Pursuing Qualitative Differences in the Flexible Use of Problem-Solving Strategy by Young Children
Kanevsky, L.
Journal for the Education of the Gifted
Vol. 13, No. 2, pp 115-140
1990

This article by Lannie Kanevsky discusses a study that was done on gifted and average intelligence children in two different age groups. The groups were given games to play to see the differences in problem-solving ability and strategy between the groups. Results indicate differences in the application of previously learned strategies.

Research literature has offered educators little guidance in their efforts to respond to demands for qualitatively differentiated education for the gifted. To address this state of neglect, an interactive assessment environment was designed to explore a number of qualitative and quantitative dimensions of young children's learning as they generalized a problem-solving strategy learned on one version of the Tower of Hanoi to a similar but different version of the task. Eighty-nine 4- to 8-year-olds were assigned to one of four groups based on their IQ and age: young average IQ (n = 20, CA = 58.3 months, IQ = 105.2); young high IQ (n = 22, CA = 59.3 months, IQ = 153.5); older average IQ (a = 22, CA = 94.5 months, IQ = 103.2); and older high IQ (n = 25, CA = 94.8 months, IQ = 155.4). Quantitative comparisons of the children's performances on the two tasks confirmed the expected benefits of intellectual ability and age on the generalization of learning. The qualitative data suggest that the high IQ children had a more accurate conception of the problem; preferred to "own" their solution to the problems; learned more from their illegal moves, and more frequently recognized the similarity of features of the two tasks. In some cases the 4 and 5 year-old high IQ children were superior to the 7 and 8 year-olds of the same mental age. Based on these findings recommendations for the design of appropriate learning opportunities for high IQ children are offered.

    To date, a great deal more is known about the extent of intellectual strength than about the essence of human intellect. But the meaning of this term as it applies to such children is not much clearer today than it was more than a half century ago when Boring defined intelligence as something that is measured by tests of intelligence. However, those who view giftedness from a psychological perspective are well aware that scores on tests of intelligence are intended to provide clues rather than understanding about superior potential. (Passow, 1985, p. 26)

When educators look to the research literature for guidance in their efforts to differentiate learning experiences for highly able children, the nature of the information they have been able to locate has reflected a dependence on quantitative orientations to the nature of intellectual ability. As a result, much of what we know about the characteristics of the gifted that differentiate them from their less able peers is quantitative in nature, (e.g., how much they can learn or how quickly). Contemporary conceptions of intelligence have become much more dynamic in their orientation with an increased emphasis on goal-directed, adaptive behavior such as problem-solving (Stemberg, 1982). Although the bulk of past research has demonstrated greater support for quantitative differences (Maker, 1986; Rogers, 1986), evidence to support the existence of the qualitative differences is accumulating (Shore, 1986; Wong, 1982). The movement toward process-oriented theories of intelligence has inspired the development of alternative assessment strategies which reflect the more dynamic nature of intelligence as it appears in problem-solving activities. Recent applications of an old, but little-used, assessment strategy appear to hold promise for educational researchers and practitioners interested in understanding the nature of the differences in learning potential related to differences in intellectual ability, rather than their size.

A dynamic assessment strategy, originally proposed by the Soviet psychologist, L. S. Vygotsky (1978) in the 1920's, provides an appropriate context for examining the nature of differences in young children's learning. It differs from traditional "static" psychometric evaluations in its tasks, procedures and outcomes. Static assessments of learning involve the presentation of one or more test items after a child has been exposed to some instructional experience. The child's response is evaluated as either correct or incorrect. A comparison of a child's performances before and after instruction reflects the amount of knowledge gained. Such assessments provide little information regarding how the child learned, only whether or not learning has occurred. Therefore the information provides little guidance on how best to design learning experiences to meet the needs of the child.

In contrast, a dynamic assessment focuses on evaluating dimensions of the process of learning, rather than simply the product. A tutor presents a complex task which is expected to be beyond the child's independent problem-solving ability. With the assistance of the tutor, the child achieves a solution. Repeated trials and tasks are offered until the child demonstrates mastery of the strategy and can solve each problem independently. The assessment can offer information related to the extent of generalization of learning, changes in the degree of assistance required, characteristics of preferred learning environments, and the nature of the child's understanding of the tasks. This data provides a much more comprehensive view of the learning process than the number of right or wrong answers provided to a set of test items. One not only gains a sense of what a child knew initially, but their ability to generalize learning from one task to the next, their ability to benefit from the information provided by the tutor, and the accuracy of their understanding of the problems and learning style preferences.

