Article

Las tareas matemáticas como instrumentos en la investigación de los fenómenos de gestión de la instrucción: un ejemplo en geometría [Mathematical tasks as instruments for research on the phenomena of instruction management: An example in geometry].

Authors:
To read the full-text of this research, you can request a copy directly from the author.

Abstract

This paper shows how the use of problems in geometry can be can be a research tool to bring to the surface some phenomena in the management of instruction. It describes and exemplifies two classes of phenomena: the adaptation of problems so that students’ initial work on them takes advantage of norms of existent instructional situations, and the transition to a different instructional situation that permits the teacher sanction the work done as valuable. The paper discusses these phenomena in the context of an analysis a priori of the problem of the angle bisectors of a quadrilateral.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the author.

... a8bkG A). This is an animated representation of a lesson that we have studied in classrooms, and reported about in other writing (Herbst, 2008(Herbst, , 2012Herbst & Dimmel, 2011;Herbst & Miyakawa, 2008). The instructional goal of the lesson is the tangent segments theorem, which states that a circle tangent to two intersecting lines has its points of tangency equidistant from the point of intersection. ...
... The problem our teacher provides in The Tangent Circle animation is different in that it also creates an opportunity for students to assert the theorem as a justification for the possibility or impossibility to do the construction requested. The sense that a mathematical claim may express the conditions under which something can be done is a paramathematical notion at stake here, and the sense that making such a statement is a desirable, unprompted response to a request, which is better than merely failing to perform that request, is a protomathematical notion at stake here (more about the mathematical values of this task can be seen in Herbst, 2012). In contrast, other ways of posing the problem might allow for a different relationship to the act of stating a theorem. ...
Article
How should we expect growing understandings of the nature of mathematical practice to inform classroom mathematical practice? We address this question from a perspective that takes seriously the notion that mathematics education, as a societal enterprise, is accountable to multiple sets of stakeholders, with the discipline of mathematics being only one of them. As they lead instruction, teachers can benefit from the influence of understandings of mathematical practice but they also need to recognize obligations to other stakeholders. We locate how mathematics instruction may actively respond to the influence of the discipline of mathematics and we exemplify how obligations to other stakeholders may participate in the practical rationality of mathematics teaching as those influences are incorporated into instruction.
... On the one hand, although problem solving is a mainstay of Beatriz's teaching, this lesson represented the pupils' first encounter with mathematical problem posing and therefore represented an interruption to their familiar classroom routines (Herbst, 2012). Her decision to orient problem posing as a purpose in itself underlies the need to encourage the understanding of the notion and nature of mathematical problems (English, 1997). ...
Article
Full-text available
This research focused on understanding the variables inherent in the design and implementation of a mathematical problem-posing task. We developed a single case study of a problem-posing lesson by an Early Childhood Education teacher in a classroom with 4-to 5-year-old children who were unfamiliar with such activities. The results of this study show the potential of considering five variables serving as critical points that pose dilemmas linked to the design and implementation of problem-posing tasks. We found that the task changed from its original design during implementation, implying that the choices the teacher made about the variables were not static and were strongly linked to the purpose of the problem-posing task as well as to the contextual characteristics of the early childhood classroom. This study provides a potentially useful framework for analyzing the design and implementation of problem-posing tasks as a dynamic process.
... Formulación de enunciados matemáticos por el proFesorado en Formación inicial y continua Las tareas son el medio de aprendizaje que construye el profesorado para asegurar la comprensión de conceptos matemáticos, haciendo uso de procedimientos como: razonar, calcular y representar diferentes experiencias en el quehacer matemático del estudiante (Herbst, 2012;Hiebert y Grouws, 2007;Martín y Gourley-Delaney, 2014), prestando especial atención al desarrollo de tareas auténticas, que implican resolver situaciones problemáticas que pueden acontecer en la vida real y cercana del estudiantado (Palm y Nyström, 2009), lo que significa que los profesores pueden reinterpretar la resolución de problemas existentes en los libros de texto en tareas que se convierten en oportunidades para plantear otros problemas y/o estrategias para hacer razonar al estudiante , destacando, el planteamiento de problemas como una parte integral de la resolución de los mismos . ...
