Children's Developing Understanding of Place Value: Semiotic Aspects

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Children's Developing Understanding ofPlace Value: Semiotic AspectsMaria Valeras & Joe BeckerPublished online: 14 Dec 2009.

To cite this article: Maria Valeras & Joe Becker (1997) Children's Developing Understanding of PlaceValue: Semiotic Aspects, Cognition and Instruction, 15:2, 265-286, DOI: 10.1207/s1532690xci1502_4

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COGNITION AND INSTRUCTION, 15(2), 265-286 Copyright O 1997, Lawrence Erlbaum Associates, Inc.

Children's Developing Understanding of Place Value: Semiotic Aspects

Maria Varelas and Joe Becker College of Education

University of Illinois at Chicago

This article is devoted to two interrelated semiotic aspects of the placevalue system: multiplicity of value and composition of values. Both aspects involve the basic characteristic of the placevalue system-the differentiation between face value and complete value. We explored whether a system that is intermediate between the written place-value system (multiplicity of value) and the base-10 manipulatives (no multiplicity of value) helps children understand the place-value system. The intermediate system that we designed, called FVCV (for face value and complete value), has a pseudomultiplicity of value. We report and discuss evidence that instruction and practice in the FVCV system helped children (a) to differentiate between the face value and the complete value of the digits in a multidigit place-value number representation and (b) to grasp that the complete values of the digits in a multidigit placevalue number representation add up to the total value.

Many schoolchildren have difficulty understanding the place-value system (Cauley, 1988; Ginsburg, 1989; Kamii, 1986; Miura & Okamoto, 1989; Resnick, 1982; Resnick & Omanson, 1987; Ross, 1989). Their difficulty lies with a basic characteristic of the place-value system: They fail to differentiate between the face value of each symbol in a number and the complete value of the same symbol. For example, when given a collection of 23 objects and the number 23, most 6- to 8-year-old children as well as some older children indicate that the 3 in 23 represents 3 objects and the 2 represents 2 objects (the face value) rather than 20 objects (the complete value of the 2 within the number representation 23). For some children, this lack of understanding persists into the fifth and sixth grades (Kamii, 1986;

Requests for reprints should be sent to Maria Varelas, College of Education, M/C 147, University of Illinois at Chicago, 1040 West Harrison Street, Chicago, IL 60607-7133. E-mail: [email protected]

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266 VARELAS AND BECKER

Resnick & Omanson, 1987). Inasmuch as this difficulty deals with the way individual sign forms (the individual digit forms) relate to their referents (quantities), one may take a semiotic perspective on this difficulty. In much of the research, this difficulty is considered only in conceptual terms.

Conceptual aspects have been stressed in two interrelated ways. Some research emphasizes the hierarchy of powers of 10: Children need to understand the inclusion relations among powers of 10 in order to comprehend the place-value system (Kamii, 1986). Most of the research, however, emphasizes grouping and regrouping: Children need to understand the different ways in which the same quantity can be grouped or divided into subsets. Thus, the number 17 may be conceptualized as seventeen 1s or as one 10 and seven 1s. This emphasis is extended to the compu- tational procedures: carrying in addition involves forming a new group of 10. For example, adding 29 and 17 in the place-value system involves understanding that the 16 resulting from 9 plus 7 is composed of one group of 10 and six units. This group of 10 then needs to be "carried over" and added to the other groups of 10. Similarly, borrowing in subtraction involves breaking up a group of 10 that already exists. As Ross (1989) noted: "Pupils need to engage in problem-solving tasks that challenge them to think about useful ways to partition and compose numbers" (p. 50). These conceptual aspects focus on children's understanding of the relations among quantities and the operations on quantities.

There is evidence, however, suggesting that children may have conceptual understanding about quantities and about operations on them, including grouping and regrouping, but still have difficulties with the place-value system. There are two kinds of evidence. The first kind of evidence comes from research establishing that children early, and without school instruction, develop an understanding of grouping and regrouping. Children use regrouping methods for adding, subtracting, multiplying, and dividing (Carraher, Carraher, & Schliemann, 1987; Ginsburg, 1989; Saxe, Guberman, & Gearhart, 1987). For example, Ginsburg quotes a 7-year-old boy adding 12 and 6. The boy gets 18 and explains that 12 is 10 plus 2, adding 6 to 10 gives 16, and adding the remaining 2 to this sum gives a total of 18. Children's facility with grouping and regrouping contrasts with the difficulties they have with the place-value system.

The second type of evidence comes from research showing that children can be competent in using base- 10 manipulatives to represent numbers and to compute, including the appropriate grouping and regrouping, and still not perform well in place-value computations. Hiebert (1984) suggested that the crux of the difficulty that children have is in mapping their conceptual knowledge about quantities onto the conventional place-value procedures. Hiebert proposed that what is needed is that "the connections between the algorithmic procedures and children's under- standings must be made explicit" (p. 509). Resnick and Omanson (1987) tried explicitly teaching children how to do this mapping. They found that "although they [children] could use blocks to represent two- and three-digit total quantities,

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PLACE VALUE: SEMIOTIC ASPECTS 267

they could not use them reliably to represent individual digits [within a multidigit number]" (p. 64) and that "children's capabilities for doing arithmetic with blocks was not reflected in a comparable ability for doing written arithmetic procedures" (p. 64). Later, Resnick (1989) noted: "It appears that children are attending to the symbols of arithmetic but not to the quantities that they represent" (p. 166). We believe that the lesson to be learned from both types of evidence is that we need to complement the analysis of the conceptual aspects of the place-value system with a careful analysis of the particular semiotic characteristics of the place-value system-that is, of the sign use, how the signs are related to their referents and to each other (Becker & Varelas, 1993).

