The February 11, 2008 Malaspina Uldarico defended the doctoral thesis,
Intuition and rigor in solving optimization problems. An analysis from the onto-semiotic approach to mathematics cognition and instruction
held at the Pontificia Universidad Catolica del Peru (Lima).
SUMMARY:
The thesis investigates a complex issue that involves three important aspects of mathematics and its teaching and learning. The first aspect has to do with what is meant by intuition and rigor in mathematics, the second with the process problem solving, and the third, with the interest that historically has had to study mathematics in situations that need to be optimized. These three aspects work together, with the major theoretical frameworks ontosemiotic Approach of Cognition and Instruction Mathematics (EOS). It makes theoretical contributions to conclude that there are reasons to affirm the existence of an intuition optimizer, based on contemporary cognitive science of mathematics, and proposing a way to fit the intuitive processes in the EOS, using a metaphor vector three components, which are three of the 16 processes considered in the EOS. It also shows how the epistemic configurations allow to consider together the concepts of problem formalization, intuition and rigor. As a practical contribution is a part of a quantitative and qualitative optimization problems in high school textbooks in Peru, and other concrete proposals to include optimization problems in primary and secondary, considered in three broad outlines.
Chapter 1 shows the relevance of the research problem, and makes clear that the fundamental objectives of the thesis are to answer four research questions:
1) Is there a optimizing intuition?, how it "fits" the term intuition in the onto-semiotic approach to mathematics cognition and instruction?, "this approach allows an integrated view of the notions of" intuition, "" rigor, "" problem "and" formalization " ?
2) What is the role of intuition and rigor in solving optimization problems in university students?
3) How are dealt with optimization problems in math textbooks high school in Peru?
4) Is it possible to propose optimization problems in basic education in Peru, so that stimulates intuition, allowing the development optimizing the functions of guessing, anticipating, and conclude and simultaneously pay attention to educating the formalization and rigor as a scientific attitude that complements the intuition?
This chapter also explains the methodology used.
In Chapter 2 the theoretical framework, reviewing the historical and epistemological mathematical optimization. The importance of problem solving in mathematics and teaching mathematics, with reference to historical facts and recent research on this aspect and explicitly what is meant by problem and optimization problem in the thesis, in a didactic perspective. Finally, we present an overview of the EOS.
Chapter 3, after reviewing different ways of conceptualizing the intuition, it answers the three parts of the first research question. It sets out the reasons why it is conjectured the existence of an intuition optimizer (primary type, in the terminology of Fischbein), of a comprehensive and understood as metaphorical projection, in the context of cognitive science of mathematics (Lakoff and Nunez , 2000; Núñez, 2000), that the mathematical structures that people construct stem everyday cognitive processes. Shows the fit of intuition in the EOS vector using a metaphor in understanding the intuition as a vector of three components: idealization, generalization and argumentation. Finally, it explains how the construct epistemic configuration allows you to see, including the concepts of intuition, rigor, problem and conclusion.
Chapters 4, 5 and 6 are answers to questions 2, 3 and 4 respectively. Explains the field work and analysis performed in response to questions 2 and 3 and detailed proposals with which the answer to question 4.
Chapter 7 is devoted to summarizing the conclusions and show some perspectives for further research with reference to the investigation.
The paper concludes with a list of references and the appendices in Chapters 4, 5 and 6.
