Because of their importance to developmental (Piaget, 1971) and cognitive-style (Turkle & Papert, 1990) explanations in problem-solving behaviors, gender and grade level distinctions are discussed in the first section. The second and third sections that follow discuss relationships between domain knowledge and problem-solving on the one hand and between cognitive styles and problem-solving on the other.

In their discussion of cognitive styles, Turkle and Papert ( 1990) relate top-down and bottom-up approaches in programming to males and females, respectively. The results of this study confirm their observations, at least with respect to males. Five of the six boys participating in the study were identified as using top-down planning, as compared to three of the five participating girls. Girls seemed more inclined than boys to represent problems globally (3 girls, I boy), and, conversely, boys seemed more inclined than girls to represent them locally (5 boys, 2 girls).

Figure 2. Domain Knowlege, Cognitive Styles, and Problem-Solving Strategies

These results lend support to Turkle and Papert's argument that programming instruction that forces students to use particular cognitive styles may be genter biased. As such, these issues clearly deserve further investigation. No other gender differences in cognitive styes were noted.

Grade-level distinctions in cognitive styles were minimal and, at that, ran counter to what might be expected friom a developmental perspective (Ginsburg & Opper, 1980). More fifth graders (3) that fourth graders (1) were categorized as concrete thinkers, and all the students characterized as bottom-up planners (3) were fifthe graders. Such findings tend to support Turkle and Papert's contention that cognitive styles do not represent stages in the evolution of formal reason but rather that each is equally valid on its own terms. No other grade-level distinctions in cognitive styles were noted.

Few gender or grade-level distinctions were found in the problem-solving strategies the students employed. Boys were more likely than girls to use extreme forward chaining strategies (4 boys, 2 girls), a preference probably related to their propensity for local problem representation, as were fifth-graders (4 fifth, 2 fourth). No ready explanation for this latter finding comes to mind. The two students who used systematic trial-and-error strategies were both fifth graders, which taken by itself might be seen as supporting developmental explanations (Ginsburg & Opper, 1980). No other gender or grade-level distinctions were noted in the problem-solving strategies utilized by these students.

There has been a long-standing controversy between cognitive psychologists who believe that problem-solving is always domain specific (Griggs, 1989; Johnson-Laird, Legrenzi & Legrenzi, 1972; Reich & Ruth, 1982) and those who believe that general problem-solving strategies are applied across domains (Anderson, 1984; Braine, Reiser & Rumain, 1984; Geis &Zwicky,1971; Henle, 1962). Developmentalists (Ginsburg & Opper, 1980; Piaget, 1971) strongly favor the latter view and relate the development of problem-solving abilities to their natural emergence with maturity. Although this author also subscribes to the latter position, it is clear that one's knowledge of a particular domain will significantly affect the ways in which he or she attempts to solve problems therein. This section accordingly examines results concerning the relationships between domain knowledge and the cognitive styles and problem-solving strategies employed by the students.

Figure 3 shows relationships between students' Logo programming skills and their mathematical abilities. The rows distinguish between "very good," "average," and "not good" Logo programming proficiency; the columns distinguish between "very good," "average," and "not good" mathematical skills. The numbers in the cells indicate the numbers of students who were judged to have the particular combination of domain knowledge designated by the rows and columns defining them. Figure 3 shows that although there seems to have been a general tendency for those who were good in one area to be good in the other, and vice versa, there was not necessarily a connection between the skills. In particular, students who were categorized as having "very good" mathematical skills exhibited a full range of Logo programming skills.

No relationships emerged between spatial discrimination skills and either Logo programming or mathematical abilities. Indeed, of the two students categorized as exhibiting poor spatial discrimination, one was also categorized as having very good mathematical and Logo programming skills, while the other was categorized as having only average skills in these areas. All other students who demonstrated a full range of mathematical and programming abilities were found to have adequate spatial discrimination skills.

Figure 3. Relationships Between Domain Knowlege and Problem-Solving Strategies

The problem-solving strategies employed to solve the Logo programming problems, however, did seem to be related to domain knowledge. Figure 4 superimposes such strategies over the domain knowledge grid shown in Figure 3 by associating the use of particular strategies with the domain knowledge combinations of students exhibiting such usage. In this representation, some interesting patterns emerge. For example, students with poor mathematical abilities used extreme forward-chaining strategies exclusively, while students with "very good" mathematical skills did not use them at all. Systematic trial-and-error strategies were used only by students with "very good" mathematical skills and "very good" to "average" Logo programming abilities. Subgoals formation strategies were used only by students with "very good" to "average" mathematical knowledge. Finally, analogy and alternative representation strategies were used by students with a broad range of domain knowledge combinations.

