Promoting Problem-Posing Explorations y fourth-grade class had just completed an exploration of pentominoes (polygonal shapes with an area of five square units). Finding all twelve shapes gives children valuable geometric problem-solving practice by highlighting transformations (flips, slides, and turns) and congruence (shapes can be differently oriented, yet congruent). Before moving on to another lesson, I realized that the students might use the same twelve shapes to examine perimeter and area. Eleven of the shapes have a perimeter of twelve units. Only one shape yields a different perimeter, ten units (see fig. 1). The children had limited experience with perimeter and area; I doubted that they understood that shapes with a fixed area could have perimeters of different lengths. Because they were so familiar with the pentominoes, I felt that this material would give them a good opportunity to address these concepts in more detail. Although I did expect them to calculate the perimeters and areas of the twelve shapes, I did not foresee that the children’s follow-up discussion would open an opportunity for problem-posing explorations. This article describes my evolving curricular decision making, the children’s investigations, and what I learned from this unanticipated experience. As part of my originally planned “last day” of working with pentominoes, I reviewed the definitions of perimeter and area and asked the children to describe their calculations and observations in their journals. As the students worked, many expressed surprise and curiosity as they compared the twelve perimeters. At the close of the period, I gathered the children together and asked, “Why

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By Phyllis Whitin Phyllis Whitin, [email protected], is an associate professor of elementary education at Wayne State University. She is particularly interested in the role of communication in developing mathematical understanding.

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does only one shape have a perimeter of ten units, and eleven shapes have a perimeter of twelve units?” What follows is a portion of their discussion of the shape with the unique perimeter: Jonathan. Four [square units] are stuck together. Mark. You’ve moved two from the bottom and put them on the side (reducing outside edges). Jessie. With the 10 [perimeter] piece, there are two squares covered up. It would probably be different if we had six squares. Lisa. There’s a line in the middle; you can’t count it. Kara. You don’t count the inner bar lines. Tricia. Why is the perimeter twelve units long for all of the other shapes? I expected that the straight line [1 × 5 rectangle] would have the most. I admitted to the children that I was surprised and intrigued by their many insights and questions, and that we might find ways to pursue their ideas.

The Tensions of Curricular Decision Making Reflecting on this conversation, I recalled a statement from Principles and Standards for School Mathematics: “Teaching mathematics well is a complex endeavor, and there are no easy recipes” (NCTM 2000, p. 17). I faced several of the perennial challenges of teaching. First was the issue of time: Do I summarize the conversation and move on to another mathematical topic, or do I provide time for further investigation? Judging from the tone in the children’s voices and the interested look on their faces, I sensed that the initial problem had served to “pique students’ curiosity and draw them into mathematics” (NCTM 2000, p. 18). Jessie’s comment— “It would be different if we used six squares”— directly suggested next steps. Furthermore, the Teaching Children Mathematics / November 2004

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Photograph by David J. Whitin; all rights reserved

children’s explanations of “bunching up” and what one can “count” in calculating perimeter suggested that many were developing an understanding of the meaning of perimeter and area. Tricia, however, courageously admitted that her results did not match her predictions, and now she wanted to know why. Spending more time exploring perimeter and area might deepen students’ understanding of these often-confused mathematical ideas. Next, I wondered what kind of follow-up lesson design would encourage the children to “think, quesTeaching Children Mathematics / November 2004

tion, solve problems, and discuss their ideas, strategies, and solutions” (NCTM 2000, p. 18). I wanted the children to take responsibility for planning and carrying out personally meaningful and challenging investigations. I knew, however, that too much choice could be overwhelming for the children and difficult for me to manage. After discussing my choices with a colleague, I decided on the following steps: • Engage the children in examining another example closely (first day). 181

• Take notes as children talk about, write about, and draw their findings. • Create problem-posing extensions from their observations. • Ask children to choose one question and predict the results (a second day). • Provide time for children to compare and discuss their findings (a portion of third day).