This methodology is sensitive to three characteristics of intellectual ability which are often cited in descriptions of gifted children: their superior problem-solving ability (Ludlow &. Woodrum, 1982); their ability to more effectively benefit from feedback and information while learning (Martinson, 1974); and their ability to quickly access and flexibly use stored knowledge (Clark, 1988; Kitano &. Kirby, 1986; Renzulli & Hartman, 1981).

Confusion has surrounded the relative contributions of age and intellect to these characteristics. Comparisons of the dynamic assessments of the performances of children of different levels of intellectual ability and age can provide insights into IQ- and age-related differences in the way children learn to solve problems and then use that learning to solve related problems. Past research, using static or dynamic approaches to learning, has not considered these variables simultaneously in order to examine their relationship to learning.

Early efforts to distinguish qualitative and age-related differences in intelligence matched children of different ability levels and chronological ages for mental age (Gallagher & Lucito, 1961; Magaret & Thompson, 1950; Merrill, 1924; Thompson & Finley, 1962). These studies provided evidence of IQ-related differences in the fruits of learning although they considered only static assessments of knowledge rather than dynamic assessments of learning. Spitz (1982) concluded his review of these and other studies comparing the performances of intellectual extremes with the observation that although the groups "had equal MAs, they arrived at these MAs by different means" (p. 171). The database and literature which supports this comment has primarily considered comparisons of below average, average and slightly above average samples. None included a group with a mean IQ above 140.

Recent studies of developmental changes in the learning and problem-solving abilities of young children (Crisafi & Brown, 1986; Klahr & Robinson, 1981) have demonstrated these youngsters are capable of much more than traditional characterizations (e.g., Pia-get, 1976) have suggested. In defense of the problem-solving capabilities of preschoolers, Gelman (1979) noted that research strategies have been "inclined to reach conclusions about what preschoolers cannot do, compared with what their cohorts can do." Individual differences in intellectual ability have not been considered in the more recent work either. Given this new credibility, it is now important to search for a better understanding of individual differences in the nature of this ability that may be attributed to intellect.

A dynamic assessment offers a number of advantages beyond past efforts to understand the qualitative and quantitative nature of differences in intellectual performances. Vygotskian evaluations of learning potential examine intellect within the context of: a) a complex cognitive activity, (e.g., problem-solving); b) a series of tasks that involve learning and the generalization of that learning, and c) an interactive learning environment. These features reflect a number of areas of current concern in educational practice and research including the need for a greater understanding of individual differences in problem-solving processes, the acquisition of a flexible knowledge base, and the examination of learning in interactive settings which are more relevant to "real" learning environments than laboratories.

Past research addressing the questions of IQ-related differences in learning have supported the "precocious development" interpretation (Scruggs & Cohn, 1983) rather than qualitative differences. The pure precocious development orientation suggest that high IQ children develop the same intellectual mechanisms and knowledge as older children sooner. In contrast, the pure qualitative differences position suggests that they develop different mechanisms and understandings. It is most likely that this is not an "either/or" phenomenon; differences in learning performances result from an interaction of the two and result in the qualitatively different products that distinguish them from their peers. No research addressing these issues has considered preschool and primary school children.

Therefore, the present study extends an old question to the realm of a younger population through the implementation of a dynamic assessment methodology. The purpose of this research is to respond to the call for an improved understanding of differences in learning that are related to differences in intellect. To do so, it explores the existence and nature of age and ability-related differences in young children's generalization of a problem-solving strategy. The implementation of a Vygotskian dynamic assessment of learning potential permitted the consideration of four dimensions of children's performance:

  1. their ability to flexibly apply a problem-solving strategy;
  2. their ability to use information provided by the tutor;
  3. their understanding of the problem; and
  4. their recognition of the similarities in two related tasks.

IQ- and age-related differences in favor of the high ability and older children were expected. Clear evidence of qualitative differences would be provided if the performances of younger, high IQ children were superior to those of older children of the same mental age.

Method
Subjects
Table 1 provides group N's and means for IQ, chronological age (CA) and mental age (MA). Eighty-nine average and high IQ 4 to 8 year-olds participated in the study. The 4 and 5 year-olds with IQs greater than 135 (YH group) were matched for MA with the 7 and 8 year-old children of average ability (OA group; IQs 90-110). The young children of average ability (YA group) were matched for chronological age (CA) with their high ability peers in the YH group. Children in the OH group were matched to the OA for CA and to the YH for IQ. Estimates of ability for the high IQ children were provided by the Stanford-Binet scores in children's school records. The IQs of the average ability children were determined through pre-treatment administrations of the Slossen Intelligence Test. All high IQ children attended the Hunter College Elementary School in New York City. Average ability 4 and 5 year-olds were enrolled in regular preschool programs in the Vancouver, British Columbia area. The average IQ 7 and 8 year-olds attended an elementary school in the Vancouver suburb of Coquitlam.

