ASSOCIATIVITY AND UNDERSTANDING OF THE OPERATION OF ADDITION IN CHILDREN WITH LEARNING DIFFERENCES.

Abstract. This study sample consisted of children from two learning categories, learning disabled (LD, n = 27, [bar]X = 9.08) and not identified as learning disabled (NLD, n = 42, [bar]X = 7.46), who were individually tested on three different mathematics tasks. The modified nonverbal task and the associativity of length task investigated the quality of students' structures of organizing activity by noting the complexity of grouping relationships abstracted among and between object sets (i.e., composite unit structures). Additionally measured in these two tasks was response accuracy. The flashcard task measured accuracy in response to the same number problems as in the modified nonverbal task as well as strategy type used. However, the strategies scored on the flashcard task were indicative of explicitly taught procedures regardless of children's structures of organizing activity. Significantly more NLD children abstracted composite unit structures suggestive of operational logic on the modified nonverbal and associativity of length tasks, although there were no significant differences in the rate of success on the modified nonverbal task. On the flashcard task, there were no significant differences between the two groups on strategy type used, although the LD children achieved greater success. These results suggest that although children state correct answers on the flashcard and modified nonverbal tasks, they may be reflecting on the tasks using thought structures that are not yet operational.

Mathematical knowledge has traditionally been thought of as the "acquisition" of problem-solving strategies that include counting fingers, verbal counting, decomposition, and retrieval. The execution of these computational strategies is thought to result in the development of an association between the problem integer and the stated answer. As memory representations between problems and answers are developed, facts can be directly retrieved (Siegler, 1986, 1988; Siegler & Jenkins, 1989). The inability to compute number facts is due to deficits in: (a) fact retrieval, (b) procedures used, and (c) spatial representation (see Geary, 1993, and Jordan, 1995, for a review of related research). Language difficulties also negatively impact the mapping of conventional mathematical symbols onto a mental model of number and number transformation (Jordan, Levine, & Huttenlocher, 1995).

We believe that the above deficits owe their existence to the information-processing perspective of mathematical knowledge. If we change our perspective of what constitutes mathematical knowledge, the source of children's difficulties also changes. From a constructivist perspective, logical-mathematical knowledge consists of biologically based forms and structures of mental activity that evolve their form through the coordinations of specific nervous activity (i.e., schemes) as objects are acted upon (Sinclair, 1990). As specialized organs regulate thinking activity between the organism and the environment while solutions to authentic problems are pondered, these cyclical structures of nervous activity evolve their internal order by their gradual extension and eventual reorganization onto more complex, higher-order levels. In turn, the coordination of more complex grouping relationships within and between object groups is made possible as the environment is acted upon (Piaget, 1971, 1985). The type of relationship coordinated within and between object groups is referred to as a composite unit structure (Behr, Harel, Post, & Lesh, 1994; Lamon, 1993, 1994, 1996; Piaget, 1987). To understand and extend children's logical-mathematical activity, it is therefore necessary to attend to the interactive relationship between students' structures of organizing activity and the type of composite unit structures that this activity coordinates while pondering solutions to problems (Bovet, 1981; Piaget, 1971, 1987). Finally, while language and equations are cultural tools that support reflective activity (e.g., von Glasersfeld, 1990), they are devoid of self-regulatory activity if taught without regard to the quality of children's schemes (Piaget, 1987).

Based on these premises, we are arguing that the current emphasis on language and equations to "transmit" facts negates attention to children's structures of coordinating activity. As a result, deficits are created that include limited regulation of means-ends relationships and the alteration of learning activity as problems are pondered (Grobecker, 1997, 1999a, 1999b). In this study we investigated these premises by observing children's performance on three different tasks. Two of the tasks (associativity of length and modified nonverbal tasks) investigated the composite unit structures children abstract as they act on objects; the third task (flashcard task) examined children's performance on an explicitly taught task.

The associativity of length task examined children's ability to conserve addition (Piaget, Coll, & Marti, 1987). Two strings were equally segmented into four colors such that each color represented a unit element of the whole string (the strings were presented as "fences"). The pieces of string were cut and then placed in various spatial orientations to represent the same cognitive complexity inherent in the operational structures of addition (see Figure 1) Specifically, the elements are the invariant line segments of each color (i.e., unit interval), the parts the various subassemblies of elements or the subsets (i.e., unit intervals that combine to form subgroups), and the whole is the sum of all elements (i.e., set of all subsets). Young children attend to the difference in number of cuts to segment length, but they fail to simultaneously attend to the effect of the cuts on the length of the segments or vice versa. In other words, children's structures of organizing activity abstract each element as a whole onto itself because they are unable to simultaneously attend to each part as nested entities within a more complex whole. This phenomenon is consistent with children's initial counting activity. Specifically, each element that precedes a count is reflected upon as a quantity that is independent of the current quantity attended to. (See Grobecker, 1997, 1999a, 1999b for a more detailed description of children's early number and operational reasoning.)

[Figure 1 ILLUSTRATION OMITTED]

The gradual expansion of children's structures of thinking activity enables them to assimilate (i.e., differentiate and integrate) the logic of the conserved cardinal number such that each element of the set is attended to as hierarchically in the number(s) that preceded it (Kamii, 1985, Piaget, 1965). However, a lack of coordination between the parts to each other and the whole and the simultaneous ability to deduce a more complex whole from the elements contained within the parts persists. In the task with the fences, children at this level perceive that the cuts both modify how each string part is related to the other and mark boundaries (i.e., create unit parts) that delimit length. The whole is not yet considered invariant, however, because children do not logically coordinate an increase in the elements of one fence part with a decrease in the others.