The Foundation: A Common Experience Many of the children’s initial comments focused on the idea that the perimeter of “bunched up” shapes were less than the perimeter of more linear shapes with the same area. To highlight this idea, I decided to focus only on rectangles with fixed areas, instead of including other types of polygonal shapes. (I realized that this decision would eliminate some potentially rich investigations.) I there-

fore gave the children centimeter grid paper and the following directions: 1. Make rectangles that have an area of 12 square centimeters. 2. Calculate the perimeter of each rectangle. 3. Analyze and explain your findings. 4. State a theory, hypothesis, or “wonder” (“I wonder if . . . ”). The children found three possible rectangles: 1 × 12 (p = 26), 2 × 6 (p = 16), and 3 × 4 (p = 14) (see fig. 2). Even though the children all calculated the same results, their written explanations provided a variety of connections and raised new considerations: Lisa. You keep taking half away [half of the square units] and putting them up against each other. My theory is that when you have more than

Figure 1 The perimeters and area of the 12 pentominoes Pentominoes with an area of five square units and a perimeter of twelve units:

One pentomino with an area of five square units and a perimeter of ten units:

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one line in the middle, you have less perimeter, but when you have less lines, you have more perimeter. Troy. Area is how much you can hold in it. Perimeter is the width all the way around it. The perimeter got shorter because we bunched the squares together. Jeffrey. When there were more groups, the perimeter got smaller. Susannah. It reminds me of a person running around a field and it’s showing you each and every one of the steps you take. I predicted that the third rectangle would be less than 16 because the rectangle is almost like a square. Libby. They’re all even numbers [perimeters]. The children were developing a deeper understanding of perimeter and area by investigating relationships involving the two for rectangles with a fixed area. In past experiences with children, I have often found that they confuse the terms perimeter and area. Susannah’s definition of “running around a field and showing each and every one of the steps” and Troy’s definition, “Area is how much you can hold in it,” incorporated “everyday, familiar language” (NCTM 2000, p. 63) that their classmates could relate to as well. I devoted a portion of the next day to an oral sharing session. I was surprised by the interest that many children expressed in Libby’s idea. I had thought it was obvious that all perimeters of rectangles are even, but raising this topic for exploration proved to be one of the most fruitful activities of the next part of the children’s followup investigations. The children were now ready to pursue problem-posing extensions.

Figure 2 Perimeters of rectangles with 12 square units

p = 26

p = 16

p = 14

tions would make the problem-posing process more explicit to them (Vygotsky 1978). I therefore listed three observations on chart paper, leaving room below each. As the lesson began, I showed the list to the children and asked, “What questions could we ask about each statement?” When completed, our chart looked like this: 1. The perimeters are all even numbers (26 – 16 – 14 units). • What about other numbers [of square units]? • Are perimeters always even? 2. The perimeter changes quickly, then slowly: -10

Problem-Posing Extensions In reviewing some of the students’ comments that evening, I developed a short representative list of observations. I wanted the list to encompass all the children’s ideas in order to preserve their sense of ownership, yet be short enough to be manageable for both the children and me. I carefully retained the children’s own descriptive language as I wrote the list. Although the children’s observations were statements, they suggested problem-posing extensions (Whitin 2004). Their statements identified important attributes of the problem (Brown and Walter 1990). The next step would be to pose questions for further investigation. I realized that involving the children in framing extending quesTeaching Children Mathematics / November 2004

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26 16 14. • What about other numbers [of square units]? • Is this always true? 3. The more the rectangles get bunched up, the less the perimeter is. • What about other numbers [of square units]? • Is this always true? Next, I gave the children the following directions, which I had written on chart paper: 1. Choose a question for your investigation. 2. Record it in your journal. 3. Make a prediction and give a reason for this prediction. 183

Figure 3 The pattern that Jenna discovered Array 1 × 12 2×6 3×4

Perimeter 26 > 10 16 >2 14

4. Investigate your problem. A sampling of the children’s plans included the following: Tricia. I’m going to investigate if all perimeters are even. I predict that they are all even because we have tried it and even if the area is odd (5) you will come out even. Troy. My plan is to see how much the perimeter drops when the square centimeters get bunched up. I predict that the perimeter will be around 30 on a 1 × 18. Lisa. I predict that 18 will drop just like 12 and the last two numbers will have just a few numbers in between them [number 2 above, using a fixed area of 18].

