Estrategias y representaciones en la resolución de problemas aritméticos de división en estudiantes con trastorno del espectro autista: un estudio de caso (Strategies and representations in the resolution of arithmetic division problems in students with autism spectrum disorder: a case study)Dra. Irene Polo-Blanco(Universidad de Cantabria)Dra. Alicia Bruno (Universidad de La Laguna) Dra. María José González (Universidad de Cantabria) Dña. Beatriz Olivera (Universidad de Cantabria)Páginas 161-180ISSN: 1889-4208e-ISSN: 1989-4643Fecha recepción: 07/04/2018 Fecha aceptación: 26/11/2018ResumenEn el presente trabajo se describen las estrategias y las representaciones que un estudiante con Trastorno del Espectro Autista (TEA) manifiesta al resolver problemas aritméticos verbales de división. El estudiante ha seguido un proceso de aprendizaje de los problemas basado en la metodología Concreto- Representacional-Abstracto que, a largo plazo, tiene como finalidad que aprenda el algoritmo de la división. Se analizan las estrategias y las representaciones que el estudiante manifiesta durante las dos primeras etapas de la secuencia (Concreto y Representacional). Se concluye que el estudiante hace uso de las mismas estrategias que alumnos de desarrollo típico, aunque adquiere finalmente una estrategia de bajo nivel (reparto uno a uno). Además, se observan representaciones pictóricas muy detalladas a lo largo de toda la experiencia que ralentizan la resolución y desvía la atención de los elementos matemáticos del problema. Dichas representaciones aparecen combinadas a lo largo de la secuencia con representaciones esquemáticas. Finalmente, se proponen posibles pautas de actuación que puedan ayudar al alumnado TEA en el aprendizaje de las operaciones aritméticas.Como citar este artículo:1089620486115Polo-Blanco, I., Bruno, A., González, M.J., y Olivera, B. (2018). Estratégias y representacionesen la resolución de problemas artméticos de división en estudiantes con trastornos del espectro autista. Revista de Educación Inclusiva, 11(2),161-180.Palabrasclave:Estrategias,problemasaritméticosdedivisión, representaciones, Trastorno del Espectro AutistaAbstractThis paper describes the strategies and representations that a student with Autism Spectrum Disorder (ASD) shows when solving verbal division arithmetic problems. The student has followed an instruction based on the Concrete- Representational-Abstract methodology, which, in the long term, aims to acquire the learning of the division algorithm. The strategies and representations that the student reveals during the first two stages of the sequence (Concrete and Representational) are analyzed. It is concluded that the student displays the same strategies as typically developing children do, although he finally acquires a low level strategy (one-to-one distribution). In addition, very detailed pictorial representations are observed throughout the experience that slow down the resolution and might divert the attention of the student from the mathematical elements of the problem. These representations appear combined along the sequence with schematic representations. Finally, possible action guidelines that may help ASD students in learning arithmetic operations are proposed.Keywords: Strategies, division arithmetic problems, representations, Autism Spectrum DisorderIntroducciónDurante las dos últimas décadas, muchos estudios se han preocupado por la enseñanza de la resolución de problemas matemáticos en estudiantes con dificultades de aprendizaje (Montague, 1992). Algunos de estos trabajos tratan la resolución de problemas de multiplicación y división en la etapa primaria (Xin y Jitendra, 1999). Sin embargo, escasean los trabajos que tratan la resolución de problemas matemáticos en estudiantes con trastorno del espectro autista (TEA), centrándose la mayoría en la estructura aditiva (Cihak y Foust, 2008, Rockwell, Griffin, y Jones, 2011), mientras que pocos de ellos ponen el foco en la estructura multiplicativa (Levingston, Neef, y Cihon, 2009, Whitby, 2012).Las investigaciones sobre los estudiantes con TEA indican que, con frecuencia, sus habilidades de cálculo están por debajo de la media con respecto a estudiantes de desarrollo típico (Estes, Rivera, Bryan, Cali, y Dawson, 2011), por lo que requieren de un proceso más a largo plazo para alcanzar la etapa final de desarrollo de hechos numéricos y algoritmos. Por otra parte, es necesario aportarles ayudas visuales y representaciones que faciliten el desarrollo de significados sobre las operaciones (Llorca, Plasencia y Rodríguez, 2009, Cihak y Foust, 2008, Rockwell, y otros, 2011).Este estudio pretende contribuir