There has been some research in children's mathematical knowledge that includes a consideration of semiotic features of the systems of representation of quantities that they use. Mitira(1987; Miura, Kim, Chang, & Okamoto, 1988; Miura Br Okamoto, 1989; Miura, Okamoto, Chungsoon, Steere, &Fayol, 1993) and Fuson and Kwon (1992) compared understanding of place value for children of different nationalities, focusing on the differences in the ways the languages they spoke named quantities. All the languages used a base-10 system, but they differed in their consistency in naming quantities. Some languages had inconsistencies, such as (a) the names eleven and twelve for the quantities 1 1 and 12, but the names sixteen and seventeen for the quantities 16 and 17, and (b) the names sixteen and seventeen for the quantities 16 and 17 (saying first the number of 1 s) but the names twenty-six and twenty-seven for the quantities 26 and 27 (saying first the number of 10s). Other languages named the quantities on a more consistent basis. For example, in Asian languages that have their roots in ancient Chinese (among them Chinese, Japanese, and Korean), the use of the base-10 system to generate numerical names is very consistent and explicit: 12 is spoken as "ten-two," and 26 is spoken as "two-ten-six." This work suffers from the difficulty of isolating the effect of this difference in languages because the children grew up in different cultures, and it is not difficult to imagine that other factors may be involved in the differences of children's performance on the tasks used (mostly representing numbers using base-10 ma- nipulative~). As Miura et al. (1993) noted:

Educational practice was not sampled in this study, and it is not clear what effects schooling might have had on these results. The children were tested before they had been introduced to Base 10 type materials and before receiving formal lessons on place value. However, the informal teaching that takes place in preschools and kindergartens may also influence these findings. (p. 29)

Fuson (1990a, 1990b) also focused on the help that uniformity and repetition mlay offer children in their construction of the units (i.e., 10s, 100s, 1,000s) used in the place-value system. Fuson examined the help children get from dealing early in their instruction in place value with multidigit numbers extending into the

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hundreds and thousands. The idea is that the repetition involved in the grouping and regrouping between successive pairs of columns helps children to grasp the powers-of-10 units and the relations between them. Fuson uses this multidigit approach in conjunction with place-value materials, such as base-10 manipulatives.

The approaches of both Miura and Fuson place importance on semiotic aspects, that is, on the form of the representation of quantities in a sign system. In this article, we extend the range of the examination of semiotic aspects of the place-value system. We will not focus, however, as Miura and Fuson have done, on the way in which regularities or irregularities in the verbal representational system and repetition in the digit notational system may help or hinder children in constructing mathematical conceptual units that they need for understanding the place-value system. Instead, we focus on the difficulties children have with the particular semiotic aspecl inherent in the place-value system itself. As Hiebert and Wearne (1992) noted:

Understanding place value involves building connections between key ideas of place value, such as quantifying sets of objects by grouping by 10 and treating the groups as units . . . [but also] using the structure of written notation to capture this information about groupings. (p. 99)

In the work presented here, the simple distinction that we are making between aspects that are purely conceptual and aspects that are semiotic is that, in the former, sign use is not a problematic issue in its own right, and in the latter, it is. The article presents an empirical examination of two specifically semiotic aspects of place value, that is, two fundamental aspects of sign use in the place-value system: multiplicity of value and composition of values. Both of these inherently involve the specific ways that signs are used in the place-value system, and both involve the basic characteristic of the place-value system that we noted earlier: the differentiation between face value and complete value. We designed a sign system that has specific semiotic characteristics intermediary between the place-value system and base-10 manipulatives and examined whether its use helps children develop an appreciation of the particular way signs are used in the place-value system.

MULTIPLICITY OF VALUE

Multiplicity of value refers to the fact that, in the place-value system, the same individual symbolic form is used to denote several different values (e.g., the sign form 1 has a value of 100 in 123 but a value of 1 in 321). This ambiguity is overcome by using spatialposition, a feature external to the individual sign form, to determine the power-of-10 value (or order of magnitude) of that sign form. Multiplicity of value and spatial position as a determiner of value are in direct contrast to the

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number 33 represented with base-1 0 manipulatives

number 33 FIGURE 1 Number representation represented in inthe base-l0manipulatives and in the the written place-value written place-value system. system 3 3 features of the base-10 manipulatives, which are often used in efforts to help children understand the place-value system for number representation and compu- tation. In the base-10 manipulatives, there is no multiplicity of value. Every piece always has the same value: a 1s piece always has a value of 1, a 10s piece always has a value of 10, and so forth. In the base-10 manipulatives, spatial position has no role in determining the value of a piece. The value of a piece is immediately known from its appearance (Figure I). Even if 10s pieces are systematically placed to the left of 1s pieces, children do not need to incorporate spatial position in the process by which they determine the value of the pieces. The base-10 manipulatives are thus similar to actual quantities and dissimilar to the place-value system. As a result, the base- SO manipulatives do not encourage children to make the distinction that the conventional place-value system requires between an individual sign form and the value it represents.

