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Editorial
Silvia Molina Roldán 1
1 ) Universidad Rovira i Virgili y Universitat Autónoma de Barcelona, España.
Date of publication: October 24th, 201 2
To cite this article: Molina, S. (201 2). Editorial. Journal of Research in
Mathematics Education, 1 (3), 21 9-221 . doi: 1 0.4471 /redimat.201 2. 1 2
To link this article: http://dx.doi.org/1 0.4471 /redimat.201 2.1 2
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REDIMAT - Journal ofResearch in Mathematics Education Vol. 1 No. 3
October 2012 pp. 219-221
Editorial
Silvia Molina Roldán
Universidad Rovira i Virgili y Universitat Autónoma de Barcelona
E
stoy encantada de poder presentar el tercer número de
REDIMAT y, con él, el último número del primer volumen que
afianza esta revista como foro para compartir
conocimiento científico alrededor de la educación matemática. La
apertura en el acceso al conocimiento y la posibilidad de compartirlo es
una realidad creciente; cada día aparecen nuevas tecnologías, programas
o redes sociales que nos ofrecen más posibilidades en este sentido. El
conocimiento científico no puede quedarse atrás. Por este motivo,
REDIMAT nació como una revista abierta, que se pone al acceso de
todos y todas, para universalizar el acceso a trabajos de calidad
científica sobre la enseñanza y aprendizaje de las matemáticas, que
puedan tomar de referencia todas y todos aquellos científicos,
profesionales o ciudadanos en general que lo deseen. Las matemáticas
son un elemento central de nuestra cultura, y al mismo tiempo un
instrumento imprescindible para comprender otros muchos aspectos de
nuestra vida, así como parte de los avances tecnológicos que día a día se
van desarrollando. Por este motivo, la educación matemática no puede
basarse en otra cosa que en evidencias científicas proporcionadas por
investigaciones que nos orienten sobre el qué y cómo enseñar y el por
qué.
En este número contamos con cuatro artículos que nos proporcionan,
desde diferentes ámbitos de estudio, contribuciones que nos ayudan a
avanzar en este sentido. El primer artículo, de Carmen Batanero, Emilse
Gómez, Luis Serrano y José Miguel Contreras, analiza la comprensión
que futuros maestros de primaria tienen del concepto de aleatoriedad.
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.12
220 Silvia Molina Roldán - Editorial
Para ello, los autores se basan en investigaciones previas para comparar
los resultados obtenidos en estas investigaciones con niños entre 11 y 16
años con los que obtienen en una muestra de futuros profesores y
profesoras de primaria, en relación a las propiedades asignadas a
secuencias de resultados aleatorios. Los resultados sirven para ofrecer
orientaciones a una formación del profesorado que ayude a profundizar
en la comprensión de este concepto, basándose en los sesgos de
razonamiento identificados, y así mejorar la labor docente de este
profesorado.
Si el primer artículo se centra en la mejora de la comprensión de
conceptos matemáticos, los artículos segundo y tercero de este número
de REDIMAT aportan interesantes contribuciones alrededor de las
matemáticas como área de enseñanza y aprendizaje en la cual avanzar
para superar desigualdades sociales. El artículo de Anna Chronaki y
Yannis Pechtelidis, se centra en la ya clásica relación entre las
matemáticas y el género, presentando el estudio de caso de una
profesora de matemáticas para discutir la interrelación de las
matemáticas y el género en la creación de la subjetividad. Chronaki y
Pechtelidis parten de la existencia de discursos hegemónicos de género
alrededor de la competencia matemática que inciden en construcción de
la subjetividad. Según su análisis, al ser las matemáticas asociadas
tradicionalmente a un carácter típicamente masculino, la competencia
matemática demostrada por mujeres no conlleva una ruptura con la
dualidad masculino-femenino y las características opuestas asociadas a
cada uno de los géneros, sino que provocan conflictos y contradicciones
en la construcción de la identidad basada en ese esquema binario, lo
que, según proponen, se podría superar mediante la deconstrucción de
categorías de género esencialistas.
En el tercer artículo, Laura McLeman y Eugenia Vomvoridi-Ivanovic
argumentan sobre la necesidad de disponer de información sobre cómo
el profesorado de matemáticas incorpora en su práctica docente
estrategias para promover la equidad en el aprendizaje de las
matemáticas para todo el alumnado, en términos de cultura, identidad o
lengua. Sus planteamientos parten de su propia experiencia docente así
como de su experiencia investigadora en el Center for the Mathematics
Education of Latinos/as (CEMELA) alrededor de la equidad en
educación. Partiendo de esta base, destacan la necesidad tanto de
REDIMAT - Journal ofResearch in Mathematics Education, 1
(3) 221
aumentar la difusión de prácticas que integran aspectos de equidad en la
formación del profesorado de matemáticas como de desarrollar una
mayor actividad investigadora alrededor de estas prácticas. El objetivo
final es promover una formación del profesorado de matemáticas que
consiga que la equidad sea un elemento central de la enseñanza de las
matemáticas.
Finalmente, este número de REDIMAT se cierra con un artículo de
Michael Voskoglou y Georgios Kosyvas, que presentan un experimento
sobre la comprensión del concepto de números reales por parte de
estudiantes. A partir de un análisis cuantitativo y cualitativo, los autores
investigan las dificultades en la comprensión de los números reales
asociadas a una comprensión parcial de los números racionales y a la
propia naturaleza de los números irracionales, que se manifiestan en
diferentes niveles educativos, también en la universidad. Los autores
apuntan algunos elementos clave en que centrar la atención del
profesorado de matemáticas para prevenir dificultades en la
comprensión de los conceptos analizados.
En conjunto, pues, los cuatro artículos de este número nos aportan
diferentes perspectivas de gran interés y actualidad en educación
matemática. Todos ellos apuntan aspectos en que la enseñanza de las
matemáticas necesita mejorar y nos proporcionan claves de cómo
hacerlo. Esperamos que su lectura sea de interés y utilidad. Disfruten
con ella.
Comprensión de la Aleatoriedad por Futuros Profesores de
Educación Primaria
Carmen Batanero 1 , Emilse Gómez2, Luis Serrano 1 , & José Miguel
Contreras 1
1 ) Universidad de Granada, Spain
2) Universidad Nacional de Colombia
To cite this article : Batanero, C., Gómez, E, Serrano, L.,& Contreras, J.L.
(2012). Comprensión de la Aleatoriedad por Futuros Profesores de Educación
Primaria. Journal of Research in Mathematics Education, 1(3), 222-245. doi:
10.4471/redimat.2012.13
The terms and conditions of use are related to the Open Journal
System and to Creative Commons NonCommercial and Non
Derivative License.
October 2012 pp. 222-245.
Understanding of Randomness
by Prospective Primary School
Teachers
Carmen Batanero
Emilse Gómez
Luis Serrano
José Miguel Contreras
Universidad de Granada
Universidad Nacional de Colombia
Abstract
Current curricular guidelines for probability at Primary school level imply the
need for a specific training of prospective teachers, which should be based on
the previous assessment of their training needs. In order to contribute to this
need, in this paper we present the analysis of responses by 157 Spanish
prospective teachers to an open question, taken from previous research on
subjective perception of randomness. The results show a mixture of correct and
wrong conceptions, some of which parallel some historical conceptions of
randomness. Teachers’ educators could start from these intuitions to help
prospective teachers advance to a broader meaning of the concept, adequate for
their future teaching responsibility.
Keywords: teacher training, randomness, assessing conceptions
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.13
October 2012 pp. 222-245.
Comprensión de la
Aleatoriedad por Futuros
Profesores de Educación
Primaria
Carmen Batanero
Emilse Gómez
Luis Serrano
José Miguel Contreras
Universidad Nacional de Colombia
Abstract
Las nuevas directrices curriculares para la probabilidad en la Educación
Primaria requieren una formación específica de los futuros profesores, que ha
de estar basada en la evaluación previa de sus necesidades formativas. Con
objeto de contribuir a dicha formación, en este trabajo se analizan las
respuestas abiertas a un problema utilizado en las investigaciones sobre
percepción subjetiva de la aleatoriedad. Los resultados muestran una mezcla de
concepciones correctas e incorrectas, algunas de las cuáles son paralelas a las
que el concepto de aleatoriedad ha recibido a lo largo de su historia. El
formador de profesores podría partir de estas concepciones y hacerlas progresar
para que los futuros profesores adquieran un significado completo del
concepto, que les capacite para su futura labor docente.
Keywords: formación de profesores, aleatoriedad, evaluación de concepciones
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.13
223
Carmen Batanero et al. - Comprensión de la aleatoriedad
A
unque la enseñanza de la probabilidad ha estado presente en la
educación secundaria en los últimos 20 años, su introducción
desde los 6 años en los diferentes ciclos de la Educación
Primaria es más reciente, y pretende proporcionar a los alumnos una
experiencia estocástica desde su infancia (MEC, 2006).
Una condición para asegurar el éxito de la enseñanza de la
probabilidad en este nivel es la adecuada preparación de los profesores
de Educación Primaria, para lo que se requiere una evaluación previa de
sus necesidades formativas (Franklin y Mewborn, 2006).
Este trabajo trata de contribuir a esta necesidad, presentando los
resultados de un estudio sobre las propiedades que una muestra de 157
futuros profesores de Educación Primaria asigna a las secuencias de
resultados aleatorios. La evaluación se realiza a partir del análisis de las
respuestas abiertas a un ítem utilizado por Green (1983) en una
investigación con estudiantes ingleses de entre 11 y 16 años y por
Cañizares (1997) en otro estudio con niños españoles de 11 a 14 años.
Se comparan los resultados con los de estos autores y se evalúan las
concepciones subyacentes sobre la aleatoriedad, siguiendo la
clasificación propuesta por Batanero y Serrano (1999).
En lo que sigue se presentan los fundamentos del trabajo, el método y
sus resultados, finalizando con algunas implicaciones para la formación
de profesores.
Fundamentos del Estudio
Significados del Concepto de Aleatoriedad a lo Largo de su Historia
La aleatoriedad se ha interpretado de forma diferente en distintos
momentos históricos e incluso en la actualidad se resiste a una
definición sencilla (Zabell, 1992; Bennet, 1998; Liu y Thompson, 2002;
Batanero, Henry y Parzysz, 2005). Exponemos a continuación algunos
significados que se le han atribuido, que nos permitirán comprender
mejor las concepciones de los futuros profesores.
Aleatoriedad y causalidad.
En la antigüedad, y hasta comienzos de la Edad Media, se usaron
dispositivos aleatorios para predecir el futuro o tomar decisiones, sin
REDIMAT - Journal ofResearch in Mathematics Education, 1 (3)
224
una idea científica de aleatoriedad. En este periodo, la aleatoriedad se
relacionó con la causalidad y se concibió como el opuesto de algo que
tiene causas conocidas (Bennet, 1998).
Liu y Thompson (2002) indican que las concepciones de aleatoriedad
y determinismo se mueven a lo largo de un continuo epistemológico,
uno de cuyos extremos corresponde a la creencia de que los fenómenos
aleatorios son reflejo de la ignorancia humana, y no tienen una
existencia objetiva. Esta visión aparece en Aristóteles, quien consideró
que el azar resulta de la coincidencia inesperada de dos o más series de
causas independientes (Batanero, Henry y Parzysz, 2005). En el otro
extremo se encuentra la creencia de que la aleatoriedad es inherente a la
naturaleza, aceptando la existencia del azar irreductible. Poincaré (1936)
ejemplifica este punto de vista citando, como ejemplo, el movimiento
Browniano, donde fenómenos aleatorios a nivel microscópico originan
una regularidad de fenómenos macroscópicos, que puede ser descrita
por leyes deterministas.
La ignorancia de las leyes que gobiernan ciertos fenómenos naturales,
sin embargo, no necesariamente involucran la aleatoriedad; como indica
Ayer (1974), un fenómeno sólo se considera aleatorio si se comporta de
acuerdo con el cálculo de probabilidades, incluso después de identificar
los factores que regulan el fenómeno.
Aleatoriedad y probabilidad.
Al comenzar el estudio matemático de las probabilidades se relacionó la
aleatoriedad con la equiprobabilidad (por ejemplo, en el Liber de Ludo
Aleae de Cardano); ello fue debido a que los primeros estudios sobre
probabilidad estuvieron relacionados con juegos de azar donde todos los
resultados elementales eran equiprobables.
Batanero y Serrano (1999) indican que también actualmente la
aleatoriedad se relaciona con la probabilidad, aunque un objeto aleatorio
se definirá en forma diferente, dependiendo de la concepción
subyacente de probabilidad. Si se defiende la asignación clásica de
probabilidad, un suceso elemental sería aleatorio, si su probabilidad es
la misma que la de cualquier otro suceso del mismo experimento
(Lahanier- Reuter, 1999). Aunque esta definición es suficiente para los
juegos de azar basados en dados, monedas, cartas, extracción de bolas
225
en urnas, etc., Kyburg (1974) indica que impone condiciones excesivas
y por ello es difícil de aplicar. Sólo podríamos decir que un suceso es
aleatorio, si el espacio muestral es finito. Si fuese infinito, la
probabilidad de cada suceso es siempre nula.
Cuando desplazamos la aplicación de la probabilidad a situaciones del
mundo físico o natural, por ejemplo, al tratar de prever el color de ojos
de un recién nacido, no siempre podemos aplicar el principio de
equiprobabilidad. Podríamos considerar en estos casos que un suceso es
aleatorio si la frecuencia relativa de ocurrencia se estabiliza a la larga,
usando la concepción frecuencial de probabilidad. Tendríamos, sin
embargo, el problema teórico de decidir el número necesario de
experimentos para considerar que, a partir de este número, habríamos
probado suficientemente el carácter aleatorio del suceso (Batanero,
Henry y Parzysz, 2005).
En estas dos acepciones la aleatoriedad es una propiedad "objetiva" de
cada elemento de una clase. Kyburg (1974) critica esta visión y propone
una interpretación de la aleatorie-dad compuesta de cuatro términos, que
son los siguientes:
• Un objeto que es miembro de un conjunto o colectivo;
• El conjunto del cual el objeto es un miembro (población o
colectivo);
• La propiedad con respecto a la cual queremos estudiar la
aleatoriedad del objeto;
• El conocimiento de la persona que emite el juicio de aleatoriedad.
En esta interpretación el mismo objeto puede ser o no considerado
como aleatorio, dependiendo de la persona; por tanto la aleatoriedad
tiene un carácter subjetivo, en consonancia con la concepción subjetiva
de la probabilidad, adecuada en las situaciones en que poseemos cierta
información que puede cambiar nuestro juicio sobre la probabilidad de
un suceso (Fine, 1973).
Formalización de la aleatoriedad.
A finales del siglo XIX, los desarrollos teóricos de inferencia estadística
y la publicación de tablas de números pseudo-aleatorios llevan a la
distinción entre un proceso aleatorio y una secuencia de resultados
aleatorios (Zabell, 1992). Aunque la aleatoriedad es una propiedad de un
226
proceso, solo se puede valorar si el proceso es aleatorio o no mediante la
observación de sus resultados (Johnston-Wilder y Pratt, 2007). Esta
discusión llevó a la formalización del concepto de aleatoriedad (Fine,
1973).
La propuesta de von Mises (1928/1952) se basó en considerar
aleatorio un proceso si es imposible encontrar un algoritmo que nos
permita predecir sus resultados. En la práctica, se considera aleatorio un
proceso si una secuencia de resultados del mismo ha pasado las pruebas
estadísticas suficientes (que tratan de probar el carácter no aleatorio del
proceso). Sin embargo, como en toda prueba estadística hay posibilidad
de error, nunca podemos estar totalmente seguros de la aleatoriedad de
una secuencia finita de resultados, sino solo tomamos una decisión con
respecto a su aleatoriedad con referencia a los resultados de las pruebas
realizadas. Esto explica por qué una secuencia aleatoria generada por
ordenador (que es producida mediante un algoritmo determinista) puede
ser considerada aleatoria si pasa las pruebas necesarias (Harten y
Steinbring, 1983).
Kolmogorov definió la aleatoriedad de una secuencia en base a su
complejidad computacional (Zabell, 1992). En este enfoque, una
secuencia debería ser aleatoria si no puede ser codificada en una forma
más simple (usando menos caracteres) y la ausencia de patrones es su
característica esencial. El número mínimo de signos necesario para
codificar una secuencia particular da una escala para medir su
complejidad, por tanto esta definición permite una jerarquía en los
grados de aleatoriedad para diferentes secuencias. Es importante resaltar
que tampoco en este enfoque existe la aleatoriedad perfecta, que es, por
tanto, sólo un concepto teórico.
Percepción Subjetiva de la Aleatoriedad
La investigación sobre percepción de la aleatoriedad ha sido muy
abundante, tanto con niños como con sujetos adultos. Falk y Konold
(1997) clasificaron las tareas propuestas en estas investigaciones en dos
grandes grupos: (a) Tareas de generación, en las que se pide al sujeto
generar secuencias que simulen una serie de resultados de un proceso
aleatorio típico, como el lanzamiento de una moneda; y (b) tareas de
reconocimiento, donde el sujeto debe elegir entre varias secuencias,
indicando cuál considera aleatoria.
227
Una de las principales conclusiones de estos estudios es que incluso
los adultos tienen dificultades para producir o percibir aleatoriedad
(Falk, 1981; Falk y Konold, 1997; Nickerson, 2002); encontrándose
sesgos sistemáticos en sus razonamientos. Por ejemplo, algunos adultos
muestran la falacia del jugador, o creencia que la probabilidad de un
suceso decrece cuando el suceso ha ocurrido recientemente, sin
reconocer la independencia de los ensayos repetidos (Tversky y
Kahneman, 1982). Estos sujetos tienden a rechazar secuencias con
rachas largas del mismo resultado en tareas de percepción, y consideran
aleatorias las secuencias con un exceso de cambios entre los diferentes
resultados (Falk, 1981; Falk y Konold, 1997).
Estos sesgos se han encontrado también en niños, a pesar de que
Piaget e Inhelder (1951), pensaron que al alcanzarse la adolescencia, se
llega a comprender la convergencia, es decir, la regularidad global y la
variabilidad local de una secuencia de resultados aleatorios de un mismo
proceso. Sin embargo, los resultados de Green (1983) en una amplia
muestra de chicos de entre 11 y 16 años contradicen esta teoría e indican
que el reconocimiento de la aleatoriedad no mejora con la edad ni en las
tareas de generación ni en las de reconocimiento de secuencias
aleatorias. El autor indica que los chicos comprenden la
equiprobabilidad de resultados en experimentos tales como lanzar una
moneda, pero no la independencia de ensayos. Basan su reconocimiento
de secuencias aleatorias en la búsqueda de patrones en los resultados,
número de rachas del mismo resultado y frecuencias de resultados, que
no siempre se asociaron en forma correcta a aleatoriedad o
determinismo. Estos resultados fueron replicados por Cañizares (1997)
con niños españoles.
Batanero y Serrano (1999) analizaron las respuestas de 277
estudiantes de secundaria (14 y 17 años) a algunos ítems sobre
percepción de aleatoriedad en secuencias aleatorias lineales y
bidimensionales sugiriendo que algunos estudiantes presentan
concepciones sobre la aleatoriedad equivalentes a algunas de las
concepciones históricas descritas en el apartado 2.1. El objetivo del
presente trabajo es analizar si dichas concepciones también se presentan
en futuros profesores, con la finalidad de tenerlas en cuenta en la
organización de su formación en probabilidad.
228
Comprensión de la Aleatoriedad por Futuros Profesores
Pocas investigaciones están relacionadas con la comprensión de la
aleatoriedad por parte de los futuros profesores, y las que existen
indican que esta comprensión es pobre. Así, Begg y Edwards (1999) en
un trabajo con 22 profesores en ejercicio, encontraron que la tercera
parte tenía dificultad con la idea de suceso equiprobable y muy pocos
comprendieron el concepto de independencia. Batanero, Cañizares y
Godino (2005) identificaron tres sesgos en el razonamiento
probabilístico en una muestra de 132 profesores en formación de
Educación Primaria: la heurística de la representatividad o confianza
excesiva en las pequeñas muestras (Tversky y Kahneman, 1982), el
sesgo de equiprobabilidad o creencia que todos los sucesos aleatorios
son equiprobables (Lecoutre, 1992) y el enfoque en el resultado o
dificultad de interpretar una pregunta de probabilidad en términos
probabilísticos (Konold, 1991).
Azcárate, Cardeñoso y Porlán (1998) analizaron las respuestas de 57
profesores de Educación Primaria a un cuestionario en que se describe
verbalmente varios sucesos y se pregunta si se consideran aleatorios. En
general, los participantes mostraron una concepción incompleta de la
aleatoriedad, que se refleja, en la mayoría de casos, en argumentos
causales y falta de reconocimiento de situaciones aleatorias cotidianas
(más allá de juegos de azar). Muchos participantes consideraron
deterministas fenómenos aleatorios, si se pueden identificar causas que
lo influyen (por ejemplo en meteorología). Entre las propiedades
correctamente percibidas se encuentran la existencia de multiplicidad de
posibilidades y la impredecibilidad de los resultados.
Chernoff (2009) analizó las respuestas dadas por 239 futuros
profesores de matemáticas (163 de primaria y 76 de secundaria) a tareas
de reconocimiento de secuencias formadas por 5 repeticiones del
lanzamiento de una moneda, todas ellas con la misma proporción de
caras. El análisis cualitativo de las justificaciones de 19 sujetos, que
aparentemente tenían una percepción incorrecta de aleatoriedad, le lleva
a concluir que dichos futuros profesores podrían razonar desde tres
interpretaciones de espacio muestral: (a) teniendo en cuenta los cambios
de cara a cruz; (b) considerando la longitud de la racha más larga, y (c)
considerando los cambios y la racha más larga conjuntamente. También
229
concluye que sus razonamientos aparentemente incorrectos con respecto
a aleatoriedad podrían ser consistentes con dichas visiones de espacio
muestral y no serían debidas a falta de razonamiento probabilístico, sino
al uso de probabilidades subjetivas personales.
A continuación, presentamos nuestra investigación, cuyo objetivo es
complementar las anteriores, analizando en profundidad las
concepciones de aleatoriedad de los futuros profesores y poniéndolas en
relación con las observadas a lo largo de la historia.
Método
La muestra estuvo formada por 157 futuros profesores de Educación
Primaria, de la Universidad de Granada, de los cuáles el 58% eran
mujeres. Los datos se tomaron como parte de una actividad práctica en
la asignatura “Enseñanza y aprendizaje de las matemáticas en la
Educación Primaria”, de contenido didáctico. Posteriormente a la
recogida de datos, se discutieron las respuestas con los futuros
profesores y se realizaron actividades de simulación para ayudarles a
reconocer sus intuiciones incorrectas.
Estos estudiantes habían estudiado probabilidad durante la educación
secundaria, así como en la asignatura “Bases matemáticas para la
Educación Primaria”, del curso anterior, donde estudiaron los conceptos
de aleatoriedad, probabilidad, asignación de probabilidades mediante
regla de Laplace, estimación frecuencial de la probabilidad, y realizaron
ejercicios sencillos de probabilidad simple y compuesta.
La tarea propuesta se presenta en la Figura 1 y se tomó del
cuestionario de Green (1983), habiendo sido también utilizada por
Cañizares (1997). Se optó por elegir esta tarea, por disponer de
respuestas de niños, que posteriormente podrían usarse para discutir con
los futuros profesores las semejanzas o diferencias de sus concepciones
con las de sus futuros alumnos. Además es una tarea semejante a otras
utilizadas en las investigaciones sobre percepción de la aleatoriedad en
sujetos adultos.
