CONSERVATION OF NUMBER AMONG FOUR AND FIVE YEAR OLD CHILDREN: SOME METHODOLOGICAL CONSIDERATIONS. Barbara B. Rothenberg

Transcription

1 î L E T I CONSERVATION OF NUMBER AMONG FOUR AND FIVE YEAR OLD CHILDREN: SOME METHODOLOGICAL CONSIDERATIONS Barbara B. Rothenberg This paper has been accepted for publication by Child Development. Any citations should be made to the Journal rather than to the Bulletin. Educational Testing Service Princeton, New Jersey October 1968

2 CONSERVATION OF NUMBER AMONG FOUR AND FIVE YEAR OLD CHILDREN: SOME METHODOLOGICAL.CONSIDERATIONS Abstract Barbara B. Rothenberg Educational Testing Service This study investigates certain methodological issues in current conservation of number assessment procedures. Concern was most particularly directed to the effects on conservation status of the focus and number of questions asked, the presentation of various transformations, and the use of justifications of prior judgments. The subjects were 210 lower and middle class preschool and kindergarten aged children, The results showed important differences in the number of conservers identified depending on the conservation question or questions asked and on the number of transformations presented. Lower class children were seen to be more likely inaccurately assessed in terms of conservation status than middle class children«the results also suggested that further study be devoted to the eliciting of and value placed on Ss' justifications. Overall, only 6$ of the S_s were actually found to be conservers. The impact of the methodological findings is discussed and evaluated.

3 CONSERVATION OF NUMBER AMONG FOUR AND FIVE YEAR OLD CHILDREN: SOME METHODOLOGICAL CONSIDERATIONS Barbara B. Rothenberg Educational Testing Service The concept of conservation, as is now well known, was formulated by Piaget and has been defined as the realization of the principle that a particular dimension of an object may remain invariant under changes in other, irrelevant aspects of the situation. The lack of realization of this principle is considered to represent a manifestation of the immature level of functioning of the child's mental processes and of their failure to conform to the operational structures of logical thought (Flavell, I963). Shantz and Sigei (1967) have suggested that alternately, the concept of conservation may be viewed as the child's increasing ability to differentiate reality and appearance, or the ability to differentiate relevant from irrelevant attributes. Acquisition of this ability would seem to foster in the growing child an increasing trust in his own reasoning as opposed to his perceptions. The understanding of number has been described as lending itself particularly well to the investigation of the development of conservation (Wohlwill & Lowe, 1962). Piaget's concern in this area has been with probing and diagnosing number relevant capabilities considerably more subtle and basic than those believed to be involved in familiar elementary operations (Flavell, 1963). Specifically, this has involved the measurement of children's ability to grasp the equivalence or nonequivalence of the elemente in a set irrespective of their arrangement. Piaget has dealt with the essential properties of the number system that show an understanding of number rather than Just the ability to use number. Previous research (Dodwell, 196.I; Wheatley, 1968) has shown that tests of number conservation may be a meaningful measure of arithmetic

4 -2- readiness. Several studies have, moreover, shown that number conservation ability and intelligence are positively related (Dodwell, I960; Elkind, I96I; Goldschmid, I967; Hood, 1962), Keasey and Charles (1967) have suggested that both chronological age and mental age be considered in evaluating children's understanding of conservation concepts. Recent studies have measured conservation attainment through standard question formats because they provide a more comparable situation for all subjects than does Piaget's more flexible, clinical method of questioning, Within the limits of standard formats, there still remain some problems which center on the language level of the vocabulary used in the questions and on the structure of the questions themselves, Some investigators (Shantz & Sigel, 1967; Wallace, 1966) have indicated that with the most common standard questions, a major confounding exists in the interpretation of the ^s' responses. This confounding is found among the Ss who fail to conserve in that it has not been possible to know whether this failure was due to inability to understand the language of the question, the concept of conservation, or both. Recent studies(griffiths, Shantz & Sigel, 1967; Shantz & Sigel, 1967) have dealt with the first aspect of the measurement problem, the language level. These studies have pointed to the importance of assessing the S's comprehension of the key words, i.e., same, more, etc., used in the conservation questions or training the S_ to criterion in his understanding of these words prior to testing his level of conservation attainment, Information gained from this prior assessment or training would then enable one to more clearly determine the reasons for any S's failure to conserve, The second aspect of the language problem, i.e.,, the structure of the questions, has been considered to a lesser extent. The problem is seen in the

