Strategy development in a block design task. (Intelligence).
This study was conducted in the general framework of the cognitive psychology of problem solving, where the processes or strategies underlying performance on psychometric tests are analyzed (Carpenter & Just, 1986; Dickes, Houssemand, & Reuter, 1996; Hunt, 1982, 1983; Hunt, Lunneborg, & Lewis 1975; Huteau, 1995; Jones & Torgesen, 1981, Lansman, Donaldson, Hunt, & Yantis, 1982; Lautrey & Huteau, 1990; Pellegrino & Glaser, 1979; Richard, 1996; Royer, 1977; Royer, Gilmore, & Gruhn, 1984; Rozencwajg & Huteau, 1996; Steinberg, 1988).
One way of analyzing the strategies used on intelligence tests consists of observing subjects' behavior while the test is underway, which generally requires adapting the situation to some extent. This approach was used to detect the strategies that subjects implemented to solve Kohs blocks (Beuscart-Zephir & Beuscart, 1988; Rozencwajg, 1991) or to solve the Passalong test (Richard, 1996; Zamani & Richard, 2000). As Royer (1977, p. 33) said, "The approach is to analyze the task directly by manipulating its structural variables rather than indirectly by factor analysing its relationship to other tests."
1.1. The block design task
The task examined here was derived from Kohs blocks, also called the block design task (Kohs, 1923), in which subjects use red and white colored blocks to reproduce two-dimensional square, red and white designs composed of geometric figures (see Fig. 1). Kohs blocks have given rise to a variety of tasks, all based on the same principle (Alexander, 1950; Bonnardel, 1971; Wechsler, 1974).
[FIGURE 1 OMITTED]
This task is interesting for several reasons. It is usually considered to be a general intelligence test which is highly saturated in factor g. Royer et al. (1984), for example, reported a correlation of .80 between Kohs blocks and IQ assessed on Binet's test. Wechsler uses it as a subtest on his child and adult scales, and its correlation to Weschler's overall score is also high. For Royer et al., "it serves, then, as a very good measure of general intelligence, as well as of performance abilities" (p. 1474).
The block design task is also saturated in the spatial factor (Vernon, 1952). In more recent models on the hierarchical organization of aptitudes, block tests are also used as indicators of a cohort factor called "visualization" (Cattell, 1971; Snow, Kyllonen, & Marshalek, 1984).
Finally -- and of particular interest to us here -- the block task is a test of development. It is sensitive to early development as well as to aging. Normative data (standardizations) obtained for the French population show that performance on the WISC-R block subtest (1981) improves until the age of 16. The results on the WAIS-R block subtest (1989) indicate a plateau between the ages of 16 and 34, after which performance starts declining.
However, the task we used to analyze subjects' strategies is not, strictly speaking, a Kohs block test, but a modified version of it. For this reason, some arguments demonstrating the theoretical validity of the modified task will be presented later in this article.
1.2. Block design task strategies
In classical research on strategies in the block design task, two major strategies are generally observed: an analytic strategy and a strategy called global, syncretic, or holistic (Beuscart-Zephir & Beuscart, 1988; Goldstein & Scheerer, 1951; Ionescu, Jourdan-Ionescu, & Toselli-Toschi, 1983; Jones & Torgesen, 1981; Schorr, Bower, & Kiernan, 1982). Schorr et al. (1982) also used the term "synthetic" to refer to the global strategy, although this term will be used here to speak of another strategy (see below). According to Schorr, in the analytic strategy, "the displayed design is mentally segmented into units corresponding to block faces, then the blocks are directly placed, one by one, to match each unit." In the global strategy, "the design is viewed as a whole and is not differentiated into units corresponding to block faces; instead, the blocks are manipulated until they match the pattern or seem to `click' with adjoining blocks to reproduce the design ... Subjects using the global strategy would attend to the gestalt appearance of the display section under consideration; they would take a block and rotate it in the construction design until it formed the desired pattern. Subjects following the `analytic strategy' would segment the focused area into its crucial sides, noticing the orientation of the colored edges (e.g. red corner north-west, white corner south-east), choose a two-colored block rotated into a matching orientation, and then place it into the target cell" (Schorr et al., 1982, pp. 479-480). Two operations are thus necessary for solving the block design problem analytically: segmentation of the design and orientation of the bicolored blocks. Subjects who use a global strategy do not carry out these operations; the design is adjusted as it is built, by trial and error. When taken to an extreme, the global strategy may not allow the subject to correctly finish reproducing the design.
In an earlier analysis of video recordings of the behavior of 17-year-olds (Rozencwajg, 1991), we found three different strategies: the two classically observed ones, global and analytic, but also a third strategy we called "synthetic." We were able to observe the synthetic strategy in that study because the designs presented to the subjects were made up of gestalts, i.e. groups of blocks that formed a simple geometric figure such as a triangle or a diamond (see Fig. 1). A gestalt is a form or structure that cannot be reduced to the mere juxtaposition of elements; it has a specific quality that is not found in any of its constituents. The modification of a single constituent can modify the entire form (Reuchlin, 1977). The shapes used here to generate the designs obeyed the gestaltist laws of proximity, similarity, continuity, and symmetry. The gestalts did not vary across designs. To our knowledge, this "synthetic" strategy had never been observed before. Unlike the analytic strategy where subjects proceed by cutting up the figure into a grid of n cells and then placing the blocks by rows or columns, the synthetic strategy involves placing the blocks in an order that conforms to the gestalts in the test design. In the synthetic strategy, then, the placement order is dependent upon the pattern in the design, whereas in the analytic strategy, the order is independent of that pattern.
There is another behavior index that differentiates these two strategies: the number of times the subject refers to the test design. In the synthetic strategy, subjects were found to refer little to the design, doing so essentially between two gestalts, whereas in the analytic strategy, subjects rely often on the design, and in the most extreme cases may even look at it after placing each block.
1.3. Developmental changes in strategies
The purpose of the present study was to contribute to the analysis of developmental changes in the performance of a block design task by examining subjects aged 12, 17, and adults. We shall focus on strategy changes.
