Research in Mathematics Education, 1 (2), 98-104. doi - CiteSeerX

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Editorial Javier Díez-Palomar1 1 ) Departamento de Didáctica de las Ciencias Experimentales y las Matemáticas, Universidad de Barcelona, España.

Date of publication: June 24th, 201 2

To cite this article: Díez-Palomar, J. (201 2). Editorial. Journal of

Research in Mathematics Education, 1 (2), 98-1 04. doi: 1 0.4471 /redimat.201 2.06

To link this article: http://dx.doi.org/1 0.4471 /redimat.201 2.06

PLEASE SCROLL DOWN FOR ARTICLE The terms and conditions of use are related to the Open Journal System and to Creative Commons Non-Commercial and Non-Derivative License.

REDIMAT - Journal ofResearch in Mathematics Education Vol. 1 No. 2 June 2012 pp. 98-104

Editorial

Javier Díez-Palomar

Universidad de Barcelona

E

s un placer introducir el segundo número del primer volumen de nuestra revista. Ya han pasado cuatro meses desde la última vez que salió REDIMAT. La andadura que iniciamos en su momento sigue con fuerza e ilusión. Me alegra comprobar la buena salud de la que goza nuestro ámbito de estudio. La investigación en educación matemática, una disciplina que a menudo se referencia como muy nueva o reciente, no cesa de crecer y crecer. Tenemos trabajos interesantes y serios que están abriendo puertas que conducen a afianzar nuestra disciplina. Revistas ya consolidadas, y nuevas revistas que aparecen en el horizonte, que ayudan a consolidar nuestra disciplina y a abrir nuevos espacios de discusión científica seria y rigurosa. Necesitamos de esos espacios para mejorar nuestro trabajo todavía más si cabe, para compartir y aprender, para que la investigación se conecte con la práctica, y la práctica se base en evidencias científicas, no en lo que autores de referencia internacional denominan como ocurrencias (Gómez, Puigvert, & Flecha, 2011). REDIMAT inició su andadura con esta voluntad muy clara. Y seguimos contribuyendo a ello. Estamos contentos de recibir propuestas, y fomentar el diálogo igualitario basado en lo que Habermas (1987) llama argumentos con pretensiones de validez. Habermas es uno de los autores más referenciados en las ciencias sociales. Su trabajo desde la teoría de al argumentación nos ha permitido clarificar las bases epistemológicas, ontológicas y metodológicas de la discusión científica. La argumentación se orienta al entendimiento. En una situación de diálogo, lo que vale es el mejor argumento, no la posición de poder de quien emita el argumento. 2012 Hipatia Press ISSN 2014-3621 DOI: 10.4471/redimat.2012.06

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Un artículo es válido en la medida que aporta evidencias sólidas sobre las que sustentar las contribuciones que hace; no por la posición que ocupe la persona que lo ha escrito. El conocimiento avanza por vericuetos a veces no del todo claros, pero el contraste de ideas, el intercambio de argumentos, el diálogo, siempre ha estado en el origen de su avance. Esto ha sido así siempre, y precisamente en matemáticas tenemos buena prueba de ello. El teorema de Pitágoras, que puede que sea uno de los temas más populares de las matemáticas, ya existía mucho antes de que Pitágoras viviera. De acuerdo a historiadores de la matemática como Boyer (1969) por ejemplo, el conocimiento de la relación particular entre los lados de un cierto triángulo cuyos lados miden 3, 4 y 5 unidades (respectivamente) es algo que ya podemos encontrar en vestigios que nos han llegado de la antigua Mesopotamia a través de piezas de arcilla como la tablilla “Yale o YBC 7289” conservada en la Universidad de Yale, o la “Plimpton 322”, que está en la Universidad de Columbia. Ambas están datadas entre el 1900 o 1600 a.C., mucho antes del periodo en el que se supone que vivió Pitágoras de Samos (Boyer, 1969; González Urbaneja, 2008). Los egipcios, también antes de que Pitágoras existiera, usaban la relación del triángulo rectángulo de lados 3,4 y 5 (o triángulos de dimensiones proporcionales). Lo hacían para trazar las lindes de los campos entre una inundación y otra, y para diseñar las bases de las grandes pirámides. Pitágoras de Samos viajó a estos territorios, y es posible que tuviera contacto con este tipo de conocimientos, cuando hizo la que se considera la primera demostración formal del teorema que lleva su nombre1 . De lo que no cabe duda es que ese conocimiento surge (y se transmite) a través del contacto entre culturas, entre personas, que comparten conocimiento. El diálogo era una de las características de la civilización helena. Fue una de las cosas que aprendieron los romanos de los griegos. El ágora clásica era espacio de debate público (político), pero también era testigo de aprendizaje a través del diálogo socrático y platónico. Quienes hemos visto Ágora, la película, seguro que recordamos la escena en la que Hipatia de Alejandría está discutiendo sobre cuestiones matemáticas con sus discípulos. La escuela de Hipatia tuvo mucha influencia en su época, y estudiantes de todo el mundo romano acudían a ella. Hipatia es la primera mujer matemática de la historia (o como mínimo, la primera de la que se tiene noticia). En un momento de declive de la civilización “occidental”, Hipatia

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dejó tras de sí importantes contribuciones a las matemáticas y al conocimiento del mundo. Después de una etapa que en los libros de historia se suele denotar como “oscura”, donde el conocimiento se encerraba en las celdas y en los pasillosde los conventos de clausura (Boyer, 1969), el esfuerzo de personas como Leonardo de Pisa (Fibonacci), Luca Pacioli, Leonardo da Vinci, Tartaglia, Cárdano, Ferrari, Recorde, o Copérnico, entre otros muchos, abrió las puertas a un Renacimiento en todos los ámbitos del saber, no solo en las matemáticas. Muchos de ellos fueron grandes viajeros: Fibonacci, por ejemplo, que era hijo de un mercader, viajó por Egipto, Siria y Grecia, y tomó clases con un maestro musulmán. Estuvo en contacto con el conocimiento matemático acumulado por generaciones. Sus contribuciones se estudian hoy en día en nuestras escuelas, y describen abundantes fenómenos de la naturaleza. La matemática es deudora de los diálogos entre clásicos de todos los tiempos, como puedan ser Newton y Leibniz, por ejemplo. El conocimiento se ha ido nutriendo de esta comunicación. Newton escribió “a hombros de gigantes”. Casi doscientos años después Robert Merton, uno de los mejores sociólogos de la ciencia, escribió A hombros de gigantes (1965), donde vuelve a recoger la importancia del diálogo en el desarrollo de la ciencia. En el siglo XIX la práctica más común eran los intercambios en los “clubs” o “sociedades” científicas que proliferaron en diversos países. Einstein en pleno siglo XX escribe sobre su pertenencia a la prestigiosa Academia Prusiana de las Ciencias (de la que más tarde renunciará, por motivos de discriminación contra los judíos). Los encuentros anuales, nuestros congresos, conferencias, simposios, jornadas, animan y profundizan en ese diálogo científico en pos de nuevas fronteras. No hace muchos años era común enviarse misivas por correo, y abundar en el género epistolar. La historia está repleta de casos de discusiones matemáticas a través del correo. Quién no recuerda la correspondencia entre Sophie Germain y Lagrange, o entre Ramanujan y Hardy. Son ejemplos claros de cómo los diálogos conducen a nuevos territorios matemáticos, y hacen que el conocimiento avance. Ahora disponemos de la plataforma que nos ofrecen las tecnologías de la comunicación, para intercambiar los resultados de nuestras investigaciones. Tenemos Internet y todas las herramientas tecnológicas que se

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sustentan sobre la red virtual, que nos permiten comunicarnos a velocidad “del pensamiento”: con un “clic” del ratón enseguida enviamos un email, una intervención en un foro de discusión, un nuevo hashtag en Twitter, o una entrada en Facebook. Pero también tenemos las revistas científicas. En este segundo número del primer volumen, REDIMAT presenta cuatro nuevos artículos. En el primero de ellos, Bill Zanher nos ofrece una interesante discusión sobre el efecto que tienen los diferentes contextos en potenciar el aprendizaje de las matemáticas. Sitúa su trabajo en un curso de matemáticas de noveno grado (chicos de 14 a 15 años), que están estudiando álgebra. Zanher analiza lo que sucede cuando facilita la actividad matemática enmarcada en dos contextos diferentes: el caso de la fabula de la tortuga y la liebre, y un problema relacionado con pupitres hexagonales. Desde una perspectiva basada en la perspectiva socio-cultural y en concreto en la teoría de la acción, Zanher analiza cómo responden los estudiantes a contextos reales para ellos, y contextos que no lo son. El caso de los pupitres resulta conocido y, por tanto, comprensible para todos ellos. Sin embargo, la fábula de la tortuga y la liebre es desconocida. En ese caso, Zanher relata como dedican una parte del tiempo de aprendizaje a entender la fábula, mientras que en el caso de los pupitres hexagonales, la discusión gira en torno a las matemáticas desde el primer momento prácticamente. La familiaridad del contexto es clave para entender la producción de diálogo matemático. Zanher concluye con interesantes implicaciones para la enseñanza y el aprendizaje de las matemáticas de este hecho. En el segundo artículo de este nuevo número de REDIMAT Nielce Lobo da Costa y Maria Elisabette Brisola Brito Prado analizan las interacciones de un grupo de estudiantes de magisterio en un entorno de aprendizaje colaborativo. Para ello construyen una herramienta de análisis cualitativo, que les permite dibujar las redes interactivas que se establecen dentro de cada grupo, según un conjunto de 15 aspectos con los que caracterizan la práctica colaborativa de aprendizaje. Su análisis es interesante por cuanto que ponen en primer término elementos clave de la interacción entre los actores del aula de matemáticas. Los mapas que construyen de cómo se relacionan dichos elementos sugieren conexiones inesperadas (y por eso mismo, interesantes) como la relación que se establece entre las acciones docentes, la reflexión sobre las prácticas y


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el desarrollo de la autonomía; y también evidencian otras que no por intuidas dejan de ser menos significativas, como es el caso de la necesidad de confianza para que haya o se produzca aprendizaje con el “otro”. Las autoras concluyen que existe una conexión entre las redes colaborativas de aprendizaje, como espacios colectivos que aparecen en el contexto de la formación de profesorado, y su potencial para potenciar el desarrollo profesional del profesorado. En el tercer artículo José Carlos Cortés Zavala presenta un trabajo sobre el uso de artefactos concretos en actividades de geometría analítica. En concreto, se centra en el caso de la elipse. Después de una contextualización histórica que sitúa al lector/a en el desarrollo de diferentes artefactos para analizar polígonos, Cortés explica cómo seleccionó dos de ellos, el elipsógrafo de palancas y colisa Inwards y el antiparalelogramo articulado de Van Schooten. El objetivo del estudio que nos presenta aquí es experimentar actividades que puedan servir como recurso didáctico para acercar a los estudiantes de bachillerato a la demostración y la construcción de conceptos en el ámbito de la geometría analítica, y en concreto en el caso de al elipse. El autor detalla todo el proceso de construcción de los artefactos, y cómo los estudiantes usaron ambas herramientas para responder a una serie de preguntas planteadas en una hoja de actividades didácticas. Los diálogos que se incluyen en la discusión de los resultados son realmente interesantes, y permiten seguir las discusiones matemáticas que se produjeron entre los estudiantes. Emergen múltiples ideas geométricas (congruencia, simetría, etc.). Cortés concluye reflexionando en el impacto que tiene el uso de una metodología de trabajo en grupos pequeños, de manera cooperativa. El cuarto y último artículo, escrito por Kai-Ju Yang continúa en el ámbito del aprendizaje de la geometría. Pero esta vez se nos remite al ámbito de las creencias y de las actitudes. Yang analiza las implicaciones que tienen sobre la preparación de programas para la formación de profesorado. El autor repasa las contribuciones de estudios previos sobre el papel que juegan las creencias y las actitudes en el aprendizaje de las matemáticas. Partiendo de dicho análisis, Yang se pregunta de dónde proceden las actitudes negativas hacia las matemáticas que tienen muchos estudiantes. Para responder a esta pregunta, recurre al modelo elaborado por McLeod (1989) para analizar las creencias y las actitudes, que distingue entre representaciones, discrepancias y metacognición.

103 Javier Díez-Palomar - Editorial Una vez establecido el marco de análisis, Yang presenta las actividades que usó para realizar su estudio, centradas en doblar piezas de papel a fin de encontrar lugares geométricos (incentro, bisectriz) y demostrar propiedades entre ellos. En el trabajo de campo Yang se centra en tres tipos de cuestiones: maneras de aprender geometría, logros en el ámbito de la geometría, y reflexiones sobre el aprendizaje de esta materia. Las citas que se incluyen de las entrevistas con los estudiantes de formación de profesorado son de gran interés para analizar el impacto sobre el aprendizaje (sobre los logros) de la geometría tanto de las creencias previas sobre la posibilidad de cada cuál de realizar correctamente las actividades, como de las emociones que produce esta materia. Yang concluye que la manera de presentar los conceptos geométricos influye de manera importante en la respuesta de los estudiantes; así como la falta de conexión entre las expectativas que uno/a tiene un los logros reales que alcanza. El análisis de la auto-evaluación del aprendizaje y la conciencia de las reacciones emocionales hacia la geometría que hace Yang le lleva a confirmar el enfoque de McLeod (1989) sobre el papel que juegan las creencias y las actitudes en el aprendizaje de las matemáticas. Además de los tres elementos del enfoque de McLeod, Yang identifica un cuarto aspecto, la comprensión, que juega un papel crucial en el aprendizaje de la geometría. Entender o no entender, marca una diferencia fundamental en el aprendizaje; y se conecta directamente con los otros elementos del enfoque basado en creencias y actitudes. Estos cuatro artículos ofrecen múltiples aportaciones interesantes, que seguro que van a animar debates que nos lleven a comprender más en profundidad los elementos que participan en la enseñanza de las matemáticas, para mejorar nuestro conocimiento en base a evidencias científicas, y sobre ello también nuestras prácticas en el aula. Dejo pues la palabra a los lectores y lectoras. Disfruten de la lectura. Referencias

Boyer, C.B. (1969). A History ofMathematics. New York: John Wiley and sons. Gomez, A., Puigvert, L., Flecha, R. (2011). Critical communicative methodology: Informing real social transformation through research. Qualitative Inquiry, 17(3), 235-245.


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González Urbaneja, M. (2008). El teorema llamado de Pitágoras. Una historia geométrica de 4.000 años. Sigma, 32, 103-130. Habermas, J. (1987). La teoría de la acción comunicativa (2 vols.). Madrid: Taurus. Merton, R. (1965). On the shoulders of giants. New York: The free press.

Notas No se han conservado documentos sobre la vida de Pitágoras, por lo que todo cuanto gira a su vida y obra está rodeado de claroscuros. Boyer (1969) recomienda atribuir los hechos y descubrimientos a los “pitagóricos”, los miembros de la escuela que creo Pitágoras, a pesar de que habitualmente se suele atribuir al “maestro” todos los descubrimientos. 1


“Nobody Can Sit There”: Two Perspectives on how Mathematics Problems in Context Mediate Group Problem Solving Discussions William Zahner1 1 ) Department of Curriculum and Teaching, Boston University. Date of publication: June 24th, 201 2

To cite this article: Zahner, W. (201 2). "Nobody can sit there": Two

perspectives on how mathematics problems in context mediate group problem solving discussions. Journal of Research in Mathematics Education, 1 (2), 1 05-1 35. doi: 1 0.4471 /redimat.201 2.07

To link this article : http://dx.doi.org/1 0.4471 /redimat.201 2.07 PLEASE SCROLL DOWN FOR ARTICLE The terms and conditions of use are related to the Open Journal System and to Creative Commons Non-Commercial and Non-Derivative License.


“Nobody Can Sit There”: Two Perspectives on how Mathematics Problems in Context Mediate Group William Zahner

Boston University

Abstract

This study examines how a group of bilingual ninth grade algebra students discussed two word problems stated in terms of "real life" contexts. Using a lens of mediated action (Wertsch, 1998), the analysis reveals two distinct ways that the problem contexts influenced the group's mathematical reasoning. In one problem, the problem context afforded particular ways of interpreting the given inscriptions, which had benefits as well as costs. In the other problem, the unfamiliar story and terminology appeared the hinder the group's mathematical reasoning. These two forms of context mediation are discussed in light of current research on the use of real life problems in mathematics education. Keywords: mathematical discussions, mediation, word problems, algebra, group discussions

2012 Hipatia Press ISSN 2014-3621 DOI: 10.4471/redimat.2012.07

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olving mathematics story problems set in an imaginary context is a common experience for school children. Prior research suggests that giving students mathematics problems set in a familiar context can promote their problem solving success by increasing their motivation and drawing upon their expertise from outside of school (Baranes, Perry, & Stigler, 1989; Moschkovich, 2002; National Council of Teachers of Mathematics, 1989, 2000). However, the semantic complexities and non-mathematical considerations that can arise when solving problems with real life context may also obscure mathematical relationships, leading students away from providing the expected responses (Boaler, 1993; Gerofsky, 1996; Martiniello, 2008; Walkerdine, 1988). This paper examines two ways that the “real life” contexts of word problems entered bilingual students’ discussions of mathematics story problems. Using the lens of mediated action (Vygotsky, 1978; Wertsch, 1991, 1998), the analysis examines how drawing upon “real life” contexts given in the mathematics problems both facilitated and hindered the group’s problem solving efforts. In particular, detailed analysis of the students’ talk during problem-based group discussions shows how the given problem contexts had a complex interaction with the reasoning and resources that students drew upon while solving two non-routine mathematical problems. This analysis shows how the problem contexts mediated the students’ problem solving discussions in at least two distinct ways. This observation leads to a discussion that can deepen the mathematics education community’s understanding of the affordances and constraints of solving mathematics problems stated in terms of “real life” context. The primary data for this paper are drawn from a study of how high school algebra students learned and reasoned about rates during group discussions. The data are used to develop and to illustrate a theoretical connection with the notion of mediated action. While the larger study investigated the relationship between group interactions and students’ mathematics learning, this paper focuses on the narrower question, How might adding “real life context” to a mathematics problem afford and constrain students’ group problem solving? The theoretical framework below introduces mediated action


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(Wertsch, 1991, 1998) from cultural historical activity theory. Then a brief literature review examines results from previous studies on how students solve mathematics problems stated with real life context. Next, a case study of how one group of students solved problems in context is introduced to develop a distinction between two influences of “problem context” that may be conflated. Finally, the discussion returns to the theoretical and practical considerations that arise from the data and theory presented in this paper. Theoretical Framework The overarching study, from which this paper was drawn, was rooted in a sociocultural approach to teaching and learning mathematics (Forman, 1996; Moschkovich, 2004). The central concept from sociocultural studies that is used here is mediation. Mediated action has been used since Vygotsky and colleagues argued that human thinking and goaldirected actions are inseparable from the cultural tools employed to reach goals (Luria, 1979; Vygotsky, 1978, 1986). Language is one essential tool for thought, and Vygotsky (1986) argued that children’s use of “egocentric speech” was evidence that children internalize a socially shared tool for thinking. However, the mediation of cultural tools in human thinking and action goes beyond the use of language in verbal thought. Wertsch (1991, 1998) showed how human actions, such as pole vaulting and doing mathematical calculations, are also instances of mediated action. He argued that goal-directed actions, for example, performing multi-digit multiplication, cannot be analyzed without accounting for the mediation of cultural tools (e.g., decimal numbers, algorithms, & calculators) used to reach those goals. Wertsch’s (1998) example of multidigit multiplication is helpful for illuminating how doing calculations is a form of mediated action that is deeply shaped by the use of cultural tools. Most adults who have learned a standard multiplication algorithm in school could compute 343 × 822 = 281 946 without the aid of a calculator. However, the multiplication algorithm, and even the decimal number system used to represent the numbers, are culturally-developed meditational tools. The affordances of these tools is made visible when the problem is stated in a different way, for example, CCCXLIII × DCCCXXII.

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Numerous mathematics education researchers who use a semiotic perspective have drawn on this framework of mediated action to analyze children’s mathematical activity (e.g., Radford, 2001; Radford, Bardini, & Sabena, 2007; Walkerdine, 1988). One important insight from these studies has been that mathematical activities of all kinds have semiotic entanglements—the notion that mathematics can happen independently of human language and sign systems is a myth. Moreover, mathematics itself is transformed as humans’ semiotic resources expand. For example Hegedus and Moreno (2011) have argued that new digital technologies are transforming the very nature of what is called mathematics, what constitutes mathematical activity, as well as the possibilities for mathematics teaching and learning. This paper follows in this tradition, but rather than analyzing high technology and digital media, it focuses on how the stories and hypothetical situations given in word problems mediated students’ mathematical activity. Wertsch’s (1991, 1998) notion of mediation provides a framework to analyze how the imagined problem context shaped students’ group discussions and influenced the mathematical conclusions they reached. This paper is drawn from a larger study of how students learned key concepts in algebra by engaging in discussions with a small group of peers. In the larger study, learning was considered as a process of appropriating and using culturally shared tools for reasoning (Forman, 1996; Moschkovich, 2004; Rogoff, 1990). For example, in the algebra classrooms where this study was situated, the students appropriated ways of reasoning about the slope of linear functions by focusing on the “rise” and the “run” between two points on the line. The definition of “problem context” for this analysis captures one meaning of “context.” This paper focuses on the context as the “cover story” in word problems (Gerofsky, 1996). In particular, the problem context is defined as the characters, objects, and relationships introduced in the problem statement. There are many other possible meanings for “context” in studies of learning. For example, the school setting, and the social composition of the group of students working together are also part of the problem-solving context. Of course, this choice of focus on problem contexts only addresses part of the overarching situation when students work together as a group to solve


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mathematics problems. However, this choice was necessary in order to make this analysis manageable, and to allow for a detailed explication of how seemingly inconsequential details of mathematics word problems can shape students’ discussions. Prior Research Stating problems with stories has been part of mathematics since antiquity (Gerofsky, 1996; Schoenfeld, 1992). Oftentimes the stories that appear in word problems are unrealistic, and the stories are carefully constructed to require the use of a recently learned algorithm (Schoenfeld, 1992). Part of a student’s task when solving such problems is to learn to attend to certain features in the problem, and to recognize what quantities and operations should be combined to produce the desire result. Another important skill that students develop through solving problems is learning when it is permissible to ignore the context. Despite the fact that the problem context in word problems is often regarded as superfluous, there is some evidence that the context within which a mathematics problem is presented influences children’s solutions and their mathematical success. Researchers who focus on the interaction between “everyday” and “academic” mathematics have found that many people and students can do certain mathematical tasks in everyday settings that they find impossible when given as a school mathematics problem (Brenner, 1998; Carraher & Schliemann, 2002; Moschkovich, 2002; Saxe, 1995). For example, in one landmark study, Carraher, Carraher, and Schliemann (1985) discovered that children who sold food on the streets in Brazil were quite adept at doing the arithmetic necessary to make change in commercial transactions, but these children could not do the “same” calculations in decontextualized form with paper and pencil. Carraher, Carraher, and Schliemann argued that changing the problem context from a selling problem to a school problem also changed the arithmetic resources that children used to do their calculations. Therefore, the statement of problems in context cued the children’s problem solving choices and success in each condition. This work has been followed by multiple studies probing the affordances of using real life contexts as a tool for teaching school

110 William Zahner - Mediation ofProblem Contexts mathematics (Boaler, 1993; Brenner, 1998; Civil, 2002; Civil & Andrade, 2002; Gerofsky, 1996; Greer, 1997; Moschkovich & Brenner, 2002; Saxe, 1995). For example Brenner (1998) studied the affordances of using coins to teach decimal numbers. Many elementary school curricula in the US have used coins to represent the decimal number system and place value concepts. Brenner conducted an ethnographic study of how Hawaiian children used money, and she found that the way Hawaiian children used money outside of school did not match the way that coins were used to teach mathematical concepts in school. One specific mismatch was that the children treated a quarter (25 cents) as the basic unit of money in their purchases, while the school curriculum treated the penny as the basic unit (Brenner, 1998). One curriculum-focused response to this work has been a push to use more realistic “real life” contexts in mathematics curricula, which focus on building meaningful connections between important mathematical concepts and the real life context of school children’s lives (Boaler, 1993; Greer, 1997; National Council ofTeachers of Mathematics, 1989, 2000). A related body of research from the cognitive framework has also examined the costs and benefits of adding context and personalizing mathematics word problems (e.g., Koedinger & Nathan, 2004; Walkington & Maull, 2010). As with the findings from the socioculturally focused research, these studies have shown that some forms of context can help students reach correct solutions, especially when a problem is stated in a way that draws on students’ resources and that motivates the student to persevere. In particular, Koedinger and Nathan found that students were able to solve more complex problems when they were stated as stories, as opposed to bare algebraic equations. In sum these studies indicate that, under felicitous conditions, adding context can aid students’ mathematical problem solving. However, there are important limitations inherent to the use of context. One critical issue is that the use of contextualized problems might interfere with the intended mathematics curriculum. While students may draw on certain aspects of the problem context as an aid to solve complex word problems, students are also expected to know when to ignore real life considerations in order to solve a problem


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using the intended mathematical algorithm. Both Gerofsky (1996) and Walkerdine (1988) observed this can be problematic for children, especially those who are not aware of the game, or whose out of school language practices are not congruent with the use of langauge(s) in school. At times, using too much knowledge about the context can actually result in students giving wrong answers. The issue of adding context to word problems can take on an added layer of complexity for students who are learning the language of instruction (Abedi & Lord, 2001; Martiniello, 2008). Since contexts for mathematics problems are usually stated in the form of a written story, adding context might also add unnecessary linguistic complexity to mathematics problems. This linguistic interference can, in turn, obscure the mathematical proficiency of students who are learning the language of instruction and result in educators making incorrect inferences about students’ mathematical knowledge. For example, Martiniello investigated how English Learners (ELs) performed on a state-wide mathematics assessment in the US that included some problems stated as stories. She found that ELs did worse than would be expected by their mathematical proficiency on questions that used unfamiliar terminology. In follow up interviews with selected students, she found that although the students knew the mathematical concept being assessed (e.g., using the counting principle to compute combinations), they were unable to answer questions on these topics because of the language used to state these questions. In light of the complexity of findings in prior research, it may be too simplistic to ask whether adding context to mathematics problems will help or hinder students’ performance. A more apposite question may be to ask how adding context to a problem might influence student reasoning. This study sets out to add to the literature by addressing this open question. Data & Methods The data for this paper are drawn from a study of how bilingual ninth grade algebra students learned to reason with linear functions. Specifically the study examined how the students generalized from data and reasoned about the relationships between the rate of change

112 William Zahner - Mediation ofProblem Contexts and slope of linear functions through engaging in small group discussions with their peers. Data collection in the larger study traced the reasoning of two groups of four students each in one algebra class across six weeks of class meetings. The data collected included video recorded observations of classroom interactions, as well as video and written work from a series of focused group problem solving sessions recorded outside of class. The in-class observations provided data on the naturalistic setting of the classroom and the types of reasoning the students did there. The out of class group discussions sessions provided more focused data on how the students reasoned through tasks, and how their reasoning developed across the six week data collection time period. This analysis focuses on the students’ out-ofclass discussions.