al conocimiento de cómo un estudiante con TEA comienza a dar significado a la operación de división cuando se le instruye usando problemas aritméticos verbales de una etapa de división partitiva. Para ello, se sigue una secuencia instruccional con tres etapas, denominada Concreta-Representacional-Abstracta (CRA), ya empleada conéxito en otros estudios sobre la enseñanza de las matemáticas en estudiantes con TEA (Stroizer, Hinton, Flores y LaTonya, 2015). No abordamos aquí la etapa Abstracta debido a que su propósito centrado en la memorización de los hechos numéricos y el desarrollo de destrezas operatorias, se aleja del objetivo central de las dos primeras etapas, orientadas al desarrollo de significados sobre la operación.Este trabajo forma parte de una investigación más amplia sobre el aprendizaje de la división de un estudiante con TEA. En el presente artículo texto se describen las estrategias de resolución de problemas de división de reparto y se analizan en profundidad las representaciones gráficas que el estudiante utilizó en la etapa de instrucción Representacional, así como el papel que éstas han jugado en dicha resolución. A partir de esto, se plantean posibles pautas a tener en cuenta en la enseñanza de operaciones aritméticas con estudiantes con TEA.Modelos de enseñanza para estudiantes con TEA. El modelo Concreta- Representacional-AbstractaPara recorrer el proceso de aprendizaje de las operaciones aritméticas, distintas metodologías de enseñanza se apoyan en plantear a los estudiantes problemas aritméticos verbales (Puig y Cerdán, 1988). En el caso de alumnado con TEA, algunas metodologías se centran en instruir en habilidades metacognitivas que ayuden al alumnado en el proceso de resolución (leer, comprender, visualizar, dar una hipótesis, estimar, calcular y comprobar) (Whitby, 2012). Otras metodologías proponen seguir estrategias basadas en el uso de diagramas esquemáticos (Schema Based Instruction) para enseñar a los estudiantes a discriminar entre distintos tipos de problemas y resolverlos (Rockwell, y otros, 2011).Una secuencia de instrucción que se ha mostrado beneficiosa a la hora de enseñar las operaciones aritméticas a niños con dificultades de aprendizaje es la denominada Concreta-Representacional-Abstracta (CRA). En esta secuencia, en la etapa Concreta se manipulan objetos físicos para resolver problemas aritméticos. En la etapa Representacional se utilizan imágenes y modelos gráficos para representar los objetos que aparecen en el problema. En la etapa Abstracta se usan números, operaciones y símbolos matemáticos. En el caso de estudiantes con TEA, Stroizer, y otros (2015) siguieron una instrucción CRA para la enseñanza de la suma y resta con llevadas y para los hechos numéricos de la multiplicación de cero a 5, con tres estudiantes que presentaban este trastorno. Los autores lograron mejoras en la compresión conceptual de estas operaciones que repercutieron en los procedimientos de resolución de las mismas. Estos autores indican la necesidad de continuar investigando sobre intervenciones que sigan el proceso CRA en alumnado con TEA.Problemas aritméticos verbales de división y estrategias de resoluciónDiversos estudios realizados con estudiantes de desarrollo típico sostienen que la acción de reparto surge de manera natural y que en los problemas de división partitiva está en el origen la comprensión de la división como operación (Bryant, 1997, Correa, Nunes, y Bryant, 1998) Para describir estos problemas utilizaremos las palabras “recipientes” y “objetos”. De esta forma, en el enunciadode los problemas de división partitiva aparecen tres cantidades: número de recipientes, número total de objetos a repartir y número de objetos por recipiente. La incógnita del problema es el número de objetos que tiene cada recipiente, por ejemplo: “Tenemos 6 caramelos y queremos repartirlos en 3 bolsas ¿Cuántos caramelos caben en cada bolsa?”.Mulligan (1992) describe los tres niveles siguientes para abordar la resolución de un problema de división partitiva: (1) modelado directo con conteo,conteo sin modelado directo y (3) hechos conocidos o derivados.En el nivel de modelado directo con conteo se hace uso de objetos concretos que permiten realizar la acción descrita en el problema. Downton (2008) incorpora en este nivel la posibilidad de que los objetos utilizados no sean concretos sino dibujos. Además, en este nivel se pueden llevar a cabo diferentes estrategias para llegar a la solución:Estrategia de reparto uno a uno: se manifiesta cuando el estudiante reparte los objetos uno por uno en cada recipiente y al finalizar cuenta los objetos que hay en uno de los