To our knowledge, except for the Jones and Torgesen (1981) study on children and the Royer et al. (1984) study on aging adults, few researchers have taken an interest in developmental patterns on the Kohs task, other than the empirical performance changes noted in standardization of intelligence scales where the block design task is a subtest (normative data). In a recent study, Akshoomoff and Stiles (1996) analyzed 4.5- to 8-year-old children's performance and errors on patterns that differed in perceptual cohesiveness, a structural variable defined by Royer and Weitzel (1977), Royer et al. (1984), and Schorr et al. (1982). The authors found differences both as a function of age and as a function of the structural variable.
Jones and Torgesen (1981) studied developmental changes in the solving behaviors of children aged 6, 8, 10, and 16 on Kohs blocks and on the Wechsler version of it (1974). They analyzed behavior on the basis of two indices, a persistence index and block placement order. Their first hypothesis was that persistence index would get better with age. Their second hypothesis was that older subjects would place the blocks in a systematic order no matter what design they were working on. Both hypotheses were invalidated, since there was no age difference on either measure. This lack of progress can be explained in several ways. First of all, since the authors were trying to measure how persistent subjects were in not leaving any mistakes in the final solution, the persistence score was calculated by discarding protocols in which all sequences were correct. They also removed all subjects who were unable to finish within 2 min. Lastly, testing was stopped after two consecutive failures. One can therefore assume that most of the variability was eliminated. Finally and more fundamentally, the lack of developmental changes on the indices may be partly due to the characteristics of the designs used, which contained only four blocks and had few gestalts (the perceptual cohesiveness of the patterns was not even mentioned until the last sentence of the article). The absence of change may also be due to the behavior indices themselves. We shall return to this topic later.
1.4. What designs should be presented to subjects?
We deliberately did not give the subjects designs that were in the earlier versions of Kohs blocks. Two criteria seemed essential: (1) the designs had to be complex enough from the information processing standpoint to still be problems for the subjects, and (2) they had to be based on a cognitive theory rather than on empirical grounds, as in traditional intelligence tests.
The question of what designs to have subjects reproduce is particularly important here, since we are attempting to analyze the development of this task at relatively advanced ages compared to the Jones and Torgesen and Akshoomoff and Stiles studies. First of all, as several authors have now shown, the designs have to be sufficiently complex, i.e. they must look to the subject like a gestalt that is difficult to break down. If the inside edges of the design are conspicuous (Royer & Weitzel, 1977; Royer et al., 1984; Schorr et al., 1982), the segmentation operation is much easier.
The amount of information to process (Royer, 1977) must also be great, which means that the designs should have at least nine blocks. Royer assessed the amount of information by calculating uncertainty (denoted H), which he defined as follows: for a design made up of four unicolored blocks, H = [Log.sub.2][2.sup.4] = 4 bits since each block has two alternatives, red or white; for a design made up of four bicolored blocks, H = [Log.sub.2][4.sup.4] = 8 bits since each block has four alternatives due to the four possible positions; for a design made up of eight bicolored blocks and one unicolored block (which will be the case in our study), H = [Log.sub.2][2.sup.7] + [Log.sub.2]2 = 17 bits. Royer found that subjects -- especially in the "uncued" condition where there was no design segmentation cue (by placing a grid on the design to delineate the blocks) -- had clearly longer solution times starting at H = 9. Solution times continued to increase until H = 18. Moreover, using sufficiently complex designs will allow us to study the block placement order in a more interesting way, and in particular, to differentiate the placement orders by gestalt and by row in a much clearer fashion than if the designs had only four blocks.
More recently, the effects of these structural variables (amount of information and perceptual cohesiveness) were found again by several authors (Akshoomoff & Stiles, 1996; Dickes et al., 1996). Their impact on performance seems undeniable.
However, we hypothesized here that these are not the only variables that account for performance. We have found that even when the amount of information and the perceptual cohesiveness are held constant, performance varies with the very nature of the gestalt to be reproduced (i.e. a triangle or a stripe) (Rozencwajg, 1991; Rozencwajg & Corroyer, 1997). Stripes appear to be much more difficult to construct than triangles or diamonds (see Fig. 1). To our knowledge, the effect of the gestalt variable has never been assessed in studies using the block design task.
The idea that a structural variable might have an effect is also based on Piaget and Inhelder's (1966) study on the development of mental images in children, where the task was to decompose and reconstruct geometric forms. To construct the triangle in this task, small triangles on bicolored blocks had to be put together to form a new triangle that was larger but of the same type. To construct the stripe, however, bicolored blocks containing small triangles had to be assembled to make a stripe, i.e. a different form. According to Piaget and Inhelder, reconstruction difficulties in this case come from a lack of kinetic mobility in the mental image. It is difficult for children to imagine that small triangles can make a figure that looks completely different.
Royer (1977) showed that one of his nine-block striped designs (see Design 14, Fig. 2, p. 37 of Royer, 1977) was particularly difficult for subjects (university students). He explained this result in terms of the very strong perceptual cohesiveness of the design. At the same time, he proposed an explanation for why the diamond was easy (see Design 7, Fig. 2, p. 37): the diamond is unique because the percept is the same even after it is rotated or flipped over to make a mirror image. In other words, the "equivalence set size" (ESS) for this gestalt is minimal.
[FIGURE 2 OMITTED]
In fact, if we analyze the designs used in earlier versions of Kohs blocks, we find out that the persons who designed the intelligence tests, being highly pragmatic, in fact empirically implemented these three complexity factors (quantity of information, perceptual cohesion, nature of the gestalts). The same factors were implemented here, but under experimental control. One could call our version of Kohs blocks the "cognitive version."
2.1. Effect of age on strategies
It was hypothesized that the global strategy would be the predominant one in the youngest children and would become less and less frequent with age.
2.2. Effect of gestalts on strategies
It was hypothesized that the nature of the gestalts in the designs would have an effect on the strategies used. More specifically, we expected designs with stripes to be more likely to trigger the global strategy than designs composed of triangles or diamonds.
2.3. Interaction effect between gestalts and age
It was hypothesized that the gestalts would have a greater effect on the youngest subjects.
The task was performed by 90 subjects from three age groups: late childhood (30 seventh graders, age 12, mean age = 12.68, S.D. = 0.44), adolescence (30 eleventh or twelfth graders, approximately 17 years old, mean age = 17.50, S.D. = 1.22), and adulthood (30 subjects with at least 2 years of higher education, aged 20 to 35, mean age = 27.00, S.D. = 6.30). Each age group had an equal number of males and females, and the three groups were from comparable socioeconomic backgrounds (mostly middle or upper-middle class). All subjects were from the Parisian area.