Examining Odd and Even Numbers After making their plans, the children worked for the remainder of the class period, and they discussed their findings on the following day. Several children investigated whether perimeters of rectangles are always even. Will and Susannah used examples for their “proof.” Will tried areas of 18 and 24. By studying the rectangles, he concluded that odd numbers added to odd numbers equaled even numbers. He explained: The perimeter is always even because when you add two odd numbers [for example, 1 + 1] you get an even number, and when you add two even numbers [24 + 24], you get an even number. 1 × 24, p = 50 2 × 12, p = 28 3 × 8, p = 22 4 × 6, p = 20 Yes, the perimeter is even! My prediction was correct!! 184

Array 1 × 18 2×9 3×6

Perimeter 38 > 16 22 >4 18

Susannah investigated rectangles with an area of 100 (for example, 50 × 2, 25 × 4) and found that their perimeters were even. She wrote, “I found that the 100s are even too [the perimeters]. I have a theory that out of all the numbers in the world, there are not many ones that you can make an odd perimeter.” She claimed that to prove her conjecture she needed to test “all the numbers in the world.” Until she did so, she was still doubtful. Her doubt hinted at the need for a proof of the generalization. Tricia’s argument better approached the generalization for adding odd numbers. She wrote, “If you add even and odd two times, you might get an even—like 6 × 3, you add 6 two times and 3 two times, you get 18, and 18 is even. Like 6 + 3 = 9, and then add 6 + 3 again and you get 9, and so 9 + 9 is 18.” Tricia was using the commutative and associative laws to show how the arrangement of odd and even numbers would still yield an even sum: 6 + 6 + 3 + 3 = (6 + 3) + (6 + 3). Jeffrey had another question about adding odd and even numbers. He noticed that because we were working with rectangles, odd numbers were always added in pairs (3 + 3). He then mused, “What if you add two odd numbers that are not the same? Three plus five is even.” Jeffrey’s comment helped the class consider that in all cases, odd plus odd equals even. By bringing all these children’s ideas to the class sharing session, the students were better able to address the generalizations for adding odd and even numbers.

Exploring Other Questions Other children pursued different questions on the problem-posing list. Jenna and Lisa were interested in looking at the numerical differences among perimeters (“Is it always true that perimeters change quickly, then slowly?”). Jenna noticed that the rectangles for an area of 12 and an area of 18 had “the same shape, but the 18s are bigger” (they have more area). In both instances, the Teaching Children Mathematics / November 2004