Children's continued ordered reflections lead to the onset of operational logic. Specifically, each unit is included in each succeeding unit such that the groups are compared successively. In the task with the fences, children conserve the whole by adding the parts as well as the elements contained within the parts despite their redistribution. Thus, conserving addition is simultaneously associative because it conserves the sum of all parts (i.e., differences between parts result only from the displacement of the elements that are considered in the process of changes in distribution) (Piaget et al., 1987).

A modified version of a "nonverbal" task was also used to investigate the type of composite unit structures children abstract. The original nonverbal task was one of three problem types used to investigate children's understanding of number and operation (Huttenlocher, Jordan, & Levine, 1994; Jordan, Huttenlocher, & Levine, 1992; Levine, Jordan, & Huttenlocher, 1992). The remaining two problem types consisted of story and number-fact problems in which simple story problems and number-fact problems were verbally presented. In the nonverbal problem, the children replicated the number of objects that were added or taken away from a total amount hidden from view. According to Levine et al. (1992), the presentation and response modes are completely nonverbal. The authors contended that the improved performance on the nonverbal task, as compared to the verbal tasks, was due to the fact that

   the operation of adding and subtracting is provided by the physical act of
   combining or separating sets. These physical referents may make it easier
   for the child to represent the terms of the problem and the operation
   involved in the calculation than on verbal problems. (Levine et al., 1992,
   p. 99)

Performance on the verbal and nonverbal calculations was found to be differentially sensitive to variation in cognitive ability in kindergarten and first-grade children (Jordan et al., 1995). Specifically, children in the low language group performed significantly worse than children in the group with NLD on story problems but not on the nonverbal problems or number-fact problems. However, when an adjustment for the use of fingers in solving the number-fact problems was made, there were significant performance differences between the language impaired and nonimpaired groups on this task. The authors concluded that language difficulties (with intact spatial skills) lead to problems on verbal but not on nonverbal calculation tasks. In contrast, spatial difficulties (with intact language skills) did not appear to lead to specific calculation problems with kindergarten and first-grade children.

We have concerns about what the nonverbal task is proposed to measure and related interpretations. First, with regard to the definition of the nonverbal task (e.g., Levine et al., 1992), one may not conclude that the presentation and response modes are completely nonverbal simply because language was not audible. Further, one may not imply that the children were transforming the sets using the logic of operational structures because they replicated the total number of disks that were presented as two separate groups. To draw such a conclusion, it is necessary to examine the quality of children's reflective activity (i.e., composite unit structure abstracted) as they act on objects, which is the modification we introduce in this task.

Second, it is possible that the children considered to have language problems performed better on the nonverbal task than story problems (Jordan et al., 1995), because counters were provided. Counting objects provides children with the opportunity to reflect on their activity using one-to-one correspondence (Baroody, 1992a, 1992b; Sinclair, Siegrist, & Sinclair, 1983). This contention is supported by the fact that these children performed as well as their non-impaired peers on the number-fact problems when the use of fingers was not adjusted for. Thus, fingers served as a tool to engage children's reflective activity because they enabled them to consider each count as independent of the count(s) that it preceded or followed (i.e., one-to-one correspondence).

In the flashcard task, we presented the same number facts as in the modified nonverbal task. Specific problem-solving procedures used by Geary and his colleagues (see Geary, 1993, for a review of this research) as well as number of correct answers were scored. The strategies scored do not investigate the quality of children's structures of logical mathematical activity because they are the outcome of explicit teaching of number facts. In fact, Geary's (1993) reference to the development of early calculation abilities as "lower-order" mathematics skills that are acquired through transmission is indicative of the lack of attention to the children's structures of coordinating activity and the complexity of early mathematic logic relative to what their structures can logically coordinate.

Based on the premises set forth, we hypothesize that although children may successfully "solve" algorithms (flashcard task) and/or replicate the number of disks on a mat that are presented as separate sets (modified nonverbal task), they may be abstracting composite units that are not representative of operational thought. Removed from the structures that guide and extend reflective thought and its evolution, children are simultaneously removed from their powers of autoregulation and generalization of learning activity. Following such logic, we predicted that although children will supply the correct answers of the flashcard and modified nonverbal tasks, many of these same children will fail abstract composite unit structures suggestive of operational structures on the fence and nonverbal tasks.

METHOD

Subjects

A total of 69 children participated in the study. Forty-two were NLD and the remaining 27 were students with LD. All children (with the exception of one seven-year-old student with LD) were from a lower to middle social-economic status in a suburban town in northern New Jersey. The other classified student was from a middle-class community in northern New Jersey and attended a private school for children with neurological problems. Consent forms were sent to the parents of all students who met the criteria for participation. The great majority of parents consented to have their child participate.

The students with LD met the following criteria: (a) classification by the child study team because IQ was in the average range but academic performance in at least one subject area was below grade-level expectancy; (b) normal vision and hearing; (c) no primary physical or emotional disorders; and (d) no lack of educational opportunities. For all of the students, reading and language arts were replaced either in the resource room or self-contained special class. Three of the students with LD who were in self-contained classes (one each at the seven-, eight-, and nine-year-old age levels) had their mathematics instruction replaced in that class. The remaining students with LD did not have their mathematics instruction replaced, but did receive support for their regular mathematics instruction in the resource room as needed or twice weekly from the basic skills instructor. The school district used a basal-based mathematics program that was explicitly taught.

