Objective:

This article describes the findings and conclusions of a research in mathematics didactics. In Chile, high school students (16 and 17) have difficulties in understanding the conics, given the weak articulation of representations of the types of theoretical and practical thoughts; they do not conceive the isolated conics of the usual metric. Given the cognitive nature of the problem, the theory of the Thinking Modes of Anna Sierpinska is used and it is proposed as a research objective to design a sequence of didactic activities that allows to move forward through different ways of thinking the conics, relating not usual metrics, so that graphic representations of them are not conceived as definitions.

Methods:

A qualitative approach was used within the interpretive-comprehensive paradigm. Questionnaires of open answers were applied to two groups made up of 17 students from third and 15 from fourth grade.

Results:

The sequence of didactic activities designed, articulates the practical and theoretical thoughts through the synthetic-geometric (practical thinking), analytical-arithmetic and analytic-structural (theoretical thinking) ways of thinking the conics, using two metrics, both non-Euclidean. The findings show that the students manage to relate the theoretical and practical thoughts, moving forward through different ways of thinking the conics, and generating understanding of the mathematical object.

Conclusion:

The evidences show that the types of thought coexist in the treatment of the conics with unusual metrics, as well as to demonstrate diverse transits between the ways of thinking, therefore, a deep (integral) understanding is achieved of the conics.

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