Cognitive maps across cultures and across sciences

Horacio J.A. Rimoldi *

* MD, and PhD in Psychology. Emmeritus Researcher of Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). Director of Centro Interdisciplinario de Investigaciones en Psicología Matemática y Experimental (CIIPME) at Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET). E-Mail: hrimoldi@conicet.gov.ar

Resumen

   En el presente trabajo se discuten aquellas investigaciones que se refieren a la forma con la cual los sujetos buscan y procesan la información. Este tema fue estudiado tiempo ha por los psicólogos de la Gestalt (Kohler, Wertheimer, Dunker) y por algunos matemáticos tales como Polya, Russel, Whitehead. Tempranas investigaciones acerca de la resolución de problemas llevaron al estudio y aprendizaje del adiestramiento en medicina. Luego, el concepto de isomorfismo fue investigado a través de diferentes sistemas de relaciones que rigen en distintas ciencias y medios culturales. El estudio de los procesos llevó a considerar el orden como un componente fundamental, que requiere nuevos métodos para apreciar cómo el orden en que se da la información, en la secuencia de acontecimientos que identifica una táctica, influye en los resultados. Así se desarrolló el sistema de puntaje que maximiza la evaluación de la eficiencia en la resolución de problemas.

Palabras clave: Imágenes - tácticas - isomorfismo.

Abstract

   Studies of how subjects search for and process information will be discussed. An approach to the problem was initiated by Gestalt psychologists (Kohler, Wertheimer, Dunker), and some mathematicians (Polya, Russel, Whitehead). Early problem solving investigation led to the study of learning and medical training. Afterwards, the concept of isomorphism was investigated for the same underlying system of relations across sciences and across cultural environments. Studied processes brought order as a main component requering new methods that considered incoming information order in the sequence of events characterizing tactics. Scoring systems, within and between subjects, maximizing problem efficiency, have been developed.

Key words: Cognitive maps - tactics - isomorphism - images.

   Few subjects have shown, across cultures and across centuries, such a widespread interest as those related to reasoning and cognition, and it was around 1950 that the cognitive revolution brought about a renewed interest in the subject. During the first half of the last century, and running the risk of ignoring other very important contributions, the studies of Wertheimer on Productive Thinking, of Kohler on The Mentality of Apes, of Polya on Mathematics and Plausible Reasoning, and of Spearman on The Abilities of Man, asked some of the questions that cognitive psychology tries to answer today.
   Our interest has centered on the investigation of the process that takes place when subjects are requested to solve a problem rather than in the final answer. In order to do so we developed, in 1950, a technique to investigate how, given a problem, subjects request and organize information to obtain an answer. Two major type of problems have been prepared: (a) those built around a known logical structure, and (b) those of unknown structure where the word structure means a set of entities, their relationship and the corresponding operations.
   After presenting a problem subjects are free to ask, in the order they wish, those questions or perform those actions that they think would help to reach a solution. After asking a question subjects are given the pertinent answer. In some cases the possible questions are presented with the problem and in other cases, the subjects themselves generate the questions.
   With problems based on a known structure, the subject performance is evaluated in terms of the structure, otherwise the performance of groups of experts is used to establish the desired norms. For instance, in the case of medical diagnostic problems, the norms given by clinicians, neurologists, surgeons, psychiatrists, among others, may be used to find out how a student's process of solution agrees with each one of the given specialty norms. And similarly with behavioral, law, physical, engineering, economic, and other kind of problems.
   The sequential arrangement of questions asked defines the subject's tactics, where the word tactics, according to the Oxford Dictionary means a procedure to gain some end. Tactics, as maps of the underlying psychological process, inform how subjects, according to their cognitive style organize information and cognitive styles, may in some cases, relate to creativity.
   The subjects we examined belonged to different cultural, educational, and age groups, and the problems they were asked to solve varied from those related to daily events to problems in different sciences. The relationship between mental pathology and problem solving, as well as the relationship between tactics and pupillary size, heart rate and visual and auditory impairments were discussed in several doctoral dissertations in our Chicago and Buenos Aires laboratories. In all cases, the order in which information is requested became one of the main variables reaching the conclusion that in the study of problem solving processes, both ordinal and cardinal components should be taken into account. Two or more tactics are equal, if and only if, they are identical in terms of the questions asked and in terms of the order in which they are asked.
   The evaluation or order has been one of the more difficult challenges we encountered in our studies. Recently, when discussing the laws of chaos, Ilyia Prigogine mentioned that when talking of thinking we need necessarily to consider a before and an afterwords, and that such consideration should be of interest in psychological research.
   In our studies, order is identified by recording in double entry tables, in which the columns correspond to questions and the rows to order, the frequency with which each question is asked in all possible orders by a group of subjects. The same question may have different ordinal spreads depending on the subjects being examined, and will provide different amounts of information depending on its placement in the tactics.
   The administration of problems and their scoring has been totally computerized. When the structure of the problem is known each question is scored in terms of the bits of information it provides according to the known structure. Otherwise, the norms of different groups of experts are used to obtain the information value of each question in terms of each group of experts.
   Each individual tactic is defined by: number of questions asked, their order in the sequence, the amount of information corresponding to each question according to the order in which it occurs, the amount of information due to each question regardless of order, the amount of information obtained at each step in the process regardless of the questions asked, the time between questions, the total time employed for the solution of the problem, the offered final solution and the alternative solutions that may occur during the process. These components define each performance curve that shows how each subject acquires information as his tactics develops.
   Whitehead stated that while facts have neither a past nor a future the events they generate vary depending on when and where they occur. Thus the same question (fact) will provide different information (events), depending on the other components of the tactics, on the order in which it occurs, and on the subject being examined. In two recently published studies, one of them together with Burger, we developed a procedure that shows the relationship between components due to subjects, and to variables, suggesting a solution for the old problem of reversing the roles.
   In Figure 1 are given the frequencies and bits of information corresponding to each one of five different questions (A, B, C, and E) depending on the order in which they were asked by a group of subjects trying to solve the same problem. For instance, given this group of subjects, question A provides maximum information when asked in second order, while question C has a maximum information value when asked in their order. The graphic presentation of three tactics BDAC, BAD, and ACDB shows that tactic BAD accumulates maximum information at each step in the process, while the other two, in spite of having a greater number of questions are clearly inferior.

