Developing knowledge and understanding of

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D

eveloping knowledge and understanding of how children learn mathematics is one of the most important aspects in teacher education for both preservice and in-service teachers of mathematics. Teachers who understand how children learn mathematics can provide higher-quality instruction. Their understanding of what children know and are able to do allows them to understand students’ strategies and determine how to respond to them, what problems to pose next, and how to manage mathematical discussions (Ball 1997, 2000; Loucks-Horsley et al. 2003; Sowder 1998).

Teachers should be able to perceive mathematics through the minds of their students (NCTM 1991). For teachers in the primary grades, understanding number acquisi- tion from the perspective of young children can be a valuable tool for becoming more aware of the challenges that students face.

As adults, we have developed familiarity with the number sequence and flexible number sense; we often have difficulty remembering what it was like when the number sequence was new to us. Young children need time and opportunities to develop a conceptual under-

standing of numbers. Children’s proficiency in reciting the numbers up to 20 or beyond is not a guarantee that they have a sense of how big a number is, what quantity it represents, or how a number such as 5 can be decomposed into 4 + 1 or 3 + 2.

This article presents an activity with the alphabet that we use in our preservice and in-service teacher training programs. This alphabet activity is often used in professional development programs such as Cognitively Guided Instruction (CGI) (see Carpenter et al. 1999). We have adapted the activity for our teacher training programs. The goal is to help teachers understand what children go through as they begin to learn the number sequence and construct strategies for solving arithmetic problems. We begin with a discussion of children’s informal strategies for simple addition and subtraction. Then we discuss the alphabet activity that we use in the teacher education programs, and compare the strategies that children and teachers use.

Children’s Strategies for Adding and Subtracting Single-Digit Numbers

A number of studies have identified three levels of strategies that children construct to solve addition and subtraction word problems with single-digit numbers and how they evolve over time (see Fuson 1992; and Kilpatrick, Swafford, and Findell 2001 for a summary of the research findings). Initially, young children employ direct-modeling strategies, using physical objects to represent quantities, actions, and relationships in a problem. (Notice that here modeling refers to what children do, not what the teacher does.) Over time, the strategies evolve to counting strategies and derived facts.

What follows is an example of each strategy level.

How Does It Feel?

Teachers Count On the Alphabet

Instead of Numbers

Jae Meen Baek, jae.baek@asu.edu, and Alfinio Flores, alfinio@asu.edu, teach at Arizona State University in Tempe, Arizona. Baek teaches preservice and in-service teachers focusing on instructional practices based on children’s mathematical thinking. Her research interest is in understanding the development of children’s mathematical reasoning in the domains of numbers, operations, and algebra. Flores teaches mathematics methods courses for prospective and in-service teachers. He is interested in how to make mathematics learning meaningful for students and teachers.

Edited by Fran Arbaugh, ArbaughE@missouri.edu, and John Lannin, LanninJ@missouri.

edu. Arbaugh and Lannin are members of the mathematics education faculty at the University of Missouri—Columbia, Columbia, MO 65203. Send submissions to this department by accessing tcm.msubmit.net.

SUPPORTING TEACHER LEARNING

J a e M e e n B a e k a n d A l f i n i o F l o r e s

A B C

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• Level 1: Direct modeling. Children use objects or fingers to represent each of the quantities in the problem, put them together, and count all the objects. For 8 + 5, they make a set of 8 objects and a set of 5 objects, push them together, and count them all: “One, two, three, . . . , thirteen.”

• Level 2: Counting strategies. In applying a counting strategy, children do not represent both quantities in the problem; they represent only the second addend to keep track of how many are added. In solving 8 + 5, for example, children say, “8” and count, “9, 10, 11, 12, 13” as they keep track on their fingers or objects to make sure they count five more. Counting back from a given number is also a strategy used at this level. For example, to subtract 2 from 8, a child may say, “8,” and then count back, “7, 6”

and keep track to make sure she counts back two numbers. Many children keep track of how many numbers they count back using their fingers; others mentally keep track of it.

• Level 3: Derived facts. Children often learn cer- tain number facts first, such as doubles and sums of ten, and use them to solve other problems. For example, children often figure out that 5 + 8 is 13 because “5 and 5 is 10, and 3 more is 13.” Some facts must be memorized to use derived facts.

Children may use strategies at different levels depending on the difficulty of the problem. For other examples of children’s strategy use and a more complete discussion, see Carpenter et al.

(1999).