We explored whether a system that is intermediate between the written place- value system and the base-10 manipulatives helps children understand the place- .value system. The intermediate system that we designed is called FVCV (for face value and complete value). There are several forms of this system, and in this article, when we refer to FVCV, we mean the digit form of FVCV. In this system, there are pieces and a board. The upper sides of all the pieces are one color; the undersides are another color. The upper side of each piece presents the face value, and the underside (usually hidden) indicates the complete value. For example, there are pieces with 3 on the upper side to indicate a face value of 3; some of these pieces have 3 on the underside, some 30, some 300, and so forth (Figure 2). Thus, in this system, sign forms having the same face value (e.g., 3) but different complete values are distinct if the underside of the pieces (e.g., 3,30,300) is taken into consideration.

In this system, quantities are represented by placing the pieces right side up (so that the face value is displayed, and the complete value is hidden) on a board divided

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Upper side -- face value Underside -- complete value

complete value) pieces.

Name: 1 FIGURE 3 Board for placing FVCV (face value and complete value) pieces.

into columns (Figure 3). For a quantity to be correctly represented in this system, both the upper side of the pieces (indicating the face value) and the underside of the pieces (indicating the complete value) must be correct, even though the latter is hidden. The quantity 33, for example, is represented correctly by two pieces arranged on the board, one in each column, with the 3s piecein the rightmost column having 3 written on the underside and the 3s piece in the next column to the left having 30 written on the underside.

FVCV occupies an intermediary position between the place-value system and the base- 10 manipulatives. As noted, in the place-value system, there is multiplicity of value, and spatial position determines the complete value of the individual sign form; in the base-10 manipulatives, there is no multiplicity of value, and spatial position is irrelevant to the complete value of an individual piece. In FVCV, there is a pseudomultiplicity of value, and spatial position can be used as a secondary indicator of the complete value of a piece. We take each of these characteristics in turn. There is a pseudomultiplicity of value because all the pieces with, for example, a face value of 3 look the sarne-they have the same upper side-but they are distinguished by their complete values on the hidden sides of the pieces. Thus, the 3s piece indicating 30 is really a different kind of piece than the 3s piece indicating 3. The user of FVCV knows this and is, therefore, in a different position than auser

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of the conventional place-value system in which the individual sign forms representing 3 in 23 and 30 in 37 are identical sign forms.

In FVCV, position can be used as a secondary indicator of the complete value of a piece. The underside of the piece can be used as the primary indicator of the complete value represented by that piece. However, because, as already noted, the pieces are used in a spatial arrangement that conforms to the conventional place- value system, the complete value is also indicated by the position of the piece within a multidigit number representation. Thus, this position can be used as a secondary indicator of the complete value; a person reading, for example, 37 in FVCV and using the value on the underside as the primary indicator of the complete value to be associated with the face value of 3 in the FVCV representation of 37, can use the position of the 3s piece as a clue to which kind of 3s piece it is, one with 3 on the underside or one with 30 on the underside, and so forth.

Experiment 1: Determining Whether Instruction and Practice in WCV Help Children Grasp the Multiplicity of Value

We compared the effects of instruction and practice in the FVCV system with the effects of instruction and practice in two systems: the base-10 manipulatives and a system representing the written place-value system. The systems and the instruction treatments using them are described in the Method section. Children's understanding of place value was tested using the bean task described in the Method section. The children were given this task several times, once as a pretest and several times after they had begun receiving instruction. Children's understanding of addition with carrying was also tested using a written addition task with 2 two-digit numbers involving carrying, which is also described in the Method section.

Method

Participants. Participants were elementary school second, third, and fourth graders in a public school in the Chicago metropolitan area. Twenty-four girls and twenty-four boys from each grade were randomly assigned to one of three treatments.

Materials. The FVCV system was used in the experimental group. The pieces in the FVCV system were made from 1-in. plastic discs, about -in. thick, colored red on one side and yellow on the other. The numbers were written in black.

The two systems used in the control groups were (a) the BTM (base-10 manipulatives) and (b) the FVO (face value only) system. The base-10 manipula-

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tives used were Dienes blocks-the 1s and the 10s pieces. The FWO system was made identical to the FVCV system except that there was no number written on the underside, that is, no explicit representation of the complete value of the piece. It is thus a form of the conventional place-value system that is most appropriate as a control for the FVCV system.

Treatments. There were three treatments, one using the FVCV system, one using the FVO system, and one using both the BTM system and the FVO system with mapping between them. In each treatment, children were individually engaged in two sessions of approximately 20 to 30 min each. All activities in the three treatments were parallel. In both sessions, children worked with two-digit numbers. We did not use larger numbers for two reasons: (a) We wanted to use the same activities for all the grades, and (b) we wanted to keep the mapping between systems manageable (in the BTM with FVO treatment).

Detailed scripts for each session of each treatment were developed in piloting and used in this study. In the first session, children were instructed and given practice in number representation and in addition without carrying using the particular symbol system (FVCV, FVO, or BTM with FVO). In each treatment, for the number representation, children worked with several types of two-digit numbers: numbers where the English language is clear and consistent with the base-10 property of our number system (e.g., 24, 45), numbers in the teens (e.g., 11, 16), numbers with double digits (e.g., 88,33), and numbers that are multiples of 10 (e.g., 20,90). There were two activities. In the first activity, we said aloud one of these types of numbers and asked the child to make this number with the appropriate materials depending on the treatment. We asked the child questions such as "How much does this chip really mean?" or 'What blocks go with this chip?" We repeated this with the other types of numbers. In the second activity, the child drew one of these types of numbers from an envelop and placed it in front of us. We then formed the number with the appropriate materials, deliberately making some mistakes. The child was then asked whether we had done it right. If the child disagreed, she or he was also asked how it should be done and why.