230
Tarea. El profesor pidió a Clara y a Luisa que lanzaran cada una de ellas
una moneda 150 veces, y que apuntaran cada vez si salía cara ó cruz. Por
cada "cara" se ha apuntado un 1, y por cada "cruz" un 0. Aquí están los dos
grupos de resultados:
Clara: 01011001100101011011010001110001101101010110010001
01010011100110101100101100101100100101110110011011
01010010110010101100010011010110011101110101100011
Luisa: 10011101111010011100100111001000111011111101010101
11100000010001010010000010001100010100000000011001
00000001111100001101010010010011111101001100011000
Una de las chicas lanzó la moneda como dijo el profesor, anotando los
resultados; pero la otra hizo trampas; no lanzó la moneda, sino que inventó
los resultados
a. ¿Qué niña ha hecho trampas?
b. ¿Por qué crees que ha sido ella?
Figura 1 . Tarea propuesta
De acuerdo a Batanero (2011), una de las estrategias que pueden seguir
los futuros profesores para resolver la tarea propuesta, es contar el
número de caras de cada una de las secuencias y comparar con el
número esperado en 150 lanzamientos de una moneda equilibrada, que
sigue una distribución binomial B(150, 0. 5) , de modo que el número
esperado de caras sería 75, con desviación típica 6.12.
Al comparar este valor teórico con el número de caras en las
secuencias de Clara y Luisa (Tabla 1), se observa que no hay
coincidencia en ninguno de los dos casos; sin embargo, en un proceso
aleatorio, habría que esperar algo de variación. Una forma de evaluar si
la diferencia entre el valor observado y esperado del número de caras en
cada caso se ajusta a la variabilidad propia de un fenómeno aleatorio
sería realizar un contraste Chi-cuadrado de bondad de ajuste.
Tabla 1
Frecuencias observadas y teóricas de caras en la tarea propuesta
Clara
Luisa
Teórica
Cara
72
67
75
Cruz
78
83
75
231
Si denotamos las frecuencias observadas como (oi) y las esperadas (ei)
para las k posibles respuestas de la variable, el valor de este
estadístico sería
, que sigue una distribución Chicuadrado con k-1 grados de libertad bajo la hipótesis de que los
datosprovienen de la distribución teórica. La aplicación de este contraste
a los datos de la Tabla 1, produce resultados que no son estadísticamente
significativos; en la secuencia de Clara χ2obs=0. 24, p=0. 6 y en la
secuencia de Luisa χ2obs=1. 71, p=0. 19.
Repitiendo el mismo procedimiento, pero analizando la secuencia
por pares (es decir, como lanzamientos sucesivos de dos monedas)
obtenemos los resultados de la Tabla 2. En este caso, al repetir el
contraste Chi-cuadrado, para la secuencia de Clara se obtiene
χ2obs=9. 84, p=0. 02 (en una distribución Chi-cuadrado con 3 g.l.), y para
la de Luisa χ2obs=4. 89, p=0. 18. En consecuencia, puesto que el
resultado de Clara es estadísticamente significativo, rechazamos la
hipótesis de que su secuencia es aleatoria, con un nivel de significación
de 0, 02. La diferencia sería todavía más evidente si se analizan los
datos como lanzamientos sucesivos de tres monedas, en cuyo caso, la
realización del contraste Chi-cuadrado para las dos distribuciones de
tripletas genera los siguientes resultados: en la secuencia de Clara χ2obs=
27. 8, p=0. 0001 (en una distribución Chi-cuadrado con 7 g.l.) y en la
secuencia de Luisa χ2obs=6=6. 33, p=0. 501 .
Tabla 2
Frecuencias observadas y teóricos de parejas de resultados en la tarea
propuesta
Clara
Luisa
Teórica
CC
12
25
19
C+
30
21
19
+C
18
12
19
++
15
17
19
Aunque los futuros profesores no tienen los conocimientos suficientes
para aplicar el contraste Chi- cuadrado, podrían contar la frecuencia de
caras y cruces (Tabla 1) en las dos secuencias y argumentar su respuesta
232
en base a la diferencia con el valor esperado, al igual que hicieron los
niños de la investigaciones de Green (1983) o Cañizares (1997),
respuesta que consideraríamos correcta, para los conocimientos que
ellos tienen. Otros participantes podrían basar sus respuestas en la
longitud de la racha más larga, que es de sólo 3 caracteres en el caso de
Clara y de 9 en el caso de Luisa. Schilling (1990) muestra que el valor
esperado de la longitud de la racha más larga en n repeticiones de un
experimento, donde el suceso de interés tiene probabilidad 0.5, se
aproxima al log2 n – 2/3 ; en este caso log2 (50) – 2/3=6. 56, por lo que la
longitud esperada para la racha más larga se aproxima a 7, de manera
que el resultado de Luisa se acerca más al valor esperado que el de
Clara. A pesar de ello, Green (1983) indica que algunos niños eligen
precisamente como aleatoria la sucesión de Clara, porque esperan
rachas cortas.
Resultados y Discusión
Recogidas las respuestas se realizó un análisis de su contenido,
estudiando separadamente las respuestas a las partes a y b de la tarea,
cuyos resultados se presentan y discuten a continuación.
Identificación de Secuencias Aleatorias
En primer lugar se obtuvo la frecuencia de futuros profesores que
consideran que Clara o Luisa hace trampas (Tabla 3). Observamos que
pocos de ellos muestran una intuición correcta, pues la gran mayoría
indica que Luisa fue quien hizo trampas.
Tabla 3
Frecuencia y porcentaje de respuestas a la pregunta a (niña que hace trampas)
Respuesta
Clara (Correcta)
Luisa
No sabe
Ninguna de las dos
No responde
Frecuencia
42
89
17
1
8
Porcentaje
26.8
56.7
10.8
0.6
5.1
233
Los resultados son incluso peores que los observados en estudios
anteriores, pues 34% de niños ingleses entre 11 y 16 años en el estudio
de Green (1983) así como 29% de niños españoles entre 10 y 14 años en
el de Cañizares (1997) indicaron que Clara hizo trampas. Las
intuiciones en este tipo de tarea parecen ser más acertadas cuando las
secuencias son más cortas, pues en el estudio de de Batanero y Serrano
(1999) con estudiantes españoles de 14 y 17 años y utilizando cuatro
secuencias de 40 lanzamientos de una moneda, dos aleatorias y dos no,
las secuencias aleatorias fueron correctamente identificadas por 54% y
59% de los estudiantes, y las no aleatorias por 40% y 64%.
Argumentos
Para profundizar el análisis, se clasificaron los argumentos de los
futuros profesores en la parte b de la tarea, en dos fases. En una primera
clasificación, se diferenciaron los argumentos que hacen referencia a la
frecuencia de caras, la longitud de las rachas, la identificación de un
patrón en la secuencia o la impredecibilidad. Seguidamente, cada una de
estas categorías se subdividió en la forma que se indica a continuación.
Respuestas basadas en la frecuencia de caras.
Algunos futuros profesores realizaron un recuento de las frecuencias de
caras en las dos secuencias, y las compararon con la frecuencia esperada
en una distribución binomial (75 caras). Estas respuestas reflejan, de
acuerdo a Serrano (1996), una concepción de la aleatoriedad consistente
con la visión frecuencial de la probabilidad, pues se espera que la
frecuencia relativa de caras se aproxime a la probabilidad teórica. Por
un lado, los sujetos que dan este argumento manifiestan la idea de
convergencia; por otro lado, han realizado un proceso de inferencia
informal (Batanero, 2011), en cuanto han usado un modelo matemático
(número esperado de caras) comparando con sus datos para rechazar o
aceptar la hipótesis de aleatoriedad de cada secuencia. En algunos casos,
esta concepción correcta se mezcla con alguna incorrecta, por ejemplo,
estimando a la baja la variabilidad de los resultados en un experimento
aleatorio. Los argumentos relacionados con las frecuencias son de dos
tipos.
234
A1. Frecuencias muy alejadas del valor teórico. Se comparan las
frecuencias observadas y esperadas, indicando que hay demasiada
diferencia entre ellas. Si el participante indica que Luisa hace trampas,
usando este argumento para indicar que su secuencia no es aleatoria,
muestra una concepción incorrecta de la aleatoriedad, pues no percibe
suficientemente la variabilidad inherente a una secuencia aleatoria. Un
ejemplo, en el que, sin embargo, el estudiante muestra concepciones
adecuadas de la equiprobabilidad de resultados, convergencia y valor
esperado, se reproduce a continuación:
Luisa hizo trampas porque la probabilidad al lanzar una moneda de
que salga cara o cruz es del 50%. Por tanto en 150 lanzamientos
estimaríamos los valores más cercanos a la media (75) y en este
caso es 78 el valor más cercano (Participante 39).
Si el participante, por el contrario, considera que es Clara quien hace
trampas, usando el argumento A1 para aceptar su secuencia como
aleatoria, ha sido capaz de reconocer la variabilidad inherente a un
proceso aleatorio, que es una capacidad constituyente del razonamiento
estadístico, de acuerdo a Wild y Pfannkuch (1999), como en el siguiente
ejemplo:
Clara hizo trampas porque la probabilidad es inexacta y da
resultados posibles, siendo más creíble el resultado de Luisa que el
de Clara (Participante 71).
A2. Frecuencias muy próximas al valor teórico. Otros participantes, una
vez realizado el recuento de frecuencias, indican que son cercanas al
valor teórico. Si el argumento se refiere a Clara, como el siguiente
ejemplo, se reconoce, como en el caso anterior la variabilidad aleatoria:
Clara hizo trampas porque le da casi un 50% de probabilidad de
caras y cruces y es muy difícil que en estos casos salga un 50%. Es
más lógico el resultado de Luisa (Participante 78).
En otros casos, se usa el argumento A2 para rechazar la secuencia de
Luisa, esperando mayor proximidad, incluso coincidencia con el valor
235
teórico, indicando una concepción incorrecta de la aleatoriedad, como
en el siguiente ejemplo; no obstante, esta concepción de la aleatoriedad
es próxima a la relacionada con la visión clásica de la probabilidad:
Luisa hizo trampas porque si los niños lanzan una moneda 150
veces y solo hay dos posibilidades, tienen que obtener cada lado
más o menos 75 veces. Solo Clara tiene este resultado pero Luisa
no (Participante 148).
Argumentos basados en la longitud de las rachas.
Otros futuros profesores analizaron la longitud de las rachas,
obteniéndose dos argumentos diferenciados, basados en dicha longitud:
A3: Rachas largas. En general, se observa la existencia de rachas
largas, como argumento para rechazar la secuencia como aleatoria,
razonamiento que también apareció en el trabajo de Serrano (1996),
quien sugiere que indica una comprensión incorrecta de la
independencia de los ensayos repetidos.
Luisa hace trampas por la combinación de 3 o más veces el mismo
resultado. Clara sólo llega a 3 repeticiones. Luisa tiene más series
repetidas, alguna de 9 repeticiones. Esto es muy improbable
(Participante 123).
Un sujeto observa la falta de rachas largas, como argumento para
rechazar la secuencia de Clara como aleatoria, mostrando una buena
percepción de la independencia de ensayos:
Clara hace trampas porque aparece de forma más aleatoria,
alternando los "0" y los "1"; en cambio lo de Luisa parece más
real, ya que hay más continuidad de resultados muchos "0" y "1"
seguidos (Participante 65).
A4: Rachas cortas. Algunos futuros profesores sugieren que las rachas
de una de las dos secuencias son demasiado cortas para un proceso
aleatorio, usándolo para rechazar la secuencia de Clara como aleatoria,
lo que indica una buena percepción de la independencia de ensayos
como vemos en el siguiente ejemplo:
236
Clara hace trampas porque en su grupo de resultados no hay más
de tres resultados iguales seguidos, y puede haber más de tres
resultados iguales seguidos porque hay la misma probabilidad de
que salga una cruz o una cara (Participante 27).
Argumentos basados en la existencia de un patrón.
La existencia o no de un patrón en la secuencia sirve a algunos
participantes para justificar quien hace trampas. También hemos
diferenciado dos tipos de argumento:
A5: Existe un patrón en la secuencia. El argumento hace referencia al
orden en que van apareciendo las caras y cruces en la secuencia y al
hecho de que parezca muy regular para ser o no aleatoria. Para algunos
la alternancia de los dos valores debe darse en experimentos con
resultados equiprobables (Luisa haría trampas). Este razonamiento
indica un enfoque en el resultado (Konold, 1989) y muestra una pobre
comprensión del significado frecuencial. Una respuesta en esta
categoría es:
Luisa hace trampas porque sus resultados se repiten mucho durante
todas las veces, es decir, por ejemplo "cruz" le sale muchas veces,
creo que hay más probabilidad que salga también cara, que se
igualen tanto cara como cruz (Participante 54).
Para otros participantes, la regularidad en el patrón de alternancias es un
indicativo de falta de aleatoriedad (Clara haría trampas), lo que indica
una concepción correcta de ausencia de patrón en las secuencias
aleatorias. Dichos participantes asociarían la aleatoriedad con ausencia
de modelo o patrón, una visión próxima a la modelización de la
aleatoriedad de von Mises (1952/1928) para quien una secuencia es
aleatoria si es imposible encontrar en ella patrones predecibles. A pesar
de que estos futuros profesores tienen una idea parcialmente correcta, de
hecho en la secuencia aleatoria se pueden identificar una multitud de
modelos; por ejemplo, la distribución Binomial o geométrica, por lo que
la aleatoriedad podría interpretarse igualmente como multiplicidad de
modelos (Serrano, 1996). Un ejemplo de esta categoría es:
237
Clara hizo trampas porque los resultados obtenidos parecen ser una
serie que se repite, ya que, aunque no se siga a la perfección es
muy parecida en sus porcentajes (Participante 56).
A6: La secuencia no sigue un patrón. Un participante usa el argumento
contrario al anterior, en este caso, rechazando la aleatoriedad, lo que
indicaría una concepción incorrecta.
Luisa hizo trampas porque apenas se intercalan valores de distinto
valor, es decir, que si tenemos una probabilidad del 50% es más
posible que tanto cara como cruz se intercalen de forma mas
sucesiva (Participante 29).
Otros tipos de argumentos.
A7: Impredecibilidad. Una característica común en diferentes
concepciones de aleatoriedad es la impredecibilidad: no poder predecir
un suceso futuro basado en un resultado del pasado (Bennet, 1998). La
comprensión del carácter impredecible de un resultado particular en un
proceso aleatorio es fundamental en la comprensión de la aleatoriedad,
pero también la de la posibilidad de predicción del conjunto de
resultados (variabilidad local y regularidad global). Sin embargo,
algunos participantes confunden el resultado impredecible y la
posibilidad de predecir las frecuencias de los diferentes resultados en
una serie de ensayos. Serrano (1996) menciona una posible relación de
este tipo de argumentos con el enfoque en el resultado, un sesgo
consistente en interpretar un enunciado de probabilidad en forma no
probabilística (Konold, 1989). Una respuesta que ilustra esta categoría
es:
No sabe quien hizo trampas porque al igual que una niña lanzó la
moneda y de forma aleatoria se obtienen los resultados, con la
chica que se los inventó ocurre lo mismo, no puede comprobarse
porque al lanzar una moneda es un caso aleatorio que no se puede
comprobar (Participante 5).
A8: Otros argumentos. Algunas justificaciones, con menores
frecuencias que las anteriores, se refieren a la equiprobabilidad de
resultados y por tanto mostrarían, según Batanero y Serrano (1999) una
238
concepción de aleatoriedad ligada al enfoque clásico de la probabilidad:
Ninguna hizo trampas porque la probabilidad de que salga
P=1/2=0.5 cara o cruz es la misma si tiras una moneda como si no
la tiras (Participante 105).
Otros argumentos expresan creencias personales poco justificadas:
Clara hizo trampas porque es más probable que una moneda caiga
por el mismo lado un mayor número de veces (Participante 7).
Finalmente, se producen algunas respuestas confusas en las que es
difícil seguir el razonamiento del futuro profesor, aunque son minoría:
Cualquiera de las dos pudo engañar, porque la probabilidad indica
sólo probabilidades, no números exactos (Participante 70).
En la Tabla 4 se cruza la respuesta a la parte a, sobre qué niña hizo
trampas, con el argumento que apoya dicha respuesta, en la parte b. Se
observa que los argumentos para indicar que Clara hace trampas se
relacionan principalmente con la existencia de un patrón (50%) o bien
con las rachas demasiado cortas (28.6%) e indicarían en los dos casos
concepciones correctas de la aleatoriedad.
Los participantes que indican que Luisa hace trampas se basan
primordialmente en el tamaño de su racha más larga (58%), por lo que
muestra una comprensión incorrecta de la independencia de ensayos
sucesivos. Otro porcentaje apreciable espera que las frecuencias
observadas debieran ser más próximas a las esperada (28.6%), lo que
sugiere falta de percepción de la variabilidad inherente a la aleatoriedad.
En general, observamos que lo más frecuente fue analizar la longitud de
las rachas, seguido por argumentar la existencia de un patrón y luego
por la proximidad de las frecuencias observadas con las esperadas. En
total el 59% de los futuros profesores da argumentos erróneos para
apoyar que Luisa hace trampas, el 27% argumentos correctos para
apoyar que es Clara la que hace trampas y el resto no es capaz de
detectar qué secuencia es no aleatoria o no da un argumento consistente.
239
Tabla 4
Frecuencias y porcentajes de argumentos en la pregunta b (n*=148)
Argumento
Niña que hace trampas
No sabe/
Luisa
Total
Clara
ninguna
Frec. % Frec. % Frec. % Frec. %
A1. Frecuencias muy diferentes 1
A2. Frecuencias muy próximas 3
1
A3. Rachas largas
12
A4. Rachas cortas
A5. Existencia de un patrón 21
A6. No existe patrón
A7. Impredecibilidad
4
A8. Otros argumentos
2.4 5 5.7
7.1 19 21.6
2.4 51 58.0
28.6
50.0 7 8.0
1 1.1
9.5
5
5.7
6
22
52
12
28
1
15 83.3 8
3 16.7 19
4.1
14.9
35.1
8.1
18.9
0.7
5.4
12.8
* Total de alumnos que dan un argumento
Este porcentaje es muy próximo al obtenido en investigaciones previas,
ya que 22% de los niños ingleses (Green, 1983) así como 29% de los
niños españoles (Cañizares, 1997) proporcionan argumentos correctos.
Una diferencia es que la ausencia de argumentos fue mayor en los niños
(14% de los ingleses y 30% de los españoles), mientras que esta
ausencia sólo se da en el 5.7% de los participantes en nuestro estudio, lo
que indica una mayor capacidad de argumentación entre los futuros
profesores.
Conclusiones y Sugerencias para la Formación de Profesores
Los resultados confirman los de otros estudios sobre aleatoriedad en
adultos (Falk, 1981; Falk y Konold, 1997; Nickerson, 2002), que
indican nuestra dificultad para percibir aleatoriedad. Al igual que en
estos estudios, se observan sesgos como la falacia del jugador o el
enfoque en el resultado, así como concepciones erróneas acerca de la
equiprobabilidad o la falta de la comprensión de la independencia.
240
Estos resultados no son sorprendentes, puesto que Bar-Hillel y
Wagenaar (1991) subrayan la dificultad del concepto de aleatoriedad,
que se resiste a una definición sencilla y que sólo puede aplicarse a
través del análisis de las secuencias de resultados. Por otro lado,
aunque expresiones como “número aleatorio”, “experimento aleatorio”
aparecen con frecuencia, tanto en el lenguaje cotidiano como en los
libros de texto, en dichos libros no se suele incluir una definición
precisa del concepto (Batanero, Green y Serrano, 1998).
Sin embargo, la comprensión de la aleatoriedad es esencial para el
aprendizaje de la probabilidad, por lo que los futuros profesores
debieran adquirir una comprensión profunda que les permita adquirir
una competencia suficiente en su futura enseñanza de la probabilidad,
como se recomienda en los nuevos currículos.
Como apunta Fernández (1990), la función principal del proceso de
diagnóstico pedagógico es la toma de decisiones sobre los cambios que
requiere el modelo de enseñanza para ayudar al alumno en su
adquisición de habilidades y competencias. Nuestra investigación no
solo sugiere la necesidad de reforzar la formación sobre probabilidad
en los futuros profesores, sino también un cambio en la aproximación
de este aprendizaje haciendo más hincapié en aquellos razonamientos
sesgados que están presentes en los futuros profesores. Una enseñanza
basada en el uso de la simulación, y la reflexión en pequeños grupos
sobre estas dificultades podrían ayudar a superar estos sesgos.
Los futuros profesores en nuestro estudio mostraron una mezcla de
intuiciones y creencias correctas e incorrectas respecto a la
aleatoriedad. Será labor del formador de profesores ayudarles a
construir una concepción más completa, partiendo de la parte correcta
de las intuiciones descritas en este estudio. Ello es particularmente
importante, debido a la dependencia, señalada por Ball, Lubienski y
Mewborn (2001), de las tareas habituales del profesor, como
evaluación de los estudiantes, u organización de la enseñanza, de su
conocimiento matemático.
Por otro lado, algunas de las respuestas de los futuros profesores a la
segunda parte de la tarea indican concepciones próximas a las
aceptadas en diferentes periodos históricos sobre la aleatoriedad. Será
importante, entonces, que el formador de profesores aproveche estas
concepciones parcialmente correctas para hacerlas progresar:
241
• La visión de la aleatoriedad como equiprobabilidad, debe hacerse
progresar pues tiene una aplicación muy restringida;
• La visión frecuencial, donde se espera una convergencia entre las
frecuencias esperadas y las observadas, ha de completarse, haciendo a
los futuros profesores conscientes de la variabilidad y la independencia
de ensayos sucesivos;
• El reconocimiento de la imposibilidad de predicción de resultados
aislados, debe también ampliarse aceptando la posibilidad de
predicción de la distribución de frecuencias de los diferentes sucesos
implicados;
• Por último, la visión de aleatoriedad como falta de modelo ha de
abandonarse a favor del reconocimiento de la multiplicidad de
modelos subyacentes en una secuencia de resultados aleatorios.
En este sentido, la tarea presentada y la discusión con los futuros
profesores de las posibles respuestas correctas e incorrectas a la
misma, puede servir para incrementar su conocimiento matemático y
didáctico sobre la aleatoriedad, ampliando la comprensión de las
propiedades de este concepto, así como de los posibles sesgos de
razonamiento relacionados con el mismo, que podrían presentarse en
sus futuros alumnos.
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245
Carmen Batanero es Catedrática de Didáctica de la Matemática en
la Facultad de Educación de la Universidad de Granada, España.
Emilse Gómez es Profesora de Estadística de la Universidad
Nacional de Colombia.
Luis Serrano es Catedrático de Escuela Universitaria de Didáctica
de la Matemática, en la Facultad de Humanidades y Educación de
Melilla. Universidad de Granada, España.
José Miguel Contreras es Profesor Ayudante Doctor de Didáctica
de la Matemática en la Facultad de Educación de la Universidad de
Granada, España.
Dirección de contacto: La correspondencia sobre este artículo
debe dirigirse a: Carmen Batanero, Departamento de Didáctica de
la Matemática, Facultad de Ciencias de la Educación, Universidad
de Granada, Campus de Cartuja 18071 Granada (España).