The materials included six 4-block designs used for familiarization with the gestalts, and four 9-block designs for the experiment proper (see Fig. 2). They differed as to the type of gestalts they contained (see Fig. 1). The nine-block designs (see Fig. 2) were constructed to be of equal complexity (from the standpoint of Royer's parameters), i.e. the amount of information was nearly maximal (H = 17 bits; maximum: 18 bits) and the number of invisible inside edges was 12, making for maximal perceptual cohesiveness.
The task was derived from Kohs blocks: it consists of reproducing a "model" design using colored squares provided. The squares are red, white, or bicolored. In order to study processes and strategies in the most reliable and precise way, we developed a software package called SAMUEL (1) (Corroyer & Rozencwajg, 1995), in memory of Samuel Calmin Kohs (see Fig. 3). The screen was divided into three main parts. On the left, the test design was displayed whenever the subject requested. It remained on the screen until the subject clicked on a block, at which point the design disappeared. Below this, the subject could select a block (an all-red one, an all-white one, or one of the four red-white bicolored ones, each oriented in a different way) and drag it onto the black reconstruction area on the right to reproduce the design. This device made it possible to record all of the subject's manipulations for later analysis.
[FIGURE 3 OMITTED]
The SAMUEL software was designed according to the following specifications.
1. The design was only displayed if and when the subject requested, so that the exact number of times and amount of time the subject needed to look at the design could be recorded.
2. Unlike the traditional version of the task, our "blocks" were in fact squares, being two-dimensional instead of three-dimensional. The purpose of this feature was to eliminate searching for the right side of the block, which in fact is information-processing "noise" (Bonnardel, 1971; Royer, 1977; Rozencwajg, 1991).
3. A bicolored square had four possible orientations. In this task, we chose not to allow the subject to turn the squares to fit with the already placed ones. The reason was that if a subject turned a bicolored square after reaching the reconstruction area, his/her behavior could not be interpreted in an unequivocal way. The turning could either mean that the subject simply did not bother to choose the square with the right orientation, knowing that he/she could do so later, or it could mean that the subject was not capable of mentally representing the orientation of the square. With our device, it was possible to distinguish between subjects capable of forming this mental image, who chose the right orientation from the start, from subjects who were not capable of doing so and had to make several attempts. Subjects were not allowed to place the squares "on end." This eliminated the type of errors observed by Akshoomoff and Stiles (1996) in younger children.
4. One of the problems with the block design task is noise-generated by the search for the right block, which is never in the same place. Here, the squares were always displayed in the same location, no matter what test design was being reproduced. The display order looked random but was in fact arranged so that no gestalts would be generated by the proximity of two bicolored squares.
5. For a 17-in. screen with 800 x 600 pixels, the squares in the designs were 1.5 x 1.5 cm, whereas those used to reproduce the designs were 2 x 2 cm. This difference in size is used in classical versions of the block design task to make it harder to find the solution. To help children for whom this task would be too difficult, Goldstein and Scheerer (1951) changed the size of the designs to make them exactly correspond to the dimensions of the squares.
Before presenting the SAMUEL screen, we held an initial familiarization phase in which the subject had to reproduce a "little man" and some multicolored "flowers." The "little man" was made of nine squares (3 x 3 matrix), with each square showing a different part of the body (head, body, right arm, left arm, right leg, left leg). For the flowers, the first flower item consisted of four flowers in a 2 x 2 matrix (gray, yellow, red, and purple). The second flower item consisted of six flowers lined up (1 x 6 matrix: green, gray, purple, green, red, and white). The last two flower items consisted of 12 and 18 flowers, respectively. This phase was used to teach the subject the procedures for clicking-dragging, design display, and deciding to stop. Familiarization with the "flowers" was achieved using four designs with an increasingly greater number of flowers. This procedure showed subjects that they could display and look at the design as many times as desired, even though the instructions did not explicitly mention this possibility.
The most characteristic feature of the instructions was their "absence." After the familiarization phase (little man and flowers), the experimenter simply said to the subject: "Now, this is the actual test. It's like the flowers; you have to reproduce the design using the squares." The test began with six 4-square designs, which were followed by the four 9-square designs of the experiment proper.
Unlike the instructions for the Wechsler scales, no demonstrations were included. Similarly, unlike the Royer (1977) instructions, the experimenter did not show the subject all possible arrangements of two bicolored squares, although the first six trials did help the subject become familiar with the different gestalts on simple four-square designs. Nor did the instructions (unlike Royer's) tell the subjects how many squares they should use or whether or not the design formed a square. It was up to the subject to analyze the dimensions of the design and to notice that it was indeed a square. In other words, the subject had to "geometrize" a mental image of the design him/herself (the black reconstruction area is composed of 6 x 6 squares).
Finally, there was no stopping criterion or time limit.
3.5. Behavior indices
All of the subject's actions as well as the time taken to execute them (in tenths of a second) were recorded automatically (looking at the design, putting such and such a square with a given orientation in a given place, or removing. Based on the recording of these actions, four indices were defined: segmentation, orientation, design consultation frequency, and square placement order. The values of the indices were determined by a computer program (Altman, Rozencwajg, & Corroyer, 1999).
Our method for calculating the segmentation index was based on Jones and Torgesen's persistence index. Jones and Torgesen used this index to determine whether subjects made corrections as they constructed the design before going on to another location or whether they left errors. This index seems to measure the segmentation operation, because subjects who do not leave behind any incorrectly placed squares have most likely correctly isolated the units in the design, i.e. they have segmented it. With this index, we were interested in the analysis of each sequence, i.e. a series of consecutive actions on the same unit of the design (one of the nine cells). A sequence did not necessarily correspond to a single action, since several consecutive actions could be performed on the same location. One point was given for a correct placement, one point was taken away every time the subject removed a correct square or left an incorrect square, and one point was added every time the subject corrected an error. The total number of points was then divided by the number of sequences. Let us illustrate with the following sequence: the subject puts an incorrect square in a given place and then takes it off and puts a correct square in the same place. This sequence is worth one point since the subject stayed in the same location (the same cell). The index varied between 0 and 1 (negative scores were changed to 0; a score of 1 meant that in the end, all cells were correctly filled, even if the subject took several tries for certain locations). Given that the strategies could not be calculated unless all indices were between 0 and 1, the segmentation score was changed to zero whenever it turned out negative. This scoring method was used by Thurstone, for example, on the spatial subtest of the PMA. A negative score is in fact very rare (even zero is hardly ever observed), so using this method only slightly reduced the variability of this index.