recipientes.Estrategia de reparto por múltiplos: tiene lugar cuando el estudiante estima una cantidad y la asigna a cada recipiente, al finalizar comprueba que todos los objetos han sido utilizados. En caso de que esto no ocurra, repite el proceso considerando otra estimación inicial sobre la cantidad de objetos.Estrategia de reparto por ensayo y error: ocurre cuando se reparten de forma aleatoria los objetos en los recipientes para, a continuación, ajustar el tamaño de cada grupo hasta conseguir la equidad.En el nivel de conteo sin modelado directo se realizan las mismas acciones que en el nivel anterior, pero no existe una utilización de objetos, sino un desarrollo de los aspectos verbales, de los procesos de resolución y de la visualización del modelo del problema. Por ejemplo, en el caso del problema anterior, utilizarían un conteo rítmico, siguiendo su estructura: 1, 2 y 3, 4, 5 y 6 para el reparto uno a uno, o bien, dirían 2, 4, y 6, para el reparto por múltiplos).Finalmente, en el nivel de hechos conocidos o derivados se realizan operaciones mentales de suma/resta reiterada (p.e., 2+2+2=6) o de multiplicación o división para alcanzar el resultado de la división (p.e., 2x3=6, 6:3=2).Los estudiantes de desarrollo típico manifiestan alguna de las estrategias anteriores con más frecuencia que otras y, en muchos casos, siguen una evolución (Downton, 2008). Por ejemplo, la mayoría de los estudios coinciden en afirmar que la estrategia reparto uno a uno es la menos frecuente, y que en los casos en los que esta estrategia se manifiesta, los estudiantes enseguida progresan a otras más eficientes basadas en la estimación y reparto por múltiplos (Carpenter, Ansell, Franke, Fennema, y Weisbeck, 1993, Mulligan, 1992).Se conoce poco sobre la manera en que las personas con TEA afrontan el aprendizaje de la división. Excepción es el trabajo de Levingston, y otros (2009) en el que se evalúa un método de enseñanza de problemas de multiplicación y división en dos sujetos, uno de ellos con TEA. Sin embargo, en este trabajo no se distingue ni profundiza en el sentido que dan los estudiantes a estos problemas según su estructura de reparto o agrupamiento, lo que da muestra de la necesidad de realizar estudios sobre esta operación. Nuestro estudio pretende contribuir a determinar qué peculiaridades tienen estas estrategias en los estudiantes con TEA.Importancia de las representaciones en la resolución de problemas aritméticos verbales por estudiantes con TEANumerosos trabajos se han centrado en estudiar el tipo de representación gráfica (dibujos o esquemas) que los estudiantes utilizan al resolver problemas aritméticos, y en cómo estas influyen en la resolución. En estudiantes de desarrollo típico, estas representaciones suelen seguir una evolución en el tiempo desde lo concreto hasta lo abstracto (dibujo realista, dibujo matemático y representación simbólica) (Hughes, 1987). Por otro lado, algunos autores argumentan que el uso de dibujos realistas puede dirigir el razonamiento del estudiante a detalles irrelevantes, desviando así la atención de los elementos esenciales del problema (Presmeg, 1986). Siguiendo esta línea, Hegarty y Kozhevnikov (1999) identificaron dos tipos de representaciones en la resolución de problemas por parte de estudiantes de 12 años de desarrollo típico: las representaciones esquemáticas (líneas, puntos, etc.), que incluyen las relaciones espaciales descritas en el problema y que son relevantes para la resolución, y las representaciones pictóricas, que muestran una apariencia real de los objetos o personajes implicados en el problema, incluso con detalles irrelevantes para la resolución. Estos autores mostraron que el uso de representaciones esquemáticas estaba relacionado con un mayor éxito en el problema, al contrario que el uso de representaciones concretas. En adelante, se utilizará esta terminología para referirnos al tipo de representación utilizada por el estudiante sujeto del estudio.Los resultados anteriores cobran especial relevancia en los estudiantes con TEA, puesto que a menudo presentan un estilo de procesamiento centrado en el detalle, denominado coherencia central débil. El interés en profundizar en cómo se manifiesta este estilo de procesamiento en distintos ámbitos ha dado lugar a numerosas publicaciones. Por ejemplo, Booth, Charlton, Hughes y Happé (2003) examinaron el procesamiento global y local en tareas de dibujo con grupos de diversas patologías. Su estudio compara estudiantes con TEA, con trastorno de déficit de atención e hiperactividad y estudiantes de desarrollo típico, todos de la misma edad y cociente intelectual. Entre sus principales conclusiones destaca