Abstract
This paper reports the results of the experimentation of a didactic engineering for the treatment of the sense of variation of functions with pre-university students. The theoretical references of the investigation are grounded in the theory of didactic situations and the methodological elements in the didactic engineering, the use of counterexample and the didactic strategy known as Scientific Debate in Mathematics Courses. As a result of the experiment, it was identified that the methodological resource allowed the students to develop construction processes of the concepts of increasing and decreasing functions; the dynamic context fostered intuitive ideas of increment and decrement, and intuition about the conditions involved; the debate and counterexample on the graphic and algebraic treatments allowed the establishment of the conditions that structure the formal definition of these concepts.
Highlights
The study of variation of functions is a mandatory content in middle-high school plans and curricula, in Mexico. The learning of this content is consolidated throughout the studies at this level, because it is an integrative content where the main concepts of differential calculus converge and are needed; the study of variation of functions may be an opportunity to strengthen the maturity of the mathematical reasoning in students for the formulation of conjectures, inductive processes, argumentation, validation and refutation of theorems related to this mathematical concept, if it is treated as a set of recipes or algorithms
The role of students-mathematicians in a scientific debate will be to produce precise counterexamples and to provide arguments recognized by the whole mini-scientific community formed in the classroom; validation will be carried out by the teacher who must establish that he has the necessary knowledge of the topic in question. As students identify this type of didactics, they will be in a position to accept or discard the theorems generated by their argumentations, considering the principle or law of the counterexample which states that: “one counterexample is enough to proof the falseness of a conjecture of universal character.”
Most of the productions of the students expressed purely intuitive ideas; the scientific debate brings these ideas closer to the conditions involved in the formal definitions
Summary
The study of variation of functions is a mandatory content in middle-high school plans and curricula, in Mexico. Several investigators (García & Morales, 2013; Hernández, Locia, Morales & Sigarreta, 2019; Klymchuk, 2012; Morales, Locia, Ramírez, Sigarreta & Mederos, 2018; Zazkis & Chernoff, 2008) agreed that the formulation of conjectures and the use of counterexamples allow students to think about the how and why of the processes used to reach conclusions, and to reduce the algorithmic and rote learning procedures; they enable progress in the structuring of the necessary logical-mathematical reasoning of students, so they can be valued and improved by the teacher and the students In this way, their reasoning can be refined or even strengthen, which, in turn, can allow the formation of a critical and analytical thought, essential to form individuals in a society. As students identify this type of didactics, they will be in a position to accept or discard the theorems generated by their argumentations, considering the principle or law of the counterexample which states that: “one counterexample is enough to proof the falseness of a conjecture of universal character.”
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