Dirección de correo electrónico: [email protected]
'Being Good' at Maths: Fabricating Gender Subjectivity
Anna Chronaki 1 and Yannis Pechtelidis 2
1 ) University of Thessaly, Volos, Greece.
2) University of Thessaly, Volos, Greece.
To cite this article: Chronaki, A. & Pechtelidis, Y. (201 2). 'Being Good' at
Maths: Fabricating Gender Subjectivity. Journal of Research in
Mathematics Education, 1 (3), 246-277. doi: http://dx.doi.org/1 0.4471 /
redimat.201 2.1 4
System and to Creative Commons Non-Commercial and NonDerivative License.
October 2012 pp. 246-277.
'Being Good' at Maths:
Fabricating Gender Subjectivity
Anna Chronaki
University ofThessaly
Yannis Pectelidis
University ofThessaly
Abstract
Current research in mathematics education places emphasis on the analysis of
men and women’s accounts about their life trajectories and choices for
studying, working and developing a career that involves the learning and
teaching of mathematics. Within this realm, the present study aims to highlight
how mathematics, gender and subjectivity become interwoven by focusing the
analysis on a single case study, that of Irene –a teacher in her early 40s. Based
on how she articulates hegemonic discourses and narrates her relation to
mathematics from the time she was a schoolgirl up till her recent work as
teacher and her endeavours as participant in a professional development teacher
training course, we argue how ‘mathematics’ becomes a mythical object for her
subjectification. Irene as a female subject appropriates through her narrative the
socially, culturally and historically constructed ideals about maths and gender
and essentialises mathematical ability. Our study reveals how dominant
discourses concerning ‘mathematics’ and ‘gender’ relate closely to subjectivity
fabrication.
Keywords: mathematics, gender, subjectivity .
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.14
October 2012 pp. 246-277.
'Ser bueno/a' en Matemáticas:
Fabricando la subjetividad de
género
Anna Chronaki
Universidad de Tesalónica
Yannis Pectelidis
Universidad de Tesalónica
Resumen
La investigación actual en educación matemática pone énfasis en el análisis de
las historias de hombres y mujeres sobre sus trayectorias y elecciones para
estudiar, trabajar y desarrollar una carrera que implica el aprendizaje y la
enseñanza de las matemáticas. En este ámbito, el presente estudio pretende
destacar cómo las matemáticas, el género y la subjetividad se interrelacionan
centrando en análisis en un estudio de caso individual, el de Irene -una maestra
que tiene poco más de cuarenta años. En base a cómo articula los discursos
hegemónicos y cómo narra su relación con las matemáticas desde que era una
joven alumna hasta su reciente trabajo como maestra, y sus practicas como
participante en un curso de desarrollo profesional de formación del
profesorado, comentamos como "las matemáticas" se convierten en un objeto
mítico para su subjetivación. Irene es una mujer que se apropia a través de su
narrativas de los ideales construidos social, cultural e históricamente sobre las
matemáticas y el genero y esencializa las habilidades matemáticas. Nuestro
estudio revela cómo los discursos dominantes relativos a las "matemáticas" y el
"genero" se relacionan estrechamente con la fabricación de la subjetividad.
Palabras Clave: matemáticas, género, subjetividad.
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.14
D
(3) 247
uring the last two decades, we have witnessed serious efforts at
the levels of both academic dialogue and policy making, to
render mathematics accessible to young children and adults. At
the same time, issues of equity in direct relation to men and women and
people from diverse communities and cultural, racial and linguistic
backgrounds have been of high priority to the field of mathematics
education (Rogers and Kaiser, 1995). Specifically, distinctive endeavors
come from varied, but at times interelated disciplinary areas such as
socio-semiotics, anthropology, sociology, psychology, critical theory
and postructural studies. Next to alerting us for a critique of hegemonic
practices, they strive towards theorising and politising alternative
perspectives on what mathematics could be and how people potentially
relate to this field of knowledge. Related theoretical discussions and
events lead to an increased awareness of mathematics as emergent and
construed through multiple sociopolitical contexts and complex
historical trajectories (see Walkerdine, 1988, 1998; Restivo, 1992;
Skovsmose, 1995; Brown, 1997; Walshaw, 2004a).
However, mathematics continues to preserve a mythologised public
image of an alien, extrinsic and inhumane subject. Mathematics is, by
and large, socially represented, as closely connected to pure reason,
absolutism and mysticism, and thus, stereotypic trancendental and
supernatural viewpoints become adhered to what mathematical practices
are (see Restivo, 2009, 1992). Αt the same time, a number of studies
reveal how prevailing discourses about mathematics, mathematicians or
even lay people who use either deliberatively, by chance or routine
mathematics as part of everyday life dealings and work permeat with
stereotypes and very limited understanding of what mathematics is and
how people relate to it (Applebaum, 1995). Such hegemonic discourses
tend to promote and perpetuate images of mathematics as hard labour,
lonely work, cold logic, and the eternal search for precision, abstraction
and absolute truth. Tied to these, prime representations of mathematical
work as correct outcome and drill are connected to the product of a
solitary, and yet, inspired mind whose nature is cast in occulitism and
uncouthness.
A number of trends in feminist research have related such a dominant
perspective on mathematical knowledge to issues of gender. Reconciling
248
Chronaki & Pechtelidis - 'Being Good at Maths': Fabricating Gender
the theories that attempt to account for the gendered subject in
mathematics education practices, one needs to acknowledge the
presence of diverse epistemological and ontological stances. Margaret
Walshaw (1999) distinguishes between the liberal approach where the
dominant discourse evolves around ‘ the woman as a problem in
mathematics’ and the reconstruction approach where ‘ women become
central to mathematics’ and their experiences across cultures, society
and history become honored and evinced. She argues that both
approaches are circumscribed by essentialist views related to subject
identity seen as rational, self-determined and stable. In accordance with
Walkerdine’s (1988, 1998) poststructural perspective, Margaret
Walshaw (1999, 2001) claims that virtues such as stability, universality
and rationality are contested as fictive. In consequence, an overemphasis
on female experiences and ways of knowing as being of a distinct nature
assume a type of commonality amongst all women. As such, women’s
life becomes an idealised singularity –a view that has been challenged
by and large through feminist postmodernist and postcolonial studies.
In the light of the above discussion around the gendered dimension in
mathematics education, these two lines of thought and research (i.e.
woman as problem and woman as central and distinct) need to be seen
as strategic approaches within the modernist regime. They both serve to
promote and perpetuate a binary optic routed in what Judith Butler
(1990) calls a masculinist construction of an essentialised self.
Accepting and remaining idle within this modernist frame of thinking
there is very little chance for developing an alternative inquiry of self
and subjectivity. Escaping the hegemony of essentialist discourses
means moving away from the discursive narratives that assist to produce
them. It is through this frame of thinking that we attempt here to
problematize and deconstruct grant narratives about gender,
mathematics and subjectivity through the case of Irene -a female
primary school teacher in her early 40s. As a first step in this long path,
we aim to map the potential effects of the essentialised meanings
produced as part of her narrative. In short, we problematise her travail to
articulate hegemonic discourses about mathematics and gender as part
of her personal struggle to fabricate subjectivity.
(3) 249
Mathematics and Gender: Articulating Discourses and Subjectivity
As far as a gendered approach to mathematics is concerned, the relative
connectivity amongst mathematics, gender and subjectivity is not a new
concern in the field of mathematics education. Valery Walkerdine (1988,
1998) was amongst the first who worked systematically towards
unravelling the tacit connections among gender, mathematics and
subjectivity. In her seminal book ‘Counting Girls Out: Girls and
Mathematics’ Walkerdine (1998) takes her readers through an
archaeology of knowledge that sketches how gendered hierarchies in the
field of mathematics education have their roots in modernist discourses
about science, childhood and education. She also discusses gender and
its relation to power and discourses of mathematical ability from
nursery, to primary and up to secondary school when adolescent girls
have to make decisions about the further studying of mathematics.
Through her meticulous longitudinal qualitative empirical research
with children, teachers and parents, she argues that there still continues
to be a huge class divide, where ‘... middle-class girls are being allowed
and pushed to achieve academically’ whilst ‘… working-class girls still
facing a huge gulf in terms of the possibilities for attainment anywhere
near matching that of middle-class girls’ (Walkerdine, 1998, p. 169).
Although the gender gap seems to be closing and girls more and more
prove their mathematical abilities at the standards of international
assessment items and examination tests, it becomes evident that, in
Walkerdine’s words, the future is still not ‘female’ in a uniform sense.
She moves on to discuss middle-class girls and boys’ anxiety about high
performance in mathematics –and academic performance in general- as
a matter closely linked to gendered subjectivity. She explains:
This anxiety often related to the conflicts between feminine
sexuality and intellectuality. While on the surface many of these
girls appeared to have a Post-Feminist dream of having one’s cake
and eating it, beneath the surface many suffered from the feeling
that they were never good enough no matter how hard they tried
and that their feminity could never ever be allowed to get in the
way of their success. (Walkerdine, 1998, p.170)
250
Walkerdine has pointed out repeatedly how female subjectivity is often
captured in essesntialist categories dictating a certain and static identity
that is biologically determined and socially situated in universal
patriarchical roles and expectations.
Margaret Walshaw (1999, 2001) follows this line of thought and argues
how the subject of the woman or girl centered research approaches is
often circumscribed by fictitious ideals that tends to romatisize the socalled female ways of knowing around very simplistic notions of
‘experience’ and ‘feeling’. Drawing on the work of Luce Irigaray and
Pati Lather she claims that engaging with the complexity of gender and
mathematics one needs to move beyond the binary logic of a unique or
singular male or female pattern of knowledge. Such analytic tools
become blind to material and discursive constraints that constitute
people as subjects and empower them to perform certain tasks and
narratives. Chronaki (2009), discussing the significance of a number of
studies concerning gender, mathematics and technology in the body of
education, denotes how binary politics of knowledge and essentialist
theorisations serve to perpetuate the old body/mind dichotomy on
several layers of how students, teachers, parents, curriculum material
and mathematics education communities interact and relate to each
other. She, along with others, stresses the importance of moving beyond
dichotomising as a political path for research in the field of mathematics
education and argues further, for the inclusion of a feminist research
optic that espouses a critique of postcolonial theorising. Such a
perspective sheds new light and potentially challenges the ethics and
morals of mathematical knowledge use and production as integral part
of our technoculture in and out of school. Heather Mendick (2005,
2006) has argued how dominant discourses serve to construct
mathematics as an experience disconnected from cultural life, emotion
and self. Based on her studies she claims that most young people reject
the possibility of a ‘mathematical’ world and resist mathematics as an
activity embedded in their imagery as an object of pleasure and joy.
Walkerdine (1988, 1998) has also drawn on the politics related to the
particular fantacy of controlling human life and the world via
mathematics. She argues that, through fiction and imagery, human
subjects position themselves in mathematical practices and construct
(3) 251
subjectivities related to either failure or success. All these studies seem
to agree on how mathematics, gender and subjectivity in the field of
education, and in particular, the mathematics classroom, influence each
other in multiple ways. Their agreement could be summarised along
three main lines: firstly, the prevealing public image of mathematics
itself is of a masculine domain of knowledge. It has been constituted
through modernist discourses of science and has been the product of
sociopolitical struggles through contigent historical localities. For
example, Walkerdine (1998) cites Charles Darwin who in 1896 claims
in his book entitled The Descent ofMan and Selection in Relation to Sex
that:
The chief distinction in the intellectual powers of the two sexes is
shewn by man’s attaining to a higher eminence, in whatever he
takes up, that can woman …if men are capable of a decided preeminence over women in many subjects, the average mental power
in man must be above that of woman. (cited by Walkerdine, 1998,
p.15).
the dominant views of girls and women’s relation to
mathematics have been theorised through discourses that represent them
as marginal and non-passionate users and producers of mathematical
knowledge. Studies in this perspective resort to direct comparisons
between men and women or boys and girls and focus on issues of
mathematical ability, skill and attitudes (Fenema, 1996). Although, an
increased closing of the ‘gender gap’ has been noted, the overtones of
such studies are still with us and are reflected upon the ways both lay
people and scientists think and discuss research outcomes and
possibilities. Very often innate and biological traits are called upon in
order to explain and interpret female ‘passive’ activity or nonparticipation. Thirdly, the espousing of a poststructuralist optic assumes
gender subjectivity as becoming fabricated and weaved discursively in
multiple sociopolitical contexts. It emphasizes the roles played by
hegemonic and marginal discourses as vital for subjectivity all way
through, but also places equal emphasis on subject agency as contigent,
multiple, local, fluid, fragile and emotional (see Weedon, 1987;
Walshaw, 2004b).
Secondly,
252
Concerning the discursive formation of subjectivity, one needs to think
about what discourse is and how it relates to human subjectivity.
Discourse refers to a certain way of structuring and organising areas of
knowledge and social practice. According to Foucault (1989), in modern
western societies the practices in the production of knowledge are
regulated and limited by certain disciplines, inside given institutional,
political and economical “regimes of the production of truth”. Foucault
dealt with the historical procedures of the construction and evolution of
various “discourses”, especially those concerning the humanities.
Specifically, he attempted to bring to the forefront the processes by
which various definitions are embodied and excluded; the principles and
the rules of hierarchal classification that define what may be taken as an
object of thought and what not; how an object of thought is constructed;
if it is legitimate or not to mention it etc. From this standpoint and
pertaining to mathematics, there is no matter of right of wrong, which
doesn’t mean, as Wittgenstein (2009) affirms, that it is necessary to
question that 1+1=2. On the other hand, doubts can be cast on the
conviction that mathematics is a series of truths exposed by
mathematicians (see Lakatos, 1976; Ernest, 1991; and Restivo, 1992).
Discourse refers to the set of rules and significations that specify what
it is possible to speak, do, and think, at a particular time. So, it is more
than a way of an attempt to provide meaning to the world; it has real,
material effect on people’s lives. It implies a particular form of social
organisation and social practices, at different historical times, which
formulates institutions and constitutes subjectivities. Rosalind Gill
(2008) claims that the focus on subjectivity:
“is relatively underexplored, with the exception of a few
groundbreaking and important studies’ and continues stressing that
‘[…] There is very little understanding of how discourses relate to
subjectivity, identity or lived embodied experiences of selfhood.
We know almost nothing about how the social or cultural “gets
inside”, and tranforms and reshapes our relations to ourselves and
others”. (Gill, 2008, p. 433)
Discussing subjectivity in relation to mathematics, one needs to
encounter how the hegemony of such prevailing discourses determines
(3) 253
subject agency on at least two levels, the level of acting locally and the
level of narrating local actions, acquaintances and feelings: As far as the
level of activity is concerned, varied discourses on either mathematics or
gender affect the decision making processes and choices for action. The
neoliberal view of the subject as an autonomous decision maker is
pertinent in the realm of a free choice discourse. In relation to the
narrating level, one needs to take into account how subject agency
(including resistance and change) becomes rationalised through events
of acting and narrating. Individual narratives are inevitably situated in
spatiotemporal localities and reflect one’s own personal attempts to
account on ways of doing things within a social context. Such attempts
heavily encompass the struggle to articulate contigent experiences by
resorting to locally embedded discourses that seem to influence and
mobilise choices, decisions, the need to innovate but also inertia or
resitance to anything new (see Laclau and Mouffe, 1985; Mouffe, 1992;
Blackman & Walkerdine 2001; Walkedine, 1997; and Walkerdine et al.
2001).
Current research related to gendered choices in studying and working
in mathematical related fields (Mendick, 2006; and Walshaw, 2005)
have brought into the fore perspectives that do not locate issues of
‘choosing’ maths merely with an ideal ‘autonomous’ individual but,
instead, refocus our attention on the social, cultural and political
complexities where men and women weave humane lives along with
study and career paths. Autonomous choice and subject agency have
been challenged as core concepts not only towards understanding but
also explaining and pursuing our relation to varied layers of a social
reality where we live as gendered, racial and aged subjectivities as we
strive to become learners and educators. Discourses concerning agency,
autonomy and choice, along with rationalism, active participation or
collaboration are central to a neoliberal agenda of politics. The
publishing of the book ‘Changing the Subject’ in 1984 was amongst the
first systematic and coherent attempts towards articulating a critique of
the ‘autonomous’ and 'self-regulated' subject ideal that mainstream
psychology discourses were producing and promoting (Henriques et al.
1984). It certainly paved the way for more studies to unravel the
multiple relational complexities amongst psychological and sociological
254
analysis and, in fact, created the space for theoretical social studies to
advance. However, the discourse of ‘free choice’ is still mobilised and
becomes the hegemonic theorisation of capturing and interpreting
behaviour, motive and change in local settings. In this realm,
mathematics seems to play a pivotal part as it is heavily connected to
power. The relation between power and mathematics is mainly
explained as symbolic, but as we reveal through our data it is also
heavily rooted and contingent to local attempts to rationalise choice for
action. The present paper starts with an outline of main claims
concerning mathematics and mathematics education as a gendered
phenomenon and aims to discuss –through the analysis of the case of
Irene as a student at school and as an adult in work life- how
mathematics becomes part of a complex performing of subjectivity. As
we shall see, Irene, our interviewee in this research study, articulates a
diffused neoliberal and essentialising discourse in order to deal with the
concealed contradictions produced through her speech, and to fabricate
an ostensible coherence in what she says.
Neoliberalism is a hegemonic discourse, and in this sense it is central
to understanding contemporary social reality or a particular aspect of
reality, such as the relation between mathematics, education and gender.
The notions ‘free’, ‘autonomous’ choice and ‘agency’ are central to this
discourse, which sees the individual as an independent actor who is
rational and solely responsible for his or her life biography (Walkerdine
et al., 2001). This discourse frequently mobilizes the concept of free and
autonomous agent in order to explain and understand behaviour.
However, we will see that these terms offer little understanding of the
complex lived experience of girls and women in relation to mathematics
education in our contemporary society and school communities. We
claim that we need to develop an understanding of subjectivity in ways
that do not complicit individuality solely with ‘inside’ or ‘interiority’
(Gill, 2008). That means we should not abandon the social, cultural,
political constraints upon the subject’s action. On these premises, we
question whether Irene is ultimately free and autonomous in her choice
of mathematics. We do this by considering how Irene deals with the
socially, culturally, historically constructed ideals about mathematics
and gender; how it is that these ideals are internalized or embodied, and
felt not as external constraints or impositions, but as her own.
(3) 255
Methodology: Research Context and Questions
This paper is part of a broader research project concerning the gendered
dimensions of mathematics and technology use at the basic levels of the
Greek educational system1 . A part of the project was the interviewing of
24 male and female teachers aged between 36 and 47, who attended a
biennial academic course aiming to offer in-service training for
teachers in affiliation to a Greek University. The aim was to consider
how they negotiate and construct their subjectivities through their
narratives. Particularly, they were asked to express how they felt about
mathematics and how these related to education and gender; whether
they had positive or negative experiences from their contact with this
field; to state the different teaching styles they had experienced as
students, as well as those they themselves used in class; to describe and
explain their professional and academic choices and their future
aspirations; to state their beliefs about gender. In other words, we asked
them to narrate their lived embodied experiences of mathematics and
education.
Drawing from this project, we focus here on the case study of Irene, a
woman in her early 40s who, even though she was good at maths, did
not manage to study in a related field; she studied to become a librarian,
she worked as a libarian for some years and currently she moved to a
teaching carreer at a primary school. Based on her narratives of a lifestory, we encounter and problematise her relation to mathematics all the
way through –from her early years as a school-girl, her time as
adolescent when crucial decisions about studying were made, her adult
life in paid work as a librarian, the shift towards becoming a teacher and
her present experiences as teacher trainee participant in a professional
development university based course. Our focusing on this particular
interview was not random. Our criteria included the fact that Irene
considered herself very competent at mathematics and on this premise
she differentiates herself from others in the course by idealising her
mathematical ability as innate. Based on our analysis, we suggest that
this idealisation offers us an opportunity to reflect upon whether such a
perception of mathematical ability as an esoteric assemblage of mindtools develops in relation with the acceptance and utilisation of gender
binaries. At the same time, we problematise the entailment and
256
reproduction of the dominant hierarchical gender order all the way
through her narrative (Connell, 1987). In addition, idealization of this
form of knowledge results in favoring mathematics at the expense of
other school subjects.
Such an articulation functions in claiming a specific dominant position
for herself (i.e. good at maths and maths makes me different to others at
school and work) and permits us to argue that Irene performs a certain
form of masculinity (Mendick, 2006; Connell, 1995). In this context, we
explored the limits and ramifications of such a performance (Butler,
1993), not only for herself, but also for the prevalent gender regime. In
other words, we examined what makes it possible for her to claim such
a positioning and whether those masculine embodiments were
connected to essentialist perceptions of gender. So, what we wish to do
in this paper is to develop a practical critique of the limits of self that
takes the form of a possible transgression (Foucault, 1984, pp. 46-54).
What we mean is an analysis of how we constitute ourselves subjects
who think and act in particular ways in order to open up new spaces for
thought and action (Wong, 2007). Foucault (1984, p. 43) describes the
‘ permanent critique’ of self-constitution as a ‘ critical ontology’ of
ourselves. Hence, from the critical ontology’s point of view we examine
ideas and principles, especially about mathematics and gender, that
denote interchangeable ways of organising discourse through alternative
narratives in order to mobilise the potentiallity to think and act
differently.
The Case of Irene: Narrating her Relation to Mathematics
Irene is 42 years old and comes from a rural area in Northern Greece
where her family is involved with farming. At school she was very good
at maths and, indeed, she expressed passion and ability for top grades.
Although she wanted to study architecture (as she was very good at
geometry) she ended up studying and working as a librarian for some
years. She, then, studied pedagogy and followed a teaching career. She
has not got married or have children until now. Recently, she completed
her dissertation for a master’s degree in Pedagogical Studies. Currently,
Irene is satisfied with her academic and professional career, and further
aspires to engage in research in the field of special education; possibly
at the level of a PhD. She claimed that her choice not to follow a maths
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related path was, more or less, random. Although, her first choice, as she
said, was architecture, mostly because of her aptitude in mathematics
and geometry, her drive to leave home was so deep that by the time she
had secured a place in librarian studies she could not think of the extra
effort needed to repeat her exams.
Irene’s case becomes an interesting one for our research as it enables
us to observe and deeper analyse how human subjectivity becomes
fabricated as people struggle to produce meaning through available
discourses in their social and cultural localities. Through her case, we
were able to denote; a) her close relation to mathematics that expands
from childhood (e.g. Irene as a schoolgirl is good at maths) up to the
current time when Irene works as a teacher, b) how mathematics
becomes narrated as part of performing her masculinity on the basis of
an inherited rationality, objectivity, accuracy and mysticism, and c) that
her choices are heavily dependent on contigency and her deep urge to
live. All the way through, Irene essentialises ‘mathematics’ as a trait that
enables her to differentiate from others ignoring how the ‘discourses’
she appropriates, articulates and re-produces, result into trapping her. In
the following sections, we will try to unpack each one of the above
issues and discuss them as part of our analysis.
Being Good at Maths: the Gift of Mathematical Ability
A core part of our discussions with Irene was her past relation to
mathematics in the school curriculum and also her current encounterings
with the subject as part of her training course and teaching practice. We
were eager to understand how she remembers herself as a school-girl
and how she talks about her relation to mathematics at school and we
wanted to identify in what ways mathematical knowledge has become
important to her. In other words, how her mathematical ability has been
inscribed at present times and how it contributes to her subjectification.