Tasks
Two versions of the Tower of Hanoi Puzzle were modified for use in this study. The Monkey Cans game (MC; Figure 1) was originally developed by Klahr and Robinson (1981). Children were introduced to the apparatus as representing three trees with a monkey family (baby, mother, father) living in them. The monkeys were positioned on one set of "trees" in front of the child in the start state. The tutor had another set of trees in front of her with the monkeys positioned in the goal state. The object of the game was to move the family to the goal state by moving one monkey at a time, from one tree to another tree, without trying to put a smaller monkey on top of a bigger monkey.

The second task, the Layer Cake game (LC; Figure 2), was a computer simulation of the traditional Tower of Hanoi puzzle. Children were asked to move the cake to the stick with the stars below it by moving one layer at a time, from the one stick to another, without attempting to put a big layer on top of a small layer. Seven moves are required to solve the problem in the minimum number of moves. Both tasks can be solved using the same strategy. Scripts for both tasks and a listing of the computer program are available on request.

Procedure
Children were presented two tasks on each apparatus. Although subjects were encouraged to do as much as possible on their own, it was made clear, at all times, that they were free to ask for a hint from the tutor if they were "stuck". Each hint was composed of the "best next move" toward the goal. All sessions were videotaped for later analysis.

Session I.
Each child was presented one 3-layer trial on LC after a 2-layer warm-up task. The 3-layer task was offered to determine baseline performance on the task. Before and during the warm-up children were introduced to the game control keys, rules and goals. On both trials, children were offered a hint after not making a move for 10 seconds. This offer could be accepted or rejected. A child was also allowed to request a hint at any time. If an illegal move was attempted, the child was told that the computer would not execute a move that would violate a game rule. Following these two trials, the children were asked if they had ever played a game similar to Layer Cake, since commercial versions are available. If so, they were not considered for further participation.

Children were then introduced to the MC apparatus, rules and goal with two abbreviated warm-up tasks. The experimental, or learning task required the child to move the Monkey family from the far left tree to the far right. The hint procedure was identical to that used in the Layer Cake game. Each child was provided as many trials as necessary to achieve the criteria of two consecutive, error-free, independent performances.

Session //.
On the following day, the session began with a review of MC rules and goal (left to right transfer of the monkey family). The task presentation, procedure and criteria for success were the same as in the previous session. The second MC task was presented once the criteria for success were achieved on the first. It was the reversal of the start and goal states of the first task. The Monkey family began resting on the far right tree and needed to be transferred to the far left. Again, the child was expected to achieve two consecutive, error-free, independent solutions.

The final two tasks were presented on LC. The first was a left-to-right transfer of the cake, and the second task moved the cake from right-to-left. Trials lasting longer than 10 minutes were considered failures and were concluded by guiding the child to the correct solution with successive hints. After two unsuccessful trials the session was terminated as was the child's participation in the study. The hint procedure and criteria for success were identical to those used in the baseline presentation of the task. Immediately after an illegal move attempt the tutor asked the child if she or he could identify the rule that had been broken. If so, they were asked to explain the rule or violation; if not, the experimenter provided an explanation.

Scoring Procedure
Analyses of the videotapes provided two sets of data: quantitative and qualitative. The quantitative counts included the number of legal, illegal and hinted moves used on each trial of all tasks. Descriptions of the three qualitative dimensions of performance considered in this study follow.

Use of hints.
While a child was learning to master LC, it was noted whether she or he either used any hints, accepted at least one hint, denied at least one hint, or requested assistance at least once. A child was counted once in each of these three categories based on his or her use of the assistance that was available.

Understanding of game rules.
A child's understanding of the game rules was explored through their ability or inability to explain the illegality of a rule violation. Based on a response to the experimenter's query after each illegal move attempted on LC, the child was classified as either a) able to explain the illegality of an attempted move or b) unable to explain so the experimenter provided an explanation.