5 -3- format of the commonly posed conservation question, "Does this row (or side) have more, or does this row have more, or do they both have the same number (amount, etc.)?" (Wallach & Sprott, 1964; Zimiles, 1966). Questions such as these, which contain a number of separate parts, present particular difficulties for young &3 V Children can be expected to encounter major problems in remembering the entire question when it is long and complex. Moreover, it seems unlikely that the te can reliably answer the three section question with a single response. Hood (I962) has, for example, noticed that with this threepart question, children tended to repeat the last thing they were asked. This suggests that the extent to which children appear to be conserving may depend on the order of the parts of the question. If the two arrays had the same number, as is most common, and the three parts of the question were ordered more, more, same, children might tend to have more conserving responses than if the three parts were ordered more, same, more.. Two-part questions, such as "Do these two rows have the same number or does one have more?" (Fleischmann, Gilmore & Ginsburg, 1966; Wheatley, I968), generally tend to present similar problems. The one further common type of question (Dodwell, I96O; Elkind, I961; Wallach, Wall & Anderson, I96Y; Wohlwill & Lowe, I962) deals only with a single event, e.g.-, "Are there the same number of eggs and egg cups?" "Which row has more pieces in it now?" While this type of question has the advantage of presenting a single phrase short enough to be remembered, it does have the disadvantage of tending to favor either a conserving response or a nonconserving response by virtue of the emphasis in the question on same or more, The use of this single phrase question can seriously affect the results of the study, particularly when the justification of the response is not considered

6 -knecessary in determining the Ss' level of conservation. This single question method can result in either many more or less conservers than might be obtained through more complete questioning of the child, Dodwell (i960), for example, when using two presentations with equal numbers in the arrays that differed only in the nature of the materials, found a much greater percentage conserving when the Ss were asked if there were the same number in the two arrays than when they were asked which array has more. Some investigators (Griffiths et al,, 19-67; Hood, 1962; Mermelstein & Shulman, I967) have suggested that further changes in the conservation question format are necessary so that one can be fairly sure of the real character of the child's thinking. Hood (I962) has suggested that this can best be accomplished by presenting standard single event questions in several forms for each problem. Gruen (1966) has noted the varying criteria used in different studies to define conservation- Primarily, the difference centers on whether or not the S_s are asked to justify or explain their response to the "conservation" question.. This criterion difference has tended to result in more conservers being selected when justification of the answer was not requested. Since most of the previous research has used either an excessively long question or a short but somewhat biased question to determine conservation status, it is understandable why these differences in the number of conservers have been found depending on whether justification was or was not included. Changes in the nature of the "conservation" questions in the direction of providing a larger and possibly more reliable sample of responses to each problem may tend to decrease the discrepancy in the results found using these two definitions of conservation, and still avoid the problem of false positives,

7 -5- noted by Shantz and Sigei (I967), that has occurred when justification was not required for determining conservation. Another aspect of the methodology that needs further consideration is the nature of the transformations used to measure conservation of number. Most previous studies have limited the transformations to only two types, expansion and collapsing, nearly always using arrays of equal number. The use of additional types of transformations should show how generalizable the results are from the more typical transformations as well as provide a larger sample of situations to more reliably determine the 13's conservation level. The major purposes of the present study are to investigate conservation of number using prior assessment of key terms and a question format that attempts to decrease some of the problems of complexity and bias present in previous studies and to study the effect on conserving status of different types and numbers of transformations. This study is moreover concerned with the importance of including justifications of responses in assessing conservation status when the revised question format is used, Possible differences in conservation status when subjects are asked to solve problems of conservation of equality, i. e,, equal numbers in the two arrays as contrasted with problems of inequality, are also explored as is the nature of the materials used to measure conservation. Previous studies (Hood, I962; Piaget, 1952) have found that provoked correspondence using complementary materials (e.g.., eggs and egg cups, flowers and vases) tends to facilitate conservation of number to a greater extent than unprovoked correspondence which uses homogeneous materials (e.g., all checkers).. Materials, then, will be investigated along dimensions similar to Piaget using unique pairs in the provoked correspondence situation which may facilitate conservation more than complementary pairs.

8 -6- Lastly, this paper will also discuss conservation of number as related to intelligence, age, social class, and sex of the S_s, Method Subjects The Ss were 210 children who were selected from two private nursery schools and from preschool groups and kindergartens in four public schools in New York City The preschool children ranged in age from four years three months to five years with a mean age of four years eight months, and the kindergarten children ranged in age from five years three months to six years with a mean age of five years seven months. The children were chosen to provide a crosssectional sample of Ss whose ages were at three month intervals., Schools were located in both lower class and middle class neighborhoods. Among the lower class children, 25$ were Puerto Rican and the remainder were Negro. The Puerto Rican children chosen were those judged by their teachers to have a basic command of English. There were no Puerto Rican children and only h^> Negro children among the middle class sample. The number of children at the two grade levels and in the two social classes can be seen in Table 1. Insert Table 1 about here Procedure The.Ss were tested on the Conservation of Number task as part of a larger study concerned with the measurement of cognitive development in preschool children. Each child was tested individually in a separate room at his school in a series of four to seven sessions during a period of not more than four weeks,