Figure 1.

Timeline of data collection

The Setting and Study Design The class that was observed was a bilingual (English-Spanish) algebra class taught by a highly regarded, experienced, and qualified teacher. The school population was over 90% Latino/a, and 77% of the students qualified for a free or reduce priced lunch, indicating that they were from low socioeconomic backgrounds. Thirty five percent of the school population was classified as English Learners. In the focal classroom, about one-third of the students were recent immigrants from Latin America who were learning English (the language of instruction), while other students were bilingual. The bilingual students, who were classified by the school as proficient in English,


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were either born in the United States but grew up in Spanish-speaking households, or they had immigrated to the United States and learned English several years before this study. The data collection time period coincided with a six week classroom unit when the students were learning to graph lines, write linear equations, and interpret linear functions. The students’ mathematics curriculum highlighted the use of applied problem solving and using real-life data to learn these concepts (Fendel, Resek, & Alper, 1996). In order to document changes in the students’ reasoning about linear functions across time, each group solved a set of three tasks at the start, middle and end of the data collection time period. These group problem-solving discussions took place outside of class, and the students were instructed to discuss each problem as a group, come to consensus, and then write one shared answer for their group. The problems that the students solved together were specially chosen to highlight key conceptual relationships, such as the relationship between the slope of a linear function and the rate of change of the dependent variable (two of the problems that will be discussed in more detail in this paper appear in the Appendix). In order to make valid inferences based on these students’ reasoning, these problems were drawn from assessments used in prior research on student reasoning about generalization, rates, and linear functions. The problems were similar to the type of problems the students solved in class, but they were chosen to highlight key conceptual relationships. These group discussions were video recorded and transcribed with a focus on capturing the propositional content of the students’ talk. Copies of each group’s agreed-upon written answers, as well as their scratch work, were collected. Finally, the author was present and recorded field notes during each discussion, but he did not intervene in the group discussions. Focal Group This case study analysis focuses on how one of the two groups used the given problem contexts as a resource while reasoning through the tasks. This group was selected for several reasons. First, the focal group engaged in extensive discussions of the problems, while the

114 William Zahner - Mediation ofProblem Contexts other group tended to have shorter discussions with less dialogue about each problem. Second, this group included four bilingual students who were all classified as “Fully English Proficient,” which decreases (but does not eliminate) the probability that the students’ level of proficiency in English would interfere with their mathematical reasoning on these questions written in English. Finally, this group made the most references to the problem context in their discussions, so its discussions help illuminate the theoretical issues discussed in this paper. Although the focus of this analysis is on one group of students, both of the groups in the study did refer to the real life contexts of the problems, and they drew upon mathematical reasoning, as they worked through these problems. The focal group consisted of two boys, Mateo and Jaime, and two girls, Krystal and Susanna. Mateo and Krystal were immigrants from Spanish-speaking countries, and all four of the students reported speaking both Spanish and English outside of school. The group members primarily spoke English when working in their group, and they were given all in-class assignments in English. The teacher selected this group to participate in this study by assembling groups of students who she thought would work well together, and who represented a range of prior achievement in her class. Mateo and Krystal had relatively high grades, while Jaime and Susanna had relatively low mathematics grades. This analysis focuses on the transcripts and written work produced during three out-of-class problem-solving sessions among the focal group. Additionally, the videos and field notes were used as a resource throughout the analysis process to clarify meanings in the transcripts and in the students’ written work. The group solved three problems about rate and accumulation across eight weeks, repeating each problem at least twice. Although the students solved the same problem twice, they did not simply recall their answers to each question from their first solution. For example, the group engaged in three sustained discussions of a problem called Hexagon Desks, using over ten minutes to discuss the problem each time they attempted it.


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Tasks This analysis focuses primarily the group’s discussion of the problem Hexagon Desks. The group’s discussion of a second problem called The Tortoise and the Hare is also presented to highlight contrasts in how the problem context can mediate students’ mathematical problem solving. Hexagon Desks was adapted from a released eighth grade item from the National Assessment of Educational Progress, and variations of this problem have appeared in numerous other forms in research and curricula. Hexagon Desks focused on using multiple representations to explore a linear relationship between the length of a chain of hexagon shaped desks and the number of students who could sit at the chain of desks. The Tortoise and the Hare was adapted from previous research on student interpretations of motion graphs, and it asked students to interpret two velocity-time graphs plotted on the same axes. In each task, the sequence of the questions was designed to elicit how students reasoned about rates in relation to slope (on Hexagon Desks) and accumulation (on The Tortoise and the Hare). This analysis focuses on the mathematical content of the students’ discussions, treating the group discussion as the unit of analysis. The students’ talk during the discussion is treated as evidence of the group’s reasoning. The design of the discussion sessions, requiring the group to agree on an answer, provided a rich source of talk because the students were forced to reconcile differences and come to an agreement before writing their final, agreed upon answer. Analysis The data analysis followed three steps. First, the students’ talk was transcribed with a focus on capturing the propositional content of the students’ talk. A total of 180 minutes of group talk was transcribed (though not all of that time was dedicated to talking about mathematics). Second, the transcripts were divided into segments corresponding with the students’ talk about each part of the problems. For example, one segment included all of the group’s talk about question two from Hexagon Desks. The third stage of analysis was identifying segments where the students made reference to the problem

116 William Zahner - Mediation ofProblem Contexts context, either implicitly or explicitly. All references that the students made to the problem context were catalogued. Finally, all of the references to context were coded according to whether referring to the context helped, hindered, or had a neutral effect on the students’ problem solving success. Problem solving success was measured by whether the group was eventually able to provide correct answers to the questions on each task. Findings Written Responses In total, the group’s written responses to Hexagon Desks task showed some development as well as a fair amount of consistency across the three times they attempted the problem. Each time the group attempted this problem, they successfully completed the table in question one which asked them to show how many people could sit around a row of three, four, five, six and seven hexagon desks arranged in a row. The only exception was that the group answered “31” in the last row of the table during Discussion 2 because they added five rather than four to 26, the previous value. On question 2, the students also successfully found the number of students who could sit at a row of 100 desks (402) each time they solved this problem. The group also successfully solved question three, and they derived an equation for how many people could sit at n desks (total = 4n + 2), during all three discussions of this problem. Question four required the students to graph a set of points, and, as expected, the group succeeded in this task each time they discussed it. The group skipped question five, which required computing the slope of a line, during their first discussion. However, during their second and third discussions, after slope had been taught in their class, the students successfully found the slope, and they wrote it in fraction form as 4/1. The group skipped question six, which asked them to interpret the slope of a linear function in terms of the problem, each time they solved Hexagon Desks. Finally, during their second and third discussions of this task, the group successfully solved question seven, which required them to generalize this relationship for octagons.


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The group produced only one set of written responses to The Tortoise and the Hare because they ran out of time when the problem was given to them a second time. They successfully answered questions one through three on the task, but their written answers had incorrect units, indicating that they struggled to reason with the velocity as a quantity on the y-axis. This response pattern makes sense because reasoning about intensive quantities is more challenging for students than reasoning about extensive quantities (Schwartz, 1988). On questions four through ten, the students agreed on incorrect answers, but they also showed signs of confusion for these questions. These relatively difficult questions required the students to reason about intervals (rather than points), and to work backwards and find the distance traveled as the product of velocity and time. Finally, in addition to their mathematical struggles on this problem, it appeared that the words “tortoise” and “hare” were unfamiliar to two of the group members. The Mediation of the Problem Context in Discussions While the students referred to the imagined story in each problem when discussing their solutions, their reliance on the context, and their relative success by using the given contexts, revealed two distinct ways that problems in context can mediate students’ mathematical reasoning. One influence of the problem context was that it drew the students’ attention to reason about the given problem in particular ways related to the story. This was evidenced by the students’ use of examples and terminology indicating that they were drawing on the context as a resource. By imagining how many students could sit at a row of desks, the context on Hexagon Desks may have helped the group reason through this problem. On the converse side, the other evidence of the mediation of context was through the evident confusion on the part of the students with unfamiliar vocabulary and an unrealistic situation in The Tortoise and the Hare. For both Hexagon Desks and The Tortoise and the Hare, the imaginary context did not always help the group reach a correct response, nor did it necessarily hinder their progress. However, the influence of the problem context, and the mediation of the imagined situation differed in important ways. One way the context mediated the

118 William Zahner - Mediation ofProblem Contexts group’s reasoning on Hexagon Desks was that the story allowed the students to reason about perimeter of a chain of hexagons by imagining a person sitting at each external segment on the figure. In this case, the imaginary context promoted a particular way of looking at the given inscriptions. A second way the context mediated the group’s reasoning was evident in The Tortoise and the Hare. For this problem, both the challenging vocabulary (tortoise and hare) as well as the implausible story appeared to hinder the group’s mathematical reasoning because it distracted from their mathematical focus. Below I illustrate how the problem context mediated the group’s discussions on both problems. The Context as a Resource on Hexagon Desks The students did not appear to have any struggles imagining the given context in Hexagon Desks, which might indicate that the idea of pushing desks together was relatively familiar. The students’ familiarity with the story in Hexagon Desks is affirmed by the contrast with how they talked about the unfamiliar and unrealistic context from The Tortoise and the Hare (see below). At key points early in their discussions of this problem the group members did use the story about seating students around desks as a resource for their mathematical reasoning. The majority of the group’s references to the story occurred as they completed the table in question one. In particular, as the group members filled in the “number of students” column in the table, they discussed whether any students could sit at the spaces represented by vertical segments in the diagrams. They agreed that “nobody” could sit at the segments where two desks meet. They also noted that “somebody” could sit at the two vertical segments at the ends of the row of hexagon desks. Excerpts 1 and 2 below contain two instances where the group referred to the context as they solved this problem. Excerpt 1 is from the group’s first discussion of this problem while Excerpt 2 is from their third discussion of the problem. In both cases they pointed to the chain of hexagons given on their paper as they made reference to the story about seating students at a row of hexagon shaped desks. Figure 2 illustrates where Mateo and his group mates were pointing as they


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used the words “somebody” and “nobody” to reason about this problem.

“Somebody” can sit here “Nobody” can sit here How the students used "somebody" and "nobody" in relation to the Hexagon Desks problem. Figure 2.

Excerpt 1 is from the group’s first discussion of the problem. The three group members who were present that day, Mateo, Krystal, and Jaime, each appeared to use slightly different methods for solving the problem. Krystal drew chains of hexagons and counted sides. Mateo appeared to be coordinating the image and the story about seating students around desks to devise pattern. Jaime derived a recursive rule for the pattern, noting that the first term was six, and subsequent terms were four more than the previous terms. In the following excerpts, clarifying comments are enclosed in double parenthesis. Square brackets are used to show the start of overlapping talk. The students used some Spanish words in Excerpt 1 and translations are in double parentheses with quotations immediately after the terms in Spanish. Excerpt 1

The group discusses the table in Hexagon Desks during Discussion 1

1. 2. 3.

Mateo It’s gonna be eighteen ((referring to row 4 of the table)) Krystal Huh? Mateo It’s gonna be eighteen because you know [nobody’s

120 William Zahner - Mediation ofProblem Contexts 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

gonna sit Krystal [but how does that work Jaime Oh it's some pattern Krystal I know its a [pattern but Mateo [no it- [[you Jaime [[its going by six y luego ((“and then”)) by [four Krystal: [four five ((continues counting silently)) Mateo six plus four is ten. Ten plus for is fourteen Krystal ((speaking louder)) Twenty twenty one twenty two twenty three twenty four. I was right Mateo No but watch Jaime xxx Krystal Watch one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen seventeen eighteen nineteen twenty twenty one twenty two three ((short pause)) damn it I was wrong its twenty [tMateo [but no cause you add another one and nobody's gonna be sitting on that one ((pointing at Krystal's paper)) Krystal I know but right now it's like yeah it works

Krystal was attempting to count the perimeter of a chain of five hexagons, but she appeared to get lost while counting to 22 in lines 11 and 14. Mateo then made an explicit reference to the problem context in line 15 to explain why Krystal’s answers were wrong. After Mateo’s comment, Krystal took up his idea and verified that it agreed with the numerical patterns she observed for this problem. Therefore, Except 1 illustrates how the story appeared to help the students agree on the correct solutions to question 1 from Hexagon Desks. Excerpt 2 is from this group’s third discussion of this problem. During their first discussion they agreed that each row of the “number of students” column in the table should be four more to the previous row, giving answers of 18, 22, 26, and 30. However, during their second discussion (which occurred between the discussions in Excerpts 1 and 2), the group agreed upon a slightly different rule: each row was four more than the previous row, except for the last row, which was five more than the previous. Thus they incorrectly wrote 31


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rather than 30 in the final row during their second discussion of this problem. For their third time working through this problem, the group returned to the question of whether to “add four” or “add five” to each row in the table. Prior to the exchange in Excerpt 2, the group agreed that the “number of students” in each row of the table should be four more than the previous row. However, Mateo also argued that the number of students in the last row of the table should again be 31—five more than in the previous row—because “somebody can sit here [on the last vertical spot].” Krystal disagreed, and Excerpt 2 shows the start of the ensuing discussion. Ultimately this group agreed that the final answer for the last row of the table should be 30 (i.e., four more than the previous row), but only after several minutes of discussion. The group came to consensus after Krystal drew out a chain of seven hexagons and counted the perimeter of all seven of them. Excerpt 2

The completes the table in Hexagon Desks during Discussion 3

1. 2. 3. 4. 5. 6. 7. 8.

Mateo You just add four to all of them Krystal Then you add four and then you add like two ((possibly referring to the general rule, perimeter = 4n+2)) Mateo The last one is five Krystal What? Mateo See six plus four is ten plus four is fourteen and the last one you just add five that's all these ((pointing to end of chain)) Susana Why add five? Krystal Really? yeah Mateo ‘Cause it that's the last one and somebody can sit on this ((points at the end of the chain of hexagons))

The question of whether “somebody” or “nobody” can sit in a particular spot in the imagined chain of hexagon desks illustrates one way the problem context and everyday language mediated the mathematical discussion among this group of students. This use of the context both afforded and constrained the students’ reasoning. The

122 William Zahner - Mediation ofProblem Contexts affordance was that by imagining students sitting at a desk, they were able to reason about the perimeter of the chain of hexagons. However, imagining people sitting at the desks also introduced a subtle problem. While it is true that “nobody” can sit at the intersection of two desks in the Hexagon Desks problem, the word nobody is slightly problematic because two spaces are removed each time two desks are pushed together. The intersection of two desks removes two sides from the available seating, so the net change in the number of seats (i.e. the perimeter) is (new perimeter) = (previous perimeter) + 6 − 2. Mateo’s use of “nobody” may have obscured this relationship because “nobody” does not quantify how many people cannot sit at an intersection. This issue may help explain why the group debated whether to “add four” spaces or “add five” new spaces with each desk, even though they quickly identified the numerical pattern “add four” in the first few rows of the table. For the remaining problems in Hexagon Desks, the group appeared to shift in their reasoning and in their reliance on the context. While they relied on the images of chains of hexagons to answer questions two and three on the task, they made relatively few references back to the problem context. For example, they did not comment on the absurdity of making a row of 100 desks. Moreover, the group also skipped question six, which explicitly asked the students to make a connection between the problem context and the slope of the linear function (defined on the natural numbers) that models this situation. The Interference of Context on The Tortoise and The Hare While the prosaic problem context in Hexagon Desks appeared to be familiar to the students, the more whimsical context in The Tortoise and the Hare was not. In this case, the story did not serve as a resource for the students; in fact, the unfamiliar context may have detracted from the students’ mathematical reasoning. The students’ distraction illustrates a second, clearly unhelpful, way that problem contexts can mediate students’ mathematical problem solving. None of the students indicated that they had heard the fable of the tortoise and the hare as they discussed this question (though knowing


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the fable was not necessary to solve this problem). The students’ lack of familiarity with the context resulted in a qualitatively different form of mediation of the problem context. First, the students used some time to discuss the meaning of the unfamiliar terms “tortoise” and “hare.” In Excerpt 3 below, Jaime struggled with pronunciation while reading “tortoise,” and Krystal asked her group mates what a tortoise was. Likewise Mateo corrected Krystal’s use of the word “bunny” for “hare” and the students discussed the Spanish and English words for tortoise and hare. Second, the students’ few attempts to use the context as a resource to reason about the mathematics were unsuccessful. For example, in line 12 of Excerpt 3, Krystal appeared to try to draw on her knowledge of rabbits to reason about the plausibility of the hare’s graph. Unfortunately, it is not clear whether Krystal’s erroneous reasoning is a result of misunderstanding the context or of misunderstanding the graph. Excerpt 3

The group reads the problem statement in The Tortoise and The Hare

1. 2. 3. 4.

5. 6. 7. 8. 9. 10. 11.

Jaime ((reading the problem)) One day tom the tortoise ((struggles with pronunciation of tortoise)) Krystal Tor tus a ((sounding out the word, pronounces incorrectly)) Mateo Tortoise ((Pronounces correctly)) Jaime Tortoise and Harold the hare race ((pause)) ran a race. Tom got a running start but they both ran across the starting line at the same moment when the times said zero seconds. They ran along a straight road for ten seconds and the graph below shows Tom and Harold's velocity during the ten second race. Krystal So oh this is tortoise the ((pointing at image)) Mateo It's Tom is a tortoise Krystal What's a tortoise? Mateo Its a [turtle Krystal [A turtle? Mateo Yeah but uh bigger [[xxx Jaime [[Who was running fas

124 William Zahner - Mediation ofProblem Contexts 12. 13. 14. 15. 16. 17. 18.

Krystal This isn't possible for a bunny ((traces pencil along the inverted V shape on the graph)) Jaime haha Mateo No this is a turtle Krystal Yeah, this is a bunny ((again makes the inverted V)) like faster and then stopped Mateo No that's a hare. Hare is bigger skinny Krystal So it's better than a bunny Mateo Yeah

After discussing the unfamiliar terms tortoise and hare, the students were able to answer questions 1-3 by reading specific values on the graph. However, the units in the students’ written answers were incorrect (e.g., they wrote “At 2 seconds Tom ran 1 second faster than Harold” in response to question 1), and the group struggled to make sense of the units throughout the remainder of the problem. They skipped questions 5, 9, and 10, and their written answers to questions 4, 6, 7, and 8 were incorrect. In terms of the mediation of problem context, the most striking contrast between the group’s discussion of Hexagon Desks in Excerpts 1 and 2 and their discussion of The Tortoise and The Hare in Excerpt 3 was that the context and terminology used in Hexagon Desks was readily accessible to the students while the context and terminology in The Tortoise and the Hare was not. Moreover, the story in Hexagon Desks was more “real life” than the story in The Tortoise and the Hare. In Excerpt 3, the mediation of everyday language was most clearly evidenced by the students’ lack of knowledge of the vocabulary used in the problem. Of course, knowing the terms tortoise and hare is not actually required to solve this problem, but that does not mean that the students were not distracted by these terms. In this sense there is a parallel with assessment items considered by Martiniello (2008) where she showed that unfamiliar terminology, even if it is unrelated to the mathematics content, can distract students during problem solving. In a strange twist, successfully solving The Tortoise and the Hare required some extra knowledge—the knowledge that the context was meant to be ignored.