Recognition of task similarity.
While working on MC and LC, each child's spontaneous comments were evaluated for indications of recognized similarities in the surface or strategic features of any of the tasks. If a child commented on the similarity of the game rules in MC and LC during the experimenter's descriptions of the games, she or he was credited for having made a "Rules Across Tasks" comment. If a comment related to the similarity of the physical appearance of the two games, it was classified as an "Apparatus Across Tasks" comment. A comment containing any reference to the reversal of the strategy which was required to solve the second task on either MC or LC was classified as a "Strategy Within Task" comment, while a "Strategy Across Task" comment contained information related to the similarity of the strategy used to solve both MC and LC. Each child was counted only once for having made a comment in any category.

Results
The quantitative and qualitative data were analyzed separately. The number of hinted, legal and illegal moves were submitted to statistical analyses to address the questions of age and IQ-related differences in the degree of generalization. Means and standard deviations are provided in Table 2. The outcome of the full analyses will not be presented in detail. Only those findings that compliment the interpretation of the qualitative data will be mentioned. Complete results have been reported and discussed elsewhere (Kanevsky, 1988; for a copy, send request to author). The data from the qualitative variables are reported as frequencies and percentages in Table 3 and discussed in depth.

Quantitative Aspects of Generalization
An index of the accuracy of a child's moves was derived by dividing the number of legal moves by the total number of moves used in each trial (legal, illegal and hinted). This proportion provided a better index of the relative accuracy of the moves each child initiated correctly and independently on each trial. The mean number of hints used and the mean number of illegal moves attempted in each trial were computed for each trial on both MC and LC. Each of these three quantitative indices of children's performance (move accuracy index, number of hints and the number of rule violations) was submitted to a 3-way repeated measures analysis of variance (IQ X Age X Task) in order to assess group differences in the degree of change in children's performance from MC to LC. Although this statistical procedure tested for three main effects (IQ, age and performance differences on the two tasks) and four interaction effects, only the significant IQ X Task and Age X Task interactions will be mentioned here. Their purpose is to demonstrate the age and ability-related differences in generalization and to provide a context for the discussion of the qualitative variables. Because LC was a more difficult task than MC, it was expected that children who generalized their learning from MC to LC more effectively would show a smaller degree of change in their LC performance levels from their MC levels due to their ability to apply their MC knowledge to LC.

Move accuracy.
The significant IQ X Task interaction [F(l,82) = 12.46, p<.01] indicated that as the children moved from the MC task to the LC the decline in the accuracy index was greater for the average IQ children than the high IQ. The same trend was apparent in the Age X Task interaction (F(l,82) = 10.12, p<.01], where a greater decline in accuracy was demonstrated by the younger children as compared to that of the older. This indicated that the accuracy of the moves selected by average IQ children and younger children suffered to a greater degree with the move from MC to LC than the accuracy of either the high IQ or older children.

Hints.
The significant IQ X Task (F(l,84) = 7.081, p<.01] and Age X Task (f(l,84) = 12.531, p<.01j interactions in the average number of hints used in each trial indicate that the degree of change in the number of hints used on each task varied with ability and age. Average ability children showed a greater increase in the need for the experimenter's assistance in each trial from MC to LC than the high IQ children. Similarly, the 4 and 5 year-olds demonstrated a greater increase in their need for hints in each trial from MC to LC tasks than the 7 and 8 year-olds.

Illegal moves.
Only the IQ X Task interaction (F(l,84j = 9.286, p<.01] achieved significance in the analysis of attempted illegal moves. This finding indicated the average ability children increased the number of illegal moves they committed in each trial from MC to LC to a greater extent than the high ability children. Average ability children appeared to have had a less complete or stable understanding of the MC task than the high IQ children.

Summary.
The significant differences in the degree of change in performance on MC and LC provide evidence that the groups generalized their learning from task to task differently, depending on their IQ or age, or both. In all three analyses, the IQ X task interactions indicated that the high IQ children generalized their learning to the more difficult computer task to a greater extent than their less able age-mates. High IQ children's performance levels changed to a lesser degree than their average ability agemates. The groups responded differently to the increase in task difficulty. The high IQ children's accuracy declined to a smaller extent and their need for assistance and illegal moves increased to a smaller degree. The significant age-related differences indicate that the 7 and 8 year-olds made more accurate move selections and needed fewer hints than the 4 and 5 year-olds on the second task as compared to the first.

Qualitative Aspects of Generalization
Hint use: Accepted, denied and requested.

The frequencies in Table 3 reflect the number of children who needed no assistance, accepted, denied or requested assistance on LC. The percentages provide an indication of the proportion of each group that the frequency includes.