Article
Full-text available
The training and experience of teachers in the creation of mathematical verbal problems is a challenge influenced by the characteristics of the statement and the person. We focus on analyzing the level of performance of the tasks formulated based on authenticity, realism, and cognitive mastery built by teachers of continuous training (or in-service) and initial training (or pre-service). Content analysis of 30 statements compiled using the content analysis technique was performed, and attending qualitative descriptions encoded according to the data and semantic sequences categorized. The results indicate a prevalence of authentic tasks by in-service teachers with an inclination to the cognitive domain as opposed to pre-service, who present a slight deviation from fictitious and plausible tasks. Findings are presented in the exercise of continuous teaching, demonstrating an advantage in the approach, understanding of tasks and redesign to generate learning opportunities and, on the other hand, the need to improve professional training in the development of cognitive skills in the formulation of complex verbal problems in future teachers.
... Al comunicar sus resultados presentan un sistema de nociones teóricas buscando describir los procesos de enseñanza y aprendizaje de las matemáticas, así como valorar la idoneidad didáctica de tales procesos desde una perspectiva global. Este aspecto es de suma importancia en la investigación en matemática educativa, pues establece bases para el análisis y evaluación tanto de instrumentos como del mismo desempeño profesoral e institucional en la formación de las personas que, próximamente serán los encargados de la formación de los jóvenes de nuestras comunidades(Herbst, 2011). Además, con este protocolo de análisis se pretende que los profesores en formación y los profesores en ejercicio reconozcan, además de los conceptos y procedimientos, los distintos registros y representaciones usados para representar un objeto, los tipos de justificaciones de propiedades y procedimientos, los procesos de argumentación y generalización. ...
Thesis
Full-text available
En este trabajo evaluó la faceta epistémica de los conocimientos didáctico-matemáticos de futuros profesores de matemáticas, de la Universidad de Sucre al hacer transformaciones de las representaciones de una función. El marco teórico tiene sus fundamentos en el modelo del conocimiento didáctico-matemático (CDM) propuesto por Godino (2009). La investigación se enmarca dentro de un enfoque metodológico mixto (Creswell, 2009) puesto que en ella se combinan técnicas y métodos de investigación cuantitativos y cualitativos. Se tomó una muestra intencional de 56 profesores en formación, de los que se recogió información durante cuatro semestres consecutivos: 28 de semestres intermedios y 28 de los semestres finales. Para analizar la información se hizo un análisis comparativo de medias y se analizaron las asociaciones entre las respuestas dadas por los estudiantes con el nivel del que éstas provinieran, utilizando tablas de contingencias y el coeficiente chi cuadrado de Pearson, y se caracterizaron las configuraciones cognitivas, procesos y elementos matemáticos primarios que emergen de los profesores en formación al dar sus respuestas a los diferentes ítems/tareas del cuestionario, las cuales fueron analizadas utilizando la noción de configuración onto-semiótica propuesta por PinoFan, Godino y Font (2015). En los participantes se encontraron rasgos distintivos del conocimiento común del contenido; mientras las configuraciones cognitivas, procesos y elementos matemáticos primarios encontrados son pobres y ligeramente heterogéneas entre grupos. Un grupo reducido mostró evidencias distintivas los conocimientos ampliado y el especializado del contenido y en otro más amplio se encontraron serias limitaciones en la producción de representaciones de una función, para establecer congruencias entre sus elementos y para decidir sobre la pertinencia procedimental (Sgreccia y Massa, 2012) y emparejar los elementos equivalentes en las diferentes representaciones, evidenciándose la necesidad de fortalecer dichos conocimientos. Además, se visionan algunas cuestiones abiertas que permitan continuar en esta línea de investigación, así mismo algunos aspectos que posibilitarían mejorar los conocimientos didácticos matemáticos del objeto función.
... Una situación problemática es un espacio para la actividad matemática, donde los estudiantes, al participar con sus acciones exploratorias en la búsqueda de soluciones a las problemáticas planteadas por el docente, interactúan con los conocimientos matemáticos y, a partir de ellos, exteriorizan diversas ideas asociada a los conceptos en cuestión (Múnera, 2011). Una tarea es una representación de la actividad matemática, encarnada en las interacciones entre personas e instrumentos culturales, que puede ofrecer una oportunidad para que los estudiantes hagan un trabajo matemático que no es habitual (Herbst, 2012). ...