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Discussion Distinctions between Two Ways Problem Contexts Mediated Discussions This analysis highlights two distinct ways that incorporating real life contexts in school mathematics problems might mediate students’ mathematical problem solving. One way that context can mediate students’ reasoning is through invoking particular semiotic resources and ways of reasoning. This form of mediation was evident in the group’s reasoning on Hexagon Desks, where they discussed the perimeter of chains of hexagons by asking whether “somebody” could sit at particular locations in the diagrams. The second type of contextrelated mediation addressed in this paper was that using unfamiliar contexts and terminology in the statement of mathematics problems might interfere with students’ mathematical reasoning. This type of problem context mediation was evident in the group’s discussion of the Tortoise and the Hare, where some students in the group were unfamiliar with both the fable of the tortoise and the hare, as well as the meaning of the words tortoise and hare. While the group also struggled with the mathematical concepts in this problem, there is evidence that the peculiar story occupied some of their attention. Together, Hexagon Desks and The Tortoise and the Hare illustrate how the mediation of problem contexts in students’ joint problem solving can operate on different levels. The interference of the problem context in the Tortoise and the Hare was readily apparent to both the students and the researcher. Because the story and the vocabulary in the Tortoise and the Hare were unfamiliar to the students, the students exerted some effort to make sense of the story and the characters, even though knowing the story did not necessarily help solve the graph analysis task. The students’ efforts to understand the story suggest that they were unfamiliar with the genre of school mathematics word problems, and the fact that the context often can—and at times must—be ignored while solving the math problem (Gerofsky, 1996). This form of mediation illustrates one way that cultural and linguistic bias enters into school mathematics tasks. As previous research has noted, when the task is an assessment, one result is that some students may suffer linguistic discrimination on assessments (Abedi & Lord,

126 William Zahner - Mediation ofProblem Contexts 2001; Martiniello, 2008). Conversely, Abedi and Martiniello’s research has also shown that with some minor linguistic adjustments, tasks can be made more comprehensible for language minority students. The students’ reasoning on Hexagon Desks shows a different, and more subtle way in which problem context can mediate students’ problem solving. In this case, the students used the context to interpret both the numerical pattern in the table and the images of chains of hexagons. This shows an affordance of the context. However, the students’ discussion also revealed that the language and metaphors used to reason about students sitting around desks may have introduced some ambiguity in the students’ mathematical problem solving (see, e.g., the problems with the term “nobody” addressed in the Findings). The students’ focus on whether “somebody” or “nobody” could sit at different spaces around the chain of desks was not inevitable. One might imagine a different situation where students were asked a similar question framed by a story about coloring the outside edges of a chain of one, two, three, and more hexagons. In such a case the mathematical pattern would be similar, but the students would likely draw on a different semiotic resources for problem solving. Research Implications This paper illustrates how Wertsch’s (1991, 1998) framework of mediated action can be used to rethink the influence of story problem contexts on students’ mathematical reasoning. While prior studies have used genre analysis, situated learning, and cognitive frameworks to examine facets of mathematics story problems, the mediated action framework helps illuminate how very subtle changes in mathematical story problems may effect significant changes in students’ reasoning. By framing the story of Hexagon Desks in terms of students sitting around a row of desks, the students were “primed” to imagine bodies arranged in physical space, and the traces of this way of thinking were evident in the students’ talk. This is closely tied to Wertsch’s (and Vygotsky’s) fundamental contention that human activity must be analyzed as a system, and the agent taking action to achieve a goal


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cannot be considered without also considering the meditational means used to achieve those goals. For mathematics education researchers, this analysis is a reminder that we must be careful to account for the meditational means when we analyze students’ mathematical reasoning. One possible follow up to this analysis would be to examine how changing the story in word problems corresponds to changes in the resources students draw upon for their mathematical reasoning. Researchers in the field of educational assessment pilot test several versions of mathematics test items with an eye toward avoiding cultural and linguistic discrimination. This consideration is critical for the development of fair tests (American Educational Research Association, American Psychological Association, & National Council on Measurement in Education, 1999). However, in the field of mathematics education, mathematics problem contexts are often treated as if they were transparent. Analyses like the one presented here, reveal that the context is highly salient for students, even if experts (such as educational researchers) know that problem solvers are supposed to ignore the story and focus on the mathematical relationships. Wertsch’s notion of mediated action provides a unique way to understand why some “equivalent” problems are more difficult than others. One caveat is that this analysis should not lead to the conclusion that experts always ignore the problem context and novices are confused by contextualized problems because they do attempt to reason based on the context. The actual differences between experts and novices may be subtler than this dichotomy. For example, ethnographic studies have shown that physicists in the university setting often draw upon multiple metaphors and imagine themselves in problem spaces as they skillfully solve abstract problems (Ochs, Jacoby, & Gonzales, 1994). Practical Implications This case study analysis does not include enough data to support definitive recommendations for teaching. However, this case does provide grist for examining how mathematics problems are used in the service of teaching and learning mathematics. Traditionally in the school curriculum, problems have been written to require the use of

128 William Zahner - Mediation ofProblem Contexts particular solution methods, and the problem context was secondary to the targeted mathematical technique (Gerofsky, 1996; Schoenfeld, 1992). With mathematics education reform, curriculum designers sought to use more “real life” problems in mathematics texts (Boaler, 1993; National Council of Teachers of Mathematics, 1989, 2000). This case study, together with Wertsch’s (1991, 1998) notion of mediated action, indicates one reason why educators should be cautious about the use of problems in context. While “real life” applications may be motivating for students, they also invite students to use alternative semiotic systems for reasoning through problems, which may result in the students providing unexpected answers. One amusing instance of this occurred when piloting items for this study. When one group of students was asked how many hexagon desks would be required to seat 44 students, the students responded 10. When the students were asked to explain their reasoning, they said that, although there would be 42 spaces at a row of ten desks, the two extra students could squeeze in somewhere on the side. In this case the students provided an answer that was incorrect from a mathematical perspective, but which would be practical in the real-life situation. Teachers may need to be aware of potential conflicts like this. Conclusion The brief analysis here shows how adding “real life” context to a mathematics problem can constrain students’ mathematical problem solving while also providing some affordances. There is little doubt the fanciful context and new vocabulary in The Tortoise and the Hare interrupted the students’ focus on the mathematical problem. The mediation of language and the problem context on Hexagon Desks is subtler. While the problem context appeared to be familiar (or did not warrant comment), the students’ use of terms like “somebody” and “nobody” indicated that imagining the context mediated the group’s mathematical reasoning. In the students’ discussions of Hexagon Desks, Wertsch’s notion of mediation can help explain the group’s responses to this task. While we cannot know for sure, it is interesting to consider whether these students would have been more successful reasoning through a problem about a chain of hexagons rather than a


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chain of desks. This study is limited by the fact that the two problems differ in terms of their mathematical difficulty. In addition to using unfamiliar terms, the Tortoise and Hare required students to interpret a velocity graph. Nonetheless, the distinction between two forms of mediation—the subtle influence of everyday language and the more overt issue of unfamiliar language—can help researchers, curriculum designers, and teachers as we consider what tasks to use for instruction and assessment. References Abedi, J., & Lord, C. (2001). The language factor in mathematics tests. Applied Measurement in Education, 14(3), 219-234. American Educational Research Association, American Psychological Association, & National Council on Measurement in Education (1999). The standards for educational and psychological testing. Washington, DC: American Educational Research Association. Baranes, R., Perry, M., & Stigler, J. W. (1989). Activation of realworld knowlegde in the solution of word problems. Cognition and Instruction, 6(4), 287-318. Boaler, J. (1993). The role of contexts in mathematics classrooms: Do they make mathematics more "real"? For the Learning of Mathematics, 13 (2), 12-17. Brenner, M. E. (1998). Adding cognition to the formula for culturally relevant instruction in mathematics. Anthropology & Education Quarterly, 29(2), 214-244. Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the streets and in schools. British Journal of Developmental Psychology, 3 , 21-29. Civil, M. (2002). Culture and mathematics: A community approach. Journal ofIntercultural Studies, 23 (2), 133-148. Civil, M., & Andrade, R. (2002). Transitions between home and school mathematics: Rays of hope amid passing clouds. In G. Abreu, A. Bishop & N. Presmeg (Eds.), Transitions between contexts of mathematical practices. Great Britain: Kulwer Academic Publishers.

130 William Zahner - Mediation ofProblem Contexts Fendel, D., Resek, D., & Alper, L. (1996). Interactive Mathematics Program: Year 1 . Emeryville, CA: Key Curriculum Press. Forman, E. (1996). Learning mathematics as participation in classroom practice: Implications of sociocultural theory. In Steffe, Nesher, Cobbl, Goldin & Gree (Eds.), Theories ofmathematics learning (pp. 115-130). Gerofsky, S. (1996). A linguistic and narrative view of word problems in mathematics education. For the Learning ofMathematics, 16(2), 36-45. Greer, B. (1997). Modeling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7(4), 293-307. Hegedus, S. J., & Moreno, L. (2011). The emergence of mathematical structures. Educational Studies in Mathematics, 77, 369-388. Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. The Journal ofthe Learning Sciences, 13 (2), 129-164. Luria, A. R. (1979). Cultural differences in thinking. The making of mind: A personal account ofSoviet psychology (pp. 58-80). Cambridge, MA: Harvard University Press. Martiniello, M. (2008). Language and the performance of EnglishLanguage Learners in math word problems. Harvard Educational Review, 78(2), 333-368. Moschkovich, J. N. (2002). An introduction to examining everyday and academic mathematical practices. In J. N. Moschkovich & M. E. Brenner (Eds.), Everyday and academic mathematics in the classroom . Reston, VA: National Council ofTeachers of Mathematics. Moschkovich, J. N. (2004). Appropriating mathematical practices: A case study of learning to use and explore functions through interactions with a tutor. Educational Studies in Mathematics, 55, 49-80. Moschkovich, J. N., & Brenner, M. E. (Eds.). (2002). Everyday and academic mathematics in the classroom . Reston, VA: National Council ofTeachers of Mathematics. National Council ofTeachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: author.


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National Council ofTeachers of Mathematics (2000). Principles and Standards ofSchool Mathematics. Reston, VA: author. Ochs, E., Jacoby, S., & Gonzales, P. (1994). Interpretive journeys: How physicists talk and travel through graphic space. Configurations, 2(1), 151-171. Radford, L. (2001). Signs and meanings in students' emergent algebraic thinking: A semiotic analysis. Educational Studies in Mathematics, 42(3), 237-268. Radford, L., Bardini, C., & Sabena, C. (2007). Perceiving the general: The multisemiotic dimension of students’ algebraic activity. Journal for Research in Mathematics Education, 38(5), 507-530. Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in cultural context. New York: Oxford University Press. Saxe, G. B. (1995). From the field to the classroom: Studies in mathematical understanding. In L. P. Steffe & J. Gale (Eds.), Constructivism in Education (pp. 287-311). Hillsdale, N.J.: Lawrence Erlbaum Associates. Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition and sense making in mathematics. In D. Grouws (Ed.), Handbook ofResearch in mathematics teaching and learning (pp. 334-370). New York: Macmillan. Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades. Reston, VA: National Council ofTeachers of Mathematics. Vygotsky, L. S. (1978). Mind in society: The development ofhigher psychological processes. Cambridge, MA: Harvard University Press. Vygotsky, L. S. (1986). Thought and language. Cambridge, MA: The MIT Press. Walkerdine, V. (1988). The mastery ofreason: Cognitive development and the production ofrationality. New York: Routledge. Walkington, C. A., & Maull, K. (2011, July). Exploring the assistance dilemma: The case ofcontext Personalization . Paper presented at the Annual meeting of the Cognitive Science Society, Boston, MA.

132 William Zahner - Mediation ofProblem Contexts Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action . Cambridge, MA: Harvard University Press. Wertsch, J. V. (1998). Mind as action. New York: Oxford University Press. Appendix Hexagon Desks

Ms. West wants to know how many students can sit around a row of hexagon shaped desks. If one desk is by If two desks are pushed If three desks are pushed itself then six together, then 10 together in a row as shown students can sit students can sit at the below, then 14 students around it. table. can sit together.

1. Fill in the following table for the number of students who can sit together for the number of desks pushed together in a row: Number of Hexagon Desks

Number of Students


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2. Imagine that 100 of the hexagon desks were pushed together in a row. How many students could sit around that row of desks? Show the work you used to find that solution. 3. If n hexagon shaped desks, were pushed together, then how many students could sit at the row of desks? Give your answer as a formula in terms of n. 4. Use the table you made in problem 1 to draw a graph showing the number of children who could sit at a row of desks

5. If you connect the dots between points in the graph to make a line, what is the slope of that line? How do you know? 6. What is the meaning of the slope of the line in terms ofthe problem about children sitting at desks? Explain your answer in terms of the problem and using words and ideas that you know from math class. 7. What if n octagon-shaped desks were pushed together? How would this problem be different? How would it be the same? Explain your answer in as much detail as possible (you may use equations, tables, graphs, words, etc.).

134 William Zahner - Mediation ofProblem Contexts The Tortoise and the Hare

One day Tom the Tortoise and Harold the Hare ran a race. Tom got a running start, but they both ran across the starting line at the same moment when the timer said 0 seconds. They ran along a straight road for 10 seconds and the graph below shows Tom and Harold’s velocity (speed) during the 10-second race:

A drawing ofTom and This graph shows the velocity (speed in meters per second) Harold’s race ofTom the tortoise and Harold the hare during the race.

1. Who was running faster at t = 2 seconds? How do you know? 2. Who was running faster at t = 4 seconds? How do you know? 3. During what time periods was Tom running faster? How do you know? 4. At what time (or times) were Tom and Harold in the same location during the race? How do you know? 5. At what time (or times) were Tom and Harold running at the same speed during the race? How do you know? 6. How far did Tom run? 7. How far did Harold run? 8. Who was ahead after 5 seconds? 9. Did Tom or Harold run backwards at some point during the race? How do you know? 10. Who won the race? How do you know?


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William Zahner is Associated professor at the Department of Curriculum and Teaching (in the program "Mathematics Education") at Boston University, USA. Contact address: Direct correspondence concerning this article should be addressed to the author at: Boston University SED, 2 Silber Way, Boston MA 02215. E-mail address: [email protected].


Mathematics Teacher Continuing Education: Fostering the Constitution of a Learning Network Nielce Meneguelo Lobo de Costa and Maria Elisabette Brisola Brito Prado 1 1 ) Universidade Bandeirante de São Paulo.


To cite this article: Lobo da Costa, N.M., & Prado, M.E.B.B. (201 2).

Mathematics Teacher Continuing Education: Fostering the Constitution of a Learning Network. Journal of Research in Mathematics Education, 1 (2), 1 36-1 58. doi: 1 0.4471 /redimat.201 2.0 8

To link this article: http://dx.doi.org/1 0.4471 /redimat.201 2.08 PLEASE SCROLL DOWN FOR ARTICLE The terms and conditions of use are related to the Open Journal System and to Creative Commons Non-Commercial and Non-Derivative License.


Mathematics Teacher Continuing Education: Fostering the Constitution of a Learning Network Nielce M. Lobo da Costa

Universidade Bandeirante de São Paulo

Maria E. Brisola Brito Prado Universidade Bandeirante de São Paulo

Abstract

This qualitative study analyzes log entries from a group of elementary school teachers aiming to understand how a collaborative learning network is set up during the continued development actions. The interpretive analysis revealed the following categories: Shared reflection, Learning, Trust, Reflection on practice, Experience Exchanging, Shared Goals and Commitment to the other, which show the features of collaborative work. Categories were analyzed using CHIC software which allowed for the relational analysis among them showing that development under this perspective can favor the practitioner’s professional development by offering him opportunities to experience his role as a learner and a teacher simultaneously.

Keywords: professional development, collaborative work, learning networks.

2012 Hipatia Press ISSN 2014-3621 DOI: 10.4471/redimat.2012.09

136 Lobo da Costa & Prado - Constitution ofa learning network

T

he paradigm for the 21st-century society brings the need to rethink education and the ways to teach and learn mathematics having as a focus the future practitioner who will work in synch with a globalized world, which has a number of different technological resources that require mastering knowledge, creativity, ethics and solidarity. In this scenario, the teacher’s role becomes evident because, being a mediator in the student’s learning process he will have to review his practices, his mathematical knowledge and teaching strategies. This implies developing new knowledge and, hence, continued education for the mathematics teacher. In Brazil, legislation makes it mandatory for children between 6 and 141 years of age to attend school, and this period is known as elementary school. The right to attend school is made available for the Brazilian population by means of federal, state and municipal schools. It is by attending them that Brazilian children have access to culture, mainly when they come from more impoverished segments of the society, since private schools provide for the families who are financially capable of paying for the costs. In short, there are two types of school: the public, free one, which offers elementary education for the population in general, in an inclusive, democratic perspective, and the private one, which provides for specific social and religious groups. It is worth noting that the majority of the Brazilian children and young adults attend public schools. Taking into account Brazil’s continental dimensions and the limited resources for education, two challenges are at stake: access to education and the quality of teaching-learning processes. The issue of access today can be considered as having been solved, but the greater challenge to promote quality, democratic teaching for public schools to provide for all kinds of students remains. When one thinks about the quality of public education, a number of issues arise: the premises, management and human resources. With regard to the latter, the first resource to come to mind is the teacher as the key mediator between students and knowledge. Among elementary school teachers, when we focus our attention on the mathematics teacher, some issues emerge, such as: (1) how can a mathematics


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teacher face the challenge to help build quality teaching in public schools? (2) With technological advances happening daily, how can the teacher keep up with them and use such innovations in the classroom? (3) How can the teacher prepare himself to deal with the challenges presented by this public school that provides for all kinds of students and teach mathematics there? Such questions immediately evoke this teacher’s continued education that should play a key role in the teaching career. How has continued education been performing? Developing processes in the continued education modality have been provided by teachers’ individual initiatives, by the government’s initiative, and by development projects linked to college researches in their mathematics education programs. When the teacher decides to pursue development individually, the courses sought are those to improve specific and/or teaching knowledge and specialization courses, while government’s proposals include training, qualifying and workshops aiming at implementing public policies. Regarding college projects, the choices for development actions are, in their majority, oriented to understand the teacher’s learning process, as well as the different development strategies. One of the development strategies recently emphasized by many authors is the one which enables the teacher’s professional development by taking into account his knowledge and a reflective and investigative attitude towards practice (Alarcão, 2001). In Brazil, there are studies about continued education for mathematics teachers showing that group work, both at colleges and development centers and also at the school itself, develops collaborative attitudes among participants (Fiorentini, 2006; Lobo da Costa et al, 2011). Besides, learning in and with the group, can become a regular practice among teachers to share what they think and do regarding the teaching and learning of mathematics in the classroom, as well as to discuss issues related to social, political, cultural and economic aspects. Such aspects, albeit not specific to mathematics, are embedded in the educational action and can interfere both with the teacher’s practice and students’ learning. This qualitative study analyzes log entries from a group of elementary school teachers aiming to understand how a collaborative learning network is set up during actions for continued development.

138 Lobo da Costa & Prado - Constitution ofa learning network Professional Development and Collaborative Work Studies have shown indicators that play a significant role in continued education processes and can lead to changes in teaching practices aiming at improving the quality of mathematics teaching. Such indicators emphasize the importance to create situations that allow for the teacher to have opportunities to reflect upon his own learning and his pedagogical practice so as to favor the perception by the individual of his own conceptions and pedagogical demeanor and the methodological strategies he uses in the classroom. (Pietropaolo et al, 2009). Hence, many authors like Imbernón (2010), Prado and Valente (2002), Campos et al (2009) among others, underscore that continued development geared to professional development should include the classroom daily issues, and integrate contextualized actions so that the teacher can revisit his practice, reflect upon it and rebuild it. … the teacher’s practical experience in the classroom should also be presented both as a study and reflective situation to the developing teacher. This situation allows for the teacher to put into practice the theoretical principles and, by doing so, notice the need to understand their relativity, considering the various elements at play in the teaching-learning process. (Prado, 2003, p. 41)

The continued education approach that prioritizes contextualized learning, and an investigative-reflective attitude from the teacher, requires a systematic follow up from the developer as the pedagogical mediator in the teacher’s acting context. In this perspective, Lobo da Costa et al (2010), Prado (2006) underscore the importance of the developer's role to promote situations that will favor interactions among teachers so that a collaborative network can be built in which everyone learns and teaches with and to one another. Collaborative work, as proposed by Fullan and Hargreaves (2000) is characterized by a number of features, of which we highlight: the attitudes and behaviors in the rapport among the teachers, which reveal trust, commitment, sharing of ideas, experience and doubts, as well as


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the recognition both of the individual and of the group to which they belong. However, it is important to emphasize that collaborative work is not immediately established among participants. According to Imbernón (2010): … collaborative work among teachers is not easy, as it is based on understanding education as a means to create spaces where both individual and group abilities can be developed, with dialog, from the analysis and discussion among all participants when exploring new concepts. (p.65)

Hence, a developing approach needs to intentionally develop strategies that favor collaboration as a practice built by group members. This is confirmed by what many researchers, such as Fiorentini et al (2002), have been finding as signs that collaborative work is essential for the professional development of teachers. Professional development for us means, according to Ponte (1997), as being formed by all the actions performed by the teacher that lead to restructuring his pedagogical practice, based on reflection, action and new reflection. It is “a process of growing competencies in terms of teaching and non-teaching practices, in self-controlling his activities as an educator and as part of the school organization” (p. 44). To enhance professional development, according to the author, it is important to consider both the collective and individual aspects, since such development is improved by collaborative contexts (institutional and associational, both formal and informal ones) where the teacher has the opportunity to interact with his peers. One research in particular conducted by Lobo da Costa (2004) identified that collaborative work involved characteristics that were present in the development process, and were defined as the following categories: (C1) Shared reflection; (C2) Learning/ learning with each other; (C3) Teacher’s actions; (C4) Development actions; (C5) Research about practice; (C6) Experience exchange; (C7) Representativeness of all participants’ thoughts; (C8) Partnership; (C9) Shared goals; (C10) Commitment to the group; (C11) Trust, (C12) Voluntary participation; (C13) Dialog/interaction; (C14) Autonomy

140 Lobo da Costa & Prado - Constitution ofa learning network development and (C15) Reflection on action. It is worth noting that researchers such as Boavida and Ponte (2002) have also pointed out that collaborative work is an interesting option in continued education processes. For these authors, collaboration happens “in cases when a number of interveners work together, not in a hierarchical relation, but in an equal basis so as to have mutual help and common goals from which everyone can benefit” (p. 45). Collaborative work has the advantage of providing multiple views about the educational situation which in turn allows for the production of consistent, interpretative frames about the issue which was researched and studied. When working in collaborative-nature groups, the relationship between developer-developee is upturned in such a way that the established belief that in a continued education process there is one developer, or team of developers, who work with a group of teachers promoting their development, is now replaced by the idea of forming a team of educators who work together with college researchers and/or institutions in charge of the projects, and teachers, in a relationship of mutual learning and developing (Lobo da Costa, 2006). Another relevant issue refers to the conclusions drawn by the GT 7 meeting in the III SIPEM as presented in the Report (2007), which pointed that: Partnership, the pursuit to build collective knowledge, meets our present needs. School teachers, very frequently, have been acting like their students when receiving knowledge that is imposed and/or meaningless: they reject it. Developing proposals that are based on the transmission of knowledge – well-meant, but foreign to the local reality of each group of teachers – have proven to be irrelevant for decades. Together, school and university teachers reflect upon their own professional knowledge and give new meaning to their own professional development3 .

The GT7 emphasizes the importance of continued education projects whose focus is not only on increasing the teacher’s mathematical knowledge, but which promote discussions of contents that relate to


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classroom daily activities. They point out that establishing partnerships among mathematics educators at the universities and the schools is essential for the development of common knowledge, required by both the academic world and the school. In this regard, the group of Brazilian researchers has established indicators that are closer to those found in a worldwide context (Jaworski, 2001). The research to which this article refers to is embedded in the Programa Observatório da Educação (Education Observatory Program, in English) whose focus is to involve academics with teachers working at elementary levels, having as a principle the development of work partnerships, closely linked to classroom reality. In other words, this partnership is made by the integration of two different types of knowledge: theory and practice. Research scenario The research which supports this article is embedded in the project “Educação Continuada de Professores de Matemática do Ensino Fundamental e Médio: Constituição de um Núcleo de Estudos e Investigações sobre Processos Formativos” (ECPMEFM) 4, linked to

the Education Observatory Program. This program is funded by the Brazilian government whose goal is to improve the teaching and learning processes in public schools in the country and is developed together with universities. The project stimulates the academic community to develop action and research oriented to the needs of teacher development for teachers that work at elementary-level education. The ECPMEFM project is being developed in a private university in São Paulo city by the mathematical education department, with a group of professors, master and doctoral students, who work and develop research together with teachers, engaged in mathematics teaching in public schools. The research and development project has as a goal to develop a continued education methodology for mathematics teachers working at elementary levels of education, and involves the creation of collaborative professional learning networks, so as to provide sustainability, deemed to be the project’s underlying concept.