9 -7- Six research tasks and the Peabody Picture Vocabulary Test were administered during these sessions and the Conservation of Number task was given as the fifth of the seven measures c The administration of the task took between 12 and 25 minutes. The examiners were women from the same city as the children. They were trained in the administration of the test and the recording of the child's responses. All the examiners spent a few days in each classroom as participant observers in order to become familiar to the children. Caucasian women worked with the middle class children and Negro women worked with the lower class children. In each individual session with a child, the examiners worked in pairs., One person presented the task while the second recorded the child's responses, Conservation of Number Technique This task was developed by modifying the procedures and materials of previously reported measures of number conservation (see Flavell, 19^3; Piaget, I952). An attempt was made in the task development to provide as unambiguous a situation as possible to the child while preserving the essential elements of the conservation of number paradigm. The first modification was the introduction of anl8" x 2k" board which was fitted over the desk top and used during the entire administration of the task. The board was painted half yellow and half blue with the two halves separated along the long axis by a narrow black line so that an array could be placed on one side and be clearly distinguishable from the other. The board was also used to minimize feelings of competition on the part of some of the children by referring to an array as the one on the blue side or yellow side rather than the one on your side or my side.

10 -8- The second modification was the assessment of and instruction in concepts basic to an adequate performance on the task. These were given in the concepts "yellow" and "blue" as pertaining to the board; "bunch" which was used throughout the task as the equivalent of row because it seemed more easilyunderstandable to the children; and "numerical equivalence, " in which thes_ had to make a bunch just like the E's and know that the two arrays had the same number as each other. The third change was the use of a warm-up item in order to give the Ss a chance to become familiar with the question format to be used in the succeeding items, as well as to provide a baseline for estimating how well the Ss understood the language of the questions apart from the difficulty of the actual test transformations. The last modification concerned the wording of the questions asked of the S_ to determine his understanding of conservation. Two consecutive questions were asked after each transformation to provide a more reliable estimate of the S_'s comprehension of each problem and to be able to separate those Ss who could not understand the language from those who did not have the concept of conservation* Both of the following questions were therefore asked regardless of the response to the first. The questions were "Does this bunch (E points to all the blocks on the blue side) have the same number of blocks as this bunch (E points to all the blocks on the yellow side)?" and "Does one bunch have more blocks?" Table 2 shows the complete procedure including the questions asked for a typical transformation. The question format shown was used for the practice item and for all the transformations. Insert Table 2 about here

11 -9- Transformation selected. Five types of transformations were selected from the many possibilities. These included various linear examples but not any nonlinear spatial arrangements because most previous work in the field had been done with only linear arrangements. The warm-up item and the first three transformations had the following basic configuration and, in their transformed state, measured conservation of equality: S: o o o o o E: o o o o o The last two transformations had the basic configuration of three objects on each side. Objects were added during both transformations, one of which measured conservation of equality in the transformed state while the other measured conservation of inequality. Table 3 depicts the warm-up item and the five transformations used to measure conservation of number. Insert Table 3 about here Variations in materials. The relative effect of two different types of materials on children's skill in conservation of number was studied by using the following sets of materials. Materials A was a set of homogeneous styrofoam blocks used to replicate the condition of spontaneous correspondence as described by Piaget (1952). Materials B was a set of matched pairs of toys encouraging the following type of provoked correspondence: S: Vase Woman Boot Dog Bead E: Vase Woman Boot, Dog Bead The instructions for the transformations designated the exact placement of the blocks or the particular toys, each of which was approximately 1 l/2" x 3A" in size. The total sample was divided into two nearly equal size groups, A and B,

12 -10- which were matched on SES, age, and sex. A t-test analysis showed no significant difference between the A and B Ss in either the raw score or the IQ of the Peabody Picture Vocabulary Test. The A group was administered the instructional section and the five test transformations with blocks and then transformations three to five were repeated using toys. Group B was administered the instructional section and the five test transformations with toys and then transformations three 2 to five were repeated with blocks. Scoring Conserving status. The possible responses to the initial conservation questions asked for each transformation, i.e., "Does this bunch have the same number of blocks as this bunch?" and "Does one bunch have more blocks?," are shown in Table k. The probable meaning, the categorization of the responses, and the score assigned to each category are also shown. Insert Table k about here Table k shows that each of the five possible types of responses to the conservation questions were categorized and scored at one of three levels of understanding These patterns of response were categorized in this manner for all the transformations in which the two transformed arrays were in fact equal For the transformed arrays that were actually unequal, the conserving answer was "No - Yes - the blue side has more"; the consistent nonconserving responses were "Yes - No and No - Yes - the yellow side has more"; and the inconsistent nonconserving responses remained the same, An S^'s total score was based on his responses to the "conservation questions" on the total of eight transformations presented to him in the second and third

