Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2019.tde-07082019-123001
Document
Author
Full name
Wilian Oliveira Rocha
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Druck, Iole de Freitas (President)
Giraldo, Victor Augusto
Melo, Severino Toscano do Rego

Title in Portuguese
Máximos e mínimos na Educação Básica: abordagens elementares sem derivadas
Keywords in Portuguese
Conhecimentos matemáticos especializados e de horizontes de conteúdo
Máximos e mínimos
Médias
Abstract in Portuguese

Title in English
Maxima and minima in Basic Education: elementary approaches without derivatives
Keywords in English
Averages
Averages' inequalities
Maxima and minima
Specialized and horizons knowledge of mathematical content.
Abstract in English
This work intends to be a contribution to the improvement of the educational action of mathematics school teachers in both initial or continuous formation. We present some elementary approaches for the study of Maxima and Minima that use final years of Elementary and High School contents only, mainly based on Ivan Nivens book Maxima and Minima without Calculus (NIVEN, 1981). We discuss the concepts of Specialized and Horizons Knowledge of Mathematical Content as a justification for the relevance of the use of this material, which were been introducted by the University of Michigans researchers, led by Deborah Ball, in the article - Content Knowledge for Teaching: What Makes it Special? (2008). We bring a critical analysis of the approach employed for the topic (maxima and minima) in some high school textbooks. We discuss the four concepts of averages - arithmetic, geometric, harmonic and quadratic - starting from problems that originated them. We also show how they can be naturally associated with measures of segments defined in squares, trapezoids and semicircles so that we can clearly visualise certains inequalities between them. Next, as an application of notable products and of second degree trinomials, we present algebraic and geometric problems of maxima or minima and discuss their solutions. We establish and prove algebraically the inequalities between the four averages (up to four positive numbers), which are applied to determine maximum or minimum points of varied functions in contextualized problems. Finally we generalize and prove the averages inequalities for n positive numbers and we develop several applications.