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Metacognition: Hypothetical Reasoning and Problem Solving

 

Authors: Jaime Iván Ullauri Ullauri

Universidad Nacional de Educación, UNAE

jaime.ullauri@unae.edu.ec

Azogues, Ecuador

 

Carol Ivone Ullauri Ullauri

Universidad Nacional de Educación, UNAE

carol.ullauri@unae.edu.ec

Azogues, Ecuador

 

Abstract

          The present work contemplates a precise and succinct theoretical reflection on the metacognitive process of hypothetical reasoning as a proper capacity of the cognitive development of children during the third childhood and how it affects the logical-mathematical problem solving, on the cognitive perspective of Flavell and Sternberg in communion with the resolving process of Polya's problems. This investigation comprised the pertinent bibliographical study, having like horizon to put in evidence the hypothetical reasoning processes in the resolution of mathematical problems, concluding with the importance of the teaching and development of basic cognitive processes in education that foment the sprouting of metacognitive processes, allowing the child will work on his task, rethink it and, as far as possible, solve it. From this perspective, this work constitutes an overview of easy access and understanding for new professionals who specialize in psychology and education.

 

Keywords: cognition; thinking; reasoning; problem solving.

 

 

 

Metacognición: Razonamiento Hipotético y Resolución de Problemas

 

Resumen

El presente trabajo contempla una reflexión teórica precisa y sucinta sobre el proceso metacognitivo de razonamiento hipotético como capacidad propia del desarrollo cognitivo de los niños durante la tercera infancia y como este incide en la resolución de problemas lógico-matemáticos, sobre la perspectiva cognitiva de Flavell y Sternberg en comunión con el proceso resolutorio de problemas de Polya. Esta investigación comprendió el estudio bibliográfico pertinente, teniendo como horizonte poner en evidencia los procesos de razonamiento hipotético en la resolución de problemas matemáticos, concluyendo con la importancia de la enseñanza y desarrollo de procesos cognitivos básicos en educación que fomenten el surgimiento de procesos metacognitivos, permitiendo al niño trabajar sobre su tarea, repensarla y en lo posible solucionarla. Desde esta perspectiva este trabajo constituye una panorámica de fácil acceso y comprensión para los profesionales noveles que se especializan en psicología y educación.

 

Palabras clave: cognición; pensamiento; razonamiento; resolución de problemas.

 

 

 

1.    Introduction

The present work leads to a theoretical approach to the cognitive and metacognitive development of children during the third childhood (7-11 years of age) in correspondence with the theoretical foundations established by Piaget (1991a) and Flavell (2000a). The ultimate goal of this work is to highlight the most relevant theoretical aspects of the development of logical reasoning as a metacognitive process in solving mathematical problems in children.

 

From the cognitive theory of Flavell (2000b): conceptual categories and evolutionary stages of thought are established, that cognitive transition comprised between the second and third childhood, so from this perspective the author considers that metacognitive knowledge can lead the child to assimilate stimuli that are in their context, assimilation that necessarily matches their interests, capabilities and goals. Sternberg (2011a): states that these abilities have to make clear the compression as the genesis of the process that the child develops to solve a problem, this problematic compression is that through the evocation-memory leads to the activation of prior knowledge, from which different solutions can be produced to the same problem.

 

2.    Theory and concepts

2.1.     Cognition

It is important to clarify basic concepts that allow building a panoramic view of cognitive development and its implications for the child's integral development. Flavell (2000c): integrates within cognitive development the development of higher mental processes "corresponding to psychological entities such as knowledge or knowledge (knowledge), consciousness, intelligence, thinking, imagine, create, generate plans and strategies, reason, infer, solve problems, conceptualize, classify and relate, symbolize and possibly fantasize and dream" (Flavell, 2000d, p.13). From another perspective Dorsch (2005): covers the conception of cognition and its development as a generic term to name all the "processes or structures that are related to consciousness and knowledge, such as perception, memory (recognition), representation, the concept, the thought, and also the conjecture, the expectation, the plan" (p.121).