13 -11- parts of the task. Conserving responses were scored k, consistent nonconserving responses were scored 2, and the inconsistent nonconserving responses received no credit. Each of the five types of transformations were weighted equally and since types 3* * > and 5 were given twice, each presentation of those types was given half the total value that transformations 1 and 2 received. The range for the total score was then Categories of Reasons Used to Explain Answers to the "Conservation Questions" The Ss' responses to the question "How can you tell?," that were given for each of the transformations, were assigned by coders to one of seven categories. These categories were developed primarily from pretest responses and covered most of the categories used in previous studies (Keasey & Charles, I967; Pratoomraj & Johnson, 1966; Shantz & Sigel, 1967 ; Wallach et al., I967). Descriptions of these categories and some examples are shown as follows. 1, Symbolic This category includes those items which indicate that the child "can tell" from what the Interviewer did, e.g., "You've moved them"; "You didn't put a tree on that side (yellow side)"; "I saw you push them over without putting anything on here (yellow side)." 2 Number This includes items that show that the child COUNTED, e,g., "there's only three here (yellow side) and three here (blue side)." 3. Matching These responses show that the child has usually matched one or more of the materials with ones on the other side, e.g., "they're both crisscrossed;" "there's a dog here and a dog here."

14 Perceptual These responses indicate that the child looked and can tell from the length of the rovs alone, e.g., "cause they're not the same"; "cause it's bigger/smaller." 5» Limited Verbal These are very brief reasons vhlch show limited verbalization by the child, e.g., "cause I look"; "I see it"; "because I can figure it out"; "because!" 6,. Don't Know Child indicates he doesn't know, e.g., "I don't know"; "I forget." 7» Magical These responses are primarily unrelated, made-up reasons, e,g.., "my mother told me"; "Jesus says so," Categories of Manipulations Used to Make Both Rows Have the Same Number As shown in Table 2,S_s who did not conserve on a transformation were given the direction, "Change the blocks (or toys) on the blue side so that they have the same number as the blocks (or toys) on the yellow side." The Ss' responses to these directions were also assigned to categories based on frequent pretest responses and are seen as follows. 1< Matching The subject matched the two arrays completely. 2, Incomplete Match The subject matched part of the row but not all of it, 3 Adding or Subtracting The subject changed his row by adding or taking away blocks, ending up with nonequivalence of the two rows,

15 -13- k. Meaningless The subject moved the blocks in some meaningless way including just rearranging those in the row, or meaninglessly shifting them.. 5* Ho Change The child didn't move the blocks or he moved them back to where they were when he started.. Verbal Comprehension Measure The Peabody Picture Vocabulary Test (PPVT), designed to provide an estimate of the S_'s verbal intelligence through measuring his comprehension of the spoken word, was administered to all the Ss as a standard measure of hearing vocabulary. Comprehension of the spoken word was considered to be a necessary skill for success in the number conservation test The PPVT, Form A, was given during the last individual session. The PPVT mean scores obtained by the Ss were as follows: four year old lower class, raw score = 32,8, IQ = 72,2; five year old lower class, raw score = 39»^> IQ = 72.5; four year old middle class, raw score 53-9, IQ = 110.6; and five year old middle class, raw score = 5^*5> IQ = For all the scores, the standard deviations were larger among the lower class S_s than among the middle class.. There were no significant sex differences for any of the groups. There was, however, a significant difference between the four and five year olds in IQ among the middle social class subjects (t = 3-03, p<".0l). Results Effect of Materials The number of conserving, consistent nonconserving, and inconsistent nonconserving responses made by Groups A and B were compared by means of a 2 X

16 -14- analysis for each transformation. The results showed no significant differences in the conserving status of the two groups. Further, there were no consistent differences in the effects of materials when the sample was divided by age, SES, or sex, There were also no significant "materials" differences in the distributions of explanations given to justify the responses to the "conservation" question. These results indicated that situations of provoked correspondence as measured by uniquely matched pairs did not facilitate conservation to a greater extent than those of spontaneous correspondence as measured by entirely homogeneous objects. Reliability of Measure Both of the groups of Ss, A and B, were administered three of the five test transformations twice, once with each set of materials as mentioned previously., Since there was no significant effect of materials on the Ss' conserving status, the total score on the first administration of the three transformations may be compared with the total score on the second administration of the three transformations as an estimate of reliability. Group A subjects (N=107), who had the first transformations administered with blocks and the second with toys, had a correlation of.73 (p<.00l) between the two administrations, Group B subjects (li=^8), who were administered the transformations first with toys and then with blocks, had a correlation between the two administrations of 59 (p< 001). These two correlations suggested an adequate degree of reliability for at least three of the five test transformations. Since the materials' differences were not significant, all further results are presented with the subjects in Groups A and B combined. These results deal primarily with age and social class differences as sex differences were consistently nonsignificant.