The orientation index is a finer measure than the segmentation index since it counts the number of trials needed to correctly fill a cell. This index represents the extent to which the subject constructed the design using trial and error or was able to correctly fill all cells on the first try. In this case, we were interested in each action and not in the sequence of actions. For each cell in the design, then, we would obtain a ratio of 1/1 if the cell was correctly filled on the first try, a ratio of 1/2 if the subject took two tries, a ratio of 1/3 for three tries, etc. The different ratios were added and then divided by the total number of tries. If a cell contained an incorrect square in the end, regardless of the number of tries, the ratio for that cell was 0. For example, for a four-square design where the subject took one, three, and two tries, respectively, to correctly fill the first three cells, and filled in the last cell with the wrong square in a single try, the calculation would be (1/1 + 1/3 + 1/2 + 0/1)/(1 + 3 + 2 + 1) = 0.26. The orientation index varied between 0 and 1 (0 if the subject ended up with only incorrectly filled cells, 1 if all cells were correctly filled on the first try).
3.5.3. Placement order
The placement order was scored once the squares were in their final locations, i.e. without taking trials and errors into account. For example, a square that had been put in one place and then taken away did not enter into the placement order index. Because of this, this index was relatively independent of the segmentation and orientation indices; only relatively though, since squares that did not belong to the right design were not counted. In other words, when subjects reproduced other designs, their placement order score was 0.
The placement orders produced by the subjects were compared to two "ideal" orders, placement by rows or columns and placement by gestalts. (1) The row or column placement index ranged between 0 and 1:0 if none of the rows (or columns) were filled in consecutively, 0.33 if 1 of the three rows (or columns) was filled in consecutively, 0.67 if two of the rows (or columns) were filled in consecutively, and 1 if all three rows (or columns) were filled in consecutively. (2) The gestalt placement index also ranged between 0 (no gestalts were constructed) and 1 (if all cells were systematically filled in by gestalt). The gestalt score was the sum of the points obtained for constructing an elementary gestalt -- a triangle or a stripe (see Fig. 1) -- and the points obtained for the construction of more complex gestalts. For example, in Fig. 4, the subject received four points, two for stripes 3-4 and 7-8 and two for the two large stripes 1-2-3-4-5 and 6-7-8. The sum was then divided by the highest possible score, which depended on the configuration of the design. For the design in Fig. 2, for example, the highest possible score was 8, which could be obtained by making six elementary stripes and two large stripes. The final score in our example was 4/8.
[FIGURE 4 OMITTED]
3.5.4. Design consultation frequency
The design consultation frequency was calculated by dividing the number of times the design was displayed, by the total number of actions (an action was defined as the placement or removal of a square). This index varied between 0 (if the subject looked at the design only once) and 1 (if the subject looked at the design before each action).
3.5.5. Total solving time
Total solving time was also noted. This index was not used in determining the strategies but was useful for studying the theoretical validity of SAMUEL.
The various analyses described below were run on DS3, Statistica (StatSoft France, 1996), and Var3 softwares (VAR3, 1988).
4.1. On the theoretical validity of SAMUEL
Before presenting the results, let us verify that using a computerized version of the task did not generate a measurement error. As Pellegrino, Hunt, Abate, and Farr (1987) said, computerization seems to be the inevitable way to gain access to the cognitive processes underlying intelligence tests. But we must nevertheless make sure that using the computer does not skew the results.
Note firstly that all 90 subjects were familiar with computers. However, some may have been more skillful than others at using it, so an extensive familiarization phase was held. Familiarization took place in two parts: the "little man" and the "flowers." For the youngest children (12-year-olds), we calculated the correlation between solving time on the little man during familiarization and mean solving time during the test phase. The correlation with the orientation index during the test phase was also calculated, since this index is dependent upon the number of trials the subject needed and is thus highly correlated with total solving time. The correlation of little-man familiarization time was -.02 with mean test phase solving time and .09 with the orientation index during the test phase. We also took the solving time on the six-flower design and calculated the correlation with total test phase solving time and with the orientation index during the test phase. The six-flower model was chosen because it seemed highly unlikely that different strategies would be used on this design. For the 12- and 18-flower items, the subjects differed in their ability to memorize the flowers (which may have also been true for the six flowers but this factor would not be as influential in any case). In other words, we chose the flower design that involved the fewest possible complex processes. The correlation of the six-flower solving time with test phase solving time was .23 (ns); with the orientation index, it was -.05. These correlations are weak, especially compared to the correlation between the total solving time during the test phase and the orientation index, which was high (- .74). We can therefore conclude that solving on SAMUEL was not linked to a higher skill level on the computer.
In the Introduction, we stated that the traditional psychometric version of Kohs blocks was saturated in factor g and in the spatial factor, and that it was also a test of development. The results of this paper will demonstrate that the strategy used evolve with age, which in itself argues for the theoretical validity of this version of the test. However, one might see this as a specific kind of development, and thereby consider SAMUEL to be a computerized task derived from Kohs blocks that lacks the general validity which would classify it in the intelligence test category. One simple method that might help us answer this question would be to calculate correlations. If SAMUEL is indeed a version of Kohs blocks and thereby retains its generality, there should be a strong correlation between solving time on the SAMUEL test and the standard score on the WAIS block test, for example. We conducted a study on a small sample of 12 subjects approximately 17 years old to compare the SAMUEL task and the WAIS performance scale. The correlation between IQ measured by the WAIS scale and solving time on SAMUEL was -.63; the correlation between the Kohs block subtest of the scale and SAMUEL solving time was -.70. Other validity studies comparing SAMUEL with various intelligence tests were conducted. In particular, out of 50 seventh graders (approximately age 12), the correlation between SAMUEL solving time and spatial aptitude on Thurstone's PMA spatial subtest was -.56; it was .45 between the SAMUEL orientation index and spatial aptitude. For another population of 39 seventh graders, the correlation between solving time on SAMUEL and Raven's PM38 test score was .44. The correlation between the PM38 and the orientation index was even better at .58. The PM38 test is widely known as a good indicator of a general or fluid intelligence factor (Horn and Cattell's Gf) (Hunt, 1990; Lansman et al., 1982). In another study out of 34 fifth graders (approximately age 10), the correlation between the SAMUEL orientation index and the Similitude subtest of the WAIS verbal scale was .66.
