Diferencias metodológicas y cognitivas en los procedimientos utilizados por estudiantes de la carrera de Sistemas Computacionales en la resolución de problemas de cálculo aplicado.

We consider our research as a comparative study, this is why we analysed three studies: Education and Society in Hong Kong and Macao: Comparative perspectives in continuity and change (Bray & Koo, 2005); Students of Engineering and Social Sciences university programs: Differences in the way they solve similar problems (Rosas & Pardo, 2009) and Comparative study of problem solving influence in students’ performance in Electro-chemist (Matute, Pérez & Di’Bacco, 2009). These studies provided clear examples of comparative studies and showed us the importance of detecting and analyzing the differences and similarities in education systems and their relations.

These studies provided knowledge on Comparative Theory and Didactic Engineering; because of they are used in a didactic activity application involving problem solving they also gave us Reif’s, Duran’s and Garcia’s solution methodologies.
In this research we compared the methodological and cognitive differences found in solving procedures used by students coursing the first three semesters of Computer Systems when they solve problems of applied calculus. In order to do this we applied Comparative Theory. This theory’s goal is to find similarities and differences, and the relations we can establish (Raventós, 1989, p.64).

Similarities found at the beginning of the research include that students are enrolled in the same school and same university program; so that there are no differences in the subjects they coursed. Students also have previous knowledge of calculus because students’ profile asks for it. Every student had coursed differential calculus in first semester, integral calculus in the second and differential equations in the third. Another similarity is the big amount of failing grades. Differences we established are that students are actually in different semester, from first to third; and they are enrolled in mathematics courses but with different topics.

Didactic Engineering was used as a research methodology (Lezama, 2003); the planning stage and epistemological dimension were started by describing the origin of the problem. In that stage we remarked the importance of problem solving in education and doubts originated by the unknown procedure used by students. After an analysis of Reif’s, Duran’s and Garcia’s’ methodologies we chose Garcia’s’ methodology as the exact research methodology for our own research. This choice was based on the fact that Garcia’s’ methodology states some stages that let detect every detail in a problems’ solution. It also let us compares the steps and procedures studied in a math class with methodologies applied by students.

To study the cognitive dimension we used Cognitive Map Theory and Feuerstein’s Cognitive Functions Theory (as cited by Zuñiga, 2005). Both theories let us understand and finish the cognitive study on mind processes used by students when they solve problems.
Didactic dimension was based in the description of the university program named Computer Systems. Description included subjects relative to computing and mathematics subjects.
Design stage included activities planning applied in the research, a set of exploration questions and problems design, descriptions of the application sessions, data collection and analysis of results.
Experimental stage included a description of participating students, the environment and observations while the activity was applied. We also described the way we collected data.
Validation stage included a variety of analysis of answers by topic, semester and knowledge area; comparing hypothesis and students’ results leaded us to conclude that there are important differences in the methodologies used by students. The different ways of representing the problems’ data and their relations with unknowns: Tables, arithmetic calculations, graphics, lines, algebraic expressions, rule of three, functions, operations and dimensions analysis. Only a few students identified problems’ objectives and they could establish what the need to find, some students translate mathematics relations from common language to mathematical language using algebraic expressions. Students operate those expressions as equations and functions. Most students said that it should be other procedures to solve the problems, but none of them could find other procedure; a few students tried to validate their answer with problem’s conditions.
Some important cognitive differences were observed with this research. Some students applied arithmetic but they also involved algebra and other pre-calculus subjects, only one student applied calculus. Some students could identify the similarities and differences on problem’s data and they generated different algorithms.
As conclusions we can say that students of advanced semesters do a better use of previous mathematics knowledge like arithmetic, algebra and more advanced courses. We also found that the involved concepts are not completely interiorized because students made mistakes when they applied those concepts.