Article
La experiencia que se presenta pretende valorar la intuición optimizadora en estudiantes de secundaria obligatoria. El problema que se aborda es, dado un conjunto de cantidades, elegir entre ellas las que sumen una cantidad exacta o lo más cercana a ella. El resultado de la experiencia de aula en un contexto específico ha permitido identificar la poca preparación de los estudiantes para este tipo de tarea. La principal conclusión es que los estudiantes están preparados para sumar cantidades, pero les resulta muy difícil elegir los sumandos que sumen una determinada cantidad, desconocen estrategias y son incapaces de inventar heurísticos que les lleve a conseguir el objetivo. La reflexión, consecuencia de la experiencia realizada, es que a los problemas de optimización se les dedica poca atención en la enseñanza obligatoria a pesar de ser de gran utilidad en la vida cotidiana.
... En esta misma línea, Zacharos y colaboradores (2014), en el área específica de las tareas matemáticas en Educación Parvularia, definen una tarea como una actividad de enseñanza con un objetivo y un contenido específicos, como, por ejemplo, la comparación de tamaño entre dos objetos. De esta forma, las tareas se convierten tanto en unidades de análisis que permiten comprender la naturaleza de la enseñanza de matemáticas en el aula, como en el reflejo de la reproducción cultural de las prácticas matemáticas en un contexto dado (Herbst, 2012). Con base en esto, se considerará a las tareas como la unidad de análisis de las experiencias de aprendizaje propuestas por la educadora, que revelan también el objetivo matemático específico propuesto en la enseñanza. ...
Article
INTRODUCCIÓN. Las habilidades matemáticas tempranas juegan un rol importante en el desempeño escolar posterior de niños y jóvenes, marcando diferencias sustanciales en las ventajas de aprendizaje futuras. Específicamente, las tareas matemáticas que ponen en práctica las docentes de educación inicial han demostrado impactar en el desarrollo del pensamiento matemático de los niños. Así, resulta de gran importancia conocer qué tipo de tareas se observan y el tiempo que en ellas se invierte en una muestra latinoamericana y en educación inicial donde estos temas han sido menos investigados. En consecuencia, el presente artículo explora las distintas tareas matemáticas que ocurren en las salas de párvulos de Chile. MÉTODO. Se analizaron vídeos de 31 clases de niños de pre-kínder para identificar y caracterizar las tareas matemáticas implementadas y el tiempo invertido en ellas. RESULTADOS. Los resultados muestran que, en las clases observadas, las profesoras de educación inicial privilegian el trabajo de tareas de contenido numérico, tales como el reconocimiento del número y la correspondencia número-cantidad, en detrimento de aquellas que requieren el dominio y comprensión de los procesos matemáticos. DISCUSIÓN. Los resultados muestran que las tareas matemáticas puestas en juego en las clases observadas de salas de pre-kínder chilenas priorizan el trabajo mecánico y procedimental, tareas que la literatura muestra que contribuyen en menor medida al desarrollo del pensamiento matemático más complejo.
Article
El presente artículo tiene como objetivo ofrecer una propuesta metodológica orientada a la gestión didáctica de la estimulación de la habilidad argumentar desde la matemática. La propuesta se estructuró en tres fases: 1) Diagnóstico de las necesidades de los docentes; 2) Planificación y organización de la superación profesional; 3) Ejecución y evaluación de la superación ofrecida. Para el estudio de la gestión didáctica de los docentes con el fin declarado, se tomaron como dimensiones e indicadores tres condiciones relevantes para lograr la estimulación deseada: estrategias comunicativas, tareas matemáticas y planificación de la clase. Se emplearon como métodos teóricos de investigación: el análisis-síntesis, la inducción-deducción y la modelación; y como método empírico, sobresalió, la revisión documental y la entrevista. La conjugación de estos métodos permitió el desarrollo exitoso del estudio propuesto. Las actividades de superación profesional se desarrollaron en forma de talleres, con el empleo de situaciones didáctico-matemática, que permitieron la reflexión crítica sobre el contenido didáctico y matemático, y el establecimiento del vínculo teoría-práctica. Los participantes reconocieron el valor teórico, metodológico y práctico de las actividades desarrolladas, que derivaron en la búsqueda y realización de nuevas acciones alternativas, a fin de mejorar la gestión didáctica para el propósito deseado.
