Rule Finding Problems

In Rule Finding problems all information available is organized according to some underlying rule and the problem is to derive the rule from the available information. This is always done by assuming or hypothizing a certain rule and subsequently applying the rule in question in order to check whether the available information (data) is consistent with the rule. If the data are inconsistent with the rule, a new rule must be hypothizied, which again must be checked against the data. The process of Rule Finding always consists of two stages: rule hypothizing and rule checking, which is equivalent to Rule Applying. The stage of rule hypothizing may also be called the stage of induction, because a rule is introduced (the latin verb 'inducere' means to introduce). Sometimes a little reasoning is required in order to discover how the rule works in the case at hand. This process of reasoning may be called deduction. Typical examples of rule finding problems are the items of the IQ-test published on INTERNET by Neeteson Internet Productions Inc. Go there if you want to try solving these problems on your own. The following intelligence tests also consist of items, which may be described as rule finding problems.

Circle Reasoning (Blakey, 1941, Guilford, p. 96)

Standard Progressive Matrices, Sets B-E (Raven, 1990)

Advanced Progressive Matrices, Sets I and II (Raven, 1965)

Form Reasoning (Blakey, 1941, Guilford, p. 182)

Figuren Sorteren, PMA

Figuren Reeksen, DAT

Note, that the process of theory building in the sciences completely corresponds to the process of Rule Finding as desribed above. In the sciences the available data are the data obtained under systematic observation or in experimentation. In the sciences, however, not only latent rules are required in order to be able to understand seeming irregularity, but also latent entities, such as latent objects. In many cases the particular theory at hand is called after the latent objects, which are assumed to exisits in order to understand the data. An example is atomic theory. The latent reality is described in terms of atoms, and sub-atomic elements, such as protons, neutrons and electrons. The data, however, consist of the so-called black lines in the spectra of all sorts of materials. It is not accidental, that one of the standard works on this topic is called "Atomic Structure and Spectral Lines (see Sommerfeld, 1923).

In order to be able to solve Rule Finding problems, the subject must have some experience in the past with the particular rule in question. For example, one of the items of the Form Reasoning test described by Blakey (1941) reads as follows (the symbols are changed):

               a  b  =  d
               c  d  =  e
               c  b  =  a
               a  d  =  f

            c  b  d  =  f  a  c  d  g
            _________________________
                        A  B  C  D  E

Which of the response options (A B C D E) is correct?

The item is displayed with the orignal symbols in Guilford (1967, p. 182). Instead of the letters a-g, the following figures were used:

 a: a circle,
 b: a star,
 c: a square,
 d: a cross,
 e: a triangle,
 f: a half moon
 g: a bar-bell.

In order to solve this items one must be familiar with

1. the addition and substraction of numbers,
2. the idea that numbers can be represented by arbitrary symbols (algebra) and
3. the idea of cancellation.

Addition, substraction, as well as multiplication and division all are special cases of the more general concatenation operation. Usually, a concatenation operation is denoted by a symbol, such as + for addition, - for substraction, * or . for multipliaction and : for division. However, sometimes the concatenation symbol is omitted, for example, in the case of multiplication: ab is read as a times b. One must have knowledge about this convention in order to be able to solve the above problem. Subjects who have some experience with algebra, especially, solving simple linear equations, will not encounter many difficulties in solving this problem. However, subjects who have no experience with algebra will find it hard to achieve the solution. It is very clear that past experience plays a paramount role in finding the correct rule.

Sometimes Rule Finding Problems may seem almost impossible to solve, while in fact the underlying rule might be amazingly simple. One is always inclined to hypothyse more complicated rules, which makes it very difficult to think of a simple rule. A nice example is the IQ-test Alphabetic Pattern Game presented on INTERNET by Jordan Schwartz. Go there if you want to try to solve the game on your own. You will experience how difficult the games looks and how simple it is.

Now you should be able to classify problems as Rule Applying or Rule Finding Problems. You could practice by classifying (and also solving) the items of the Psycom IQ-test edited by Carriere Contact in The Netherlands. Go there if you want to do so. You should also try to solve the problems, in order to experience that all problems can be solved if all time is allowed and to use aids and tools such as paper, pencil, scissors, etc. You may also become aware of the difficulties, which the subject may encounter in trying to solve the problems.

