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Aprendizaje de álgebra con herramienta de modelado y solución de problemas
- 1. Consideraciones para un producto de software educativo para el aprendizaje del álgebra. Eugenio Jacobo Hernández Valdelamar Febrero, 2007
- 3. Antecedentes – planes de estudio de nivel medio superior Números y operaciones básica Ecuaciones lineales Sistemas de ecuaciones lineales Variación directamente proporcional y funciones lineales Ecuaciones cuadráticas y factorización CCH Conjuntos, sistemas de numeración y números reales Productos notables y factorización Operaciones con fracciones y radicales Operaciones con monomios y polinomios Ecuaciones y desigualdades ENP Sistemas de ecuaciones y desigualdades Aritmética: una introducción al álgebra Ecuaciones: modelos generalizadores Función lineal y ecuaciones de primer grado con 2 variables Lenguaje algebraico: operatividad Funciones polinomiales CB Análisis de funciones Estos temas corresponden a los 2 primeros semestres
- 4. Antecedentes – PISA Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen. Espacio y forma Cambio y relaciones Cantidad Incertidumbre Contenidos Thinking and reasoning: Argumentation: Communication: Modelling: Problem posing and solving: Representation: Using symbolic, formal and technical language and operations: Use of aids and tools: Competencias
- 9. Ecuaciones lineales – planes de estudio de nivel medio superior Números y operaciones básica Ecuaciones lineales Sistemas de ecuaciones lineales Variación directamente proporcional y funciones lineales Ecuaciones cuadráticas y factorización CCH Conjuntos, sistemas de numeración y números reales Productos notables y factorización Operaciones con fracciones y radicales Operaciones con monomios y polinomios Ecuaciones y desigualdades ENP Sistemas de ecuaciones y desigualdades Aritmética: una introducción al álgebra Ecuaciones: modelos generalizadores Función lineal y ecuaciones de primer grado con 2 variables Lenguaje algebraico: operatividad Funciones polinomiales CB Análisis de funciones Estos temas corresponden a los 2 primeros semestres Un área interesante es la de funciones/ecuaciones lineales . Hay muchos problemas que involucran estas construcciones.
- 11. Elementos del juego Personaje Máquinas / dispositivos / software para una PDA Pistas, mensajes, trampas, cofres Escenario Contactos de intercambio/comercio Peligros, trampas
- 17. Ejemplo del proceso de planteamiento y solución de un problema v.0.1 Enunciado del problema: María tiene una cubeta con 5 litros de agua; usa dos para lavar los trastes y pone 1 de nuevo en la cubeta; ¿cuantos litros de agua tiene? 2. Identificar valores y representarlos como magnitudes (positivas y negativas) 3. Organizar valores, identificar variables y representar relaciones + 4. Organizar valores con respecto a un sistema de referencia 0 + - + x 1. Establecer el enunciado del problema (lenguaje común); las cantidades quedan resaltadas 5. Opera los valores para obtener el resultado 0 + - 5. Resaltar y comentar el resultado =
- 19. Ejemplo del proceso de planteamiento y solución de un problema v.0.2 Enunciado del problema: María tiene una cubeta con 5 litros de agua; usa dos para lavar los trastes y pone 1 de nuevo en la cubeta; ¿cuantos litros de agua tiene? 2. Identificar valores y representarlos como magnitudes (positivas y negativas) 3. Organizar valores y variables con respecto a un sistema de referencia 1. Establecer el enunciado del problema (lenguaje común); las cantidades quedan resaltadas 5 1 2 0 + - x 4. Resaltar y comentar el resultado x 4 = Enunciado del problema: dada la expresión 7x+2=-54 , encuentre el valor de x 2. Identificar valores y representarlos como magnitudes 3. Organizar valores y variables con respecto a un sistema de referencia 1. Establecer el enunciado del problema (lenguaje común); las cantidades quedan resaltadas 2 -54 0 + - 7x 4. Resaltar y comentar el resultado x -8 = 7x = -54 2 x x x x x x x x x =
- 21. Ejemplo del proceso de planteamiento y solución de un problema v.0.2 a Enunciado del problema: dada la expresión 8x-1=23-4x , encuentre el valor de x En este caso, ambos lados de la ecuación tienen una variable por lo que pueden graficarse 2 rectas. La intersección está en x=2
- 25. Ejemplo del proceso de planteamiento y solución de un problema – visualización de procesos de cambio Enunciado del problema: Un caracol cayó a un pozo de 6 metros de profundidad al iniciarse el día; durante el día trepaba 3 metros, pero por la noche descendía 2. ¿Cuántos días tardó en salir del pozo? Objeto/móvil Reglas para cada instante de tiempo: Unidades de tiempo: día 3 2 Meta (condición de salida): distancia = 6 t d 6 0 + - Toda esta información se usa para iterar las reglas hasta alcanzar el objetivo. 4 Valida meta Valida meta
Notas del editor
- Propuesta originalmente presentada a Tecnología Educativa Galileo (FAR), 2007.