For students with LD, intelligence was measured by the WISC-R, which had been individually administered by the school psychologist. All children were within the low-to-high average ability range (range = 90-115).(1) The 27 children with LD were distributed across four age levels (7 to 10 years of age). The mean ages for these four age groups (youngest to oldest) were 7.7, 8.5, 9.4, and 10.6 with a combined mean age of 9.1 (see Table 1). There were 5, 8, 8, and 6 students in the 7-, 8-, 9-, and 10-year-old age groups, respectively. The percent of students in the first through fifth grades at each of the four age levels, respectively, were: (a) 60% grade 1, 20% grade 2, 20% ungraded; CO) 38% grade 2, 50% grade 3, 12% ungraded; (c) 12.5% grade 2, 50% grade 3, 25% grade 4, 12.5% ungraded; and (d) 16.5% grade 3, (twice retained), 16.5% grade 4, 67% grade 5.

Table 1 Mean Ages(1) WRAT Reading and Mathematics Scores, and Number of Males/Females for Each Student Group

Children with NLD

Mean Age     WRAT-R      WRAT-M      Male   Female       IQ

 6.5 (.34)     --      108.9 (9.8)    8       6          --
 7.4 (.27)     --      101.8 (7.3)    7       7          --
 8.5 (.23)     --      101.7 (4.1)    5       9          --

Children with LD

Mean Age

 7.7 (.32)   75 (11)   87.0 (9.8)     3       2      102.8 (9.0)
 8.5 (.22)   83 (17)   91.6 (10.4)    6       2      102.4 (5.1)
 9.4 (.31)   86 (10)   94.5 (6.6)     5       3      103.3 (7.5)
10.6 (.23)   91 (10)   86.3 (9.0)     4       2       96.7 (7.1)

(1) Standard deviations in parentheses.

As detailed on Table 1, the mean standard scores (SC) on the Wide Range Achievement Test (WRAT-Second Edition) in reading are below one standard deviation of the SC mean of 100 for the seven-year-old age (SC = 75). The remaining students at the 8-, 9-, and 10-year-old age levels achieved approximately one SD below the SC mean (youngest to oldest respectively): 83, 86, and 91. The SC in mathematics tended to be higher, although they were consistently below the mean (youngest to oldest respectively): 87.0, 91.6, 94.5, and 86.3.

Forty-two children with NLD were evenly distributed across three age groups (six to eight years of age) with 14 children at each of the three age levels. The mean ages for the three age groups were: 6.5, 7.4, and 8.5 with a combined age mean for all three age levels of 7.5 (see Table 1). This mean age was significantly less than that of children with LD (t = 46, p [is less than] .0001) because delays in this type of composite unit structures abstracted have previously been noted in LD children (Grobecker, 1996, 1999b). If the ages between the groups were the same, we would have tapped ceiling levels of performance in children with NLD and basal performance levels in children with LD. The percent of students in kindergarten through third grade at each of the age levels, respectively, for the NLD children were: (a) 50% kindergarten, 50% grade 1; Co) 71% grade 1, 29% grade 2; and (c) 50% grade 2, 50% grade 3. The students with NLD were not receiving special education services and school personnel indicated that NLD students were not experiencing academic difficulties or emotional problems.

Because the children did not take the cognitive test of abilities until the end of grade 2, no cognitive scale indexes are reported. However, the children's classroom teachers judged them to have average mathematical abilities for their grade. The mean SC performance on the WRAT indicated that most children achieved close to the expected mean in mathematics (youngest to oldest, respectively): 108.9, 101.8, and 101.7.

Materials and Apparatus

Associativity of length task. Children's construction of operational structures was examined in a task assessing associativity of length as previously described (Piaget et al., 1987). The length of the four color segments was 5.5, 12.5, 9.0, and 7.5 centimeters, respectively. Each piece of the string segments was compared until the child agreed that the length was the same. As each segment piece was compared, the other segments were pushed aside so that the entire length of the string would not be seen together as one unit. The strings were then placed on a board with pins to make identical fences (23.5 cm in length) for the cat to walk on (see Figure 1). The experimenter then walked the cat on both fences asking, "Did the cat walk the same length on both fences or was the walk on one fence longer than on the other fence?" The children justified their answers.

For problem 2, one cut was made on each fence at different segments while the child watched. The cat then walked each fence and the question was asked again. In problem 3, the pieces of the fence used in problem 2 were placed in different spatial orientations. Again, the cat walked the fence and the same question was asked. If the child included the space between the cuts as adding to the length of the fence in his or her explanation, the child was told to consider only the pieces of the fence. For problems 4-6, two new pieces of string were compared as described in problem 1 and three more fences were constructed. In the fourth problem, the two fences were presented without cuts and in the same spatial orientation. In the fifth problem, two cuts were made in the experimenter's fence and one cut made in the child's fence at a different segment. Finally, the fence segments were placed in different spatial orientations for the sixth problem.

Because this task involves understanding length equality, a preliminary task was administered to assess children's construction of this notion. Specifically, they were first shown two standard, unshaped pencils (7.5 inches) that were placed next to each other and asked, "Are the pencils the same length or is one pencil longer than the other" and "why?" One pencil was displaced relative to the other and the children were then asked the same question and to justify their choice. The pencil was then returned to its initial position and the child was again questioned about the length equality or was asked, "Can anything be done to make them the same length again?" Each child was asked to provide a justification for the response given.

Flashcard task. Fourteen addition number problems were written on 3 x 5 index cards in vertical form. Each numeral was approximately 1" long and written with a black marker. There were seven number problems whose sums were less than 10 and seven number problems whose sums ranged from 11-20. The cards were randomly selected from a shuffled presentation with the provision that if there was repetition of either the augend or the addend across consecutive problems, the cards were put at the end of the pile and moved until there was no repetition in either augend or addend across consecutive problems.

Each child was told, "You will be shown some number problems. Some of the problems may be hard for you. Just do the best you can to give me an answer." The card was then held in front of the child until the child supplied an answer or stated he or she did not know. When an answer was supplied, the child was asked, "How did you know that?" The experimenter noted anything the child did to assist in solving the number fact.

