Academic journal article Focus on Learning Problems in Mathematics

# The Development of Children's Understanding of Equality and Inequality Relationships in Numerical Symbolic Contexts

Academic journal article Focus on Learning Problems in Mathematics

# The Development of Children's Understanding of Equality and Inequality Relationships in Numerical Symbolic Contexts

## Article excerpt

A fundamental concept in arithmetic and algebra is the equivalence relationship represented by the '=' symbol. In order to understand equations such as 7+8 = 7+2+6 or 2x+2 = 3x-6, children have to understand that the term 'equals' as represented by the '=' symbol is used as a relational symbol. As a relational symbol, '=' denotes a relationship of quantitative sameness between the two members of an equality sentence. There is considerable evidence, however, that children, especially those in the age range 5-8 years, consider the equals sign to be an operator rather than a relational symbol (Behr, Erlwanger, & Nichols, 1980). As an operator symbol, '=' is interpreted as a command to perform an arithmetical operation. For example, children tend to view the equals sign in the sentence 5+3 = [] as indicating that the addition should be performed. As a first grader in Baroody and Ginsburg's (1983) study explained, 'equals' in a sentence such as 5+3 = [] "means to put another number that they would add up to" (p. 207).

Children also appear to acquire the notion that the operator is one-directional or left-to-right; they tend to insist that a "math problem" must precede the equals sign and that the answer to the problem must come after the equals sign (Denmark, 1976). Thus children generally tend to view sentences like 4 = 1+3 as "backwards" and sentences like 1+3 = 2+2 as incorrect. They tend to reject atypical equations such as 5 = 5 and rewrite the sentence to include an operation (e.g., 5+0 = 5, 5-0 = 5) (Behr, Erlwanger, & Nichols, 1976). In only a few interviews with children in grades 1-6, and rarely in the lower grades, did Behr et al. (1976) encounter a child who could explain the meaning of sentences such as 5 = 5 and 1+3 = 2+2 in terms of a relation.

As children progress through school, they are increasingly exposed to equations where the interpretation of the '=' symbol as an operator is meaningless (e.g., 2 + (3+7) = (2+3) + 7, 2 x (10+3) = 2 x 10 + 2 x 3,47 = [] tens + [] ones, 3x + 7 = 9) (Anderson, 1976). However, it appears that many students in higher grades, even at the college level, do not develop a concept of an equals sign as an indication of a relation. This appears to affect algebra learning and the learning of other mathematical subjects that require algebraic competence (Falkner, Levi, & Carpenter, 1999; Frazer, 1976; Herscovics & Kieran, 1980; Renwick, 1932; Weaver, 1973).

The finding that children tend to interpret the equals sign as an operator is a robust one (Baroody & Ginsburg, 1983; Behr et al., 1980; Denmark, 1976; Falkner et al., 1999; Kieran, 1981; Renwick, 1932). This phenomenon has not been fully explained, however. Children's failure to interpret the equals sign as a relational symbol may reflect a lack of requisite understandings about equality and inequality relationships between symbolically represented numerical amounts. Alternatively, children may have the requisite conceptual understandings but may fail to associate the understandings with the symbol that represents an equality relationship. It is also possible that both explanations are valid but for different age groups. The level of instructional support that is required to assist children in grades 1-7 in acquiring a relational interpretation appears to decrease with age (Anderson, 1976; Denmark, 1976; Falkner et al., 1999; Herscovics & Kieran, 1980; Saenz-Ludlow & Walgamuth, 1998). This change may reflect a lack of requisite conceptual understandings for younger children whereas the only role for instruction for older children is to help them associate their understandings about equality relationships with the equals sign. These explanations cannot be assessed, however, because prior research has not established what children of different ages know about equality relationships in the context of symbolically represented numerical amounts or the relationship between children's interpretation of the equals sign and their conceptual understandings about equality relationships.

Another important question that remains unanswered is the following: When instruction is successful in developing a relational interpretation, why is the instruction successful? There are several studies that suggest that a relational interpretation of the equals sign is not ordinarily achieved in the absence of extensive and explicit instruction (e.g., Anderson, 1976; Baroody & Ginsburg, 1983; Denmark, 1976; Falkner et al., 1999; Herscovics & Kieran, 1980; Saenz-Ludlow & Walgamuth, 1998) and there is some evidence that, once an operator interpretation develops, it is difficult to discard (Baroody & Ginsburg, 1983). Thus it is important to develop a relational interpretation of the symbol '=' when it is first introduced and to identify the features of successful beginning instruction. The symbol is typically introduced in grade 1. Various instructional interventions have been carried out to develop first graders' relational conception of the equals sign and these had differing levels of success (Baroody & Ginsburg, 1983; Denmark, 1976; Davydov, 1982; Falkner et al., 1999). It is unclear what features of the instruction facilitated learning; for example, in all of the interventions, children were exposed to a variety of atypical number sentence forms as part of the instruction. The critical features of successful instruction cannot be identified because it is not clear what younger children understand about equality relationships, how this conceptual understanding is related to their interpretation of the equals sign, and how the instruction and the difficulties are related to what they understand.

The current research examined (a) children's knowledge about equality relationships between symbolically represented numerical amounts, (b) the relationship between this understanding and their interpretation of the equals sign, and (c) changes with age in (a) and (b). Three research questions (RQs) were investigated:

RQ1. In the context of symbolic numerical tasks, do children recognize principles related to the properties of equality and inequality relationships, and can they apply these principles to solve problems? Are there changes in these understandings over the age range 6- to 11-years?

RQ2. In the context of symbolic numerical tasks, can children select an appropriate procedure to solve various types of tasks that involve reasoning about equality relationships? Are there changes in this ability over the age range 6- to 11-years? In particular, do young school-aged children distinguish comparative situations that require reasoning about equality relationships and quantifying exact numerical amounts within the same task, and use an appropriate procedure to solve the task?