These correlations seem satisfactory to us and show that SAMUEL is not merely a specific problem-solving task but is indeed an intelligence test with a larger scope.
4.2. Age-related changes in behavior indices
To measure the magnitude of the effects from a descriptive standpoint, we calculated calibrated effects (Corroyer & Rouanet, 1994; Rouanet, 1996). Basically, this index is the ratio of the between-variance to the within-variance. The higher it is, the greater the difference of means between the groups compared to the individual variations within each group. A calibrated effect is called small when the ratio is below 1/3 and large when it is above 2/3.
CE = [S.sub.between]/[S.sub.within] with [S.sup.2.sub.between] = (k/k - 1) [V.sub.between] and [S.sup.2.sub.within]
= (n/n - k) [V.sub.within]
where k is the number of groups and n is the total number of subjects. The usual F ratio of the analysis of variance (ANOVA) may be expressed as a function of the sample size and the calibrated effect: F = (n/k)(C[E.sup.2]).
4.2.1. Segmentation index
The mean value of this index was high (0.84). Fig. 5 shows that the low segmentation values were obtained mainly by the youngest subjects. The mean increased with age (see Table 1) but the spread decreased. The magnitude of the overall effect of age, assessed by a calibrated effect (CE) (Rouanet, 1996), was medium (CE = 0.47). The ANOVA yielded a significant result [F(2,87) = 6.77, P<.05].
[FIGURE 5 OMITTED]
4.2.2. Orientation index
The mean was 0.72. Table 1 shows that this mean also increased with age]. The increase was considerable (CE = 0.76). The F test was significant [F(2,87) = 17.55, P<.001], although the spread decreased with age. Fig. 6 shows that this index was always above 0.60 for the oldest subjects, whereas the youngest subjects exhibited a great deal of variability.
4.2.3. Design consultation frequency
The design consultation frequency index averaged 0.42 (meaning that subjects looked at the design nearly every other action). This index did not vary much across age. We shall see below that in fact, it differed substantially on another factor: the solving strategy.
4.2.4. Row/column placement
The mean was 0.47, which means that the subjects constructed an average of half of the figure by rows or by columns. This index did not vary much across age [CE = 0.32; F(2,87) = 2.29, P = .055].
4.2.5. Gestalt placement
The mean value of this index was 0.63, which means that subjects constructed an average of nearly 2/3 of the figures based on gestalts. It thus seems that the gestalt components of the designs were more salient than the row and column components, at least from the standpoint of the indices and types of designs used here. The difference between the groups was also very small (CE = 0.15; F<1).
The methodology was checked by making sure the numerical variations in the placement order indices were in fact determined by a given cognitive representation of the design (rows/ columns vs. gestalts) and not by chance. To do so, we drew 90 placement orders at random and then calculated the row/column placement index and the gestalt placement index for each design (note that the placement order was scored once the squares were in their final locations, i.e. without taking trials and errors into account). If the indices are indeed relevant, then their mean values on the random orders should be low. The results were as follows: for the four designs averaged, the random gestalt placement index was low (0.21), while the observed gestalt placement index was 0.63. The random row/column placement index was also low (0.19) whereas the observed row/column placement index was 0.47. Clearly, then, the subjects' placement orders did rely on a solving process that corresponded to a specific representation of the design.
In order to study strategy changes with age, we need a method for combining the various indices. Below, we first analyze the relationships between the indices and then present a method for combining them.
4.3. Relating the indices to each other
The design consultation frequency and the gestalt type of placement are negatively correlated (r = -.54). This negative correlation indicates that when subjects constructed the design by gestalts, they did not look at it very often. The segmentation and orientation indices are highly correlated (r = +.78). The row/column placement index is correlated to both of the above indices (r = +.45 and r = +.42). Placement by rows or columns thus appears to be linked to efficient segmentation and orientation. The segmentation and orientation indices were only weakly linked to the design consultation frequency index (r = +.19), because the design consultation frequency index is linked to whether the strategy used is analytic or synthetic. They were not linked at all to the gestalt placement order (r = -.11), because some subjects construct the design by gestalts without errors (synthetic strategy) and others construct the design by gestalts using trial and error (global strategy).
4.4. Identifying the strategies
There are several possible ways of identifying the strategies implemented by the subjects. In an earlier study (Rozencwajg, 1991), we conducted a multiple correspondence analysis on the cross-tabulation of the subjects and the indices. This was followed by an automatic classification procedure that divided the subjects into different groups on the basis of the strategy used.
For the present study, another method was chosen. Based on previous research, we now know the characteristics of the various strategies (analytic, synthetic, and global). We know, for example, that a synthetic strategy is characterized by high segmentation and orientation indices, a low design consultation frequency, and a gestalt placement order. An analytic strategy also has high segmentation and orientation indices, but unlike the synthetic strategy, it has a high design consultation frequency and a row/column placement order. Finally, a global strategy is characterized above all by low segmentation and orientation indices.
It is thus possible to define a theoretical profile for each strategy. These theoretical profiles are given in Table 2.
For the synthetic strategy, the theoretical value for the design consultation frequency was set at 0.20. Subjects who used this strategy referred to the gestalts in the design and thus only looked at the design between two gestalts. Since the designs usually contained two gestalts, the theoretical value should therefore be two consultations for every nine actions, or approximately 0.20.
For the analytic strategy, the theoretical value of the gestalt placement index was set at 0.40. Given the structure of the designs, subjects who systematically constructed the design by rows or by columns reproduced -- sometimes even involuntarily -- the gestalts that we called elementary (stripes and triangles). This gave them a nonzero gestalt placement index (0.50 for Design 7, 0.38 for Design 8, 0.43 for Design 9, and 0.38 for Design 10), making for an average of approximately 0.40.
For the global strategy, defining the theoretical values was not as straightforward. For the segmentation index, the theoretical value was set at 0.50, which was the mean observed in the previous study (Rozencwajg, 1991) for subjects who used this strategy. This value would be obtained for a four-square design when the subject made a single error that was later corrected: (3 - 1 + 1/2)/5 = 0.50.
For the orientation index, a value of 0 is highly improbable (the figure produced in the end would have to be entirely incorrect; cf. Fig. 6). A value of 0.50 for a four-square design with two cells not correctly filled on the first try would be (1/2 + 1/2 + 1 + 1)/6 = 0.50.