Nonverbal calculation task. The materials for the nonverbal calculation task (Levine et al., 1992) consisted of two 12" x 12" white cardboard mats with black horizontal lines to divide the mats in half, a set of 40 black buttons (3/4 of an inch in diameter), a box for the buttons, and a cover for the buttons. One side of the cover had an opening so the buttons could easily slide under it. The experimenter and child sat at a small table facing each other, with a mat in front of each.

The experimenter placed two buttons on her mat in full view of the child and pushed them under the opened side of the cover. Three buttons were then placed on the horizontal line on her mat and slid under the cover one by one. Five buttons were then placed on the horizontal line on the child's mat. The experimenter then lifted the cover on her mat to show the two previously hidden buttons while saying, "See, yours is just like mine." This demonstration item was presented again following the same procedure, except this time the child was asked to "make yours just like mine" after the experimenter had made the transformation. In contrast to the study design of Levine et al. (1992) in which no verbal response was required, the child was asked to justify what was laid out if he or she did not count verbally. If the child placed the wrong number of buttons on the mat, the response was corrected and the item was repeated. Each succeeding problem was given after the child made clear how he/she was counting the buttons.

The nonverbal problems were presented immediately following the demonstration problem. For each additional problem, the experimenter placed the set of disks comprising the augend on the mat and then pushed them under the cover. The experimenter then put the set of disks comprising the addend in a horizontal line next to the cover and slid them under the cover one by one. As in the demonstration procedure, the children indicated how many disks were hiding under the cover by placing the appropriate number of disks on their mat. The children were asked to justify their solutions.

The same number facts that were used on the flashcard task were used for the nonverbal task but with changes in the order of presentation. The order of problem difficulty was determined in the same manner as for the flashcard task. In addition, the order of the nonverbal and flashcard tasks was switched for every other child to control for any effects of memory. None of the children commented that they remembered a number problem from the previous task, although some remembered problems within tasks. The children sometimes used this information to help solve other problems in the flashcard task.

Scoring

Associativity of length task. The associativity of length task was used to judge levels of operative thinking (Piaget et al., 1987). Three levels of behavior were scored in this task. A child was judged to be at level IA if there was an absence of any quantitive length concept in the sense of additivity. For example, children at this level commented on the difference in the number of cuts between the fences but did not attend to how the cuts affected the equality of the elements within the parts or the equality of the parts to each other. Often these children commented on the shape of the fence, stating that one is longer because of its shape (e.g., "My fence is longer cause it's shaped like a triangle and is flat out").

Children at level 1B perceived the cuts as affecting length, but were unable to apprehend the notion of compensation between parts where an increase in one part of the fence leads to a decrease in another part (e.g., "My fence is longer `cause the white one is shorter than the green one," "Mine is longer `cause you cut different colors on mine and each color is a different length than the other") in at least one of four problems where the strings were altered in some way.

Children at level IIA got all six problems correct. Thus, they were consistently able to simultaneously coordinate the parts with changes in the distribution of the elements such that the two were apprehended as necessarily equal in length (e.g., "If you keep cutting until a million pieces, it would still be the same," "Cause well, if they were together and added up in a straight line they would be the same").

The pencil task, which was introduced prior to the associativity task, was judged only in terms of correctness of response as its purpose was to help delineate children's understanding of the terms "same length," which was the language used in the associativity of length task.

Thus, nine different performance criteria were generated in relation to the associativity of length task. These criteria consisted of a stage variable and dichotomous correct/wrong variables for each of the specific problems tested. A reliability check was performed on 20 protocols by a student familiar with Piagetian theory. The obtained interrater reliability was r = .88.

Flashcard task. Six strategies were coded for the type of answers provided based on scoring schemata identified in past research (Baroody, 1987; Carpenter & Moser, 1984; Geary, Brown, & Samaranayake, 1991; Siegler, 1986, 1987; Svenson & Sjoberg, 1983). The strategies were (a) don't know (DK) -- no attempt to solve the problem; Co) counting fingers (CF) -- fingers used to represent problem integers and then counting fingers to derive the total; (c) fingers (F) -- using fingers to represent integers but not recounting the fingers prior to stating the sum; (d) verbal counting (C) -- observation of audible counting or lip movement without the use of the fingers; (e) decomposition (D) -- decomposition of a problem into an easier problem or using a previous problem solution to help derive the total; and (f) retrieval (R) -- no counting observed. If children were not observed counting but stated that they counted (subvocalization), they were coded as having used the verbal counting strategy.

According to Siegler (1986, 1987), memory retrieval is the preferred strategy because it places fewer demands on the resources of the working memory and requires less time to execute. The other four strategies serve as backup strategies. However, because decomposition involves the ability to relate the elements into various subsets around an expanded whole, decomposition and retrieval strategies are considered the more advanced strategies for the purposes of this study. Finally, we also scored whether the children stated the correct answer.

Nonverbal calculation task. Levine et al. (1992) coded children's strategies with respect to four of the flashcard strategies listed above (counting fingers, fingers, counting, and unobserved) and added the strategy of imitating the experimenter. These authors suggested that although children infrequently used overt counting with or without the use of fingers on this task, it is plausible that the children were frequently counting silently (Jordan et al., 1992; Levine et al., 1992). In the present sample, it was evident that children were consistently counting the buttons (subvocally or vocally). Further, the manner in which the buttons were counted differed between children. Because the counting strategies appeared indicative of the quality of children's logico-mathematical activity, five different counting strategies were coded to assess task performance: (a) no attempt and/or consistent approach to the problem solution (NA); Co) interval count using fingers (IF) -- counting two sets as one whole with the assistance of fingers; (c) interval counting (I) -- counting two sets as one whole without the assistance of fingers; (d) sequenced group count (G) -- counting two separate groups from one using the terms first/then when combining groups; and (e) summing group count (A) -- counting two separate groups from one while using additive terms that included "in all," "I added them," or "plus." Two children used their fingers to assist them with their count in a small number of problems. Because they directed their counting activity on two separate groups and used additive terms, this strategy was coded as level 4. Strategies G and A were considered more complex because they consist of assimilating the two sets as a homogeneous whole. As in the flashcard task, performance criteria were generated for correctness of an answer and for type of strategy generated.

