Sequentially Ordered Steps: Multiple Steps

Rule Applying problems sometimes require sequentially ordered steps to achieve the solution. Each step is a transition from one state to another. The sequence of states is usually not immediately apparent, and, therefore, the task involves either trial and error, going from one state to the other in a random way, or systematic search of all possible states, and subsequently, searching for the minimum number of steps. Examples are

The Tower of Hanoi Task

The standard version of the Tower of Hanoi Task consists of three pegs and a pyramide of three disks decreasing in size from bottom to top. The disks start out on a specific peg and the goal is to move the disks to another selected peg without putting a larger disk on top of a smaller one, and without moving more than one disk at a time. A demonstration of the Tower of Hanoi problem on INTERNET has been made by FWU-SHAN SHIEH. Go there if you want to try solving the problem on your own.

Luchins' Water Jar Test

In the Luchins' Water-Jar Test (Luchins & Luchins, 1955), the problem task is to use three different sized jars to measure out a specified amount of milk (or any other liquid). The jars don't have measuring marks on the side. Usually, the largest jar is fully filled in the initial state of the task. It is only allowed to pour out the milk from one jar into the other untill the latter is full or the former is empty. Luchins' Water Jar problems are also referred to as the water-jug volume-measuring problems. Luchins' Water Jar test should not be confused with the Water Jar test used by Gladue & Beatty, 1990).

Alexander's Passalong Test

The Passalong Test is part of the Alexander Performance Scale (19..). This test requires the execution of a planned series of moves from an initial to a final pattern of coloured wooden blocks. According to Jones (1936) it was designed to assess "practical ability", operationally defined as "that ability necessary for success in a technical school."

The Planning a Circuit Test

The test has been developed by Guilford and Lacey (1947). An example is given in Guilford (1967, p.105). Sould be suited as "a measure of forsight" (Guilford. 1967, p.104).

The Ship Destination Test

The test has originally been proposed by Christensen and Guilford (1955). An example is given in Guilford (1967, p. 98).

The solution always consists of passing through a number of states, starting from a initial state and ending in the end state, that is, the required solution state. At each state there is always the possibility to go to a next state, however, this might be a state, which was encountered before. In this way one may enter a loop. This should be avoided. Therefore another next state should be looked for. It could be possible that for some states there is no new next state. In that case one enters into a dead end. If there is a final solution, then one can always go some states back, and follow another path. In that way one can still arrive at the required solution state. So, in principle, Rule Applying problems can always be solved. For the actual solution enough memory space should be available, as the subject has to keep track off the intermediate results, that is to keep track of the states that were already encountered. If the subject is allowed to to make use of paper and pencil, or some other physical devise, such as a computer, then he can alwasy achieve the solution, although it may take a lot of time.

In the case of multiple steps Rule Applying problems, it frequently occurs that subjects cannot solve the problem, only because they think that, when they cannot instantaneously see the solution, they cannot solve the problem. They have never realized that in the case of these problems it is just a question of going from one state to the next. They have to be told that it is sufficient to ask oneself at each state: What can I do next?, without bothering about the question whether that might be the solution or not. Here a failure in general education becomes manifest. Children have to be taught about this type of problem (Rule Applysing) and how one always can achieve a solution if it exists. When the subject knows about this general strategy, and he is not allowed to use pencil and paper, that is, he has to solve the problem mentally, then the only limitation is memory space. However, if one is interested in measuring memory space, then one should use specific memory tests, in this case short term memory tests and not tests which contain problem solving items. If the subject is allowed to use pencil and paper and knows about the general solving trategy (at each state you can always do something) and if they are told to continue until the solution is found, then they will always arrive at the solution, however, the time needed will differ from subject to subject. Now one could argue that the time needed to solve the problem could be used as a measure of some relevant psycholgical construct. However, the time needed to solve the problem, for short the solution time, can be long for some subject just because that particular subject accidently went through some paths, which did not lead to the solution state. Hence the fact that the subject used more time then others does not necessarily say anything about some mental ability. In that perspective one should consider the solution time is a random variable T. As the problem involves a random search process, the variation across subjects is partly determined by variation within subjects. This relation can be mathematically be expressed in terms of the usual analysis of variance breakdown of the total variance into the sum of (1) average within-subject variance and (2) among-subject variance:

Var(T) = E(s)[VarT(s)] + Var(s)[ET(s)]

The derivation of this equation is given by Freeman (1963, pp. 54-57) and in many other texts. Another within subjects variance component may be the fact that some subjects take long and/or frequent rest pauses during the execution of the task and some not. By rest pauzes is meant deliberately choosen rest pauzes, rest pauzes, which the subjects in principle may be aware of. A third within subject random component stems from the fact that subjects, when being in a given state, do not immediately see wat a possible next new state might be, while that state actually exists. They think they have encountered a dead end, while in fact they have not. Such an arrest may be caused by the fact, that the subject is not yet sufficient familiar with the rule. It also can be caused by a temporarily mental breakdown induced by the inclination to work too fast. This, however, is also a matter of getting to know, that working fast is counter productive and that periods of loss of attention are quite normal and not any reason to be nervous about.