Problems to Deepen Teachers’ Mathematical Understanding: Examples in Multiplication or children to learn to reason, represent, communicate, and build connections among mathematical ideas, their teachers need a strong mathematical foundation (CBMS 2001; Ma 1999; NCTM 2000). Laying the groundwork for such teacher education includes supporting teachers in developing mathematical concepts meaningfully, seeing unifying ideas, learning to reason and justify, solving problems strategically, and making connections among ideas in multiple forms. We believe that teachers’ learning mathematics themselves in these ways increases their capacity for guiding similar learning for children. As one of our preservice teachers has said very simply, “In order for me to teach for understanding, I need to understand myself.” What is needed to prepare teachers in this way? In our work of reforming our curriculum at the University of Michigan—Dearborn, we have found that selecting, modifying, and orchestrating worthwhile tasks and problems for future teachers is highly challenging. We see problems as a major tool for our work in curriculum and instruction. We are becoming increasingly convinced that problems need to engage, motivate, and challenge

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teachers while providing a springboard for discussions that develop and deepen their mathematical understanding. A particularly powerful approach is to use problems that are unfamiliar to all. When problems challenge everyone, then the playing field is leveled, engagement is heightened, and a wide range of responses can be brought to the fore. Starting with mathematics education research, exemplary K–8 curriculum projects, and our own teaching experience, we have been developing an array of problems for use in class work, homework, projects, writings, and examinations. We have been identifying problems in three categories, as shown in the selected examples in figure 1. Whereas we have used problems from the first two categories quite heavily in our content courses, we have usually reserved problems in the third category (problems related to classroom teaching) for our methods course. This approach has both advantages and disadvantages. For example, one advantage is that problems in the third category elicit pedagogical issues that are better addressed in a methods course. One disadvantage of delaying these classroom-related problems is that they appeal to future teachers’ interest in seeing how their mathematics learning will relate to future practice.

By Judith Flowers, Angela S. Krebs, and Rheta N. Rubenstein Judith Flowers, [email protected], Angela Krebs, [email protected], and Rheta Rubenstein, [email protected], are colleagues in the Department of Mathematics and Statistics at the University of Michigan—Dearborn. They are interested in helping preservice teachers understand mathematics in ways that support their guiding meaningful mathematics learning for children.

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Developing Problems for Whole-Number Multiplication In the following sections we present some examples of problems we have been developing in a unit on whole-number multiplication for a first course in mathematics for elementary school teachers. Teaching Children Mathematics / May 2006

Copyright © 2006 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM.

(Related work on subtraction can be found in Flowers, Kline, and Rubenstein 2003.) The examples are drawn from the first category in figure 1. In the remainder of the article we identify problems by type and sequence them as they might appear in class activities, revealing in each instance the objectives and some of the flavor of the class discussion. A major thrust of our work has been to emphasize future teachers’ learning the reasoning behind mathematical procedures and helping them see the potential within the realm of calculation for understanding meanings of operations, properties, and number relationships. As Carpenter (1985) and Hiebert (2003) have noted, reasoning related to calculation is mathematical problem solving.

Create a story We have been preceding the unit by having the future teachers do some reading about meanings of Teaching Children Mathematics / May 2006

multiplication. In class we then ask them to talk with one another about the key ideas they have gleaned. Then we pose a task, such as the following: “Write a story problem or think of a context for the calculation 12 × 9. Represent your situation with a diagram.” We find they often fall into stories with common meanings, such as those adding a value repeatedly (equal groups). To move them further, we encourage them to think of another context or story that has a different “feel.” We have them post several different stories and representations on the chalkboard, making sure to select ones in which the factors, 9 and 12, play different roles. Typical stories are “I have 12 bags of 9 cookies each. How many cookies are there?” or “I have 9 boxes of donuts. Each box has 12 donuts. How many donuts are there?” The representations in figure 2 illustrate variations that may arise from different stories. 479

Figure 1 Categories of mathematics problems and examples Problems centered on teachers’ own problem solving • Create a story to fit a given calculation • Complete a calculation from a given “starter” • Create equivalent problems • Use a cluster • Solve in more than one way • Explain another person’s reasoning • Justify the reasoning in a solution • Compare representations, strategies, or meanings in identified solutions • Solve with representations, using two or more different meanings of an operation • Explore the effects of changing the numbers in a problem • Analyze the mathematics underlying an error or misconception • Map nonstandard procedures to standard procedures • Classify strategies • Identify and analyze relationships among problems (current and former) • Prove that a strategy always works • Generalize a strategy Problems using children’s work as a source • Analyze student work (What was the child thinking? What does he or she know?) • Evaluate student work (which may contain misconceptions or creative solutions novel to teachers) • Speculate on sources of misconceptions • Solve another problem with a child’s strategy • Decide whether a child’s strategy is generalizeable Problems related to classroom teaching • Identify the mathematics embedded in a task • Anticipate a range of student responses to a task • Revise a task to make it more challenging • Select or modify tasks to fit certain learning goals • Choose numbers for a task to make it challenging enough to be engaging but simple enough to be elucidating • Sequence problems by difficulty level • Propose a reasonable justification that a student could make • Compare the affordances and limitations of particular models