142 Lobo da Costa & Prado - Constitution ofa learning network The project is under development and extends for four years, involving various groups of mathematics teachers at elementary schools in the city of São Paulo, and development actions are performed on-site at the university campus, using practical and theoretical activities related to mathematical concepts and their implications to the process of teaching and learning. The group reports and discusses practices developed by the teachers in the context of their classroom together with their students. The group uses a virtual learning environment - (AVA) specially customized for the project - to improve the interaction and dialog among participants for them to share ideas and experiences about each one’s own learning experience in the project and in their practice teaching mathematics. One of the guidelines for mathematics teachers’ development in the ECPMEFM project is the constitution of collaborative groups between school and university teachers, among school teachers only, and between teachers and students. We seek to investigate to what extent such groups improve the professional development of the teachers involved. This is about using qualitative research, of a co-generating nature, as proposed by Greenwood and Levin (2000), that is, a particular kind of action-research that is developed by the partnership between researchers and teachers who create knowledge together. Both types of knowledge, the practical and the academic, are key for the research development. Continued development is designed through a number of strategic actions linked to contextualizing learning and building a collaborative network among peers, including the possibilities of virtual interactions as one way to allow for recording in writing the participants' reflective logs (Prado, 2003; Bairral, 2003; Lobo da Costa et al, 2008). This is the scenario in which the present study was developed. The goal was to understand how a collaborative learning network is set up during continued development actions. For that, we analyzed data collected from the first group of participants in the project, which included thirty elementary school teachers working at the public school network in São Paulo. The development actions performed with this group aimed to: • Approach contents based on the official mathematics syllabus of the state of São Paulo, starting at Sequences and followed by Plane


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Geometry. • Develop activities involving: narratives, identification of teachers’ expectations and demands, as well as life histories and reflection upon their own learning. • Discuss the purposes of mathematical education. • Discuss issues related to practices performed at the schools with the students and the theories studied, as well as reflections and studies carried out in the classroom. As for data collection, one of the strategies used at ECPMEFM project was to request, after one school-term interaction with the group, that a reflexive log be produced. The log was individually written and made available in the virtual environment used as a support for development actions. The reflexive log was a key element for data collection for two main reasons. First, because it enables the teacher to register his learning path in such a way that he can reflect and become aware of what he experienced with the group in this process of reconstitution. Second, to take developers and researchers to know the actions, reactions, feelings, impressions, interpretations, explanations, hypotheses and concerns in the experiences lived by the group of teachers and also to redirect future development actions. The log entries analysis was interpretative and used as categories the characteristics found in the research conducted by Lobo da Costa (2004), previously mentioned. Besides this interpretative analysis of the entries, a statistics treatment was applied to the categories using the software CHIC, 2004 (Coercive and Hierarchical Implication Classification) which allows us to have an overview of similarities and variable classes mapped on the hierarchical levels of a tree. From these indicators, we identified and analyzed the categories that were present in the entries made available in the virtual environment, which came from the participating teachers’ reflexive logs. Results The interpretative analysis of the logs stored in the AVA, showed the following characteristics of collaborative work:

144 Lobo da Costa & Prado - Constitution ofa learning network Code C1 C2 C3 C6 C9 C10

Category Shared reflection Learning/Learning with each other Teaching actions Experience exchange Shared goals

C11

Commitment to the group Trust

C13

Dialog/Interaction

C14

Autonomy development Reflection on action

C15

Description Reports expressing thoughts and queries to the group. Reports stating own learning (specific and content syllabus). Reports involving classroom experiences. Reports involving syllabus contents and practical activities. Reports involving the search process to achieve joint goals. Reports stating commitment to one another. Reports indicating a feeling of belonging and comfort. Reports showing recognition of group dialogs’ worth. Reports indicating more confidence in decision-making. Reports showing the reconstitution of pedagogical practices applied.

Hence, out of the fifteen categories listed by Lobo da Costa (2004), ten were found in this study. As shown by the excerpts of the participants’ reflexive log entries, they exemplify the different categories of collaborative work as described below: C1 – Shared reflection C2 – Learning C13 – Dialog/interaction C11 – Trust

Sharing with this group was a great help, because many of my doubts were clarified and the themes discussed here were of great importance. For instance, studying Parsysz’s levels was very interesting, because we thought a little more about how children’s minds work. The interaction was intense in this study group. (Teacher_A’s log)


C1 – Shared reflection C2 – Learning C6 – Experience exchange C13 – Dialog/interaction

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The meetings in which we did practical exercises were more profitable, we managed to exchange experiences and learned how to solve the exercises in various ways, because different ways to solve them came up.

(Teacher_B’s log)

In these two logs we notice that mastering the mathematical content is essential and is the first step to lead the teacher to rethink his pedagogical practice, although we know that having the mathematical knowledge does not ensure reflection upon action or changes in the classroom. C1 – Shared reflection C2 – Learning C15 – Reflection on action

I could notice that talking about geometry to my students is not that complex, it’s possible and real, the geometry presented in the modules is a beautiful geometry, easy to be developed, and worked as an incentive to be applied in the classroom. For instance, about the Pythagorean theorem, when we made the drawing on the poster paper and explored various issues (as one leads to another), building the tangram, using it as a puzzle, gee, that was awesome! Soon after that we tackled another topic about the trapeze area (… ) and went deeper into the Pythagorean theorem. (Teacher_C’s log)

In the above entry, it is clear that when the teacher realizes there are new possibilities to approach mathematics which he finds meaningful, he becomes elated and concludes that it can be adapted to his students; from then on, a motion is set and it can have an impact in the classroom.

146 Lobo da Costa & Prado - Constitution ofa learning network C1 – Shared reflection C2 – Learning C11 – Trust

I learned a lot, it’s hard to tell what was more meaningful, because everything and every subject were meaningful. I learned about the different types of trends in mathematical teaching and fitted in some of them, I learned how we can use geometry in the classroom by just using concrete materials in a simple, constructive way, I learned a lot about the Pythagorean theorem and its contextualization. The most meaningful content was the development of geometry exercises, the part about triangles similarity, in which I had the opportunity to clear doubts and learn… (Teacher_D’s log)

The word that appears most frequently in the entry above is “learned”. We found that for this to happen it is essential the group of teachers feel at ease and confident to take an open attitude as a “learner”, which allows him to establish relations between what is being discussed and studied in the group and his daily actions in the classroom. It can be noticed in this entry that the learning of mathematical contents did not happen as an isolated event, but rather related to the context and in a reflexive way. C1 – Shared reflection C2 – Learning C3 – Teaching actions C6 – Experience exchange C11 – Trust C15 – Reflection on action

The demonstrations for the Pythagoras theorem and others performed in the triangles were presented exactly when I was teaching similarity to my 9th grade students. I had to make a few changes, doing practical applications instead ofusing technical terms. I made good use of some of my colleagues’ ideas that were used in the demonstrations and they worked very well. (Teacher_E’s log)


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What can be noticed in this entry is the openness to learn with each other and that the learning in the group has catalyzed changes in classroom practice. We noticed that, as there was a coincidence between what was being discussed in the meetings and the class syllabus, it was possible to develop teaching actions using materials and methodologies previously discussed and analyzed in the group’s work. C1 – Shared reflection C9 – Shared goals C10 – Commitment to the group

What attracts me most in this and other courses is to set up groups who have the same focus ofinterest: improving the ways to approach contents in the classroom. And, for that, I realized that undergraduate education for teachers has shortcomings, because many colleagues show, and admit having, a lot ofdifficulty in understanding many ofthe contents, which worries me, because I don’t know how to go round this situation.

(Teacher_F’s log)

The teacher recognizes having the same focus of interest as one of the most important characteristics of collaborative work in a group. His view of his peers, identifying their conceptual shortcomings deriving from deficiencies in their undergraduate education, gives us the feeling that in many cases the teacher is the casualty of an educational system which will continue to lead to a vicious circle that hinders improvements in education, unless changes are made to it. C1 – Shared reflection C2 – Learning C9 – Shared goals C10 – Commitment to the group C15 – Reflection on action

In the second module, which was a positive continuity of the first, we had more approaches to geometry, Broadening our understanding. There was one extremely interesting lecture conducted by teacher Serrazina that made me think and startle when she said, ‘If I teach and my students don’t learn, it’s because I'm not teaching’.

(Teacher_G’s log)

148 Lobo da Costa & Prado - Constitution ofa learning network The effect produced by the phrase “if I teach and my students don’t learn, it’s because I'm not teaching” shows the teacher’s moment of awareness to recognize that, although teaching and learning are two distinct processes, they are interrelated in educational actions. Teacher and student constitute one system and while interacting, one teaches and learns how to teach while the other learns and teaches how he learns, being both accountable for each other’s development. C1 – Shared reflection mudar C2 – Learning C11 – Trust

. . . I refreshed my knowledge, I learned certain subjects that I had not studied before, either in high school or at college. Certainly, what I learned has helped me a lot, both to my personal and professional growth. (… ) I'm thankful for the opportunity to be part of this group, with wonderful teachers and colleagues. I learned a lot with all of you.

(Teacher_H’s log)

The feeling of accomplishment expressed by the teacher as soon as he recognizes his learning potential, shows that the educator, regardless of his specialty, must feel prepared in terms of syllabus contents to perform his trade autonomously. Hence, a deep revision of role of the educational institution is required. C1 – Shared reflection C2 – Learning C3 – Teaching actions C14 – Autonomy development C15 – Reflection on action

Geometry undoubtedly contributed a lot for classroom teaching. (. . . ) It was easy to clarify students’ doubts, because we had discussed exactly the same content with the group. I worked on problems using everyday situations, like the ladder and the kite we had used in our meetings. I found the group assignments interesting. Usually, while in the classroom, I apply exercises to be solved individually, but from then on I started to use group work study and the students like it very much. (Teacher_I’s log)


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This entry confirms what we have previously discussed and brings some complementary information related to group work practices; he underscores that what has been experienced in the project - such as situations where they can learn with each other by interacting, clarifying ideas and debating viewpoints -, have enriched his learning. C1 – Shared reflection mudar C2 – Learning C11 – Trust

Ways to solve exercises. I wish we had moreIn Module 2: I was dazzled by the many diverse meetings with geometry, because I still have many doubts, I have a lot to learn. As Einstein said, ‘I know that I know nothing’. (Teacher_J’s log)

The desire and openness, that is, the internal motivation to learn presented by the teacher shows that it was possible to create a relationship of trust among group members, allowing for each of them to genuinely recognize themselves in terms of what they know and what they need to know. C1 – Shared reflection mudar C2 – Learning C3 – Teaching actions C6 – Experience exchange C13 – Dialog/Interaction C15 – Reflection on action

At first I thought it would be yet another course which, at our level oflearning, would be useless, but I noticed in the subsequent meetings that it wasn’t just a course, but a lot more: it was about exchanging ideas and experiences lived by other teachers in their daily teaching practices in the classroom. (. . . ) I learned that calculating a simple area in a figure like a square, is something that we can teach at least in three different ways, thus creating a more interactive environment between the content and the students.

(Teacher_K’s log)

The entry hints that throughout time, a more flexible relationship was established, with a less hierarchy than the one established in continued education courses. In other words, more than college professors going

150 Lobo da Costa & Prado - Constitution ofa learning network to conduct a course for mathematics public school teachers, we became a group of mathematics educators, discussing and exchanging experiences about learning and teaching mathematics. C1 – Shared reflection C2 – Learning C11 – Trust

The group helped me solve difficulties in. At first, when requested that we geometry developed activities about certain subjects, I felt a little insecure sometimes; but as the course developed, I noticed that my difficulties were the same as some of my colleagues'. I also noticed that the way we had our group meetings helped us solve doubts that everyone had. (Teacher_L’s log)

C1 – Shared reflection C2 – Learning C11 – Trust C6 – Experience exchange

I learned a little with each one of the participants and with the professors’ interventions bringing in viewpoints I had not yet conceived, (… ), it’s an area which I found difficult to work with: the theoretical field of demonstrations and proofs of some axioms. I profited a lot from all the explanations given by the colleagues (… ). It was enough for me to generalize this knowledge. (… ) with the group’s help, I improved. (Teacher_M’s log)

C1 – Shared reflection mudar C2 – Learning

I could tell I’m different from when I started, I have a ‘little’ more knowledge, but I believe it serves to improve my teaching practice.

(Teacher_N’s log)


C1 – Shared reflection mudar C2 – Learning C15 – Reflection on action C13 – Dialog/Interaction

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I learned to reflect more about questions to be worked on with the students, thus developing a better perception in the thinking, emphasizing the researches and commentaries in group. Valuing everyone’s opinion for a better understanding of mathematical applications. (Teacher_O’s

log) C1 – Shared reflection C15 – Reflection on action C11 – Trust C6 – Experience exchange C3 – Teacher actions

What is interesting is that we can bring situations that happen in the classroom to be discussed by the group, the way the students behave, how they learn and their behavior in relation to the various subjects ofthe content applied. (Teacher_P’s log)

These last entries show that teachers recognize the existence of a new way to learn based on the exchange of experiences in a development context that provides collaborative work, thus establishing and strengthening an atmosphere of trust to teach and learn with each other. A wider lookout allows us to notice the constant presence of the characteristic Shared reflection (C1) in the entries. This is due to the fact that all participants had access to the logs, which were available in the virtual environment at any time. This opportunity for the participating teacher to write and re-write their logs, as well as to read and re-read his and his fellow teachers’ logs as many times as he wished and make comments, is what encourages the sharing of ideas, reflections, experiences and queries among group members. The second characteristic that was most frequent in the logs was Learning (C2), showing that openness to learn in collaborative work is imperative. Some of the entries clearly show that the teacher

152 Lobo da Costa & Prado - Constitution ofa learning network recognizes the importance of the fellow teacher’s role in his own learning process. The characteristic Trust (C11) is related with learning, as teachers experiencing a collaborative task must feel confident to expose their shortcomings to peers without being afraid of judgments but rather with the courage and expectations of his personal and professional development. Besides the above interpretative analysis of the logs, we also treated the resulting categories using CHIC, which allowed us to carry out a relational analysis among them. The next figure shows the similarity tree produced:

Figure 1 . Category Similarity Tree

In figure 1, the similarity tree, two classes are found: Class-1, formed by categories (C1) Shared reflection, (C2) Learning/learning with each other, (C11) Trust, (C9) Shared goals and (C10) Commitment to the group, and Class-2, composed by categories (C3) Teaching actions, (C15) Reflection on practice, (C14) Autonomy development, (C6) Experience exchange and (C13) Dialog/interaction.


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Class-1 was named Interaction and contains a sub-class, constituted by categories (C1, (C2, C11) and by a cluster formed by categories (C9, C10), as shown in the figure below:

Figure 2.

Class 1 - Interaction

It can be observed that the cluster (C9, C10) shows a significant level of similarity indicating a high probability that some interaction occurred among the teachers in the group in terms of sharing goals and creating attitudes of commitment. This possibility of interaction among peers is paramount and should be one of the goals for developers, since this experience can contribute for the establishment of commitment with one another in the learning context provided by the Education Observatory Program. The sub-class formed by the set of categories (C1 (C2, C11), shows a discreet level of similarity, but still hints that the teachers recognize the fact that an atmosphere of trust must be created to enable learning with each other so that they can expose their conceptual frailties. Such trust allows the teacher to feel accepted by the group, and he can change his behavior towards learning with his colleagues’ experiences and sharing his reflections about practical and theoretical questions studied in group, supported by the developers’ pedagogical mediation. Class-2 was named Teacher's work and is composed by the chaining of categories {((C3, C15), C14), (C6, C13)}, as shown in the figure below:

154 Lobo da Costa & Prado - Constitution ofa learning network

Figure 3 . Class 2 - Teaching Activity

This chain of categories clearly demonstrates the existence of a higher degree of similarity between the categories (C3, C15), showing that the experiences lived by the teachers participating in the Education Observatory Project, allowed for the teacher’s classroom practice to be reported, reflected upon and understood. Thus the importance for development courses to incorporate the teacher’s actions, those experienced in his school context. However, such actions should be reflected upon and understood. This probably happened through experience exchanging between the teacher and his peers and through the dialogs established among group members, as well as between the teachers and the studied theoreticians, the ones who clarify their understanding, providing better conditions for the development of intellectual autonomy. Such autonomy can propel the mathematics teacher of elementary schools to pursue professional self-development, taking into account that this learning process should be continuous and dynamic for him to interact with the students, the future professionals in a new society. We can notice that these characteristics are interrelated: the teacher’s professional development is linked to reflecting on his practice to reconstruct it, and it is within this process that exchanging experiences becomes encouraging, because it shows new possibilities which allow them to dare to change their teaching actions concerning learning


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strategies. Hence, the acknowledgement of the role played by dialog and interactions established in collaborative work on continued development for teachers. The characteristics which appeared more discreetly in these logs were “Shared goals” and “commitment to each other”, which indicate that such characteristics develop as group members feel they have more autonomy to share their goals in search of knowledge, as well as to develop commitment to their peers' learning. We insist that, for the developers, understanding this process is essential to design development strategies that incorporate dynamic actions, so as to establish a movement between analysis and deepening of mathematical contents, and almost simultaneously, incorporate various aspects from the teacher’s pedagogical practice. It is in this movement between action and reflection, between mathematical contents and their re-contextualization in the school practice, that the knowledge of the teacher’s praxis will be developed towards a learning spiral. Conclusion This study showed the connection between collaborative network as a collective learning space in the context of continued education and the potential it has to propel the teacher’s professional development. The network is created through a process in which the development actions are designed based on experiences that reinforce characteristics that are typical of collaborative work. The indication is that this network includes the use of the contributions provided by virtual environments, as the latter allow for breaking barriers of space and time among group members and also enable dialogs/interactions which are established by means of writing using various communication resources in the virtual environment. This type of interaction, involving the sharing of experiences, knowledge, reflections and queries, helps build a collaborative learning-reflection space among the teachers. This form of learning, in turn, makes every participant able to experience simultaneously being a learner and a teacher to the others, and to move towards the sustainability of learning throughout life.

156 Lobo da Costa & Prado - Constitution ofa learning network References Alarcão, I. (2001). Escola Reflexiva e nova racionalidade. Porto Alegre: Artmed. Almouloud, S. (1992) L’ Ordinateur, outil d’aide à l’apprentissage de la démonstration et de traitement de donnés didactiques. These de Docteur. U.F.R. de Mathematiques. Rennes, França: Université de Rennes I. Bairral, M.A. (2003). Dimensões de Interação na Formação a Distância em Matemática. Erichim (RS): Revista Perspectiva, 27 (98), 33-42l. Boavida, A. M. e Ponte, J. P. (2002). Investigação colaborativa: Potencialidades e problemas. In GTI (ed.), Reflectir e investigar sobre a prática profissional, 43-55. Lisboa: APM. Campos, T.M.M.; Pietropaolo, R.C.; Prado, M.E.B.B; Campos, Silva, A.C. (2009). Uma abordagem de educação a distância em um processo de formação continuada de professores de Matemática. VI CIBEM - Congreso Iberoamericano de Educación Matemática. Puerto Montt, Chile. Ferreira, A. C. (2003). Metacognição e desenvolvimento profissional de professores de matemática: uma experiência de trabalho colaborativo . Tese de Doutorado em Educação. Campinas:

FE/Unicamp, SP. Fiorentini, D.; Nacarato, A.M.; Ferreira, A. C.; Lopes, C.A. E.; Freitas, M. T. M.; Miskulin, R. G. S.(2001). Formação de professores que ensinam Matemática: um balanço de 25 anos da pesquisa brasileira. Educação em Revista, 36, 137-160. Fullan, M. & Hargreaves, A. (2000). A escola como organização aprendente: buscando uma educação de qualidade. 2ª ed. Porto Alegre: Artes Médicas,135p. Gras R. (2000). Les fondements de l’analyse statistique implicative, Quaderni di Ricerca in Didattica del Gruppo di Ricerca sull’Insegnamento delle Matematiche (G. R. I. M. ), 9, 189-209.

Greenwood, D. & Levin, M.(2000). Reconstructing the relationships between universities and society through action research. In: Norman D. and Yvonna L. (ed.) Handbook for Qualitative Research , 85-106, 2nd ed. Thousand Oaks, California: Sage Publications Inc.


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Imbernón, F. (2010). Formação continuada de professores. Trad. Juliana dos Santos Padilha. Porto Alegre: Artmed, 120 p. Lobo da Costa, N.M. (2004). Formação de professores para o ensino da matemática com a informática integrada à prática pedagógica: Exploração e análise de dados em bancos computacionais. Tese de Doutorado em Educação. PUSP.

Lobo da Costa, N. M. (2006). Formação continuada de professores: uma experiência de trabalho colaborativo com matemática e tecnologia. In: Nacarato, A.M., Paiva, M. A. V. (orgs) A formação do professor que ensina matemática: perspectivas e pesquisas, 67196. Belo Horizonte: Autêntica. Lobo da Costa, N. M., Prado, M.E.B.B., Pietropaolo, R. C. (2010). Currículo e Mediação Pedagógica online. IX Colóquio sobre Questões Curriculares / V Colóquio Luso-Brasileiro , Porto, Portugal. Disponível http://www.fpce.up.pt/ciie/publs/Actas_ IX_Coloquio_QuestoesCurriculares_Junho2010.zip (acesso em 20/01/2011). Lobo da Costa, N. M., Prado, M.E.B.B., Campos, T.M.M. (2008). Formação do professor de Matemática: Uma abordagem pedagógica usando recursos de ambientes virtuais In: 6o Congreso Internacional de Educación Superior Universidad, Havana, Cuba. Lopes, C. A. E. (2003). O conhecimento profissional dos professores e suas relações com Estatística e Probabilidade na Educação Infantil. Tese de Doutorado em Educação Campinas: FE/Unicamp.

Pietropaolo, Ruy, C.; Lobo da Costa, N., M.; Prado, M. E. B.B. (2009). Análise da constituição de um grupo de pesquisa sobre formação de professores de matemática In: Anais do IV Seminário Internacional de Pesquisa em Educação Matemática, Taguatinga, DF. Ponte, J. P. (1997). O conhecimento profissional dos professores de matemática. Relatório final de Projecto “O saber dos professores: Concepções e práticas”. Lisboa: DEFCUL. Prado, M.E.B.B. (2003). Educação a distância e formação do professor: redimensionando concepções de aprendizagem . Tese de Doutorado em Educação. São Paulo: PUCSP.

158 Lobo da Costa & Prado - Constitution ofa learning network Prado, M.E.B.B. (2006). A Mediação Pedagógica: suas relações e interdependencias. Anais do XVII do Simpósio Brasileiro de Informática na Educação . SBC – Sociedade Brasileira de Computação. Brasília. Prado, M.E.B.B. & Valente, J.A. (2002). A educação a distância possibilitando a formação do professor com base no ciclo da prática pedagógica. In: Moraes, M.C. (org.) Educação a Distância: fundamentos e práticas. Campinas, SP: NIEDUNICAMP. SBEM - Relatório do GT7 – Formação de Professores que Ensinam Matemática. (2007). Coordenação: Adair Mendes Nacarato e Maria Auxiliadora Vilela Paiva. III SIPEM – Águas de Lindóia. Disponível em: http://www.sbem.com.br/files/RelatorioGT7.pdf. (acesso em 01/10/2011).

Notes According to the reference document available at: portal.mec.gov.br/arquivos/pdf/conae/documento_referencia.pdf 2 SIPEM: International Research Seminar in Mathematical Education, Águas de Lindóia, São Paulo, Brazil, 2006.GT7: Research Group about Teacher Development. 3 GT7 report, available at: http://www.sbem.com.br/files/RelatorioGT7.pdf 4 In English, “Continued Education for Mathematics Teachers of Elementary and High Schools: Establishing a Study and Research Group about Development Processes”. 5 For more details about CHIC, see Gras (2000) and Almouloud (1992). 1

Nielce M. Lobo da Costa is Professor at the Program of postgraduate studies in Mathematics Education, at the Universidade Bandeirante de São Paulo, Brazil. Maria E. Brisola Brito Prado is Professor at the Program of post-graduate studies in Mathematics Education, at the Universidade Bandeirante de São Paulo, Brazil. Contact address: Direct correspondence concerning this article should be addressed to the authors at: Av. Braz Leme, 3029, São Paulo - SP - CEP: 02022-011, Brazil - E-mail address: [email protected], [email protected].


Uso de artefactos concretos en actividades de geometría analítica: una experiencia con la elipse José Carlos Cortés Zavala & Héctor Arturo Soto Rodríguez1 1) Universidad Michoacana de San Nicolás de Hidalgo. Morelia. México.