Thesis
Full-text available
El presente trabajo de investigación tiene por objetivo analizar el desempeño y la percepción de estudiantes de segundo año de bachillerato al resolver tareas no auténticas y sus versiones auténticas. Inicialmente se construyó un cuestionario con cuatro tareas cuya autenticidad fue valorada por un grupo de expertos. Con ayuda de sus comentarios y sugerencias se realizó la modificación de estas tareas para volverlas auténticas. Se aplicaron las ocho tareas a estudiantes de segundo año de bachillerato y a través de sus producciones escritas y un cuestionario con una escala tipo Likert se pudo analizar cualitativa y cuantitativamente que los estudiantes tienen un mejor desempeño cuando resuelven tareas auténticas en contraste con las no auténticas, corroborando así que un ligero cambio en la formulación de la tarea, incluso en la escritura, provoca cambios significativos en el desempeño de los estudiantes.
Article
Full-text available
El uso de nuevos instrumentos para la evaluación de conocimientos y competencias es un reto en la Educación Matemática a nivel universitario. Dentro del marco teórico de este artículo se abordan las rúbricas como una de las herramientas más adecuada para realizar una evaluación formativa. Esta adquiere sentido si se realizan tareas que faciliten tanto el aprendizaje como la evaluación. En este trabajo, se expone el perfil de las tareas de aprendizaje-evaluación a través de la resolución matemática de problemas en el contexto económico. A partir de las relaciones entre los conocimientos y las competencias a evaluar, se diseña una rúbrica en la que se detallan los criterios de evaluación y sus respectivos indicadores de logro para su uso como herramienta pedagógica. La ventaja de esta propuesta es que puede ser adaptada a cualquier otro contexto. Disponible en la siguiente dirección http://revistas.um.es/reifop/article/view/277981/
Article
Full-text available
This study aims to understand the student???s position in instruction. I conceptualize instruction as interactions between the teacher, students, and mathematics, in educational environments (Cohen, Raudenbush & Ball, 2003; Lampert, 2001). In the three manuscripts contained in this dissertation, I look at the position (Harr?? & van Langenhove, 1999) of student from the perspective of the teacher, the student, and the mathematics. ???Mathematical Arguments in a Virtual High School Geometry Classroom??? looks at the position of the student from the perspective of mathematics. It examines the mathematical arguments that could be made by learners in response to a virtual classroom discussion by comparing arguments made by a learner who had taken a geometry class to arguments made by a learner who had not. It shows the virtue of the two-column proof in its affordance to support chains of implications in arguments. However it also shows the drawback of the two-column proof in its lack of flexibility to support backings and rebuttals in arguments. ???Teachers??? Perceptions of Geometry Students??? looks at the position of the student from the perspective of the teacher. It examines teachers??? perceptions of students that are instrumental in the work of teaching. It shows that while ???making conjectures??? teachers perceive students in terms of engagement, ignoring the mathematical value of students??? work. While ???doing proofs??? teachers perceive students in terms of the mathematical content at stake. These different perceptions of students crucially influence how students are supported in their mathematical work. ???The Work of ???Studenting??? in High School Geometry Classrooms??? looks at the position of the student from the perspective of the student. It examines the work that students do in instruction and the tacit knowledge that could guide this work. A theoretical model that describes ???studenting??? is developed as well as a model for the rationality that supports ???studenting.??? Each group of participants involved in this study responded to the same scenario of geometry instruction, depicting a geometry class working on an open ended mathematical problem. These data sets provide three points of view on instruction. Together they serve to inform the instructional position of students.
Article
Full-text available
The research program described in this article has focused on the work students do in classrooms and how that work influences students' thinking about content. The research is based on the premise that the tasks teachers assign determines how students come to understand a curriculum domain. Tasks serve, in other words, as a context for students' thinking during and after instruction. The first section of this article contains an overview of the task model that guided research. The second section provides a summary of findings concerning the properties of students' work in classrooms, with special attention to work in mathematics classes. I conclude with a brief discussion of implications of this research for understanding classroom processes and their effects.