 

In conjunction with these definitions, Ullauri conceptualizes cognition as:

Cognition is not just a process or a set of processes that enable the human being to solve their problems, cognition is a set of skills and competencies that allow the human being to be 'capable of', this ability translates into being able to outline and establish logical relationships, which can only be made by the human being as such (Ullauri, 2013a, p.18).

 

In this way "capable of" it is determined that the child can logically establish strategies for solving problems "cognition must be understood not only as a mental unit of intelligence but as a general unit of thought, which allows to the human being to build and rebuild" (Ullauri, 2013b, p.18).

 

2.2.     Metacognition

For Bruner (1995): metacognition is first of all a skill, which allows the child to think about his or her thinking, making it possible to be aware of the situation in order to solve it, also indicating the classification by levels of thought: in the first level there are basic thought processes that are innate, in the second level we find the ability to remember and adequate strategies to be literate culturally and in the third level the thought processes and strategies that are consciously evoked.

 

From the perspective of Flavell (2000e): considers metacognition as the fourth level, the "highest level" of the mental activity of the child, which involves the most complex processes of thought, typical of the conscious mind. For the author in the third childhood the basic aspects in which he frames the metacognition are: a). own cognitive abilities, b). the tasks, c). the metacognitive strategies. Also for Hacker (1998a): metacognition integrates: a). knowledge and b). Metacognitive regulation, understood as the first knowledge that is possessed on knowledge itself, which also involves their cognitive and affective states, interpreting and reinterpreting them, while metacognitive regulation is the process by which the child is able to control and regulate cognitive processes.

 

While from the Wellman theory of mind (1985), postulates stages in which knowledge and metacognitive regulation are constructed are: i). existence, ii). process distinction, iii). integration, iv). knowledge of the variables and v). cognitive monitoring.

 

3.    Cognitive development in the second and third childhood

The main characteristic of the child's thinking in this stage of development is the advancement to a logical thought, in which he abandons intuition and self-centeredness. Thus from this perspective Piaget, includes at this stage the concrete operational stage, a space of development in which children manage to master the notions of conservation, transitivity, class inclusion, multiple classification and seriation, as well as initiating the understanding of the reversibility, identity and notions and understanding of logical concepts (Piaget, 1991b, p.54).

 

Flavell (2000f): in the stage of concrete formal operations, he visualizes the cognitive processes that occur during the second and third childhood in the following way: in the second childhood cognitive processes are presented as i). Perceived appearances, ii). centrations, iii). states and iv). irreversibility, while in correspondence to each of them in the third childhood they are already developed as: i). inferred reality, ii). decentration, iii). transformations, iv). reversibility, determining that the nature of the hypothetical reasoning process is metacognitive and that it occurs from the third childhood.

 

During the second childhood (2-7 years), a development in the basic competences of a progressive way with presence of the intuition and egocentrism is evidenced, in the third childhood (7-11 years), the concrete operations and the logical thought are potentialized , as Flavell (2000g) states: "the development of these that advances from the total absence of competencies to the presence of the most advanced" (p.74). Around this aspect Flavell also indicates that the centering, the perceived appearances and the irreversibility are conditions of the second childhood, whereas the decentration, the inferred reality and reversibility are conditions of the third childhood.

 

3.1.     Metacognition

The development of metacognitive processes leads the individual to work on meta-knowledge, which Flavell (2000h) defines as "the ability to control and evaluate one's current memory capabilities" (p.145); and as the capacity of the "cooking on knowledge". So it is in the third childhood the space in which metacognitive activities are developed.

 

For Flavell (2000i): it is necessary to understand that meta-knowledge involves the knowledge and metacognitive experiences that the child internalizes about the cognitive contexts in which it develops, such knowledge has more the characteristic of being declarative than procedural, subdividing itself into knowledge about the people, tasks and strategies:

The knowledge about people involves any type of knowledge that the child can internalize and these differ between one person and another, the knowledge of the task includes the information that is available in the context in which the child is to be assimilated, this information is generated from the metacognitive experience, while the knowledge about the strategies involves the ability of the child in our case to identify the cognitive processes involved in the problem-solving process (Flavell, 2000j, p.147).

 

Thus, it is understood that a cognitive strategy contributes to the realization of the cognitive activity in question, while the metacognitive strategy is the one that will generate the information about the work that is being developed or in itself the progress of the own strategy.