RQ3. What is the relationship between children's understandings about equality relationships between symbolically represented numerical amounts and their interpretation of the equals sign? Are there changes in the nature of this relationship over the age range 6- to 11-years? Do children fail to associate their understandings of equality relationships with the equals sign?

In prior research, children's understanding of equality relationships in symbolic numerical contexts was assessed by examining children's performance on symbolic tasks that included the equals sign; children's interpretation of the equals sign was used as the indicator of their conceptual understanding of an equality relationship. Hence it remains unclear whether the difficulties reflect a lack of requisite conceptual understandings or a failure to link conceptual understandings with the equals sign. The current research addressed this question by comparing children's performance on two types of symbolic tasks that involved reasoning about equality relationships-tasks that did, versus tasks that did not include the equals sign (RQ3). In comparison with prior research, the current research also assessed children's conceptual understanding using a fuller range of tasks. Children's conceptual understanding was assessed by examining their knowledge of principles related to the properties of equality and inequality relationships, their ability to apply the principles to solve problems and to make deductive inferences, and their ability to select an appropriate procedure to solve a range of problem types that involved reasoning about equality relationships (RQ1 and RQ2). All of these understandings are involved in full competence (see, for example, Baroody & Ginsburg, 1986). However, there has been little research on children's knowledge and application of principles in symbolic contexts, or ability to select an appropriate procedure in response to several types of equality tasks.

Prior research has not established whether younger children's difficulties are conceptual or notational, or the nature of young children's conceptual understandings about equality relationships in symbolic contexts. There is, however, an extensive body of literature on young children's understanding of equality relationships between physical amounts. This research suggests more competence than their symbolic performance, but also a specific pattern of competencies and deficiencies. In an attempt to provide a fuller understanding of the nature of young children's understandings about equality as a relationship, the current research investigated the hypothesis that the same pattern of competencies and deficiencies may also describe younger children's performance in symbolic contexts (RQ1 and RQ2); the research that motivated this hypothesis will now be reviewed.

Research on children's understanding of equality relationships in concrete contexts

The nature of younger children's understandings about equality relationships between concrete, physical amounts suggests some hypotheses about the nature of younger children's understandings and difficulties in symbolic contexts. In the context of collections of objects, children appear to understand that small numerosities satisfy a numerical equivalence relation or a numerical ordering relation by around age 4 (Gelman & Gallistel, 1978; Sophian, 1996) and appear to have at least an implicit understanding of the following principles pertaining to the effects of additions and subtractions on numerical equivalence relationships by around age 6-7-years:

(1) a + c - c = a - c + c = a ("inversion") (Brush, 1978; Cooper, Starkey, Blevins, Goth, & Leitner, 1978; Smedslund, 1966)

(2) If a = b and c = d, then a [+ or -] c = b [+ or -] d ("compensation") (Cooper et al., 1978; Smedslund, 1966)

(3) If a = b and c > d, then a + c > b + d ("complex addition") (Brush, 1978)

(4) If a = b and c > d, then a - c < b - d ("complex subtraction") (Brush, 1978)

Cooper et al. (1978) presented similar tasks involving initial inequality relationships (see also, Cooper, 1984). For example, the initial numerosities of the two arrays differed by two and then one item was added to (subtracted from) the smaller (larger) array. In both cases, many children judged that the outcome was an equal number of objects in the two sets (Cooper, Campbell, & Blevins, 1983). Solutions of this type were referred to as qualitative solutions because they appear to be based only on the initial relationship and the type of transformation (addition or subtraction) and fail to quantify numerical amounts. In general, children are not successful on this type of problem until 6 or 7 years of age (Ginsburg, Klein, & Starkey, 1998).

In Cooper et al.'s, Brush's, and Smedslund's tasks, children are asked to reason about different combinations of initial states and transformations. Table 1 shows the types of situations used in the tasks. In the cells marked "indeterminate," the relationship between the final amounts cannot be determined solely on the basis of the information indicated in the row and column headings. For example, if X and Y are the starting amounts in the two arrays with X < Y, and z is added to the array containing X objects and w is added to the array containing Y objects and z > w, then the relationship between the final amounts X' and Y' cannot be determined on the basis of this information alone; X' (i.e., X + z) could be less than, equal to, or greater than Y' (i.e., Y + w). For the remaining cells, the result is logically deducible on the basis of information about relative numerosity alone; if one knows the relative numerosity of the initial amounts and the relative numerosity of the amounts involved in the transformations, then the relative numerosity of the final amounts can be deduced logically.

The conclusion is logically deducible because the situations involve principles. Any task that involves a principle can be solved by encoding the relative numerosity of the starting amounts and the amounts involved in the transformation in ordinal terms (i.e., as "equal" in number, more numerous, or less numerous). For example, in the complex addition task, the child can reason that if two sets have the same amount, and more is added to one set than to the other, then the sets are not equal. It is sufficient to encode the amounts in ordinal terms because this is the only feature of the numerical amounts that must be encoded to solve a task that involves a principle.

Problems that involve initial inequalities and unequal additions or subtractions, however, often do not allow an ordinal encoding of amounts. In Cooper et al.'s task, for example, it is not valid to reason that if X is initially less than Y, and more is added to X than to Y, then the resulting sets will have the same number of objects; the result is ambiguous with this encoding. The child must determine how much more one set is than the other and determine whether the transformation compensates for the initial difference.

The relative difficulty of the tasks presented by Brush (1978), Smedslund (1966), and Cooper et al. (1978) suggests that a logic pertaining to the effects of additions and subtractions on equality and inequality relationships may develop prior to much ability to integrate the logic and the necessary quantifications. In other words, the logic of Table 1 and appropriate solution procedures for the different problem types are more fully developed over time. In the context of collections of objects, the majority of 4-6-year-old children appear to have the requisite competencies to deal with the problems that involve principles. They can encode numerical amounts as same, more, and less, and understand that addition to a set increases numerosity and subtraction decreases numerosity (Gelman & Gallistel, 1978; Ginsburg et al., 1998). Their knowledge appears to include principles pertaining to the effects of additions and subtractions on equality and inequality relationships (Brush, 1978; Cooper, 1984; Gelman & Gallistel, 1978; Smedslund, 1966).

