### Problem Solving Standard for Grades 3–5

 Instructional programs from prekindergarten through grade 12 should enable all students to— build new mathematical knowledge through problem solving; solve problems that arise in mathematics and in other contexts; apply and adapt a variety of appropriate strategies to solve problems; monitor and reflect on the process of mathematical problem solving.

Problem solving is the cornerstone of school mathematics. Without the ability to solve problems, the usefulness and power of mathematical ideas, knowledge, and skills are severely limited. Students who can efficiently and accurately multiply but who cannot identify situations that call for multiplication are not well prepared. Students who can both develop and carry out a plan to solve a mathematical problem are exhibiting knowledge that is much deeper and more useful than simply carrying out a computation. Unless students can solve problems, the facts, concepts, and procedures they know are of little use. The goal of school mathematics should be for all students to become increasingly able and willing to engage with and solve problems.

Problem solving is also important because it can serve as a vehicle for learning new mathematical ideas and skills (Schroeder and Lester 1989). A problem-centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion. Good problems can inspire the exploration of important mathematical ideas, nurture persistence, and reinforce the need to understand and use various strategies, mathematical properties, and relationships.

#### What should problem solving look like in grades 3 through 5?

Students in grades 3–5 should have frequent experiences with problems that interest, challenge, and engage them in thinking about important mathematics. Problem solving is not a distinct topic, but a process that should permeate the study of mathematics and provide a context in which concepts and skills are learned. For instance, in the following hypothetical example, a teacher poses these questions to her students:

 If you roll two number cubes (both with the numbers 1–6 on their faces) and subtract the smaller number from the larger or subtract one number from the other if they are the same, what are the possible outcomes? If you did this twenty times and created a chart and line plot of the results, what do you think the line plot would look like? Is one particular difference more likely than any other differences?

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Initially, the students predict that they will roll as many of one difference as of another. As they begin rolling the cubes and making a » list of the differences, some are surprised that the numbers in their lists range only from 0 to 5. They organize their results in a chart and continue to mark the differences they roll (see fig. 5.24). After the students have worked for a few minutes, the teacher calls for a class discussion and asks the students to summarize their results and reflect on their predictions. Some notice that they are getting only a few 0's and 5's but many 1's and 2's. This prompts the class to generate a list of rolls that produce each difference. Others list combinations that produce a difference of 2 and find many possibilities. The teacher helps students express this probability and questions them about the likelihood of rolling other differences, such as 0, 3, and 5.

 Fig. 5.24. A chart of the frequency of the differences between the numbers on the faces of two dice rolled simultaneously

The questions posed in this episode were "problems" for the students in that the answers were not immediately obvious. They had to generate and organize information and then evaluate and explain the results. The teacher was able to introduce notions of probability such as predicting and describing the likelihood of an event, and the problem was accessible and engaging for every student. It also provided a context for encouraging students to formulate a new set of questions. For example: Could we create a table that would make it easy to compute the probabilities of each value? Suppose we use a set of number cubes with the numbers 4–9 on the faces. How will the results be similar? How will they be different? What if we change the rules to allow for negative numbers?

Good problems and problem-solving tasks encourage reflection and communication and can emerge from the students' environment or from purely mathematical contexts. They generally serve multiple purposes, such as challenging students to develop and apply strategies, introducing them to new concepts, and providing a context for using skills. They should lead somewhere, mathematically. In the following episode drawn from an unpublished classroom experience, a fourth-grade teacher asked students to work on the following task:

Show all the rectangular regions you can make using 24 tiles(1-inch squares). You need to use all the tiles. Count and keep a record of the area and perimeter of each rectangle and then look for and describe any relationships you notice.

When the students were ready to discuss their results, the teacher asked if anyone had a rectangle with a length of 1, of 2, of 3, and so on, and modeled a way to organize the information (see fig. 5.25).

 Fig. 5.25. The dimensions of the rectangular regions made with 24 one-inch square tiles

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The teacher asked if anyone had tried to form a rectangle of length 5 and, if so, what had happened. The students were encouraged work with partners to make observations about the information in the chart and their rectangular models. They noticed that the numbers in the first two columns of any row could be multiplied to get 24 (the area). The teacher noted their observation by writing "LW = 24" and used the term factors of 24 as another way, in addition to length and width, to describe the numbers in the first two columns. Some students noticed that as the numbers for one dimension increased, those for the other dimension decreased. Still others noted that the perimeters were always even. One student asked if the rectangles at the bottom of the chart were the same as the ones at the top, just turned different ways. This observation prompted the teacher to remind the students that they had talked » about this idea as a property of multiplication—the commutative property—and as congruence of figures.

The teacher then asked the students to describe the rectangles with the greatest and smallest perimeters. They pointed out that the long "skinny" rectangles had greater perimeters than the "fatter" rectangles. The teacher modeled this by taking the 1-unit-by-24-unit rectangle of perimeter 50, splitting it in half, and connecting the halves to form the 2-unit-by-12-unit rectangle (see fig. 5.26). As she moved the tiles, she explained that some tile edges on the outside boundary of the skinny rectangle were moved to the inside of the wider rectangle. Because there were fewer edges on the outside, the perimeter of the rectangle decreased.