 

3.2.     Metacognitive experience

The metacognitive experience translates as "the bulk of metacognitive knowledge, really refers to combinations or interactions between two or three of the categories" (Flavell, 2000k, p.148), and by its nature this can be short or long, as also of a simple or complex nature depending on the content that these entail, these cognitive experiences can occur throughout the cognitive process that is undertaken for the construction of a task or the resolution of a problem.

 

3.3.     Problem resolution

Solving problems in the daily life of the human being is necessarily daily, in any context that is found, so it is necessary that the child can establish a resolution plan, in which not necessarily the main objective is the effective solution or not of the problem, but the process that the child develops within that plan that he established with the intention of solving it, in other words, to be aware of the different steps he takes to solve the problem, the conscience that the child has of this process allows you to go back over each of the phases of the work you have developed with the aim of correcting it so you can consider that:

any obstacle situation that facilitates the activation of a cognitive process, which to be solved requires the interaction of actions that can be executed, based on skills such as: observation, inference, assumption, analysis, etc., that the human being undertakes as such, allowing the acquisition of new skills that enable the development of a process that creates the way for children to establish logical and real solutions to the problem (Ullauri, 2013c, p.46).

 

Next, we address two problem solving models, the first proposed by Polya (1984a), and the second one defined by Sternberg (2011b). The Polya model establishes four phases for solving mathematical problems that are: i). understand the problem, ii). set up a plan, iii). execute the plan, iv). look back (Polya, 1984b, pp.51-53):

 

      I.         Understanding the problem: this understanding of the problem does not only imply its literal reading, its words, its signs and symbols, but also the implicit, which at first glance is not seen, such as the relationships, categories and variables that compose it, helping the child to understand the problem in its entirety. (Polya, 1984c; Ullauri, 2013d).

 

The genesis of the process implies reasoning about the task that is presented to solve it, recognizing and understanding the problem in question, this implies that the child must start from the critical analysis of relationships, categories and variables to determine them in a first phase, it will not be possible that the child reaches this stage if he has not been able to establish in sufficient depth the characteristics of the problem.

 

    II.         A second moment is the configuration of the plan, which includes the mapping of the problem situation that needs to be solved, the structuring of this plan requires the staging of the skills and competences pertinent to the type of problem that is trying to be solved, in this case the metacognitive process of hypothetical reasoning as a means that allows to look in the future towards the construction of a visualization about the resolution of the problem:

that as a video player can be reviewed forward and backward, so that you can perceive the smallest details, which will allow you to decide which strategy is the most appropriate for resolving the problem, going back and reviewing again the scene and hypotheses (Ullauri, 2013e, p.57).

 

So it is understood that the nature of hypothetical reasoning is metacognitive.

 

   III.         After the configuration of the plan the execution becomes. The execution of a plan does not guarantee the resolution of the problem, which can not be classified as an error or failure but rather as an opportunity to rethink the strategy, assess errors and start planning the new plan and its execution.

 

The dynamics of these phases as containing cognitive processes necessarily involves the awareness of these by the child, which enables him to correct errors about the situation. During the execution stage of the plan, it will be able to stop on the fly and rethink the strategy that is not working correctly to continue.

 

  IV.         In the end, looking back makes it possible to discriminate whether the solution achieved is the correct solution or not, this exercise that the child makes generates the space to be able to understand and understand the problem, achieving the verification that satisfies him.

 

Polya (1984d): estimates that the process of solving problems includes two types of reasoning; regressive reasoning and projective reasoning, these two types of reasoning should be reflected in the understanding of the problem, the constitution of the plan to solve it and especially when the resolution of the problem is not generated to reformulate the strategies to try to solve it (p.134).

 

From this perspective, the author considers that Polya:

we look for from what background the desired result could be deduced; then we look for what could be the antecedent of this antecedent, and so on, until, passing from one antecedent to another, we finally find something known or admitted as true. We call this process analysis, backward solution or regressive reasoning. In the synthesis, on the contrary, reversing the process, we start from the last point reached in the analysis, of the element already known or admitted as true. We deduce what in the analysis preceded it and we continue like this until, retracing our steps, we finally reach what we were asked. We call this process synthesis, constructive solution or progressive reasoning (Polya, 1984e, p.134).