17 -15- Understanding of Basic Concepts and Question Format Warm-up item. The warm-up item was administered partly to provide base line data on the Ss' understanding of the necessary concepts, same and more. Subjects who answered in the "conserving" category were those who appeared to understand both essential phrases. The percentage of Ss that answered in each of the response categories is shown in Table 5. Insert Table 5 about here The lower class four year olds had a very small percentage of correct responses, clearly less than the three other groups of subjects (z = 6.25, p<c«0ol), Moreover, fewer than half of the lower class five year olds correctly answered both of the "conservation" questions.. The lower class S_s further showed deficiency in their basic understanding of the language with a greater percentage of inconsistent nonconserving responses, particularly of type 3 (i.e., Yes, Yes), than among middle class Ss (z = 6.57, p<.00l). It can also be seen that the younger children within each social class had more inconsistent nonconserving responses than the older children which showed more difficulty in the understanding of the "conservation" questions for the younger Ss (z = 3 02, p<c.0l). For the sample as a whole, then, only 58% seemed to understand the necessary concepts in a practice situation using an item that did not actually require the understanding of conservation in its solution, The percentage correct for each category of >s would appear to be the maximum percentage possible that would be likely to conserve on any of the test transformations.

18 -16- The difference in the number of "conservers" on the practice item when both questions were answered correctly about the transformation as compared to when only one question was answered correctly can be seen in Table 6, Insert Table 6 about here These results show that when correct responses to only the question "Does this bunch have the same number of blocks (or toys) as this bunch?" are considered, there is a much greater percentage considered to be right than when correct responses to both questions are required (z = 5.01, p<00l).. This increase in percentage correct is, however, much smaller when just the response to the other question, "Does one bunch have more blocks (or toys)?," is necessary for the accurate solution of the problem (z = 1.11, n.s,). Test transformations. The differences in the percentage of actual conservers on each of the five test transformations, depending on the number of responses required to be correct, are shown in Table 7- Insert Table 7 about here It can be seen in the conservation of equality items, one to four, that nearly twice as many conservers are identified when only their accurate response to the question of "same" is required to be a conserver than when correct responses are required for both the questions "same" and "more" (z = 2,72, p<-ol). The results are generally similar for the fifth transformation, conservation of inequality, and show that when conservation is determined only by accuracy in response to the question of "more," there are clearly more conservers identified than when a correct response to the question "same" is also required (z = 2,65,

19 -17- p<,ol), For all the conservation of equality transformations, there was also a somewhat higher percentage of conservers determined when a correct response to the question of "more" alone was considered as compared to when correct answers to both questions were necessary for conservation (z = 1.81, n,s ).- These percentage increases tended to be lower, however, than those found for just the questions of "same," This was also true for the conservation of inequality item with the results found for the reverse words (z = 1,09, ns,). The problem of determining conservation through a correct answer to one versus both of the "conservation questions" is most pronounced among the lower class children. These children tended to have much higher percentages of Yes, Yes (inconsistent nonconserving) responses (four year olds, 40$; five year olds, 19$) than the middle class children (four year olds, 2$; five year olds, 4$) on all five test transformations (z = 5.27, p<;.00l), The differences in the frequencies of answering yes to both conservation questions reveal that the lower class Ss had a greater chance of being considered correct in response to one or the other question than did the middle class Ss, Due to the clear differences in the percentages of conservers depending on the criteria used, conserving responses or judgments for the remainder of the paper are defined as correct answers to both "conservation" questions asked after each transformation as previously shown in Table 4, Measurement with Various Transformations Comparison of difficulty level. The level of difficulty of the various transformations was determined by comparing the percentage of conserving responses across transformations and is shown in Table 8 along with the percentages of consistent nonconserving and inconsistent nonconserving responses,