These understandings will be referred to as knowledge about numerical relationships. These competencies can be distinguished from quantification skill-i.e., skill in carrying out procedures that quantify exact numerical amounts (e.g., sets, sums, differences) (Cooper, 1984). These procedures include counting, estimating, subitizing, and computing. Cooper et al.'s (1978) tasks are examples of the indeterminate cases in Table 1, and a correct solution procedure requires the coordinated application of knowledge about numerical relationships and quantification skills.

Researchers and teachers often present the following types of tasks to assess or to develop a relational interpretation of the equals sign: (a) Is the following number sentence true: 4+3 = 5+2? (or 13 = 7+6, etc.); (b) Which number will make the number sentence true: 3+4 = [] + 5? (or 1 + [] = 3, [] = 1+1, etc.) (e.g., Anderson, 1976; Baroody & Ginsburg, 1983; Denmark, 1976; Falkner et al., 1999). These types of tasks require the coordinated application of knowledge about numerical relationships and quantification skills at appropriate points in the solution process.

For example, to judge the equality of 4+3 and 5+2, the child can compute the value of 4+3 to obtain 7, compute the value of 5+2 to obtain 7, and determine the relationship between 7 and 7. Alternatively, the child can decompose and recompose the parts. "One of the ones from 3" can be shifted to 4, for example, to yield 5+2 on the left side; 5+2 on the left can then be compared with 5+2 on the right. Both solutions require the application of skills that quantify exact numerical amounts-to compute the value of both expressions or to determine the numerical amount that must be shifted from one quantity to another in order to yield identical or commuted expressions-and the application of knowledge about numerical relationships-to judge the relationship between the resulting two computed values or the two obtained expressions. To solve task (b), the child can apply his/her quantification skills to compute the value of 3+4 to obtain 7, apply knowledge about numerical relationships to judge the equivalence of 5 and 7, determine how much more 7 is than 5 using his/her quantification skills, and determine that 2 must be added to the right side in order to establish an equality relationship. Again a correct solution requires the ability to recognize that exact quantification is necessary, that conceptual knowledge about equality and inequality relationships should be applied, and the ability to deploy these skills and understandings appropriately at each stage in the solution.

If there is some continuity in children's reasoning across symbolic and concrete contexts, the pattern of results from Cooper et al. (1978), Brush (1978), and Smedslund (1966), and other evidence suggests that three interrelated characteristics of young children's reasoning may interfere with their developing a correct solution process: (1) Young children may develop and use invalid qualitative rules to compare numerical amounts. For example, young children appear to develop the following invalid rule to solve Cooper et al.'s tasks: If a < b and c > d, then a+c = b+d (Starkey & Gelman, 1982). (2) Young children often fail to apply their quantification skills to reason about relationships among numerical amounts; their ability to quantify the value of a set, sum, or difference may be poorly developed (Gelman & Gallistel, 1978; Riley & Greeno, 1988) and there also appears to be a developmental increase in the ability to recognize situations in which information about exact numerical amounts and differences is needed (Cooper et al., 1983; Hunting & Sharpley, 1991; Resnick, 1992; Siegler & Robinson, 1982; Sophian, 1996; Tollefsrud-Anderson, Campbell, Starkey, & Cooper, 1992). (3) There appears to be a developmental increase in the ability and tendency to bring both types of competencies-knowledge of numerical relationships and quantification skills--to bear on a single task (Resnick, 1992; Resnick & Greeno, 1990). Cooper et al. (1983) found that the percentage of children who gave quantitative, rather than qualitative, solutions increased from 5-7-years; this suggests that children become increasingly capable of carrying out a procedure that integrates reasoning about relationships with the necessary quantifications, and of recognizing situations that require such a procedure.

It was posited that many children in the age range 5-7-years may understand some principles pertaining to the properties of equality relationships. However, they may not distinguish comparative situations that require the application of both their knowledge of numerical relationships and quantification skills, or have much competence in carrying out a procedure that integrates reasoning about relationships with the necessary quantifications. This may be one source of younger children's difficulties on symbolic tasks. The tendency to bring only adding and subtracting skills to bear on symbolic tasks (e.g., to regard the sentence '3+4 = 5+2' as incorrect and to correct it by replacing 5+2 with 7), and not conceptual knowledge about quantitative sameness, may be another manifestation of the tendency to bring one type of competency or the other to bear. If the quantification aspect is most salient, young children may tend to apply their quantification skills but not their knowledge about comparing quantities; on the other hand, if the most salient feature of a task is the requirement to compare quantities, then young children may apply their knowledge about numerical relationships but not their quantification skills. The research questions investigated these possibilities.

Method

Sample

Twenty first graders (mean age = 7.25 years, SD = 0.34 years, ranging from 6 years-9 months to 7 years-10 months), 24 second graders (mean age = 8.02 years, SD = 0.38 years, ranging from 7-4 to 8-8), and 20 fourth graders (mean age 10.28 years, SD = 0.42 years, ranging from 9-6 to 11-3) from two public schools in an eastern state volunteered to participate. The participants were drawn from 11 classrooms. Table 2 shows the mathematics curricula that were used in the participants' classrooms during the year of the study. The schools drew from lower and middle-class neighborhoods. The research was conducted in April and May of the school year.

Instruments and Procedure

Each child was interviewed individually. Interviews were audiotaped and transcribed.

Set 1 tasks. Set 1 tasks were taken from Baroody and Ginsburg (1983). The interviewer said: "Cookie Monster did some math homework last night. He wrote out some math sentences and was wondering if you would be his teacher and correct them for him" (Baroody & Ginsburg, 1983, p. 202). Eight sentences were then shown to the child, with the order of presentation randomized across children. The sentences were as follows: 7+6 = 13, 13 = 7+6, 7+6 = 6+7, 7+6 = 4+9, 7+6 = 6+6+1, 7+6 = 14-1, 7+6 = 6, 7+6 = 0. For each sentence, the interviewer asked:

Q0: "What does that say?"