We recognize that Levine et al. (1992) had designed this task in an effort to assess calculation abilities that were independent of verbal labels. However, in the current research the labels were generated by the children, thus demonstrating a predilection to use these terms to justify their answers.

Procedures

Children were individually tested by the first author for approximately 40 minutes. The associativity of length task was administered first followed by either the flashcard task or the nonverbal task with the order of flashcard and nonverbal tasks changed for each child. Finally, the WRAT (second edition) test of mathematics was administered. The testing was administered during the latter part of the school year (late April into early May).

RESULTS

In evaluating the relationships among group membership, task, strategy, problem difficulty, and the likelihood of a correct response, the data were measured on nominal and ordinal scales. Therefore, the majority of the analyses were done with nonparametric statistics (Green, Salkind, & Akey, 1997). On a few occasions where we employed a parametric analysis, we have noted the change along with the results. Our analytic strategy included using a global test with several follow-up tests. We used a global test to preserve our predetermined Type 1 error rate at 0.05. The global test was designed to evaluate the effect of group membership on response across all tasks and was significant. Therefore, we proceeded to examine the effect of group membership on response and strategy (high and low levels reported below) controlling for task. These tests were also significantly significant.(2) Subsequently, we examined all pairwise comparisons of group membership on response and strategy within task. We also evaluated the relationship between problem difficulty and response and strategy for the flashcard and modified nonverbal tasks. For the associativity of length task, we appraised the relation between problem difficulty and group membership and response. Results are reported below.

Pencil Task

The percent (and number) correct on the pencil task are found in Table 2. The children with and without LD achieved equally well when the pencil was in the same position ([chi square] (1, N = 69) = 10, p = .75) and when one pencil was moved up from the other ([chi square] (1, N = 69) = .42, p = 0.51).

Table 2 Percent (and Number) of Students Achieving Success on the Pencil Task (Same and Different Positions) and the Six Problems on the Associativity or Length Task(a)

                                     Problem

Age   Pencil-S   Pencil-D      1        2        3

Children with NLD

  6    93 (13)    29 (4)    79 (11)   43 (6)   29 (4)
  7   100 (14)     0 (0)    86 (12)   36 (5)   36 (5)
  8   100 (14)   71 (10)    93 (13)   64 (9)   57 (8)

Children with LD

  7   100 (5)      0 (0)     60 (3)    0 (0)    0 (0)
  8   100 (8)     13 (1)     88 (7)   25 (2)    0 (0)
  9    88 (7)     25 (2)     75 (6)   13 (1)   25 (2)
 10   100 (6)     67 (4)    100 (6)   33 (2)   33 (2)

                 Problem

Age       4         5        6

Children with NLD

  6    79 (11)   21 (3)   14 (2)
  7    86 (12)   50 (7)   14 (2)
  8   100 (14)   64 (9)   57 (8)

Children with LD

  7    60 (3)     0 (0)    0 (0)
  8   100 (8)    13 (1)   13 (1)
  9    88 (7)    38 (3)   25 (2)
 10   100 (6)    17 (1)   17 (1)

(a) See Figure 1 for the six problems on the associativity of length task.

Associativity of Length Task

Response. The percent (and number) correct on the six problems of the associativity of length task are also presented on Table 2. The problems with no cuts, same spatial orientation, were the easiest for both groups. The level of difficulty of the altered fence configurations in ascending order for all levels combined was: (a) 1C, SO (NLD = 48%, LD = 19%); (b) 2C, SO (NLD = 45%, LD = 19%); (c) 1C, DO (NLD = 40%, LD = 15%); and (d) 2C, DO (NLD = 29%, LD = 15%). Significant group differences on the associativity of length tasks across the six problems were found ([chi square] = (1, N = 414 = 10.25, p = 0.001), with the NLD children achieving greater success (see Table 3 for a summary of these statistics and those to follow). A significant interaction between group membership and problem difficulty ([chi square] = (5, N = 414 = 4.18), p = 0.524) was not found, although problem difficulty explained a significant amount of variation in the children's ability to correctly respond to the question over and above the child's group membership ([chi square] (5, N = 414 = 117), p = 0.001). Thus, while group membership determines if a child is likely to get a problem correct, children in both groups found the same problems to be most difficult.

Table 3 Analysis of Task Type by Response, Strategy, and Level Employed

Task            Type       Test                  chi-square     P

Associativity   Response   Group Response          10.56      0.001
of Length                  Group Difficulty         4.18      0.524
                           Difficulty Response     117.0      0.001
                Level      Group Level             17.67      0.001

Modified        Response   Group Response          0.007      0.934
Nonverbal                  Difficulty Response      35.7      0.001
                Strategy   Group Strategy           14.0      0.001
                           Response Strategy       18.36      0.001
                           Difficulty Strategy       7.8      0.005

Flashcard       Response   Group Response           14.2      0.001
                           Difficulty Response      43.6      0.001
                Strategy   Group Strategy           2.49      0.115
                           Difficulty Strategy      89.0      0.001
                           Response Strategy       153.0      0.001

We categorized the six fence problems into two levels: low (problems 1 & 4) and high (problems 2, 3, 5, and 6). The high-level problems were significantly more difficult than the low-level problems (t = 5.4, p [is less than] 0.001). For the less difficult problems, there were no group differences in response ([chi square] (1, N = 138) = 0.081, p = 0.78). However, there were statistically significant group differences on the combined four problems requiring conservation ([chi square] (1 N = 276) = 17.30, p = 0.001). On the more difficult problems, the odds of a correct response for a child with NLD were twice that of their LD peers.

