Beyond Answers

(excerpt from Making Problems, Creating Solutions: Challenging Young Mathematicians, Stenhouse Publishers, York, Maine c. 1999)

by Jill Ostrow

Six-year-old Kyle was about twenty  minutes into drawing the "guys" in his picture to look exactly the way he was imagining them in his mind. I looked over his shoulder. His picture was typical for his artistic talent-precise, detailed, colorful-but he was using crayon, which was unusual; he usually drew with colored pencils. Although it was a wonderful piece of art, this wasn't art workshop; the kids were working on solving a mathematical problem.

"Kyle, this doesn't have to be a masterpiece, you know! I really love your drawing, but I'd like to see you try and solve the problem too, okay?" I felt terrible disturbing him like that, but I really wanted to see him solve the problem. "You can go back to the coloring after you have a solution. How's that?"

"Okay," he muttered. I watched as he got out of his chair, marched over to get a basket of Unifix cubes, and in seconds came up with a solution to the problem. The kids know that I always ask them to write how they solved problems, so Kyle wrote, "15. 1 GASH YOSD U NAFKS KOWBS." [15. 1 just used Unifix cubes.] He put the Unifix cubes back in the basket, marched them back to the shelf, and sat back down to continue his picture.

"So, how many people are you going to draw?" I asked him.

"Fifteen. Look at this guy flying with the balloons."

"That's cool. Kyle, before you used the Unifix cubes, how did you know how many people to draw?"

"Oh, I was counting after I did each person. But you wanted to see me solve it, and I wasn't done drawing, so that's why I got the Unifix cubes."

"Oh." What more could I say?

Over the years I have realized the importance of drawing as a way for children to solve mathematical problems, but it wasn't until I watched Kyle that I grasped all the connections. Kyle was able to use two strategies for solving problems--drawing and using manipulatives-just as many children do. And drawing is as valid a strategy as using manipulatives. If I hadn't questioned Kyle, I would have seen his picture simply as a piece of art, not as a problem he was trying to solve.

Kyle never did complete the drawing of the fifteen people. It took him two days to do four, and by then we were already on the next problem. But it was important for him not only to be able to draw as he did, but also to demonstrate that he could solve the problem.

I teach a class of first through third graders. The age range is four years, and the intellectual or academic range is even greater. In math workshop our class explores mathematical concepts-each child in ways that match her or his understanding. Just as I see every child at a different level of understanding during writing workshop, I see this difference during math workshop. Children learn to write by practicing and experimenting and exploring writing; children learn mathematical concepts the same way. Even if everyone is investigating the same concept, each child may have a very different experience.

The problems here are ones that I have written over the past couple of years. I don't get them from workbooks: I integrate them with what we are studying. One year, we became an island community, and our curriculum for the entire year grew out of that idea. The following year, we set up a time travel chamber. Two of the places the kids chose to explore were 1863 (to learn about the Civil War) and the contemporary Arctic. Many of the problems discussed here relate to one or another of these topics.

In math workshop, kids also write their own problems, create their own math games, share what they are learning with one another, record what they are learning, take part in minilessons, and explore and investigate various concepts. I make spaces for children to move beyond answers, to understand concepts, to make explorations, and to investigate the mathematical process.

Understanding Concepts

Children use manipulatives not only to help them solve a problem, but also to help them understand the concepts inherent to a particular problem. Manipulatives are not the only vehicle for gaining understanding, however. Kyle, for instance, who is now a second grader in my class, often chooses pictures to help him solve problems. He has now learned to draw quick symbols to represent what he is trying to solve. In one problem, he was proving that three twenty-sixths of the class were wearing glasses. He started out by emphasizing the three children who wore glasses, drawing them in detail. He then went on to draw the remaining twenty-three members of the class as stick figures, thus showing his understanding of the concept of fraction (see Figure 5-1). Using manipulatives might have been more confusing for him.

Many younger children need to use their own drawings because that's the way they organize their thinking. Melissa, one of the youngest and most inexperienced problem solvers in my class, was working on a problem that asked, If there are 8 Union soldiers on a battlefield, how many legs would there be? She drew eight stick figures, and then counted up the legs. I knew from her drawing that she could organize her thinking well enough to represent eight soldiers and count to sixteen. Another student solved the same problem by getting out two cubes eight times, showing me her beginning, understanding of multiplication. Yet another child solved it by writing that 8 X 2 = 16, demonstrating her solid understanding of multiplication. It is important for me as a teacher to know that the concepts individual children use to approach a particular problem are often very different.

Division lends itself very nicely to pictorial representations. Jordan was solving a problem that required the concept of division:

A team of travelers is having a problem. They have 6 kayaks that hold 2 people each. But there are 13 people on the team. How many people will be able to travel in the kayaks? Will there be any extra? What should they do?

Drawing pictures as part of a story helped Jordon reach a solution, as his math work shows (see Figure 5-2).

Dave wrote a division problem based on his knowledge that a snowflake has six sides: There are 34 sides. How many snowflakes? Are there any extra? Carly solved Dave's problem by drawing thirty-four lines and then circling each group of six. I knew she had an understanding of division by her drawing; just seeing the answer would have been meaningless.

When children  in my class do solve a problem in their heads and just write the answer, they know they will have to explain their thinking; that sometimes comes in the form of a picture. Morgan knew the answer to one problem very quickly. She wrote her explanation, and when I asked her to go back and show me what her thinking "looked like," she drew a picture and a chart (see Figure 5-3). She knew how to organize the problem in her head, she had a solid understanding of multiplication, and she could also show her thinking visually.

I have begun having the children create presentations for some of their solutions and share them with the whole class. These presentations serve two purposes. One, I am able to observe what the kids know about problem solving on a broader scale. They are not only solving the problem mathematically, they are also determining what materials to use, what will show up well in front of an audience, and how they can explain the problem well enough so that the audience will understand the concepts they are trying to get across. Two, the visual nature of a presentation usually forces them to solve the problem in a different way, thus gaining further understanding of the concept.

Anna chose to present a problem that she had originally solved by using Unifix cubes:

There are 36 dogs on a team of dog sleds and 6 people. Each person needs to take a group of dogs. To make I tfair, each person will take care of the same number of dogs. So, how many dogs will each person have to take care of?

Her written explanation was, "There are 6 dogs in each group. I used cubes and put them into groups." Her presentation was much more detailed. She cut out six people and glued them onto a piece of poster board. She then cut out thirty-six dogs and divided them evenly between each cutout person. It was clear that she understood what she had done: "Well, first I cut out these people.

Those are the ones that need to take care of the dogs. Then. I cut out these dogs. Thirty-six of them. Then, I said to myself, One for you, one for you, one for you all the way 'til there wasn't any more left." When she was finished, the audience offered comments or asked questions. Many students commented about how easy it was to understand the picture, saying that is was neat and her explanation was clear.

Carly and Tessia ,vorked together on their presentation for this problem:

11 hunters each killed 2 seals. Then, they put all of their seals in a pile and divided them up equally among 6 families. How many seals would each family get? Will their be extra? How many extra? What should the hunters do with the extra?

The display for their presentation was great. They used black poster board for the background and bright colors for the pictures. They made the eleven hunters out of green paper and put two orange circles representing the seals next to the hunters. Then they drew six families and showed that each family would get three and there would be four extra seals. They also made a key explaining their symbols. They presented the problem together, and each had a different idea about what would happen to the extra seals.

Tessia: I thought that they could split the seals in half and then each family could get another half and then the extra seal could go to the dogs."

Carly: "I thought that they could use the seals for clothes and stuff like that."