- CCH- Colegio de ciencias y humanidades ENP- Escuela nacional preparatoria CB- Colegio de bachilleres
- R = Read for understanding, P = Paraphrase – in your own words, V = Visualize – draw a picture or diagram, H = Hypothesize – make a plan, E = Estimate – predict the answer, C = Compute – do the arithmetic, C = Check – make sure everything is right
- Tools While working on activities, we used, or could have used, the following manipulative, electronic, and traditional math tools: grid tools (cubes or tiles, graph paper, geoboards, dot paper); visualization tools (function diagrams, the Lab Gear, Cartesian graphs); computational tools (calculators, graphers); and paper-and-pencil tools (tables of values, symbol manipulation). We see four main advantages to a tool-based approach: Access: By providing immediate feedback, these tools make it possible for all students to get involved with significant mathematical concepts: data analysis; using tables, looking for patterns; generalizing and using variables; linear, quadratic, and rational functions; squares, square roots and operations with radicals; the distributive law and factoring; optimization and equation-solving; not to mention connections with many other topics. Discourse: The tools also facilitate the transition from a traditional class format into one where discovery learning, problem solving, and cooperative work are the norm. They are not only objects to think with, but also objects-to-talk-about. Instead of the teacher's authority being the sole arbiter of correctness, tools make it possible for students to use reasoning and discussion about a concrete reference as a way to judge the validity of mathematical statements. Independence: As students work with tools over time and develop more and more understanding of the concepts of algebra, they have less and less need of certain tools (such as manipulatives or function diagrams), which have merely served as a bridge to understanding abstract ideas. On the other hand, they become more sophisticated users of other tools, (such as calculators and electronic graphers), which will remain useful throughout their mathematical careers. In both cases, the students are more self-reliant, and therefore more self-confident. Multiple Representations: There is a synergy in the interaction of math tools. A student who has thought about square roots in a multidimensional way, with the help of geoboards, dot paper, radical gear, calculators, and graphing calculators, has much more depth of understanding than one who has only practiced disembodied operations with radicals, particularly if the relationships among the representations have been made explicit. Top Themes The twenty-four activities were all inspired by the theme of area. The various tools, themes and concepts we touched on are all inter-related, as can be seen on the map of these connections ( Figure 19 ). Naturally, we were not able to address everything shown on the map in this paper, but it is not unrealistic to expect many of these topics to be among the ones addressed in a one-year course. Note that the numerous connections between algebra topics, and between them and geometric topics are revealed by the thematic approach, while it is obscured in the traditional curriculum which assumes an arbitrary sequence within algebra, and fails to make the geometric connections. (The corresponding map for the traditional algebra course can be seen in Figure 20 .) As shown above, geometric themes such as area and distance are a gold mine of exciting work in Algebra. However, since an introductory Algebra course needs to include topics such as exponential functions, laws of exponents, equations and inequalities, and so on, other themes will be needed. Here are some examples of themes which we have found to be rich with mathematical content at the appropriate level: motion; optimization; growth and change; satisfying constraints; making comparisons. Well-chosen themes offer: Connections within algebra and to other branches of mathematics. Motivation for specific topics and for learning algebra in general. Applications to other fields. Spiraling , since a core idea can be previewed, then studied, and later on reviewed, each time in the context of a different theme. Top Concepts While symbol manipulation is a useful tool, accurate and/or speedy manipulation is no longer defensible as a central goal of the new algebra. Instead the goal should be understanding of concepts . Tools and themes are the means, not the end: their purpose is to help create a course where students can learn algebra concepts such as function, numbers, variables, operations, equations, and more generally mathematical structure . Tools and themes create an environment where students are empowered and motivated, where problem solving, discovery and cooperative learning can thrive, and where skills can develop naturally and in context. While they are necessary, even well-chosen tools and themes are not sufficient to guarantee that students will generalize and transfer their understanding from one context to another. Essentially, this will come from our students learning how to think mathematically, which is best achieved through group work, teacher-led discussion, and writing, and best monitored through a range of assessment techniques such as reports, projects, portfolios, and tests. These changes, which are well presented in the NCTM Standards (1989, 1991), must be made in conjunction with the implementation of the contextual tool-based approach. As students' understanding of algebra concepts deepens, they are gaining symbol sense : an appreciation for the power of symbolic thinking, an understanding of when and how to apply it, and a feel for mathematical structure. Symbol sense is a level of mathematical literacy beyond number sense, which it subsumes. It is the true prerequisite for further work in math and science, and the real purpose of a new Algebra course.
- CCH- Colegio de ciencias y humanidades ENP- Escuela nacional preparatoria CB- Colegio de bachilleres