20 -18- Insert Table 8 about here Results for the total sample showed that the conservation of equality transformations including the frequently studied, row collapsing, all had approximately the same percentage of conserving responses» The conservation of inequality item did tend to be easier with a noticeably greater percentage of conserving responses than any of the other test transformations (z = 2.-78, p<ol). Generally, the lower class children had fewer conserving (z = 3-^2, p<- 001 ) and more inconsistent nonconserving responses (z = 5 ΐ6, p < 001) than the middle class children as had also been seen in the responses to the practice item. Although the differences in the number of conserving responses between the two age groups in each social class were not marked, there was a difference between the age groups in the nature of the nonconserving responses. The older children within each social class tended to give a greater percentage of consistent nonconserving responses (z = 2.33, P < 05 ) while the younger children gave more inconsistent nonconserving responses (z = 2..81, p<..0l) A comparison of the CONS responses on the five transformations for the 58$ of the Ss who were correct on the practice item with the 42$ who were incorrect on the practice item showed that those S_s who succeeded on the practice item had an average of 13$ more conserving responses than those that were incorrect on that item (z = 1.97, P< 05) Further, the former group also had an average of 20$ fewer INC responses on the test transformations than did the latter group (z = 2.42, p<.0l). The percentage of!s correct on the practice item was moreover greater than found for any of the test transformations.- These results suggest that Ss are more likely to be conservers if they can correctly answer questions of same and more in a nonconservation situation,

21 -19- Conservation status across transformations. The total group of S_s tended to vary in the number of transformations conserved with only a small number conserving on all the transformations as shown in Table 9 Insert Table 9 about here Most of the Ss who conserved on only one item tended to do so on item 5, a conservation of inequality item, which has previously been seen (Table 8) to be less difficult than the other transformations.. In general, the results in Table 9 tend to show that when a variety of transformations were presented, most four and five year old children who gave conserving responses were not consistent in giving such responses for each item. Therefore, the true conservation status of a child appears to be not reliably determined on the basis of one or even two types of transformations, Subjects 1 Justifications of Conservation Judgments The explanations given to justify responses to the "conservation" questions were categorized into seven types as previously described. These seven types of explanations were separated into adequate versus inadequate reasons such that categories one through three were considered adequate and four through seven considered inadequate. Shantz and Sigel (196î) This dichotomy generally followed that described by Table 10 shows the numbers and percentages of categories of explanation as compared with categories of judgments made on the "conservation" questions, Insert Table 10 about here

22 -20- These results show that Ss with conserving judgments had 59$ of their explanations categorized as adequate which contrasts with 3^#> adequate reasons for the consistent nonconservers (z = 2.78, p<0l) and 19$ for the inconsistent nonconservers (z 3 62, p<..00l). Nonconservers as a group then had 30$ adequate reasons. Among the Ss who gave conserving judgments, there was a clear social class difference in the number that gave adequate justifications., Thirty-nine percent of the lower class group had good explanations as compared to 6k% of the middle class children (z = 2-T9, p<..0l)> Symbolic (z = 5-12, p<..00l) and number oriented explanations (z = 2 8l, p<..0l) appeared to be more common than the use of matching among all Ss who gave adequate justifications, Conserving Ss also tended to give explanations in most of the inadequate categories with the exception of the Magical or irrelevant category, Nonconserving Ss depended particularly on the Perceptual category in their justifications. Inconsistent nonconservers tended to give slightly more explanations in the Limited Verbal, Don't Know, and Magical categories than the consistent nonconservers did, although none of these differences were significant. The number of conserving judgments as compared to the categories of explanations can also be viewed according to each subject's conserving status across transformations. Table 11 shows the numbers of conserving judgments and adequate levels of explanations and gives the percentage of conserving judgments that the Ss had justified with adequate reasons Insert Table 11 about here Complete conservers, those who gave conserving judgments on all five transformations, showed the highest percentage of adequate justifications for their

23 -21- judgments No clear pattern emerges from the consideration of the percentages for subjects who conserved on one to four transformations. However, even when just "complete" conservers are considered, only three out of every four of their conservation judgments were "adequately" justified. Manipulations to Make Both Rows Have the Same Number On each transformation that an S did not conserve, his response to the direction "Change the blocks (or toys) on the blue (S's) side so that they have the same number as the blocks (or toys) on the yellow (E's) side" was categorized- The percentage of responses occurring in each of the five categories for each of the transformations can be seen in Table 12 Insert Table 12 about here The first category, "Matching," was clearly the most common response for most subjects to the conservation of equality transformations On the conservation of inequality item, unequal addition, many subjects tended to match all of the row while a large number of others were categorized as incomplete matchers usually because they failed to remove the one block or toy that made the two rows unequal. The one other category that a noticeable number of S_s' responses were scored in was the "Meaningless" category, in which subjects appeared to respond to the direction, "Change,..." by moving the blocks around on their side of the board but not with any discernible plan. It should be noted that all categories of responses were actually "correct" for the conservation of equality transformations, in terms of the direction given to the S, except for the "Adding or Subtracting" category. That the majority of nonconserving Ss had to match the objects so that the two rows looked identical provides, of course, further corroboration of the nonconserving child's concept of "same number "

