Sequentially Ordered Steps: Single Step

The smallest number of steps in Rule Applying problems is equal to one. This implies two states: the initial state and the end state. In the case of this type of problems, usually, the subject's task simply is to apply the rule and to check whether a correct end state is obtained or not. Well-known examples are:

Three Dimensional Space, GATB

Tool Matching, GATB

Form Matching, GATB

Figuren Aanvullen, PMA

Gedraaide Figuren, PMA

GATB is an abbreviation for the General Aptitude Test Battery, form B-1002 (United States of Labor, 1962) and PMA is an abbreviation for the Primary Mental Abilities (Thurstone, 1938). The tests 17-29, which were used by Thurstone (1938) in his search for the primary mental abilities also belong to this type of problems. The question arises why Single Step problems can be difficult. The above mentioned tests all require the manipulation of objects in in imaginitive or virtual space. These tests become extremely easy when the problems are presented in such a way, that manipulation in physical space would be possible. For example, in the case of the test Three-dimensional Space, one could offer the unfolded figures as physical objects, using paper or cardboard, and allow the subject to fold the figures physically along the dotted lines. In the case of the Form Matching and Tool Matching one could copy the figures into separate physical objects, using plastic or some other kind of solid material, and allow the subject to match the figures physically. In the case of test Figuren Aanvullen from the PMA one could copy the figure parts into separate physical objects, and allow the subject to lay the figure parts against each other physicallly. Similar methods can be used in the case of the other tests. One may expect that, when these tests are administered in this way, even nursery school children are able to solve the test items involved. In the actual tests, the manipulation of objects in real space is not allowed. The solution can only be found by manipulation in virtual space. What is needed in order to respond to the items could be described as spatial visualization. However, that is not necessarily, what is measured by the test. If sufficient time is given, and if the subjects are allowed to practice before the actual administration of the test, then these problems can always be solved. Under the latter circumstances spatial visualization is still needed, but certainly not measured. The problem with these kind of tasks can be desribed as follows. It is quite clear what should be done, it is not quite clear how it should be done. Especially, in the beginning of the test the subject might feel quite uncomfortable not knowing how to apply the rule or what kind of strategy to use. As is always the case with rule applying problems, he must know that the only way to find the correct answer is simply to start and to persist. The only way to get used to the task is to look for possible strategies, which, after some practice, can be performed in an automatic way. Naturally, it will take some time before a certain stage of practice is attained. However, when the task is overlearned, applying the rule does not seems very difficult anymore. Whether a rule is difficult to apply is a matter of being familiar with the rule or not. Familiarity may be already be obtained in the past. In that case the task will be experienced as easy. When familiarity must be obtained during the test, then the task will be experienced as difficult. In the actual tests only the number of correct answers is registered. That is no information is obtained about the process of learning, which is unavoidable in order to get a satifactory test score (see also further down).

The number of possible initial states may be more then one and part of the problem may be, to think up all possible initial sates. A typical example is the Match Problems Test. This test was originally proposed by Wilson, et al. (1954) and consisted of 12 items. Examples are given in Guilford (1967, p. 153). Most items are composed of collections of adjacent squares with each side representing a removable matchstick. The problem is to remove a given number of matchsticks, leaving a certain amount of complete squares and no excess lines. The initial state consists of the intact collection of adjecent squares and the end state consist of the collection of adjecent squares with a cerain number of matchsticks removed. The subjects has to check how many squares have been left and whether this number is equal to the number which was asked for. This has repeatedly to be done for all possible collections with the required number of matchsticks removed. The latter is a combinatorial problem. When the subject is not familiar with combinatorics, he may easily overlook some of the possibilities. However, this is a matter of experience and patience, and not of any ability, whatsoever. Sometimes the solution requires the unusual resort to a square of larger-than-normal size. A later version (see Berger et al., 1957) included problems each of which could be solved in four different ways. In another version, which was also proposed by Berger et al. (1957), the multiple-solution principle was used again, but problems were also emphasized that could have unusual solultions, with final squares of different sizes, one within another, and overlapping each other. Functional Fixedness might play a role here. For example, the subject could think that only squares of normal size (consisting of four matchsticks) are allowed. But, again, this is not a matter of some quantitative ability, but a matter of idea. The most important reason for the emergence of functional fixedness might be test instruction. It is not explicitly mentioned in the instruction, that squares of larger-than-normal size should be taken into account. In that case one should not be surprised when the subjects disregards these possibilities. In this respect it might be worthwhile to mention a well-known riddle, in which one has to build four equilateral triangles, each triangle made up by three matchsticks, by using six matchsticks. In trying to solve this riddle, subjects implicitly assume that the solution should be given in a two-dimensional space. Naturally, they conclude that there is no solution.