Irene, quite proudly referred back to her school days denoting her high
ability in doing mathematics. In particular, she exclaimed:
Irene: At school I was really good at maths. […] In high-school I
had top grades in mathematics and writing. […] Really good
grades! […] I had a gift for maths.
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The above interview extract sums up, in a representative way, Irene’s
endeavors to articulate her relation to mathematics as a school girl. As
we can see, she develops her argument along two lines; first,
emphasizing excellence in maths at both primary and secondary school
and second, interpreting her excellence as a gift. Drawing on the first
line, Irene, proudly emphasizes her excellence in mathematics as
curricular knowledge at a continuum from primary to secondary levels
of schooling. By stressing her mathematical skills in primary and
secondary school, she wishes to denote that she could cope well not
only with arithmetic and practical problem solving (i.e. as taught in
primary school) but also with more abstract mathematics such as
theorems, proof and argumentation (i.e. as taught in secondary school).
At the same time, the act of distinguishing among primary and
secondary, rises the prominence of her continuous performance in
mathematics as ceaselessly good. In relation to the second line of her
argument, but also interweaved to the first, Irene refers to her
mathematical knowledge and skills as not something really possessed or
controlled by herself, but as an external fairing. She characterises her
own mathematical ability as the ‘gift’ of a mathematical mind –a trait
given to her by birth or God- and thus adhering supernatural powers to
it.
Concerning Irene’s accounting of her mathematical ability as a ‘gift’
coming from external sources, Valery Walkerdine’s reference to
attribution theory as explained by Weiner (1972) or Bar-Tai (1978)
might be useful here so as to take a deeper look at her positioning.
According to this theory an essential gendered difference exists between
boys and girls as far as their ways of talking about failure and success
are concerned. Specifically, boys tend to attribute their success to
internal and stable causes (ability) and their failure to external, unstable
causes (e.g. lack of effort), whereas girls tend to reverse this pattern
taking personal responsibility for failure but not for success (as referred
by Walkerdine, 1998, p. 22). But, this was not the case with Irene. On
the contrary, Irene breaks this gendered pattern and performs the
‘brilliant academic male’. Such positioning serves to some extend
women’s struggles to prove themselves equal to men by performing
intellectual masculinity recognized as rationality, logic, ability, talent
and competition. This interpretation reflects the liberal ‘woman as
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problem’ feminist discourse.
In parallel, and in connection to the above, Irene characterises her
mathematical ability as a ‘gift’ with mystical connotations to a net of
supernatural powers coming from heaven. Mendick (2006) narrates her
personal experience of studying mathematics at a prestiguous college in
the UK marked by a competitive and masculine cultural context. Her
colleagues, besides all being male, were not open to disclosing
processes and personal paths of learning in doing mathematics. As such,
construction of mathematical knowledge was represented as an
individual, mystical, innate, closed task relevant only to the chosen few
(see Mendick, 2006, p. 8). Irene, in a similar way talks about her talents
in mathematics as having almost the magic touch of gifts. In this way
she unconsciously creates barriers for any potential to unlock the
material and social assemplages that afford her success in doing
mathematics and permit the construction of her mathematical ability.
She shuts and occludes any personal and collective efforts for becoming
better, accomplishing effective strategies, and even sustaining success.
Articulating success as a matter of magic signifies success as closed,
mystical and, ultimately, inaccesible. Restivo (2009) argues how
mathematics ‘ has been shrouded in mystery and halos for most of its
history’ making it ‘ impossible to account for the nature and successes of
mathematics without granting it some sort of transendental status’
(Restivo, 2009, p.39). He goes further to explain that such a sacred way
of viewing mathematics assists mainly to conceal the complex
geopolitical scientific networks that serve to create the history of
becoming a subject. For example, the persisting monolithic view that
the development of non-Euclidean geometry was a remarkable
phenomenon that occurred simultaneously in distinct scientific
laboratories fails to acknowledge that scientists had already formed
social and scientific networks and ideas circulated amongst them. This
perspective conceals the construction of mathematical knowledge as a
social assemblage that mobilises people for further action and, at the
same time, becomes mobilised by human agency. Whilst it is relevant
here to ask why the idea of mathematical knowledge as absolute and
mystical strenously persists, we also need to denote how Irene’s struggle
for articulating her relation to mathematics as a school girl in such
essentialising genre enables her to perform a masculine subjectivity.
260
Being Good at Maths Is Not Enough: the Urge to Live
As Irene admitted, being good at school maths was not enough to safeguard her
enrollment to a mathematically related study-course at higher education. She
explains:
Irene: My first choice was architecture […]; I didn’t pass the
admission exams… Eh… I studied to be a librarian, which was my
20th choice… I liked it along the way. But, it was not my first
choice.
Interviewer: And why didn’t you insist in order to study
architecture or something related to mathematics?
Irene: At eighteen I just wanted to leave home. Yes. I was accepted
at the university in Athens; I had friends and acquaintances there,
so I went and I never had any regrets. I worked as a librarian for
eight years and liked it a lot. I liked the structure of this field. It
was something completely new to me.
Taking into account Irene’s pride in being good at maths and its
significance for performing the mathematically talented school girl, it is
difficult to see how she, at the stage of planning her studies at higher
education, so easily chooses to abandon mathematics and give in to her
twentieth option. Instead of insisting on pursuing a mathematically
related field that was closer to her heart and abilities she opted for
library studies that, at the time, was something entirely different from
her interests. Irene, quite honestly, admits that it was her urge to live an
independent life away from her parents and her village that motivated
her for any option that could take her away from the rural home
community of her upbringing and closer to the cultural urban capital.
So, Irene’s urge to study at higher education is closely connected to her
urge to escape from a culturally deprived community heavily dependent
on traditional and patriarchical values.
As has been argued, the farming sector of labor and work in rural
Greece during the early 80s was highly gendered. Female status in the
context of family, community and work practices was marginalised
–even though women and girls were a major part of working labor. By
and large, women in rural communities were working at several fields
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such as household, farming and are primarily responsible for raising
children among others. However, their work was mostly unpaid or very
low-paid and their subject identity was regulated and restricted to follow
specific sociocultural norms and ethical codes of their community (for
more details see Strategaki, 1988, Papataxiarchis, 1995). This is the
context, where Irene as an adolescent in rural Greece of the early 80s
was raised. A cultural context where young women’s attendance in
public spaces was limited and their life was restricted to home and
school. As such, she had to perform a lifestyle closely regulated by
family and community values and customs. Patriarchy and religion were
among the pelars for raising and bringing up children, and especially
girls. For Irene, but also for other women in the rural country, the
opportunity to enrol in a study course in higher education was, almost,
the only chance for freedom. Leaving home was an escape from a
highly controled and gendered cultural context and way of living.
Papadopoulos, Stephenson, and Tsianos (2008) discuss ‘escape’ as a
route for facing deadends in an oppresive life that is obeisant and
subdued to regimes of subversion, oppression and marginalisation.
Resorting to Nietzche’s ideas from his early book ‘Birth of the Tragedy’
they argue how ‘[t] he exodus from the lived life is to be found in life
itself’. Nietzche argues that the promise of a better future to come has a
series of actions such as revolutions, innovations, occupations and
discoveries amongst others as its object. Promise and object seem
logically and inextricably connected despite the fact that they rarely
fulfill each other. In other words, as in Irene’s case, women in rural
countryside cannot easily bring any straightforward change how life is
experienced through local forms of resistance. Nietzche tries to break
this logical connection between promise and object by suggesting that
life itself is ‘ the solution to the problem of life’. They continue arguing
that ‘[w] ith Nietzche the lived life and the logic of life come together’
(Papadopoulos et al, 2008, p.85)
In a similar vein, Irene’s choice to leave home at the cost of
abandoning her thirst for mathematics was inexorably connected to her
urge for exploring life. Being female in a rural community she had faced
processes of close regulation of her everyday encounters, behaviour and
wherabouts in an environment more or less culturally deprived. Her
urge to live mobilized her to risk the safety of a stable identity
262
embedded within the discourse that fullfiled her subjectification as the
female mathematical genius. Related to how the sociocultural context
determines women’s choices in mathematically related fields of study,
Mendick (as cited in Chronaki, 2008) refers to the case of Anelia, a
Turkish adolescent who lives in the UK with her family and, who,
although good in science and mathematics, resolves not to study this
subject but to abstain due to being in the presence of many male
students. By declining a favourable option, she preserves herself from
any possible seduction that might make her risk her family values.
Summing up, being good at mathematics proves not enough when
adolescent girls confront the need to balance existing possibilities in
their material contexts. Irene’s choice to study anything that would
enable her to escape home, was not an autonomous free choice
according to the prevailing liberal discourse, but, for her, it was an
escape from a socio-cultural regime of control that oppresses her. Within
such contexts, for Irene, the urge for exploring life seems to win.
Returning to Maths: Developing Status Quo As a Teacher
Having worked for eight years as a librarian, Irene moved to a higher
education course in pedagogy by enrolling to a teacher training course.
Her pedagogical studies ensured her with a teaching post and the last
five years she has been working as a primary school teacher. She talked
rather enthusiastically about her carreer change from a librarian to a
teacher. Although her work as a librarian was beneficial and useful, the
teaching profession fullfilled her more. She, recently, had the chance to
participate in a University based training course contributed to her
professional development through courses, seminars and project-work.
She felt that her teaching skills and status could benefit the most
through new terrains of knowledge in specialised topics related to
pedagogy, didactics, technology and mathematics. Moreover, through
her teaching experience at school and her further training at the
university, Irene had the chance to get in touch with mathematics in
more depth once again. For her this was almost like a return to
mathematics – the object that in many ways determined her life as a
school girl. In the course of our interview the discussion, thus, turned
towards unravelling how she, at present, perceived her relation to
mathematics as part of her current experiences in teaching and learning
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the subject. Does it currently function in similar ways for her as it did in
the past?
Her continuous delight for mathematics was evident in the enthusiam
she had shown when speaking about the specific seminars on
‘mathematics didactics’ offered at the university based professional
training course. Despite the fact that most of the teacher trainees
evaluated this seminar as too difficult due to its austere focus on
mathematics, Irene held a positive attitude. Mathematical austerity was
for them problematic as they were not provided with opportunities to
connect such a high and abstract level of pure mathematics offered at
the seminar with the mainstream practices as required by the school
Irene: This year, that we have a course in math, I notice it again
[she refers to her competence in math]; although my other
colleagues complain about the course, and despite the fact that it is
difficult and all the concepts are new. We are taught stuff I hadn’t
heard about in school. I am fascinated by it and if I didn’t have so
many other obligations right now (I am focusing on my
dissertation), I would like to investigate this new field further.
Interviewer: So do you believe that your current training in math
has been beneficial?
Irene: Eh… yes, because it gives me stimuli and contact with fields
of knowledge I was oblivious of. And I reckon that I might become
involved with them in the future.
Interviewer: Do you think that this knowledge might be applicable
when teaching at school?
Irene: No. Not as such, because it involves a higher level of
mathematics. But, as our professor tells us, to teach something
simple, you have to understand the philosophy of mathematics; it
is not enough to simply be familiar with the material presented in
class. You need to possess comprehensiveknowledge, in order to
communicate it.
Interviewer: Do you agree with your tutor’s point of view?
Irene: I think this is the case in any field of knowledge. Otherwise
it would be possible for…eh, say, a high-school graduate to teach
primary school students. Knowledge certainly needs to be
profound in order to be properly communicated.
264
Irene undoubtedly celebrates a way of teaching mathematics during the
seminar that has caused a number of problems to many of her
colleagues both male and female. It’s of importance here to take into
consideration what is actually happening during the course.
Characteristically, the training course was attended by 24 teachers of
whom exactly half are female and six of them had proved to be very
competent at mathematics. Besides, it is worth mentioning that the
highest score at the final exams for the seminar ‘mathematics didactics’
was achieved by a female teacher trainee. In addition, two male students
had expressed their negative disposition towards mathematics all the
way through and complained for the abstract way of delivering the
seminar. Albeit this, it is worth mentioning that all interviewees,
including Irene, complied with the stereotypical view that male teachers
were more competent, skilled and hold positive attitudes as compared to
female teachers. In this way most teachers tended to reproduce a
prevailing image of female incompetence and insufficiency, thus
fuelling the sense of stress and unease many women experience
concerning those fields.
Irene was an indicative case of appropriating dominant discourses.
Her unquestioned acceptance of the way class was organised and taught
during the training course may be interpreted on two levels. At one
level, we might argue that Irene becomes fascinated by a subject that is
considered difficult and challenging by most of her colleagues. At
another level, we might construe her preference for this abstract way of
working with mathematics as a pleasurable challenge. Although pure
mathematics has little to do with the actual requirements for teaching
and learning school mathematics she expresses creativity and
contentment. In parallel, Irene’s narrative reveals how her resort to
mathematics supports her efforts towards differentiating herself from
other teachers and denoting her superiority. Mathematics, and her
mathematical competence in particular, is instrumentally used towards
augmenting her status quo as a teacher. She performs the supreme
teacher who, although female, dares to do the maths required for maths
at the primary school class. Instead of dreading, she masters the subject
on both the basic arithmetic and the high or abstract level. Being female
constitutes her certainly as exception. In this manner, she attempts to
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provide herself with high regime, since mathematics is considered to be
a field of considerable status as such. The above become even clearer
when we consider how she talks about her colleagues at primary school
and their relation to school mathematics.
Interviewer: Do you think there is a difference in the way men and
women engage in mathematics?
Irene: I realised that here, on the course, my male colleagues are
quicker to respond to questions asked by the professor. Usually
male colleagues teach older children –fifth or sixth grade classes-,
where mathematics is at a higher level. Female colleagues usually
take on younger ones, and there math is basic.
Interviewer: How come? Why is it that men teach higher grades?
Irene: There is a status quo… not that it is standard, but it usually
works this way.
Interviewer: And why does this happen?
Irene: I have met female colleagues who didn’t want to teach
higher grades because they felt insecure about math. I think that
sometimes teachers “fall short” when it comes to the material they
need to teach (in mathematics) in fifth and sixth grade.
Interviewr: Both, male and female?
Irene: Females more often, yes. Because I remember helping some
female colleagues prepare for the exercises they had to teach the
following day.
Irene refers to a gendered division of labour at the primary school where
male teachers become more often responsible for higher grades whilst
female teachers take the lower ones. She explains that this is due to the
fact that younger children are taught basic skills (i.e. arithmetic) whilst
older ones require more advanced mathematical knowledge. For Irene,
school mathematics at higher grades is challenging and argues that,
unlike her, most female teachers cannot take this risk. To sum up,
Irene’s argument is founded on a bipolar perception, according to which,
maths is divided into complex or basic, difficult or easy and becomes
accordingly appropriated to high and low grades in primary school. It is
therefore implied that male subjects are more familiar with complex and
difficult math. Throughout her narrative, Irene reproduces the
patriarchal order of mathematics as a male domain that is carried
266
through to a controlled and patriarchal division of labor between men
and women as teachers.
Mathematics is thus being reproduced as a masculine field of practice.
Specifically, we realise how Irene considers readiness to understand and
solve a problem to be the cornerstone of mathematical thought, and on
these premises she claims that her male colleagues have undoubtly a
better and more effective understanding than their female colleagues,
and, therefore, they are better at maths as compared to female primary
school teachers. What is of interest here is that although Irene adheres to
this essentialist position, she differentiates herself so as to stress her
resemblance to male and not to female behavior. In other words, whilst
most females are prone to dislike or fear mathematics she takes a
different position. It becomes evident again how she uses mathematics
to perform her masculine subjectivity. Mendick (2006) argues:
One of the main tensions that I have experienced in thinking and
writing about gender […] is between equality and difference. The
idea that women are different was the starting point for feminist
political struggle. However, it is always double-edged, being prone
to political misuse as a defence of discriminatory practices and
status quo. As discussed in ch1, explanations based in gender
difference so easily become self-perpetuating; indeed, when I have
presented material from this book I have met the view that work
such as mine, which seeks to explain gender differences, is actually
part of the problem. Perhaps without all this talk about gender
differences there wouldn’t be any… (Mendick, 2006, p. 101)
At an additional layer, which is nevertheless linked to all others, Irene is
reproducing the prevalent gender regime to the extent that she both
idealises the dominant male-orientated status of mathematics and
conceals how it becomes constituted. In this way, she ignores the fact
that this discursive strategy might provide some students -especially
girls and women- with ostensible obstacles in the appropriation of such
knowledge. She does so in a way that gender becomes a technology of
self, in Foucault’s words, for re-producing old knowledge politics by
means of ethical or moral evaluations (Foucault, 1978). We could also
denote how Irene uses maths to subjectify as a successful, competent
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teacher who is able to cope with difficult and challenging arenas of
knowledge such as mathematics. Since mathematics has been conceived
as a male culture, Irene positions herself as a masculine teacher in the
gendered field of education. Her relation to maths supports her efforts to
perfom a particular teacher identity that could compete even her male
colleagues and she performs –through and with maths- a power position.
In, some ways, she re-lives her success story as a school girl who was
gifted in mathematics and now is the master of mathematics.
Returning to Mathematics: Essentialising Mathematical Ability
Perceiving mathematics as an essential body of knowledge is even more
obvious in the way she narrates her handling of school mathematics in
the classroom as a teacher. Specifically, Irene argues that she often alters
the official school curriculum by stressing and expanding the teaching
of mathematics at the expense of other subjects. She admitted paying
less attention to subjects such as music, arts and religion evaluating
them as secondary. Her vission as teacher was to advance her pupils in
mathematics. In this manner, she clearly reproduces a hierarchal
classification of school knowledge where mathematics comes at the top
and the arts follow.
Irene: Between Greek (language) and mathematics I suppose it is
math I am best at. In Greek I only teach what is mandated by the
curriculum. In mathematics, it is different. When I was teaching
second grade last year, all the children (14 of them) learned how to
multiply. The teacher who took over the class this year told me that
she had taught sixth grade the previous year and that those kids
(who went off to junior high this year) still couldn’t multiply
properly. […]But I taught mathematics at the expense of other
subjects such as music or art, which also isn’t right.
Her example concerning the emphasis she placed on pupils’ training on
multiplication signifies a particular perspective of mathematics. Asked
about her views on mathematics and the potential connotations
mathematics brought to her mind, she talked about 'organisation, order,
method, eh… one step above, structured thought and affection. [… ] I
feel a special kind of affection towards mathematics'. And she added
268
that; ‘ Mathematical thought makes
you more precise. It helps you get
straight to the point providing you with a framework’. Irene connects
mathematics to order, precision, structure, rationalism, and superiority
to other types of knowledge. In her own words, mathematics is ‘one step
above’. However, through her unquestioned acceptance of the
hegemonic bipolar optic of mathematical knowledge, she embodies
equally hierarchal gender binaries according to which mathematical
thought is socially and culturally linked to the male-mind. Asked
whether she could discern any gendered differences on mathematical
competence among her students at school she claimed that boys have
certainly a special flair for mathematics.
Interviewer: How do you explain that?
Irene: I believe it lies in the structure of the psyche of each sex.
[…] I have often thought that men have greater technical dexterity
and skills. But it is not something I can explain scientifically. […]
And this fuelled my curiosity because as a child I had a special
ability in and affection for maths. And the more I was praised for
my aptitude, the harder I tried, I …played that part. It’s the
motivation; I was good at math ever since I was a child and I was
encouraged by my parents and teachers.
Interviewer: So your parents encouraged you.
Irene: Yes. And the more they did, the harder I studied, because I
knew I was going to be praised.
Almost forgetting –or not being conscious- about her own efforts for
developing knowledge and becoming a female success in mathematics,
Irene narrates her effective mathematics ability as an innate trait that
resembles naturally the male-mind. Although her learning was highly
dependent upon the social conditions of her close environment and the
support provided by teachers and her parents, she assigns her skills the
magic gift of innate motivation and flair. Irene’s interpretation probably
draws on popular psychology and pedagogy where emphasis is
generally given to the individual or to special social categories such as
women and their distinct temperament. This optic tends to account for
their deficiency in certain fields, like maths or technology, instead of
focusing on social factors such as the nuts and bolts of education. This
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way the dominant male-orientated structure of those fields remains
largely unchallenged (see Walshaw, 1999, 2001).
Irene seems trapped in mythologies about maths as an absolute body
of knowledge to such an extend that she becomes blind even to her own
personal experience. As said above, she cannot consider how she as a
female has managed to move forward, to be able to do mathematics and
develop motivation and affection for mathematics. In consequence, she
cannot also see how some of her collagues at the training course did not
relate to pure mathematical knowledge. For her, mathematics is an
absolute power. It is a matter of right and wrong, black and white and
indispituble answers. Within the frame of thinking, rationality is directly
related to pure maths method of proof and claims for mathematical
certainty. Despite efforts for challenging this absolutist knowledge and
truth in mathematics by seminal philosophers in the mathematics
education field (see Lakatos, 1976; Ernest, 1991; Skovsmose, 1994; and
Burton, 1995) their work, although appealing, has not had yet great
impact on teachers’ values and practices. Mendick (2006) problematises
the appealing status of mathematics as absolute and objective. Both,
appeal and pleasure take us to discuss our relations to discourses and in
particular how discourses position people within networks of power.
Foucault (1989, 1979) alert us to presuppose not an idealised discourse
foisted upon the individual but also the formating power of specific
disciplining, regulating and controling practices on self.
As a Way of Conclusion
As already mentioned, οur focus here has been on discussing possible
interrelations amongst representations of mathematics, education,
gender and subjectivity. We intended to explore through Irene’s
narrative the specific discourses in which these representations are
inscribed; the subject positionings that their articulation could make
possible; and their potential effects for subjectivity fabrication. Based on
Irene’s case, as presented in the previous sections, we wish to stress
three main issues as a matter of concluding our analysis; firstly,
performing success in mathematics contributes towards fabricating a
gendered masculine subjectivity as a self-formating power, secondly,
gendered subjectivity depends heavily on appropriating an essentialist
ideal of both mathematics and gender through a struggle of articulating
270
available discourses, and, thirdly the essentialist appropriation of
hegemonic discourses on gender and maths do not liberate but trap the
subject in contradictory and conflicting discourses and practices.
Becoming masculine: As far as the first issue is concerned, Irene’s
case was an exemplification of performing success in mathematics both
as school girl and as teacher. Her subject positioning of the gifted,
talented and charismatic in maths at school time secured her a very
positive and celebrated socialisation. Her abilities and skills were
praised by parents and teachers. In this way, she was able to perform not
only the gifted one in a difficult domain such as mathematics, but also
the female subject who breaks the norms and stereotypes of a patterned
male subject through her success in maths. In other words, she was able
to perform a male who, according to Irene’s resorting on prevailing
discourses, excels naturally in mathematics. It was evident that her
subjectification with mathematics was an attempt to perform
masculinity. Bob Connell’s perspective of masculinity (1995, p.71)
allows us to claim that masculinities are not inherently limited to men
(or femininities to women). Male or female experiences are not uniform
or homogeneous, overlapping is not excluded, and actions that do not
correspond to the person’s gender are not silenced. Under this
perspective a broad band of options need to be available for any variety
of people. Thus, opening up activities conditioned by gender seem
important as they facilitate reflection and recognition of the effects
gendered classifications have on social life, in order to become less
influenced by these. In this realm, Irene’s gendered engagement with
mathematics can be seen as having direct effects on her social life.