Level achieved. The percent (and number) of students achieving at each level for both groups is shown in Table 4. For both groups, the number of students achieving at level IIA increased as the age of the student increased, especially for students with NLD. Analysis of differences in behavior levels indicated that these group differences were statistically significant ([chi square] (1, N = 414) = 17.67, p = 0.001). The odds of being in level IIA were 3.19 times higher for a student with NLD.

Table 4 Percent (and Number) of Students Achieving at Each Level for the Associativity of Length Task

                            Level Achieved(a)

Children with NLD

  Age               Level IA   Level IB   Level IIA

   6                 22 (3)     64 (9)     14 (2)
   7                 22 (3)     64 (9)     14 (2)
   8                     --     43 (6)     57 (8)

Children with LD
   7                 80 (4)     20 (1)         --
   8                 25 (2)     62 (5)     13 (1)
   9                 25 (2)     62 (5)     13 (1)
  10                     --     84 (5)     16 (1)

(a) See text for a description of levels.

Modified Nonverbal Task

Response. Percent correct responses are shown in Table 5. We wanted to investigate the relationship between group membership and likelihood of a correct response. No relationship was found between the probability of a correct response and group membership [chi square] (1, N = 966) = .007, p = 0.934). We then proceeded to investigate the relationship between the likelihood of a correct response and problem difficulty. When the number problems are dichotomized on the basis of difficulty (sums less than 10, more than 10), problem difficulty does affect the likelihood of a correct response ([chi square] (1, N = 966) = 35.7, p = 0.001). As problem complexity increased, the likelihood of a correct response decreased. In fact, the children in this sample were 2.5 times more likely to get a correct response if the problem sum was less than 10.

Table 5 Percent of Students Using the Five Strategies and Achieving Success on the Nonverbal Task for Each of the Number Problems

                Percent of Students Using Strategies
                      for Each Number Problem

Strategy(a)

Children with NLD

              3+1   1+4   2+4   2+5   6+1   5+3   6+3

NA              2     2     2     2     2     2     2
IF             --    --     2     2     2    --    --
IC             36    31    33    38    36    36    33
G              17    17    22    15    19    26    24
A              45    50    41    43    41    36    41

Success

               98    88    95    81    88    81    86

Children with LD

NA             --    --    --    --    --    --    --
IF             --    --     4    --     4     4     4
IC             44    56    41    52    48    52    48
G              26    26    30    30    26    22    30
A              30    18    25    18    22    22    18

Success

              100    96    85    85    93    85    81

                 Percent of Students Using Strategies
                       for Each Number Problem

Strategy(a)

Children with NLD

              2+9   4+8   5+7   5+8   9+5   9+6   9+7

NA              2     2     2     2     2     2     2
IF              2    --    --    --     2    --     5
IC             45    41    45    41    43    50    41
G              22    26    22    26    24    17    19
A              29    31    31    31    29    31    33

Success

               71    74    83    83    71    76    79

Children with LD

NA             --    --    --    --    --    --    --
IF              4     4     7     4     7     4    --
IC             48    52    52    56    60    60    60
G              26    26    30    22    22    18    22
A              22    18    11    18    11    18    18

Success

               81    63    78    78    74    63    89

(a) See text for a description of strategies.

Strategy type. Table 5 also shows the percent of times students used each of the five different strategy types for each of the 14 number problems. For the purpose of statistical analysis, strategies 1 through 3 (NA, IF & F) formed the lower-level strategy of counting intervals (CI) and strategies 4 and 5 (G & A) made up the higher-level strategy of counting groups (CG). A significant relationship between combined strategy type and group membership was found ([chi square] = (1, N = 966) = 14, p = 0.001). NLD children used the higher-level combined strategy of counting groups approximately 57% of the time whereas the LD children used it 45% of the time. Put another way, the odds of using the combined higher-level strategy (CG) were approximately 1.5 times greater for a child with NLD.

When controlling for differences in group membership, we found the relationship between response and strategy was statistically significant ([chi square] (1, N = 966) = 18.36, p [is less than] 0.001). The odds of a correct response were 2.1 times higher when the combined higher-level strategy was employed. There was a relationship between strategy and problem difficulty ([chi square] (1, N = 966) = 7.8, p = 0.005). The NLD group used the higher-level combined strategy 62% of the time on problem sums less than 10 and 53% of the time on problem sums more than 10. For children with LD, the higher-level strategy was used 49% of the time on the less difficult number problems (less than 10) and 41% of the time on the more difficult number problems.

Flashcard Task

Response. Percent correct responses on each of the 14 number problems on the flashcard task are presented in Table 6. As shown, the percent correct is higher for the LD children, particularly on the number problems over 10. Analysis of differences in number correct on the combined responses indicated that differences in group performance were statistically significant ([chi square] = (1, N = 966) = 14.2, p = 0.001). The odds of a correct response were double (2.2 times higher) for a member of the LD versus NLD group.