Date of publication: June 24th, 2012

To cite this article: Cortés, J.C.; Soto Rodríguez, H.A. (2012). Uso de Artefactos Concretos en Actividades de Geometría Analítica: Una Experiencia con la Elipse. Journal of Research in Mathematics Education, 1(2), 159-193. doi: 10.4471/redimat.2012.09 To link this article: http://dx.doi.org/10.4471/redimat.2012.09

PLEASE SCROLL DOWN FOR ARTICLE The terms and conditions of use are related to the Open Journal System and to Creative Commons Non-Commercial and Non-Derivative License.

REDIMAT Journal of Research in Mathematics Education Vol. 1 No. 2 June 2012 pp. 159193

The Use of Concrete Artifacts in Analytic Geometry: the Ellipse Experience

José Carlos Cortés Zavala & Héctor Arturo Soto Rodríguez Universidad Michoacana de San Nicolás de Hidalgo

Abstract The purpose of this article is to provide the results of a research on the use of specific artefacts (two different ellipsographs) for the learning of Mathematics, specifically in Analytic Geometry on the topic of Ellipse. It was implemented one activity for each one of the artefact, by means of the use of working sheets that guided the students for tem to build the formal concept of Ellipse, answe ring to the corresponding questions in each activity as well as manipulating the ellipsograph. All with the aim to facilitate to the students the understanding and learning of the mathematic concepts.

Keywords: analytic geometry, ellipsograph, collaborative learning

2012 Hipatia Press ISSN 20143621 DOI: 10.4471/redimat.2012.09

REDIMAT Journal of Research in Mathematics Education Vol. 1 No. 2 June 2012 pp. 159193.

Uso de Artefactos Concretos en Actividades de Geometría Analítica: Una experiencia Con la Elipse José Carlos Cortés Zavala & Héctor Arturo Soto Rodríguez Universidad Michoacana de San Nicolás de Hidalgo

Abstract El propósito de éste articulo, es dar a conocer los resultados de una investiga ción relacionada con el uso de artefactos concretos (dos elipsógrafos distintos) para el aprendizaje de las Matemáticas, específicamente en Geometría Analítica con el tema de Elipse. Se implementaron una actividad por cada uno de los ar tefactos, a través del uso de hojas de trabajo las cuales guían a los estudiantes para que ellos puedan construir el concepto formal de Elipse, respondiendo las preguntas correspondientes en cada actividad así como manipulando el elipsó grafo en cuestión. Lo anterior con el objetivo de facilitar la comprensión y el aprendizaje de conceptos matemáticos por parte de los alumnos.

Keywords: geometría analítica, elipsógrafos, aprendizaje colaborativo.

2012 Hipatia Press ISSN 20143621 DOI: 10.4471/redimat.2012.09

E

161 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse n el siglo XVII el concepto de Geometría Analítica poseía un significado diferente a nuestra noción moderna. La principal diferencia radica en que antes las ecuaciones no representaban curvas, sino que las curvas daban origen a una ecuación y en la actuali dad la curva es dada a partir de un análisis de propiedades algebraicas. Durante mucho tiempo se han creado una infinidad de artefactos físi cos con la finalidad de trazar algunas de las cónicas, se les han llamado, Parabológrafos (trazan parábolas), Elipsógrafos (trazan elipses) e Hiper bológrafos (trazado de hipérbolas). Revisando el libro de Dyck (1994) encontramos artefactos articulados para el dibujo de curvas desde la an tigua Grecia. Meneachmus (~380 ~320 A.C.) tenía un dispositivo mecánico para construir cónicas; Proclus (418485) también menciona a Isidoro de Mileto quien tenía un instrumento para trazar una parábola (Dyck, 1994, p.58). Leonardo Da Vinci (14521519) inventó un elipsó grafo con un movimiento invertido de la conexión fija. Los dispositivos mecánicos para dibujar curvas fueron utilizados también por Albrecht Dűrer (14711528). René Descartes (15961650) publicó su Geometría (1637) libro en el cual daba métodos geométricos para dibujar cada cur va con algunos aparatos, y estos aparatos eran a menudo articulados. En el año de 1657, Van Schooten publicó su “Exercitationum mathematica rum libri quinque”. Como el título sugiere, la obra se divide en cinco "libros" de un centenar de páginas cada uno. El libro I es una revisión bastante estándar de la aritmética y la geometría ordinaria. El libro II contiene construcciones con regla. En el Libro III, van Schooten trata de reconstruir algunas de las obras de Apolonio en lugares geométricos. Este fue un importante tema de investigación de la época. El libro IV contiene la obra más conocida de Van Schooten. Su título es " Orgánica conicarum sectionum ", o "Los instrumentos de las secciones cónicas." La palabra "orgánica" está más estrechamente relacionada con el órgano como instrumento musical que a la "orgánica" a veces encontramos en la química o la agricultura. Como sugiere el título, el capítulo describe una variedad de bellos artefactos para la elaboración de las diferentes secciones cónicas. En 1877 A. B. Kempe publicó un pequeño libro: Cómo dibujar una línea recta: Una conferencia sobre artefactos articulados. Mencionó a J. Watt (17361819) y también el trabajo de J.J. Sylvester (18141897), Richard Roberts (17891864), P.L. Chebyshev

REDIMAT Journal of Research in Mathematics Education, 1 (2) 162 (18211894), Harry Hart (18481920), William Kingdon Clifford (1845 1879), Jules Antoine Lissajous (18221880), Samuel Roberts (1827 1913), y Arthur Cayley (18211895)1. Por otro lado, Franz Reuleaux (18291905), quién a menudo es llama do "el padre del diseño de las máquinas modernas”, tenía muchos meca nismos de líneas rectas en su colección del modelo cinemático. La Universidad Cornell tiene una colección con cerca de 220 modelos ci nemáticos distintos de F. Reuleaux y 39 de ellos son mecanismos sobre el movimiento rectilíneo2. Finalmente, también tenemos el caso de Iván Ivánovich Artobolevski, (1905 – 1977). Fue un ingeniero mecánico, y científico ruso en el campo de la Teoría de Mecanismos y Máquinas. Fue miembro de la Academia de Ciencias de la Unión Soviética desde 1946. Artobolevski propuso una clasificación de los mecanismos espaciales y desarrolló métodos para su análisis estrucural, cinemático y cinetostático. Recopiló en “Les mécanismes dans la technique moderne” (Artobolevski, 1975) varios artefactos mecánicos cuya finalidad era trazar alguna cónica. Marco conceptual

Resultados recientes de investigación constatan la importancia del uso de nuevas tecnologías en la enseñanza de las matemáticas y las ciencias, y en la incorporación al trabajo científico por parte de los estudiantes. En sesiones de trabajo dirigido, los alumnos son capaces de desplegar recursos matemáticos que se desencadenan por medio de la compren sión de nociones (Hoyos, Capponi y Génèves, 1998), o se promueve la creatividad y el ingenio en el diseño científico mediante el uso de nue vas tecnologías (Verillon y Rabardel, 1995; Jörgensen, 1999). Por otro lado, perspectivas teóricas y prácticas alternativas complementarias en didáctica de las matemáticas (Mariotti et al 1997; Bartolini et al 2003, 2004; Bartolini 2007, Boero et al., 1996, 1997; Arzarello, Robutti 2004, Jill et al 2002), argumentan a favor de la introducción en el salón de cla ses de contextos históricos de recreación de la experiencia científica, en particular aquéllos que tienen que ver con la práctica de la geometría y que utilizan modelos mecánicos o articulados de máquinas para dibujar o trazar, como un medio de generación de ideas o nociones matemáticas complejas. Por otro lado, como menciona Duval la articulación de

163 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse registros semióticos de representación, en este caso el gráfico y el al gebraico, deben ser necesarios para la aprehensión conceptual (Duval 1995). El trabajo de investigación realizado tuvo como propósito experi mentar actividades que puedan servir como recurso didáctico, para acercar a los estudiantes de bachillerato hacia la demostración y cons trucción de conceptos en geometría Analítica, específicamente el de Elipse, utilizando dos elipsógrafos los cuales son el Elipsógrafo de pa lancas y colisa de Inwards y el Antiparalelogramo articulado de Van Schooten. En las figuras siguientes se presentan los 2 elipsógrafos: el Elipsó grafo de palancas y colisa de Inwards y el Antiparalelogramo articula do de Van Schooten, los cuales fueron realizados con acrílico (modelo físico) y construidos también con Geogebra (modelo virtual).

Figuras 1 y 2 . Elipsógrafo de palancas y colisa de Inwards construcción física y construc ción virtual. Figura 3 y 4. Antiparalelogramo articulado de Van Schooten construcción fí sica y construcción virtual.

Para cada uno de los modelos desarrollados, tanto el físico como el virtual, se diseñó una hoja de trabajo que servía de guía para encontrar el modelo matemático inmerso. Es decir a través de la manipulación guiada de los artefactos se espera que el estudiante descubra el modelo matemático inmerso en el artefacto, esto permitirá la reversibilidad del conocimiento (Piaget, 1950).

REDIMAT Journal of Research in Mathematics Education, 1 (2) 164 Metodología

La investigación se llevó a cabo en un ambiente de trabajo colaborati vo, que incentiva la cooperación entre individuos para conocer, com partir y ampliar la información que cada uno tiene sobre el tema. Es de carácter cualitativo, es decir las conclusiones y comentarios que se desprenden del análisis de datos obtenidos, no son producto de las relaciones numéricas que se puedan obtener de las hojas de trabajo aplicadas, sino emergen del análisis de las cualidades asociadas tanto a las preguntas de las hojas de trabajo, como a los comportamientos de los estudiantes en cada una de ellas. En cada uno de los Elipsógrafos seleccionados se analizaron sus ca racterísticas para así realizar las hojas de trabajo y definir lo que se es peraba lograr con ellos. Las hojas de trabajo con las respuestas de los estudiantes, las videograbaciones realizadas durante el trabajo de cada uno de los equipos, así como las observaciones en el trabajo de campo, conforman el cuerpo de datos para llevar a cabo el estudio. Nos proponemos observar el potencial que tiene los Elipsógrafos para promover el aprendizaje en el salón de clases. Por ello, el proceso para la implementación de los Elipsógrafos consta de seis etapas: (1) Selección de artefactos (2) Descripción y Demostración matemática de los artefactos (3) Construcción de los artefactos (4) Realización de actividades didácticas (5) Etapa de Aplicación (6) Análisis de los datos Resultados del cuestionario por ítems Se revisó la obra de Iván Ivanovich Artobolevski “Les mécanismes dans la technique moderne” (Artobolevski, 1975), la cual es un com pendio de varios artefactos que trazan diferentes cónicas. Después se prosiguió a seleccionar cuáles de ellos eran factibles para poderse construir, puesto que había artefactos interesantes, pero físicamente complicados de realizar. De esta manera, los artefactos seleccionados fueron el Elipsógrafo de palancas y colisa de Inwards y el Antiparale logramo articulado de Van Schooten.

165 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Descripción y demostración matemática de los artefactos

Elipsógrafo de palancas de Inwards:

Figura 3. Elipsógrafo de palancas y colisa de Inwards

Las longitudes de los segmentos del artefacto satisfacen que EB = BC = CD = DE, es decir, la figura EBCD es un rombo. Los puntos A y C se mantienen fijos (focos). El segmento BD es la diagonal del rombo. El punto K es la intersección entre la corredera 2 y 3. Cuando el seg mento AE gira alrededor del punto fijo A, el punto k describe una elip se. Existe una condición para que dicho artefacto pueda describir una elipse. La condición es que 1/2 AE > AO (AO = OC) es decir, que la distancia entre los focos sea menor que el segmento AE. Por construcción sabemos que: BC = CD = DE =EB. Ahora tracemos el segmento KC:

Figura 4. Trazado de segmento KC

Los triángulos KDE y KDC son congruentes por LAL por tener: (a) DE = DC (por construcción) (b) KD lado común (c) Ángulo KDE = Ángulo KDC puesto que BD es diagonal

REDIMAT Journal of Research in Mathematics Education, 1 (2) 166 De la congruencia anterior podemos deducir que: KC = KE Observemos que: AE = AK + KE AE = AK +KC (por ser KE KC) Pero AE es constante AK + KC = constante Se cumple la condición para que el punto K describa una elipse según su definición.

Antiparalelogramo articulado de van Schooten: La base de este artefacto es el Antiparalelogramo ABCD. Cabe men cionarse que: AB = CD y AD = BC. Se dice que es Antiparalelogramo, puesto que en el movimiento, el artefacto sigue manteniendo la pro piedad de no tener dos pares de lados paralelos. Los puntos fijos (fo cos), son los puntos A y B. Con la intersección de la corredera 1 y 2 obtenemos el punto E. Al mover el artefacto, el punto E tiene despla zamiento por los segmentos AD y BC, describiendo durante el movi miento una elipse.

Figura 5. Antiparalelogramo articulado de van Schooten

Por construcción conocemos que: AB = CD AD = BC Ahora tracemos los segmentos CA y DB:

167 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse

Figura 6. Trazo de los segmentos CA y DB

Ahora tracemos el eje de simetría al cual llamaremos t: Sea M el punto de la intersección de t con CA y N el punto de la intersección entre t y BD:

Figura 7. Trazo de los segmentos M y N

Nótese que el trazo del eje de simetría tiene algunas implicaciones in teresantes en nuestra figura como son las siguientes: (a) t es perpendicular a BD y AC y los corta en su punto medio (mediatriz). (b) t bisecta a los ángulos DEB y CEA. Ahora tenemos todo lo necesario para afirmar que ∆BEN es congruen te con ∆DEN por el criterio LAL: (a) Tienen a EN como lado común. (b) Angulo BNE = Angulo DNE (porque el eje de simetría es perpendicular a BD formando ángulos rectos). (c) BN = ND (puesto que t bisecta a BD por ser eje de simetría).

REDIMAT Journal of Research in Mathematics Education, 1 (2) 168 ∆BEN

∆DEN

También ∆MEA es congruente con ∆MEC: (a) EM lado común. (b) Angulo CME = Angulo AME (porque el eje de simetría es perpendicular a BD formando ángulos rectos). (c) CM = MA (puesto que t bisecta a AC por ser eje de simetría). ∆MEA

∆MEC

De las dos congruencias anteriores obtenemos que el ∆AEB es con gruente con ∆CED por el criterio LLL (ya que de las congruencias an teriores deducimos que EB ED y EA EC además de que por construcción conocíamos que AB CD). De allí vemos que: AE + EB = AE + ED = AD = cte. Se cumple la condición para que el punto E describa una elipse según su definición. Construcción física de los artefactos

Se trató de adecuar las medidas para cada uno de los artefactos con el fin de que no fueran muy grandes y que funcionaran adecuadamente de acuerdo a sus propiedades. En esta etapa se realizaron varios bosquejos utilizando diferentes materiales. Los primeros prototipos se construyeron en papel ilustración, el cual es utilizado para elaborar maquetas. Se cortaban las barras correspon dientes y después se perforaban en los extremos para por allí unirlas mediante postes metálicos y obtener el artefacto deseado. Este material nos proporcionaba una idea de lo que se podría lograr sin embargo sufrí demasiadas deformaciones durante la manipulación. Después de éste se trabajó con tireno, el cual nos brindaba un mejor prototipo. Se decidió que los artefactos tuvieran menor deformación durante el mo vimiento por lo que se volvieron a construir en tireno de mayor calibre. Una vez obtenido un prototipo de calidad, se decidió mejorar el

169 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse material de construcción. Fue difícil encontrar a una persona que los realizara en metal, una vez que se logró lo anterior, surgieron distintos problemas al trabajar los distintos metales, por lo que los artefactos quedaban muy bromosos y pesados. Después del fracaso en metal, se optó por llevarlos a cabo en Acríli co, obteniendo de esta manera los artefactos que se probaron ante es tudiantes de bachillerato.

Figura 8. Elipsógrafos en diferentes materiales

Realización de actividades didácticas

Se comenzó tomando en cuenta las características de cada artefacto así como su demostración matemática. A partir de allí se decidió que di chas actividades deberían de consistir en una serie de preguntas que invitaran a los estudiantes a manipular cada artefacto, que conocieran cada parte que los conforma, que observaran las figuras que se forma ban entre sus barras así como las longitudes de las mismas, también que verificaran su comportamiento mientras dicho artefacto tenía mo vimiento así como en estado inmóvil. Lo que se deseaba era que los alumnos, a partir de esa guía y de la manipulación del artefacto pudie ran construir por sí mismos el concepto de elipse, deduciendo el mo delo matemático involucrado en cada artefacto. De esta manera se hizo una actividad didáctica por cada artefacto (ver annexos). Etapa de aplicación

Las actividades didácticas desarrolladas fueron aplicadas primeramen te en una prueba piloto dividida en dos sesiones. Auxiliándonos de los estudiantes se pretendía con esta prueba, afinar detalles, cambiar el or den de las preguntas de ser necesario, quitar o agregar alguna de ellas

REDIMAT Journal of Research in Mathematics Education, 1 (2) 170 así como mejorar su redacción, etc. Así finalmente haciendo las modificaciones pertinentes a las hojas de trabajo se hizo la prueba formal, la cual fue dividida en dos sesiones. Se contó con dos cámaras de video grabación en cada una de ellas. Primero se aplicó la actividad del Elipsógrafo de palancas y colisa de Inwards, ésta se llevó a cabo el día jueves 6 de Mayo del 2010, en el CBTIS 149 en Morelia Michoacán, con una duración aproximada de 150 minutos. En esta ocasión se realizó con ocho alumnos del cuarto semestre de la especialidad de Administración, cuatro hombres y cua tro mujeres. Se formaron cuatro equipos de dos integrantes cada uno, formados por un hombre y una mujer. En la segunda actividad se aplico el Antiparalelogramo articulado de van Schooten, se llevó a cabo el día miércoles 13 de Mayo del 2010, en el mismo lugar donde se realizo la primera sesión con una duración aproximada de 120 minutos y con los equipos conformados en la se sión anterior. Análisis de los datos

Se digitalizaron las hojas de trabajo de todos los equipos y se analiza ron las mismas. Posteriormente se revisaron las notas de observación realizadas durante las actividades así como los videos de las mismas y la información obtenida de ellos se organizó en tablas que contenían columnas para el episodio, tiempo y explicación de los equipos. Esto nos permitió seleccionar los diálogos más importantes y luego resca tarlos para analizarlos más a fondo. Sin embargo, por ser muy extensa la información sólo se muestran algunas de las respuestas más rele vantes. Discusión

A continuación, primero se discuten los resultados obtenidos respecto del uso del elipsógrafo de palancas y colisa de Inwards, y luego se ex ponen los resultados obtenidos en el caso del antiparalelogramo arti culado de van Schooten.


Resultados de la hoja de trabajo del elipsógrafo de palancas y colisa de Inwards Los resultados que se muestran a continuación corresponden a las pre guntas 7 y 8, presentando en esta última el razonamiento de dos equi pos participantes. En cada diálogo presentado se indica el nombre del equipo participante así como el nombre de sus integrantes. De la mis ma manera se indica con el nombre de Héctor al profesor investigador. Se espera que los alumnos puedan responder la pregunta 7 en fun ción de alguna de las características obtenidas en la pregunta anterior, el enunciado es: ¿El segmento BD, siempre pasa por la mitad del seg mento CE? Justifica tu respuesta. Observemos lo que el equipo de Ga bi y Uriel respondió así como en que se basaron para llegar a ello: Participantes: Gabi y Uriel (Gabriela, Uriel) y Héctor. Introducción: Los alumnos tratan de dar la respuesta de la pre gunta 7 y la comienzan a escribir en sus hojas de trabajo. Des pués de eso se les pide que mencionen su respuesta a dicha pregunta. Se les hacen más preguntas al respecto y ellos tratan de responder. Gabriela: ¿Por qué sería? Uriel: Es un eje de simetría. Gabriela: Es la diagonal… ¿así nada más?, ¿BD es eje de si metría de CE? Uriel: Pues sí, el punto K no cambiaría de distancia entre el EC también. Gabriela: No influye, K no influye (señala el punto K), nada más te está preguntando de estos (señala los puntos D y B). En tonces por qué es su eje de simetría… siempre va a ser su eje de simetría. Uriel: Pues… sí. Héctor: ¿En qué pregunta van? Gabriela: En la siete. Héctor: ¿En la siete?, a ver ¿qué respondieron? Ga briela: Que si porque EC es el eje de simetría (señala el seg mento EC en su artefacto).

REDIMAT Journal of Research in Mathematics Education, 1 (2) 172 Héctor: ¿Por qué? Gabriela: Este… se supone que nosotros estamos tomando que es un rombo, ¿no? Uriel: Rombo. Gabriela: Entonces tiene dos ejes de simetría, lo que viene sien do éste (señala el segmento CE) y viene siendo éste (señala el segmento BE). Uriel: BD (corrigiendo a Gabriela que señaló BE). Gabriela: Ajá, éste (señala el BD) y éste (señala el CE). Enton ces se supone que aquí, quedaría hacia la mitad si quedáramos en éste (simula con sus manos que dobla el artefacto por CE). Entonces al doblarlo (ahora por BD) automáticamente me esta formando su eje de simetría… y es una figura que siempre lo va a tener allí y aquí (señala el segmento CE). Héctor: ¿Siempre en distintas posiciones se cumple que la dis tancia de C al segmento BD y de E a BD es la misma? Uriel: Sí. Gabriela: Cuando es un eje en el rombo sí porque sus cuatro la dos son iguales. Héctor: Ok sigan adelante…

Observaciones

(1) En la pregunta 6, se les pedía que observaran la figura formada por su artefacto cuando este estuviera fijo, que dijeran de qué figura se trataba así como algunas de sus características. Allí se dieron cuenta de que se trataba de un rombo y entre sus características mencionan que el rombo tiene dos ejes de simetría.

Figura 9. Respuesta 6 del equipo 3 en sus hojas de trabajo

173 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse (2) Ellos ya se habían dado cuenta de que su rombo tenía dos ejes de simetría y que uno de ellos era el segmento BD y el otro eje lo era el segmento CE, por lo que no tuvieron dificultad alguna para contestar.


Una buena observación al momento de tratar de responder la pregunta 6 les dio de inmediato la respuesta de la pregunta 7. De antemano ellos conocían que su rombo tenía dos ejes de simetría y de esa manera no dudaron en responder que BD siempre pasa por la mitad del segmento CE. En la pregunta 8 se espera que los estudiantes lleguen a que los triángulos que se les pide que comparen son congruentes y de allí pue dan responder sin mucha dificultad las preguntas posteriores. Su enun ciado dice lo siguiente: Deja fijo el artefacto y traza con tu pincelín el segmento KC. ¿Cómo son los triángulos KED y KCD entre ellos? Jus tifica tu respuesta. A continuación se mostrará el diálogo correspon diente a la respuesta mencionada por parte del Equipo 1. Participantes: Equipo 1 (Moisés, Ayla) y Héctor. Introducción: Los integrantes del Equipo 1 y el profesor inves tigador tratan de dibujar con ayuda de un pincelín los triángulos KED y KCD (pregunta 8). Después de hacerlo Moisés trata de explicar cómo son dichos triángulos entre ellos, usando la ayuda obtenida por respuestas anteriores y sus observaciones. Héctor: Ya márcale y después hacemos lo demás, para que vean los triángulos que ya marcaron o algo así, pero bueno, si no se puede (trazar adecuadamente dichos triángulos) pues nada mas obsérvenlos a través del artefacto. Bueno te piden el KCD (triángulo), ¿verdad? Ayla: Sí.