All of the 4 and 5 year-olds needed some degree of assistance while an IQ-related difference arose in the percentage of 7 and 8 year-olds who needed no assistance (OA = 9.1%, OH = 44%). In addition, 86.4 percent of the OA children accepted at least one hint while only 48 percent of the OH children did. Strong IQ-related differences were apparent in the percentages of children denying a hint. More high than average ability children denied hints in both age groups. In fact, a greater percentage of the YH children denied a hint than the OA children (22.7% versus 9.1%). In the number of children requesting hints, only age differences arose, with a smaller proportion of older children asking for assistance on the LC tasks.

Rule violations.
Three OA and eight OH children committed no rule violations. No clear age or ability-related trends were apparent in the percentages of children who were able to explain at least one rule violation or of children who required at least one rule violation to be explained by the experimenter. However, a single group stands out from the other three on each of these two variables. First, almost 60% of the high ability 4 & 5 year-olds were able to describe the illegality of their moves, while less than one-third of each of the other three groups understood their errors well enough to explain them. Second, a noticeably smaller percentage of high ability 7 & 8 year-olds needed the experimenter to explain the illegality of a rule violation than was found in the other three groups.

Recognition of task similarity.
Although the number of children who offered spontaneous comments on the similarities of tasks was small, overall the numbers suggest developing age and ability-related trends. None of the 4 & 5 year-olds of average ability expressed any comments related to the task similarities and only a few of the YH and OA children did. The OH children commented on the common features of the task much more frequently. Although small, the number of YH children expressing recognition of the task similarities equaled or exceeded that of their older mental agemates (OA) on all of the variables except one. The OH group substantially exceeded all of the other groups on all variables.

Discussion
Quantitative Aspects of Generalization
The significant interactions found in the accuracy index indicated the benefits of increasing age to a child's ability to select a legal move from all possible alternatives. The high IQ children's accuracy was relatively stable across the change in task (from MC to LC) in comparison to the drop in accuracy demonstrated by the average IQ children. The same can be said of the older children when they are compared to the younger. These results suggest that the high IQ and older children generalized the information gained through their move selections on MC more accurately to their moves on LC than their less able to younger peers. The performance of the average IQ and younger children suffered due to the cognitive demands created by the substantial change in the nature of the tasks.

The number of hints was sensitive to a child's need for factual support, as well as reflecting differences in risk-taking behavior and self-confidence. Significant age- and IQ-related group differences in generalization were found here as well. Although all children needed more help on the computer task, the brighter or older children were less dependent on the experimenter with the increase in task difficulty than their less able or younger peers.

If the significant findings in the number of hints and accuracy index are considered simultaneously, they provide a more complete picture of ability- and age-related differences in the children, in terms of greater relative dependence or independence. On the MC task, average ability and younger children relied more on the experimenter and initiated a smaller proportion of legal moves than the high ability and older children. The average ability and younger children substantially increased their dependence on the experimenter from MC to LC and showed a decrease in accuracy while their high IQ and older peers showed a slight increase in hint use and a slight decrease in accuracy. This is clear evidence of the advantage of higher age and IQ on generalization.

The rule violation analysis provided insights into IQ-related differences in the accuracy of a child's internal problem representation and the degree to which these groups were able to understand and learn from their mistakes. No significant age-related group differences were found. Because all children received the same description of the tasks, goals and constraints, this variable reflected IQ-related differences in retention and operationalization of this information as well as the effect of their experience on subsequent similar tasks. The extent of the increase in the number of illegal moves attempted by the high ability children was minimal (from .26 to .30 in each trial), while the average ability group rose substantially as they learned the second task (from .36 to .79). The high ability groups were not as confused by the change in demands from MC to LC, as the average ability children. The latter broke more rules in their efforts to acquire the LC solution strategy. This finding suggests that greater intellectual ability allowed the high IQ children to apply their understanding of the task more accurately from MC to LC.

These findings serve three purposes. First they provide an empirical basis for extending the quantitative evidence of IQ-related differences in the process of generalization into the high end of the distribution of intellect. Second, it provides further support for the movement to reconceptualize young children as active, complex problem solvers rather than as weak, simple learners. Finally, they go far beyond the right/wrong, static comparisons of learning products in the information they provide about IQ- and age-related differences in the learning process.

The quantitative analysis indicates the separate and powerful influences of ability and age on generalization and, due to the nature of the experimental procedure and variables, provides some insight into the nature of these differences. The qualitative data revealed a variety of findings which address this issue.