Q1: "Did Cookie Monster write that correctly-like you would in math class?" If the child responded "yes" to Q1, the interviewer asked:

Q2: "Is it correct? Does it make sense?"

If the child responded "no" to Q1, the interviewer asked:

Q2: "Is there any way in which it is correct? Does it make any sense?"

Q3: "Should we put this in the wrong pile or the right pile?" If the child hesitated, the interviewer said, "If you had to choose, which pile would you put it in? If it makes sense, put it in the right pile; if not put it in the wrong pile" (Baroody & Ginsburg, 1983, p. 203).

The interviewer then said: "Cookie Monster did some more math sentences last night and would like you to correct them again." The interviewer then handed the child a list of equality sentences: 7 = 5+2, 4+3 = 3+4, 6+4 = 5+5, 6+3 = 4+4+1, 5+1 = 7-1, 8 = 8, 2+2 = 2,4+2 = 42, 3+1 = 1+1+1, 9 = 9. The interviewer said: "Put a C next to his sentence if his math sentence is right-if it makes sense. Put an X if his math sentence is wrong-if it does not make sense." When the child had completed the list, he/she was asked to explain his/her responses. The Set 1 tasks were used to examine whether the child applied an operator interpretation of the equals sign only, a relational interpretation of the equals sign only, or both interpretations (RQ3).

Set 2 tasks. Following the Set I tasks, the interviewer said, "Now we're going to do something different. I want to see how you solve these problems and you can do them in any way that you want." Paper and a pencil and counters were available to the child. The tasks appeared in a notebook. Each page had two printed lines (hereinafter referred to as the "top line" and the "bottom line"). The interviewer placed the notebook in front of the child. A sample problem was presented:

```
3      3
3+4-4  3
```

The interviewer concealed the bottom line so that only the top line was visible. The interviewer read the top line, saying "3 and 3. Are these amounts equal, or is one amount more? Is this one more [pointing to the '3' on the left-hand side], are both amounts equal [finger waved back and forth], or is this one more [pointing to the '3' on the right]?" Following the child's response, the interviewer requested the reason for the response. The interviewer then revealed and read the bottom line: "Three plus four take away four [pointing to the left side, pauses] and [pointing to the right side] 3. Are these amounts equal, or is one amount more? Is this one more [pointing to the '3+4-4' on the left-hand side], are both amounts equal [finger waved back and forth], or is this one more [pointing to the '3' on the right]?" The reason for the response was requested. No assistance was offered. The interviewer ended the treatment of the sample problem by saying, "Great. Do you understand what we're doing now? Okay let's start."

The Set 2 tasks were then presented. The tasks appeared on 38 pages in the notebook; thus there were a total of 76 printed lines. Four of the lines were not target tasks, leaving a total of 72 target Set 2 tasks. For each page, the procedures were identical to those in the sample problem.

In each task, the top line consisted of two numerical amounts, X and Y, on the left and right sides. On 30 pages, X and Y were two equal or unequal numbers (e.g., 3 and 3, 5 and 8). On eight pages, X and Y were two expressions of the form a+b and c+d, where a+b was equal to c+d (e.g., 4+5 vs. 3+6). On all 38 pages, the bottom line consisted of two symbolic expressions of the form X+z and Y+z, X-z and Y-z, X+z-z and Y, X and Y-z+z, X+z and Y+w, or X-z and Y-w. In all cases, w and z were numerals and w [not equal to] z, and X and Y were the quantities appearing on the top line.

Thus 30 tasks simply involved comparing two numbers. For the remaining 42 target tasks, there were two types of tasks: there were 28 tasks that involved principles and 14 tasks that required quantification. The tasks that involved principles embodied the logic underlying the inversion, compensation, complex addition, and complex subtraction principles and principles pertaining to the properties of inequality relationships. These tasks are shown in Table 3. For example, the task on the bottom line is an example of a task that involved a principle:

```
8    8
8+1  8+3
```

On the top line, children judged that 8 is equal to 8. On the bottom line, unequal amounts are added to the equal amounts on the top line; the problem embodies the logic underlying the complex addition law. Children can reason that because 8 is equal to 8 and 1 is less than 3, 8+1 is less than 8+3. It is sufficient to encode the initial amounts as the "same," to encode one of the added amounts as "more," and to apply knowledge about the effects of additions on an equality relationship. In general, the problems that involved principles could be solved by encoding the initial amounts (X and Y on the top lines) and the "transformations" (the amounts w and z that were added and/or subtracted on the bottom lines) in ordinal terms and then applying conceptual knowledge about the effects of additions and subtractions on equality and inequality relationships; they did not require the quantification of a sum or difference. For all tasks that involved principles, the question of interest was whether children recognized principles related to the properties of equality and inequality relationships, independently recognized problems in which the principles were applicable, and spontaneously applied valid principles to solve the problems (RQ 1).

For the remaining 14 target tasks no principle was applicable. These tasks appear in Table 4. These tasks are nearly identical to the "traditional" kinds of tasks that are used to develop a relational interpretation of the equals sign; i.e., they require judging the equality of two sums a+b and c+d (e.g., 3+6 and 4+5) and performing some kind of computation of exact amounts and a comparison. The following problem is an example of a task that requires quantification.

```
3    5
3+4  5+2
```

All tasks that required quantification had the same form as this task: They required comparing amounts X + z and Y + w; X, Y, z, and w were numerals with X < Y and z > w and no principle was applicable. For the 14 problems that required quantification, the question of interest was whether children would distinguish comparative situations that required reasoning about equality relationships and quantifying exact numerical amounts within the same task and use an appropriate procedure to solve the task (RQ2). The 72 tasks were presented in random order. Hence they examined younger children's ability to respond appropriately to different types of tasks (RQ2).