In the second session, children were instructed and given practice in addition with canying in the same symbol system. In addition without or with carrying, each type of two-digit numbers as explained earlier was used. All the sessions were videotaped. These videotapes were primarily used to code the children's performance on the tasks described next. Additionally, they allowed us to monitor the administration of the treatments to individual children for conformity to the scripts.

Pretest and posttests. Two tasks were used: the bean task and the written arithmetic task. The bean task was administered before the first session (pretest), at the end of the first session (first posttest), and at the end of the second session (second posttest). The written arithmetic task was administered before the first

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session (pretest) and at the end of the second session (posttest). Different numbers were used in the various administrations of the same task. For the bean task, we used the following numbers: 23 (pretest), 12 (first posttest), and 15 (second posttest). For the written arithmetic task, we used 37 + 25 (pretest) and 46 + 17 (posttest).

Bean task: We placed in front of the children a card with a two-digit number (X) written on it and an empty cup. The children read the number, and pointing to the number on the card, we told the children, "Now, I'm going to give you X beans." We then poured X beans into the empty cup from a bag containing the exact amount and told the children, "Now remember all the beans together are right for this number. The number says X, and there are X beans." We told the children that the number has two parts, pointing to each digit. Then, we placed an empty cup by the side of the 1s digit, and pointing to that digit, we said to the children, "I want you lo put in the empty cup the right amount of the X beans for this part." After they had finished putting beans in the cup, we asked the children how many beans they put in the cup. We repeated this procedure for the 10s digit using another cup (Figure 4).

If children were correct for both digits, we gave them a countersuggestion. We removed the beans from the cup for the value of the 10s digit and put in the cup the number of beans equal to the face value of the 10s digit saying, "Another child did this. Is that a good way of doing it? Why or why not?"

If children were incorrect for either digit, we gave them a hint. We asked them 1.0 read the number on the card. Then, we lifted the two cups with the beans the children had placed in them-the cup for the value of the 1s digit and the cup for the value of the 10s digit-and we asked the children to say how many beans there were altogether in the two cups. Children who corrected themselves on being given the hint were not given the countersuggestion on that administration of the bean task because the majority of the children who were given the hint had themselves initially given the response described in the countersuggestion and corrected it.

Written arithmetic task: We placed in front of children an index card with an addition of 2 two-digit numbers written on it in vertical form. The addition involved carrying, and we asked children to do the addition as they wanted. When children

FIGURE 4 The bean task.

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were done, we asked them to tell us how they did it. If necessary we used prompts, such as "Why did you do . . . ?'

Coding. In the bean task, we coded children's responses for the 1s and the 10s digits. In each case, we coded whether the response was incorrect or correct. For a response to be coded as correct, children both needed to put the right number of beans in the cup and needed to say correctly how many beans they put in the cup. Nearly all children succeeded spontaneously for the 1s digit in each administration of the bean task. For the 10s digit, we distinguished between correct responses given spontaneously and correct responses given after the children received the hint. When presenting results and analyses, we specify in each case whether, for the 10s digit, we refer to total correct responses (with or without the hint) or only to spontaneous correct responses (without the hint). Additionally, we coded whether children gave correct judgments in response to the countersuggestion, and examined the explanations they gave for their judgments.

For the written arithmetic task, we coded whether children's written symbols for the 10s and the 1s digits of the sum of the two numbers were correct or incorrect. We also examined children's explanations about how they did the addition, coding them separately for each digit of the sum.

Responses of 57 children were coded by two coders. There were 868 cases of such coding. There was agreement in 858 cases, giving an interrater reliability of 98.8%.

Results and Discussion

We first examine children's performance on the written arithmetic task. The majority of the children in each treatment succeeded in the written arithmetic task; that is, they wrote down the correct digits in the 1s place and 10s place of the sum. Table 1 shows the number of correct responses for the 1s and the 10s digits in the written arithmetic task in the pretest and posttest as a function of grade. Sixty-eight

TABLE 1 Number of Children Giving Correct Responses for the Two Digits

in the Written Arithmetic Task as a Function of Grade

Pretest Posttest

Digit Grade 2 Grade 3 Grade 4 Grade 2 Grade 3 Grade 4

1s 30 39 47 35 45 48 10s 27 37 47 33 41 48

Note. N = 48 for each grade.

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TABLE 2 Number of Children Giving Correct Responses for the 10s Digit

in the Written Arithmetic Task as a Function of Treatment

Treatment Pretest Posttest

FVCV FVO BTM with FVO

Note. N = 48 for each treatment. FVCV = face value and complete value; FVO = face value only; BTM = base-10 manipulatives.

percent of the children who gave the wrong answer for the 1s digit did not provide any verbal explanation when asked how they did the addition. Another 25% corrected themselves when they were asked to explain how they did the addition. The remaining 7% added 7 and 5 to get 12 and wrote down 12 in the sum (obtaining a total of 512 for 37 + 25). The most predominant incorrect answer for the 10s digit was due to forgetting to carry. There was a total of 55 incorrect responses for the 10s digit of the sum for both administrations of the written arithmetic task, and 43 (78%) were of this type. Children who gave this type of written response either neglected to carry in their verbal explanation or did not provide any verbal explanation.

The distribution of the correct responses for the 10s digit of the sum across treatments and task administrations is shown in Table 2. In each of the FVCV and W O treatments, 6 children progressed from failure to success in the 10s digit from the pretest to the posttest. In the BTM with FVO treatment, 3 children progressed from failure to success, but 3 other children changed from success to failure.