Qualitative Aspects of Generalization
Both the number of children accepting at least one offer of assistance and the number of children requesting assistance demonstrate more substantial differences related to age than IQ. Because all of the 4 and 5-year-olds, regardless of IQ, accepted some aid, it appears that immaturity was a greater factor than IQ in the ways children used external support for the younger children. The influence of intellectual ability became more apparent in the older children's acceptance of hints. Almost all of the average ability 7 and 8-year-old children accepted one or more offers of aid while this was true of less than half of the high ability children. In addition to an age difference, a slight ability effect became apparent in the number of children who requested aid.

The group differences in the number of children denying hints provide the first glimpse of a behavior which qualitatively differentiated some high IQ children from their chronologically older mental agemates. Regardless of age, the high IQ children were more likely to refuse an offer of help. In fact, a greater percentage of the YH children declined aid than the older, but intellectually less able children (YH = 22.7% as compared to OA = 9.5%). This reflected the high IQ children's preference for achieving an independent solution which increased with age. The OH group, who accepted fewer hints than any other group, also refused aid most often. They appeared to prefer testing the next move they felt was best rather than relying on an externally provided, "best next move." They would say, "No. No, I wanna do it myself." They enjoyed the personal satisfaction of an autonomous solution, in contrast, the group that needed the greatest degree of aid, the YA, was also least likely to deny an offer of help. These children appeared relieved when a hint was offered and accepted it gratefully. Ownership of a successful solution was not as important to them as it was to their high ability agemates.

The high ability children appeared to have more confidence in their ability to find a solution and were more able to judge their need for aid. This finding is consistent with other recent research (Jacobson & Crammond, 1988) and the literature describing the behavioral characteristics which are expected to differentiate gifted children from their less able peers. The gifted are frequently characterized as having a high level of self-confidence and independence (Davis & Rimm, 1989), and as developing "an internal locus of control and satisfaction" earlier (Clark, 1988). It appears that these nonintellective attributes become influential in the learning and problem solving behavior of bright children as early as 4 and 5 years old, and increase with age.

When considering the results of the analysis of children's understanding of their rule violations, the YH children were again superior to the OA children. More of the YH were able to correctly explain the illegality of their violations than their mental age-mates, in fact more than any other group. They clearly had a different conception of LC. This allowed the YH children to express their understanding more often than those of average ability.

The number of OH children able to explain a rule violation provided a false indication of the quality of their understanding of the task. When asked to explain their rule violations, some of the OH children would realize the nature of their difficulty, exclaim, "Oh, I get it now!" and carry on with the task without stopping to respect the experimental procedure. This also prevented the experimenter from explaining. Therefore, it is doubtful that the OH numbers accurately reflect the true extent of their understanding (i.e., the degree of understanding is underestimated). The quantitative analysis of the number of illegal moves attempted also supports this as the OH children committed the fewest rule violations of all of the groups. Therefore, overall, it appears that the high IQ children not only broke fewer game rules, but understood them better after violating one. Differences also appeared in their interpretation of the experience of breaking a rule. They were less shaken by the computer's refusal to execute their move and often smiled or nodded once a rule violation was understood, as if it were pleasing to understand and learn from their mistakes. The average ability children were often more concerned with having been wrong than they were with understanding. A factor that appeared to contribute to these responses was whether or not the child believed he or she could solve the task. If the child perceived LC to be beyond his or her level of competence, a rule violation was simply further evidence of weakness. When children felt the problem was within their level of competence, it was fun, a challenge; they just needed to think about it in order to understand.

This "give me time to understand" attitude was highlighted by the hint procedure. If children began the learning sequences with doubts about their learning, the supportive atmosphere provided by the dynamic assessment environment allowed them to rely on the experimenter for guidance until they felt competent enough to rely on their own understanding of the task. The number of children in each group who denied a hint provides clear evidence of group differences in the confidence children had, not only in their learning and problem-solving ability, but in their ability to monitor their learning, i.e., when they needed help and when they didn't.

Because the proportion of YH children who were able to explain an illegal move was almost twice that of their mental agemates who were two years older, this provides further evidence of qualitative differences in learning. Purely cognitive explanations of this finding include: (a) the high IQ children are encoding the information provided in the introduction to the game differently so that they develop a different sense of the task initially; (b) while learning, the high IQ children develop a different, more accurate problem representation which lends itself to easier verbalization; or (c) the high IQ children are better able to search through their prior knowledge in order to deal with current task demands. Any combination of these, and other unrecognized factors may be contributing to the findings.

The affective component in learning cannot be ignored. The issue of confidence was raised earlier. Due to their superior self-confidence, the high IQ children respond to "bad" moves by learning from them. This may not be a causal relationship but an interactive one. Superior self-confidence and learning potential result in a better attitude toward problem solving challenges and better learning performance which result in greater confidence…and so on.

