Essay On The Concept Of Problem Solving

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Essay on the concept of problem solving

The concept of problem solving can be defined in several ways, depending on the perspective of the author that is analyzed. However, there are several ideas in which they agree and show it as a strategy both to teach mathematics, based on the analysis of their main concepts, paradigms and models, and to learn mathematics.

The development of the transverse problem solving competition is one of the objectives of the school system and that tries to be worked through all learning sectors. This is how problem solving can go from facing and solving very explicit problems contextualized in the real world, and direct to comparing and evaluating different resolution strategies.

Both in basic and secondary education, it is considered that, through each of the learning sectors, they contribute to the integral formation of individuals, in order to train people with creative and critical thinking, capable of facingand solve various problems posed by current society,

In the field of thought development, the following skills must be promoted among others:

• Of research: identify, process and synthesize information from various sources;Organize information about a topic or problem;review approaches and perspectives;suspend judgments when not possess enough information.
• Of analysis: interpretation and synthesis of information and knowledge, which lead to students being able to establish relationships between the different learning sectors, to compare, to understand the systemic character of processes, to design, plan and carry out projects and evaluate theirown learning.
• Of communication: that are linked to exposing ideas, opinions, convictions, feelings and experiences in a consistent and grounded way, making use of various forms of expression.

Similarly, the Maps of Progress of Learning of the Areas of KnowSelection, application and evaluation skills of strategies for problem solving, as well as the argumentation and communication of strategies and results. 

From the aspects that we have been able to relate between the curricular framework and the progress maps, we can affirm that when we talk about problem solving, we are referring to a competition that is being developed from an early age in the school system and in its differentLearning levels and sectors. This is why we must consider this competence as a central component in the development of students in an integral way as proposed in the ODT.

In this way, for example, Kilpatrick considers that a problem is a situation in which you want to get a goal and the direct path to achieve it is blocked. Mayer raises three elements of a problem:

•  Data: conditions, objects, information, which are present in the problem.
•  Objectives: Desired or terminal status of the problem, which must be reached from the initial state.
•  Obstacles: He who thinks (he who solves) has certain possible routes to bring the problem from the initial state to the desired state, however, initially he does not know the problem’s response, then his resolution is not immediate or obvious.

Puig describes that the study of problem solving, in the beginning, was focused on the product of the activities of the resolutors, in how effective methods could be taught for problem solving. Then, the axis is changed focusing on the resolution process and the subject that solves it. This is how Isoda et al. (2007) cite some authors, who from both mathematics and psychology, have been considered as precursors in theories that try to describe the problem solving process. Among these authors we highlight:

• Poincaré (1908) establishes three phases in which he describes how he solves a problem:

1.  A conscious work period
2.  An unconscious work period
3.  A second conscious work period

•  John Dewey (1910) raises five phases:

1.  Experience a difficulty
2.  Define the difficulty
3.  Generate a possible solution
4.  Try the solution reasoning
5.  Verify the solution.

• Graham Wallas (1926) raises four phases:

1.  Preparation and preliminary attempts
2.  Incubation or leave the problem aside, rest
3.  Lighting or appearance of the key idea for the solution

Verification, the solution is proven

• George Polya (1945) raises four phases:

1.  Understanding the problem
2.  Traced an action plan
3.  Plan execution
4.  Reconsideration and retrospection

Also, Toulmin, cited by Couso (2008), raises three necessary mechanisms in problem solving, which are:

1.  Improve representation (theoretical models)
2.  Introduce new communication systems (new languages, graphic or mathematical symbology)
3.  Refine the methods of experimental intervention in phenomena (applications, procedures, technology).

In addition, he adds that these resolution mechanisms are related to the fact that, to answer a question (a problem), the context in which it is generated, characterize how the phenomenon is represented, determine what the language with the language with thewhich intervention is expressed in it and, finally, the applications that can be given to these interventions. The three resolution processes referred to by Toulmin provide us with authentic ideas and resources to establish problems to consider in teaching and learning processes.

Under this look a large number of researchers have recognized the importance of these strategies, but many consider them general, and do not show school activity, which the concrete students do. Schoenfeld, establishes that to understand the way in which students solve problems, as well as to propose activities that go to the aid of these, it is necessary to discuss problems in different types of context and consider the categories or knowledge factors that are involved in theexercise:

• The mastery of knowledge: corresponds to the resources related to the discipline that the student has to be used in the problem, such as definitions, propositions, theorems, procedures and conceptions.
• Cognitive strategies: it is related to heuristic methods such as drawing diagrams, investing the problem, using manipulable material, testing and error, etc.
• Metacognitive Strategies: They are related to control and monitoring, involving decisions regarding actions such as proposing, evaluating, etc.
• The belief system: it is composed of the vision of the discipline that the resolution has and of itself. These beliefs determine the way in which a person approaches the problem, the techniques that he uses or avoids, time, in effort among others.

Some authors establish the steps that the resolution must follow in problem solving, while Schoenfeld aims to explain the behavior of the problem sol. According to the belief system that he defines, everything related to school activity is a cultural, social and constructive experience that is reduced in the community in which the activity takes place. In the particular case of Schoenfeld mathematics (1992), he points out that, “if you want to understand how the mathematical perspective develops, research in terms of mathematical communities in which students and teachers live together, and in thepractices that are carried out in those communities ". From this comment we can rescue the importance of considering the environment in what and how students learn since it is in the educational community where they develop the activity.

In this way, we can see that there is a difference between the concept of ‘exercise” and “problem’. The first involves the use of an algorithm in a mechanical way, avoiding the difficulties that occur by using increasingly complex rules, and the second requires delivering a coherent explanation to a set of related data within a context. This answer is usually unique, but the strategy used to solve is determined, mostly, by maturity factors and this is where the opportunity is presented for students to develop the skills to investigate, infer, validate and argue, it issay, they have the opportunity to apply their knowledge and look for strategies that allow them to solve them.