24 -22- Total Conservation Score Age, social class and sex differences. Differences due to age, SES, and sex were determined through analyses of the Ss' total score, which included scores on five types of transformations, three of which were administered twice.. The Ss' conserving status for these results does not include the justifications given to explain answers to the "conservation" questions because the two questions asked after each transformation and the number of transformations used seemed to be sufficiently stringent criteria for the assessment of conservation status. The results of a 2 x 2 analysis of variance among the lower class Ss (N=77) showed that there were noticeable differences in the total conservation score due to age (p =,02) but not to sex. Among the middle class Ss (N = 106), an analysis of variance showed no significant main effects for age or sex. For the total sample (N = I83), there was a social class effect (p< 00l) but no significant age effect found. A significant age by social class interaction (p<,05) for the total sample showed an age increase among the lower class Ss but not among the middle class Ss. This lack of age increase in total conservation score was possibly due to the similarity in vocabulary level between the two middle class age groups as found on the picture vocabulary test. Vocabulary test relationship.. The correlation between the PPVT raw score and the Conservation of Number total score was.52 (p<.-001, N = I70). This relatively strong correlation between vocabulary comprehension and conservation contrasts with the noticeably lower correlation of.17 (p<,02) between chronological age and conservation for the same sample..

25 -23- Discussion The major concern of this study has been the method of measuring conservation of number in young children. Problems in the nature of the assessment within the framework of standard questions have been the focus, The understanding of the key words, same and more, was shown to be sufficiently less than complete when assessed through a nonconservation practice item, particularly among the lower class children, which further substantiates the need for the continued use of assessment or teaching basic understanding of these concepts before one can be sure that conservation is being measured and not simply understanding of vocabulary, However, regardless of whether assessment or teaching is used, it has become clear that nonconserving children continue to make certain irrelevant interpretations of the words same and more when these are used in conservation of number problems Braine and Shanks (1965a, 1965b) have suggested that nonconserving children interpret "same" to mean look alike in terms of appearance rather than really alike as in the similarity in numbers of each array. The former interpretation of same number was also seen in this study in the extensive use of the "matching" category when the nonconserving Ss were asked to make the two arrays have the same number of objects, Hood (1962) has suggested that the word "more" may mean to the nonconserving child only that the shape of the set is changed and the space it occupies is greater than it was before. These interpretations of the words, same and more, appear in solving conservation problems even after appropriate understanding of these words in other nonconservation situations has been demonstrated. Comparisons of the percentages of identified conservers across the various test transformations, depending on the number and types of questions asked to determine conservation, showed that about half as many conservers were identified

26 -2*1 on each transformation when correct responses to questions of "same" and "more" were required for conservation than when just one correct answer was necessary. Based on the number of correct answers obtained to these questions separately, it appears that a correct answer to one or the other is clearly more likely to be fortuitous than a correct answer to both questions, That iss tended to be correct more frequently when answering questions of "same" than questions of "more" on conservation of equality items, and vice versa for the conservation of inequality item, was most probably due to the tendency of young children to answer yes when in doubt This tendency seems to be most apparent when there is an inability to understand what is being requested by the adult and was obvious in the high percentages of Yes, Yes responses given by the younger lower class children, the group with the lowest verbal comprehension.. It can be seen that when questions of same were asked on the conservation of equality items, the yes answer was correct, whereas for the question of "more," the no answer was correct, On the conservation of inequality item, where the question of more is answered correctly by yes and the question of same, by no, there were noticeably more correct responses to more alone than there were to same. This tendency to give answers that are correct because of the structure of the question further emphasizes the need for a series of questions with different structures to be asked about each conservation problem. These findings suggest that stage 1, defined as the absence of conservation by Piaget (1952), might be more meaningfully divided into substages. These substages would be: (l) lack of understanding of the "conservation" questions as seen among the inconsistent nonconservers and (2) understanding of the language of the questions with, however, no conservation of any transformations, as seen among the consistent nonconservers.