We used a theoretical value of 0.50 for the design consultation frequency index also. This value would indicate referral to the design every two actions. Since the consultations depended upon the succession of adjustments, it is difficult to determine a theoretical value. In our previous study, subjects who used a global strategy scored in between those with an analytic strategy and those with a synthetic strategy.
For the gestalt placement order, subjects with a global solving strategy tried to reproduce the gestalts, but they did so by trial and error. A theoretical value of 1 would not take into account a typically global solving mode consisting, for example, of reproducing gestalts that resemble those in the design but are larger. Nor would it take into account the 3 x 3 matrix or the dimensions of the white spaces between the red gestalts (the most frequent case). A theoretical value of 0.50 would correspond to constructing half of the figure using the gestalts in the design.
Once these theoretical profiles were defined, we could calculate the distance, for each subject, between his/her profile and each of the theoretical profiles. A classical distance, the "city-block" was used. Each subject thus had three new scores, representing his/her distance from (1) the theoretical analytic strategy, (2) the theoretical synthetic strategy, and (3) the theoretical global strategy. To illustrate, the distance calculations for a 12-year-old subject on Design 8 (see Table 3) are given below.
Distance from the theoretical global strategy:
|1 - 0.50| + |0.52 - 0.50| + |0.14 - 0.50| + |0.75 - 0.50| + |0.67 - 0| = 1.80
Distance from the theoretical synthetic strategy:
|1 - 1| + |0.52 - 1| + |0.14 - 0.20| + |0.75 - 1| + |0.67 - 0| = 1.46
Distance from the theoretical analytic strategy:
|1 - 1| + |0.52 - 1| + |0.14 - 1| + |0.75 - 0.40| + |0.67 - 1| = 2.02
These indices can be used as they are, but they can also serve to characterize each subject according to the strategy closest to him/her (in the above example, the subject would be classified in the synthetic strategy category). Using this method, we assigned each subject a strategy category for each design, and for all designs pooled (average of the distances on the four designs).
The distribution of strategies thus obtained for all 90 subjects was as follows: 39 subjects (43%) used a synthetic strategy, 22 (24%) used an analytic strategy, and 29 (32%) used a global strategy. Of course, although this distribution is dependent upon the theoretical strategy profiles defined above, our main goal was to analyze the extent to which the distribution changed with age.
4.5. Strategy changes with age
Remember that the scores on the three strategies are distances, so the lower the score, the closer the subject to the theoretical strategy.
4.5.1. Synthetic strategy changes
The mean was 1.52. Table 4 shows a mean decrease with age, but not a large one (CE = 0.32). The F test approached significance [F(2,87) = 2.99, P = .055]. In contrast, the 12-year-olds were significantly different from the 17-year-olds and adults taken together [F(1,87) = 5.84, P<.02].
4.5.2. Analytic strategy changes
The mean was 1.82. Table 4 indicates a moderate mean decrease with age (CE = 0.48). The overall F test was significant [F(2,87) = 6.79, P<.002], as was the comparison between the 12-year-olds, and the 17-year-olds and adults pooled [F(1,87) = 11.14, P<.001]. The spread was greater at age 12.
4.5.3. Global strategy changes
The mean was 1.62. Table 4 shows a moderate increase in the means with age (CE = 0.47). The overall F test was significant [F(2,87) = 6.53, P<.002], as were the 12-year-old to 17-year-old/adult comparison [F(1,87) = 6.10, P<.02] and the 17-year-old to adult comparison [F(1,58) = 6.76, P<.01].
4.5.4. Age-related changes in the strategy distribution
We shall now present the strategy changes by comparing the subject distributions as a function of age and preferred strategy (defined as the one with the shortest distance).
We can see that there was an age-related change in strategy use. At the age of 12, the global strategy was the most frequent (16 subjects out of 30 or 53%). The frequency of this strategy decreased markedly with age, leaving only 30% of the 17-year-olds (9 out of 30) and 13% of the adults (4 out of 30) to still be using it. For the 17-year-old and adult subjects, the synthetic strategy became the most frequent (47% and 53% for these two age groups, respectively). In summary, in line with our initial hypothesis, the subjects' strategies indeed changed with age. The chi-square test was significant ([chi square] = 11.25, df = 4, P<.02). Analysis of the relative contributions to chi-square (see Table 5) confirmed that the change was essentially caused by a decrease in the frequency of the global strategy, from age 12 to adulthood, to the benefit of the synthetic and analytic strategies, both of which increased with age.
4.6. Effect of type of gestalt on strategies
One of our hypotheses pertains to the effect of the type of gestalt on the strategy used. Remember that Designs 7, 8, 9, and 10 were presented to the subject in that order, and that Designs 7 and 9 consisted mainly of diamonds, while Designs 8 and 10 were made up of stripes. For each design, the type-of-gestalt effect on the synthetic, analytic, and global strategy scores (distances) are presented below for all ages pooled (see Table 6).
For the synthetic score, there was a significant main effect [F(3,261) = 22.64, P<.0001] of the design factor. More specifically, the striped designs (Designs 8 and 10) were found to be more difficult than the diamond designs (Designs 7 and 9) [F(1,87) = 60.78, P<.001]. There was no difference between Designs 7 and 9 on the overall synthetic score (F<1) but there was a difference between designs 8 and 10 [F(1,87) = 10.12, P<.002]. This result can be interpreted as a learning effect between the processing of two striped designs (both difficult).
For the analytic score, we again observed a significant main effect [F(3,261) = 4.11, P<.01] of the design factor. Designs 8 and 10 (stripes) were always more difficult than Designs 7 and 9 (diamonds) [F(1,87) = 12.25, P<.001]. The differences between designs of the same type were nonsignificant, whether diamonds (Designs 7 and 9) or stripes (Designs 8 and 10) were involved.
For the global score, there was a significant main effect [F(3,261) = 24.30, P<.0001] of the design factor. Designs 8 and 10 (stripes) were more difficult than Designs 7 and 9 (diamonds) [F(1,87) = 77.70, P<.0001]. The global score difference between Designs 7 and 9 was not significant, but it was so between Designs 8 and 10 [F(1,87) = 4.93, P<.03]. Once again, this result can be interpreted as a learning effect between the two designs composed of stripes.
