackets,” writes Spinelli. “You could barely make out the individual strands. It was grimy, moldy, crusted over. Here and there a loop stuck out, maybe big enough to stick your pinky finger through….”
This is what confronted me that day on the beach, crusted with sand, and as large as my fist. Any ordinary person would have just cut the string off above the knot and called it a day. But to me it was a challenge crying out to be tackled. I began just as Maniac Magee had begun, applying the first part of Math Practice 1, trying to understand the problem that lay in my lap. “For the first hour, all he did was turn it around, study it, and delicately test a few of the loops and strands.” I’ll admit that Maniac was a bit more patient than I was; I studied the kite string for about five minutes before I started poking at it.
Maniac then displayed the second part of the Practice: perseverance. He worked steadily and calmly, straight through the entire day, stopping only briefly for lunch (orange soda and butterscotch Krimpets) and a fifteen minute mid-afternoon nap. By dinner, Cobble’s Knot was no more. My own perseverance on the kite string wasn’t nearly as epic, though it did take me the better part of three hours to finally get it undone.
So why don’t students feel this way about solving math problems? A good mathematical problem presents the same opportunities for sense-making and perseverance as Cobble’s Knot. Part of it is we take away all of the fun and leave just the drudgery. Instead of giving students experiences where they puzzle through something and find their own way, we do all the interesting work for them, present them with a prefabricated procedure, and ask them to do just the mechanical part. There’s no sense-making because there’s nothing left they have to make sense of. And instead of a sense of accomplishment there is only a sense of finally being done. (Hint: it’s not the same thing.)
Jerry P. King in his book, The Art of Mathematics, says that we need to consider the beauty and wonder of math as an artform. Those who have experienced only traditional math instruction “will no more believe mathematics can be lovely than they will believe the sea is boiling hot or that pigs have wings.”
To change this in your own math classroom, try these three strategies.
1. Use nonsense problems
Consider another section of my favorite poem, where the protagonists walk along a beach not unlike the one where we spent that summer:
The Walrus and the Carpenter
Were walking close at hand;
They wept like anything to see
Such quantities of sand:
“If this were only cleared away,”
They said, “it would be grand!”
“If seven maids with seven mops
Swept it for half a year.
Do you suppose,” the Walrus said,
“That they could get it clear?”
“I doubt it,” said the Carpenter,
And shed a bitter tear.
This is, of course, nonsense. But what if you were to present this to students as a question they actually had to answer. Could in fact seven people sweeping a beach for six months clear away all the sand? How would you begin to tackle a problem like that? What other information would you need to know? How would you know whether the sweepers were successful? What does that even mean? Wrestling with those kinds of questions starts to get at the real meaning of “making sense of problems.”
I call these and problems like them iWonders, since making sense of them and solving them often involves doing some kind of Internet research to find evidence supporting your mathematical assertions and estimates: How much sand is on a typical beach? How fast can a single person sweep? How many hours a day is it reasonable for them to work?
These problems are easy to invent. “iWonder if you scrambled all the eggs laid by all the chickens in Iowa on October 18, 2016, how big an omelette it would make.” or “iWonder how much all the pavement on I-95 would weigh if you turned it into gold.” Try writing your own and share it in the comments.
2. Teach mathematics as a language
Give students the opportunity to work with math vocabulary and symbols in ways similar to learning a second language. Write and read mathematical prose frequently, and practice translating back and forth between symbolic language and English. For example, how many different ways can you think of to express this in English:
3 + 5 = 8
Certainly the literal translation would be “Three plus five equals eight.” But it is also equivalent to “Eight is the sum of three and five,” or “If you add the values three and five, the total will be eight.” To an adult, these may be obvious parallel statements, but to a student there is enough variance in the vocabulary they may not see the relationships. Make them explicit.
Likewise, we should borrow strategies from language arts teachers to support better comprehension of math. Robert Marzano details six steps for teaching new vocabulary that work well for mathematical terms. If you think of mathematics as a second language, then all of your students are actually Mathematical Language Learners (MLLs); Larry Ferlazzo gives great tips for supporting English Language Learners which are easily adapted to the math classroom.
As students develop their vocabularies, they will have the language to discuss and share and think about problems more deeply, promoting more perseverance. Share your own thoughts about math as a second language in the comments.
3. Create a math makerspace
Instead of always solving problems you provide, give students the opportunity to construct their own. The process of creating, testing, refining, and solving problems will promote perseverance and deeper engagement.
One way to begin building a math makerspace is through a problem-finding box. Collect lots of interesting objects that can provoke mathematical ideas: measuring cups, expired coupons, coins, blocks, sheet music, a computer technical manual, and so on. Then have students individually or in pairs select two items at random from the box and develop an interesting math problem around them.
Encourage students to collect their ideas in a common space, whether it’s a notebook, an online digital document, or a dry erase board in the classroom. Even if the ideas aren’t fully formed, they can go back later and develop them further. Students can also borrow problem ideas from others and take them in new directions.
This common space also provides an area for long-term problem solving where partial solution ideas can be posted and others can add to them or give feedback and suggestions.
What other innovative ways can you think of to build a mathematics makerspace in your classroom? Share the ideas below.
Ultimately, the key to Math Practice #1 is in avoiding shortcuts until students have had the opportunity to struggle with the problem for a while. Allow them the chance to figure things out for themselves, to poke at the strings and loops for a while like Maniac Magee, looking for a loose end. Much learning comes from recognizing when an approach isn’t working and backing up to try a different tack.
This Mathematical Practice is listed first for a reason. Problem solving is the reason for learning math, and the satisfaction of solving a tough problem is what makes it enjoyable.
What other strategies do you have for making problem solving joyful and fun while still promoting sense-making and perseverance?