Didáctica de las matemáticas para niños con síndrome de down a partir de una visión integrada de la aritmética y de la geometría elementales

Abstract

This thesis presents a lesson-plan proposal for the introduction of mathematics to children with Down syndrome between the ages of 3 and 6. It integrates elementary arithmetic and geometry, allowing these students to overcome the well-known obstacles they tend to have when the utilitarian aspects of the discipline are emphasized. In the first part of the thesis the three theoretical foundations for the proposal are introduced. First, Édouard Séguin’s works (1812-1880), a French educator and doctor whose pioneer contributions to the role of geometry in the intellectual development of the so-called idiot children are analyzed. Second, the papers written by Italian professor Elisabetta Monari (1995-2005) which pose the convenience of avoiding arithmetic-based teaching methods in the learning process of people with Down syndrome. Finally, we present insights into the ideas of Giorgio Israel and Ana Millán Gasca which are exposed in their book Pensare in matematica (2012), about the role of the historical origins and the axiomatic foundations of arithmetic and geometry (Peano, 1899; Hilbert, 1889) in children´s initiation into mathematics. These theoretical foundations lead to a teaching proposal tested in a math workshop with eight children aged 3 to 8 over a ten-session period and a case study of an eight-year-old boy over a school year described in detail in the second part of the thesis. The following steps are proposed: First, one should begin with the exploration of children’s own naive conceptions in order to introduce the idea of number and shape (Millán Gasca, 2015). Secondly, one should choose and gradually introduce geometric teaching contents including primitive concepts such as point, line, and plane and relationships such as, to be between, to pass through, and other concepts like segment and angle; solids such as a sphere, a cylinder, and a cone whose important role in the construction of abstract geometry was stressed by Poincaré; plane figures through which the concept of number is introduced thanks to the counting of sides and vertices and measurements which reinforce the connections between arithmetic and geometry through the repetition of equal elements (Lafforgue, 2007). Thirdly, one should use appropriate educational methods based upon the individual strengths of Down syndrome children related to their competence in visual learning, ability to learn by imitation and good social skills. Thus, mimesis (Scaramuzzo, 2013) is used as a way of learning by assimilation to others, involving body and movement as well as the elements of motor, visual and tactile representative space proposed by Poincaré (1902). The methodology used in the analysis of the data is included in Research for Practice proposed by Faraguer (2014). It shares objectives with Van Manen’s hermeneutic phenomenology (1997) such as focusing on the learning/teaching method and the singularities of each child’s personal experience: classical ethnographic methods for observation, reflection and narration of what happened are also used. From the achievements of children in the two research groups we conclude that children with Down syndrome are able to assimilate genuine geometric knowledge if this approach is adopted, that no mechanical activities involving understanding contribute to the development of abstract thinking allowing them a better understanding of their environment and the ability to enjoy the challenges posed by mathematics. Some light on the general mathematics education for children and the need to base it on the number and shape are also obtained Future research aims to develop a complete program to develop the didactic proposal and the elaboration and testing of activities for development of abstract thinking through mathematical symbolic thought.