Table 6 Percent of Students Using the Six Strategies and Achieving Success on the Flashcard Task for Each of the Number Problems

              Percent of Students Using Strategies and
              Achieving Success for Each Number Problem

Strategy(a)

Children with NLD

              3+1   1+4   2+4   2+5   6+1   5+3   6+3

DK              5     5     7    10     2     5     7
CF              2     2     5     5     2     7    10
F               5    --    10    10    --     5     5
C               7     7    21    23     5    17    29
D              --    --     5    --    --     5     2
R              81    85    52    52    91    61    47

Success        95    95    91    86    95    88    88

Children with LD

DK             --    --    --    --    --    --    --
CF              7     4     4     4    --     4     7
F              --     4     7    15     0    18    22
C              11     7    30    15    15    22    58
D              --    --    --    --    --     4    --
R              82    85    52    67    85    52    33

Success       100   100   100   100   100    89    93

              Percent of Students Using Strategies and
              Achieving Success for Each Number Problem

Strategy(a)

Children with NLD

              2+9   4+8   5+7   5+8   9+5   9+6   9+7

DK             10    12    10    10    12    10    12
CF              2    12    14    16    12    14    14
F               7    19    26    29    24    19    26
C              26    31    24    21    26    29    19
D               7     9    17    12    14    11    19
R              48    17     9    12    12    17    10

Success        90    74    74    74    71    74    67

Children with LD

DK             --    --     4     7     7     4     4
CF             11     7    11    15    11    15    15
F               4    26    37    30    33    22    18
C              15    18    22    15    11    11    15
D               7     7    11    18    23    26    30
R              63    42    15    15    15    22    18

Success       100    85    81    85    85    81    81

(a) See text for a description of strategies.

Controlling for group membership, a statistically significant relationship existed between sum values on the flashcard task and the ability to obtain the correct answer ([chi square] (1, N = 966) = 43.6, p = 0.001). The odds of a correct response were 3.96 times higher when the problem sum was less than 10.

Strategy type. Table 6 also shows the percent of times students used each of the six strategy types for each of the 14 number problems. Strategy type was dichotomized such that the first four strategies (DK, CF, F, & C) made up the lower-level strategy of counting (C) and the remaining two strategies formed the retrieval (R) strategy. There was no relationship between group membership and strategy used ([chi square] (1, N = 966) = 2.49, p = 0.115).

Second, a relationship existed between flashcard sum and the combined level strategy employed ([chi square] (1, N = 966) = 89, p = 0.001). The odds of selecting a retrieval strategy were 4.2 times higher when the flashcard sum was 10 or less.

Finally, there appeared to be a relationship between combined level strategy and probability of correct response ([chi square] (1, N = 966) = 153, p = 0.001). The odds of a correct response were 10.4 times higher if a retrieval strategy was used. Even when controlling for differences in level of problem difficulty, the odds of a correct response were more than seven times higher when the child used a retrieval strategy for the flashcard task.

WRAT

The mean scores on the WRAT mathematics test for all children at each age level are found in Table 1 and have been previously reported for each group. The combined WRAT score in mathematics for the children with NLD was 104.1 (8.02) and 90.44 (9.11) for the children with LD. These means are statistically significant (t = 6.56, p [is less than] 0.001).

DISCUSSION

Because the age mean of the children with LD ([bar]x = 9.1) was higher than that of their NLD peers ([bar]x = 7.5), we were not surprised to find that the children with LD outperformed their peers with NLD in number correct on the flashcard task. Specifically, the children with LD were drilled in basic facts over a longer period of time. The finding that both groups achieved greater success with number problems less than 10, which they had practiced the most, further supports the contention that drill and repetition lead to a greater rate of success in stating the correct answer (Siegler, 1986, 1988; Siegler & Jenkins, 1989).

In contrast to number of problems correct, analysis of the type of strategy used showed that the two groups did not differ in the use of higher- (R) or lower-level combined strategies (C). Both groups used the higher-level combined strategy (R) significantly more when the sums were less than 10 and performed significantly better when this higher-level combined strategy was used. Thus, our results are consistent with past research showing that children appear to use the retrieval strategy when they feel confident and use the lower-level combined strategy of counting when they are unsure of their answers (Geary et al., 1991; Siegler, 1988). The decomposition strategy, which was part of the combined higher-level strategy, was used by a small percent of children in both groups mainly when the number problems exceeded a total of 10.

What appeared to contribute to the poorer success rate using the lower-level combined strategy (C) compared to the higher-level combined strategy (R) is children's limited understanding of number. Specifically, the six-year-old children with NLD who provided incorrect answers to number problems more than 10 ran out of fingers to support a one-to-one correspondence. As a result, the number of counts that exceeded the number of fingers they had became meaningless. Children at the seven-year-age level were better able to use fingers to assist with the counting because they did not need an object to represent each separate count. In other words, they began to dissociate amount from the objects it represented such that the final count represented all the objects that preceded the count (i.e., cardinal value). For children with LD, half of those in the seven-year-old group were also unable to provide the total when the amount exceeded 10. The eight-year-olds were much more efficient in their counting.

The above performance patterns are consistent with the notion that young children do not understand the relationships inherent in cardinal value when using one-to-one correspondence even if they are able to supply an appropriate label (Baroody, 1992a, 1992b; Sinclair et al., 1983). Thus, knowledge of referents (i.e., use of fingers) is not the stumbling block as suggested by previous authors (e.g., Levine et al., 1992). Nor would we agree with the position that children with mathematics difficulty demonstrate a delay in understanding essential and unessential features of counting (i.e., principled knowledge) (Geary, Bow-Thomas, & Yao, 1992). Instead, we argue that the delay observed in counting skills implies less coordinated forms and structures of logical-mathematical activity that regulate what is attended to as objects are reflected upon.