Now the question what intelligence tests measure or what they should measure has invariably been a matter of lengthy debate. One thing, however, has always been clear. It should not be experience or anything related to experience. The use of these kind of problems will ceratinly discriminate subjects, but the discrimination is on past experience, more specific, on school education, and not necessarily on mental ability as such, which should be independent of experience.

Another difficulty, which the subject may encounter in trying to solve ` this problem is caused by the shape of the figures, which were used. One would expect that the shape of the figures plays a role in one way or another. However, this is not the case. In order/applying the problem, the subject has to desert the obvious presupposition, that the shape of the figures has something to do with the solution of the problem. However, it is precisely this idea, which creates functional fixedness. To be fixed by a certain notion has nothing to do with some underlying quantitative ability, but it has everything do to with some underlying idea. An idea is a qualitative structure and not a quantitative variable.

The Hidden Figures test (Guilford, p.179) is a typical exmple of a Rule Applying test, in which difficulty depends on the fact that applying the rule requires imaginative ability.

In the case of the GATB subtest Three Dimansional Space and the PMA subtest Space, the faculty of spatial visualization is certainly needed for these tests, however, it is not allowed to conclude, that, consequently, the test measures something like Spatial Ability. That, what is needed for a test is not necessarily that, what is measured by the test. For example, something like the faculty of sight is also needed for most intelligence tests, but no one would claim that it is the faculty of sight which is measured. A similar situation occurs in the case of the well-known Snellen test. This test has been the most popular clinical measurement of visual acuity for over a century. No one would argue, that this test measures letter reading ability, only because the chart consists of letters. The discriminative factor is distance. Some subjects need a larger distance to be able to read the letters of a certain line, whereas other subjects need a shorter distance or, stated differently, at a certain distance, some subjects can read letters of a smaller size, whereas other subjects need letters of a larger size. When the subjects were allowed to choose their own distance, then they always are able to read the letters of a given line, irrespective of the line they might choose. In that case the discriminative power of the test would be zero. The same reasoning holds for so-called intelligence tests. If the subjects would be given sufficient practice to get used to the task and if these tests would be administered with an unlimited time, then all items would be completed and a correct response would always be given to all items. Naturally, the subjects should be told in advance, that it is allowed to take all the time they might need and only to give an anwser when they are absolutely certain about the correctness of the answer. Under these circumstances, all subjects would have all items correct and, as a result, no differences would be observed between the subjects. Consequently, the test would not measure anything, although it still would be true, that in order to accomplish the test, certain capacities, such as spatial visualization, would be necessary. If one wants to know what it is, that is measured by a test, than it seems, that explanations, which take into account test time and test instruction ("Work as fast and accurate as possible!"), are far more important then explanations which refer to necessary capacities. For example, when a time limit is used, the number of items correct is mainly dependent on the number of completed items, that is, on the individual item completion times. In tests, which have no time limit, the test score may still be dependent on the respective item times, such as is the case in the test Block Design from the Wechsler Adult Intelligence Scale, in which 'bonuses are given for rapid performances' (Wechsler, 1981, p. 48). This test has also a speed instuction: 'Try to work as quickly as you can.' (Wechsler, 1981, p. 74).

In the realm of ophtalmology, the Snellen test is a fully accepted test and, therefore, it could be used as a kind of prototype for intelligence tests. The question arises then, what it is, what the Snellen test has and intelligence tests have not. The answer is simple. The Snellest test is a measure for focal distance, which is a variable, which has theoretical as well as practical meaning. Focal distance is a theoretical concept (construct) and it came into existence with the discovery of the lens. The concept obtained practical meaning, as soon as it was known, that the lens is part of the visual system. The Snellen test has only meaning within a theory on visual acuity. Without the ideas about the lens and the concept of focal distance, the Snellen test is completely meaningless. At the same time, it is clear, that a mere description of the test in terms of what is needed for the test, in order to accomplish the task when distance restrictions are abandoned, does not give any information about the question what is really measured by the test. In comparing the Snellen test with current intelligence tests it becomes quite clear, that, in the case of intelligence tests, one is faced with a complete lack of theoretical reference. But, even in the case of a complete lack of theoretical knowledge (lens, focal distance, etc.), one could still perform meaningfull research with the Snellen test by varying the limiting factor of the test, which is visual distance. This would at least give some information about the relation between visual acuity and distance, although, without optical theory, one would still be in complete darkness regarding the why of this relation.