Essentialising strategies: Concerning the second issue, Irene
articulates hegemonic and essentialising discourses about mathematics
and gender to speak either for herself or for her colleagues and pupils.
She assumes a series of ideals and dichotomies that represent hers and
others’ experiences in relation to mathematical ability and success.
Mendick (2006) has argued that the subject position ‘good at maths’ is
inevitably a performance of masculinity as it evolves through the
acceptance and utilisation of a set of binaries such as competitivecooperative, active-passive, naturally competent-hard working, always
appointing the inferior term to women. Such false categories according
to which the feminine is conceived as exclusively and essentially
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representing nature, emotionality, sensuality and irrationality. This
negative representation of the feminine emanates from the mind/body
dichotomy which has dominated western science and philosophy.
Within this grant dichotomy, which was clearly and powerfully
expressed by Cartesian thought, mind and rationality has gained priority
over the body (for more details see Chronaki, 2009). Through this
viewpoint, the mind is customarily correlated with public space (i.e.
politics, economy, warfare, science) and masculinity, while the body
connected with the private sphere (i.e. home, children upbringing, labor,
arts) and femininity. On those grounds, mathematical competence has
been constructed as inherently natural, individual and male, withholding
their social, symbolic and historical nature, thus concealing the fact that
such skills are a product of practice and social construction (Bordo,
1993; Walkerdine, 1988; and Mendick, 2006). The process of
dichotomising and at the same essentialising constructs mathematics as
oppositional to femininity and, thus, makes it difficult for many women
to identify as capable, effective or successful and even to invest within a
related field of study or work. What is of interest here is how Irene
through such dichotomising and essentialising use of available
discourses fabricates subjectivity. By means of her natural and
gifted ‘mathematical ability’ Irene constructs for herself, all the way
through, a superior position that entails power and provides her high
status as a school girl and as a teacher.
Being trapped: Irene’s talent at mathematics opens for her the
opportunity to engage in a gendered domain. From this perspective,
performing a masculine gendered subjectivity might entail the dynamics
to challenge the prevalent gendered order and trouble oppressive
practices or the established gender binaries. On the contrary, she seems
trapped through espousing the essentialising strategies of narrating
relations about maths. Irene, by and large, attempts to manage the
contradictions inherent in her speech by invoking a personal explanation
that stems from a diffused neoliberal discourse informing an ontological
individualism. Her individualist explanation lends an ostensible
coherence to what she says, and covers up tensions that result from
conflicting roles and aspirations. For example, her own performance in
mathematics belies her conviction that men are superior in this field.
272
This contradiction causes confusion and seems to be resolved through
her invocation of maths as talent and charisma. One of the crucial issues
tackled by critical ontology is what Foucault calls the ‘ paradox of the
relations of capacity and power’ (1984, p. 47). The question Foucault
(1984, p. 47) raises is, ‘ how can the growth of capabilities be
disconnected from the intensification of power relations’? Individuals
become autonomous agents through the development of capacity for
thought and action (Tully, 1999, p. 93). However, such capabilities are
developed within disciplinary regimes of pedagogical, medical and
punishing institutions where the subject becomes also normalized and
hierarchized (Foucault, 1984, 1986; Wong, 2007, p. 73). Hence, drawing
from Foucault (1984: 45), we should search for the points ‘ where
change is possible and desirable, and to determine the precise form this
change should take’.
Adherence to the discourse of essentialising the mathematical mind as
a God’s gift is a formittable barrier to ending the hegemony of absolute
and pure reason in mathematics education practices. If we want to
seriously undermine tendencies to purify and essentialise the categories
and classifications that inevitably and universally organise our social
and moral orders and produce differences and distinctions, we urgently
need to reject transcendetalism and supernaturalism. The essentialised
articulation of discourses effects in producing an equally essentialist
subjectivity and in particular a ‘ masculinist construction of an
essentialised self’ in Judith Butler’s words. The positioning of
mathematics as 'natural gift' does not allow her to perceive the
contigency of doing school mathematics and in consequence, the hard
work invested in this practice. Thus, Irene, and any other subject as
Irene, cannot disclose the fact that mathematics as well as gender is
constantly constructed and reconstructed from and within discursive
articularions as part of their social relations and practices (Mendick,
2006, p.18, and Restivo, 1992, p. 102).
As a final comment, we would like to affirm that an alternative
approach seems necessary. The goal of such an approach would be the
systematic deconstruction of essentialist gendered categories in order to
show how woman and man are constructed as categories within
discursive formations (Mouffe, 1992) even in the field of mathematics
education. Therefore, we can claim that the deconstruction of gender
(3) 273
categories in mathematics education aims to challenge traditional
objectified classifications of certain qualitative features attributed to
each one of them, and thus renders the nature of every attempt for
fabricating subjectivity contingent and precarious. In this manner,
qualitative features are liberated from essential classifications leaving
them floating and available for everyone.
Notes
The research reported here is part of the project ‘Mathematics and Technologies in
Education: The Gender Perspective’ EPEAEK Pythagoras I [co-funded by the Greek
Ministry of Education and the EU] period 2004-2007. Project Director: Anna Chronaki,
Professor ECE, University ofThessaly ([email protected]).
1
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Understanding the Knowledge and Practices of Mathematics
Teacher Educators Who Focus on Developing Teachers'
Equitable Mathematics Pedagogy
Laura McLeman 1 and Eugenia Vomvoridi-Ivanovic 2
1 ) University of Michigan-Flint.
2) University of South Florida.
To cite this article: McLeman, L., & Vomvoridi-Ivanovic, E. (201 2).
Understanding the Knowledge and Practices of Mathematics Teacher
Educators Who Focus on Developing Teachers' Equitable Mathematics
Pedagogy. Journal of Research in Mathematics Education, 1 (3), 278-300.
doi: http://doi.dx.org/1 0.4471 /redimat.201 2.1 5
October 2012 pp. 278-300.
Understanding the Knowledge
and Practices of Mathematics
Teacher Educators Who Focus
on Developing Teachers'
Equitable Mathematics
Pedagogy
Laura McLeman
University ofMichigan-Flint
Eugenia Vomvoridi-Ivanovic
University ofSouth Florida
Abstract
Most mathematics teacher educators (MTEs) would agree that teachers must be
prepared to provide equitable mathematics instruction to all their students.
However, to date, there is not a wide database regarding the practice of MTEs
who play an integral role in this preparation. In this paper we argue that
additional information is needed about the approaches in which MTEs have
addressed or incorporated equity issues such as race, identity, language, and
culture as a core part of the preparation of teachers. We further argue for the
importance of developing a research agenda that examines the practices of
MTEs who teach through this lens of equity, the goal of which would be to
build models of professional development that prepare and support other MTEs
to develop this specialized knowledge.
Keywords: mathematics teacher educators, practice, equity, research.
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.15
October 2012 pp. 278-300.
Comprender el Conocimiento y
las Prácticas del Profesorado de
Matemáticas Centrado en el
Desarrollo de una Enseñanza
Equitativa de las Matemáticas
Laura McLeman
Universidad de Michigan-Flint
Resumen
Eugenia Vomvoridi-Ivanovic
Universidad de South Florida
Muchos maestros/as de matemáticas (MTEs) estarían de acuerdo que el
profesorado tiene que estar preparado para proveer una instrucción matemática
equitativa para todos los estudiantes. Sin embargo, hasta la fecha, no hay una
base de datos amplia referente a prácticas del profesorado que juega un papel
integral en su preparación. En este artículo sostenemos que se necesita más
información sobre los enfoques que el profesorado ha utilizado para abordar o
incorporar temas de equidad tales como la raza, la identidad, el idioma, o la
cultura en el núcleo de sus formación como maestros/as. Reclamamos la
importancia de desarrollar una agenda de investigación que examine las
prácticas del profesorado que enseña a través del enfoque de la equidad, con el
objetivo de construir modelos de desarrollo profesional para preparar y apoyar
otros maestros y maestras para desarrollar este conocimiento especializado.
Palabras Clave: profesorado de matemáticas, prácticas, equidad,
investigación.
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.15
O
(3) 279
ver the past two decades, the field of mathematics education has
paid considerable attention to understanding and confronting
differential mathematics achievement (DiME, 2007).
The nature of this attention has been on issues of race, class, gender,
language, culture, and power in mathematics education and on how to
promote achievement among culturally, linguistically, and
socioeconomically diverse students. Researchers have described the
knowledge and skills necessary to teach in this manner (e.g. Gay, 2009;
Gutiérrez, 2009; White, 2002), specifically arguing that learning to
teach mathematics for equity should be central to the teacher
preparation curriculum.
Several mathematics teacher educators (MTEs1 ) have described their
approaches to preparing teachers to incorporate equitable instructional
strategies that focus on issues such as race, identity, language, and
culture within mathematics (e.g., Aguirre, 2009; Bartell, 2010; Chval &
Pinnow, 2010; Drake & Norton-Meier, 2007; Dunn, 2005; Gutiérrez,
2009; Kitchen, 2005; Turner et al., 2012; Vomvoridi-Ivanovic, 2012).
However, there is a need for more dialogue regarding the instructional
practices of MTEs (Strutchens et al., 2012). In this paper, we echo this
call by arguing for an increased public dissemination about the
approaches in which MTEs have addressed issues of equity such as
race, identity, language, and culture as a core part of the preparation of
teachers. Moreover, we further argue for the development of a research
agenda that focuses on learning from and about how to develop these
practices.
To support our argument, we first share parts of our personal narratives
and explain why we have chosen to use the term equity to frame our
position. We continue by discussing findings from relevant literature,
sharing what MTEs have argued are some necessary instructional
practices and possible challenges when preparing teachers to develop
equitable mathematical pedagogy. Through this discussion we
demonstrate the need for further dissemination on how issues of equity
are integrated into mathematics teacher preparation. We conclude by
calling for the development of a research agenda and offering
recommendations for future research.
We acknowledge that the practice of MTEs is ever changing and
280
McLeman & Vomvoridi - MTEs' practices regarding issues ofequity
and that we have not identified all such complexities. Nevertheless, our
intent is to promote a dialogue that encourages MTEs to share and
discuss the elements of their practice that focus on issues of equity.
Further, we wish to initiate a discussion about areas of research within
the field of mathematics teacher education for the purposes of working
toward the development of a framework of MTEs’ knowledge base for
teaching through a lens of equity. It is through the identification of and
examination of current practices that we can begin to build models of
professional development that prepare and support other MTEs to
develop this specialized knowledge.
Our Positionality
What we as researchers attend to in our work, including the questions
we ask and the interpretations we draw, is shaped by our own
knowledge and background, among other things. However, sharing how
we are situated within our work is not a widely held practice in
mathematics education (Foote & Bartell, 2011). Foote and Bartell argue
that doing so will enrich and impact how audiences receive our work.
With this consideration, we share what motivated our desire to
understand how MTEs integrate issues of equity into their instructional
practice with the goal of providing anecdotal evidence that supports our
call for dissemination and research.
Each of our educational journeys has led us to question how
mathematics instruction is equitably provided to all students. The
beginning of Laura’s journey occurred while she was a secondary
school mathematics teacher in a low-income, urban area with AfricanAmerican students whose lived realities were very different from her
own. Eugenia’s journey began at an early age when she wondered why
some of her classmates in Greece succeeded in mathematics, while
others fell through the cracks. Her journey continued when as a
mathematics teacher in the United States she wondered if race played a
factor in who was placed in advanced mathematics courses and who was
placed in remedial ones. As we progressed through these phases of our
lives, we both saw that our methods of teaching (generally a mix of
teacher-centered lectures and cooperative learning activities) did not
help all of our students succeed. We realized that something more than
just applying generically “good” teaching strategies was needed, but we
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did not know what.
Our journeys led us to pursue doctorates in education, where we were
Fellows in the Center for the Mathematics Education of Latinos/as
(CEMELA2) and acquired knowledge, theories, and frameworks related
to equity and diverse populations. After receiving our degrees, we each
took positions in universities where we prepare teachers to teach
mathematics in a way that not only draws on their students’ lives, but
also uses the students’ lives as a basis to critique the world. However,
we could not find published research regarding established models of
practice or curriculum that we could use to frame our teaching. As we
pondered the lack of research on the various ways that MTEs integrated
issues of equity into their practice, several questions arose. How do
MTEs teach their courses in a manner that does not unwittingly promote
or reinforce deficit views of certain populations of students? What is the
knowledge base of those MTEs? How can more MTEs develop the
knowledge and disposition to make issues of equity such as race,
identity, language, and culture central to their work? Discovering
answers for these types of questions in order to improve the preparation
of MTEs, mathematics teachers, and ultimately students in mathematics
classrooms is what motivates our work.
What Do We Mean by “Equity?”
In recent years there has been an increased attention on providing
equitable mathematics instruction to all students (Hart, 2003). For
example, the National Council of Teachers of Mathematics (NCTM,
2000) identified equity as one of its core principles. In particular, they
argued that equity requires schools and teachers to set high expectations
and provide the necessary resources and support for all students to
achieve, while acknowledging and accommodating the inherent
differences that exist among learners (NCTM, 2000). Although NCTM’s
definition of equity does not focus on the sociopolitical context in which
teaching occurs and does not offer specific approaches in which to
achieve equitable instruction (Kitchen, 2005), it does frame the teaching
of mathematics in a way that is accessible to many different individuals
associated with mathematics education (e.g. teachers3 , administrators,
policy-makers). Therefore, we have deliberately chosen to use the term
equity, even though there has been a shift among researchers to move
282
away from this terminology (Burton, 2003). We posit that our choice to
use the term equity, as opposed to other terms such as social justice or
culturally responsive teaching, makes our work more accessible to a
broader range of individuals. For example, an MTE who focuses on
issues of language and culture might identify more with the construct of
culturally responsive teaching rather than that of social justice. An MTE
who focuses on using mathematics as a tool to take action upon social
inequalities, however, might identify more with the construct of social
justice. Yet we assume that both MTEs would agree that their work
prepares teachers to develop equitable mathematics pedagogy.
As with any broad idea, equity can mean different things and be used
in different ways. Indeed, many scholars have presented varied
definitions of equity (e.g., Aguirre, 2009; Crockett & Buckley, 2009;
Gutiérrez, 2002; 2009; Secada, 1989). We take the stance of Gutstein et
al. (2005) that the existence of different definitions for equity is not
inherently problematic, as certain definitions can serve specific
purposes. However, to make clear how we conceive of equity, we view
it in a way similar to Gutiérrez (2002). We see equity as an inclusive
construct in which characteristics such as race, class, gender, language,
culture and/or sexual orientation should not determine the level of
mathematics achievement that one attains. Further, as others have
argued (e.g., Bartell, 2011; Crockett & Buckley, 2009; Gay, 2009;
Matthews, 2003), we believe that these characteristics should be an
integral part of the mathematics curriculum so that students can use
mathematics to “examine one’s own lives and other’s lives in
relationship to sociopolitical and cultural-historical contexts” (Gutstein,
2006, p. 5).
Why "Equity"?
Mathematics has acted as a “Gate-Keeper” and is not something that is
available for all students (Bishop & Forgasz, 2007; Silva, Moses,
Rivers, & Johnson, 1990; Stinson, 2004). Students of color, low-income
students, and language minority students have received a subpar
mathematics education, as their mathematics instruction has been
disproportionally focused on acquiring rote memorization of formulas
and facts instead of on a deep and flexible understanding of concepts
(Becker & Perl, 2003; Leonard, 2007; Oakes, 2005). While there is a
growing body of research on equitable mathematics education (e.g.,
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Adler, 2001; Apple, 1992; D’Ambrosio, 1985; Fennema & Sherman,
1977; Gutiérrez, 2002; Povey, 2002; Secada, 1989; Setati, 2005;
Zevenbergen, 2000) much of this work has focused on the pre-collegiate
level. This is understandable considering the ultimate goal is to affect
positive change in the outcomes of K-12 students’ mathematics learning
and achievement.
In comparison, little emphasis has been placed on research regarding
the preparation of mathematics teachers to focus on issues of equity.
Although studies at the general teacher education level have examined
teachers’ beliefs/attitudes/values toward teaching for equity (Hollins &
Guzman, 2005), we know very little about mathematics teacher
preparation in particular. Moreover, as is the case in general teacher
education (Zeichner, 2005), even less is known about the practices of
MTEs who prepare teachers to teach for equity. Most of the research on
MTEs’ practices is comprised of small-scale self-studies and/or
reflections on practice (e.g., Bartell, 2011; Bonner, 2010; Kitchen,
2005). Even then, this literature focuses mainly on teachers’ learning
through the MTEs’ practices as opposed to how the MTEs themselves
acquired and developed the knowledge, skills, and dispositions to enact
these practices.
Recommendations for research have called for a better understanding
of the role that mathematics teacher education programs play in
preparing teachers to teach mathematics for equity (Gutstein et al.,
2005). We posit that in order to develop this understanding, MTEs who
teach through a lens of equity must disseminate elements of their
instructional practices. It is likely that in addition to what has already
published, others have and are attempting this work but have not made
their experiences public. With an increased attention placed on the
practices of MTEs, the field can develop theories about best approaches
to engage in this work. Moreover, all MTEs might begin to recognize
the importance of helping teachers understand the inherent inequities
that might take place when teaching mathematics.
What Do We Know?
While there has not been a systematic, broad scale, examination of the
population of MTEs who frame their work through equity, some MTEs
have reflected on their personal experience of infusing equity into their
284
courses or have undertaken small-scale studies to examine teachers’
learning. To gain an understanding of what literature existed, we first
looked to the work of prominent researchers whose work we already
knew centered on issues of equitable mathematics teacher instruction.
Next we searched databases such as ERIC and websites such as Google
Scholar to find literature (e.g., journal articles, conference presentations,
books) that focused on equity within mathematics teacher education.
For any sources we identified, we also examined the references that
were cited in order to broaden our literature base.
An examination of this literature provides insight regarding two
overarching themes: (1) necessary components of MTEs’ instructional
practice and (2) challenges that MTEs may face when preparing
teachers to develop equitable mathematical pedagogy.
Theme 1: Necessary Components of MTEs’ Instructional Practice
Building relationships with students. A number of MTEs, such as
Gutiérrez (2009) and Kitchen (2005), have shared their desire to help
teachers become advocates for their students. This advocacy can only be
developed, though, through the creation of a respectful and trusting
community of learners (Gay, 2009). Kitchen shares how he starts each
semester with his students by discussing his personal narrative and why
this work is meaningful to him. Kitchen also details how he makes a
conscious effort at the start of every class to help his preservice teachers
acquire concrete methods of teaching. Kitchen argues that by situating
himself within his teaching and by attending to his students’ most
immediate needs of learning about specific strategies to teach
mathematics, preservice teachers will then be willing to engage in
activities that challenge their perceptions and thinking about the world.
Examining activities from others’ perspectives. Some researchers (e.g.,
Bonner, 2011; Drake & Norton-Meier, 2007; Gutiérrez, 2009) have
addressed the preparation of teachers to develop equitable mathematics
pedagogy by challenging their students to examine mathematical
activities through the perspective of others. Stocker and Wagner (2007)
share the importance of allowing teachers who participate in a “culture
of power” (Delpit, 2006, p. 24) to experience the viewpoint of some
underserved and underrepresented students. Oftentimes, these
opportunities are presented in the form of field experiences that have
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preservice teachers focus on issues of equity such as observing which
students have a “voice” in the classroom (e.g. Bonner, 2011; Drake &
Norton-Meier, 2007). Some researchers (e.g. Gallego, 2001) have
argued that to be meaningful, though, such field experiences need to
occur in settings that reflect student diversity and must not perpetuate
teachers’ misconceptions, stereotypes, and assumptions about various
groups of students. Instead, the experiences need to occur in settings
that model the ideals proposed in teacher education coursework (White,
2002). Doing so supports the need for teachers to explore the
mathematical knowledge of a variety of students, to learn how to select
and utilize mathematics tasks that draw on students’ experiences, and to
interact and work with individuals who are culturally different from
themselves (White, 2002; Vomvoridi-Ivanovic, 2012).
Engaging teachers in critical reflection. Other mathematics educators
(e.g. Dunn, 2005; Kitchen, 2005; Rousseau & Tate, 2003) have argued
that critical reflection also needs to play an important part in teachers’
preparation. Critically reflecting on issues of equity in mathematics,
which Rousseau and Tate argue is absent from mainstream mathematics
education, involves having teachers consider social, political, and
cultural contexts while they examine their assumptions, beliefs, and
values about mathematics teaching and learning. Over a span of four
years, Dunn found that by engaging teachers in critical reflection in her
mathematics methods courses some teachers transformed their view
about the mathematics education of underserved and underrepresented
students. For example, one teacher shared her amazement at what
different students could achieve when they had ownership of the
classroom. Further the teacher revealed that students with various
backgrounds know and can do mathematics. The challenge for MTEs
then is to help teachers continually critique and reflect on their views
and attitudes about who can achieve and participate (Kitchen, 2005).
Equity as a central component of instructional practice. A final
component of the necessary components of instructional practice
discussed in the literature is the incorporation of equity issues as a
central focus throughout the curriculum. Specifically, researchers have
argued that MTEs should model through their own instruction how
equity can be woven throughout their instruction (Bonner, 2011; Gay,
286
2009) while also providing opportunities for teachers to grapple with
this integration as well (Bartell, 2011; Bonner, 2010). This is an
important consideration, as some issues of equity do not occur outside
the realm of teaching mathematics; rather they occur within the context
of mathematics teaching and learning (Crockett & Buckley, 2009). For
example, debating mathematical ideas with peers is a discourse practice
that is valued in reform classrooms. However this form of discourse
may not be aligned with the norms of some cultures. For example, in
many Native American tribes non-verbal communication is a highly
valued skill. Additionally, many children are taught to intently listen and
observe until they feel they are prepared to participate or until they feel
there is a real-world practical application (Grant & Gillespie, 1993).
MTEs need to be mindful of such issues in their own instructional
practice and provide opportunities for teachers to grapple with similar
issues as well.
While the literature points to some elements of an MTE’s instructional
practice that are necessary for preparing teachers to develop equitable
mathematics pedagogy, it also elucidates some challenges that MTEs
may face when doing so.
Theme 2: Challenges that MTEs May Face
Finding the balance between mathematical and equity concepts. Some
researchers (e.g., Aguirre, 2009; Bartell, 2011; Stocker & Wagner, 2007)
have discussed the challenges they or other mathematics teachers have
faced when trying to balance the focus on mathematical content with the
focus on equity. Gutiérrez (2009) acknowledges this tension of how to
“cover” mathematics content while also addressing issues of equity by
arguing that teachers should embrace the notion that they teach much
more than mathematics. Instead, Gutiérrez argues, teachers first and
foremost teach students, and at times it is important to focus on issues
that do not seem to directly relate to mathematical concepts. As Aguirre
notes, the issue becomes which of equity or mathematics takes
precedent and when.
Lack of formal equity education . Another challenge facing MTEs is
the lack of formal preparation in making equity a priority in instruction
(Taylor & Kitchen, 2008; Zaslavsky & Leikin, 2004). In their
recommendations for integrating issues of diversity and equity in
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doctoral programs in mathematics education, Taylor and Kitchen share
how large numbers of U.S. doctoral students exit their programs and
take positions in institutions around the world with limited to no
experience in examining issues of equity. In addition to this lack of
formal preparation, there also exists a lack of formal professional
development experiences to help MTEs integrate issues of equity into
their instructional practice (Zaslavsky & Leikin, 2004). Indeed we
could find only the “Teachers Empowered to Advance Change in
Mathematics” (TEACH) project that specifically focuses on the
development of instructional modules for MTEs to prepare teachers to
teach mathematics for equity (Turner et al., 2012). This dearth of
relevant preparation and support speaks to the under-preparedness of
many colleagues to meaningfully include issues such as race, language,
identity, or culture within their mathematics teacher preparation courses.