Counting with the Alphabet

As adults, we tend to underestimate the depth of understanding of numbers that children need in order to apply their knowledge in problem-solving situations. We often overlook how confused children feel when we expect them to make a big leap as soon as they start to recite the number sequence.

Children’s fluency in reciting the number sequence often leads us to assume that they have the same depth of understanding of quantities that we do.

To get a sense of what children go through as they develop number sense and problem-solving strategies, we introduce an activity in which preservice and in-service teachers must use the alphabet sequence instead of the number sequence to solve arithmetic problems. The letters A, B, C, D,

. . . , recited in order, serve to keep track of the number of objects. Thus, a collection such as ♣ ♣ ♣ would be described as having C objects. All the problems used in the activity involve only small collections, so there is no need to worry about what comes after Z in this activity. Teachers are instructed not to translate the letters to numbers.

We chose letters because we all know the alphabet sequence well, just as many young children know the number sequence, but we may not have mas- tered the quantity that each letter represents.

In this activity we present the following set of word problems and provide each teacher with Unifix cubes or other counting objects. The word problems are from Fennema et al. (1999). We replaced the numbers in the problems with letters for our activity.

• Lucy has H fish. She wants to buy E more fish.

How many fish would Lucy have then?

• Tee Jay had M chocolate chip cookies. At lunch she ate E of them. How many cookies did Tee Jay have left?

• Janelle has G trolls in her collection. How many more does she have to buy to have K trolls?

• Max had some money. He spent I dollars on a video game. Now he has G dollars left. How much money did Max have to start with?

• Willy has L crayons. Lucy has G crayons. How many more crayons does Willy have than Lucy?

• K children were playing in the sandbox. Some children went home. There were C children still playing in the sandbox. How many children went home?

At first, teachers work individually on the problems without using numbers. They may use the cubes or their fingers. After teachers solve the problems, they share their strategies. The examples of strategies presented in the next section are from teachers who participated in the alphabet activity.

Teachers’

Strategies

For the problems with letters, both preservice and in-service teachers in our classes tended to use the same kinds

of strategies that young children use for

problems with numbers. Most teachers

X _Y Z

Teaching Children Mathematics / September 2005 55

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used direct-modeling strategies. For example, teachers were given the following problem: “Lucy has H fish. She wants to buy E more fish. How many fish would Lucy have then?” Most teachers represented each set as they counted: “A, B, C, D, E, F, G, H, and A, B, C, D, E.” Then they counted them all: “A, B, C, . . . , K, L, M” (see fig. 1). This strategy is identical to the direct-modeling strategy that young children use that we described in the previous section.

A relatively small number of teachers used strategies more sophisticated than direct modeling. For example, teachers were given the following problem: “Janelle has G trolls in her collection.

How many more does she have to buy to have K trolls?”

Ms. Jacobson (all names are pseudonyms) did not repre- sent the first set of G but rep- resented only the number of trolls to be added to make the sum of K. She said, “G” and counted, “H, I, J, K,” as she was pointing to each counter for each letter (see fig. 2). In order to find out how many were added, she counted the number of cubes added on. Ms.

Jacobson’s strategy is classified as a counting strategy, which is a second-level strategy that children use for addition or subtraction problems. Figure 3 illustrates a modeling strategy that is closer to this counting-on strategy.

A few teachers tried to use “letter facts.” For example, Ms. Perry knew right away that the dif- ference between M and E would be H in the second problem because in the first problem, the sum of H and E was M. Mr. Wilson tried to use E and J, rep-

resenting 5 and 10, respectively, as benchmark quantities. In solving the last problem, he said, “K children were playing in the sandbox. Some went home. There were C children still playing in the sandbox. How many children went home? I know that the problem is like K – ■■= C, and K = J + A = E + E + A = E + C + C. So ■■= E + C, which is [counting E, F, G, H] H ” (see fig. 4). This example shows how difficult simple number decomposi- tions can be to follow when quantity representa- tions are foreign to us.

Initially, many teachers in the class did not understand what Mr. Wilson was trying to do. After some discussion, they came to understand that he decomposed the numbers K, J, and (E + A) into facts that he already knew, which were J = E + E and E + A = C + C. If we think of his strategy in numbers, Mr. Wilson was decomposing 11 into 10 and 1, and again into 5, 5, and 1, and again into 5, 3, and 3. This helped teachers realize that developing derived-fact strategies by decomposing a number into smaller and more manageable numbers is not a simple task when you are developing concepts of numbers. It is interesting to note that Mr.