In the pretest written arithmetic task, 77% of the children carried correctly and gave the correct digit for the 10s place. This high performance constitutes a ceiling effect showing that the children were generally able to solve correctly a two-digit addition with carrying even before the interventions. Thus, the written arithmetic task was not a good indicator of effects that the treatments may have had on children's understanding of place value.

We now examine children's performance on the bean task. The number of children placing the correct number of beans in both cups in the bean task is shown in Table 3 for each treatment and task administration. The numbers reported represent the total number of children in each treatment and task administration who gave a correct response with or without the hint (Total columns) and the number of children who changed from an incorrect response to a correct response when given the hint (After Hint columns). The majority of the children in all treatments and grades gave a correct spontaneous response for the 1s digit. There were only 10 incorrect spontaneous responses across treatments and grades out of a total of 432 responses in the pretest and the two posttests. In 8 of these 10 cases,

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TABLE 3 Number of Children Placing the Correct Number of Beans in Both Cups in the Bean Task

Pretest Posttest I Posttest 2

Treatment Total Afer Hint Total Ajter Hint Total Afer Hint

FVCV 14 1 21 5 28 5 FVO 16 4 17 1 23 3 BTM with FVO 18 6 20 3 22 4

Note. N = 48 for each treatment. FVCV = face value and complete value; FVO = face value only; BTM = base-10 manipulatives.

children also gave an incorrect response for the 10s digit and did not correct their response for either digit when given the hint. In the remaining two cases, children gave an incorrect response for the 10s digit but corrected their responses for both digits when given the hint. There were 147 correct spontaneous responses for the 10s digit in the pretest and the two posttests. Of these responses, 140 (95.2%) were accompanied by sensible explanations regarding why the countersuggestion (i.e., putting in the cup for the value of the 10s digit the number of beans equal to the face value of the 10s digit) was wrong. There were two predominate reasons that children gave for rejecting the countersuggestion: (a) The left digit was in the 10s place so, for example, if the digit was 2, it should be worth 20 and not 2; or (b) the beans in the two cups did not add up to the total number written on the card, for example, if the number was 23 and there were 2 beans in one cup and 3 in the other, then there were only 5 beans in both cups and not 23. The most predominant incorrect response for the 10s digit across treatments and grades was for children to give the face-value quantity for the digit in the 10s place. There were 285 incorrect spontaneous responses in the pretest and the two posttests, and 275 (96.5%) were of this type.

It might be thought that children who put the face-value quantity in the cup when asked to put in the cup the right amount of the X beans for the 10s digit considered at that time that the cup was a "10s cup." That is, it might be thought that those children considered that, analogously to the 10s position in the multidigit number, each unit in the cup represented a quantity of 10, so that, for example, two beans in that cup would represent a quantity of 20. In this respect, it is relevant to emphasize that the cups were never referred to as a 1s cup or a 10s cup and did not contain any marking to that effect. More important, the following observations make it unlikely that children took this approach: First, no child who put the face value in the cup for the 10s digit, and did not change on being given the hint, used this way of considering the beans in this cup to justifl that the number of beans in the two cups did not add up to the total number of beans. Second, no child who changed from face value to complete value on being given the hint indicated that

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she or he originally considered the beans in this cup in this way. Third, no child who put the complete value (20 beans in this example) in the cup agreed to the countersuggestion on the basis that the beans in this cup could be considered in this way.

Table 4 shows the number of children succeeding (or failing) the bean task-with or without the hint-as a function of succeeding (or failing) the written arithmetic task. These numbers are presented by treatment and task administration. We compared the pretest bean task with the pretest written arithmetic task and the second posttest bean task with the gosttest written arithmetic task because these were administered at the same time. These results indicate that those who succeeded in the bean task had a very high probability (nearly 100%) of succeeding in the written arithmetic task. In the pretest, 48 of 144 children in all groups succeeded in the bean task, and of these, 42 (88%) also succeeded in the written arithmetic task. In the posttest, 72 of 144 children in all groups succeeded in the bean task, and of these, 67 (93%) also succeeded in the written arithmetic task.

Furthermore, these results indicate that the bean task did discriminate among those who succeeded in the written arithmetic task. That is, those who succeeded , in the written arithmetic task did not necessarily succeed in the bean task; in fact, in the pretest, they tended not to succeed in the bean task. In the pretest, 110 of 144 children in all groups succeeded in the written arithmetic task, and of these, 42 (38%) succeeded in the bean task. In the posttest, 122 of 144 children in all groups succeeded in the written arithmetic task, and of these, 67 (55%) succeeded in the bean task. Therefore, overall in the posttest, children who succeeded in the written arithmetic task had about an equal chance to succeed or fail the bean task. This overall finding is also apparent in each of the FVO and BTM with FVO treatments. In the FVCV group, however, 39 children succeeded in the written arithmetic task, and 25 (64%) of these children also succeeded in the bean task. This difference

TABLE 4 Number of Children Succeeding (or Failing) the Bean Task With or Without the Hint

as a Function of Succeeding (or Failing) the Written Arithmetic Task

Pretest Bean Task Posttest Bean Task

Treatment Arithmetic Task Correcf Incorrect Correct Incorrect -- -

FVCV Cmect 11 22 25 14 Incorrect 3 12 3 6

FVO Correct 14 24 23 21 Incorrect 2 8 0 4

BTM with FVO Correct 17 22 19 20 Incorrect 1 8 3 6 -

Note. N = 48 for each treatment. FVCV = face value and complete value; FVO = face value only; BTM = base10 rnanipulatives.