Teacher resistance. In detailing some of the challenges they face, some
MTEs (e.g., Aguirre, 2009; Drake & Norton-Meir, 2007; Ensign, 2005)
discuss the issue of teacher resistance. One type of resistance that MTEs
may face from the teachers with which they work is similar to the
resistance mathematics teachers face from students, parents, and/or
administrators when they attempt to integrate issues of equity in the
mathematic curriculum. Specifically, students (or parents/
administrators) may perceive the mathematics in students’ lives as not
constituting “real” mathematics since it might not align with the
mathematical knowledge found it textbooks (Ensign, 2005). Similarly,
as Aguirre (2009) notes, teachers as well as other members of the
mathematics education community might not feel that teaching through
a lens of equity is “mathematical” enough. She further draws on
Rodríguez’ (2005) notions of resistance to ideological and pedagogical
change to describe mathematics teachers’ resistance to teaching through
the lens of equity. Ideological resistance (RIC) refers to teachers’
reluctance to change their beliefs and values, while pedagogical
resistance (RPC) refers to teachers’ reluctance to embrace instructional
practices that differ from their experiences. Specific to the preparation
of teachers for mathematics through a lens of equity, Aguirre shares her
experiences with some teachers holding on to cultural deficit models
(RIC) while others were skeptical about teaching mathematics through a
lens of equity with young children (RPC).
288
Another type of resistance focuses on the cultural background of both
MTEs and their students. At times, an MTE may be part of the dominant
culture while the preservice teachers in the class are not. In this
situation, the MTE’s credibility to address issues such as race and
identity within the mathematics classroom, as well as the MTEs’
membership in the dominant culture, may be challenged. Howard
(2006) describes how this resistance can manifest if individuals do not
confront the elements of dominance that are ingrained in their actions
and perceptions. This then may result in another form of resistance to an
MTE’s instructional practice of helping teachers develop equitable
mathematics pedagogy, namely when the students in an MTE’s
classroom are part of the dominant culture. As Landsman (2011) shares,
white teachers can be willing to examine biases present in curriculum or
against students and parents. Yet, even with this willingness, they might
still be resistant to examine how they are afforded advantages in the
world based on the way they look.
While resistance based on cultural, pedagogical, or ideological
differences is to be expected and at times may be encouraged, it has the
potential to develop into an adverse learning environment. One
consequence of this is that MTEs may receive negative course
evaluations, which may act as a deterrent for some MTEs. Indeed,
Aguirre (2009) shares how her focus on equity in her mathematics
methods courses has at times resulted in course evaluations that were
less than favorable. While Aguirre did not let these evaluations dissuade
her from teaching in a manner that was so much a part of her identity,
other MTEs, especially tenure-track faculty, may be reluctant to risk the
possibility of receiving negative feedback from students. Since course
evaluations are a large part of most, if not all, tenure files, this fear is
understandable.
Next Steps
A Call for Public Dissemination
Even though we know some about the instructional practices of MTEs
and the possible challenges they may face when teaching through a lens
of equity, there are many elements of this practice that remain elusive
especially to those MTEs who are new to this line of work. For
example, how do MTEs work with the teacher resistance they may face
(3) 289
when integrating issues such as race within their instruction? How do
they negotiate the tensions that may arise in classroom discussions
about testing biases against African-American students?
More information is needed about approaches in which MTEs have
addressed issues of equity such as race, identity, language, and culture
as a core part of the preparation of mathematics teacher educators. In
addition to work that has been published in this area, it is likely that
others have attempted to tackle this issue but have not made their
experiences public. However, it is only through a public dissemination
of experience that the field can begin to develop theories about best
approaches to prepare teachers to develop equitable mathematics
pedagogy. These theories will then allow other MTEs to develop an
understanding about how to enact their own instructional practice
centered on equity. Moreover, these theories will continue to bring the
issue of equity to the forefront of mathematics education, an important
consideration if all MTEs are to begin to recognize the importance of
helping teachers understand the inherent inequities that might take place
when teaching mathematics.
The Development of a Research Agenda
With the field of mathematical education only beginning to recognize
the importance of investigating MTEs’ impact in mathematics teacher
education (Jaworski, 2008), an increased attention to the practice of
MTEs who focus on issues of equity will also help the field of
mathematics education begin to develop a research agenda around this
area. Such a systematic examination does not currently exist on a broad
scale (Gutstein et al., 2005) and will contribute to building of models of
professional development that prepare and support other MTEs to
develop this specialized knowledge. Since MTEs play an important role
in mathematics teacher education programs, an explicit focus should be
placed on researching those MTEs who teach through a lens of equity.
There are a myriad of potential lines of inquiry to pursue. In the
following sections, some possibilities are shared. At times, specific
questions/ideas are shared; other times, some general areas that are
worthy of consideration are discussed. By no means is this section
meant to be comprehensive; rather it is intended to serve as a starting
point to promote further discussion and examination.
290
Who is Doing This Teaching and What and How are They Teaching?
As discussed earlier, the available information on the topic of equity and
mathematics teacher education has typically been documented in
published self-studies and self-reflections. As a result, only what these
individuals have chosen to share about themselves and their individual
practice is known. What is further needed is a broad scale understanding
of what Gutiérrez (2002) called the “core characteristics” (p. 175) of
this population of MTEs. This would include investigating their
knowledge, lived experiences, beliefs, attitudes, values, and
dispositions, among other things. An understanding of these core
characteristics will support the need to develop a working model of
MTEs’ knowledge base for teaching through a lens of equity.
There is also the need to develop a greater understanding of the
instructional practices of MTEs who teach through a lens of equity
across geographic (urban, suburban, rural), departmental (mathematics,
education), and grade level (elementary, secondary) contexts. This
would entail analyzing the curricular choices of these MTEs and
determining how they model what they advocate in their instructional
practice (Zeichner, 2005). Moreover, a critical examination of the role
that MTEs may play when they encounter teacher resistance is needed.
While it may be pointless to try to determine causality of why resistance
occurs, it is important to consider how an instructor’s actions might
unknowingly prompt resistance to occur.
Each of these areas alone is worthy of investigation. Yet it will not
suffice to simply examine them independently of each other; we must
understand how each impacts the other. Some specific questions that
would attend to deepening this understanding include:
1. How do MTEs’ beliefs, knowledge (be it mathematical, linguistic,
and/or cultural) and backgrounds influence their curricular choices
and how they are enacted in the classrooms (Zeichner, 2005)?
2. How do MTEs draw upon their resources (e.g. their lived
experiences, their knowledge) when they teach mathematics through
an equity lens (Gutiérrez, 2005)?
3. How might MTEs’ teaching practices impede teachers’ abilities to
teach mathematics for equity (Dunn, 2005)?
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Another important component of how MTEs teach with a focus on
equity is the support the MTEs do or do not draw upon. Therefore, an
investigation into the support that they need (regardless of whether it
exists or not) is also critical. In particular, in light of some of the
possible challenges that exist for MTEs of color and other members of
underserved and underrepresented populations to teach mathematics
with an equity focus, explicit attention must be paid to the current
support structures for this population of MTEs.
Finally, previous calls for research have indicated a need to understand
the preparation of teachers to teach mathematics to those students whose
home language is different from the official language used in schooling
(e.g., Lucas & Grinberg, 2008; Zeichner, 2005) as well as the
preparation of mathematics teachers of color (e.g., Villegas & Davis,
2002; Vomvoridi-Ivanovic, 2012). This focus on mathematics teacher
preparation would necessarily include the active participation of MTEs.
Thus, there is a need then to understand better the population of MTEs
who specifically attend to these areas in their work.
How Might We Increase This Focus on Equity?
To make teaching mathematics through a lens of equity a priority in all
teacher education programs around the globe, we need to look beyond
the existing population of MTEs who are doing this work (or have the
propensity to do so). Instead, we need to examine how MTEs in general
can develop the needed dispositions/beliefs and/or acquire the necessary
knowledge. Is it, as Aguirre (2009) or Taylor and Kitchen (2008)
proposed, through doctoral programs of studies? Or is it rather through,
or in conjunction with, ongoing MTE professional development that
involves collaboration among multiple members of the community of
mathematics education as discussed by Zaslavsky and Leikin (2004)?
We also need to look beyond methods courses and student teaching
practicum to investigate how other faculty associated with the
preparation of mathematics teachers can or do teach mathematics
content for equity. For example, what does it mean to teach a content
course through an equity lens? Researchers such as Felton, SimicMuller, and Menéndez (2012) have begun to examine the challenges and
successes involved with teaching mathematics to K-8 preservice
teachers through a sociopoliticial lens. We further ask what moves could
292
a mathematician make to teach Calculus 1 or Linear Algebra (required
courses for many secondary mathematics teachers) through an equity
lens? What mechanisms must be in place in order to support those
individuals who wish to teach these content courses in such a manner?
How might they develop the skills, knowledge, and dispositions to do
so?
Conclusion
In this paper we have presented an argument for the necessity of
expanding the public dissemination of practices that focus on integrating
issues of equity into mathematics teacher education and for developing
a research agenda around these practices. In particular, we discussed
how research on mathematics teacher preparation has not addressed the
population of MTEs as important roles in preparing teachers to teach
mathematics through this lens. We further shared how current literature
has informed the field about some of the necessary components of
MTEs’ practice and the challenges that accompany this practice.
Through this call for an increased public dissemination and the
development of a research agenda, we hope to expand the discussion on
preparing teachers to teach diverse students by focusing explicitly on
the MTEs who are involved in this preparation. In order to make equity
a priority in mathematics education, we need to move beyond the
examination of mathematics teachers and learners. We need to also
critically examine the population of MTEs who prepare teachers to
teach through a lens of equity.
Notes
Jaworski (2008) defines MTEs as “professionals who work with practicing teachers
and/or prospective teachers to develop and improve the teaching of mathematics” (p. 1).
For us, this includes all levels of faculty (tenure-track, tenured, graduate students, and
adjuncts) in undergraduate and graduate mathematics teacher preparation programs.
2 CEMELA is a Center for Learning and Teaching supported by the National Science
Foundation, grant number ESI-0424983. Any opinions, findings, and conclusions or
recommendations expressed in this document are those of the authors and do not
necessarily reflect the views of the National Science Foundation.
3 We use the term teachers to refer to both practicing teachers of mathematics and those
individuals who are preparing to become mathematics teachers.
1
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education (pp. 737-759). Mahwah, NJ: Lawrence Erlbaum
Associates.
Zevenbergen, R. (2000). “Cracking the code” of mathematics
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300
Laura McLeman is Assistant Professor in the University of
Michigan Flint, United States ofAmerica.
Eugenia Vomvoridi-Ivanovic is Assistant Professor in University
of South Florida, United States ofAmerica.
Contact Address: Direct correspondence to Laura McLeman at
303 E Kearsley Street, Flint MI 48502, USA or at
[email protected].
Analyzing students' difficulties in understanding real numbers
Michael Gr. Voskoglou 1 and Georgios D. Kosyvas 2
1 ) Graduate Technological Educational Institute, Patras, Greece.
2) Varvakio Pilot Lyceum - Palaio Psychico, Athens, Greece.
To cite this article: Voskoglou, M.G., and Kosyvas, G.D. (201 2). Analyzing
students' difficulties in understanding real numbers. Journal of Research in
Mathematics Education, 1 (3), 301 -336. doi: http://dx.doi.org/1 0.4471 /
redimat.201 2.1 6
October 2012 pp. 301-336.
Analyzing students' difficulties
in understanding real numbers
Michael Gr. Voskoglou
Graduate Technological
Educational institute
Georgios D. Kosyvas
Varvakio Pilot Lyceum Palaio Psychico
Abstract
This article reports on a study of high-school and of technologist students
(prospective engineers and economists) understanding of real numbers. Our
study was based on written response to a properly designed questionnaire and
on interviews taken from students. The quantitative results of our experiment
showed an almost complete failure of the technologist students to deal with
processes connected to geometric constructions of incommensurable
magnitudes. The results of our experiment suggest that the ability to transfer in
comfort among several representations of real numbers helps students in
obtaining a better understanding of them. A theoretical explanation about this is
obtained through the adoption of the conceptual framework of dimensions of
knowledge, introduced by Tirosh et al. (1998) for studying the comprehension
of rational numbers. Following in part the idea of generic decomposition of the
APOS analysis (Weller et al. 2009) we suggest a possible order for
development of understanding the real numbers by students when teaching
them at school. Some questions open to further research are also mentioned at
the end of the paper.
Keywords: real, rational, irrational, algebraic and transcendental numbers,
fractions, decimals, representations of real numbers.
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.16
October 2012 pp. 301-336.
Analizando las Dificultades de
los Estudiantes con la
Comprensión de los Números
Reales
Michael Gr. Voskoglou
Instituto Técnico Superior
Resumen
Georgios D. Kosyvas
Instituto Piloto Varvakio Facultad de Psiquiatría
Este artículo presenta un estudio realizado a estudiantes de un instituto y de una
escuela técnica superior (ingenieros y economistas) sobre su comprensión de
los números reales. Nuestro estudio se basó en las respuestas escritas a un
cuestionario diseñado cuidadosamente, y a entrevistas realizadas a los
estudiantes. Los resultados cuantitativos de nuestro experimento muestran un
fracaso prácticamente absoluto de los estudiantes de ingeniería para lidiar con
procesos que conecten construcciones geométricas y magnitudes
inconmensurables. Los resultados de nuestro experimento sugieren que la
habilidad para transmitir una cierta soltura en el uso de ciertas representaciones
de los números reales ayuda a los estudiantes a obtener una mejor comprensión
de los mismos. Una explicación teórica de esto se obtiene a partir del uso del
marco conceptual de las dimensiones del conocimiento, introducido por Tirosh
et al. (1998) para estudiar la comprensión de los números racionales. Siguiendo
en parte la idea de la descomposición genérica del enfoque APOS (Weller et al.,
2009), sugerimos un posible orden para el desarrollo de la comprensión de los
números reales de estudiantes cuando se les enseña este tema en la escuela. Al
final se mencionan algunas preguntas que quedan abiertas para futuras
investigaciones.
Palabras Clave: números reales, racionales, irracionales, algebráicos y
transcendentes, fracciones, decimales, representación de números reales.
2012 Hipatia Press
ISSN 2014-3621
DOI: 10.4471/redimat.2012.16
302 Voskoglou et al. - Analyzing students' difficulties with real numbers
T
he empiric approach of numbers starts from pre-school age, when
children distinguish the one among many similar objects and
count them (Gelman, 2003). This first acquaintance with
numbers helps significantly in understanding the structure of the set N
of natural numbers. For example, it contributes in clarifying the
principle of the “next natural number” leading to the conclusion that N
is an infinite set (Hartnett & Gelman, 1998). Further it supports the
development of strategies for addition and abstraction that are based on
counting (Smith et al., 2005), the comparison between natural numbers
(definition of order in Ν), etc. All these are strengthened during the first
two years of primary school, where pupils are usually studying the
natural numbers and their operations up to 1000.
Fractions and decimals are also introduced in primary school at a later
stage, while negative numbers are usually introduced at the first class of
the lower secondary education (i.e. at the 7th grade). It is well known
that students face significant difficulties in understanding rational
numbers (e.g. Smith et al., 2005). Many of these difficulties are due to
the improper transfer of properties of natural numbers to rational
numbers (Yujing & Yong-Di, 2005; Vamvakousi & Vosniadou, 2004,
2007). For example, many students believe that “the more elements a
number has, the bigger it is” (Μοscal & Magone 2000), or that
“multiplication increases, while division decreases numbers” (Fischbein
et al. 1985). They also believe that the principle of “next number” holds
for rational numbers as well (Malara, 2001; Merenluoto & Lehtinen,
2002).
Another characteristic of rational numbers that possibly affects
negatively their understanding is their multiple representations (e.g. we
can write 1/2 = 2/4 = ... = 0.5). In fact, novices tend to categorize
objects in terms of their surface rather, than their structural
characteristics (Chi et al., 1981), therefore they face difficulties in
understanding that different symbols may represent the same object
(Markovitz & Sowder, 1991). Consequently many of them think that
different representations of a rational number represent different
numbers (Khoury & Zazkis, 1994; O’Connor, 2001) and even more that
decimals and fractions are two disjoint subsets of the set Q of rational
numbers. We notice that the above false conception is taken roots as a
REDIMAT - Journal ofResearch in Mathematics Education, 1 (3) 303
habit even to many adults, who consider in all cases fractions as parts of
the whole (e.g. 1/4 of something), while there also exist other
considerations for fractions, e.g. as a ratio, as an operator, as the
accurate quotient of a division, etc. On the contrary, they consider
decimals only as quotients of divisions (e.g. 1÷4=0.25) that have a
resemblance with natural numbers.
An essential pre-assumption for the comprehension of irrational
numbers is that students have already consolidated their knowledge
about rational numbers and, if this has not been achieved, as it usually
happens, many problems are created. It has been observed that pupils,
but also university students at all levels, are not able to define correctly
the concepts of rational and irrational numbers, neither are in position to
distinguish between integers and these numbers (Hart, 1988; Fischbein,
et al., 1995). It seems that the concept of rational numbers in general
remains isolated from the wider class of real numbers (Moseley, 2005;
Toepliz, 2007).
Several reports document students’ difficulties on the topic of
repeating decimals, particularly confusion over the relationship between
0.999… and 1 (Tall & Schwarzenberger, 1978; Hewitt, 1984; Hirst
1990; Sierpinska, 1987; Edwards & Ward, 2004; Weller, Arnon, &
Dubinsky, 2009, 2011, etc.). Students in the above reports were
expected to realize that converting 0.999… to a fraction (or in some
other way) one finds that 0.999… = 1. However, mathematically
speaking, there exists actually a confusion even among the
mathematicians concerning the truth or not of the above equation. (see
Appendix 2)
Research focussed on the comprehension and proper didactical
approach of irrational numbers is rather slim. It seems however that,
apart from the incomplete comprehension of rational numbers, they are
also other obstacles (cognitive and epistemological) that make the
comprehension of irrational numbers even more difficult (Herscovics,
1989; Sierpinska, 1994; Sirotic & Zazkis, 2007a; etc).
Fischbein et al. (1995) assumed that possible obstacles for the
comprehension of irrational numbers could be the intuitive difficulties
that revealed themselves in the history of mathematics, i.e. the existence
of incommensurable magnitudes and the fact that the power of
continuum of the set R of real numbers is higher than the power of Q,
which, although being an everywhere dense set, can not cover all points
of a given interval. Their basic conclusion resulting from their
experiments with school students and pre-service teachers is that school
mathematics is generally not concerned with the systematic teaching of
the hierarchical structure of the various classes of numbers. As an effect,
most of high school students and many pre-service teachers are not able
to define correctly the concepts of rational, irrational and real numbers,
neither to identify various examples of numbers. They also found that,
contrary to their initial assumption, the concept of irrational numbers
does not encounter in general a particular intuitive difficulty in students’
mind. Hence they assumed that such difficulties are not primitive ones
and they express a relatively high level of mathematical education.
However they suggest that for a better understanding of irrational
numbers teacher should turn students’ attention on these difficulties
rather, than ignore them.
Peled & Hershkovitz (1999) when performing an experimental
research observed that pre-service mathematics teachers being at their
second and third year of studies, although they knew the definitions and
basic characteristics of the irrational numbers, they failed in tasks that
required a flexible use of their different representations. Further, Sirotic
& Zazkis (2007b) focusing on the ability of prospective secondary
teachers in representing irrational numbers as points on a number line
observed confusion between irrational numbers and their decimal
approximation and overwhelming reliance on the latter. They also used
(Zazkis & Sirotic, 2010) the distinction between transparent and opaque
representations of concepts (Lesh et al., 1987) as a theoretical
perspective in studying the ways in which different decimal
representations of real numbers influenced their responses with respect
to their possible irrationality.
According to Lesh et al. (1987) a transparent representation has no
more and no less meaning than the represented ideas or structures. On
the contrary, an opaque representation emphasizes some aspects of the
ideas or structures and de-emphasizes others. For a practical approach of
transparent and opaque representations of real numbers we give the
following examples:
The rational numbers 3/5 = 0.6, 1/3 = 0.33..., 281849/99900 =
2.821131131131... have transparent decimal representations, since one
can foresee their decimal digits in all places; but the same is not
happening with 144/233 = 0.61802575107..., which, possessing a period
of 232 digits, has an opaque decimal representation. The decimal
representations of irrational numbers are opaque in most cases due to
their complex structure, but there are also irrational numbers having
transparent representations. This happens for example with the numbers
2.001313113111311113111113... where 1, following 13, is repeated one
more time at each time, and 0.282288222888222288882... where 2 and
8, following 28, are repeated one more time at each time. We shall
return with more examples on transparent and opaque representations of
real numbers and their important role for the understanding of real
numbers by students.
The Experimental Research
Janvier (1987) describes the comprehension of a concept in general as a
cumulative process mainly based upon the capacity of dealing with an
ever- enriching set of representations. In particular, an extended research
has been developed on the role of representations for the better
understanding of mathematics (Goldin & Janvier, 1998; Goldin, 2008;
Godino & Font, 2010, etc.). Reflecting on the results of this general
research as well as on findings of experimental researches on real
numbers already mentioned above (Peled & Hershkovitz, 1999; Sirotic
& Zazkis, 2007b) we led to the hypothesis that the main obstacle for the
understanding of real numbers is the intuitive difficulties that students
have with their multiple semiotic representations, i.e. the ways in which
we describe and we write them down. In constructing the theoretical
framework of our research we put the following targets:
• To check our basic hypothesis that students’ difficulties to deal
successfully with the multiple semiotic representations of real
numbers is the main obstacle towards their understanding.
• To verify the existence of other obstacles mentioned by other
researchers, e. g. the incomplete understanding of the rational before
studying the irrational numbers, the intuitive difficulties with the
perception of incommensurable magnitudes and the “property of the
continuum of R”, etc.
• To investigate if other factors like the age, the breadth of the
mathematical material covered by students, etc, affect the
comprehension and the better use of real numbers.
Our basic tool in our experiment was a questionnaire of 15 questions
(see Appendix 1) designed with respect to the above targets. In fact,
with question 1 we wanted to check if the students were in position to
distinguish the category in which a given number belongs. Questions 25 were designed to check the degree of understanding of rational umbers
by students. Further, questions 6-8 and 13 were designed to check if
students were able to deal in comfort with the square roots of positive
integers, while questions 9-12 were connected with the density of Q and
R. Finally with question 14 we wanted to investigate students’ ability to
deal with geometric representations of real numbers and with question
15 we wanted to check if they realize that the set of irrational numbers
is not closed under addition.
Notice that the two authors studied carefully analogous questionnaires
of similar experiments performed earlier by other researchers (see back
in section 1), they had extensive discussions on the choice and
suitability of the questions involved and they attempted (together and
separately) several pilot experiments in a smaller scale before reaching
the final form of the above questionnaire. At ay case, there is no claim
that our final was the best possible. For example, it seems that there was
a problem with the choice of question 14 (explained at the end of this
section) in favor of the students of Gymnasium. Nevertheless, in general
lines the questionnaire was proved in practice to be useful in
investigating the above mentioned targets of our research.
