¿A dónde va la investigación sobre la prueba?

La traducción de la tesis de Nicolás Balacheff al castellano es un hecho que debe celebrarse, particularmente por lo que esta obra significa para quienes nos interesamos en investigar la problemática de la validación de los cono- cimientos matemáticos en la clase. Como resulta evidente de la lectura del libro, el trabajo reconoce tres filiaciones intelectuales importantes. Hay una filiación epistemológica con las obras de Imre Lakatos (1976, 1978), de quien Balacheff obtiene la noción fundamental de que pruebas y refutacio- nes están necesariamente ligadas a las concepciones de los objetos mate- máticos —las pruebas sirven a la construcción de objetos matemáticos (véanse también Balacheff, 1991a y Balacheff, en preparación) y por lo tanto son irreducibles a la lógica formal. Hay una filiación antropológica con la obra de Pierre Bourdieu (1990), que le permite a Balacheff estable- cer una relación fundamental entre la prueba y las prácticas matemáticas de los alumnos —las pruebas se adaptan a las necesidades de gestión de los objetos matemáticos dentro de una cierta práctica de los conocimientos (o una racionalidad). Y fundamentalmente, hay una filiación didáctica con la obra de Guy Brousseau (1997) de quien Balacheff toma la noción de situa- ción de validación como modelo para pensar en situaciones donde la pro- ducción de pruebas y refutaciones constituya el sentido de la demostración matemática (que se vuelve la solución óptima a un problema de producir una prueba)

Las tareas matemáticas como instrumentos en la investigación de los fenómenos de gestión de la instrucción: un ejemplo en geometría

Este trabajo demuestra cómo el uso de problemas en clase de geometría puede ser utilizado para traer a la superficie fenómenos en la gestión de la instrucción. Se describen y ejemplifican dos clases de fenómenos, en particular: la adaptación de los problemas para que su trabajo inicial por parte de los alumnos se beneficie de las normas de situaciones de instrucción existentes en la clase, y la transición a otra situación de instrucción que permita al maestro adjudicarle valor a la tarea realizada. El trabajo discute estos fenómenos en el contexto de un análisis a priori del problema de las bisectrices de un cuadrilátero

How Theory-Building Research on Instruction can Support Instructional Improvement: Toward a Modeling Perspective in Secondary Geometry

How can basic research on mathematics instruction contribute to instructional improvement? In our research on the practical rationality of geometry teaching we describe existing instruction and examine how existing instruction responds to perturbations. In this talk, I consider the proposal that geometry instruction could be improved by infusing it with activities where students use representations of figures to model their experiences with shape and space and I show how our basic research on high school geometry instruction informs the implementing and monitoring of such modeling perspective. I argue that for mathematics education research on instruction to contribute to improvements that teachers can use in their daily work our theories of teaching need to be mathematics-specific.The ideas expressed in this paper have been developed with the support of the National Science Foundation through grants REC-0133619, ESI-0353285, and DRL- 0918425 (P. Herbst, PI), and DRL- 1316241 (D.Chazan, PI). All opinions are those of the author and do not necessarily represent the views of the Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/134508/1/Herbst-MERGA_plenary.pdf-1Description of Herbst-MERGA_plenary.pdf : Main articl

Studying Professional Knowledge Use in Practice Using Multimedia Scenarios Delivered Online

We describe how multimedia scenarios delivered online can be used in instruments for the study of professional knowledge. Based on our work in the study of the knowledge and rationality involved in mathematics teaching, we describe how the study of professional knowledge writ large can benefit from the capacity to represent know-how using multimedia representations of practice and alternatives to it. These instruments can be used to study what professionals notice and decide to do in practice in ways that improve upon earlier uses of written representations of professional scenarios or videorecorded episodes. In particular, storyboards and animations of nondescript cartoon characters can be used to explore professional knowledge variables systematically while the multimodal representation of human activity in context ensures the face validity of questions.The work reported here has been done with the support of the U.S. National Science Foundation (NSF) grants ESI-0353285 and DRL- 0918425 to the authors. All opinions are those of the authors and do not necessarily represent the views of the Foundation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/110693/1/Multimedia Scenarios and Professional Knowledge.pdf-1Description of Multimedia Scenarios and Professional Knowledge.pdf : Main Articl