REDIMAT Journal of Research in Mathematics Education, 1 (2) 174

Héctor: Tienes que tenerlo fijo (el artefacto) para que no se te quede así (indicándoles como deben fijar el artefacto). ¿Y cuáles son los triángulos que se piden? Moisés: KED y KCD. Es que si esta recta (señala el segmento BD) siempre está dividiendo al cuadrilátero en dos partes igua les, cualquier punto de aquí (se refiere a cualquier punto que es te sobre el segmento BD), pues será la misma distancia de E al punto K y de C al punto K. Entonces sí son distancias iguales, esta distancia (CD) es igual a esta (DE), y la distancia (DK) será igual para los dos. Héctor: Pero bueno ¿KD qué sería entonces si lo divide en 2 partes iguales (al cuadrilátero)? Moisés: ¿Cómo?, ¿Qué sería que? Héctor: ¿Qué es BD por ejemplo? En la pregunta 7 se te pide que justifiques si el segmento BD siempre por la… a ok (mues tra Moisés la justificación de la pregunta 7). Entonces ¿por qué son congruentes? (señalando los triángulos en el artefacto). Moisés: Este lado siempre será igual (señala el segmento KD) porque el segmento BD siempre está dividiendo en dos al cua drilátero. Entonces si está aquí o aquí (señala el recorrido que hace el punto K) o en el punto que sea siempre va a ser igual en los dos lados, a los dos triángulos. Entonces si se pone por ejemplo aquí, como es a la mitad, pues el segmento EK y KC siempre serán igual, según el movimiento en que se dé, siempre estarán de la misma longitud y el ED y CD miden 12 centíme tros, es decir lo mismo. Héctor: Ah ok. Moisés: Siempre serán congruentes en cualquier movimiento. Héctor: Pues anota tu justificación allí y si te hace falta espacio puedes anotarla atrás, sólo pon que la respuesta la anotaste atrás en caso de que no te llegue a caber.

175 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Observaciones

(1) En preguntas anteriores, los alumnos tuvieron la posibilidad de medir los segmentos del artefacto y darse cuenta que algunos de ellos son iguales entre sí. (2) En otras palabras, Moisés logra darse cuenta de que BD es una dia gonal del rombo y que por esa razón siempre va a pasar por la mitad del segmento CE. (3) Otro aspecto importante es que Moisés logra darse cuenta que no importa el movimiento que se le dé al artefacto ni la posición en que este quede, que siempre va a mantenerse que los segmentos CK y KE van a ser iguales, puesto que los dos segmentos se relacionan con el punto K y si este se mueve afecta de la misma manera a ambos seg mentos. (4) Él justifica que los triángulos KED y KCD son congruentes puesto que CK = CE, CD = ED puesto que los midió y ambos lados miden 12 cm, y que el lado KD va ser el mismo para ambos triángulos, es decir encontró una congruencia LLL. (5) Se pudo observar que los alumnos conocen el concepto de con gruencia pero no lo recuerdan completamente y les cuesta trabajo ar gumentar el por qué los triángulos mencionados son congruentes.

Figura 11. Respuesta escrita de la pregunta 8 del equipo 1

Así pues, a través del análisis de estos datos se pone de manifiesto que los integrantes de dicho equipo conocen el término “congruencia” y saben en qué situaciones pueden utilizarlo, así mismo se pudo observar que no pueden recordar los distintos criterios de congruencia, y no obtienen una congruencia distinta a LLL.

REDIMAT Journal of Research in Mathematics Education, 1 (2) 176 Cabe mencionar que en esta actividad, los alumnos cuentan con una regla para medir los segmentos. A partir de la evidencia recabada, se logró deducir que les hizo falta más observación y ver los segmentos involucrados en dicha pregunta para obtener la congruencia solicitada. Ahora veamos lo que contestó el equipo de Gabi y Uriel en la misma pregunta y comparemos la respuesta entre ambos equipos: Participantes: Gabi y Uriel (Gabriela, Uriel) y Héctor. Introducción: Los alumnos tratan de responder la pregunta 8, donde Gabriela menciona que los triángulos KED y KCD son congruentes. Ella trata de convencer a su compañero del por qué dichos triángulos son congruentes así como explicárselo tam bién al profesor investigador. Gabriela: ¿Cómo son los triángulos QED?...pero si de este lado no se hace triángulo (señala los puntos mencionados). Uriel: KED (corrigiendo a Gabi la cual dijo QED. Señala tam bién los puntos). Gabriela: Y KCD pero jamás me dijo que trazáramos este lado. Bueno se supone que son… ¡espera espera!, ¿cómo se llaman cuando son?... son congruentes. Son dos triángulos congruentes porque has de cuenta que… Uriel: No, pero mira este lado está más chico (señalando el lado EK). Gabriela: Pero por aquí va la línea esta, esta de aquí que parte de la línea de en medio, es la línea fija (traza el segmento KE). En tonces si marcas esto de aquí (segmento ED). Gabriela: Este lo doblas y quedan iguales (simula doblar los triángulos KED y KCD por BD), entonces los dos triángulos son congruentes. Uriel: No, es que no serían congruentes porque si comparáramos los triángulos quedarían así (señala que los triángulos no que darían uno sobre otro). Gabriela: Al levantarlos, los dos picos quedarían arriba, que darían igual (con sus manos indica que dichos puntos

177 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse se tocan). Uriel: Sí pero, ¿qué te están diciendo? (refiriéndose a la pregun ta). Gabriela: Deja fijo el artefacto, ¿cómo son los triángulos KCD y KDE? (etiqueta los triángulos que previamente trazó). En este con este (indica los triángulos a comparar). Héctor: ¿Por qué me dices que son triángulos congruentes? Gabriela: No me acuerdo como se dice cuando los dos son iguales, que están opuestos sólo por una diagonal. Héctor: A ver, ¿este lado cómo es para los dos triángulos, el KD? Gabriela: El KD es igual. Héctor: ¿Es igual para los dos? Gabriela: Sí. Uriel: Para los dos. Héctor: ¿Por qué? Uriel: Porque mide lo mismo. Héctor: Concéntrense sólo en el segmento KD, ¿qué pasa con KD? Gabriela: Es la diagonal media. Héctor: En estos dos triángulos ¿qué pasa con KD (señalo el segmento)?, ¿es igual? Gabriela: Es un cateto de ambos. Héctor: ¿Cómo es el lado KC con el KE? Gabriela: ¿KC con KD? Uriel: Con KE Gabriela: Son… ¿cómo se dice? Héctor: A ver ¿por qué no los mides? Uriel: Son iguales Gabriela: Mide 5 (CK), se supone que también mide 5 (KE)…son iguales (mide los segmentos). De CK y KE es igual también (afirmando). Por lo tanto quedaría que de C a D y de D aE… Uriel: Deben de medir lo mismo. Gabriela: Miden lo mismo (CD=DE), entonces los dos triángu los son iguales solamente que KD es un eje de

REDIMAT Journal of Research in Mathematics Education, 1 (2) 178 simetría para los dos. Es el eje de simetría de una figura y es por eso que se forman dos triángulos. Héctor: Por eso son ¿cómo me dijiste? Gabriela: Congruentes…

Observaciones (1) Gabriela usa su artefacto para trazar los triángulos KED y KCE. Remarca los lados de los triángulos con su pincelín por debajo del artefacto.

Figura 12. Integrantes del equipo 3 remarcando los triángulos KED y KCE

(2) Una vez qué quedan remarcados los lados de sus triángulos opta por quitar su artefacto de la hoja de dibujo. Prefiere trabajar con los triángulos que dibujó previamente, etiquetando los puntos correspon dientes con los que tenía el artefacto en dicha posición.

Figura 13. Triángulos remarcados y etiquetados en la hoja de dibujo del equipo 3

179 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse (3) Los integrantes de este equipo sabían por su respuesta de la pre gunta 7 que el segmento BD es un eje de simetría. Tratan de justificar que los triángulos KED y KCD son congruentes porque dicho seg mento (BD) los separa y al doblarlos por allí los triángulos van a pe garse el uno con el otro. Al analizar las evidencias recabadas se deduce que intuitivamente trataban de responder de esa manera, al saber que la figura era un rombo, que tenía como ejes de simetría a los segmentos BD y EC y que ED y CD son barras del artefacto que miden lo mismo (esto lo conocieron en la respuesta de la primera pregunta). Además también conocían que el punto K se desplazaba sobre el segmento BD. Su razonamiento tuvo que ver con todo lo anterior y se dieron cuenta de que al medir lo mismo las barras ED y CD y saber que dichos seg mentos cambiaban de longitud por estar conectados al punto K, los segmentos EK y CK cambiaban de la misma manera y podían medir lo mismo. (4) Intuitivamente y por observación pudieron darse cuenta de que los segmentos EK y CK eran iguales, antes de sugerirles que midieran di chos segmentos Uriel respondió inmediatamente que dichos segmentos eran iguales, aún cuando Gabriela media los segmentos, Uriel seguía mencionando que deberían de medir lo mismo. (5) Les costó mucho trabajo tratar de justificar su respuesta, aún cuan do habían contestado bien las preguntas anteriores a esta.

Figura 13. Respuesta escrita del equipo 3 en sus hojas de trabajo

REDIMAT Journal of Research in Mathematics Education, 1 (2) 180 Así pues, podemos ver que los integrantes de dicho equipo conocen el término “congruencia” y saben en qué situaciones pueden utilizarlo, pero también, a través de la evidencia recabada se observó que no re cuerdan los criterios de congruencia distintos al LLL, se deduce que esto ocurre ya que en la actividad los alumnos pueden medir los seg mentos con regla. Resultados del antiparalelogramo articulado de van Schooten

Las primeras preguntas tienen que ver con las figuras que se forman con los segmentos del artefacto así como que segmentos cambian de longitud durante el movimiento y cuales no lo hacen. Analizaremos lo que ocurrió en las preguntas 7, 10 y 14. El enunciado de la pregunta 7 es: Trace los segmentos AC y BD. ¿Cuál es la figura que se forma con los segmentos AC, BD, AB y CD? Menciona algunas de sus características. A continuación veremos lo que contestaron Gabriela y Uriel así co mo las dificultades y aciertos que tuvieron en la pregunta 7. Participantes: Gabi y Uriel (Gabriela, Uriel) Introducción: En la pregunta 7 los integrantes de este equipo afirmaron que la figura formada por su artefacto fijo era un tra pecio. Entonces a partir de ello comienzan a mencionar algunas de sus características y Gabriela las comienza a escribir mien tras algunas de ellas son dictadas por Uriel. Gabriela: Estos dos no son paralelos en sí (señala los segmentos AB y CD). Uriel: No, no son paralelos. Gabriela: Entonces se forma un trapecio y luego sus caracterís ticas… tiene cuatro lados. Uriel: AC y BD son paralelos. Gabriela: ¿AC y qué? Uriel: AC y BD son paralelos, ¿qué más? Gabriela: AB y CD tienen la misma longitud Uriel: Pues si quieres (anotarlo). Gabriela: ¿Y cuáles son las otras características?...tiene un

181 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse eje de simetría aquí, este de aquí (lo señala en su artefacto fijo), si ¿verdad? Uriel: Uju. Gabriela: ¿Encuentras otra, otra característica? Uriel: Que su perímetro siempre será igual aunque cambie a re cuadro (se refiere a que si mueve el artefacto su trapecio puede cambiar y ser un cuadrado, esto lo dice mientras observa el mo vimiento del artefacto). Sí, aunque cambie (ahora observa el ar tefacto fijo). Gabriela: Son las únicas características porque aquí ya lo tene mos de una manera inmóvil (el artefacto) y no se te pide que lo vuelvas a mover. Tiene cuatro lados, AC y BD son paralelos, AB y CD tienen la misma longitud y tiene un eje de simetría.

Observaciones (1) Es muy importante que los alumnos se den cuenta que la figura formada es un trapecio (isósceles) puesto que esto es la base para lle gar a la demostración. (2) Los alumnos mantuvieron fijo el artefacto, mencionaron algunas características y después trazaron los segmentos AC y BD. Después de esto quitaron su artefacto y lo colocaron en otra parte tratando de po nerlo en la misma posición que tenía antes de cambiarlo de lugar. Des pués de esto continuaron observándolo y mencionando características.

Figura 14. El equipo 3 trazando su trapecio

(3) Los integrantes de este equipo mencionan que el trapecio tiene un eje de simetría y Gabriela lo señala en su artefacto. Este hecho es im portante, puesto que es fundamental para responder la siguiente

REDIMAT Journal of Research in Mathematics Education, 1 (2) 182 (4) En su respuesta mencionan todas las propiedades del trapecio isós celes aunque no se dan cuenta de ello y sólo nombran a su figura como trapecio.


(5) Fue importante que los estudiantes se dieran cuenta que al mover el artefacto y mantener paralelo el segmento AB con el CD se forma un cuadrado. Cabe mencionar que las indicaciones sugeridas antes de res ponder la pregunta 7 se hacen con la finalidad de evitar llegar a dicha figura, puesto que de esa forma puede haber confusiones para respon der las siguientes preguntas. Al analizar estos resultados podemos ver que la buena observación, conocimiento y acordarse un poco de sus clases de geometría de los estudiantes, fue clave para darse cuenta de la figura que se formaba en las condiciones pedidas en la pregunta 7, así como para rescatar pro piedades del trapecio. Lo más importante fue que lograron darse cuen ta de que su trapecio tiene un eje de simetría, ya que de allí podrían responder sin ningún problema la pregunta 8. Ahora, el siguiente diálogo, mostrará el razonamiento usado por los Sexys para responder la pregunta 10 la cual dice lo siguiente: Compara los triángulos MEA y MEC. ¿Cómo son entre ellos? Justifica tu res puesta. Participantes: Los Sexys (Luz, Jorge) y Héctor. Introducción: Los integrantes de este equipo se encuentran jus tificando la pregunta 10, en donde responden que los triángulos MEA y MEC son congruentes. Tratan de explicar su respuesta.

183 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Jorge: ¿Cuáles son congruentes aquí? Luz: Por eso, allí te va. Ponle…el segmento AE es congruente (observa con atención el artefacto). Jorge: ¿Cuál? Luz: AE es congruente con el segmento CE… y luego el seg mento AM es congruente con el segmento CM. Jorge: Y ambos comparten (lo interrumpe Luz)… Luz: Ambos triángulos comparten el segmento ME… Héctor: ¿Cuáles triángulos estás tomando en cuenta para res ponder la pregunta 10? Jorge: Éste y éste, estos dos (señala los triángulos MEA y MEC). Luz: Estos dos (los señala) y aquí nosotros decimos que los triángulos también son congruentes. Héctor: ¿Sí?, ¿por qué? Luz: Porque … (Jorge la interrumpe) Jorge: Porque su lado, su segmento AE y el EC son congruentes, igual que el W, digo el M. Luz: El MC y el MA tienen la misma longitud. Jorge: Y comparten el ME (lo señala en su dibujo). Héctor: ¿Y por qué me dicen que el segmento MC y MA tienen la misma longitud? (señalo dichos segmentos) Jorge: Pues porque… Luz: Pues porque… Jorge: El punto M…(es interrumpido por Luz) Luz: Está a la mitad del segmento AC. Héctor: ¿Por qué es la intersección del eje de simetría? Luz: Sí. Héctor: Ok. En la siguiente pregunta quiero que se auxilien de estas dos que acaban de responder para tratar de contestarla.

Observaciones (1) A diferencia del equipo anterior, Jorge y Luz contestaron de una mejor manera la pregunta 9. Respondieron que los triángulos BEN y DEN eran congruentes puesto que todos los lados eran iguales, indi cando que lados correspondientes medían lo mismo. De esta manera

REDIMAT Journal of Research in Mathematics Education, 1 (2) 184 lograron darse cuenta de que la respuesta a esta pregunta (pregunta 10) era análoga a la anterior.

Figura 16. Respuesta escrita del equipo 4

(2) Luz le dictaba la respuesta a Jorge para que él la escribiera aunque Jorge también observaba el artefacto y hacía comentarios. Su respuesta quedó muy parecida a la de la pregunta 9.

Figura 17. Respuesta escrita de la pregunta 10 del equipo 4

En este caso vemos que los integrantes de este equipo demostraron sa ber lo que es una congruencia, aunque posiblemente no recuerden los distintos criterios. Algo importante fue que se auxiliaron de la medi ción de segmentos y por esta razón, en esta pregunta y en la anterior respondieron que los triángulos en cuestión eran congruentes por LLL. Los dos diálogos siguientes muestran la respuesta del equipo deno minado Los Sexys a la pregunta 14, la cual es la siguiente: Reflexiona sobre tus respuestas anteriores y escribe con tus palabras que es una Elipse. Aquí los alumnos deben ver las respuestas anteriores y unirlas para construir por sí mismos la definición formal de elipse a partir de la manipulación y observación que los alumnos hicieron del artefacto, así como de la medición de algunos de sus segmentos.

185 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Participantes: Los Sexys (Luz, Jorge) Introducción: Tratan de dar con sus palabras la definición de elipse (pregunta 14). Ambos aportan buenas ideas y al final Luz dice la posible definición que podrían anotar en sus hojas de trabajo. Luz: Es una figura… formada por… bueno que tiene dos focos y que tiene… un punto… no sé ¿pues qué?... que la suma de los dos segmentos de… Jorge: Una elipse es una figura ovalada. Luz: No. Jorge: Que tiene dos focos, ¿Cómo no? (le reclama a Luz quien duda acerca de la afirmación de Jorge). Luz: Bueno síguele. Luz: Pues podemos decir que la suma de los segmentos relacio nados con el punto que gira y que forma la elipse siempre es constante (dice eso mientras mueve el artefacto). O sea en cual quier punto que lo muevas la suma de este a este (de A a E) más la de este a este (de E a B) va a ser constante. Eso para mí es elipse. Jorge: Sí (hace una seña de aceptación). Luz: Escríbelo tú.

Observaciones (1) Este equipo también tuvo algunas complicaciones para definir con sus palabras el concepto de elipse. (2) Identificaron rápidamente los elementos del artefacto que partici pan en la definición de elipse. (3) No usaron la nomenclatura del artefacto para escribir su definición de elipse. En este ejemplo final se puede ver que los alumnos pasan desapercibidos en más de una ocasión lo que sucede cuando mueven el artefacto y no toman en cuenta las respuestas que dieron anteriormente para tratar de ayudarse y dar una definición de elipse a partir de allí.

REDIMAT Journal of Research in Mathematics Education, 1 (2) 186 Conclusiones

El aprendizaje cooperativo es un enfoque de enseñanza, en el cual se procura utilizar actividades en las cuales es necesaria la ayuda entre estudiantes, ya sea en pares o grupos pequeños, dentro de un contexto enseñanzaaprendizaje. El aprendizaje cooperativo se basa en que cada estudiante intenta mejorar su aprendizaje y resultados, pero también el de sus compañeros. Cabe mencionar que a los alumnos se les dificulta trabajar en equipo puesto que en ocasiones no pueden asimilar las opiniones de sus com pañeros. En determinadas partes de las actividades así como en deter minados equipos se observaron fragmentos de debate científico. Para cada uno de los instrumentos, los estudiantes hicieron uso de re cursos matemáticos como la utilización de representaciones algebrai cas, lenguaje geométrico y transformación del lenguaje cotidiano al lenguaje matemático, como parte fundamental en la construcción del conocimiento. Trabajar con instrumentos concretos en el aprendizaje es viable ya que, se observó que la mayoría de los estudiantes se motivan trabajan do con ellos, es una dinámica muy distinta a la de una clase cotidiana, además el concepto en cuestión es construido por ellos mismos por lo que este puede ser más duradero. En cuanto a la manipulación del artefacto, al principio les costó tra bajo moverlo, lo cual se vio reflejado en sus trazos, los cuales queda ban muy chuecos. Después de realizar varios trazos, los alumnos obtenían práctica y las elipses les comenzaron a quedar bien. Un pro blema para los alumnos fue que en ocasiones se salían las tachuelas con las que se fijaban los puntos fijos (focos) y esto les perjudicaba en la estética de sus dibujos. Referencias

Artobolevski, I. (1964). Mechanisms for the Generation of Plane Cur ves. New York: Pergamon Press. Artobolevski, I. (1975). Mecanismos en la técnica moderna. Tomo 2, parte 1. Moscú: Ed. Mir. Arzarello, E. & Robutti, O. (2004). Approaching functions through motion experiments. Educational Studies in Mathematics, 57(3). Special issue.

187 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Bartolini, M. et al. (2004, July). Learning Mathematics with tools. Pa per presented at the 10th International Congress on Mathematical Education (ICME10), Copenhagen, Denmark. Bartolini, M. (2007) Experimental mathematics and the teaching and learning of proof. Research funded within the PRIN 2005019721 on "Meanings, conjectures, proofs: from basic research in mathe matics education to curricular implications. Bartolini, M. et al. (2004, September). The MMLAB: a laboratory of geometrical instruments. Paper presented at the minisimposio Ap plicazioni della Matematica all’industria culturale, Venezia, Italy. Boero, P., Garuti, R. & Mariotti, M. (1996). Some dynamic mental processes underlying producing and proving conjectures. Procee dings of PMEXX, Vol. 2 (pp. 121128). Valencia. Boero, P., Pedemonte, B. & Robotti, E.: (1997). Approaching Theore tical Knowledge Through Voices and Echoes: a Vygotskian Pers pective. En E. Pehkonen (1997, Ed.) Proceedings of the 21st Conference of the International Group for the Psychology of Mat hematics Education (pp. 8188). Lahti Research and Training Center. Finland: University of Helsinki. Dennis, D. (1995). Historical Perspectives for the Reform of Mathe matics Curriculum: Geometric Curve Drawing Devices and Their Role in the Transition to an Algebraic Description of Functions. Unpublished doctoral dissertation, Cornell University. Duval, R. (1995). Semiosis et Penseé Humain. New York: Peter Lang. Dyck, W. (1994). Katalog matematischer und matematischphysikalis cher Modelle, Apparate und Instrumente. New York: Georg Olms Verlag, Zurich. Hoyos, V. (2006). Funciones Complementarias de los artefactos en el aprendizaje de las transformaciones geométricas en la escuela se cundaria. Enseñanza de las ciencias, 24(1), 31–42. Hoyos, V. & Falconi, M. (2005). Instrumentos y matemáticas: historia, fundamentos y perspectivas educativas. México: Ed. UNAM. Hoyos, V., Capponi, B. & Génevès, B. (1998). Simulation of drawing machines on CabriII and its dual algebraic symbolization. En In ge Schwank (Ed.), Proceedings of CERME1, Alemania: Universi dad de Osnabrueck. Kempe, A. B. (1877). How to Draw a Straight Line. London, England: Macmillan and Co.

REDIMAT Journal of Research in Mathematics Education, 1 (2) 188 Mariotti, M.A., Bartolini Bussi, M., Boero, P., Ferri, F. y Garuti, R. (1997). Approaching Geometry Theorems in Contexts: From His tory and Epistemology to Cognition. E. Pehkonen (1997, Ed.) Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education (pp. 180195). Lahti Research and Training Center. Finland: University of Helsinki. Piaget, J. (1950). Introducción a la Epistemología Genética. “El pen samiento matemático”. Buenos Aires: Paidós. Schooten, F. van (1657). Exercitationum mathematicarum liber IV, sive de organica conicarum sectionum in plano descriptione. Lugd. Batav ex officina J. Elsevirii. Verillon, P. & Rabardel, P. (1995). Cognition and Artifacts: A Contri bution to the Study of Thought in Relation to Instrumented Acti vity. European Journal of Psychology of Education, 10(1), 77101. Vincent, J., Chick, H., & McCrae, B. (2002). Mechanical linkages as bridges to deductive reasoning: A comparison of two environ ments. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mat hematics Education, 4 (pp. 313 – 320). Norwich, UK: PME.

Notas 1 2

Más sobre esto ver en [Kempe]. Vea [Denis, D (1995)].