Out of a subsequent discussion of similarities and differences among their stories and representations comes an awareness of the different roles that the 9 and the 12 may play: A factor may represent the number of items in each group, or it may represent the number of equal groups. The distinction makes them think more deeply about multiplication. It also lays the groundwork for later work with division, in which a divisor may be the number of groups (partitive division) or may be the size of each group (measurement or quotitive division). As future teachers explore the results of their multiplication work and their labels for factors and products, they discover that the units change. For 480

example, in the donuts problem, we begin with boxes and donuts per box and end with donuts. When they are encouraged to compare this outcome with that from other operations, the future teachers come to realize that the units do not change for addition or subtraction, whereas they can for multiplication. We have speculated that recognizing this distinction may help them discriminate multiplicative from additive situations. By asking for stories that “feel” different, a few teachers usually produce a situation involving the area of a rectangle. (If no one proposes such a story, we ask them to think of geometric uses of multiplication, or, because this view of multiplication is so powerful, we offer such a story ourselves.) We see multiple benefits to bringing this image to the table. First, it broadens the teachers’ view of multiplication and its uses. Second, it provides a means for illustrating commutativity. Unlike in the “equal groups” stories, which produce different images when factors play different roles, in the area model one rectangle is produced regardless of which factor is length or width. This representation provides a basis for generalizing and seeing why multiplication is commutative. Although the teachers have known this fact before and have explored it inductively with many examples, here they see the concept in a single convincing image.

Justify the reasoning in a solution Multiplying by 10 or its multiples is easy for preservice teachers. Explaining what is really going on is not. They know rules, but are not used to unpacking the mathematics undergirding these rules. They are also uncomfortable when they realize that they were never encouraged to develop the kind of knowledge necessary to understand these memorized procedures. Helping teachers make sense of these operations and realize that this understanding is needed for teaching continues to be a challenge for us. Here are some strategies that we are using to make inroads. We have begun by asking teachers to calculate 28 × 400 using mental-mathematics strategies, then to explain and justify their thinking. Many teachers begin with 28 × 4. To the product, 112, they “add two zeroes.” They are challenged to explain why the “add zeroes” rule works. Their beginning explanations are often based on carrying out the standard algorithm and their knowledge that a number times zero equals zero. For example, “There’s a zero in the ones place, so when I multiTeaching Children Mathematics / May 2006

Figure 2

Figure 3

Two representations of 12 × 9

Student strategy for the problem 28 × 400 and class analysis 1. Althea was using reasoning to find 28 × 400. Analyze the logic of her steps. 30 × 400 = 12,000 2 × 400 = 800 12,000 – 800 = 11,200

12 groups of 9

9 groups of 12

ply by that I get a zero in the ones place of my answer. It’s the same for the zero in the tens place. So I have to get two zeros.” The description is accurate, but it does not reflect an understanding of the base-ten structure of the number system. We encourage the teachers to dig deeper. We persist in asking, “Why do we ‘add on’ the zeroes? What is really going on mathematically?” What we are learning from this activity is that future teachers need a more robust understanding of, and flexibility with, place value. For example, few of them realize that 11,200 has many meanings: 10 thousand + 1 thousand + 2 hundred, or 112 hundreds, or 1120 tens, and so on. One way we are addressing teachers’ needs is to ask them to analyze others’ justifications. For example, we ask them to explain Althea’s thinking for 28 × 400 and justify her steps for themselves (see fig. 3, problem 1). Teaching Children Mathematics / May 2006

2. Students in another class were asked to justify Althea’s first step. Evaluate their justifications: • Student A: 3 times 4 is 12; add three zeroes to get 12,000. • Student B: 30 × 4 = 120 × 100 = 12,000. • Student C: 30 times 4 is 120. 400 is 100 times greater than 4, so 30 times 400 is 120 hundreds. I can write 120 hundreds as 12,000. • Student D: 30 times 400 is 30 4-hundreds, which is the same as 120 1-hundreds, which is the same as 12 1-thousands, and that is 12,000. • Student E: I am thinking that I have 30 packs with 400 dollars in each pack. If it were only 10 packs, I would have 4000 dollars total. So for 30 packs I would have 3 times as much, or 12,000 dollars total.