Proof, proving, and teacher-student interaction: theories and contexts

This chapter focuses on the role of the teacher in teaching proof and proving in the mathematics classroom. Within an over-arching theme of diversity (of countries, curricula, student age-levels, teachers' knowledge, and so forth), we review three carefully-selected relevant theories: socio-mathematical norms, teaching with variation, and instructional exchanges. Each of these theories starts from the abstraction of observations in existing school mathematics classrooms and uses those observations to probe into the rationality of teachers in order to understand what sustains those classroom contexts and also how such contexts might be changed. We argue that each theory offers insight into the role of the teacher in the teaching and learning of proof and proving. In informing future research, this chapter provides support for meeting the challenge of theorising about the role of the teacher in the teaching and learning of proof and proving in mathematics classrooms across the diverse contexts worldwide

ANALYZING THE DIAGRAMMATIC REGISTER IN GEOMETRY TEXTBOOKS: TOWARD A SEMIOTIC ARCHITECTURE

Diagrams are key resources for students when reasoning in geometry. Over the course of the 20th century, diagrams in geometry textbooks have evolved from austere collections of strokes and letters to become diverse arrays of symbols, labels, and differently styled visual parts. Diagrams are thus multisemiotic texts that present meanings to students across a range of communication systems. We propose a scheme for analyzing how geometric diagrams function as resources for mathematical communication in terms of four semiotic systems: type, position, prominence, and attributes. The semiotic architecture we propose draws on research in systemic functional linguistics (Halliday, 2004; O’Halloran, 2005) and suggests a framework for analyzing how geometry diagrams function as mathematical texts.http://deepblue.lib.umich.edu/bitstream/2027.42/91288/1/DiagrammaticRegisterJKDPH.pdf-

Evaluating Teachers’ Decisions in Posing a Proof Problem

When Geometry teachers pose proof problems to students, it is the teacher who provides the givens and the statement to be proven; we hypothesize that teachers of geometry recognize this to be the norm. This study examined teachers’ decision-making in regards to the posing of a proof problem, and whether recognition of this norm accounted for the decision made. Results of a multinomial regression indicated that the more participants recognized that norm of posing proof problems, the less likely they were to select an action that breached the norm.Research reported had the support of the National Science Foundation through grant DRL-0918425 to P. Herbst. All opinions are those of the authors and don’t necessarily reflect the views of the Foundation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/91282/1/PMENA_2012_Decision_finalweb.pd

Research on Practical Rationality: Studying the justification of actions in mathematics teaching

Building on our earlier work conceptualizing teaching as the management of instructional exchanges, we lay out a theory of the practical rationality of mathematics teaching—that is, a theory of the grounds upon which instructional actions specific to mathematics can be justified or rebuffed. We do that from a perspective informed by what experienced practitioners consider viable but also in ways that suggest operational avenues for the study of instructional improvement, in particular for improvements that enable students to do more authentic mathematical work. We show how different kinds of experiments can be used to engage in theory building and provide examples of initial work in building this theory

Using cases as triggers for teachers’ thinking about practice: A comparison of responses to animations and videos

The AERA proposal is preserved here as summary. The full paper presented in AERA 2012 (Vancouver) is in the Main Article.This study compared conversations among groups of teachers of high school geometry that had been triggered by either a video or an animation representation of instruction and managed with an open-ended agenda. All triggers represented scenarios that departed from what was hypothesized as normative. We used as dependent variable the proportion of modal statements about instructional practice made by a group, which we argue is a good quantitative indicator of the presence of tacit group knowledge about the norms of practice. Animations and videos produced similar proportion of modal statements, but that the types of modal statements differed—with animations being associated with more statements of probability and obligation and videos being associated with more statements of inclination.National Science Foundation grants ESI-0353285 and DRL- 0918425 to Patricio Herbsthttp://deepblue.lib.umich.edu/bitstream/2027.42/89606/1/Modality_Comparison_of_Animations_and_Videos.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/89606/4/Modality_Comparison_AERA_2012.pd

Depict: A Tool to Represent Classroom Scenarios

A functional version of Depict can be found at www.lessonsketch.orgThis document describes design features of Depict, a web based software that allows users to represent classroom scenarios using comics. The document provides the conceptual bases of the design and a description of the user interface. The document also sketches out a direction for further development.This work has been done with support from NSF grants ESI-0353285 and DRL- 0918425 to Patricio Herbst.http://deepblue.lib.umich.edu/bitstream/2027.42/87949/1/Depict_2011.pdf-