27 -25- The differences in the number of conservers found among those Ss who were and were not correct on the practice item further supports the substage notion. Asking at least two separate questions about each transformation then was found in this study to be a more reliable way of assessing the child's understanding of each problem.. This approach is suggested then as preferable to the single question method, which tends to favor correct or incorrect answers depending on the structure, and also preferable to the more complex questions dealing with two and three events in each question- The variety of transformations used to measure conservation of number showed that among the conservation of equality items, there were approximately the same percentage of conservers on all four types of transformations, which showed that the frequently used, row collapsing, was as difficult as other possible conservation of equality itemsr. The conservation of inequality item was the only transformation that differed in the percentage of conservers and was found to be noticeably easier^ Another study (Rothenberg & Orost, I968) has shown similar results using other unequal transformations with a different group of Ss This is in contrast to findings by Zimiles (1966) who reported no differences between the two types of transformations in terms of the Ss' conserving status. Since most of the items were found to be similar to each other in terms of difficulty level, it was surprising to find that most Ss who were conservers on one or more problems did not tend to be conservers on all of them, Only 6$ of the Ss conserved on all the transformations, while more than half of thes_s did not conserve on any of the items. These results tend to indicate that although any single transformation may be typical of others in terms of the percentage of conservers that it yields, a particular subject should probably not be considered a conserver of number unless he is

28 -26- able to demonstrate his conserving ability through a variety of different problems The most likely explanation for this inconsistency among many Ss is that they were at a transitional age, usually seen between the ages of five and seven and designated as stage 2 by Piaget «All the Ss in this study were six years or less Since the transformations in this study generally used arrays of five objects in each row, it may be argued that the arrays were too small to actually be assessing conservation In the previously mentioned study (jrothenberg & Orost, 1968) no difference was shown in the percentages of Ss in each response category when arrays of five as compared with 10 were used Similar evidence was reported by Wohlwill and Lowe (1962) who found no change in the number of conservers when the array size was increased from seven to 12 These results contrast with Zimiles (1966) who has shown somewhat more conserving responses using aggregates of 3 or 5 than with 7 or 9 He did conclude, however, that on the whole, the effect of the size of the aggregate upon conservation of number was not great The importance of including the justifications of the conservation judgments in the determination of thes_s r conserving etatus was also investigated- The results showed that about three-fourths of the judgments made by complete conservers, i»e*, those that conserved on all five transformations, were justified with adequate reasons while only slightly more than half of the reasons given across all conserving judgments were classified as adequate- Results in this study showed more noticeable differences in the number of conservers identified when one as compared to two conservation questions were asked about each transformation and when the number of problems conserved on was considered than when justifications of judgments were considered Since

29 -27-30$ of the reasons given to justify nonconserving responses were also considered to be adequate, this implies that there is room for revision in the usual manner of obtaining justifications from children.. It is suggested that more attention be placed on the obtaining of justifications through the use of additional probes of the ^ε' initial responses to "How can you tell?" or "Why is that?" than 1ε now commonly done These probes should be aimed at getting the child's most complete answer as well as some of his more secondary explanations. The obtaining of explanations from young children will, however, be less than complete because of the many low verbal ability children. If in determining conservation, equal emphasis is placed on the justification and on the original judgment, many less verbal children will continue to be considered as nonconservers when they can accurately and reliably answer, however briefly, differently posed questions about a wide range of conservation of number problems. There is still then the problem of what criteria to use in deciding whether or not a child has attained the concept of number conservation or other types of conservation as well.. The question is whether this attainment must depend on adequate explanations being given by the child or if accurate judgments to a number of conservation questions and transformations may be sufficient. The results of this study also show the general extent of conserving present among a sample of children from four years three months to six years of age representing both the lower and middle social classes. Among the total sample, social class differences, rooted primarily in the nature of the Ss' language comprehension, ;were most apparent. Increases with age from four to six years were shown in the total scores for the lower class children.. This increase was, however, primarily due to changing from inconsistent nonconserving

30 -28- to consistent nonconserving responses which showed language development rather than conservation development- It was unexpected to find no significant increase in the total score from four years to six years old in the middle class sample particularly because neither the four nor the five year olds were approaching the ceiling for the total score One possible explanation is that significant increases in conservation attainment do not really begin to take place until age 6 and that both age groups had similar vocabulary understanding so that there was no noticeable increase in total score Another possibility, however, is that because the two age groups were similar in general mental age as seen on the PPVT, rather than being a year apart, they obtained a similar total score It would then follow that if the groups differed by a year in mental age in addition to chronological age, they would not have had essentially the same total score on conservation of number- The finding of no sex differences supports most previous studies The correlation between the PFVT raw score and the conservation total score (r = -52) was noticeably higher than the correlation between chronological age and the total score (r = 17) This descrepancy is due, at least in part, to the lack of age differences in the total score for the middle class sample It does appear to be important, however, to determine the relationship between both CA and MA and conservation status so that the importance of the two factors in the development of conservation can continue to be evaluated The major findings of this investigation suggest important methodological implications A question format using more than one question about each transformation and possibly varying the order of the questions asked, and the inclusion of a variety of transformations measuring both conservation of equality and inequality, seem essential for gaining an accurate picture of the