   Depending on the kind of representation or images used, where the word image is defined as the representation of something, problems having the same underlying logical structure can be prepared. The images used may be: common or technical words, drawings, colours, concrete objects and so forth. The structure and image components may be used to: (1) investigate the part played by different types of images when solving isomorphic problems, and (2) the part played by the structure when non isomorphic problems are presented using the same kind of presentation.
   The same structure may be used to build problems in different languages and in different contexts like we have done when comparing the solution of everyday problems in subjects of different age, different countries (USA, Canada, Israel, Uruguay, Argentina) and ethnic groups, different sex, and educational levels and so forth. It would be impossible to list the results obtained in all of them, but they show that in order to conclude that a subject is unable to solve a problem, it is necessary to show that this may be due to either the difficulty of the structure, or to the images used to present it.
   In 1962 we investigated, in Chicago high school students, how training in problem solving related to learning, finding that by comparison with a control group, their score in mathematics, had increased significantly.
   To investigate aspects of mathematical talent using the symmetries of the square and of the rectangle we built total of 51 problems that were administered to 120 subjects of both sexes and three educational levels. Each subject was given a white cardboard square or rectangle with the four angles painted in different colours. Starting from a given position the subject had to rotate the given cardboard square until reaching a position different to the original one (inequality problems) or to reach the original one after going through a different one (equality problems) (see Figure 2).

   The results were evaluated in terms of usual statistical procedures and information theory, showing that inequality problems are easier to solve than those involving equality, that they have less relative entropy and that there are no differences either in terms of sex or to eduactional level. Subjects discovered the non-commutativity of some groups of transformations and the irreversibility concept.
   The described technique makes it possible to compare how a subject performs in terms of different norms. We have used this approach to evaluate medical student's processes in terms of different norms showing how the student performance relates to each one of the norms.
   The experimental procedure is as follows: the student is given a brief account of the patient complaints and a set of possible questions that he may wish to ask in any desired order to reach a diagnosis. The answer to each question is shown in the computer's screen and the whole tactic is evaluated in terms of the bits of information corresponding to each norm. The results are graphically shown by the performance curves correponding to each norm. Therefore while in terms of one norm the performance curves of different subjects may be obtained, each subject's tactic may be evaluated in terms of as many norms as desired.
   In Figure 3 are shown the performance curves for the same subject (same tactic) who asked 15 questions in order to diagnose the given case. According to surgeons no information is accumulated after the four question asked, while in the case of clinicians this occurs after the seventh question. Besides, the performance of this subject accumulates more information when evaluated in terms of clinicians norms, than in terms of surgeons of general prectitioners.

   I do not desire to enter into the actual controversy about IQ scores but it is well known that a given subject will show different scores depending on the ability being tested. It seems reasonable to think that the presentation of the material does not change the subject's intelligence, but that the obtained differences reflect the way in which subjects, whatever their cognitive ability, respond differentially depending on the presentation or kind of image being used.
   Thurstone stated long ago that each of the primary factors can be regarded as a composite of an independent primary factor and a general factor which it shares with other primaries, and in a study we published in Psychometrika entitled The Central Intelective Factor, we found that in a second order analysis different abilities converge in a factor that could be identified with Spearman's G adding that: The dynamic aspects underlying factors are more meaningful than their simple description. Perhaps, we should examine again some of the solutions offered before our time by those who faced problems similar to the ones we face today. Here Spearman's noegenetic principles may play a part.
   In cognitive processes the function of both logic and language (images), as well as their interaction should be properly identified. To reduce one to the other leads to dead ends. The manyfold interplay of modalities and logic within a goal-directed performance, makes of problem solving a dynamic open system that in some cases is rich in invention and discovery.
   Not being able to solve a problem may result from inability to understand the relational system involved, from lack of familiarity with the images used, or from inability to establish the rules of correspondence between images and structure. Reducing levels of thinking to the familiarity with images may be as one-sided as reducing it to pure logic. Both interact in ways that may be observed when exploring the cognitive maps in terms of culture and in terms of science.

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