 Fig. 5.26. Forming a 212 rectangle from a 124 rectangle

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The "24 tiles" problem provides opportunities for students to consider the relationship between area and perimeter, to model the commutative property of multiplication, to use particular vocabulary (factor and multiple), to record data in an organized way, and to review basic number combinations. It reinforces the relationship LW = A. It also allows the teacher to help students with different needs focus on different » aspects of the problem—building all the rectangles, organizing the data, looking for patterns, or making and justifying conjectures.

Reflecting on different ways of thinking about and representing a problem solution allows comparisons of strategies and consideration of different representations. For example, students might be asked to find several ways to determine the number of dots on the boundary of the square in figure 5.27 and then to represent their solutions as equations (Burns and Mclaughlin 1990).

 Fig. 5.27. The "dot square" problem

Students will likely see different patterns. Several possibilities are shown in figure 5.28. The teacher should ask each student to relate the drawings to the numbers in their equations. When several different strategies have been presented, the teacher can ask students to examine the various ways of solving the problem and to notice how they are alike and how they are different. This problem offers a natural way to introduce the concept and term equivalent expressions.

In addition to developing and using a variety of strategies, students also need to learn how to ask questions that extend problems. In this way, they can be encouraged to follow up on their genuine curiosity about mathematical ideas. For example, the teacher might ask students to create a problem similar to the "dot square" problem or to extend it in some way: If there were a total of 76 dots, how many would be on each side of the square? Could a square be formed with a total of 75 dots? Students could also work with extensions involving dots on the perimeter of other regular polygons. By extending problems and asking different questions, students become problem posers as well as problem solvers.

 Fig. 5.28. Several possible solutions to the "dot square" problem

#### What should be the teacher's role in developing problem solving in grades 3 through 5?

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Teachers can help students become problem solvers by selecting rich and appropriate problems, orchestrating their use, and assessing students' understanding and use of strategies. Students are more likely to develop confidence and self-assurance as problem solvers in classrooms where they play a role in establishing the classroom norms and where everyone's ideas are respected and valued. These attitudes are essential if students are expected to make sense of mathematics and to take intellectual risks by raising questions, formulating conjectures, and offering mathematical arguments. Since good problems challenge students to think, students will often struggle to arrive at solutions. It is the teacher's responsibility to know when students need assistance » and when they are able to continue working productively without help. It is essential that students have time to explore problems. Giving help too soon can deprive them of the opportunity to make mathematical discoveries. Students need to know that a challenging problem will take some time and that perseverance is an important aspect of the problem-solving process and of doing mathematics.

As students share their solutions with classmates, teachers can help them probe various aspects of their strategies. Explanations that are simply procedural descriptions or summaries should give way to mathematical arguments. In this upper elementary class, a teacher questioned two students as they described how they divided nine brownies equally among eight people (Kazemi 1998, pp. 411–12):

 Sarah: The first four we cut them in half. (Jasmine divides squares in half on an overhead transparency.) Ms. Carter: Now as you explain, could you explain why you did it in half? Sarah: Because when you put it in half, it becomes four ... four ... eight halves. Ms. Carter: Eight halves. What does that mean if there are eight halves? Sarah: Then each person gets a half. Ms. Carter: Okay, that each person gets a half. (Jasmine labels halves1 through 8 for each of the eight people.)
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 Sarah: Then there were five boxes [brownies] left. We put them in eighths. » Ms. Carter: Okay, so they divided them into eighths. Could you tell us why you chose eighths? Sarah: It's easiest. Because then everyone will get ... each person will get a half and (addresses Jasmine) ... how many eighths? Jasmine: (Quietly) Five-eighths. Ms. Carter: I didn't know why you did it in eighths. That's the reason. I just wanted to know why you chose eighths. Jasmine: We did eighths because then if we did eighths, each person would get each eighth, I mean one-eighth out of each brownie. Ms. Carter: Okay, one-eighth out of each brownie. Can you just, you don't have to number, but just show us what you mean by that? I heard the words, but... Jasmine: (Shades in one-eighth of each of the five brownies that were divided into eighths.) Person one would get this ... (points to one-eighth) Ms. Carter: Oh, out of each brownie. Sarah: Out of each brownie, one person will get one-eighth. Ms. Carter: One-eighth. Okay. So how much then did they get if they got their fair share? Jasmine & Sarah: They got a half and five-eighths. Ms. Carter: Do you want to write that down at the top, so I can see what you did? (Jasmine writes 1/2 + 1/8 + 1/8 + 1/8 + 1/8 + 1/8 at the top of the overhead transparency.)

In this discussion, the teacher pressed students to give reasons for their decisions and actions: What does it mean if there are eight halves? Could you tell us why you chose eighths? Can you show us what you mean by that? She was not satisfied with a simple summary of the steps but instead expected the students to give verbal justifications all along the way and to connect those justifications with both numbers and representations. This particular pair of students used a strategy that was different from that of other students. Although it was not the most efficient strategy, it did reveal that these students could solve a problem they had not encountered before and that they could explain and represent their thinking.

Listening to discussions, the teacher is able to assess students' understanding. In the conversation about sharing brownies, the teacher asked students to justify their responses in order to gain information about their conceptual knowledge. For any assessment of problem solving, teachers must look beyond the answer to the reasoning behind the solution. This evidence can be found in written and oral explanations, drawings, and models. Reflecting on these assessment data, teachers can choose directions for future instruction that fit with their mathematical goals.