Tasks like the following embody the compensation principle, but appear to be trivial.

```
3    3
3+4  3+4
```

Therefore tasks that involved the compensation principle always took the following form: the task on the top line was a task that required quantification and always required comparing two expressions of the form a+b and c+d that had equal sums. The task on the bottom line involved the compensation principle; equal amounts were either added to or subtracted from the two equal expressions on the top line.

There were four large number problems that involved the compensation principle. The top line quantities therefore also involved sums. These top lines were not target tasks; it was anticipated in advance that some children might not be able to perform the computations necessary to find the four large number sums and they were included only to examine children's ability to apply the compensation principle in the context of large numbers. Therefore these problems were not included to investigate RQ2, but only RQ1. If children were unable to compute the sums on the top line, the interviewer provided the sums; the point of interest was whether the child could judge the equality of the amounts on the basis of the information about the sums, and then use the information about the equality relationship between the top line quantities and the compensation principle to deduce the equality of the bottom line quantities.

Scoring

Set 1 tasks. Two coders determined whether the child used an operator interpretation of the equals sign only, a relational interpretation only, or both interpretations at some point during the tasks. Both coders coded all Set 1 responses; agreement across the two coders was 100%.

Set 2 tasks. For the 30 tasks that involved comparing two numbers (e.g., 2 and 3, 4 and 4), responses were coded as 'equal' or 'not equal.' Children's responses on the remaining 42 tasks were classified using four mutually exclusive categories. The categories were developed on the basis of children's responses. Examples of participants' responses are provided for each category.

1. Applied a valid principle: Tasks that involved principles always appeared on the bottom lines. The child determined the relationship between the quantities on the bottom line by making a logical inference about the effects of equal or unequal additions or subtractions on an equality or inequality relationship. The child's explanation included no calculation of amounts. For tasks that involved the compensation principles, if the child incorrectly evaluated the top line quantities as unequal, the child was considered to have applied a valid principle on the bottom line if the response was consistent with the error; the child determined the relationship between the quantities on the bottom line by making a logical inference about the effects of equal additions or subtractions on an inequality relationship.

For responses that fell into the category "applied a valid principle," responses ranged from more explicit statements about the properties of an equality relationship and the properties of the operations of addition and subtraction to implicit use of the properties. However, the response had to clearly indicate that the child was reasoning on the basis of her knowledge of relationships among quantities and effects of additions and subtractions on these relationships, as opposed to numerically quantifying amounts and then comparing them. For example, for the principle that had to do with the effects of unequal additions on an equality relationship, in more explicit responses, relationships and operations were treated as nouns and their properties were discussed. The child focused on the equality relationship and not the specific values, the properties of relationships and operations, and the effect of adding unequal quantities on an equality relationship. In less explicit responses, the child applied knowledge about the effects of adding unequal quantities to equal quantities to determine whether the expressions were equivalent without computing the values of the sums. However, 'plus' was used as a verb, not a noun. The child focused on the properties of the numerical results (i.e., whether they were equal or not equal) and the properties of the numbers (e.g., 27+56 is higher than 27+14 because 27 and 27 are the same amount and 56 is a bigger number than 14), not on explaining properties of operations and relationships in general, although s/he used these properties to judge the equality of the expressions.

Examples of responses that fell into this category are given below. In each case, the examples first give the child's judgement of the relationship between the two expressions, and then the child's justification for the judgement.

8+2 < 8+3: "8 and 8 are the same but it's plus 2 and plus 3 and 3 is more than 2 so it's plus more." (first grader)

Complex subtraction principle:

56-34 < 56-27: "56 and 56 are the same. Because if you take away more, then it's the lesser number you get. The lesser you take away the more you have. 56 take away 27 is bigger because that [56-27] takes away lesser than that [56-34]." (first grader)

50+22+60 = 33+39+60: "Because they [50+22 and 33+39] both have the same number [i.e., the child is indicating that it was already established on the top line of this page that 50+22 and 33+39 had equal sums], and they add the same thing, so they have to equal the same thing." (first grader)

Compensation (subtraction) principle:

7+3-2 = 4+6-2: "Equal the same thing because this number [pointing to 7+3 and 4+6] is the same thing [i.e., the child had established on the top line that 7+3 and 4+6 were equal] and if you take away the same thing, this number is the same thing still." (first grader)

22+39-15 = 23+38-15: "They're [22+39 and 23+38] both equal. But if you minus equal amounts from each one, they both have to be equal." (first grader)

22+39-15 = 23+38-15: "I know 22+39 is equal to 23+38. They're the same. And 15 is equal to 15. These two up here [22+39 and 23+38 on the top line] equal the same amount and you're taking an equal amount from both of them. It would still be the same because you're taking an equal amount from equal amounts." (fourth grader)

Inversion principle:

25 = 25-17+17: "25 minus 17 goes down. Then you get the 17 back you get 25 again. 25 and 25 are the same things." (first grader)

25 = 25-17+17: "Because you take away 17 and you add 17, take away 17 equals whatever and you put 17 in, it equals the same thing back." (first grader)

1048+987-987 = 1048: "You get it and then they took away the same amount. Both have to be ten-forty-eight." (first grader)

1048+987-987 = 1048: "Because if you minus the number that you add you're just back at the number you had first." (second grader)

1048+987-987 = 1048: "This is adding and then subtracting the same amount and then it will be the same number as in the beginning so they will be equal." (fourth grader)

If a < b, then a+c < b+c:

2+5 < 3+5: "I knew because 3 is more than 2 and 5 is equal to 5." (first grader)

5+3 < 8+3: "8 is more than 5 and then it's plus the same amount of number." (first grader)

If a

6-3 > 5-3: "Six is greater than five and you're subtracting the same amount so 6-3 would be the larger amount." (fourth grader)

If a < b and c < d, then a+c < b+d:

5+2 < 8+3: "If you add 3, it's higher, 3 is more than 2 and 8 is more than 5. This [5+2] is higher and you add higher and this [8+3] is lower and you add lower." (first grader)

5-3 < 6-2: "If you take away lesser and you have a higher number [points to 6-2] and here [points to 5-3] you have a lower number and take away more." (first grader)

5-3 < 6-2: "If you subtract a lower number then you have a greater difference than if you subtract a bigger number. And the starting amounts are not equal. Five is less than 6 and 3 is greater than 2. So 5-3 must be less." (fourth grader)

2. Compute and compare method: The child calculated or recalled the sums or differences, and then compared the resulting numerical values; if the sums or differences were equal, the child responded "equal," and if one was more, then the child identified the larger quantity.