Of the children who commented on the similarities of the problem solving tasks, the high and average ability children distinguished themselves from each other within each age group. The lack of similarity comments by the YA children is not surprising when one considers their 50% failure rate. In fact, it raises doubt regarding whether or not any of the YA children actually generalized at all. Those who completed both tasks may have learned them as unrelated tasks, without applying knowledge from their first experience as they were not aware of its appropriateness within the context of the second task.

Ability-related differences between the 7 and 8-year-old groups were much more substantial than those in the younger groups. This suggests that the IQ-related differences in childrens' sensitivity to task similarity increases with age. These trends became more apparent with the increasing complexity of the knowledge reflected in the content of the comment. The slightly greater number of YH than OA children offering comments on the rules and apparatus suggests that high IQ 4 and 5 year-olds were already becoming more sensitive to patterns of similar features in tasks than their average ability mental agemates. The OH children seemed more attentive to similarities in strategic aspects of the game than in the simpler, more obvious aspects of the task (rules and apparatus).

Spontaneous comments on the similarity of the problem solution or strategy reflected an ability to search for and find patterns in the procedures and knowledge children developed as they learned the games. The children expressing such comments may have developed different conceptions of the games that lend themselves to making connections between tasks. This may be a factor contributing to the superior generalization of the high IQ children which was found in the analyses of the quantitative variables. Not only do these children recognize the similarity but they verbalize it without prompting from the experimenter. As mentioned earlier, it is possible that their strategies for executing internal searches of their prior knowledge are more efficient. This would allow them to search for a greater number of familiar features simultaneously than their less able peers.

The information that a child must understand before making a comment on the similarity of the strategy developed to solve MC and LC is more complex and abstract by far than that which would enable children to comment on the likeness of the rules or apparatus. The children's behavior in this study suggests that with age: (a) the high IQ children became increasingly more able to or prefer to think abstractly than their less able agemates; (b) the high IQ children became more sensitive to the strategic characteristics of a problem solving task than their less able agemates; and (c) IQ-re-lated differences in young children's ability to generalize complex knowledge increase.

The analysis of the qualitative variables provides a glimpse of qualities of problem-solving related to intellect and maturation that had eluded researchers in the past. This may have been due to the 50 IQ point spread between the means of the average and high ability groups. This permitted clearer distinctions between the ways these two groups of children oriented themselves to a problem to emerge. In comparison to their average peers, high ability children prefer to plan independently, have a better understanding of the problem and the rules they break, are quicker to integrate learning from their mistakes and more sensitive to increasingly complex similarities in the tasks.

Although educators have suspected these differences for many years, they had not been supported in the research literature on learning. The data reported here demonstrate that qualitative differences in learning exist, and hint at their nature, but do not make them explicit. A number of potential explanations have been offered but these must be investigated in future research.

Before they can be asked to modify their instructional materials and techniques into qualitatively different programs for children of different levels of intellectual ability, educators must be provided information regarding the nature of the differences they are attempting to nurture. The results of this study indicate that the generalization of problem solving may be a suitable starting point for the pursuit of educationally relevant qualitative differences. Flexible access and application of knowledge are fundamental characteristics of effective learning, problem solving, and cognitive development. It is therefore imperative that "appropriate" educational programming respond appropriately to a child's need for breadth, depth, and abstraction of knowledge as well as speed and volume. Classroom learning can and should offer opportunities to develop skills and strategies applicable in real-world settings with methods appropriate to each child's learning characteristics. The results of this study suggest that even young children, particularly those of high ability levels, can be challenged with problem-solving tasks that will employ mechanisms that underlie and promote the development of complex cognitive skills which we have traditionally associated with older learners.

High IQ children appear to approach tasks differently than their average ability agemates. The quantitative data demonstrate their greater learning efficiency. These results suggest that an accelerated program is one appropriate strategy for meeting the educational needs of the gifted but it must be considered in the context of other learning characteristics found in both the quantitative and qualitative analyses, i.e., children's ability to make broader generalizations of their knowledge, their superior understanding of a task, their desire to "own" their solutions, their desire to learn from their mistakes, and their interest in more complex, abstract, richer levels of knowledge. Therefore, asking them to move more quickly through tasks of the same level of complexity as their average ability age-mates may not be sufficient to promote cognitive development. High IQ children need to be offered learning opportunities that also stretch the application of their learning into more diverse contexts and complex knowledge. These experiences, combined with the nature of their learning abilities, may optimize their development of a more flexible, sophisticated knowledge base, rich with complexity, and the skills necessary to apply it.