4.6.1. Strategy distribution by design
The same question can be analyzed by considering the distribution of the strategies as a function of design (see Table 7).
We can see that striped designs (Designs 8 and 10) were more often reconstructed using the global strategy than were diamond designs (Designs 7 and 9). Designs with diamonds were more often solved using the synthetic strategy than the analytic one. This result supports the idea that the gestalts were more salient than the 3 x 3 matrix. The chi-square test calculated on this table was significant ([chi square] = 42.99, df = 6, P<.0001).
In summary, these analyses clearly confirm the conclusion, in line with our hypothesis, that stripes affect the strategy used: designs with stripes are more difficult, and more often lead to the use of a global strategy.
4.7. Effect of type of gestalt on strategies, by age
A gestalt effect was found for every age group (see Table 8): the scores were always better for diamond designs (7 and 9) than for striped ones (8 and 10).
For the distance to the synthetic strategy, the difference between age 12 and the older ages was nonsignificant on diamond designs (Designs 7 and 9), whereas it was significant for striped designs (Designs 8 and 10) [F(1,87) = 8.82, P<.004]. The stripe effect was thus more pronounced at the younger ages, while the age groups were not significantly different on the diamond designs. The differences between the 17-year-olds and the adults were nonsignificant.
For the distance to the analytic strategy, the difference between the 12-year-olds and the older subjects was significant for both the diamond [F(1,87) = 5.68, P<.02] and striped [F(1,87) = 12.45, P<.001] designs. The difference between the 17-year-olds and the adults was significant on the striped designs [F(1,58) = 4.996, P<.03].
For the distance to the global strategy, there was a significant difference between the 12-year-olds and the older subjects for the diamond designs [F(1,87) = 8.63, P<.04] but not for the striped ones. The difference between the 17-year-olds and the adults was significant for the diamonds [F(1,58) = 7.73 (1,58), P<.01] but not for the stripes. This result means that the global score increased on striped designs at all ages.
4.7.1. Strategy distribution by age and design
The same question can also be examined by looking at the strategy distribution as a function of age and design (see Table 9).
The greatest contributions to chi-square occurred for the 12-year-old subjects, who used the global strategy the most on Designs 8 and 10, the striped ones ([chi square] = 73.25, df = 22, P<.0001). As hypothesized, there was not only an age effect but also a stripe effect on the strategies used: striped designs were more difficult, and more often led to the use of a global strategy, especially for the youngest subjects.
5. Discussion and conclusion
What strategy changes were observed here in block-design problem solving? When the indices were considered separately, we saw that the segmentation index evolved to a moderate extent. Remember that Jones and Torgesen found no strategy changes in subjects aged 6, 8, 10, and 16. For the orientation index, we found a strong age effect. No individual age differences were found on design consultation frequency or placement order, although these indices are tightly linked to the strategy used.
Regarding the strategies -- which give us an overall picture of the different behaviors observed -- we have seen that they did in fact evolve with age. The global strategy was more frequent in the younger subjects. This finding is consistent with the fact that the block design task is considered to be a test of development. Our findings thus differ from those obtained by Jones and Torgesen. The discrepancy can be explained not only by the fact that our designs were more complex (in terms of amount of information and perceptual cohesiveness), but also by the fact that we controlled the types of gestalts in the designs. Another tactic that allowed us to observe the developmental change was the fact that the strategies were not only measured by the segmentation and placement order indices, as in the Jones and Torgesen study, but also by an orientation index, a refinement of the segmentation index.
Analysis of the effects of the designs and their gestalt components pointed out the greater difficulty of striped gestalts, no matter how old the subjects were, with the most difficulty being experienced by the youngest subjects. The 12-year-olds made stripes that were too long or too wide, so the reproduced design was not a square of the right size (3 x 3 matrix). In contrast, the presence of the diamond facilitated the solving process. It gave the subjects the proportions of the design.
How can we account for this phenomenon? Design uniformity does not appear to be a likely explanation (Royer, 1977). Indeed, note that the (more difficult) striped designs involved fewer possible orientations of the bicolored square. For example, in Design 8 (see Fig. 2) the bicolored square was only in two different positions, whereas in Design 7, there were four such positions. We can see, then, that this factor had little effect on the solution process. The complexity appears to stem more from the particular configuration generated by two bicolored squares next to each other i.e. either a triangle or a stripe. This finding supports the idea that the difficult part of the block design task is indeed breaking down the design into units, and that the subject performing this cognitive operation is "tricked" or "misled" by gestalts with certain shapes. The nature of certain spatial relationships between the squares makes them harder to separate. Another explanation for the easiness of the diamond relative to the stripe may lie in the perceptual phenomenon of field dependence-independence (Huteau, 1987; Witkin & Goodenough, 1981). In the case of the triangle (see Fig. 1), the subject must place the two bicolored squares together horizontally, and the design must be perceived horizontally. In contrast, in the case of stripes, the subject must always join the two bicolored squares horizontally, but the design must be visually perceived with a diagonal orientation, which does not help. The two directions are thus conflicting and this may account for why the stripe is so difficult. A recent review of mental image development (Bialystok & Jenkin, 1998) confirms this hypothesis. The stripe is more difficult to construct because the standard axes used in spatial representations (horizontal and vertical) are not congruent with the diagonal direction of the stripe. Note, however, that this spatial stimulus is complex on the average; certain subjects do not experience any difficulty. Either these subjects are not disrupted by the perceptual conflict (this is true of field-independent subjects), or they literally "divide up" the difficulty by mentally placing a standard horizontal-vertical grid on the design, cutting up the stripes.
[FIGURE 1-2 OMITTED]
In conclusion, it seems that the block design task, initially designed by Kohs in 1923, is still providing fruitful data. But also and more generally, the study of strategies in intelligence tests may help us gain insight, for a given task, into the information-processing operations now being studied in cognitive psychology. At the same time, tests that have proven their "adaptability" to various theoretical frameworks should be reserved for analyzing intellectual functioning. Initially designed as a general intelligence test and still employed as such in Wechsler's or Alexander's batteries, Kohs blocks can be applied to the analysis of cognitive processes as they are conceptualized today, in an information-processing framework. The Kohs block task seems to be a very robust intelligence test, perhaps because it requires "the ability of an individual to select the functions to be used in attacking a particular problem, and to coordinate the execution of those that are selected" (Hunt, 1980, p. 457).