que el estilo de dibujo centrado en el detalle era propio únicamente del grupo TEA.Objetivos de la investigaciónEn esta investigación se aborda el aprendizaje inicial de la división de un estudiante con TEA a través de la resolución de problemas de división partitiva. El estudiante sigue una secuencia de instrucción que respeta la metodología Concreta-Representacional-Abstract (CRA). En este trabajo se analiza lo que ocurre durante las etapas Concreta y Representacional, orientadas al desarrollo de significados sobre la operación aritmética.El objetivo del estudio que se presenta es describir los procedimientos de resolución de los problemas de división reparto que lleva a cabo el estudiante, especificando:Los niveles y las estrategias de resolución que manifiesta.Los tipos de representaciones que realiza, referidas a las cantidades y estructura del problema.Metodología de investigaciónSe ha seguido una metodología de un estudio de caso descriptivo, realizada con un estudiante que presenta TEA, de 11 años y 8 meses de edad en el comienzo del estudio. A partir de ahora nos referiremos al estudiante con el pseudónimo de Tom.Las características del estudiante son las siguientes. Tom es un varón diagnosticado a la edad de 6 años con TEA por un neuropediatra. En ese momento, según el inventario de desarrollo Battelle (BDIST) presentaba una edad de 5 años en aspectos cognitivos y de 4 años en aspectos comunicativos. En la actualidad sigue mostrando importantes alteraciones del lenguaje y dificultades en la relación social, principalmente con sus iguales.Tom está escolarizado en un centro de educación especial. Comparte aula con otros cuatro alumnos de edades comprendidas entre 7 y 13 años que presentan distintos trastornos de personalidad o deficiencia cognitiva. Recibe clases ordinarias de las diferentes materias escolares, dedicando 4 horas a la semana al trabajo en matemáticas. Tom presenta un lenguaje funcional con un nivel de comprensión elevado, tanto oral como escrito, y con un amplio vocabulario. Se comunica por medio de frases cortas gramaticalmente correctas. Su capacidad lectora es equiparable a un nivel de tercero de Primaria (8-9 años). Ello le lleva a leer y comprender los enunciados de los problemas, aunque presenta dificultades de comprensión de determinadas palabras. Como estudiante, es disciplinado y muestra especial interés en tareas manipulativas y de dibujo.Tom sigue un currículo adaptado de matemáticas y recibe clases de apoyo y refuerzo de lenguaje y matemáticas. Los contenidos numéricos que ha trabajado antes de la experimentación son: sumar y restar con llevadas (no tiene memorizados los hechos numéricos), resolución de problemas de suma y resta de manera escrita, utilizando dibujos y utilizando el formato de los algoritmos tradicionales por columnas, problemas de multiplicación, aunque no tenía las tablas memorizadas, lo que le llevaba a resolver estos problemas con dibujos y a obtener los resultados de las multiplicaciones como suma reiterada, o con un recuento de objetos sobre los dibujos realizados. En un estudio inicial sobre los conocimientos previos del estudiante (Polo-Blanco, González, Olivera, y Bruno, 2017) se observó que sabía resolver problemas de división por agrupamiento a través de dibujos (por ejemplo, problemas del tipo: “Tenemos 6 caramelos, si en cada bolsa ponemos 2 caramelos, ¿cuántas bolsas necesitamos?”). Sin embargo, no fue capaz de llegar a una respuesta en los problemas de división partitiva, los cuales afrontaba con estrategias erróneas, tanto de tipo multiplicativo como a través de dibujos. Por esa razón, esta investigación se centra en los problemas de división partitiva.InstrucciónLa instrucción se realizó en clases de apoyo escolar, durante una sesión de trabajo semanal, habitualmente de una hora de duración. A Tom se le plantearon entre 2 y 5 problemas en cada sesión, en función del grado de la receptividad y concentración que mostrase en la sesión. Durante tres meses, en 15 sesiones de trabajo se realizaron un total de 49 problemas.Diferentes investigaciones realizadas con estudiantes con TEA han mostrado los beneficios del uso de materiales manipulativos y visuales para el aprendizaje de las operaciones aritméticas (Cihak y Foust, 2008, Rockwell, y otros, 2011). Por otra parte, la utilización de lenguajes aumentativos, como pictogramas, beneficia y facilita la comunicación con las personas con este trastorno (Mirenda, 2003). Ambos resultados nos llevaron a elaborar un material concreto que al mismo tiempo toma el aspecto de un pictograma, que denominamos “Pictomaterial”. Este material consiste en cartulinas en la que aparecen