Analyzing the work of others and overlaying their own reasoning prepares them for evaluating others’ justifications. We believe that doing so builds better understanding of the reasoning presented, helps them evaluate their own justifications, and supports their clarification of their own ideas. Next we present a set of possible explanations (see fig. 3, problem 2). The future teachers are asked to interpret and analyze the justifications offered. This activity helps them identify symbolization errors (for example, in student B’s work) and discern among strong justifications, partial explanations, and unjustified descriptions. The good examples provide models that help them develop place-value language and connect it with the notation for meanings of numbers.

Use a cluster Standards-based elementary curriculum materials are resources that resonate with future teachers and that have strong potential for helping them reason

Figure 4 Cluster problems prompt multiple solution paths. Use problems in the cluster or others you create to help you solve the bolded last problem. 8 × 25 40 × 25

30 × 25 38 × 25

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Figure 5 Area models for 25 × 38 38 38

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Figure 6 General task using clusters 2 × 72

10 × 72

5 × 72

20 × 72

200 × 72

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210 × 72

1. How are the problems related? How can relationships be used to solve the individual problems? 2. How do the problems in the cluster help solve the final problem in bold?

Figure 7 “Equal group” problems promote multiplicative reasoning. Without calculating, find values that make each sentence true. Explain why your values work.

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recognize how decomposing one of the factors produces subproblems that are easier to solve. In the language of equal groups, teachers might say, “38 groups of 25 is the same as 8 groups of 25 and 30 groups of 25.” When they are ready to translate their words into symbols, they see that 38 × 25 = (8 × 25) + (30 × 25), an illustration of the distributive property. Someone could also reason that 38 groups of 25 is the same as 40 groups of 25 minus 2 groups of 25, or 38 × 25 = (40 × 25) – (2 × 25), illustrating that multiplication distributes over subtraction as well as over addition. Interestingly, we generally find that future teachers produce explanations based on “equal groups.” When asked for other ways to represent these multiplications, teachers help one another see that an area model fosters insight (see fig. 5). A more general task using clusters is shown in figure 6. This example illustrates a course theme of having teachers find relationships and look for multiple solution paths.

■ groups of 12 = 10 groups of 6

Create equivalent problems

30 groups of 2 = 10 groups of ■

To promote multiplicative reasoning we have been using a set of problems, such as the one shown in figure 7, that focus on an equal-groups meaning of multiplication. In each of the first two problems, a factor that plays a different role is missing. In explaining their reasoning, teachers begin to say such things as, for the first problem, “If groups of size 6 are changed to groups of 12, there have to be half as many groups, so 10 groups of 6 equals 5 groups of 12”; or, for the second problem, “If the number of groups is changed to one-third the original, then there need to be three times as many things in each group, so 30

■ groups of 7 = ■ groups of 21 Adapted from Lampert (2001)

mathematically. As one example, we have been using Investigations in Number, Data, and Space (Russell et al. 1998), a program that uses cluster problems for multiplication and for developing number sense and appreciation of number relationships, such as the distributive property. For example, in the cluster for 38 × 25 in figure 4, teachers 482

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Figure 8 Student’s representation that 5 groups of 12 equals 10 groups of 6

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Figure 9 Area model

groups of 2 equals 10 groups of 6.” Explanations necessitate the language needed for multiplicative comparisons, for instance, half as many or three times as much, and they pave the way for proportional reasoning. We also press for representations that justify the reasoning, and we ask for connections among words, symbols, and pictures. The representation in figure 8 illustrates the assertion that 5 groups of 12 equals 10 groups of 6. This visual representation helps teachers judge the assertion that doubling the size of a group results in half as many groups when the total, or product, stays the same. An area model can also be used, as shown in figure 9. The third problem helps future teachers generalize the relationship. “If the size of the group is three times larger, the number of groups is divided by 3.” Some of them may generalize the relationship of equivalent multiplications: a × b = an × b/n. The underlying idea of equivalence is a major Teaching Children Mathematics / May 2006