189 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Annexos Hoja de trabajo para la manipulación del elipsógrafo de palancas y colisa de Inwards Nombres de los integrantes del equipo: Nombre del equipo: Grado: Institución: Instrucciones: 1) Se te proporcionará un artefacto, un lápiz, un pincelín y una regla. 2) El punto K es el punto de la intersección de los segmentos AE y BD. 3) Coloca el lápiz en el punto K, puesto que este es el punto que reali za el trazo. 4) Podrás mover el artefacto mediante el punto E o bien mediante el lápiz que se inserta en el punto K. Analiza con mucha atención el lugar geométrico trazado por el punto K, así como el movimiento en general del artefacto. 5) Puedes medir la longitud de los segmentos con la regla para contes tar algunas de las preguntas que se te piden. 6) Responde las preguntas planteadas lo más detallado posible, ha ciendo uso de la manipulación del artefacto. 7) Al hacer referencia a un segmento, escríbelo de la siguiente forma. AB ( )

Elipsógrafo de palancas y colisa de Inwards

REDIMAT Journal of Research in Mathematics Education, 1 (2) 190 1. ¿De cuántas barras está conformado el artefacto? Escribe la longitud de cada una de ellas, nombrando a cada barra con los puntos de cada uno de sus extremos (por ejemplo barra AE). 2. Mueve el artefacto y observa sus diferentes puntos durante el movi miento. Cuando moviste el artefacto, ¿Cuáles fueron los puntos que se mantuvieron fijos? 3. Al mover el artefacto ¿Cuáles son los segmentos que no cambian de longitud durante el movimiento? 4. Mueve el artefacto y observa sus segmentos. ¿Cuáles fueron los segmentos que cambiaron su longitud durante el movimiento? 5. De los segmentos que cambiaron de longitud durante el movimien to, ¿Cuáles tienen la misma longitud entre ellos? 6. Mueve el artefacto y observa la figura formada por los segmentos BC, CD, DE y EB. ¿Qué figura se forma?, ¿qué características tiene dicha figura? 7. ¿El segmento BD, siempre pasa por la mitad del segmento CE? Jus tifica tu respuesta. 8. Deja fijo el artefacto y traza con tu pincelín el segmento KC. ¿Cómo son los triángulos KED y KCD entre ellos? Justifica tu respuesta. 9. Suma la distancia de los segmentos AK+KC. ¿A cuánto equivale la distancia AK+KC? , ¿Siempre se cumple la equivalencia en cualquier posición del artefacto? Justifica tu respuesta. 10. Auxiliándote de la respuesta de la pregunta 1, ¿Hay alguna barra del artefacto cuya longitud sea igual a la suma de AK + KC?, ¿cuál es dicha barra? 11. Acerca el punto A al punto C, mueve el artefacto y observa el trazo. Después coloca el punto A sobre el punto C, mueve el artefacto y ob serva el trazo. ¿Cómo se comportaron estos trazos en comparación con el primer trazo realizado en la actividad? 12. Reflexiona sobre tus respuestas de las preguntas anteriores y escri be con tus palabras la definición de Elipse. 13. Solicita a tu coordinador el Anexo 1 para responder lo siguiente. Explica si el instrumento cumple la definición de elipse y ¿por qué?

191 Cortés Zavala y Soto Rodríguez Una experiencia con la elipse Annexos Hoja de trabajo para la manipulación del antiparalelogramo (van Schooten) Nombres de los integrantes del equipo: Nombre del equipo: Grado: Institución: Instrucciones: 1) Se te proporcionará un artefacto, el cual está etiquetado en ciertos puntos. Así mismo, un lápiz, un pincelín y una regla. 2) El punto E, es la intersección entre los segmentos AD y BC. Este punto se tomará sobre las deslizaderas, en donde se colocará el lápiz que previamente se te entregó. 3) Para manipular el artefacto (una vez fijo), tendrás que girarlo a par tir del punto C y D o bien, sólo debes manejarlo con el lápiz. 4) Observa con atención el artefacto durante el movimiento, principal mente el lugar geométrico trazado por el punto E. 5) Puedes medir la longitud de los segmentos con la regla para contes tar algunas de las preguntas que se te piden. 6) Responde las preguntas planteadas lo más detallado posible, ha ciendo uso del artefacto. 7) Al hacer referencia a un segmento, escríbelo de la siguiente forma. AB ( )

Antiparalelogramo articulado de van Schooten

REDIMAT Journal of Research in Mathematics Education, 1 (2) 192

1. ¿De cuántas barras está conformado el artefacto? 2. Mueve el artefacto y observa atentamente lo que ocurre con todos sus segmentos. Durante el movimiento del artefacto, ¿qué figuras geométricas forman dichos segmentos (incluyendo los segmentos AC y BD)? 3. ¿Cuáles son los puntos que están fijos sobre el plano durante el mo vimiento del artefacto? 4. Al mover el artefacto ¿Cuáles son los segmentos que no cambian de longitud durante el movimiento? 5. Mueve el artefacto y observa sus segmentos ¿Cuáles son los seg mentos que cambian de longitud durante el movimiento? 6. De los segmentos que cambiaron de longitud durante el movimien to, ¿Cuáles segmentos son iguales entre sí? Ahora deje el artefacto inmóvil de manera que los segmentos AB y CD no sean paralelos y responda lo siguiente: 7. Trace los segmentos AC y BD. ¿Cuál es la figura que se forma con los segmentos AC, BD, AB y CD? Menciona algunas de sus carac terísticas. 8. Lee el anexo1 que se encuentra al final de las hojas. Observa la figura de la pregunta anterior ¿Tiene algún eje de simetría? Traza el o los ejes de simetría de dicha figura. ¿Cómo es el eje o los ejes de simetría respecto a los segmentos AC y BD? 9. Llama punto M a la intersección del eje de simetría con el segmento AC y N al punto de la intersección del eje de simetría con el segmento BD. Ahora, compara el triángulo BEN con el triángulo DEN. ¿Cómo son entre ellos? Justifica tu respuesta. 10. Compara los triángulos MEA y MEC. ¿Cómo son entre ellos? Justifica tu respuesta. 11. Auxiliándote de las 2 preguntas anteriores, ¿Qué puede decirse de la comparación entre los triángulos AEB y CED? Justifica tu respuesta. 12. Realiza la suma de los segmentos AE + EB. ¿A cuánto equivale la suma de AE + EB?, ¿siempre se cumple dicha suma en cualquier posi ción del artefacto? 13. Mide las distintas barras que conforman el artefacto. ¿Hay alguna barra del artefacto cuya longitud sea igual a la suma anterior?, ¿cuál es dicha barra?


14. Reflexiona sobre tus respuestas anteriores y escribe con tus pala bras que es una Elipse. 15. Solicita a tu coordinador el Anexo 2 para responder lo siguiente. Explica si el artefacto cumple la definición de Elipse y ¿Por qué? Eje de simetría Una línea que atraviesa una fi gura de tal manera que cada la do es el espejo del otro. Si dobláramos la figura en la mitad a lo largo del Eje de Simetría, tendríamos que las dos mitades son iguales, quedarían parejas. El eje de simetría es la mediatriz del segmento cuyos extremos son puntos simétricos.

José Carlos Cortés Zavala es Profesor del área de Matemática Educativa de la Universidad Michoacana de San Nicolás de Hi dalgo, Morelia, México. Héctor Arturo Soto Rodríguez es Profesor del área de Matemá tica Educativa de la Universidad Michoacana de San Nicolás de Hidalgo, Morelia, México. Dirección de contacto: Para correspondencia directa con el autor, diríjanse a Universidad Michoacana de San Nicolás de Hidalgo, Avenida Francisco J. Mújica s/n, Ciudad Universitaria, 58030 Morelia, Michoacán, México. Dirección de Email: [email protected]


How do elementary preservice teachers form beliefs and attitudes toward geometry learning? Implications for teacher preparation programs Kai-Ju Yang 1 1 ) Indiana University, USA.


To cite this article: Yang, K-J. (201 2). How Do Elementary Preservice Teachers Form Beliefs and Attitudes Toward Geometry Learning? Implications for Teacher Preparation Programs. Journal of Research in Mathematics Education, 1 (2), 1 94-21 3. doi: 1 0.4471 /redimat.201 2.1 0 To link this article: http://dx.doi.org/1 0.4471 /redimat.201 2.1 0 PLEASE SCROLL DOWN FOR ARTICLE The terms and conditions of use are related to the Open Journal System and to Creative Commons Non-Commercial and Non-Derivative License.


How Do Elementary Preservice Teachers Form Beliefs and Attitudes Toward Geometry Learning? Implications for Teacher Preparation Programs Kai-Ju Yang

Indiana University

Abstract

In my personal interactions with elementary preservice teachers (EPSTs) at a large Midwestern university in United States, many EPSTs held negative beliefs and attitudes about geometry learning. Although finding ways to help EPSTs change their negative beliefs and attitudes is an important issue, it can be best addressed by first investigating how they are formed. This study sought to document how EPSTs’ beliefs about and attitudes toward geometry were formed prior to and during a mathematics and pedagogy course at a large Midwestern university in United States. McLeod’s (1989) theoretical framework of influencing one’s beliefs and attitudes toward specific action events, and objects –Representation, Discrepancy, and Metacognition– was used to analyze data from two interviews with each of four EPSTs. The results of the analysis confirmed McLeod’s framework but also identified a fourth factor, understanding, as playing an important role in affecting EPST’s beliefs about and attitudes toward geometry. Keywords: beliefs, attitudes, geometry, elementary preservice teachers 2012 Hipatia Press ISSN 2014-3621 DOI: 10.4471/redimat.2012.10

195 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes

I

n my personal teaching experiences and interactions with elementary preservice teachers (EPSTs) at a large Midwestern university in United States, I found that many EPSTs held negative beliefs and attitudes about geometry learning. This negativity concerned me; if these EPSTs continued to hold these beliefs and attitudes, they would not be well-prepared to teach geometry to their future students. Although finding ways to help EPSTs change their negative beliefs and attitudes is an important issue, it can best be addressed by first investigating how they are formed. The purpose of this study, therefore, explores the origins of EPSTs’ beliefs and attitudes toward geometry learning. Conceptual framework and reseach questions Mathematics educators (Lester & Garofalo, 1982; Schoenfeld, 1983 and 1985; Charles & Lester, 1984; McLeod, 1994; Leder, Pehkonen & Törner, 2002; Maaß & Schlöglmann, 2009) have investigated student beliefs about and attitudes toward mathematics and how they influence students’ mathematics performance. Schoenfeld (1985), having found that students were not able to make use of the necessary mathematical knowledge gained from their coursework to solve problems, attributed this failure not to misunderstanding or forgetting mathematical knowledge but rather to not believing that it would be useful to them. The beliefs and attitudes that the students held, then, limited their understanding of mathematics and their ability to solve mathematical problems. In a similar study, Törner (2001) analyzed students’ ad hoc answers to mathematical questions and concluded that the mental net of “knowledge” is dominated by beliefs, raising the question: How did students form these negative beliefs about and attitudes? The formations of one's beliefs and attitudes McLeod (1989) proposed a theoretical framework for investigating beliefs about and attitudes toward specific actions, events, or objects as they are affected by three factors: representation, discrepancy, and metacognition (see table 1).


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Table 1

McLeod's (1989) framework for forming beliefs and attitudes

Representation * The format of the objects or events determines one's beliefs and attitudes toward those particular objects or events. * The order of the objects or events determines one's beliefs and attitudes toward those particular objects or events.

Representation .

Beliefs and attitudes

Discrepancy * Error: One expects the action to be correct but in fact it produces unexpectedly negative results, causing negative beliefs and attitudes. * Success: One's particular action produces unexpectedly positive results, producing positive beliefs and attitudes.

Metacognition * One reflects on one's own cognitive processes. * So one should be aware of one's emotional reactions toward the things experienced. * Next, one uses this awareness to control one's cognitive processes. * Then, one rethinks and possibly changes one's beliefs and attitudes.

McLeod argued that representation plays a crucial role in problem solving, because it influences how students learn, which in turn affects how they view what they are learning. Representation that promotes mathematical understanding, therefore, might positively change their view of doing math problems (Fennell & Rowan, 2001). The format of the mathematical concepts and problems (e.g., written statements alone or written statements with pictures) and the order in which those concepts and problems are presented (e.g., moving from concrete to abstract concepts) may be assumed to affect students’ beliefs about and attitudes toward mathematics. In sum, representation may give students useful tools for building understanding, communicating information, and demonstrating reasoning (Greeno and Hall, 1997; NCTM, 2000). Yang (2008) investigated the effects of Cognitive Tutor, a math software program, on the problem solving behaviors of 12 tenth-grade students. The program presented linear algebra word problems with

197 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes simultaneous verbal and visual representations that moved from familiar, concrete problem situations (e.g., a truck averages 12 miles per hour and has already traveled 70 miles. In two more hours, how many total miles will the truck have traveled?) to more abstract, symbolic forms (e.g., to write an expression, define a variable for the additional time traveled and use this variable to write a rule for the total distance the truck has traveled). The results showed that the combined verbal and visual representations helped students grasp the target math concept. Students also reported that Cognitive Tutor functioned in the same way as a human tutor, guiding them step-by-step from the concrete through the abstract problems to develop their mathematical thinking. These experiences helped students begin to regard learning mathematics as less difficult than they had previously thought, confirming that how concepts are presented and organized affects how students think about mathematics. Discrepancy. Referring to Mendler’s (1989) work, McLeod stated that any discrepancy between an expected outcome and the actual outcome in the course of problem solving in general, and in mathematical reasoning in particular, affects students’ beliefs and attitudes. Discrepancies can be experienced as either errors or successes. An error occurs when students engage in actions that they believe to be correct but in fact are incorrect, resulting in a mismatch between an expected outcome (“I thought I did what would solve the problem”) and the reality (“It didn’t work”). An error can create a negative evaluation of the current situation, which may result in negative beliefs about and attitudes toward the subjects in general. A success occurs when actions produce unexpectedly positive results (“I just tried. I am not sure if the method I used is the correct way to solve the problem. But it works”). This success is linked to a positive evaluation, which may positively orient the learners’ beliefs about and attitudes toward the subject. Metacognition . Schoenfeld (1985) described metacognition as referring to a cognitive process in which one plans a strategy to solve a task, monitors the comprehension of task-related knowledge, evaluates the progress towards the completion of the task, and makes a decision about whether the strategy is appropriate to apply in performing the task or he needs to select a new strategy. Brown, Bransford, Ferrara, and Campione (1983) suggested that metacognition includes two


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phases: (a) awareness of one’s own cognitive process, which is the knowledge of cognition, and (b) use of this awareness to make a decision, which is the regulation of cognition. In both Schoenfeld’s and Brown et al.’s views, metacognition is a cognitive activity that leads to reflection on one’s own thought process in problem solving. In addition to playing a significant role in students’ success in problem solving, McLeod argued that metacognition is closely tied to students’ beliefs and attitudes. Learning to reflect on their own cognitive processes, therefore, can not only help students realize how much they have learned (derived from their successes) as well as how much they still need to learn (indicated by their errors), but also increase their awareness of their emotional reactions to learning endeavors. Being aware of their emotional reactions toward learning mathematics will give students greater control over their cognitive processes, thus affecting their beliefs about and attitudes toward mathematics. For example, students who receive low scores on a math quiz initially feel sad, even angry, and then conclude they cannot learn mathematics, resulting in a negative attitude. But if they can be helped to understand the relevant mathematics knowledge, they may at least temporarily suspend their negative reactions and reflect on their errors in light of this knowledge. Such reflection gives students a sense of control over their learning and, at the same time, raises their awareness of their previous negative emotional reactions, so they may begin to think that mathematics is not as hard as they had believed and develop more positive attitudes. Is McLeod’s (1989) theoretical framework for determining beliefs about and attitudes toward specific actions, events, or objects applicable to elementary preservice teachers’ (EPSTs) experiences with geometry? What other factors might influence their beliefs and attitudes? Those are the research questions that I would like to study. Description of four proposed approaches This study took place in a mathematics and pedagogy course, focusing on geometry, at a large Midwestern university, in order to investigate how EPSTs’ beliefs and attitudes toward geometry were formed. Following is an overview of how the instructor introduced the geometry

199 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes concept of an Inscribed Circle Within a Triangle (ICWT) through the four learning approaches: a paper-folding activity, making a construction with a compass and a straightedge, determining proof, and operating Geometer’s Sketchpad (GSP), a dynamic geometry software. Hands-on activity: folding paper Each EPST was given a white sheet and asked to draw a triangle ABC. Next, s/he folded the paper by finding the angle bisector of each angle of triangle ABC. After folding the paper, s/he located the point at which the three angle bisectors (the three folds) intersected and then used a compass to draw a circle that just touched each side of triangle ABC (see figure 1). After completing the activity, the geometry instructor led a whole-class discussion by asking such questions as “What is the mathematical name of point P, where the three bisectors intersect? What relationships have you discovered concerning the distance from point P to each side of triangle ABC?”

Folding angle bisectors from each angle of a triangle to investigate the inscribed circle within a triangle

Figure 1.

Making constructions with a compass and a straightedge Each EPST was given a blue sheet and asked to draw a triangle ABC. Next, a compass was used to construct an angle bisector for each angle of this triangle to produce an intersecting point P where the three angle bisectors intersected. Then this intersecting point P was used as a center to make a circle that touched each side of the triangle ABC (see figure 2). After the activity, EPSTs were asked, “Do you see any relationship between the distances from the incenter to each side of the


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angle?” By now, many EPSTs were able to understand that the distance from the incenter to each side of the triangle is the same since it is actually the radius of the inscribed circle. Next, the EPSTs were asked, “Can you prove it?”

Constructing angle bisectors for each angle of a triangle with a compass and a straightedge to explore the inscribed circle within a

Figure 2.

Constructing proof In the proof activity EPSTs (in pairs or groups) were first given some time to discuss how to prove that the distance from the incenter (point P) to each side of the triangle is the same. After 10-15 minutes, geometry instructors led a whole-class discussion to facilitate EPSTs’ thinking by asking several questions, such as, which two triangles could be used to solve this proof problem? What are the prerequisite (given corresponding congruent parts) you could find from the two triangles you have chosen? Based on what condition (SSS, SAS, or ASA) of triangle congruence, you could say these two triangles were congruent (see figure 3)? With proof, EPSTs were able to gain a deeper understanding of what the inscribed circle within a triangle is. More specifically, they would learn the triangle property that any point at angle bisector to the sides of the triangle will be the same. Too often, these two math ideas (inscribed circle within a triangle & property of angle bisectors) are considered separate concepts. With proof, EPSTs were able to make the fundamental connections.

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Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes

Figure 3.

Proving that the distance from the incenter to each side of the triangle is the same

Geometer’s Sketchpad (GSP) EPSTs also work modeling real life situations such as the shipwreck survivor problem using GSP, dynamic geometry software (see figure 4), to understand property of angle bisectors. In this problem one needs to find the place where a survivor could set camp in an island that closely approximates the shape of a triangle. Specifically, EPSTs (two EPSTs worked as a pair) were asked to construct an inscribed circle within a triangle through angle bisectors with GSP. After 10- 15 minutes, geometry instructors led a whole-class discussion to talk about how to make the construction via GSP. In order to accurately make the construction, EPSTs need to have good understanding about the inscribed circle within a triangle related to angle bisector property from previous activities and apply what they have learned. After the construc-


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tion with GSP, EPSTs were able to “see” that no matter where and how they drag the vertex of the triangle, the inscribed circle always touches each side of the triangle and that the distance from the incenter to each side of the triangle remains the same (see figure 4). This GSP activity reconfirmed their understanding about the property of angle bisector as well as inscribed circle within a triangle. It also allows EPSTs see the fundamental connections among those representations.

Figure 4.

Finding the incenter to solve the shipwreck survivor problem with GSP

Methodology Data for this study were collected during the Fall 2007 semester in a mathematics and pedagogy course, focusing on geometry, at a large Midwestern university. After the Study Information Sheet approved by the university’s research office had been distributed, four EPSTs volunteered to participate: John, Karen, Becky, and Carrie (pseudonyms). The geometry instructor, Mr. Grow (pseudonym), gave lectures every Monday and Wednesday from 9:30 a.m. to 10:45 a.m. Each EPST participated in two interviews, each lasting sixty to ninety minutes. The first interview questions focused on the EPSTs’ geometry learning experiences prior to their current geometry-related mathematics and pedagogy course. Examples of the first interview questions included: When did you learn geometry? What kind of geometry knowledge had you learned before? How did you learn geometry? Do you think the ways you learned geometry were effective for you and why? What did you do when you were learning geometry inside or outside the

203 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes classroom? Did you like geometry during or after learning geometry back then and why? The second interview questions focused on the EPSTs’ current geometry learning experiences in their geometry-related mathematics and pedagogy course. Examples of the second interview questions included: What kind of geometry knowledge have you learned from Mr. Grow’s class? How do you learn geometry from Mr. Grow’s class? Were the ways of learning geometry from Mr. Grow’s class different from the ones you had experienced before? Do you think the ways you learned geometry from Mr. Grow’s class were effective for you and why? Questions in the first and second interviews were similar but used different verb tenses in order to investigate how the EPSTs’ previous and current geometry learning experiences affected their beliefs and attitudes toward geometry. The prepared interview questions were used as guiding questions and then, based on individual responses, the interviewees were asked sub-questions related to their initial answers. Such now-and-then investigation offered the researcher the opportunity to confirm how the four EPSTs’ beliefs and attitudes toward geometry were formed. Data analysis and results Three sets of questions and sub-questions –ways of learning geometry, geometry performance, and reflection on geometry learning– emerged as most useful for understanding how the four EPSTs’ beliefs about and attitudes toward geometry were formed. Table 2 shows those three interview question sets. Table 2

Three interview question sets for investigating the origins ofthe four EPSTs' beliefs and attitudes toward geometry learning

Ways of learning geometry

* How did/do you learn geometry? What did/do you learn about geometry? * Did/do you think the ways you learned geometry were effective for you and why? * Did/do you like geometry after taking the course?


Geometry performance

Reflection on geometry learning

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* Did/do you think your beliefs and attitudes toward geometry were influenced positively or negatively by the ways oflearning geometry? * Did/do you have any geometry homework? * How was/is your homework performance? How about the midterm or final exam? * Did/do those performance influence what you think about geometry? * During the class, did/does the instructor help you reflect on what you have learned and what you have missed? * After the instructor has returned homework or an exam, did/does the instructor help you reflect on what you have learned and what you have missed?

Ways of learning geometry During the first interview, all EPSTs mentioned that they first experienced geometry when they were elementary school students. Although geometry was not a specific subject, they learned basic geometric shapes and their characteristics, such as the four equal sides of a square. In the ninth or tenth grade, they took courses on geometry concepts, relationships, and operations such as the properties of parallel lines, their relationship to angles, and calculation of the areas of geometric shapes, making basic constructions with a compass and determining proofs. Most instruction was teacher-and textbook-centered with students listening to lectures and observing projected or written demonstrations, followed by worksheets and textbook assignments, but with little group interaction, whole-class discussion, or opportunity to explore the concepts being taught. The EPSTs regarded this approach as an ineffective way for them to learn geometry, which they came to believe it was too abstract and difficult to understand, leading to negative attitudes toward the subject. Becky said:

205 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes …When he [the geometry instructor] taught the criteria for congruent triangles, he used an overhead projector to give us lectures about triangle congruence. He selected a question from the textbook to demonstrate how to prove it. After he taught the lesson, he gave us a worksheet to practice it until he thought that we had grasped the concept…. No, I don’t think it [the way of learning geometry] was effective because I did not connect the concept and questions very well. I mean I understand the concept but I don’t know how to apply these concepts to questions…. I did not get much understanding from that class. I hate geometry.

During the second interview, the EPSTs explained that the geometry concepts they learned from Mr. Grow in current mathematics and pedagogy course were similar to those they had learned in high school, so they were relearning but in ways that helped them understand the concepts better. They felt that learning geometry concepts through the four approaches in a sequence moving from the easiest one (folding paper) to the more complicated ones (determining proof or using GSP), helped them develop a deeper understanding of geometry a step at a time. These positive experiences encouraged them to rethink their ideas about geometry. Again, Becky said: …In Mr. Grow’s class, we used different ways to learn a geometry concept and I especially like hands-on activity because I am experiencing the concept. I am actually learning by doing it.... Yes, I think it [the way of learning geometry] was effective because I am able to practice the concept by myself, not watching or listening to [an explanation of] the concept from the instructor. Mr. Grow introduced the concept by engaging us in an activity that we can “see” or “touch” the concept and later he helped us generalize the concept in a mathematical term…. I think I engaged in Mr. Grow’s class more, compared to my previous geometry learning experiences. Mr. Grow made me feel geometry learning is easier.