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between the posttest for the FVCV group versus the two other groups is due to the combination of two factors: the ceiling effect in the written arithmetic task and the larger number of children succeeding in the bean task in the posttest than in the pretest in the FVCV group. Children's success in the written arithmetic task in conjunction with their struggles in the bean task points once again to a finding that many researchers in math education have documented: that children who successfully complete procedures for adding numbers do not necessarily understand the place-value system.

The benefit to children of instruction aimed specifically at helping them deal with multiplicity of value was evaluated by comparing the effect of the three different treatments on the children's performance on the 10s digit in the bean task. A 3 x 3 (Treatment x Grade) analysis of variance (ANOVA) with the bean task pretest scores (correct with or without the hint vs. incorrect) as the dependent variable showed that there were no significant differences among the mean pretest scores of the three treatments, the three grades, or all nine groups.

A 3 x 3 (Treatment x Grade) analysis of covariance (ANCOVA) with the first posttest bean task scores (correct with or without the hint vs. incorrect) as dependent variable and pretest scores as a covariate did not show any significant difference among the three treatments. The adjusted mean scores were .47, .35, and .38, for FVCV, FVO, and BTM with FVO, respectively. The ANCOVA showed a significant difference among grades, F(2,143) = 3.22, p < .05. The adjusted mean scores for the three grades in increasing order of grade were .35, .36, and .49.

A 3 x 3 (Treatment x Grade) ANCOVA with the second posttest bean task scores (correct with or without the hint vs. incorrect) as the dependent variable and pretest scores as a covariate showed a significant difference among the three treatments, F(2, 143) = 3.28, p < .05. The adjusted mean scores for the three treatments were .62, .48, and .43, for FVCV, FVO, and BTM with FVO, respectively. The Bry- ant-Paulson post hoc test for differences between two groups showed that the FVCV treatment was significantly more effective (p c .05) than the BTM with FVO treatment. The FVO treatment, however, was not significantly different from either of the other two treatments. The 3 x 3 ANCOVA did not show any significant difference among grades. The adjusted mean scores in increasing order of grade were .47, .52, and .54.

Children in the FVCV treatment (the experimental treatment) performed better in the second posttest than the children in either control treatment. Thus, instruction and practice in the FVCV system helped children differentiate between the face value and the complete value of the digits in a multidigit place-value number representation. In this respect, instruction and practice in the FVCV system was more helpful to the children than instruction and practice in mapping from the BTM to the FVO system (and than instruction and practice in just the FVO system). If we take into consideration that, in the course of their regular schoolwork, the

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children have instruction and practice with the BTM system, the relative effectiveness of the short FVCV intervention used in this study seems even more striking. Apparently, despite the earlier effect of grade, when the two sessions are considered as a whole, there was no difference across grades for how much children gained from each treatment. The contribution of the second session to the effectiveness of the FVCV treatment may be due to more time spent with the FVCV system, or to using that system for computing-in particular for adding with carrying--or to both. Further work is planned to determine which factors are operating here.

At this point, we have evidence that instruction and practice with the FVCV system helps children differentiate between face value and place value. However, there is a second aspect of the place-value system that we have not specifically attended to in the preceding analysis, an aspect we refer to as composition of values.

COMPOSITION OF VALUES

Composition of values refers to the fact that the complete value of a multidigit place-value representation of a quantity consists of the sum of the complete values of its parts. Thus, the complete value of 37 is the addition of the complete value of the 7 and the complete value of the 3. We emphasize that we are not focusing here on the ability to understand addition (i.e., in structural terms, the ability to grasp, for example, the 10 and the 7 both as separate subgroups and together as composing the total of 17). Continuing with our semiotic perspective, we are focusing on the sign use. The place-value system for representing quantities involves this property of the sign system: The complete value, or meaning, of a composite sign form such as 37 is determined by the values of the individual sign forms 3 and 7.

Although this aspect of the place-value system is also found in the base-10 manipulatives (considering the pieces as sign forms, the addition of the values of the individual sign forms used to represent a quantity gives the total quantity represented), there are two ways in which the situation is different in the place-value system. First, in the base-10 manipulatives system, there is no distinction between face value and place value. As a result, with these manipulatives, summing the individual sign forms to give the total quantity does not involve differentiating the complete value of an individual sign form from its face value. Second, the individual sign forms of the place-value system resemble the letters used in alphabetic writing more than do the base-10 manipulatives, and this similarity with alphabetic writing may be significant. In the place-value system, the total sign form for a given quantity greater than nine somewhat resembles a word. However, the meaning of a word represented in alphabetic writing is not composed of the meanings of the individual letters. Although the sound of the word bird is made up of the sounds of the individual letters, the meaning of the word bird is not made up of the meanings of

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the individual letters. Therefore, the resemblance of a multidigit number to a word may hinder children from grasping the composition of values in the place-value system.

In the Multiplicity of Value section, we analyzed children's performance in the three administrations of the bean task in terms of whether children differentiated between face value and complete value. If children offered correct responses for both digits, we considered that an adequate response. On this basis, we presented evidence that children are helped to understand the multiplicity of value aspect of the place-value system by instruction and practice in the FVCV system. Another question that arises involves whether these children also grasp the composition of values. This question leads to further analysis of children's performance on the bean task. There are two distinct ways that children may respond correctly concerning the 10s digit: counting or dumping. Having placed the correct number of beans in the cup for the value of the 1 s digit, children may either count out the correct number of beans for the cup for the value of the 10s digit and place them in the cup, or they may simply dump the remaining beans into the cup for the value of the 10s digit. We consider each strategy in respect to children's grasp of composition of values. We start with the counting strategy and consider separately spontaneous counting and counting after children were given the hint.