idea in all of mathematics and arises repeatedly in arithmetic contexts. For example, in the realm of subtraction, teachers in our classes have talked about the age of an adult and that of a child, and the fact that the difference remains constant as each person’s age increases with time. This fact is related to the idea that, in general, the difference of two numbers is unaffected if the same value is added to both numbers being subtracted. With multiplication, we ask the future teachers to find equivalent problems for such products as 36 × 25. They can build on the work with proportional reasoning discussed previously. They tend to use doubling and halving as initial strategies (producing, for example, 18 × 50), but later realize that other factors can be used. When explaining what is going on, some use equal groups or area explanations like those described in the previous section. Some connect what is happening when multiplying and dividing by the same number as equivalent to multi483

plying by 1 or multiplying 1/n by n. Some see an implied factoring, for example, 36 × 25 = (9 × 4) × 25 = 9 × (4 × 25). In general, we find that emphasizing the topic of equivalence has many benefits in helping teachers see how operations are related.

Solve in more than one way Selecting appropriate strategies is an important aspect of computational proficiency (National Research Council 2001, p. 122), a major goal of our course. To this end, we regularly ask teachers to solve problems in more than one way. Moreover, we strongly encourage them to use methods with which they are less comfortable, so that they broaden their understanding and develop a wide range of strategies. As an example we pose the problem “Find 25 × 364. Use more than one strategy.” We want future teachers to recognize the features of a computation that make using specific strategies advantageous. Accordingly, in the example given, we would hope that some recognize that the numbers lend themselves to using an equivalent multiplication (100 × 91). Producing and analyzing a range of strategies help teachers recognize the role that number properties play in different approaches. They need to realize that when they decompose one of the factors and multiply each of the parts by the second factor, they are applying the distributive property, as in 25 × 364 = 25 × 300 + 25 × 64. When they factor the factors to find “nice” factor pairs, they are using the associative property, for example, 25 × 364 = 25 × (4 × 91) = (25 × 4) × 91 = 100 × 91.

Reflections We have found that when we focus the content of a preservice mathematics course on reasoning, justification, and connections, different types of problems emerge as productive. We are striving to create problems and tasks that share qualities with tasks of higher cognitive demand. According to Smith and Stein (1998), tasks at the highest level of cognitive demand require— • complex, nonalgorithmic thinking; • exploring and understanding the nature of mathematical concepts, processes, and relationships; • accessing relevant knowledge and experiences and making appropriate use of them in working through the task; and • considerable cognitive effort and may involve some level of anxiety for the (learner) because of 484

the unpredictable nature of the solution process. We believe that teachers’ experiencing these struggles and the consequent joy of learning deepens their understanding and increases the likelihood that they will challenge their own students in comparable ways. We hope the analysis and examples of problem types provided here are useful in other programs aiming to develop, support, and deepen teachers’ mathematical knowledge for teaching.

References Carpenter, Thomas P. “Learning to Add and Subtract: An Exercise in Problem Solving.” In Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, edited by Edward A. Silver, pp. 17–40. Hillsdale, NJ: Lawrence Erlbaum Associates, 1985. Conference Board of Mathematical Sciences (CBMS). The Mathematical Education of Teachers. Providence, RI: American Mathematical Society, 2001. Flowers, Judith, Kate Kline, and Rheta N. Rubenstein. “Developing Teachers’ Computational Fluency: Examples in Subtraction.” Teaching Children Mathematics 9 (February 2003): 330–34. Hiebert, James. “Signposts for Teaching Mathematics through Problem Solving.” In Teaching Mathematics through Problem Solving: Prekindergarten–Grade 6, edited by Frank K. Lester Jr. and Randall I. Charles, pp. 53–61. Reston, VA: National Council of Teachers of Mathematics, 2003. Lampert, Magdalene. Teaching Problems and the Problems of Teaching. New Haven, CT: Yale University Press, 2001. Ma, Liping. Knowing and Learning Mathematics for Teaching: Teachers’ Understanding of Fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates, 1999. National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, VA: NCTM, 2000. National Research Council. Adding It Up: Helping Children Learn Mathematics, edited by Jeremy Kilpatrick, Jane Swafford, and Bradley Findell. Washington, DC: National Academy Press, 2001. Russell, Susan Jo, et al. Investigations in Number, Data, and Space. White Plains, NY: Dale Seymour Publications, 1998. Smith, Margaret Schwan, and Mary Kay Stein. “Selecting and Creating Mathematical Tasks: From Research to Practice.” Mathematics Teaching in the Middle School 3 (February 1998): 344–50.

This article is based on work supported by the National Science Foundation under grant DUE 0310829. Any opinions, findings, conclusions, or recommendations expressed herein are those of the authors and do not necessarily reflect the views of the National Science Foundation. ▲ Teaching Children Mathematics / May 2006