Although the domain knowledge grid in Figure 4 does not show spatial discrimination, a very interesting relationship between poor spatial discrimination and the other knowledge domains may have affected the problem-solving strategies employed by at least one student. As previously noted, two students exhibited poor spatial discrimination skills. One of these students, Kathy, was characterized as having "very good" mathematical and Logo programming knowledge. Spatial discrimination problems seem not to have affected her choice of problem-solving strategies. The other student, Lilly, was characterized as having "average" mathematical and Logo programming knowledge. In Lilly's case, spatial discrimination problems did seem to affect her choice

Figure 4. Relationships Between Domain Knowlege and Problem-Solving Strategies (Superimposed Strategies)

Figure 5. Relationships Between Domain Knowlege and Cognitive Styles

The patterns that emerge in Figure 5 are very similar to those shown in Figure 4. They tell us that in addition to the relationship between top-down planning and "average" to "very good" mathematical ability previously noted, all students who were categorized as abstract thinkers were also categorized as having "average" to "very good" mathematical abilities. All the students who represented problems globally were judged to have "very good" mathematical skills, while all the students who were categorized as concrete thinkers had "average" to "not good" mathematical skills. However, Figure 5 does not show that the division between students who exhibited local problem representation, abstract thinking, and top-down planning and the student who demonstrated local problem representation, concrete thinking, and bottom-up planning found in the center square of the grid ("average" domain knowledge on both dimensions) is most likely related to that student's spatial discrimination problems.

Thus, we can see a general tendency for better domain knowledge combinations to be related to particular cognitive styles (i.e., global problem representation, abstract thinking, and top-down planning) and vice versa (i.e., bottom-up planning and concrete thinking) and for better domain knowledge combinations to be related to the use of specific problem-solving strategies (i.e., systematic trial and error and subgoals formation) and vice versa (i.e., forward chaining). Such findings suggest that something more than either development (Piaget, 1971) or disposition (Turkle & Papert, 1990) was at work here. What is most interesting, however, is how similar those tendencies are. This finding is discussed in the following section.

When Figures 4 and 5 are considered together, one is struck by the fact that the patterns of problem-solving strategy usage and cognitive style are nearly identical. These relationships are detailed in Figure 6, which gives the problem-solving strategies used by students with varying cognitive styles. The three lines indicate the global versus local problem representation, abstract versus concrete thinking, and top-down versus bottomup planning distinctions. The angles created by those lines thus enclose the various possible cognitive style combinations. For example, the bottom right corner of Figure 6 encloses the local problem representation, concrete thinking, bottom-up planning combination. Notice that no students were characterized as both representing problems globally and thinking concretely or as being bottom-up planners and using global representations.

Figure 6 tells us that only the students who represented problems globally also employed systematic trial-and-error strategies; that only the students who represented problems locally also used extreme forward-chaining strategies; that only abstract thinkers also used subgoals formation strategies; that only top-down planners also used analogy and alternative representation strategies; and that only bottom-up planners also used extreme forward-chaining strategies exclusively. We can conclude, then, that cognitive style combinations were more closely related to problem-solving strategies than were domain knowledge combinations. Such finding would tend to support arguments for the existence of general problem-solving skills that are applied across domains (Anderson, 1984).

Figure 7 represents the relationships between cognitive styles and problem-solving strategies found among students participating in the study. In this representation, the

Figure 6. Problem-Solving Strategies Used by Students With Varying Cognitive Styles

ellipses designate the various cognitive style distinctions—the thin solid lines enclose areas representing the global/local problem representation distinction, the striped lines enclose areas representing the abstract/concrete thinking distinction, and the thick solid lines enclose areas representing the top-down/bottom-up planning distinction. The numbers in parentheses indicate the number of students who exhibited each particular cognitive style. The numbers not in parentheses indicate the number of students who exhibited particular cognitive style combinations, namely, those combinations indicated by the overlapping ellipses in which they are found.