Spontaneous Counting

Spontaneous counting does not require that children appreciate that, if the quantity represented by the digit in the 1s place of a two-digit number is removed, the remaining quantity is the complete value of the digit in the 10s place. However, children who spontaneously succeeded in the bean task by counting (and did not need the hint) were given an incorrect countersuggestion. Nearly all these children rejected the countersuggestion, and analysis of their explanations sheds light on their grasp of composition of values when they counted to succeed in the bean task. As mentioned in the Multiplicity of Value section, children who were given the (incorrect) countersuggestion gave primarily two reasons for rejecting it: (a) The left digit was in the 10s place so that, for example, when the digit was 2 it should be worth 20 and not 2, which we call specifying the place value; and (b) the beans in the two cups did not add up to the total number written on the card, which we call specifying the composition of values.

Children who spontaneously counted to get to the correct answer in the bean task gave either or both of these reasons. Table 5 shows the number of children who gave each type of response over all grades, treatments, and administrations of the bean task. About half of the children who counted gave a reason for rejecting the countersuggestion indicating that they understood the composition of values.

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TABLE 5 Number of Children Giving Each Type of Response

to the Countersuggestion in the Bean Task

Response - --

No. of Children

Specifying the place value 20 Specifying the composition of values 19 Both responses 4

Counting After Being Given the Hint

Children who succeeded in the bean task after they were given the hint were not given the countersuggestion on that administration of the bean task. Therefore, we do not have a further means for evaluating whether children who counted after being given the hint grasped the composition of values. The hint that these children received, however, was to direct their attention to the question of whether the sum sf the beans in the cup for the value of the 1s digit and in the cup for the value of the 10s digit was the same as the number on the card. Thus, that they corrected themselves after receiving the hint indicates their grasp of composition of values. We find support for this interpretation by examining the performance of these children in subsequent administrations of the bean task. Fourteen children in the first or second administration of the bean task took the hint and corrected their initially incorrect response for the 10s digit and, in subsequent administrations, succeeded spontaneously. These children received the countersuggestion in those subsequent administrations of the bean task. Of these children, 13 (93%) either dumped or counted and specified the composition of values in rejecting the countersuggestion.

Dumping

As a reminder, children who used the dumping strategy put the correct number of beans in the cup for the value of the 1s digit and then dumped the remaining beans into the cup for the value of the 10s digit. It seems probable that children who use this strategy know that, after taking out the correct number of beans for the cup for the value of the 1 s digit, the number of beans remaining must be correct for the cup for the value of the 10s digit. That is, they understand the composition of values. It is possible, however, to dump the remaining beans into the cup for the value of the 10s digit without grasping the composition of values. To help determine whether children who use the dumping strategy grasp the composition of values, we examined this question empirically. We describe this experiment and its result and

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then return to the question of whether instruction and practice in the FVCV system help children grasp the composition of values.

Experiment 2: Determining Whether Children Who Succeed in the Bean Task by Dumping Grasp the Composition of Values

To determine whether children who responded by dumping the remaining beans into the cup for the value of the 10s digit understood the composition of values, we gave the children a variation of the bean task. In this variation, which we call the unknown-quantity bean task, the total number of beans was not known to the children. For clarity, in the remaining part of this article, we refer to the original bean task as the known-quantity bean task. For the known-quantity bean task, counting and dumping are both correct. For the unknown-quantity bean task, counting is correct, and dumping is incorrect.

Method

Participants. The unknown-quantity bean task was given to children who dumped with or without the hint in any administration of the known-quantity bean task in Experiment 1. Of the 53 children who satisfied this criterion, one child was unavailable for further testing, and 52 children were given the unknown-quantity bean task.

The unknown-quantity bean task, The first part of the unknown-quantity bean task is the same as the known-quantity bean task described earlier, except that, on giving the cup of beans to the children, the interviewer said, "Now I'm going to give you a cup of beans. I don't know how many beans there are." The interviewer gave the children a cup with 30 beans, and the number written on the index card was 18. When children were correct for both digits, we gave them the same countersuggestion as in the known-quantity bean task. When children were incorrect for either digit, we gave them the same hint as in the known-quantity bean task. If children, after putting the correct number of beans in the cup for the value of the 1s digit, dumped the remaining beans in the cup for the value of the 10s digit but responded to the question "How many did you put in the cup?" by saying "10," the interviewer asked them, "How do you know?" and, if necessary, she said, "I don't know how many beans I had altogether in the cup, do you?"

Coding. Children's successful performance on the known-quantity bean task in the pretest and two posttests discussed earlier was coded for counting or dumping. Children's performance on the unknown-quantity bean task was also coded for counting, dumping, or incorrect other responses.

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Results and Discussion

Of the 52 children who were given the unknown-quantity bean task, 36 spontaneously counted, 3 spontaneously gave a face value response but gave a correct response after the hint, 6 gave incorrect responses and remained with them after being offered the hint, and 7 dumped. All 7 children who dumped changed their response to counting to give the correct response as soon as the interviewer told them that she did not know how many beans she had altogether in the cup. A chi-square test of uniformity of the distribution of the children's responses to the unknown-quantity bean task reveals that children's performance in this task was significantly skewed toward the correct response of counting, x2(3, N= 52) = 54.92, p c .001. This result indicates that children who responded by dumping in the known-quantity bean task understood the composition of values.

