Aporte Teórico

Rutina, concepto, conocimiento

Con la rutina también se aprende, pero NO es válido creer que con SÓLO rutina se aprende.

Cuando los niños y jóvenes sólo ejecutan rutinas operativas o recitan correctamente propiedades simplemente están dando muestras de aprender esas rutinas y enunciar esas propiedades, y esto es demasiado pobre para el tiempo escolar de los alumnos.

Lo que proponemos analizar es cómo lograr una eficaz equilibrio entre rutina y análisis conceptual de los contenidos, que hagan comprender a nuestros niños y jóvenes que ellos son capaces de construir un conocimiento y posteriormente aplicarlo eficientemente.

Para confirmar nuestros principios los invito a recordar la primer frase de 2011 inserta en esta página: “Quien sabe donde localizar el conocimiento está en paridad de condiciones con quien lo posee” Miailow; pero a su vez transcribiremos un fragmento de un libro de Algebra, escrito por uno de los más notables algebrista de los tiempos que corren, Irving Herstein: “…..en nuestra educación matemática se nos introdujo muy pronto – generalmente en los primeros años de secundaria- al estudio de los polinomios. Durante una temporada que parecía que no iba a tener fin se nos obligaba hasta el punto de aburrimiento insoportable a factorizarlos, multiplicarlos, dividirlos y simplificarlos. La facilidad en factorizar un polinomio de segundo grado se interpretaba como una muestra de genuino talento matemático”. ¡que estafa! ,agrego yo.

La teoría, aún vigente, que la rutina de ejercicios semejantes logran el aprendizaje de un contenido ha dado sobradas muestras de su fracaso dado que, está absolutamente demostrado, los aprendices después de resolver decenas de ejercicios casi iguales entre sí fracasan ante un nuevo ejercicio del mismo tema pero con distinta presentación y, más fracasan aún, cuando se enfrentan a situaciones donde deben aplicar esas rutinas "aparentemente" bien resueltas previamente.

Esta insistencia en mantener la metodología del ejercicio ha logrado, entre otras tantas malas cosas, la pérdida de posibles matemáticos.

Demostrado está que quien no interpreta el “primer modelo” de la rutina no podrá hacer los otros y quien hizo una interpretación correcta y personal no necesita de otras situaciones casi calcadas.

Veamos un caso rutinario y, si usted es docente póngalo a prueba con sus alumnos.

Los jóvenes definen correctamente D.C.M. entre dos números naturales.

Seguramente han hallado el DCM entre 180 y 450, o entre 84 y 660, o entre 300 y 140, por lo tanto , sin ninguna duda, podrán hallar el DMN de 630 y 108.

Ahora bien, pidamos a esos jóvenes que encuentre el DCM entre 385 y 286.escuchemos sus respuestas, sus comentarios y analicemos sus dificultades.

A esta altura se justifica aplicar el algoritmos de Euclides, hagámoslo:

385 286 286 99 99 88 88 11
99 1 88 2 11 1 8

Por lo tanto 11 es el DCM.

Pero, tomando en cuenta que un alumno de escuela secundaria tiene derecho a dudar del algoritmo y si nadie duda el docente será el incrédulo, podemos demostrar sencillamente la certeza de su aplicación.

Imaginemos que los dos números son A,B para nosotros 385 y 286.

Por lo tanto si A:B = c con un resto r será A= B.c + r (1)

B = r.k + m (2)

r = m.t + h (3)

m = h.d (4) y este tal h es el DCM

veamos:

de (4) en (3) r = t.h.d + h ⇒ r = h.(t.d + 1) (5)

de (5) en (2) B= h.(t.d+1).k + h.d ⇒ B= h.((t.d+1).k + d) (6)

de (5) y (6) en (1) A = h.((t.d+1).k + d).c + h.(t.d + 1) ⇒ A = h.(((t.d+1).k+d).c+(t.d+1)) (7)

por lo tanto h es divisor de A según (7), lo es de B según (6), lo es de r según (5)h es DCM

¡algoritmos demostrado, Euclides fue genial!

Comentarios

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