5+2 = 4+3: "Same. 5+2 is 7. 4+3 is 7. The same number." (first grader)

9 = 9-3+3: "9 is on one side. Then 9-3 is 6, plus 3 is 9. Both 9. Equal." (second grader)

5+2 = 4+3: "Both equal 7. Equal the same thing, 7, just different equations." (second grader)

3. Decompose and recompose the parts method: The child "shifted" some precise numerical amount from one number to another in order to obtain two identical or commuted expressions on the left- and right-hand sides of the page. S/he then concluded that the two identical or commuted expressions were equal.

3+5 = 6+2: "Same because take one off 3 and add it to 5, and they would both have 2 and 6." (first grader)

5+4 = 6+3: "Equal. Six is one more but if one of the six went to the three, 5+4. The one from six jumped over to the three." (first grader)

5+4 = 6+3: "Four gives five one, it will be a 3, and the 5 will be a 6, and they both equal the same." (second grader)

4. Applied an invalid strategy: The child applied an invalid rule or strategy to compare the quantities. The most frequently used invalid rules are described below.

a. First number rule: The child compared the first number of each expression and selected the expression with the largest first number as the largest quantity.

5+4 < 6+3: "5+4 less because 6 is higher than 5." (first grader)

b. Last number rule: The child compared the last number of each expression and selected the expression with the largest last number as the largest quantity.

3+5 > 6+2: "3+5 is higher than 6+2 because if you plus in 5, it would be higher than plussing in 2." (first grader)

c. Highest number rule: The child selected the expression with the largest number as the largest.

3+5 < 2+7: "Because 7 is bigger than the rest of all the numbers. It would make it much bigger than the other one." (first grader)

d. Qualitative decompose and recompose the parts rule: In comparing a+b and c+d, if an addend in one expression was less than an addend in the other expression, whereas the reverse relationship held for the remaining two addends, then the expressions were judged equal (i.e., If a < b and c > d, then a+c = b+d).

7+2 = 4+5: "Both the same. That's [7+2] more and gets less and that's [4+51 less and it gets more." (first grader)

e. Invalid complex subtraction principle: In comparing a - b and a - c where b < c, the child claimed that since b < c, a - b must be smaller than a - c.

37-19 vs. 37-26: "37 is equal to 37 but 19 is less than 26 so 37 take away 19 has to be less." (first grader)

56-34 vs. 56-27: "56 and 56 are same and you're taking more away from this one [56-34] so 56-34 is bigger." (first grader)

Whenever the invalid complex subtraction principle was applied, the interviewer repeated the child's response and requested verification; the interviewer emphasized that the operation was subtraction to ensure that the response did not just reflect a failure to attend to the sign.

f. Other invalid reasoning strategies: All other invalid approaches were categorized as "other invalid reasoning strategies."

25 < 25-17+17: "That one is bigger [25-17+17] because it has more numbers." (first grader)

25 < 25-17+17: "They're both 25 and that one's subtracting 17 and then adding 17 so 25-17+17 will be bigger." (fourth grader)

The child's responses were coded as a set. Two coders coded the 30 responses involving the comparison of two numbers for all children; agreement was 100%. They also coded the remaining 42 Set 2 responses for all children using the four categories. If the child applied an invalid strategy to compare quantities, the type of invalid strategy was also identified using the six categories. Agreement across the coders for these questions was 96.7%.

Results

An alpha level of .05 was used for all statistical tests. The first research question was as follows:

RQ1. In the context of symbolic numerical tasks, do children recognize principles related to the properties of equality and inequality relationships, and can they apply these principles to solve problems? Are there changes in these understandings over the age range 6- to 11-years?

The Set 2 tasks were used to investigate this question. On the 30 Set 2 tasks that involved comparing two numbers (e.g., 2 and 3), all children responded correctly for every task. For the remaining 42 target tasks, there were 28 tasks that involved principles. A total score for application of principles was calculated for each child. For each Set 2 task for which a principle could potentially be applied, children were given a score of 1 if a principle was appropriately applied to solve the task--i.e., if the child's response was coded as "applied a valid principle"--and 0 otherwise. Scores could range from 0 to 28.

Table 5 shows mean total scores for application of principles by grade level. A single factor ANOVA revealed no statistically significant differences across grade levels in mean total scores for application of principles (F(2, 61) = 2.05, p = . 138). Figure 1 shows the percentages of children at each grade level who received various scores for application of principles. The figure shows, for example, that 60%, 50%, and 65% of the first, second, and fourth graders respectively had scores of 8 or more; i.e., they appropriately applied principles to solve eight or more Set 2 tasks.

[FIGURE 1 OMITTED]

Table 6 shows the percentages of children who applied each principle on at least one Set 2 task. For each of the nine principles, there were no statistically significant differences across grade levels in the number of children who applied each principle with two exceptions. Significantly more fourth graders than first graders applied the inversion principle ([chi square](1, N = 40) = 6.67, p = .010). In addition, significantly more fourth graders than second graders applied the complex addition principle ([chi square](1, N = 44) = 5.52, p = .019). However, this principle appears to be well-known by first grade; 85% of the first graders applied the principle.