Does Instruction and Practice in the FVCV System Help Children Grasp the Composition of Values?

In light of this discussion regarding children's grasp of composition of values, we extend our examination of the effectiveness of the new experimental FVCV system in helping children understand glace value. Our discussion indicates that there are three responses in the bean task that can be taken as indicating grasp of the composition of values: putting the correct number of beans in the cup for the value of the 10s digit by (a) dumping (with or without the hint), (b) counting spontaneously and specifying composition of values in rejecting the countersuggestion, or (c) counting after taking the hint. In this analysis, we consider only these responses to be adequate. That is, in any given administration of the known-quantity bean task we recoded children's responses to the 10s digit as incorrect if they succeeded in this task by counting and if, in their reason for rejecting the countersuggestion, they failed to specify composition of values. A 3 x 3 (Treatment x Grade) ANOVA with the known-quantity bean task pretest

scores as the dependent variable showed that there were no significant differences arnong the mean pretest scores of the three treatments, the three grades, or all nine groups. A 3 x 3 (Treatment x Grade) ANCOVA with the known-quantity first posttest

bean task scores as the dependent variable and pretest scores as a covariate did not show any significant difference among the three treatments. The adjusted means were .44, .34, and .33, for FVCV, FVO, and BTM with FVO, respectively. The ANCOVA showed a significant difference among grades, F(2, 143) = 3.09, p < .05. The adjusted mean scores for the three grades in increasing order of grade were .34, .31, and .46. A 3 x 3 (Treatment x Grade) ANCOVA with the known-quantity second posttest

bean task as the dependent variable and pretest scores as a covariate, however,

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showed a significant difference among the three treatments, F(2, 143) = 4.64, p < .02. The adjusted mean scores for the three treatments were .61, .42, and .38, for FVCV, FVO, and BTM with FVO, respectively. The Bryant-Paulson post hoc test for differences between two groups showed that the FVCV treatment was significantly more effective (p < .05) than the BTM with FVO treatment. The FVO treatment, however, was not significantly different from either of the other two treatments. The 3 x 3 ANCOVA did not show any significant difference among grades. The adjusted mean scores in increasing order of grade were .42, .48, and .5 1.

These results indicate that instruction and practice in the FVCV system helped children to differentiate between the face value and the complete value of the digits in a multidigit place-value number representation and also to grasp that the complete values of the digits in a multidigit place-value number representation add up to the total value.

Progression From Counting to Dumping in the Bean Task

We have presented evidence (from Experiment 2) that children who succeed in the bean task by dumping grasp the composition of values. Additionally, we have found (through their reasons for rejecting the countersuggestion) that at least some children who succeed in the bean task by counting grasp the composition of values. Nevertheless, dumping does seem to be a more advanced strategy. Nineteen children changed their method of succeeding at a certain point throughout the three administrations of the known-quantity bean task from counting to dumping and vice versa. Seventeen (89%) of these children changed from counting to dumping, and only 2 children changed their responses in a contrary direction, from dumping to counting, ~ ' ( 1 , N = 19) = 11.84, p < .002. On one hand, this consistency in the direction of change may be due to the greater efficiency of dumping. On the other hand, from a semiotic point of view, this direction of change may be interpreted as indicating that dumping involves a higher level of integration of conceptual understanding of addition and subtraction with the composition of values in the place-value system. When children justify rejecting the countersuggestion by explaining that the cups for the values of the 1 s and 10s digits together need to give the correct total quantity, they demonstrate a grasp of composition of values. In this response, however, they do not reveal a full integration of the part-whole relation underlying addition and subtraction with the place-value system. That is, they do not reveal a full integration of "Part A + Part B = Total and Total - Part A = Part B" with the meaning of a multidigit number in the written place-value system. Whether or not they give this particular justification for rejecting the countersuggestion, further work is needed to determine whether children who count have achieved this integration. For example, further work might determine whether children who use the counting strategy accept or reject the dumping strategy when it is presented to them and what their justifications are.

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CONCLUSIONS

Existing research indicates that children can have facility with conceptual issues, such as grouping and regrouping of quantities and still have difficulties with the written place-value system. Furthermore, children may successfully use base-10 manipulatives to represent quantities and to compute and still have difficulties with the written place-value system. This led us to focus on semiotic aspects of the written place-value system and to design a new symbol system (FVCV) with

. properties intermediary between the written place-value system and the base-10 manipulatives. The results reported here show that instruction and practice in the FVCV system helped children to differentiate between the face value and the complete value of the digits in a multidigit place-value number representation and also to grasp that the complete values of the digits in amultidigit place-value number representation add up to the total value. We view this as a step toward complement- ing the frequent emphasis on conceptual issues with appropriate emphasis on semiotic issues of the written place-value system. In this work, we have used only two-digit numbers because this sufficed to examine the value of the approach and to demonstrate the specific semiotic issues that were the focus of this work. Clearly, the work should be extended to larger multidigit numbers, especially to examine children's use of the semiotic features of the place-value system to indicate the constant ratio between adjacent columns in larger multidigit numbers. Furthermore, this work can also be expanded to contribute to understanding the difficulties children have with, for example, fractions and decimals, which constitute increas- ingly complex sign use in the written place-value system.

ACKNOWLEDGMENTS

This research was supported in part by a grant from the Campus Research Board of the University of Illinois at Chicago. We thank Jean Marie Grant for all her invaluable contributions to this research. We also thank the children, teachers, and principal for all their time and effort.

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Documents

Children's Developing Understanding of Place Value: Semiotic Aspects