Wilson picked letters that correspond to 5 and 10 as benchmarks—the same numbers that young children often use as benchmarks in primary grades. We also observed that other teachers learned more quickly the value of letters that cor- respond to very small collections, such as A, B, C, and the value of letters that correspond to one full hand (E), and two full hands (J), than they learned the letters that correspond to other quantities.

Teachers’ Reactions

Most participating teachers shared that the alphabet activity was helpful for them to understand children’s learning experiences. One preservice

Figure 1

Teachers’ use of direct-modeling strategy for the problem H + E =■■

Counting each set “A,

Counting all to find the sum

B, C, D, E, F, G, H.” “A, B, C, D, E.”

“A, B, C, D, E, F, G, H.” “I, J, K, L, M.”

Developing

derived-fact

strategies is

not simple when

you are developing

concepts of numbers

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teacher selected this activity as one of the five most valuable activities that helped her growth in understanding how to teach mathematics. She wrote the following in her end-of-semester portfolio about the activity:

It exemplified how students may feel when learning a new concept. This activity was very different than any I had encountered before. It was a great activity to show me how students may think, act, and feel when learning a new concept.

Many teachers commented that the problems were indeed much more difficult to solve when they had to use the alphabet instead of the corresponding numbers. They noted that conceptual differences between direct modeling, counting, and derived facts were much greater than they initially thought. They talked about how the alphabet activity helped them realize the importance of paying close attention to children’s strategies. Many teachers said that they could not think of any strategies other than direct modeling for the given problems. This also helped them better appreciate the power of direct-modeling strategies for young children. The teachers shared that when they did not have a firm grasp of the size of the quantities that letters represented, they could clearly see the importance of

tools to represent quantities in the problem, and this experience allowed them to gain better understanding of the importance of letting young children be direct modelers when they started to learn the concepts of numbers and operations.

Many teachers were surprised to observe that only a very small number of teachers used counting strategies to solve the problems. For example, teachers were given the following problem:

“Lucy has H fish. She wants to buy E more fish.

How many fish would Lucy have then?” Most teachers represented both the first set, H, and the second set, E, and counted all of them. When a few teachers shared a counting strategy, repre- senting only the second set, E, many teachers were surprised to discover that they did not have to represent the set H because acting on the first set is unnecessary, so the number of items in the first set need not be counted again.

The teachers also noticed that they do not automatically memorize “letter facts” such as H + E = M just by working on several problems. Even though the problem M – E = ■■was given imme- diately after the teachers had solved H + E = M, very few teachers noticed the connection.

That is, very few used the relationships of H, E, and M in H + E = M when trying to find M – E.

Teachers also discussed how frustrating it would be if they were asked to memorize facts such as H + E = M, P – I = G, or K – H = C when they

Figure 2

Teachers’ use of counting strategy for the problem G +■■= K

“H, I, J, K.”

Say “G,” then count

Figure 3

Teachers’ use of direct-modeling strategy for the problem G +■■= K

Count the first set.

Continue counting until the total is reached.

Count to find the number of cubes added on.

“A, B, C, D, E, F, G.” “H, I, J, K.”

“A, B, C, D.”

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did not have a solid understanding of numbers.

The activity helped them be more aware of the fact that children need many opportunities to phys- ically and mentally represent numbers in order to internalize the corresponding number facts.

Connecting the Alphabet Activity to Current Research

Research on professional development of teachers supports the use of activities such as the alphabet activity to facilitate teachers’ understanding of the processes of children’s learning.

In order to transform the way in which they teach mathematics, teachers must develop an understanding of what it means for children to understand and learn mathematical concepts (Franke, Fennema, and Carpenter 1997). One effective way to do this is through learning new mathematics. As Schifter and Fosnot (1993) note, “It is the insight into the learning process yielded by their own self-observation” that allows teachers to see issues clearly (p. 30). Finding ways to help teachers learn the mathematics in a fresh way, as if they were learning it for the first time, will help them understand how children learn.

The alphabet activity is one of the components in our preservice and in-service teacher programs designed to help teachers learn how young children develop understanding of numbers and operations. Recent research programs have developed other tools to help teachers understand children’s mathematical thinking. For example, collections of video clips of students solving

problems (Carpenter et al. 1999; Philipp and Cabral 2005), transcriptions of students’ dia- logues, or students’ written work serve as tools for teachers and researchers to understand the processes of students’ sense making.