With this understanding of the nature of the high IQ learner, the following suggestions can be made for the development of problem-solving environments for extremely high IQ children:

  1. They can be asked to generalize knowledge further, to related tasks in different subject areas and formats. Encourage them to search for similarities in the task feature, demands, and solutions. In the early stages of developing competence, generalization seldom occurs unless tasks are similar in meaningful ways, instruction focuses children's attention on features that are common to the tasks and later trains them to search for common features.
  2. Encourage children not only to use higher-level thinking processes, but to consider deeper or more abstract levels of knowledge (i.e., instead of facts, address the rules, principles, and relationships inherent in a problem, correct solutions, incorrect solutions, etc.).
  3. Select tasks that the children will perceive as challenging, but within their range of competence, with assistance if it can be provided. If cognitive growth is a desired outcome, the task should be slightly beyond current levels of competence.
  4. Encourage children to learn from their mistakes by allowing them to happen, and then asking the children, to explain the nature of the error.
  5. Create settings where guidance is available, but not imposed. Encourage children to monitor their need for support so they are able to ask for it when needed. Assure the children that you realize the task is going to be something they will not know how to solve immediately, and that is why help, not a solution, will be provided. It is important for them to realize that they are working in a supportive environment and may ask for help when they have exhausted their resources. At the same time it respects the preference the high IQ children showed for "owning" their solution, i.e., they wanted to work without the interference of the experimenter rather than acquiring a solution from her.

As children's cognitive abilities develop, the nature of their intellect and learning will change. The rate of change is quite impressive and increases with IQ. The restricted age groups in this study leave questions of longterm developmental differences relatively unanswered. The results indicate trends within a limited age range (4 to 8 years) but focus on the differences between the groups rather than the issues related to the continuum of change over time. Future research will need to involve children at all points along an extended age continuum.

The same comment can be applied to the ability ranges compared here. It must be demonstrated that the ability to generalize learning is correlated with intelligence at all points along the distribution of ability as well. It is doubtful that either goal can be achieved until a suitable series of well-analyzed, analogous problem-solving tasks is developed for this purpose. The foremost concern of any future investigations of the generalization of problem solving strategies must be the development of a series of analogous tasks that allow the controlled manipulation of task complexity (Crisafi &. Brown, 1986).

The well-defined nature of the Tower of Hanoi task is both an asset and a liability. It permits clean performance assessments and analyses. However, it is a far cry from the ill-defined problems children learn to solve in day-to-day life. This must be kept in mind when attempting to generalize trends found in these results to academic or real-world problems.

The issue of qualitative differences in cognition will haunt gifted educators until their nature can be specified. For many years difficulties experienced in attempts to find evidence of their existence had been due to the application of inappropriate, product-oriented methodologies and psychometric tools. More recent learning strategy research (Scruggs & Mastropieri, 1988; Wong, 1982), studies of metacognition (Shore, 1986), applications of information-processing approaches (Davidson & Stemberg, 1984, Marr & Sternberg, 1986), and dynamic assessment methods (Campione, Brown, Ferrara, Jones, & Steinberg, 1985; Crisafl & Brown, 1986; Ferrara, Brown, & Campione, 1986) are proving more fruitful.

The results of this study indicate that there are differences in learning, generalization and problem solving related to intelligence which cannot be explained as the early development of the cognitive mechanisms and structures characteristic of older children of the same mental age. The high ability children develop a different sense of "the game," learn more from their mistakes, and prefer to "own" their solutions than their mental agemates. In these areas, the high IQ 4 and 5 year-olds most clearly distinguished themselves from the average ability 7 and 8 year-olds. In other areas, where their performance fell between their chronological (YA) and mental agemates (OA), (e.g., the amount of assistance needed when asked to generalize their MC solution strategy to LC) there is insufficient evidence to indicate whether precocious development or qualitative differences contributed to their superiority over the YA children. Nevertheless the differences do exist and can be considered when developing appropriate educational experiences.

Educators have patiently awaited a response to their pleas for guidance from research for the curriculum and program development for high ability learners. Investigations that search for the nature of qualitative differences in dimensions of the learning process will provide more complete answers than past research which has focused on learning products. The recommendations offered here are offered cautiously as they are based on promising, but preliminary findings. The pursuit of qualitative differences must continue with refined questions and tools in order to provide educators with the information they need to guide their efforts to appropriately nurture the intellect and affect of all children.


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