A printed copy of the above questionnaire (in Greek language) was
forwarded to 78 students of the second class of 1st Pilot Gymnasium of
Athens (13-14 years old), one of the good public schools of lower
secondary education in Greece, by the end of school year 2008-09, i.e. a
few months after learning about real numbers. At the same time the
above questionnaire was also forwarded to 106 students of Graduate
Technological Educational Institute (T.E.I.) of Patras, from two
departments of the School of Technological Applications (prospective
engineers) and one department of School of Management and
Economics, being at their first term of studies (18-19 years old). The
students of T.E.I. had of course much more mathematical experiences
than the 14 years old students of Gymnasium, but, according to the
marks obtained in the exams for entering tertiary education, they are
considered to be moderate graduates of secondary education in general.
The choice of the subjects of our experiment was not made by chance,
neither because we had an easier access to them. Our purpose was to
compare the data obtained from two groups of different ages hoping to
obtain some conclusions about the possible effects of age and of
individual’s mathematical background in understanding better and
making a correct use of real numbers. As far as we know, a similar
experiment was performed in past only by Fischbein et al. (1995),
whose study concerned, apart from high school students, prospective
teachers of mathematics, who logically must had studied more carefully
the system of real numbers that would be in future one of their basic
objects of teaching. On the contrary, students of T.E.I. are using
mathematics as a tool for studying and better understanding their
sciences (prospective engineers and economists).
The time given to students to complete in writing the questionnaire
was one hour. The students’ answers were characterized as correct (C)
and wrong (W). In few cases of incomplete answers the above
characterization created some obscurities, which nevertheless didn’t
affect significantly the general image of student’s performance. In
Voskoglou & Kosyvas (2011) we reported in detail the percentages
(with unit approximation) of the correct and wrong answers given by
students for each question, separately for Gymnasium and T.E.I.
Therefore here we shall give only two examples of coding in order to be
understood how exactly the data of the experiment were analyzed.
The following matrix gives the percentages of wrong answers given
by students in question 1:
Table 1
Gymnasium
T.E.I.
0W
0
0
1-2 W
5
11
3-5 W
22
33
6-10 W > 10 W
21
52
36
20
The most common mistakes were the identification of the symbol of
fraction with rational and the symbol of root with irrational numbers.
The failure of many students to recognize that all the given numbers
were real numbers was really impressive. Notice that no students gave
correct answers for all cases.
Table 2
Gymnasium
T.E.I.
3C
27
0
2C
20
1
1C
13
1
3W
40
98
With this question we wanted to investigate the students’ ability to
construct incommensurable magnitudes and to represent irrational
numbers on the real axis. The answers of students of T.E.I. were really
an unpleasant surprise. Nobody constructed the length √3 correctly, only
two of them constructed √2 and only one found the point corresponding
to it on the real axis! On the contrary, the high-school students, recently
taught the corresponding geometric constructions, had a much better
performance.
A similar analysis was attempted for all the other questions. Next and
in order to obtain a statistical image of students’ performance the
completed by them questionnaires were marked in a scale from 0 to 20.
A number of units was attached to each question according to its
difficulty and the mean time required to be answered (see Appendix). In
the diagram of five numbers’ summary [maximum (xmax) and minimum
(xmin) graduation, median (M), first (Q1 ) and third (Q3 ) quarter] of the
total sample the median is lying to the left part of the rectangular
formed, which indicates accumulation to low marks (Figure 1).
Figure 1 . Five number's summary of total sample
The means obtained, 9.41 for Gymnasium, 9.49 for Τ.Ε.Ι. and 9.46 in
total, show that students’ general performance was insufficient with
regard to dexterities and cognitive capacities for real numbers evaluated
by the questionnaire. Nevertheless, from both samples becomes evident
that students possessed some basic abilities and therefore a great part of
the deficiencies observed could be corrected.
Figure 2. Frequencies of marks for Gymnasium
The diagrams of frequencies of marks separately for Gymnasium
(Figure 2) and T.E.I. (Figure 3), where marks are shown on the
horizontal axis and the numbers of students obtained the corresponding
marks on the vertical axis, give a descriptive view of our experiment’s
data.
Figure 3 . Frequencies of marks for T.E.I.
Finally, the diagram of the percentenges of marks for Gymnasium and
T.E.I. together (Figure 4) gives to the reader a better acess in making the
necessary comparisons.
Figure 4. Comparative diagram of percentages of marks of Gymnasium and
T.E.I.
The general conclusions obtained through the evaluation of our
experiment’s data are the following:
• The understanding of rational numbers was proved to be
incomplete by many students (questions 1-5 and 9-12). In general
students worked in more comfort with decimals rather, than with
fractions (questions 11, 12, etc). Further, students who failed to give
satisfactory answers to questions 1-5 and 9-12, failed also in
answering satisfactorily the rest of the questions. This obviously
means that, the incomplete understanding of rational numbers is in
fact a great obstacle for the comprehension of irrational numbers.
• Our basic hypothesis that the main obstacle for the understanding
of real numbers has to do with students’ intuitive difficulties with
their multiple semiotic representations was verified in general
(questions 5, 8, 13, etc) with a characteristic exception: The students
of T.E.I. showed an almost complete weakness to deal with
processes connected to geometric constructions of incommensurable
magnitudes and to the representation of the irrational numbers on
the real axis (question 14). However this didn’t prevent them in
answering satisfactorily the other questions.
• The density of rational and irrational numbers in a given interval
doesn’t seem to be embedded properly by a considerable number of
students, especially by those of high-school (questions 9-12).
• It seems that the age and the width of mathematical knowledge
affect in a degree the comprehension of the real numbers. In fact,
although the majority of the T.E.I. students corresponded to
mediocre graduates of secondary education, the superiority of their
answers was evident in most of the questions (apart from 3, 7, 8 and
14).
The negligible difference of means of students’ marks between
Gymnasium and T.E.I. does not represent the real situation (evident
superiority of T.E.I. students’). In fact, the means have been formed at
this level for two reasons: First, because of the total failure of the T.E.I.
students’ in answering question 14, which had the highest graduation
(2.5 units). Second, because of the high marks (16-18) obtained by a
number of students of Gymnasium in contrast to the students of T.E.I.
whose marks were below 16.
There are reasonable explanations about these facts: The total failure
of T.E.I. students’ in constructing geometrically irrational lengths by
using the Pythagorean Theorem is probably related to the low attention
given today in Greece to the teaching of Euclidean Geometry at the
higher level secondary education (Lyceum). On the contrary, the
students of Gymnasium, who recently had taught the corresponding
geometric constructions, had a better success on this topic. This is an
example of the impact that instruction could have on the students’
performance.
Also, the fact that a number of students of Gymnasium, which is one
of the good (pilot) high schools of Greece, obtained high marks (16-18)
in contrast to the students of T.E.I. is not surprising, since the students
of T.E.I. correspond to mediocre graduates of the secondary education
in general.
A New Qualitative Research
Reflecting on the answers appeared in the completed questionnaires of
our experimental research we considered useful to penetrate deeper to
the reasons that urged students in giving these answers. Therefore we
decided to make a complementary qualitative research by taking some
interviews from students. We conducted 20 in total interviews, 10 for
the students of T.E.I. by the first author and 10 for high school students
by the second author. The choice of students was based on the type of
their written answers (answers needing a further clarification) and on
their will to participate. The interviews were conducted by appointment
in the offices of the researchers and type recorders were used to save
them. The two researchers worked together to study and analyze the
interviews. Many of the students’ answers given during the interviews
were similar and therefore they were grouped. We present and analyze
below the most representative parts of the interviews separately for the
students of Gymnasium and of T.E.I.
Gymnasium
Question: Why did you answer that - √5/2 is a rational number?
Answer: Because it is a fraction.
In this case we have a classical misinterpretation of the definition of
rational numbers. The student focused her attention on the symbol of
fraction without realizing that, in order to be a rational number, its terms
must be integers, with non-zero denominator.
Question: Why did you answer that -√4 is an irrational number?
Answer: Because it has the root.
Here the student identifies the symbol of the root with an irrational
number. He does not think that the given number is equal to -2, which is
an integer. Distinction among several types of numbers remains muddy
in general, each time depending on their semiotic representations.
Question: Which is the exact quotient of the division 5÷7?
The student answered that the exact quotient of the division 5÷7 is
0.714285714285 and that he found it with a calculator. Only when he
was asked by the teacher to perform the division by hand he realized
that it never ends and that the result is a periodic decimal. In general, the
identification of a real number with its given rational approximation
(e.g. identification of π with 3.14) is a common mistake in students’
responses.
Question: Why did you answer that 2.0013131131113111131111…
is a periodic number?
Answer: Because the decimal digits following 00 are repeated in a
concrete process: 13, 131, 1311, 13111, 131111, etc. (he explains it
orally).
Question: And does it mean that this is a periodic number?
Answer: Of course!
Question: Why is this so?
The student referred to the mathematics text book of his class, where (p.
187) we read that π is not a periodic number, since its infinite decimal
digits are not repeated in a concrete process.
Our initial impression was that no student had observed the regularity
appearing in the decimal digits of the above number (Voskoglou &
Kosyvas, 2011, question 5). The explanation given from the (very good)
student for his answer has to do with a superficial definition of irrational
numbers. Frequently in text books irrational numbers are defined
(correctly) as non rational numbers (they cannot be written as fractions
with integer terms), but there is no attempt to identify them with the
incommensurable decimals, which are not defined explicitly. Some
examples are simply given for the approximate calculation of square
roots having no exact values and it is reported that, apart from such
roots, there exist other types of real numbers as well, like π.
Consequently, if the teacher does not make the necessary interventions
urging students to think on these things, children will probably remain
with the doubt: What is the form of irrational numbers in general?
Question: Why did you answer that there is no rational number
between fractions 1/10 and 1/11, as well as between decimals
10.20 and 10.21?
Answer: Because 1/11 is the next fraction of 1/10 and 10,21 is the
next decimal of 10.20.
Student’s belief in this case is that both fractions and decimals have a
next number, which is a classical case of improper transfer of a property
of natural numbers to rational numbers. Taking such opportunities,
teacher could point out (although this cannot be easily understood
without the notion of equivalent sets) that, if in the everywhere dense set
of rational numbers we characterized a number as the “next” of another
one, we should have omitted as many numbers as the whole set Q has.
Question: Why did you answer that x=√3 is the unique root of the
equation x2=3?
Answer: Because we know that the square root of 3 is a positive
number, such that x2=3.
Question: But (-√3) 2=3, is n’t it?
Answer: You are right, x=-√3 is also a root of the above equation.
Although enough hints are contained in student’s text book concerning
solution of equations of the form x2=α, the restriction imposed that the
square root must be a positive number it seems to create some confusion
to students. By accepting that for each x>0 there exist two square roots,
one positive denoted √x and the negative one - √x, this confusion could
be overcome. Using the previous notation we have no problem in
considering the relation f(x) =√x as a function, which is the basic
argument of the supporters of definition of square root as a positive
number. The rejection of the negative root, although it focuses in
keeping the one-valued property of the above function, is a restriction
that, among the others, does not permit students in understanding roots
as the inverse process of the raising to a power. Furthermore, we believe
that it is unnatural to accept, extending the restriction to roots of any
order (as it usually happens in the text books of mathematics), that 3 √-8
does not exist, despites to the fact that (-2) 3 =-8. This in a later stage
forces us to accept that the domain of the function f(x) = 3 √x is the set of
positive numbers, which intuitively cannot be easily accepted. However,
for the moment that in school books of mathematics the square root is
defined as a non negative number, teacher could be better to give
emphasis to the reasons of adopting this definition rather, and not to its
mechanical use.
T.E.I.
Question: Which is the exact quotient of the division 5÷7?
Αnswer: There is no exact quotient, since the division’s result is
an infinite decimal.
Question: What is the result and how did you find it?
Αnswer: It is 0.71428571428… and I found it by using a
calculator.
Question: Could you carry out the division by hand?
Αnswer: Yes, but why?
Αnswer: I will explain in a while.
In fact, student starts performing the division and after 6 steps he
finds 5 as remainder. At this point the instructor asks:
Question: Are you observing anything now?
Αnswer: (Rather perplexed): No.
Question: The last remainder that you found is the same with the
initial dividend. What does it mean?
Αnswer: (After thinking for a while): The same process will be
repeated again.
Question: How many times?
Αnswer: As much as we want (he is thinking…), infinitely many.
Question: Consequently what will happen with the quotient ?
Αnswer: The decimal digits 714285 will be repeated
continuously. Oh! I remember now. We have found a periodic
decimal, which is the exact quotient of the division.
Question: Correct. Nevertheless, there is no other way to
represent the exact quotient of the division, apart of writing it as a
periodic decimal?
Αnswer: (He is thinking): I don’t think so.
Question: What about the fraction 5/7?
Answer: (Surprised): Oh, yes! This is in fact the exact quotient of
the division.
An evident difficulty is revealed here in distinguishing between
different semiotic representations of rational numbers. Student had not
clarified that the exact quotient is the fraction 5/7, or alternatively the
periodic decimal 0.714285714285... He simply agreed condescendingly
with teacher’s view about the first following a question that disclosed
the correct answer. We must notice that frequently in text books is not
given emphasis to the fact that a fraction represents, among the others,
the exact quotient of the division of two integers. On the other hand,
student performing the division 5÷7 by a calculator, as it usually
happens today, was not helped in recognizing the periodicity of the
quotient, since the result obtained happened to be an opaque decimal
representation of it. On the contrary, performing the division by hand he
had the chance to realize, with the help of the teacher, that from the
moment where the same remainder was reappeared, the same process
would be repeated infinitely many times and therefore we shall have a
continuous appearance of the six digits’ period. In other words, students
performing the division between integers by hand could be exercised
better in recognizing the periodicity of the quotient:
Question: Why did you consider the equality as
= 1-√17
a correct one?
Answer: By applying the property √x2 = |x|.
Question: So √(-3) 2 = -3?
Α. No, square root is always a positive number. (He is
thinking…). I am sorry, I made a mistake. The correct is that
√x2=|x|.
Question: That is |1-√17| = 1-√17?
Αnswer: Yes.
Answer: But in this case we should have that 1>√17.
Αnswer: Ops! I am sorry. My answer was wrong. The above
equality is not correct.
A superficial application of properties of roots is appeared n this case
and of the definition of the absolute value that have not been properly
assimilated.
Question: Why did you answer that (√3 + 2) (√3 - 2) is an
irrational number?
Answer: Because it is a product of two irrational numbers.
Question: Could you make the multiplication?
Αnswer: Of course (she performs the corresponding operations by
using the distributive law). Ops, I am wrong! The result is -1.
The student had in this case the wrong belief that multiplication is a
closed operation in the set of irrational numbers. Teacher should turn
students’ attention about this illusion earlier in high school, when they
learn the real numbers for first time.
From many of the above dialogues (questions 1-4 for Gymnasium
and 1 for T.E.I.) it becomes evident the students’ difficulty in dealing
successfully with the multiple representations of real numbers
(fractions, periodic and non-periodic decimals, roots, etc.). A
theoretical explanation about this can be obtained through the
conceptual framework of dimensions of knowledge, introduced by
Tirosh et al. (1998) for studying the comprehension of rational
numbers. Their basic assumption is that learners’ mathematical
knowledge is embedded in a set of connections among the following
dimensions (types) of knowledge:
• Algorithmic dimension, concerning individual’s ability
in applying rules and prescriptions to explain the
successive steps involved in various standard operations.
• Formal dimension, concerning the ability of recalling
and applying definitions of concepts, theorems and their
proofs in problem-solving situations.
• Intuitive dimension, composed of learner’s intuitions,
ideas and beliefs about mathematical entities and
including cognitive models used to represent number
concepts and operations. This is the type of knowledge
that we tend to accept directly and confidently. It is selfevident and psychologically resistant (Fischbein, 1985).
It seems that people tend to adapt their formal and algorithmic
knowledge to accommodate their beliefs (i.e. the conclusions of their
intuitive knowledge), perhaps as a natural tendency towards
consistency. Therefore, when their beliefs are not clear and/or accurate,
it is very possible to lead to mistakes and/or inconsistencies. This is
exactly what happens with the multiple representations of real numbers,
In fact, as we have already seen above, students are frequently thinking
that different representations of the same fraction are different numbers,
that fractions and decimals or roots and decimals are sets of numbers
disjoint to each other, that infinite decimals are equal to their given
finite approximations (e.g. π=3.14, e=2.71, 144/233 = 0.6180257, etc.)
and so on. All these wrong beliefs, when they have been formed in the
individual’s cognitive structures, it is very difficult, according to the
explanations provided by the conceptual framework of the dimension of
knowledge, to be changed later.
Teaching Real Numbers at School
Weller et al. (2009, 2011) report on the mathematical performance of
pre-service elementary and middle school teachers who completed a
specially designed experimental unit on repeating decimals that was
based on APOS (Action, Process, Object, Schema) theory and
implemented using the ACE (Activities in the computer, Classroom
discussion, Exercises done outside of class) teaching style. The
quantitative results of their experiments suggest that the students who
received the experimental instruction made considerable progress in
their development of understanding the relation between a rational
number (fraction or integer) and its decimal expansion.
The implementation of APOS theory as a framework of learning and
teaching mathematics involves a theoretical analysis of the concepts
under study in terms of the mental constructions a learner might take in
order to develop understanding of the concepts, called a generic
decomposition (GD). It comprises a description that includes actions,
processes and objects, which describe the order in which it may be best
for learners to experience them. While we do not fully employ the idea
of a GD here, the construct is a useful one and helped in suggesting a
possible order for the development of the understanding of real
numbers. The following suggestions were based not only on the
outcomes of our experimental research presented above, but also on our
many years didactical/pedagogical experience in secondary and tertiary
education.
There are several methods known for the construction of the set of
real numbers (Voskoglou & Kosyvas 2011, section 3). Apart of their
representation as infinite decimals (where a finite decimal can be
written as an infinite one, with period equal to 0 or to 9, e. g. 2.5 =
2.500... = 2.499...) the rest of these methods are too abstract to deal with
in a regular curriculum for school mathematics.
Two prerequisites seem to be indispensable for a successful
presentation of real numbers as infinite decimals at school:
• First, students must have realized that periodic decimals
and fractions are the same numbers written in a different
way.
• Second, the definition of non periodic decimals must be
given in a strict and explicit way, so that it could not give
rise to any misinterpretations: An infinite decimal is a non
periodic decimal not because its decimal digits are not
repeated in a concrete process (this in fact could happen, as
the relevant examples given in our introduction show), but
because it has not a period, i.e. its decimal digits are not
repeated in the same concrete series.
The first of the above prerequisites helps students to realize the
equivalence between the two definitions of irrational numbers given at
school: As non rational numbers (i.e. they cannot be written as fractions
µ/v, µ,v Z, v ≠ 0) and as incommensurable decimals on the other hand.
For this, students must have clearly understood that, for each fraction
µ/v, µ, v Z, v ≠ 0, the quotient of the division µ÷v is always a
periodic decimal. The probability to be a finite decimal is small enough,
since a fraction, whose denominator is not a product of powers of 2
and/or 5, cannot be written as a finite decimal. In case of an infinite
decimal, students must be in position to observe that, since the
remainder of the division µ÷v is smaller that v, performing the division
and after a finite number of steps the same remainder will reappear at
some step. This means that from this point and so on the same digits
will appear periodically in the quotient again and again, infinitely many
times. Conversely, students must be in position to convert periodic
numbers (either simple ones, or mixed) to fractions. We recall that a
standard method for doing this (although they are others as well) is by
subtracting both members of proper equations containing multiples of a
power of 10 of the given number. For example, given x= 2.75323232…,
we write 10000x = 27532.3232… and 100x = 275.3232…., wherefrom
we find 9900x = 27532 - 275, or x = 27257 / 9900.
An instructional treatment for the definition of non periodic decimals
could be to ask students to calculate the finite approximations of square
roots of non quadratic positive integers. For example, √2 is written as
√2 and is constructed as the limit of the sequence 1, 1.4, 1.41, 1.414,
1.4142... of its finite (rational) approximations.
The concepts of a sequence of rational numbers and of its limit (i.e.
what it means to “tend” to a number) should be presented in a practical
way by teacher (the detailed study of these topics is a didactic object in
an upper level of studies) and explained to students through the above
examples. The dots at the end of the number indicate that the sequence
of its decimal digits is continued. Students must understand that the
acceptance of this symbolic representation of an infinite decimal does
not mean that we can see written all its decimal digits. We can only see
the digits of its given decimal approximation each time.
For students it is difficult in general to understand a number if they
don’t know an explicit way of writing it down. Therefore it is very
important to give frequently opportunities to them to rethink critically
about the decimal representations of real numbers. For example, let us
consider the following (vertical) pairs of numbers:
The rational numbers of the first row have a period of 6.232 and 1860
digits respectively, while the irrational numbers of the second row have
not any regularity concerning the appearance of their decimal digits. As
most of the decimal digits of all the above numbers remain unknown,
given only their decimal representations you cannot be sure where they
are, or not, rational numbers . In fact, although a number of digits of the
above vertical pairs of numbers coincide, the rest of them remain
unknown. As a result their possible rationality or not depends upon the
completion of their decimal representations with their opaque parts. In
converting a fraction to a decimal a long and laborious division is
reached, if the quotient obtained is an infinite decimal having a long
period, which is not possible to be determined soon.
We also observe that, if we restrict the decimal representations of the
above numbers to their digits written in bold only, then they take the
following form:
Now the decimal representations of the corresponding vertical pairs of
numbers coincide to each other. Consequently it is completely
impossible to conclude whether they are rational numbers, or not.
Problems however are increasing when we arrive to the expected
(since students already know that fractions can be written as periodic
decimals) question: Which numbers can be written as incommensurable
decimals? Firstly, students realize that this happens with the square
roots of non quadratic positive rational numbers. Later they learn that
the same happens with the roots of any order whose value is not a finite
decimal. Nevertheless they are also irrational numbers having not this
form, or, in a more general context, numbers which are not roots of a
polynomial equation with coefficients in Q, i.e. which are not algebraic
numbers. In this way we approach the concept of transcendental
numbers, with π and e being the better known examples. It can be
shown that the set of algebraic numbers is a denumerable set, while
Cantor proved that the set of transcendental numbers has the power of
continuous. This practically means that transcendental are much more
than algebraic numbers, but the information that we have about them is
very small related to their multitude. That is why we have characterized
them as a “black hole” (with the astronomical meaning of term) in the
“universe” of real numbers (Voskoglou, 2011). The instructor could give
to students a brief description of algebraic and transcendental numbers,
so that to obtain a complete view of the whole spectre of real numbers.
A good opportunity is given in reviewing the basic sets of numbers at
Lyceum before studying the complex numbers. However references at
earlier stages are not excluded, since this new kind of numbers usually
activates students’ imagination and increases their interest by creating a
pedagogical atmosphere of mystery and surprise.
The quantitative results of our experimental research (section 2) show
that the complete failure of the students of T.E.I. to deal with processes
connected to geometric constructions of incommensurable magnitudes
didn’t prevent them in answering satisfactorily the other questions about
the real numbers. However our didactical/pedagogical experience
suggests that the teaching of geometric representations of real numbers
at school helps in general their better understanding by students. We
shall close this section with some comments on it.