Conclusion

We do not suggest that teachers use this activity with their students. The purpose of the activity is to help teachers understand what children experience when they are at the beginning stage of developing number concepts and strategies for addition and subtraction. This activity also helps teachers understand the importance of opportunities that children need in order to develop funda- mental concepts of numbers and adding/subtracting strategies, opportunities that often take considerable time.

The reactions from participating teachers con- firm that this alphabet activity could assist them in developing a better understanding of children’s learning processes of numbers and arithmetic operations or strategies. Many teachers shared that before the alphabet activity they wanted children to use faster, more efficient strategies, such as counting strategies or derived facts, rather than direct-modeling strategies.

Teachers often expressed perplexity about how some children, even after having been taught to count on, revert to direct modeling as soon as the teacher’s instruction stops. After the activity, teachers were more aware of the difficulties in developing counting and derived-facts strategies.

They realized that many of them initially could not even make sense of Mr. Wilson’s derived-fact strategy. In addition, we observed that teachers

Figure 4

Teacher’s use of derived-fact strategy for the problem K –■■= C

K = J + A

= (E + E) + A

= (E + C) + C K – ■ ■ ^{= C}

(E + C) + C – ■ ■ ^{= C}

Therefore, ■ ■ = E + C = H

■

■ ^{= H}

M N S

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were more inclined to pay closer attention to the examples of children’s strategies in video clips or written work during class, not only in the context of numbers and operations, but also in other con- texts, such as place value. They also developed a better understanding of how to use research on children’s mathematical thinking in practice. The alphabet activity gave teachers a great opportu- nity to understand the process of learning mathematics through the eyes and minds of children.

References

Ball, Deborah Loewenberg. “From the General to the Particular: Knowing Our Own Students as Learners of Mathematics.” Mathematics Teacher 90 (Decem- ber 1997): 732–37.

———. “Bridging Practices: Intertwining Content and Pedagogy in Teaching and Learning to Teach.” Jour- nal of Teacher Education 51 (2000): 241–47.

Carpenter, Thomas P., Elizabeth Fennema, Megan L.

Franke, Linda Levi, and Susan B. Empson. Chil- dren’s Mathematics: Cognitively Guided Instruction.

Portsmouth, NH: Heinemann, 1999.

Fennema, Elizabeth, Thomas P. Carpenter, Linda Levi, Megan L. Franke, and Susan B. Empson. Children’s Mathematics: A Guide for Workshop Leaders.

Portsmouth, NH: Heinemann, 1999.

Franke, Megan L., Elizabeth Fennema, and Thomas P.

Carpenter. “Teachers Creating Change: Examining Evolving Beliefs and Classroom Practice.” In Mathe- matics Teachers in Transition, edited by Elizabeth Fennema and Barbara Scott Nelson, pp. 255–82.

Mahwah, NJ: Lawrence Erlbaum, 1997.

Fuson, Karen. “Research on Whole-Number Addition and Subtraction.” In Handbook of Research on Math- ematics Teaching and Learning, edited by Douglas A.

Grouws, pp. 243–75. New York: Macmillan, 1992.

Kilpatrick, Jeremy, Jane Swafford, and Bradford Findell.

Adding It Up: Helping Children Learn Mathematics.

Washington, DC: National Academy Press, 2001.

Loucks-Horsley, Susan, Nancy Love, Katherine E.

Stiles, Susan Mundry, and Peter W. Hewson. Design- ing Professional Development for Teachers of Sci- ence and Mathematics. 2nd ed. Thousand Oaks, CA:

Corwin Press, 2003.

National Council of Teachers of Mathematics (NCTM).

Professional Standards for Teaching Mathematics.

Reston, VA: NCTM, 1991.

Philipp, Randolph A., and Candace Cabral. IMAP Inte- grating Mathematics and Pedagogy to Illustrate Chil- dren’s Reasoning. San Diego, CA: San Diego State University Foundation, 2005. CD-ROM.

Schifter, Deborah, and Catherine T. Fosnot. Reconstruct- ing Mathematics Education. New York: Teachers College Press, 1993.

Sowder, Judith, Barbara Armstrong, Susan Lamon, Mar- tin Simon, Larry Sowder, and Alba Thompson. “Edu- cating Teachers to Teach Multiplicative Structures in the Middle Grades.” Journal of Mathematics Teacher Education 1 (1998): 127–55. ▲