Table 6 also shows mean scores for each principle by grade level; children were given a score of 1 if the principle was applied to solve the task and 0 otherwise. For each principle that involved equality relationships, scores could range from 0 to 4 (there were a total of four tasks where the principle could potentially be applied). For the principles that involved inequality relationships, scores could range from 0 to 2 (there were a total of two tasks where the principle could potentially be applied). For the nine principles, the only statistically significant difference across grades was in the mean scores for application of the compensation (addition) principle between second and fourth graders (p = .042).

The second research question was as follows:

RQ2. In the context of symbolic numerical tasks, can children s elect an appropriate procedure to solve various types of tasks that involve reasoning about equality relationships? Are there changes in this ability over the age range 6- to 11-years? In particular, do young school-aged children distinguish comparative situations that require reasoning about equality relationships and quantifying exact numerical amounts within the same task, and use an appropriate procedure to solve the task?

Table 7 shows the number of children by grade who used a correct strategy on all 38 tasks and on 35 or more of the 38 tasks. Table 7 also shows mean scores for application of a correct strategy (out of 38). Second graders' scores (p <.001) and fourth graders' scores (p <.001) were significantly higher than first graders' scores. Figure 2 shows the percentages of children who received various scores by grade level.

[FIGURE 2 OMITTED]

Table 9 shows percentages who applied each of the two valid strategies (compute and compare or decompose and recompose the parts) on at least one of the 14 small number tasks that required quantification, and the mean number of problems on which children applied each strategy by grade level. There were significant differences in the number of children who applied the compute and compare strategy between first and second graders ([chi square](1, N = 44) = 5.28, p = .022) and first and fourth graders ([chi square](1, N = 40) = 4.44, p = .035). There were significant differences in the mean number of problems on which children applied the compute and compare strategy between first and second graders (p = .001) and first and fourth graders (p = .001).

Table 10 shows the invalid strategies that children applied on the Set 2 tasks and the percentages of children who applied the invalid strategies on at least one task. All of the invalid strategies were "qualitative strategies"; i.e., the child did not compute any exact numerical amounts in his/her comparison of quantities. The description of the strategies appears in the Scoring section.

There were statistically significant differences between first and second graders in the number of children who applied the first number rule ([chi square](1, N = 44) = 8.61, p = .003), last number rule ([chi square](1, N = 44) = 8.61, p =.003), highest number rule ([chi square](1, N = 44) = 6.77, p =.009), and other invalid strategies ([chi square](1, N = 44) = 7.59, p = .006). There were significant differences between first and fourth graders in the number of children who applied the first number rule ([chi square](1, N = 40) = 10.00, p = .002), last number rule ([chi square](1, N = 40) = 10,00, p = .002), highest number rule ([chi square](1, N = 40) = 5.7 1, p = .017), invalid complex subtraction principle ([chi square](1, N = 40) = 7.06, p = .008), and other invalid strategies ([chi square](1, N = 40) = 10.00, p = .002).

An invalid strategy score was calculated for each child. A child was given a score of 1 on a task if he/she applied an invalid strategy in one of the six categories in Table 10, and 0 otherwise. Table 11 shows the mean scores for application of invalid strategies by grade. First graders' invalid strategy scores were significantly higher than second graders' scores (p <.001) and fourth graders' scores (p <.001).

The third research question was as follows:

RQ3. What is the relationship between children's understandings about equality relationships between symbolically represented numerical amounts and their interpretation of the equals sign? Are there changes in the nature of this relationship over the age range 6- to 11-years? Do children fail to associate their understandings of equality relationships with the equals sign?

The Set 1 tasks were used to examine children's interpretation of the equals sign. The interpretations of the equals sign that children applied during the Set 1 tasks, and the percentages of children who applied each interpretation, are shown in Table 12. There were significant differences in the number of children who applied an operator interpretation only between first and second graders ([chi square](1, N = 44) = 9.43, p =.002), first and fourth graders ([chi square](1, N = 40) = 29.57, p <.001), and second and fourth graders ([chi square](1, N = 44) = 10. 18, p = .001). The number of children who applied both an operator and a relational interpretation also differed significantly across first and second grade ([chi square](1, N = 44) = 6.94, p = .008), first and fourth grade ([chi square](1, N = 40) = 17.14, p < .001) and second and fourth grade ([chi square](1, N = 44) = 4.23, p = .040). There were significant differences in the number of children who applied a relational interpretation only during the Set 1 tasks between first and fourth graders ([chi square](1, N = 40) = 5.71, p = .0 17).

In order to examine the relationship between children's understandings about equality relationships between symbolically represented numerical amounts and their interpretation of the equals sign, the relationship between performance on the Set 1 and Set 2 tasks was examined. Criteria were established for understanding of equality and inequality relationships as measured by the Set 2 tasks. Both a strict and a lax criterion for understanding were established. In the strict criterion, a child was classified as having less than a full understanding of equality and inequality relationships in the context of numerical symbolic tasks if he/she had a total correct strategy score of 32 or less on the 38 Set 2 tasks (see the last column of Table 7) and a corresponding invalid strategy score of 6 or more (see Table 11). A child was classified as understanding the concepts of an equality and inequality relationship if he/she had a total correct strategy score of 33 or more and an invalid strategy score of 5 or less. Fifty percent of first graders, 92% of second graders, and all fourth graders met the strict criterion for understanding.

In the lax criterion, a child was classified as having less than a full understanding if he/she had a total correct strategy score of 28 or less on the 38 Set 2 tasks and a corresponding invalid strategy score of 10 or more. A child was classified as understanding the concepts of an equality and inequality relationship if he/she had a total correct strategy score of 29 or more and an invalid strategy score of 9 or less. Sixty percent of first graders, and all of the second and fourth graders met the lax criterion. In sum, using the different criteria, 50% to 60% of first graders, 92% to 100% of second graders, and 100% of fourth graders were classified as understanding the concepts of an equality and inequality relationship on the basis of their Set 2 performance.