It seems that within the culture of ancient Greek mathematics the
geometric figure was the basis for unfolding mathematical thought,
since it helped in obtaining conjectures, fertile mathematical ideas and
justifications (proofs). In fact, convincing arguments are built by
drawing auxiliary lines, optical reformations and new modified figures,
and therefore mathematical thinking becomes more completed in this
way. For example, the invention of the existence of incommensurable
line segments by the Pythagorean philosophers was the starting point for
the discovery of irrational numbers.
Most of irrational numbers, like 3 √2, π, e, etc., correspond to lengths
of line segments that cannot be constructed by ruler and compass only.
Nevertheless, at school level we correspond all these numbers to points
of the real axis in an axiomatic (or approximate, if you prefer to call it
so), way, which usually is not clearly understood by students (actually it
is based on the principle of nested intervals). At the 27 th Panhellenic
Conference on Mathematics Education of the Greek Mathematical
Society that took place in Chalkida (2010) we had the opportunity to
hear the description of an experienced colleague, who is teaching for
years in a very good private school (Gymnasium) and who became
embarrassed when she was asked by a student the following question:
“Are there any circles whose length of circumference is a rational
number? For example does it happen for the circle of radius 1/π? ”
Algebraically speaking the student’s remark was logical. The problem
however is that the length 1/π and therefore the corresponding cycle also
cannot be geometrically constructed!
In contrast to the ancient Greek mathematics, numerical thought is the
most frequently used at school today. This is logically expected, since
numerical excels geometrical culture in our contemporary world and
therefore it plays the main role in representations that students build at
school. Nevertheless, we have the feeling that the excessive use of
numerical arguments wounds the geometrical intuition. In fact, we
believe in general that a rich experience of students with geometric
forms, before being introduced to numerical arguments and analytical
proofs, is not only useful, but it is indispensable (Arcavi et al., 1987).
The geometric representations of real numbers enrich their teaching,
connecting it historically with the discovery of existence of
incommensurable magnitudes and the relevant theory of Eudoxus.
Activities of geometric constructions of irrational numbers could
be organized in classroom combining history of mathematics with
Euclidean Geometry, like the problem of doubling the volume of a cube
(Delion problem), which is appeared in Plato’s dialogue “Menon”
(Kosyvas & Baralis, 2010).
Based on those discussed in the present section, we conclude that,
since the probability of appearance of opaque representations of rational
and irrational numbers is high from one hand, and because of the
existence of transcendental numbers on the other hand, some voids,
inconsistencies, or misconceptions remain often to students, but even to
adults after finishing school, concerning the understanding of real
numbers. Therefore teacher’s attention is necessary in preventing such
phenomena.
We finally ought to clarify that all that we have discussed here are
simply some ideas aiming to help the instructor towards the difficult
indeed subject of the didactic approach of real numbers at school level.
However, by no means they could be considered as an effort to
introduce, or even more to impose, a model of teaching, because our
belief is that the effort of introducing such a model is actually a utopia!
In fact, teacher should be able to make a small “local research”,
readapting methods and plans of the teaching process according to the
special conditions of his (her) class (Voskoglou, 2009).
In general lines our didactic proposition includes: A fertile utilization
of already existing informal knowledge and beliefs about numbers,
active learning through rediscovery of concepts and conclusions,
construction of knowledge by students individually, or as a team, in
classroom. Construction of knowledge follows in general student’s
optical corner, while teacher’s role is limited to the discussion in the
whole class of wrong arguments and misinterpretations observed. The
teaching process could be based on multiple representations of real
numbers (rational numbers written as fractions and periodic decimals,
irrational numbers considered as non rational ones and as
incommensurable decimals, which are limits of sequences of rational
numbers, roots, geometric representations, etc.) and on flexible
transformations among them.
Conclusions
The understanding of irrational numbers is fundamental for students of
secondary education in reestablishing and extending the notion of
numbers. Nevertheless, the transition from the set of rational numbers to
the set of real numbers strikes against inherent difficulties, connected to
the incomplete understanding of rational number and to the nature of
irrational numbers.
According to the mathematics curricula of secondary education and
the restricted abilities of students’ at this age in understanding abstract
and difficult concepts, the only suitable method for presenting the real
numbers at school is by using their decimal representations.
Our basic hypothesis for our experimental research reported in this
article was that the main obstacle for the understanding of real numbers
is the difficulties that students face in dealing with their multiple
semiotic representations, i.e. the ways in which we describe and we
write them down.
The first part of our research was based on students’ written response
to a properly designed questionnaire. The novelty of this study has to do
with the choice of the subjects of our experiment, consisting of highschool students (13-14 years old) a few months after learning about the
real numbers and students of a graduate technological institute (18-19
years old) using mathematics as a tool for studying and better
understanding their sciences (prospective engineers and economists). As
far as we know a similar experiment was performed in the past only by
Fischbein et al. (1995) with high-school students and prospective
teachers of mathematics, while analogous experiments performed by
other researchers with prospective or pre-service teachers only.
The quantitative results of our experiment showed an almost complete
failure of the technologist students to deal with processes connected to
geometric constructions of incommensurable magnitudes. However, and
contrary to our hypothesis about the role of their semiotic
representations for the understanding of real numbers, this didn’t
prevent them in answering satisfactorily the other questions. In fact,
although the majority of them correspond to mediocre graduates of the
secondary education, the superiority of their correct answers with
respect to those of high-school students was evident in most cases. This
is a strong indication that the age and the width of mathematical
knowledge of the individual play an important role for the better
understanding of the real numbers. This is crossed by the findings of
Fischbein et al. (1995), which however were more or less expected,
since they concern prospective teachers of mathematics.
In general (with the exception of the geometric representations) our
basic hypothesis was verified by the experiment’s results, since students’
performance was connected to their ability of flexible transformations
among the multiple representations of real numbers. Apart from the
above contributions to the research literature, the results of our
experiment verified also findings of experiments performed by other
researchers, connecting students’ difficulties in understanding the real
numbers with the incomplete understanding of rational numbers, the
incommensurability and nondenumerability of irrational numbers, the
frequently appeared opaque representations of rational and irrational
numbers, etc.
Reflecting on certain characteristic answers appeared in the completed
questionnaires of our experimental research we considered useful to
penetrate deeper to the reasons that urged students in giving these
answers. Therefore we decided to make a complementary qualitative
research by taking some interviews from students. From the dialogues
of these interviews presented above it becomes (among the others)
evident again the students’ difficulty to deal successfully with the
multiple representations of real numbers. A theoretical explanation
about this was obtained through the adoption of the conceptual
framework of dimensions of knowledge, introduced by Tirosh et al.
(1998) for studying the comprehension of rational numbers.
Following in part the idea of generic decomposition of the APOS
analysis (Weller et al., 2009) we suggested a possible order for
development of understanding the real numbers by students when
teaching them at school. Based on those discussed we concluded that,
since the probability of appearance of opaque representations of rational
and irrational numbers is high from the one hand, and because of the
existence of transcendental numbers on the other hand, some voids,
inconsistencies, or misconceptions remain often to students, but even to
adults after finishing school, concerning the understanding of real
numbers. Therefore teacher’s attention is necessary in preventing such
phenomena.
In general terms, our didactic proposition includes a fertile utilization
of the already existing informal knowledge and beliefs about numbers,
active learning through rediscovery of concepts and conclusions,
construction of knowledge by students individually, or as a team, in
classroom. The teaching process could be based on multiple
representations of real numbers and on flexible transformations among
them.
Open Questions - Epilogue
The discussion made in this article marked out the following open to
further study and research questions concerning the understating and
teaching of real numbers at school:
• How useful is for their better understanding the enrichment
of teaching of real numbers with geometric representations?
The data of our classroom experiment did not permit us to
obtain an explicit conclusion about this, since the almost
total failure of T.E.I. students’ in constructing geometrically
incommensurable lengths and/or in corresponding them to
points of the real axis did not seem to prevent them in
answering successfully the other questions.
• How students could understand better the approximate/
axiomatic correspondence of incommensurable magnitudes
that cannot be geometrically constructed to points of the real
axis? For example, for the construction of length 3 √2
(doubling the volume of a cube with edge equal to the unit of
lengths) we could use the graph of function f(x) = 3 √x (or
f(x) = 3 √x - 2) constructed in absolute exactness (Sirotic &
Zazkis, 2007b). Nevertheless this could be succeeded only
by the help of a computer, which means that it will be a
distance between the theoretic and the practical approach of
the problem.
• Which is the proper way, for each level of education, to
study the continuum of R in contrast to the density of Q? In
other words how students could be persuaded that in a given
interval (of numbers, or of points, if we consider the real
axis) it is possible to have an infinity of elements of a certain
type (rational numbers, or points) when this is not
compatible with usual logic and our intuition?
• How we could communicate to students the image of
mathematics as an organized whole, where the systems of
numbers play an important role? In this way students could
get the feeling of the grandeur, the beauty of mathematics as
a fundamental human achievement, not only its utility for
practical matters (Fischbein et al. 1995).
In answering last question it is heard faintly to suggest a turn to “new
mathematics”, where the whole teaching is based on theory of sets,
algebraic structures and mathematical logic, like it happened with
educational reform of 1960’s, that was proved to be a complete failure.
Nevertheless, it could be useful to be examined, if and how much the
teaching, in a simple and practical approach, of some elements from
theory of algebraic structures at the last class of the secondary
education, could help for a better and deeper understanding of real
numbers. More explicitly, that Q and R with respect to the known
properties of addition and multiplication (subtraction is defined in terms
of addition, and division in terms of multiplication) have the structure of
a field (it is not necessary to give the definitions of a group and a ring
before and the corresponding axiomatic foundations), and the concept of
isomorphism as a 1-1 correspondence between fields “preserving” the
properties of operations. For example, the concept of isomorphism
could help students to understand why the set of all series kn/10n, with
k0 Z, and k1 , k2, ... kv, ... natural numbers less than 10, not all equal to
9, coincides in practice with the set of real numbers (see Appendix 2),
and, later on, why the same happens with R2 and the field C of complex
numbers. All these could be taught either in parallel with reviewing the
basic sets of numbers, that usually precedes the teaching of complex
numbers, or as part of a voluntary, experimental course, together with
other mathematical regularities.
We are aware that the above idea will possibly give rise to critiques of
the form: “When constructivism is today the predominant theory for
learning, such formalistic approaches are out of place and time”.
Nevertheless our belief is that in matters like this we must not be
absolute. In fact, none of epistemological/philosophical trends in
mathematics and its didactics could be considered as the perfect one.
Each one of them has its advantages and its weak points that affect in an
analogous way the march of mathematical science. Therefore the
required thing is to find a kind of “balance” among them (Voskoglou
2007, section 5), so that to be able to drive forward more effectively a
combined scientific and didactic vision for research and teaching of
mathematics.
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Appendix 1
List of questions of our experimental research
1. Which of the following numbers are natural, integers, rational,
irrational and real numbers?
(Units 2)
2. Are the following inequalities correct, or wrong? Justify your
answers.
3. Which is the exact quotient of the division 5÷7? (Unit 1)
4. Convert the fraction 7/3 to a decimal number. What kind of decimal
number is this and why we call it so? (Unit 1)
5. Are 2.8254131131131… and 2.00131311311131111… periodic
decimal numbers? In positive case, which is the period? (Units 1,5)
6. Find the square roots of 9, 100 and 169 and describe your method of
calculation. (Unit 1)
7. Find the integers and the decimals with one decimal digit between
which lies √2. Justify your answers. (Units 1,5)
8. Characterize the following expressions by C if they are correct and by
W if they are wrong: √2 = 1.41, √2 =1.414444…, √2 1.41, there is no
exact price for √2. (Units 1,5)
9. Find two rational and two irrational numbers between √10 and √20.
How many rational numbers are there between these two square roots?
(Unit 1)
10. Find two rational and two irrational numbers between 10 and 20.
How many irrational numbers are there between these two integers?
(Unit 1)
11. Are there any rational numbers between 1/11 and 1/10? In positive
case, write down one of them. How many rational numbers are between
the above two fractions? (Unit 1)
12. Are there any rational numbers between 10.20 and 10.21? In
positive case, write down one of them. How many rational numbers are
in total between the above two decimals?
13. Characterize the following expressions as correct or wrong. In case
of wrong ones write the corresponding correct answer.
, the unique solution of the
equation x2 = 3 is x = √3,
= 1 - √17. (Units 2)
14. Construct, by making use of ruler and compass only, the line
segments of length √2 and √3 respectively and find the points of the real
axis corresponding to the real numbers √2 and -√3. Consider a length of
your choice as the unit of lengths. (Units 2,5)
15. Is it possible for the sum of two irrational numbers to be a rational
number? In positive case give an example. (Unit 1)
Appendix 2
Discussion on the decimal representations of the real numbers and the
equation 0. 999… =1
In most books on Number Theory and Number Systems (e.g. Hardy &
Wright 1993, Sierpinski 1988, Feferman 1989, etc) it is argued that a
non negative real number, say x, is expressed as a decimal, or
equivalently it has a decimal representation, if
In the above expression [x] denotes the integral part of x (i.e. the largest
integer not exceeding x) and ci , i=1,2,3,…, are integers such that 0 ≤ ci
≤ 9. We write then x=[x],c1 c2c3 … A negative real number can be
expressed as a decimal by using the decimal expansion of its opposite
number in the obvious way.
It is well known that any non negative real number x has a decimal
representation of the form (1) (e. g. Kalapodi 2010; Theorem 3.2). More
specifically, if x has a finite decimal representation, then it has exactly
two decimal representations (Kalapodi 2010; Theorem 3.7); e.g. 2.5 =
2.5000… = 2.4999… On the other hand, if x has no finite decimal
representation (infinite decimal), then it has a unique decimal
representation, in which there exist infinitely many ci’s different from 9
(Kalapodi, Theorem 3.5 and Theorem 4.5). We recall that a decimal
representation of the form (1) is called finite, if there exists an index i0
such that ci = 0, for all i ≥ i0. Notice that, in any decimal representation
of the form (1) at least one of the ci’s must be different from 9. In fact,
assume that x = [x] + 9/10i (2) is a decimal representation of the form
(1). Then, since 9/10i is a decreasing geometric series with common
ration 1/10, we get that 9/10i = (9/10) / (1-(1/10)) = 1. Thus x=[x] +1,
which is impossible, since, according to its definition, [x] is the largest
integer not exceeding x. Consequently, all the expressions of the form
(2) cannot be accepted as decimal representations of real numbers in the
sense of definition (1). In particular, although the series
9/10i
converges to 1, we can not accept the form 0.999... as a decimal
representation of 1. The question arising under the above data is what is
actually the meaning of the symbol κ0.999…, with κ0 a non negative
integer. Having in mind that instead of saying that the sum (i.e. the limit
of the sequence of its partial finite sums) of a given series, say Σ, is
equal to α, we usually write Σ=α, where the symbol “=” has not the
usual meaning of equality in this case the answer could be that the
above symbol represents the series κ0 + 9/10n and not its sum, which
is equal to the real number κ0+1. A number of colleagues believe that,
for reasons of mathematical consequence, we must accept in general
that all symbols of the form κ0,κ1 κ2… κn…, with κ0 a nonnegative
integer and κ1 , κ2,..., κn,... natural numbers less than 10, represent the
series κn/10n and not its sum, which is equal to the corresponding real
number. Consequently the representation of real numbers as infinite
decimals has no meaning at all!
Fortunately the results obtained when using these representations are
conventionally correct, because the corresponding operations could be
performed in an analogous way among the sequences of the partial sums
of the corresponding series. This allows us to pass through this sensitive
matter at school level without touching it at all. However, from the
above analysis it becomes evident that all the above problems (let me
characterize them as pseudo problems, because, as we’ll see below, they
can easily be solved) are created due to the fact that the definition of the
decimal representations of real numbers is given in the form (1). In fact,
one can extend definition (1) by accepting that any positive integer, say
k, apart from its usual (let us call it main) decimal representation, has
also another one (let us call it secondary) of the form x = k-1.999…,
where [x]= k. In particular the secondary decimal representation of 1 is
x=0.999…, with [x] = 1.
Probably, an easy way to avoid giving all these explanations at school
level (which obviously could create confusion to students) is to define
the set R of real numbers in terms of their decimal representations. In
fact, from the above analysis it becomes evident that in order to consider
each real number only once, one must take into account only the
decimal expressions (and their opposite numbers) of the form
a,c1 c2c3 …, with a and ci (i=1,2, 3,…) integers, a ≥ 0, 0 ≤ ci ≤ 9, where
there exists an index i0 such that it is not ci = 9 for all i ≥ i0 .
The first author wishes to thank his colleague at the Graduate
Technological Educational Institute of Patras, Greece, Dr. Aleka
Kalapodi for the useful discussions on the decimal representations of
real numbers that helped him in enlightening some important details
contained in this Appendix.
Michael Gr. Voskoglou is Professor of Mathematics and
Sciences in the School of Technological Applications, Graduate
Technological Educational Institute, Patras, Greece.
Georgios D. Kosyvas is Teacher of Mathematics and teachers'
advisor in the Varvakio Pilot Lyceum - Palaio Psychico, Athens,
Greece.
Contact Address: Direct correspondence to Michael Gr.
Voskoglou at T.E.I., 26334 Patras, Greece or at
[email protected]
Opening the Cage. Critique and Politics of Mathematics
Education.
Yuly Vanegas Muñoz1
1) Universidad de Barcelona, España.
Date of publication: October 24th, 2012
To cite this article: Vanegas, Y. (2012). Opening the Cage. Critique and
Politics of Mathematics Education. REDIMAT - Journal of Research in
Mathematics Education, 1 (3), 337-339. doi: 10.4471/redimat.2012.17
To link this article: http://dx.doi.org/10.4471/redimat.2012.17
REDIMAT Journal of Research in Mathematics Education Vol. 1 No. 3
October 2012 pp. 337339
Review
Skovsmose, O., and Greer, B. (Eds.) (2012). Opening the Cage.
Critique and Politics of Mathematics Education. Rotterdam – Boston –
Taipei: Sense Publishers.
Opening the Cage es un libro revelador, que realiza una crítica profunda
a las matemáticas y su enseñanza, desde la clásica concepción
occidental de la matemática. Los autores hacen un llamamiento a las
raíces del pensamiento crítico en la didáctica de las matemáticas, y traen
al centro del debate ideas que ya empezaron a aportar personas como
Claudia Zaslavsky, Peter Damerow, Ulla Elwitz, Christine Keitel,
Jürgen Zimmer, Ubiratan D’Ambrosio, Dieter Volk, Stieg MellinOlsen,
Marilyn Frankenstein, o Alan Bishop. Estos autores y autoras, desde sus
diferentes puntos de partida, y desde sus diferentes enfoques y
tradiciones de trabajo, nos han dejado preguntas tales como si es posible
la equidad en la enseñanza de las matemáticas, sobre las que hoy en día
existe un debate que goza de una gran salud y profusión de ideas.
Espacios como el congreso anual CERME (Congress of the European
Society for Research in Mathematics) o el MES (Mathematics,
Education and Society) son ejemplos que muestran la actualidad de los
análisis de corte social (o incluso sociológico) en la didáctica de las
matemáticas.
Este libro es una denuncia clara que seguro que no dejará a nadie
indiferente. Pone la figura del docente de matemáticas en el centro de la
palestra, y claramente resalta su papel “político” como responsable de
muchas de las decisiones que se producen en torno al currículum de
2012 Hipatia Press
ISSN 20143621
DOI: 10.4471/redimat.2012.17
338 Yuly Vanegas Opening de Cage
matemáticas que se implementa dentro del aula y otros factores que
intervienen en los procesos de enseñanza y aprendizaje de las
matemáticas.
El libro se organiza en cuatro bloques de contenidos. En primer lugar,
Eric (Rico) Gutstein, Alexandre Pais, Munir Jamil Fasheh y Brian
Greer discuten sobre la idea de la educación matemática y su conexión
con las políticas. Usando referentes cruciales como es el caso del
trabajo y de la obra de Paulo Freire, estos autores a lo largo de sus
respectivos capítulos ponen ejemplos claros de cómo las matemáticas
aparecen en medio de las luchas de diferentes comunidades para tener
la oportunidad de aprender. Ya sea a través de las calles del barrio
Latino en Chicago, o visitando los entresijos de un organismo como el
National Mathematics Advisory Council en Estados Unidos, pasando
por lugares frontera como el caso de Palestina, los autores sitúan las
matemáticas, y especialmente su didáctica, en el centro mismo de las
luchas. La idea de leer y releer el mundo críticamente de Freire emerge
de estas viñetas por su propio peso.
Más adelante, Marta Civil, Sikunder Ali Baber, Mamokgethi Setati,
Nuria Planas, Gelsa Knijnik, Fernanda Wanderer, Danny Bernard y
Maisie Gholson en el segundo bloque de contenidos analizan
situaciones extremas por lo delicado del juego de fuerzas entre los
actores implicados. Ya sea el caso de las personas inmigrantes Latinas
en zonas fronterizas en Estados Unidos, el caso de la comunidad
tremendamente diversa de Sudáfrica, los campesinos sin tierra en
Brasil, las niñas pakistaníes en Barcelona, o la comunidad
afroamericana en Chicago, en donde encontramos ejemplos claros que
ilustran cómo la enseñanza de las matemáticas se enfrenta a múltiples
situaciones del contexto, ajenas quizás a lo que es la matemática
propiamente dicha, pero no a su didáctica, ni por supuesto al impacto
sobre los colectivos de personas más vulnerables.
Estas reflexiones nos llevan a considerar, entre otros aspectos, la
relación entre la enseñanza de las matemáticas y el poder. Sobre este
tema se centra el tercer bloque del libro. Aquí Brian Greer, Swapna
Mukhopadhyay, Keiko Yasukawa, Ole Skovsmose y Ole Rvan discuten
ideas clásicas del pensamiento crítico, como es la reflexión sobre la
hegemonía de las matemáticas. Esta discusión de corte gramsciano deja
REDIMAT Journal of Research in Mathematics Education, 1 (3) 339
paso a consideraciones más pedestres: las matemáticas en entornos
cotidianos como el laboral, por ejemplo, y el impacto de la
alfabetización numérica en el acceso a oportunidades de vida en la
sociedad actual.
Finalmente, el libro se cierra con un conjunto de capítulos donde los
autores y autoras hacen el esfuerzo de plantear posibilidades de
desarrollos futuros. Eva Jablonka, Uwe Gellert, Annica Anderson, Ole
Ravn, Bill Atweh, Ole Skovsmose y Brian Greer buscan inspiración
para llevar la democracia y el acceso a las ideas matemáticas más allá
de los límites impuestos por las estructuras sociales. El programa de
Educación Matemática Crítica aparece como una respuesta natural a
esta necesidad de ampliar los horizontes de la matemática, reconsiderar
los fines de su enseñanza y aprendizaje, en búsqueda de lo que los
propios autores denominan una "educación matemática socialmente
relevante".
Para acabar, nada mejor que una reflexión de uno de los grandes
investigadores que ha inspirado muchas de las ideas expuestas en este
libro, Ubiratán D’Ambrosio: la matemática tiene un tremendo
potencial, tanto para liberar a las personas, como para ser la más
cruenta de las herramientas. Nuestra reflexión sobre las matemáticas,
nos debería llevar a verlas y usarlas como un instrumento para obtener
una paz global. Me atrevería a decir, incluso, que las matemáticas, y la
investigación en este ámbito, tendrían que servir para que todas las
personas tengan oportunidades de ser libres y decidir sobre sus vidas.
Yuly Vanegas Muñoz
Universidad de Barcelona
Email: [email protected]

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