Article
Full-text available
In this theoretical paper we take an exercise to be a collection of procedural questions or tasks. It can be useful to treat such an exercise as a single object, with individual questions seen as elements in a mathematically and pedagogically structured set. We use the notions of ‘dimensions of possible variation’ and ‘range of permissible change’, derived from Ference Marton, to discuss affordances and constraints of some sample exercises. This gives insight into the potential pedagogical role of exercises, and shows how exercise analysis and design might contribute to hypotheses about learning trajectories. We argue that learners’ response to an exercise has something in common with modeling that we might call ‘micro-modeling’, but we resort to a more inclusive description of mathematical thinking to describe learners’ possible responses to a well-planned exercise. Finally we indicate how dimensions of possible variation inform the design and use of an exercise.
Article
Two questions are asked that concern the work of teaching high school geometry with problems and engaging students in building a reasoned conjecture: What kinds of negotiation are needed in order to engage students in such activity? How do those negotiations impact the mathematical activity in which students participate? A teacher's work is analyzed in two classes with an area problem designed to bring about and prove a conjecture about the relationship between the medians and area of a triangle. The article stresses that to understand the conditions of possibility to teach geometry with problems, questions of epistemological and instructional nature need to be asked not only whether and how certain ideas can be conceived by students as they work on a problem but also whether and how the kind of activity that will allow such conception can be summoned by customary ways of transacting work for knowledge.
Article
Edited and translated by Nicolas Balacheff, Martin Cooper, Rosamund Sutherland and Virginia Warfield. Excerpts available on Google Books (link below). For more information, go to publisher's website :http://www.springer.com.gate6.inist.fr/education+&+language/mathematics+education/book/978-0-7923-4526-8
Article
This article focuses on mathematical tasks as important vehicles for building student capacity for mathematical thinking and reasoning. A stratified random sample of 144 mathematical tasks used during reform-oriented instruction was analyzed in terms of (a) task features (number of solution strategies, number and kind of representations, and communication requirements) and (b) cognitive demands (e.g., memorization, the use of procedures with [and without] connections to concepts, the “doing of mathematics”). The findings suggest that teachers were selecting and setting up the kinds of tasks that reformers argue should lead to the development of students’ thinking capacities. During task implementation, the task features tended to remain consistent with how they were set up, but the cognitive demands of high-level tasks had a tendency to decline. The ways in which high-level tasks declined as well as factors associated with task changes from the set-up to implementation phase were explored.
Article
In order to develop students' capacities to "do mathematics," classrooms must become environments in which students are able to engage actively in rich, worthwhile mathematical activity. This paper focuses on examining and illustrating how classroom-based factors can shape students' engagement with mathematical tasks that were set up to encourage high-level mathematical thinking and reasoning. The findings suggest that when students' engagement is successfully maintained at a high level, a large number of support factors are present. A decline in the level of students' engagement happens in different ways and for a variety of reasons. Four qualitative portraits provide concrete illustrations of the ways in which students' engagement in high-level cognitive processes was found to continue or decline during classroom work on tasks.
Article
In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students' understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study.
Article
In this article we examine students' perspectives on the customary, public work of proving in American high school geometry classes. We analyze transcripts from 29 interviews in which 16 students commented on various problems and the likelihood that their teachers would use those problems to engage students in proving. We use their responses to map the boundaries between activities that (from the students' perspective) constitute normal (vs. marginal) occasions for them to engage in proving. We propose a model of how the public work of proving is shared by teacher and students. This division of labor both creates conditions for students to take responsibility for doing proofs and places boundaries on what sorts of tasks can engage students in proving. Furthermore we show how the activity of proving is a site in which complementarity as well as contradiction can be observed between what makes sense for students to do for particular mathematical tasks and what they think they are supposed to do in instructional situations.
Article
What features of a mathematics classroom really make a difference in how students come to view mathematics and what they ultimately learn? Is it whether students are working in small groups? Is it whether students are using manipulalives? Is it the nature of the mathematical tasks that are given to students? Research conducted in the QUASAR project, a five-year study of mathematics education reform in urban middle schools (Silver and Stein 1996). offers some insight into these questions. From 1990 through 1995, data were collected about many aspects of reform teaching, including the use of small groups; the tool that were available for student use, for example, manipulatives and calculators; and the nature of the mathematics tasks. A major finding of this research to date, as described in the article by Stein and Smith in the January 1998 issue of Mathematics Teaching in the Middle School , is that the highest learning gains on a mathematicsperformance assessment were related to the extent to which tasks were et up and implemented in ways that engaged students in high levels of cognitive thinking and reasoning (Stein and Lane 1996). This finding supports the position that the nature of the tasks to which students are exposed detennines what students learn (NCTM 1991), and it also leads to many questions that should be considered by middle school teachers.