 

The problem solving process proposed by Sternberg (2011c): it is not necessarily oriented towards solving mathematical problems. This process has a cyclical nature that starts from: 1). identification of the problem, 2). definition and representation thereof, 3). formulation of strategies, 4). organization of information, 5). location of resources, 6). monitoring, 7). evaluation (p.430).

 

Figure 1. Sternberg problem resolution cycle.

SEE IN THE ORIGINAL VERSION

Source: Sternberg (2011, page 430).

 

The process of problem solving of Sternberg (2011d): after the second phase does not maintain a sequential order, this particular does not indicate that the ordering of these phases negatively or positively affects the resolution of the problem, that is why from the The third phase up to the sixth phase allows the creation of new metacognitive processes that facilitate the restructuring of strategies to solve problems, that is, the reversibility of thought.

 

In correspondence with the cognitive processes determined by Flavell (2000l), inferred reality, decentration, transformations, reversibility and those indicated by Polya (1984f): the regressive and projective reasoning are corresponding, identifying these as necessary to be able to solve a problem with complete satisfaction the child must have developed these basic processes of metacognitive thinking to understand the problem, configure the plan, establish resolution strategies, develop the plan to solve it and in case of not being able to solve it, the regressive reasoning will enable you to return to the plan and strategies , to establish improvements or configure a new plan that allows you to solve the problem, in this sense in a mathematical problem are involved all the cognitive processes that we have reviewed, from this point the mathematical logical thinking plays a decisive role.

 

3.4.     Mathematical logical thinking

For Castaño (2010): "the development of logical-mathematical thinking is the development of the ability to establish relationships and to operate with them" (p.96), understanding that the development of this type of thinking involves the involvement of cognitive processes that allow the establishment of relationships. In this sense Piaget (1991c): argues that "the child's thinking does not become logical only through the organization of systems of operations that obey laws of common sets that are; composition, reversibility, direct operation and its inverse, association of operations" (p.71), from this perspective we can understand" mathematical logical thinking as the cognitive and metacognitive process generated by the interactions of experiences and actions in the solution of problem" (Ullauri, 2013f, p.46).

 

According to Saguillo (2008a): the mathematical logical thought is based on the nature and objective reality that is expressed in propositions that acquire a mathematical value that can be false or true, thus "the classical mathematical logical thought is articulated also presupposing certain epistemic capacities of the Humans". (Saguillo, 2008b, p.6), which are necessarily binding with cognitive and metacognitive processes, allowing different relations to be established in relation to situations, objects and concepts, allowing the child to structure and restructure reality, using cognitive processes and metacognitives such as the approach of projective solutions to a specific task, in other words, the hypothesis approach.

 

The resolution of a problem of any kind requires the establishment of resolution strategies that are configured in a plan, which is necessarily built on the basis of cognitive and metacognitive processes.

 

4.    Methodology

The present work contemplated the pertinent bibliographical study on the development of logical reasoning in the third childhood and how this metacognitive process is immersed in the resolution of problems. The bibliographic inquiry was conducted between the months of September and October 2017, for which some descriptors have been used that include: cognitive development in the second and third childhood, metacognition and metacognitive processes, hypothetical reasoning, mathematical logical thinking and problem solving. From the sources consulted, those that report on the basic aspects that show the cognitive and metacognitive development of the children were selected and in a spatial way those that implicitly are related to the resolution of mathematical problems..

 

5.    Conclusions

The realization of this work has allowed to reach some key conclusions in the process of development of hypothetical reasoning as a metacognitive process for the resolution of problems, these conclusions are concretized in:

·       Metacognitive processes are the product of the sum of cognitive processes, thus the emergence of hypothetical reasoning is inherently linked to the sum of basic cognitive skills such as perception, attention, memory, thinking, reasoning and language.

 

·       It is evident that the treatment of the tasks faced by the children are considered by them as problems, which are not necessarily known by them and that will most likely be solved by the development of some cognitive skills such as: problem identification, capacity for definition and representation, formulation of strategies, organization of information, location of resources, monitoring the process, establishment of inferences, evaluation and in a special way for the development of metacognitive dexterity of hypothetical reasoning as a mathematical logical thinking process.