Discussion

This research examined (a) children's knowledge about equality relationships between symbolically represented numerical amounts, (b) the relationship between this understanding and their interpretation of the equals sign, and (c) changes with age in (a) and (b). Tables 5 and 6 and Figure 1 show that the majority of the children at each grade level recognized principles related to the properties of equality and inequality relationships and could apply these principles to solve problems; these data show that the majority of children spontaneously applied some valid principles, recognized particular symbolic numerical situations in which the principles were applicable, and spontaneously applied the appropriate principle to deduce the nature of the relationship between two symbolically represented expressions. There were no significant differences across grades in mean total scores for application of principles. For the nine principles, the only statistically significant difference between first and fourth graders was in the application of the inversion principle; significantly fewer first graders than fourth graders applied the principle.

Thus the results provide some support for the hypothesis underlying Research Questions 1 and 2. There may be some continuity in children's performance across symbolic and concrete contexts. In both contexts, children show at least an implicit understanding of some valid principles early on. However, this knowledge may initially include both valid principles and invalid rules for comparing amounts. As counting skill and computational proficiency increase, children may become increasingly likely to apply their quantification skills in situations that involve comparison; on the other hand, with the increased use of quantification skills, the invalidity of qualitative approaches is more likely to be discovered. There may be a related increase in the ability to recognize comparative situations in which no principle is applicable and consequently, in which information about exact numerical amounts and the application of quantification skills is needed. The results also support the conclusion that by second grade, virtually all children form stable conceptions of the basic properties of equality and inequality relationships in symbolic contexts, can apply principles to solve problems, and can select and carry out appropriate procedures to determine the equality relationship between two expressions.

These data suggest that instruction should and could do a better job of helping children to develop a relational interpretation and to associate conceptual understandings about equality relationships with the equals sign from the beginning of instruction. One possible way to accomplish this is to wait until grade two to introduce the equals sign. The findings support the conclusion that, under current forms of instruction, almost all children have the requisite understandings only in grade two. More children would be able to bring the requisite competencies to bear on the task. For example, it would be easier for them to make sense of the atypical forms that are often used to develop a relational interpretation of the equals sign (see e.g., Denmark, 1976; Falkner et al., 1999). They would be more likely to be able to carry out an appropriate procedure that brings both types of competencies to bear on the task in an integrated way.

If the equals sign is introduced in K-1, however, the current research suggests some guidelines for instruction. The meaning that we want children to acquire for '=' is an abstraction of the notion of quantitative sameness. Children's notion of quantitative sameness appears to arise from experiences that involve comparing equivalent sets of objects or continuous quantities (see for example, Cooper, 1984). Thus the problem is how to help children apply their notions of sameness and ideas about comparison to symbolic contexts given children's level of understanding in K-1, and how to associate their notions about the comparison of quantities with '=' from the very beginning (as opposed to leading them to interpret the sign as meaning "the answer is").

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(1) Performance on the four large number tasks that involved the compensation principles was not included in this analysis because, as explained in the Method section, children sometimes received assistance on the four large number sums on the top lines; this analysis focuses only on those tasks for which all children independently selected a strategy on both the top and bottom lines. In addition, the 30 tasks that required comparing two numbers (e.g., 8 and 8, 3 and 4) were not included in this analysis. Therefore scores could range from 0-38.

```
Table 1 Types of comparisons that were used in Brush (1978), Smedslund
(1966), and Cooper et al. (1978)

Initial relationship
Type of
transformation     X=Y                     X>Y            XY'          X'Y'          X'Y'                   X'>Y'          ?
to Y where z>w     (complex addition law)                 indeterminate
Subtract z from X  X'=Y'                   X'>Y'          X'w          law)

Table 2 By Grade, Percent of Sample Who Experienced Various Mathematics
Curriculums

Curriculum used in the child's current mathematics class  1   2   4

Investigations in Number, Data, and Space                 30  25  60
(Economopoulos et al., 1998; Akers et al., 1998; Akers
et al., 1998)
Both Investigations in Number, Data, and Space            70  75   0
(Economopoulos et al., 1998) and Saxon Math 1 (Larson &
Matthews, 1998)
Math (Charles et al., 1998)                                0   0  40

Table 3 Tasks that Involved Principles

Equality Relationships
Addition         (TL) 8 (TR) 8               (TL) 27 (TR) 27
(BL) 8+2 (BR) 8+3           (BL) 27+56 (BR) 27+14
(TL) 6 (TR) 6               (TL) 54 (TR) 54
(BL) 6+4 (BR) 6+2           (BL) 54+39 (BR) 54+27
Subtraction      (TL) 5 (TR) 5               (TL) 37 (TR) 37
(BL) 5-3 (BR) 5-2           (BL) 37-19 (BR) 37-26
(TL) 9 (TR) 9               (TL) 56 (TR) 56
(BL) 9-2 (BR) 9-4           (BL) 56-34 (BR) 56-27
(Addition)       (TL) 4+5 (TR) 3+6           (TL) 50+22 (TR) 33+39
(BL) 4+5+2 (BR) 3+6+2       (BL) 50+22+60 (BR) 33+39+60
(TL) 3+5 (TR) 6+2           (TL) 63+49 (TR) 56+56
(BL) 3+5+2 (BR) 6+2+2       (BL) 63+49+17 (BR) 56+56+17
(Subtraction)    (TL) 7+3 (TR) 4+6           (TL) 22+39 (TR) 23+38
(BL) 7+3-2 (BR) 4+6-2       (BL) 22+39-15 (BR) 23+38-15
(TL) 5+4 (TR) 6+3           (TL) 45+31 (TR) 40+36
(BL) 5+4-1 (BR) 6+3-1       (BL) 45+31-14 (BR) 40+36-14
(TL) 4 (TR) 4               (TL) 25 (TR) 25
(BL) 4+2-2 (BR) 4           (BL) 25 (BR) 25-17+17