biggest drop in perimeter occurred when the long array was divided in half. The third rectangle in both examples showed a less dramatic change. Even though she did not reach a mathematical generalization, she discovered a pattern that occurred across several rectangles. Jenna helped show that after this initial halving, the perimeter did not change as dramatically (see fig. 3). Lisa, on the other hand, was most intrigued with the 6 × 3 array. She noticed that the number of linear units (p = 18) was the same as the number of square units (a = 18). Her writing opened a new question to pursue: “I wonder if any other number [rectangle] has the same number for perimeter and area.” (Of course, it is important to clarify that the same “number” applies to two different units of measure.) Lisa’s speculation demonstrated once again that raising new questions is part of the problem-posing cycle. Troy and Jessie were interested in the relationship between the shape of a rectangle and its perimeter (“Is it always true that the more rectangles get bunched up, the less the perimeter is?”). They each found that an area of 18 yielded three different rectangles with decreasing perimeters. Troy explained his discovery by writing, “I noticed that the more bunched up the square cm get, the littler the perimeter gets because all the square cm are inside and you aren’t able to count part of their perimeter” (see fig. 4). Jessie used the square as a reference point for her description: “The closer a shape is to a square, the smaller the perimeter.” These insights are important because they build a foundation for discovering that a square is the rectangle that yields the least perimeter for any given area. Kara also chose the “bunched up” question, but her observations addressed an issue that I had taken for granted. She made one 1 × 18 rectangle, and one 3 × 6, but decided to show the 2 × 9 twice— one oriented vertically on the page, and the other horizontally. She carefully marked each segment of the 2 × 9 arrays in order to calculate the perimeter of 22, and she wrote consecutive numbers in the squares to show the area of 18. Then she wrote, “I noticed that when I put 2 × 9 two times but in different directions, they might have the same perimeter. So I counted it and they were the same!” Kara’s writing gave me a window into her thinking. I had assumed that all the children understood that orientation does not affect size and shape, yet apparently Kara had not been so sure. I realized that as teachers, we cannot assume what is clear to children. Problem posing allows children to test out Teaching Children Mathematics / November 2004

Figure 4 Troy’s work

what is intriguing to them. In this context, Kara had the freedom to make sense of this fundamentally important idea for herself. In reflecting on the children’s work and the discussions of odd and even numbers, I considered the benefits of this kind of problem-posing investigation. This work involving perimeter and area incorporated several mathematical ideas: factors and products, odd and even numbers, mathematical patterns, properties of rectangles, spatial orientation, and, of course, the skill of calculating perimeter and area. These choices gave students of various abilities the opportunity to challenge themselves by pursuing what was most intriguing to them.

Reflecting on the Experience This extended experience with perimeter and area helped me clarify several aspects of my role as a teacher. First, that the children wrote about their observations was important. Providing some openended prompts such as “What do you notice?” and “What do you predict?” encouraged these fourth graders to identify a range of attributes that described relationships involving perimeter and area for different types of rectangles. Their oral and written statements gave me an opportunity to assess their understandings and doubts, as well as their interests. Their observations, such as “Perimeters are even,” raised issues that I had not envisioned and guided my curricular decisions. Second, this experience highlighted the role of 185

teachers in planning, organizing, and developing tasks with their students. It is teachers who must decide “what aspects of a task to highlight, how to organize and orchestrate the work of the students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminating the challenge” (NCTM 2000, p. 19). The children’s initial conversation indicated that continued work with perimeter and area would be worthwhile. They showed interest in the topic, and their observations highlighted several interconnected mathematical ideas. Engaging them in the systematic process of turning their observations into problems could strengthen the disposition to “analyze situations carefully in mathematical terms and to pose problems based on situations they see” (NCTM 2000, p. 53). This experience also illustrated that by highlighting only certain attributes of the problem, I eliminated other potentially rich investigations. In the opening conversation, Tricia expressed a keen interest in the fact that most of the different shapes (with an area of five square units) had the same perimeter.

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Jessie’s observation that six square units would “probably be different” also suggested work with hexominoes or even other types of polygonal shapes. I was aware that by having the children work only with rectangles, I would sacrifice these excellent ideas, but I felt unable to manage too many choices. The explorations that the class did pursue within these parameters encouraged me to look for future opportunities in which I would be able to support an even broader range of student investigations.

References Brown, Stephen, and Marion Walter. The Art of Problem Posing. Hillsdale, N.J.: Lawrence Erlbaum, 1990. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. Vygotsky, Lev. Mind in Society. Cambridge, Mass.: Harvard University Press, 1978. Whitin, David J. “Building a Mathematical Community through Problem Posing.” In Perspectives on the Teaching of Mathematics, 66th Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Rheta N. Rubenstein, pp. 129–40. Reston, Va.: NCTM, 2004. ▲

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