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This question set, ways oflearning geometry, was related to the role of representations in affecting EPSTs’ beliefs and attitudes toward geometry. The results of both interviews showed that the way of presenting geometry concepts strongly affected how EPSTs formed their beliefs and attitudes. Specifically, EPSTs’ beliefs and attitudes about geometry were influenced by the format of how the geometry concepts were presented, for example, in written or diagram form only (EPSTs’ previous geometry learning) or in a combination of visual presentation and interactive experiences (EPSTs’ current geometry learning). EPSTs’ beliefs and attitudes were also influenced by the order in which approaches were presented. In Mr. Grow’s class, EPSTs first experienced geometry concepts by performing operations and then later Mr. Grow helped them make generalizations about the concepts. In this way, EPSTs’ learning progressed from concrete examples to abstract ideas. This result confirmed McLeod’s (1989) position that representation strongly affects the formation of one’s beliefs and attitudes toward specific actions, events, or objects. Ways of learning geometry During the first interview, the EPSTs mentioned that in high school, their geometry instructors assigned them homework consisting of approximately 20 questions selected from the textbook, mostly short, discrete questions such as definitions or area calculations and a few more complex questions such as proof or construction problems. Because they did not have a good understanding of what was taught in class, they sometimes had to ask a tutor or their parents for help. Although they felt they had tried to prepare themselves to the best of their ability, their performance on the geometry tests often did not meet their expectations, which made them dislike geometry. Carrie said: …Doing homework was not an easy job for me. I had to hire a tutor to re-teach me in order to finish the assignment because I did not understand geometry at all in the class…. When the midterm was approaching, I needed to meet with my tutor several times to go over the geometry concepts that were needed for doing the midterm…. Even though I spent so much time at preparing for the midterm, the midterm results I got were very disa-

207 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes ppointing. I was very frustrated. I felt that I made a lot of effort to prepare for the exam but the results were not very good…. I did not like geometry before I took the midterm. But after the midterm result came back, I just disliked geometry more.

During the second interview, the EPSTs explained that answers for the approximately 10 homework questions Mr. Grow gave them every week could not be found in the textbooks or on worksheets but required students to comprehend what was taught and consult their class notes. None needed to hire a tutor because Mr. Grow’s four learning approaches, especially the folding activity, helped them grasp the geometry concepts. To prepare for their midterm, they studied the textbook and reviewed their homework, hand-outs, and in-class practice with problems or other activities. They even searched for websites with information about the geometry concepts being taught in order to better understand them. Overall, they felt well-prepared for the midterm and thought they had performed well on it, at least, according to their own criteria. Again, Carrie said: …Now there is no need for me to hire a tutor for my geometry class. Mr. Grow helped me gain a deeper understanding about geometry by actually “doing” the concept…. I used the notes, textbook, homework, or worksheets to prepare the midterm…. When I received my midterm I was happy with that…. That makes me realize that “Wow! Actually I can do well in geometry”.

This question set, geometry performance, was associated with the role of discrepancy in affecting EPSTs’ beliefs and attitudes. The results of both interviews revealed the effects of discrepancy between expected actual and performance on exams. In high school, because they had put a lot of effort into preparing for geometry exams, they believed they would perform well, but, in fact, they didn’t, resulting in negative beliefs and attitudes. This finding resonates with McLeod’s (1989) error discrepancy, a mismatch between an expected outcome and the actual outcome resulting in negative beliefs and attitudes. Because of their previous geometry learning experiences, the EPSTs thought they might


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not be able to perform well on Mr. Grow’s geometry midterm although they continued to practice the geometry concepts. In reality, the results showed that they were able to do well in geometry. This discrepancy led to more positive beliefs and attitudes, which, resonates with McLeod’s (1989) success discrepancy, when particular actions or thoughts produce unexpectedly positive results. In short, both geometry learning experiences confirm McLeod’s (1989) beliefs and attitudes framework. Reflection on geometry learning The EPSTs recalled that in high school they spent little time reflecting on their geometry test results to assess what they had learned and what they needed to learn. The instructors just checked the correct answers with them to make sure that the grade was accurate, leaving them disappointed and with negative feelings about geometry learning. Compared to their previous geometry learning experiences, the participants in this study commented that Mr. Grow spent more time going over geometry concepts, especially when they received back homework assignments, quizzes, major exams. This process helped them realize which parts of the concepts they had mastered and which parts they still needed to work on. Although when they first saw low grades they might feel frustrated and averse to geometry, Mr. Grow’s guidance helped them understand the nature of their mistakes and reconsider their initial emotional reactions toward geometry, which they could change by paying closer attention to problems and, with their instructor’s help, improving their comprehension of target concepts. This greater sense of control, in turn, motivated them to study geometry harder and rethink their initial perceptions of geometry. Karen said: …Sometimes it [geometry performance results] did bother me and made me so frustrated when I saw the grade… I even thought to quit learning geometry because I thought I understand the concepts and I should get a good grade, better than the one I received…. When Mr. Grow explained the questions, I realized that I did not think though the concepts completely. I did not grasp the concepts totally…. I realized there was no need for me to feel frustrated. Instead, I should study geo-

209 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes metry harder so I will not miss the points next time when I see the similar questions.

This question set, reflection on geometry learning, was linked to the role of metacognition in affecting EPSTs’ beliefs about and attitudes toward geometry. The results of both interviews indicated that metacognition –a cognitive process which helps EPSTs think about their own thinking and be aware of their emotional reactions to a subject– strongly affects how EPSTs form their beliefs and attitudes. Specifically, Mr. Grow helped these EPSTs reflect on what they knew and what they still needed to master, overcome negative emotions, and acquire a sense of control, which positively affected how they viewed themselves as learners of mathematics. This reflection process confirms McLeod’s (1989) theoretical claim that metacognition plays a significant role in formation of one’s beliefs and attitudes. Another important insight derived from analysis of the interview data is that another factor, understanding, is important in affecting the formation of the EPSTs’ beliefs and attitudes toward geometry, as reflected in the following statements: …I know that Mr. Grow used different ways to help us understand geometry concepts…. I think the way Mr. Grow taught geometry is more influential for me in learning geometry. He made geometry simple and I understand more. (John) … Mr. Grow made me think about geometry from different ways such as hands-on activity, constructions, or proof. Those ways helped me master geometry concepts. Now I understand geometry more compared to previous geometry learning…. I enjoyed learning geometry. (Karen) … Basically, I used geometry performance to tell how much I have learned. If my geometry performance is good, that means I understand more, then my beliefs and attitudes about geometry will be more positive. But if I get low geometry performance, that means I understand less, then my beliefs and attitudes about geometry will be negative. (Carrie)


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… The four geometry learning approaches helped me understand geometry concepts, hands-on activity particularly… Although I still don’t like geometry, at least I don’t dislike it that much. (Becky)

From these statements, it suggested that Mr. Grow’s use of four learning approaches, including a paper-folding activity, making a construction with a compass and a straightedge, determining proof, and operating Geometer’s Sketchpad (GSP), a dynamic geometry software, provided more learning opportunities for EPSTs in exploring the geometry concepts being taught, and the better understanding that resulted was an important link in the chain leading to the participants’ development of more positive beliefs about mathematical learning and their attitude toward geometry learning as a subject to be both learned and taught. Conclusions and discussion In this study, I explored how EPSTs both formed their beliefs about and attitudes toward geometry by analyzing data from two interviews as different points in the learning period. This analysis revealed the strong influences of (1) the format and the order of presenting geometry concepts, (2) mismatches between expected and actual performance on the geometry midterm, and (3) reflection leading to self assessment of learning and awareness of emotional reactions to geometry, confirming McLeod’s (1989) beliefs and attitudes framework explicating the roles of representation, discrepancy, and metacognition. In addition, the analysis identified understanding as another important factor affecting beliefs and attitudes (see table 3). If in fact we can understand how EPSTs beliefs about and attitudes toward learning geometry are formed, we will be able to find out ways to help EPSTs change negative orientations, which can have positive implications for designing mathematics and pedagogy courses for EPSTs that will better prepare them to be effective mathematics teachers.

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Table 3

Results confirming McLeod’s (1989) framework for forming beliefs and attitudes through representation, discrepancy, and metacognition and the emergence ofa fourth factor: understanding

Representation * The format of the objects or events determines one's beliefs and attitudes toward those particular objects or events. * The order of the objects or events determines one's beliefs and attitudes toward those particular objects or events.

Beliefs and attitudes

Discrepancy * Error: One expects the action to be correct but in fact it produces unexpectedly negative results, causing negative beliefs and attitudes. * Success: One's particular action produces unexpectedly positive results, producing positive beliefs and attitudes.

Metacognition * One reflects on one's own cognitive processes. * So one should be aware of one's emotional reactions toward the things experienced. * Next, one uses this awareness to control one's cognitive processes. * Then, one rethinks and possibly changes one's beliefs and attitudes.

Understand * One learns about the objects, events, or persons from multiple perspectives. * One masters the ideas about the objects, events, or persons. * One, then, increases the level of understanding of the objects, events, or persons. * Thus, one rethinks previous beliefs and attitudes or forms new ones.

Toward this end, further inquiry might investigate how each of the different instructional approaches affected EPSTs’ beliefs and attitudes as well as the cumulative effects of the four approaches in the sequence in which they were taught. Similarly, further research could pursue deeper understanding of the roles that representation, discrepancy, metacognition, and understanding play in changing EPSTs’ beliefs about and attitudes toward geometry learning.


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References Brown, A.L., Bransford, J.D., Ferrara, R.A., & Campione, J.C. (1983). Learning, remembering, and understanding. In J. Flavell & E. Markman (Eds.), Mussen’s handbook ofchild psychology (Vol. 3, pp.77-166). Somerset, NJ: Wiley Charles, R. I., & Lester, F. K., Jr. (1984). An evaluation of a processoriented instructional program in mathematical problem solving in grades 5 and 7. Journals for Research in Mathematics Education, 15, 15-34 Fennell, F., & Rowan, T. (2001). Representation: An important process for teaching and learning mathematics. Teaching Children Mathematics, 5, 288-292 Greeno, J. G., & Hall, R. B. (1997). Practicing representation: Learning with and about representation forms. Phi Delta Kappan, 79, 361-367 Leder, G. C., Pehkonen, E., & Törner, G. (2002). Beliefs: A hidden variable in mathematics education? Dordrecht: Kluwer Academic Publishers. Lester, F., & Garofalo, J. (1982). Metacognitive aspects ofelementary school students’ performance on arithmetic tasks. Paper presented at the annual meeting of the American Educational Research Association, New York. Maaß, J. & Schlöglmann, W. (2009). Beliefs and attitudes in mathematics education: New research results. Rotterdam/ Taipei: Sense Publishers. Mandler, G. (1989). Affect and learning: Causes and consequences of emotional interactions. In D. B. McLeod & V.M. Adams (Eds.), Affect and mathematical problem solving: A new perspective

(pp.3-19), NY: Springer-Verlag. McLeod, D. B. (1989). The role of affect in mathematical problem solving. In D.B. McLeod & V.M. Adams (Eds.), Affect and mathematical problem solving: A new perspective (pp.20-36), NY: Springer-Verlag. McLeod, D. B. (1994). Research on affect and mathematics learning in the JRME: 1970 to the present. Journal for Research in Mathematics Education, 25(6), 637–647

213 Kai-Ju Yang - Elementary preservice teachers' beliefs and attitudes National Council ofTeachers of Mathematics (2000). Principle and Standards for School Mathematics. Reston, VA. Schoenfeld, A.H. (1983). Beyond the purely cognitive: Belief system, social cognitions, and metacognitions as driving forces in intellectual performance. Cognitive Science, 7, 329-363 Schoenfeld, A.H. (1985). Mathematical problem solving. Orlando, FL: Academic press. Törner, G. (2001). Mentale Repräsentationen –der Zusammenhang zwischen ‘Subject-Matter Knowledge’und ‘Pedagogical Content Knowledge’– dargestellt am Beispiel der Exponentialfunktionen in einer Fallstudie mit Lehramtsstudenten. In G. Kaiser (Ed.), Beiträge zum Mathematikunterricht (pp. 628–631). Hildesheim: Franzbecker. Yang, K. (2008). Learning oflinear algebra word concepts by using cognitive tutor software: An implication for in-service teachers. Paper presented at the annual meeting of the Society for Information Technology & Teacher Education Conference, Las Vegas, Nevada. Kai-Ju Yang is doctoral candidate in Curriculum Studies (Mathematics focus), at Indiana University, USA. Contact address: Direct correspondence concerning this article should be addressed to the author at: 1179, Cooper River Drive, San Jose, CA, 95126, USA. E-mail address: [email protected].


Crossroads in the History of Mathematics and Mathematics Education. Joan Cabré1 1) Universitat Rovira i Virgili, España.

Date of publication: June 24th, 2012

To cite this article: Cabré, J. (2012). Crossroads in the History of Mathematics and Mathematics Education. REDIMAT Journal of Research in Mathematics Education, 1 (2), 214218. doi: 10.4471/redimat.2012.11 To link this article: http://dx.doi.org/10.4471/redimat.2012.11

PLEASE SCROLL DOWN FOR ARTICLE The terms and conditions of use are related to the Open Journal System and to Creative Commons NonCommercial and NonDerivative License.

REDIMAT Journal of Research in Mathematics Education Vol. 1 No. 2 June 2012 pp. 214218.

Review Sriraman, B. (Ed.) (2012). Crossroads in the History of Mathematics and Mathematics Education. The Montana Mathematics Enthusiast. Monograph series in Mathematics Education. Charlotte, NC: Information Age Publishing. El uso de la historia de la matemática como recurso didáctico no es nuevo. Existen varios trabajos muy interesantes que nos recuerdan cada día las estrechas conexiones entre el desarrollo de las ideas matemáticas, y su enseñanza. En este libro Bharath Sriramann recopila trabajos que nos llevan desde el teorema fundamental del cálculo, hasta el libro de los sólidos regulares de Euclides, pasando por la aportaciones matemáticas procedentes del antiguo Egipto. La interacción entre la historia de la matemáticas y la educación matemática viene ya de lejos. Me vienen a la memoria libros fundamentales como la historia de la matemática de Boyer, o la de Collette, o los trabajos clásicos ya de Kline, por solo citar algunas obras que me han marcado. Gratos recuerdos. Este libro se presenta como un recurso para estudiantes de grado, tanto de futuros matemáticos, como de futuros maestros y maestras de matemáticas. A lo largo de sus páginas, los diferentes autores aportan reflexiones interesantes en base a episodios y conceptos que forman parte del conocimiento matemático que hemos ido acumulando a través de los siglos. El libro se organiza en tres partes bien diferenciadas: historia y 2012 Hipatia Press ISSN 20143621 DOI: 10.4471/redimat.2012.11

215 Joan Cabré Crossroads in the History of Mathematics didáctica del cálculo, por un lado; de la geometría y la numeración, por otro; y finalmente el papel que juega la historia de la matemática en la matemática educativa. El primer bloque comienza con un capítulo de Eva Jablonka y Anna Klisinska. Las autoras usan la referencia al teorema fundamental del cálculo para discutir sobre la institucionalización del conocimiento matemático. A través de entrevistas a once matemáticos, repasan el desarrollo histórico de este concepto. El cálculo, que en todo el mundo se ha convertido virtualmente en al piedra de toque que da entrada, o no, a los estudios superiores de las diversas ingenierías, hunde sus raíces históricamente en la Grecia antigua de Eudoxo o Arquímedes, utilizando un método que mucho más tarde, en el siglo XVII Gregoire de St. Vicent bautizaría como el método de la exhausción. Este “cálculo” nada tiene que ver con el concepto de “cálculo” que procede de los trabajos de Newton sobre las fluxiones, y de Leibniz y las integrales, cuando trataron de someter el infinito a sus algoritmos de cálculo. Los trabajos posteriores de Cauchy, las “sumas de Riemann”, o las presentaciones de l’Hospital y Johan Bernouilli han ido poco a poco definiendo lo que hoy en día conocemos como “cálculo”, esa disciplina fundamental de toda disciplina técnica. Las respuestas de los once matemáticos a las preguntas de las autoras sobre las diferentes versiones del teorema fundamental del cálculo, no dejan de ser interesantes y llaman la atención sobre la importancia de la recontextualización de un conocimiento matemático como el cálculo, que ha ido superando “obstáculos epistemológicos” a medida que iba pasando de un enfoque geométrico al enfoque algebraico que le dieron los matemáticos del siglo XVIII. Nicolas Haverhals y Matt Roscoe continúan en el ámbito del cálculo. Su capítulo relata la transición de los estudiantes al cálculo usando la historia como libro de ruta. La primera frase que leemos nos remite a la asignatura de “Precálculo.” A lo largo de las siguientes páginas, los autores analizan el trayecto histórico que ha conducido a conceptos como la definición del límite de la derivada, o el estudio del cálculo diferencial e integral. Se presentan los métodos históricos para encontrar la pendiente de la recta tangente a una función en un punto desarrollados por Descartes y Hudde, Fermat y Barrow.

REDIMAT Journal of Research in Mathematics Education, 1 (2) 216 Tras el capítulo de Mikhail Katz y David Tall sobre la tensión entre los infinitesimales intuitivos, y el análisis matemático formal, y el capítulo de Brating, Kallem y Sriraman sobre la naturaleza didáctica de algunos ejemplos históricos poco conocidos de la historia de la matemática, en el ámbito del cálculo infinitesimal, Jeff Babb y James Currie usan el problema de la curva braquistócrona, también conocida como “curva del descenso más rápido” para reflexionar sobre la contextualización de los problemas matemáticos, a fin de acercarlos a una audiencia más amplia. El problema de la curva braquistócrona es uno de los problemas más antiguos del cálculo de variaciones. Matemáticos como Leibniz, L’Hospital, Newton, o Jakob y Johann Bernoulli encontraron la solución (que es un segmento de la cicloide). Los retos entre Leibniz y Newton, y las disputas entre los hermanos Bernoulli, sirven a los autores para contextualizar el concepto de una curva tal que la formada por una partícula que cae deslizándose por ella, de un punto a otro situado debajo del primero, pero sin que esté en la vertical, en el mínimo tiempo posible. Tras una incursión en el terreno de los logaritmos de la mano de Rafael VillarrealCalderon, que discute sobre los usos históricos que han tenido esas herramientas matemáticas, Nicolas Haverhals y Matt Rocoe cierran esta primera parte dedicada al cálculo con una referencia a la proyección de Mercator como herramienta para enseñar la integral de una recta secante. Mercator, cuyo verdadero nombre (no latinizado) era Kaufmann, trabajó en el cálculo de series infinitas, usando un método de cuadraturas igual que hacía Gregory, con el que obtiene resultados análogos a los que Newton ya había obtenido Newton en De analysi con el desarrollo binomial infinito, pero que no había publicado. El siguiente bloque de contenidos, sobre la historia de la geometría y su impacto en la enseñanza de las matemáticas, comienza, como no, con una referencia clara a Euclides, el autor del manual de matemáticas más antiguo que se conserva, y el libro más editado de la historia, después de la Biblia. Michael N. Fried discute el valor educativo del libro de los sólidos regulares. El libro XI de Los elementos es el que contiene las proposiciones relativas a la geometría tridimensional. Aquí es donde encontramos definiciones familiares para todos los que hemos estudiado matemáticas en la escuela, como que un sólido es lo que

217 Joan Cabré Crossroads in the History of Mathematics tiene “longitud, anchura y profundidad.” Michael Fried, en este capítulo, se concentra en el libro XIII de Los elementos, donde Euclides escribe sobre los cinco poliedros regulares, los conocidos como “sólidos platónicos”, y que el matemático griego se esforzó en inscribir en sendas esferas. A través de sus comentarios, Freid nos ofrece elementos de reflexión para conectar nuestras clases de geometría con un referente clásico como es el trabajo de Euclides, que ha sido y es maestro de tantas y tantas personas a lo largo de las generaciones. Siguiendo con el estudio de la obra euclidiana, a continuación Jade Roskam centra su trabajo en el libro X de Los elementos. Este libro está dedicado, como es harto conocido, a los “inconmensurables”, que hoy en día denominamos como números irracionales. Euclides consideraba este libro como una parte de la geometría, y no tanto de la aritmética (como cabría esperar hoy), y de hecho, las proposiciones 2 y 3 son indudablemente de interpretación geométrica. Roskam nos lleva a un viaje que ilumina con claridad la obra de este clásico de la matemática, para beneficio de las personas que nos dedicamos a la enseñanza. A continuación Mark Beintema y Azar Khosravani se trasladan al origen del concepto de género, tan usado en la topología para referirnos a la invarianza de los objetos. Estos autores nos invitan a visitar el trabajo de Gauss, el llamado “príncipe de las matemáticas”, y su teoría sobre las formas cuadráticas binarias. A lo largo de las páginas de su capítulo, estos dos autores repasan el desarrollo histórico de las expresiones cuadráticas binarias del tipo f(x,y)=ax2+bxy+cy2 desde la formulación de Gauss en Disquisitiones Arithmeticae, hasta las aportaciones de Euler, Legendre o Lagrange. Este bloque de contenidos se cierra con un capítulo de Gabriel Johnson, Bharath Sriraman y Rachel Saltzstein sobre las matemáticas del antiguo Egipto. Tal y como dicen, la mayor parte de las matemáticas que enseñamos hoy en día en las escuelas ya se inventaron en el mundo antiguo. Este hecho abre la posibilidad de enfocar el currículum desde un punto de vista sociocrítico e histórico. En este capítulo, los autores retroceden hasta los tiempos del antiguo Egipto, donde encuentran las raíces de un modelo semítico de la matemática actual, frente a la tesis que defiende el anclaje en la Grecia clásica el

REDIMAT Journal of Research in Mathematics Education, 1 (2) 218 origen de las matemáticas occidentales. A través de los ojos de personajes históricos como Heródoto, John Greaves, Napoleón Bonaparte, y otros “visitantes” que “descubrieron” las grandes pirámides, los autores van reconstruyendo el significado matemático de esta antigua cultura, y su legado en nuestro tiempo. El último bloque temático del libro está dedicado al análisis de la historia de las matemáticas en la educación matemática. El primero de los capítulos que compone esta parte, escrito por Constantinos Tzanakis y Yannis Thomaidis, habla sobre la clasificación de argumentos y los esquemas metodológicos que proponen ambos autores para integrar la historia en la educación matemática. Con ellos aprendemos que las matemáticas, que se nos presentan en un formato tan pulido, formalmente impecable, deductivo, son resultado de procesos más humanos, desordenados, de confrontación de ideas, de avances y retrocesos constantes. Tal y como afirman los autores, no es hasta que una idea o una teoría alcanza un cierto grado de madurez, que se formaliza. Por tanto, la historia de las matemáticas aparece como un recurso didáctico para comprender de dónde salen las ideas y las nociones matemáticas que pueblan nuestros libros de texto en las escuelas. A continuación Uffe Thomas Jankvist nos presenta un capítulo en el que intenta identificar y clasificar estudios empíricos en el ámbito de la historia y la educación matemática. A lo largo de estas páginas el autor danés revisa el papel de la historia [matemática] en la enseñanza y el aprendizaje de las matemáticas. El capítulo que le sigue, escrito por Tinne Hoff Kjeldsen, analiza los beneficios de usar la historia de esta manera. Para ello recurre a dos ejemplos concretos del trabajo de los estudiantes, en un instituto de bachillerato. El libro concluye con un trabajo de Shirley B. Gray y Libby Knott el progreso en el siglo XVII, de las matemáticas. Con este capítulo se cierra el libro, pero se abre un amplio espectro de preguntas interesantes, y motivos de reflexión, para incorporar la historia de las matemáticas en nuestra práctica docente como maestros y maestras. Joan Cabré, Universitat Rovira i Virgili [email protected]