Figure 7 also relates particular problem-solving strategies to the cognitive with which they were exclusively associated. Analogy and alternative represen strategies are linked to top-down planning because only students who were categorized. as top-down planners employed those strategies. Extreme forward-chaining strategies are linked to local problem representation because only students categorized as sensing problems locally used extreme forward-chaining strategies. Global problem representation and systematic trial-and-error strategies and abstract thinking and subgoals formation strategies were found to be similarly linked.

Several quite interesting observations can be made based on the diagram in Figure 7. As previously noted, particular problem-solving strategies were linked with specific cognitive styles. This suggests that the concept of "cognitive style" might be instructionally important. Individualizing problem-solving instruction might be for example, by focusing on the development of particular problem-solving strategies

Figure 7. Relationships Among Cognitive Styles and Problem-Solving Strategies

appropriate to individual students' specific cognitive styles. Conversely, trying to develop problem-solving strategies might be useless, if not harmful, to students whose cognitive styles are not suited to them. This seems to be in part what Turkle and Papert (1990) suggest.

Second, although certain cognitive style/problem-solving strategy associations seem relatively obvious, others do not. The linking of local problem representation with extreme forward-chaining strategies, for example, makes a certain kind of sense, in that it seems logical that only students who represented problems locally would use a problem-solving strategy so locally grounded. Likewise, it makes sense that only students who consider a problem as a whole (top-down planning) could relate that problem to previous problems (analogy) or alternatively represent it. On the other hand, systematic trial-and-error strategies seem more logically related to abstract thinking than to global problem representation. Indeed, the use of systematic trial-and-error strategies is one of the criteria Piaget (1971) gives for abstract thought. Similarly, subgoals formation strategies seem more logically related to top-down planning.

Third, all of the cognitive style distinctions shown in Figure 7 overlap. Thus, although most of the students who were categorized as top-down planners were also categorized as abstract thinkers, one was not. This indicates that the distinctions do not refer to identical behaviors, suggesting that cognitive styles cannot be unilaterally described. In particular, the distinctions between concrete and abstract thought made by Piaget ( 1971 ) and between top-down and bottom-up planning made by Turkle and Papert (1990) may describe different dimensions of cognition and should, therefore, be considered as such. To these unique cognitive style dimensions, the current study adds a third, that of global/local problem representation.

Finally, Figure 7 shows that although the cognitive styles identified are not identical' they are related. In particular, all the students identified as representing problems globally were also characterized as abstract thinkers, and all the students categorized as abstract thinkers were also found to be top-down planners. Similarly, all the students identified as bottom-up planners were also identified as concrete thinkers, and all the students categorized as concrete thinkers were also found to represent problems locally. These relationships are outlined in Figure 8. In this representation, the cognitive style distinctions are paired horizontally and the arrows indicate cognitive style combinations exhibited by the students who participated in the study. In addition to the exclusionary relationships just noted, it shows that students who represented problems locally were found to be both abstract and concrete thinkers, and that students who were identified as concrete thinkers were categorized as both top-down and bottom-up planners. Likewise, students who were categorized as top-down planners were found to be both abstract and concrete in their thinking, and students found to be abstract thinkers represented problems both globally and locally.

The relationships described in Figure 8 suggest the possibility of a kind of hierarchical emergence of cognitive styles (and their corresponding problem-solving strategies) related to the development of domain knowledge. In such a view, total novices would exhibit local problem representation, concrete thinking, and bottom-up planning (and extreme forward-chaining problem-solving approaches) because they lacked the domain knowledge to structure their problem-solving otherwise. As they gained domain expertise, they would move from bottom-up to top-down planning (and incorporate alternative representation and analogy strategies into their problem-solving approaches) as they accumulated enough knowledge to consider a problem as a whole; then they would move from concrete to abstract thinking as they gained enough experience to generalize (incorporating subgoals formation into their problem-solving repertoires); and finally they would move from local to global problem representation (abandoning extreme forward-chaining strategies in favor of the more global strategies, including, at this juncture, systematic trial and error). Such a view offers a third explanatory perspective to be considered along with development (Piaget, 1971) and dispositions to particular cognitive styles (Turkle & Papert, 1990). Such a view is, moreover, most consistent with the domain knowledge, cognitive styles, and problem-solving strategies found among the students who participated in this study. As such, it deserves further investigation.

Figure 8. Relationships Among Cognitive Styles