Our argument is supported by the finding that the type of combined strategy used on the modified nonverbal task was significantly different between groups; that is, the students with NLD used the combined higher-level strategy (CG) approximately 57% of the time whereas the NLD students used the same strategy approximately 45% of the time. Thus, the majority of students with NLD perceived each of the elements (buttons) as nested within a subgroup (parts) that was, at the same time, nested within a new, expanded whole. In contrast, the majority of the students with LD abstracted each subset (part) or individual element (button) as an independent whole, rather than as inclusive of an expanded whole (elements nested in parts, which are nested within the whole). Related to this contention is the observation that the NLD children who achieved success on problem 2 of the associativity of length task (1C, SO) also used the higher-level combined strategy (CG) on the modified nonverbal task 68% of the time.

Thus, while children can be taught to "compute" addition problems through repetition and drill, thinking is disengaged from its powers of self-regulation if the specialized organs that regulate this activity (i.e., logical-mathematical structures) are not evolved to the degree necessary to abstract these operational groupings. Unfortunately, Gersten and Chard (1999) noted that mathematics instruction in special education continues to focus on computation rather than on activities that develop number sense (i.e., fluidity and flexibility with numbers and meaning of numbers). They argued that computation and number sense should be taught simultaneously. We would add to this recommendation that number sense needs to evolve from children's structures of logical-mathematical activity, which guide the alteration and generalization of thinking activity as authentic problems are pondered. When children's thinking activity has evolved to the degree necessary to abstract operational groupings, then the equations that represent and help to exercise this logical activity can be introduced. If equations are introduced that children's thought structures cannot support, they will require external focus to support their attention (Sinclair & Sinclair, 1986), which creates "context- and problem-specific routines and skills rather than insight, self-confidence, flexible strategies, and autonomy" (Bauersfeld, 1988, pp. 37-38).

It is important to note that while the type of combined level strategy used between the two groups on the modified nonverbal task was statistically significant, the number correct did not differ between the groups. Thus, not all children who achieved the correct end-state answer reflected upon the problem using operational logic. This finding is further evidence of our contention that one may not imply that children were using operational structures of addition because they replicated the total number of disks that were presented as two different groups.

Of additional interest in the modified nonverbal task is the finding that the likelihood of a correct response decreased as the sum increased for both groups. This finding makes sense in light of the fact that as quantity increases, it becomes necessary for the child to perceive more elements as inclusive of each preceding element, which, in turn, is inclusive of each succeeding unit that the elements are embedded in (Kamii, 1985). When children begin to experiment with counting, it is therefore important to keep the quantity small. The number of objects should increase as children's readiness to coordinate increasingly greater object quantities is observed. Thus, observations of the quality of composite unit structures abstracted when operating on number quantities serve to define appropriate questions to help extend understandings (Bovet, 1981; Sinclair & Sinclair, 1986). These questions should serve to stabilize a newly abstracted order or to create conflict, which fuels the extension and gradual reorganization of schemes onto a higher-order level. It is necessary to embed these questions in meaningful activities with relevant materials that simulate reflection because such activities afford the child the opportunity to exercise and extend his or her logical orders.

Evidence of more coordinated structures of logical-mathematical thought in children with NLD was again noted in the associativity of length task. Specifically, a significantly greater number of children with NLD conserved the whole across the six number problems to achieve at level IIA (29%). Although fewer children with LD achieved success on all six problems (11%), students in both groups generally found the problems with the cut(s), same spatial orientation, easier than the problems with the cut(s), different spatial orientation. The most difficult problem was two cuts, different spatial orientation (2C, DO). Thus, with an increase in the number of cuts, the children were increasingly deprived of the perceptual facilitation afforded by the continuity of the line segments. As a result, task difficulty increased when both the elements and the parts in total figure changed orientation (Piaget et al., 1987). The small degree of variability from this order of difficulty within each group can be explained by a degree of disturbance in the children's thinking as they progressed from problem to problem. Specifically, if children are transitional in their level of thinking, conflict can result in the reorganization of schemes which, in turn, leads to the abstraction of more complex composite unit structures.

In summary, we argue that strategic behaviors that are explicitly taught to children to compute number problems negate attention to children's structures of organizing activity, which guide and constrain the logical orders that can be coordinated within and between object groups. Thus, while children may "solve" addition problems, they could be acting upon the problem using thought forms that are not yet operational. Because thinking is disengaged from its powers of self-regulation and generalization, children become dependent on explicit procedures to "solve" problems that have very limited meaning and that cannot be generalized. To prevent these learning dynamics, teachers need to better understand the developmental evolution of logical-mathematical thought structures and to pose questions that stimulate thinking as children engage in problem number solving. We hypothesize that such teaching methodology will serve to evolve "number sense." While this study supports previous research demonstrating less expanded logico-mathematical structures (Grobecker, 1997, 1999b) in children with LD, the sample in this group was small (see Grobecker, 1998, for an explanation of how these structures evolve as well as the nature of the differences of these structures in children with LD compared to same-aged peers). More empirical investigation is necessary to better understand the quality of thought structures in children with LD as well as teaching methodologies that investigate how to best facilitate the expansion of logical-mathematical activity.

NOTES

(1) Standardized tests of IQ and intelligent activity evaluated on the associativity of length task are derived from different constructs of what constitutes intelligent activity. Specifically, Piaget argued that cognition consists of open systems of hierarchically organized energy structures whose nature cannot be evaluated on standardized measures of IQ (Piaget & Garcia, 1989). Standardized IQ scores are derived from mathematical abstractions and inappropriately reified as a measure of intelligence (Gould, 1981).

(2) The specific statistics are available by writing to the first or second author.

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BETSEY GROBECKER, Western Michigan University, Michigan.

FRANK LAWRENCE, University of Alabama, Birmingham.