Article
The abstract for this document is available on CSA Illumina.To view the Abstract, click the Abstract button above the document title.
Article
This paper presents a model of teachers' construction of mathematics curriculum in the classroom or their curriculum development activities. The model emerged through a qualitative study of two experienced, elementary teachers during their first year of using a commercially published, reform-oriented textbook that had been adopted by their district (Remillard 1996). The aim of the study was to examine teachers' interactions with a new textbook in order to gain insight into the potential for curriculum materials to contribute to reform in mathematics teaching. The resulting model integrates research on teachers' use of curriculum materials (cf. Stodolsky 1989) and studies of teachers' construction of curriculum in their classrooms (cf. Doyle 1993). The model includes three arenas in which teachers engage in curriculum development: design, construction, and curriculum mapping. Each arena defines a particular realm of the curriculum development process about which teachers explicitly or implicitly make different types of decisions. The design arena involves selecting and designing mathematical tasks. The construction arena involves enacting these tasks in the classroom and responding to students' encounters with them. The curriculum mapping arena involves determining the organization and content of the entire curriculum into which daily events fit. Through articulating each piece of the model, the author highlights the complex and multidimensional nature of teachers' curriculum processes, identifies significant characteristics of each arena that have implications for textbook use and instructional change, and indicates areas that call for further understanding and research.
Article
We outline a theory of instructional exchanges and characterize a handful of instructional situations in high school geometry that frame some of these exchanges. In each of those instructional situations we inspect the possible role of reasoning and proof, drawing from data collected in intact classrooms as well as in instructional interventions. This manuscript is part of the final report of the NSF grant CAREER 0133619 “Reasoning in high school geometry classrooms: Understanding the practical logic underlying the teacher’s work” to the first author.All opinions are those of the authors and do not represent the views of the National Science Foundation.
Article
Typescript. Thesis (Ph. D.)--George Mason University, 1996. Includes bibliographical references (leaves 135-148). Vita: leaf 188.
Games people play: The basic handbook of transactional analysis
  • E Berne
Berne, E. (1996). Games people play: The basic handbook of transactional analysis. Nueva York, NY, EE.UU.: Ballantine.
The role of labels in promoting learning from experience in teachers and students
  • J Mason
Mason, J. (1999). The role of labels in promoting learning from experience in teachers and students. En L. Burton (Ed.), Learning mathematics: from hierarchies to networks (pp. 187-208). Londres, Reino Unido: Falmer Press.
The neglected situation The Goffman reader (pp. 229-233)
  • E Goffman
Goffman, E. (1997). The neglected situation. En C. Lemert & A. Branaman (Eds.), The Goffman reader (pp. 229-233). Malden, MA, EE.UU.: Blackwell.
Examining teachers' use of (non-routine) mathematical tasks in classrooms from three complementary perspectives: teacher, teacher educator, researcher
  • R Tzur
  • O Zaslavsky
  • P Sullivan
Tzur, R., Zaslavsky, O., & Sullivan, P. (2008). Examining teachers' use of (non-routine) mathematical tasks in classrooms from three complementary perspectives: teacher, teacher educator, researcher. Proceedings of the 32nd PME and 30th PME-NA Annual Meeting 1, 121-123.
Students' geometry toolbox: How do teachers manage students' prior knowledge when teaching with problems? Documento presentado en el Annual Meeting of AERA
  • G González
  • P Herbst
González, G., & Herbst, P. (2008, March). Students' geometry toolbox: How do teachers manage students' prior knowledge when teaching with problems? Documento presentado en el Annual Meeting of AERA, Nueva York, EE.UU.
Students' geometry toolbox: How do teachers manage students' prior knowledge when teaching with problems?
  • G González
  • P Herbst
González, G., & Herbst, P. (2008, March). Students' geometry toolbox: How do teachers manage students' prior knowledge when teaching with problems? Documento presentado en el Annual Meeting of AERA, Nueva York, EE.UU.