Table 1 Segmentation index and orientation index by age Mean (S.D.) Mean (S.D.) Age n segmentation orientation 12 30 0.76 (0.23) 0.59 (0.22) 17 30 0.85 (0.13) 0.74 (0.12) 25 30 0.91 (0.10) 0.84 (0.11) Mean N=90 0.84 (0.17) 0.72 (0.19) Table 2 Theoretical profiles for each strategy Design consultation Index strategy Segmentation Orientation frequency Synthetic 1 1 0.20 Analytic l 1 1 Global 0.50 0.50 0.50 Gestalt Row/ Index strategy order column order Synthetic 1 0 Analytic 0.40 1 Global 0.50 0 Table 3 Profile of a 12-year-old subject on Design 8 Design consultation Segmentation Orientation frequency Observed strategy 1 0.52 0.14 Gestalt Row/ order column order Observed strategy 0.75 0.67 Table 4 Synthetic, analytic, and global strategy by age Mean (S.D.) Mean (S.D.) Mean (S.D.) Age n synthetic analytic global 12 30 1.66 (0.39) 2.11 (0.68) 1.52 (0.27) 17 30 1.47 (0.36) 1.80 (0.45) 1.58 (0.30) 25 30 1.43 (0.40) 1.57 (0.55) 1.77 (0.27) Mean N=90 1.52 (0.40) 1.82 (0.61) 1.62 (0.30) Table 5 Distribution of subjects by age and by strategy Synthetic strategy Analytic strategy Age 12 9 (-10.95) 5 (-6.60) Age 17 14 (+0.68) 7 (-0.13) Adult 16 (+6.16) 10 (+8.62) Total 39 (17.79) 22 (15.36) Global strategy Total Age 12 16 (+36.90) 30 (54.45) Age 17 9 (-0.41) 30 (1.23) Adult 4 (-29.54) 30 (44.32) Total 29 (66.85) 90 (100) Relative contributions to [chi square] in parentheses, with the sign representing the difference between the observed values and the theoretical values. Table 6 Strategy means by design, for all ages pooled Design 7 Design 8 Mean (S.D.) Mean (S.D.) Synthetic strategy 1.32 (0.60) 1.83 (0.63) Analytic strategy 1.74 (0.86) 1.92 (0.89) Global strategy 1.83 (0.45) 1.52 (0.49) Design 9 Design 10 Mean (S.D.) Mean (S.D.) Synthetic strategy 1.33 (0.50) 1.58 (0.57) Analytic strategy 1.69 (0.66) 1.95 (0.80) Global strategy 1.76 (0.48) 1.38 (0.39) Table 7 Distribution of subjects, by design and by strategy (signed relative contributions to [chi square] are in parentheses) Design strategy Design 7 Design 8 Design 9 Synthetic 48 (+6.12) 25 (-10.34) 51 (+10.34) Analytic 28 (+1.55) 26 (+0.39) 24 (+0.00) Global 14 (-16.28) 39 (+10.05) 15 (-14.04) 90 (23.95) 90 (20.78) 90 (24.38) Design strategy Design 10 Total Synthetic 28 (-6.12) 152 (+32.93) Analytic 18 (-3.49) 96 (+5.43) Global 44 (+21.27) 112 (+61.64) 90 (30.88) 360 (100) Table 8 Mean score for the different age groups by strategy, for designs with diamonds (D7 and D9) vs. designs with stripes (D8 and D10) Strategy Synthetic Design D7 D8 D9 D10 Age 12 1.32 2.08 1.49 1.74 Age 17 1.32 1.75 1.30 1.51 Adult 1.33 1.68 1.21 1.50 Strategy Analytic Design D7 D8 D9 D10 Age 12 2.16 2.25 1.73 2.31 Age 17 1.58 1.87 1.77 1.96 Adult 1.48 1.65 1.56 1.57 Strategy Global Design D7 D8 D9 D10 Age 12 1.65 1.51 1.63 1.28 Age 17 1.79 1.45 1.70 1.36 Adult 2.04 1.60 1.94 1.50 Table 9 Cross-tabulation of strategies, designs, and ages (signed relative contributions to [chi square] in parentheses) Age x Design Age Age Age Age Age strategy 12 D7 12 D8 12 D9 12 D10 17 D7 Synthetic 17 (+2.0) 6 (-4.8) 13 (+0.0) 6 (-3.5) 16 (+1.2) Analytic 4 (-2.7) 6 (-0.7) 10 (+0.7) 3 (-4.3) 11 (+1.5) Global 9 (-0.0) 18 (+11) 7 (-0.8) 20 (+16.6) 3 (-5.9) 30 (4.8) 30 (16.5) 30 (1.5) 30 (24.4) 30 (8.6) Age x Design Age Age Age Adult strategy 17 D8 17 D9 17 D10 D7 Synthetic 8 (-2.4) 19 (+4.3) 9 (-1.5) 15 (+0.6) Analytic 9 (+0.2) 6 (-0.7) 6 (-0.7) 13 (+4.3) Global 13 (+2.0) 5 (-2.8) 15 (+4.7) 2 (-7.9) 30 (4.5) 30 (7.8) 30 (6.8) 30 (12.7) Age x Design Adult Adult Adult strategy D8 D9 D10 Total Synthetic 11 (-0.3) 19 (+4.3) 12 (0.0) 152 (24.9) Analytic 11 (+1.5) 8 (0.0) 9(+0.2) 96(17.4) Global 8 (-0.3) 3 (-5.9) 9(0.0) 112(57.8) 30 (2.1) 30 (10.2) 30 (0.2) 360 (100)
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Paulette Rozencwajg (a),*, Denis Corroyer (b)
(a) Laboratoire Cognition et Developpement, Equipe Cognition et Differenciation, Universite Paris V Rene Descartes, CNRS UMR 8605, 71 Avenue Edouard Vaillant, 92774 Boulogne-Billancourt Cedex, France
(b) Laboratoire Psychologie Environnementale, Universite Paris V Rene Descartes, CNRS ESA 8069, Boulogne-Billancourt Cedex, France
* Corresponding author.
E-mail address: email@example.com (P. Rozencwajg).
Received 23 September 1998; received in revised form 20 July 2000; accepted 15 October 2000
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|Author:||Rozencwajg, Paulette; Corroyer, Denis|
|Date:||Jan 1, 2002|
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