(Page 1 of 7) - Manners of understanding and solving mathematic problems of high school students authored by Sep�lveda, Armando., Garc�a, Roberto. and Guerrero, Lourdes.

Manners of understanding and solving mathematic problems of high school students

Armando Sepúlveda, Lourdes Guerrero, Roberto García

Universidad Michoacana de San Nicolás de Hidalgo

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Abstract. In this article we report on the work realized by high school students when
confronted with a combination of problems that involve different methods of solution in a
scenario of instruction based on problem solving. During the process of implementation, the
students had the opportunity to work in small groups, to present and defend their ideas to the
whole class, and to constantly revise their work as a result of the criticisms and opinions they
were given during their presentations and discussions in class. In this context, the students
exhibited different cycles of understanding that permitted them to comprehend the
fundamental ideas associated with the solution, and eventually, they resolved the tasks.
Introduction. Recent proposals for a mathematics curriculum (NCTM, 2000: Balanced
Assessment Package for the Mathematics Curriculum, 1999; 2000) suggest the organization
of the teaching and learning of mathematics around the resolution of problems. In these it is
recognized that the experiences of the students are enriched when they work with attractive
problems or tasks that are posed in real contexts wherein they have the opportunity to apply
and extend the basic mathematical relationships. The convenience is also recognized of
implementing a manner of working in the classroom in which collective work in the whole
class and in small groups is combined with individual work, and wherein the students can
present and defend their ideas before the others. This also permits them to invigorate their
comprehension of the mathematical contents and fortalice their abilities in the resolution of
problems. In this study we were interested in documenting the strategies, representations and
resources that the students used when confronted with a combination of problems that were
interesting for them and were therefore easy to understand, that contained fundamental
contents of the curriculum and that, because of their design, permitted the recuperation of the
processes utilized in the attempts at solution. Some of the questions that guided the
development of the study are: What manners of comprehension and methods for solution
appear during the processes of problem resolution? What is the role of the professor during
the development of the sessions of application? What signifies that the students are learning
mathematics?
Conceptual framework. Mathematics learning involves the development of a disposition on
the part of the students to: explore and investigate mathematical relationships, employ distinct
representations in order to analyze particular phenomenon, to use distinct types of arguments
and to communicate results (NCTM 2000). This disposition permits them to better their initial
attempts because the students exhibit cycles of understanding in the distinct phases of
problem resolution (Lesh et al. 2000) that permits them to constantly refine their models of
solution and advance their mathematic comprehension.
Furthermore, it is recognized that learning mathematics is a continual process that is favored
in an atmosphere of problem resolution (Schoenfeld 1998) wherein the students have the
opportunity to develop manners of thinking that are consistent with the routine work of the
discipline. In this context, the students conceptualize mathematics in terms of problems that
they must examine, explore and resolve through the use of distinct mathematical strategies
and resources (Hiebert et al. 1997).
To bring about teaching and successful, significant learning, the Balanced Assessment
Package for Mathematics Curriculum (1999; 2000) group proposes the utilization of tasks that
are designed in a manner which is easy to understand and interesting for the students. These
involve fundamental curriculum concepts and ideas presented in a manner in which the work