A husband-and-wife team of Dutch educators, Pierre van Hiele and Dina van Hiele-Geldof, noticed the difficulties that their students had in learning geometry. These observations led them to develop a theory involving levels of thinking in geometry that students pass through as they progress from merely recognizing a figure to being able to write a formal geometric proof. Their theory explains why many students encounter difficulties in their geometry course, especially with formal proofs. The van Hieles believed that writing proofs requires thinking at a comparatively high level, and that many students need to have more experiences in thinking at lower levels before learning formal geometric concepts.
This paper is an attempt to provide some background on the various approaches to understanding how people learn and the application of this understanding to teaching. Please note that I only briefly mention the discussion of the theory and associated framework that I personally espouse since it is the topic of another session. I draw a distinction between learning theories and frameworks used for teaching. Learning theories attempt to explain all the factors involved as an individual grows in knowledge and understanding, while epistemological frameworks guide the researcher in investigating certain aspects of learning. The theories of Piaget, Skinner, Vygotsky, and Lakoff and Nunez are presented briefly to set the stage for the epistemological frameworks. In general, these frameworks are based on (and possibly extend) an existing theory of learning. I describe the frameworks of the leading researchers in mathematics education and, as frameworks tend to evolve, try to present them in a fairly current state. A version of this paper was used as part of a preliminary examination for doctoral research in 1996, so many of the references may seem out of date. While more current citations might be available, I only updated those portions that have been modified in recent years. An additional resource is Theories of Mathematical Learning (1996), edited by Leslie P. Steffe and Perla Nesher, which contains the proceedings of Working Group 4 from the 1992 International Congress on Mathematical Education. That book has a broader scope in addressing learning from pre-kindergarten through post-secondary educational situations. Thus its intersection with the following is rather small.
1. CONSIDERACIONES GENERALES Sabemos que los errores forman parte de las producciones de la mayoria de los alumnos, y constituyen, generalmente, un elemento estable en los procesos de ensenanza y aprendizaje de la Matematica en todos los niveles del sistema educativo. Asimismo, si tenemos en cuenta que el correcto aprendizaje de la Matematica es un objetivo comun en los procesos de ensenanza de la misma, es claro que las respuestas incorrectas a las cuestiones que se les plantean a los estudiantes seran consideradas – por parte de quienes estan a cargo de su instruccion – como senales de serias deficiencias, e incluso fracaso en el logro de los objetivos propuestos.
Una breve revision de la literatura de investigacion sobre el conocimiento del profesor de matematicas de secundaria y el papel que este conocimiento juega en la ensenanza sugiere la importancia de considerar al profesor como agente cognitivo y de estudiar la ensenanza desde la perspectiva de los principios constructivistas sobre el desarrollo del conocimiento. Discutimos el modelo Simon (1995) que sigue estos parametros y sugerimos maneras de elaborar este modelo a traves de las ideas de analisis didactico y conocimiento didactico. Hacemos una elaboracion de la nocion de organizadores del curriculo y mostramos su relacion con el analisis didactico.
Abstract This presentation addresses critical issues in the use of information technology in Mathematics Education. By reflecting on human thinking processes, it will consider developments of enthusiastic researchers using technology to teach mathematics as new facilities develop, the possible gains shown by this research and the realities that may be achieved in the classroom. Resumen Esta presentacion aborda aspectos fundamentales en el uso de tecnologias de la informacion en Educacion Matematica. Al reflexionar sobre los procesos de pensamiento humano, va a examinar la evolucion de los investigadores entusiastas utilizando la tecnologia para ensenar matematicas como una nueva facilidad de desarrollo, los posibles beneficios muestran las realidades que se pudieron lograr en el aula.