 

·       The theoretical studies of Piaget (1991d), Flavell (2000m), Sternberg (2011e) and Hacker (1988b) are contrasted in relation to Polya's (1984g) problem-solving perspective: it can be stated that the function that the cognitive and metacognitive processes such as hypothetical reasoning allows you to configure a plan and its strategies in a projective way to solve problems of different kinds.

 

6.    References

Bruner, J. (1995). Escuelas para pensar. Madrid: Paidós.

 

Castaño, J. (2010). La matemática en Transición y Primer Grado de Escuela Nueva. Manual de implementación escuela nueva, generalidades y Orientaciones Pedagógicas para Transición y Primer Grado. Tomo I. Bogotá, Colombia: Ministerio de Educación Nacional. ISBN: 978-958-8712-41-3, págs. 212. Recuperado de: https://www.mineducacion.gov.co/1759/articles-340089_archivopdf_orientaciones_pedagogicas_tomoI.pdf

 

Dorsch, F. (2005). Diccionario de Psicología. Barcelona: Heder.

 

Flavell, J. (2000a,b,c,d,e,f,g,h,i,j,k,l,m). El Desarrollo cognitivo. Madrid: Visor.

 

Hacker, D. (1998a,b). Metacognition: Definitions and empirical foundations. En Metacognition in educational theory and practice. EE. UU.: The University of Memphis, pp. 1-23. Recuperado de: http://vcell.ndsu.nodak.edu/~ganesh/seminar/Hacker_Metacognition%20-%20Definitions%20and%20Empirical%20Foundations.htm

 

Piaget, J. (1991a,b,c,d). Seis estudios de Piscología. Barcelona: Labor.

 

Polya, G. (1984a,b,c,d,e,f,g). Cómo plantear y resolver problemas. México: Trillas.

 

Saguillo, J. (2008a,b). El pensamiento lógico-matemático. Madrid: Akal.

 

Sternberg, R. (2011a,b,c,d,e). Psicología cognitiva. México: Thomson.

 

Ullauri, J. (2013a,b,c,d,e,f). Proceso metacognitivo del pensamiento lógico matemático: razonamiento hipotético. Cuenca, Ecuador: Universidad de Cuenca, págs. 207.

 

Wellman, H. (1985). The origins of metacognition. In D.L. Forrest-Pressley, G.E. MacKinnon, & T.G. Waller (Eds.), Metacognition, cognition and human performance, pp. 1-31. Orlando, FL: Academic Press.

 

 

Jaime Iván Ullauri Ullauri

e-mail: jaime.ullauri@unae.edu.ec

 

Born in Cuenca, Ecuador. Lcdo. in Educational Psychology, Master in Education and Development of Thought (Universidad de Cuenca-Ecuador), Master in Educational Guidance (UNED-Spain) and PhD student of the Psychology of Education program of the Universitat de Barcelona. Professor-Researcher in the Chair of Human Learning and General Leveling at the Universidad Nacional de Educación, has participated in similar processes at the Universidad de Cuenca. Teacher of Basic General Education and Bachillerato in Social Studies and Development of Philosophical Thought. Academic Facilitator within the programs of Advisory and Educational Audit and Continuing Education "Yes Profe" of the Ministry of Education of Ecuador.

 

 

Carol Ivone Ullauri Ullauri

e-mail: carol.ullauri@unae.edu.ec

 

Born in Cuenca, Ecuador. Bachelor in Educational Psychology Specialization in Professional Guidance, Master in Educational Psychology and Human Development in multicultural contexts (Universidad de Valencia-Spain). Professor-Researcher in the Chair of Neurosciences at the Universidad Nacional de Educación. Professor-Researcher in the Theories of Learning, Psychology of Human Development, Theories and Psychological Systems and General Psychology at the Universidad de Cuenca. Professor of Leveling knowledge at the Universidad de Cuenca. Teacher of Basic General Education and Bachillerato in Language and Literature.

 

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- Original Version in Spanish -

DOI: https://doi.org/10.29394/Scientific.issn.2542-2987.2018.3.8.6.121-137