dibujadas unas casillas rectangulares vacías y fichas que deben ser repartidas en dichas casillas, además, aparecen dibujadas unas flechas que representan la acción de repartir (ver Figura 1). Mediante este material se aporta una ayuda visual y manipulativa para que el estudiante asocie la acción de reparto con la de situar las fichas en las casillas.253873014351000Figura 1. Pictomaterial para la división 5:5Este material tiene un aspecto similar a los esquemas utilizados en trabajos previos realizados con alumnos con TEA (Rockwell et. al, 2011) en los que se ha utilizado la metodología Squema Based Instruction, pero aporta además la posibilidad de manipulación de los objetos implicados en el reparto. Se diseñaron distintas cartulinas, variando el número de casillas dibujadas, en función del divisor. En los problemas en los que se aportó este material, se proporcionó al estudiante la cartulina correspondiente al divisor del problema y tantas fichas como indicaba el dividendo.Como ya se ha indicado, en esta investigación, utilizamos la secuencia de enseñanza CRA para la enseñanza de la división a través de problemas de división partitiva. Aunque en este trabajo solo describiremos lo ocurrido en las etapas Concreta y Representacional, la instrucción completa incluyó las tres etapas, de la siguiente forma:Concreta: Se presenta el problema al estudiante mediante un texto escrito que va acompañado del material concreto que se menciona en el enunciado (por ejemplo, coches y cajas) o del pictomaterial. El pictomaterial se utilizó especialmente en las primeras sesiones y fue aceptado de modo natural por el alumno. Posteriormente, el material concreto y el pictomaterial se fueron intercalando, con la intención de que el pictograma permitiese al estudiante asociar la idea de reparto con un esquema visual que pudiera usar como referencia.Representacional: El texto del problema se acompaña de un dibujo que representa al divisor (ver Figura 2). No se aporta material concreto ni pictomaterial. La intención es ayudar al estudiante a que identifique yrepresente por sí mismo el dividendo y el divisor, lo que nos permite ver de qué manera evolucionan las representaciones que utiliza.2452425214828Figura 2. Enunciado de un problema con dibujo de representación del divisorAbstracta: En la hoja con el enunciado del problema se acompaña la simbología del algoritmo de la división a través de una imagen con las casillas en las que debe colocar los números (dividendo, cociente y divisor).Cada enunciado del problema se proporcionó en formato escrito en un folio. Los enunciados eran sencillos e implicaban objetos conocidos para el estudiante (bolígrafos, piruletas, fichas etc.). Las cantidades que representaban el dividendo variaron entre 6 y 28.Las sesiones fueron dirigidas por la tutora. En la primera sesión, se explicó a Tom el método de resolución de los problemas, focalizando la acción de reparto por medio del pictomaterial. De esta forma, Tom comenzó a asociar esta acción con el significado de la palabra “repartir”. En sesiones sucesivas, la tutora siempre dejaba a Tom resolver por sí mismo cada problema con el fin de observar sus propias estrategias sin intervenir en ellas. Cuando Tom encontraba dificultades que no podía resolver por sí mismo, la tutora le ofrecía alguna ayuda, bien sugiriéndole el uso de material, bien haciéndole observar que la solución obtenida no era correcta. En este sentido, cabe señalar que el pictomaterial resultó de gran ayuda ante las dificultades que surgieron en los primeros momentos de la etapa Representacional: por un lado, la tutora lo utilizó como apoyo para ayudar al estudiante a desarrollar la estrategia de reparto uno a uno en los problemas enunciados sin material, por otro lado, el estudiante lo solicitó en algunas ocasiones en las que no sabía qué hacer en dichos problemas.En la instrucción se le proporcionaron 49 problemas distribuidos en 15 sesiones (S1 a S15): 23 de ellos con material concreto (pictomaterial o material cotidiano) y 26 problemas en formato escrito (presentados con o sin dibujo del divisor). Durante las tres primeras sesiones solo se realizaron problemas de la etapa Concreta. A partir de la sesión 4 se introdujeron los problemas escritos sin apoyo del material (etapa Representacional). Al cabo de 6 semanas de finalizar la instrucción se llevaron a cabo ocho sesiones de mantenimiento (M1 a M8) en las que se le proporcionaron 22 problemas (11 con material y 11 en formato escrito).Recogida de datosLa recogida de datos se realizó durante las sesiones descritas. Se recopilaron las respuestas a los problemas que Tom realizó por escrito y se grabaron todaslas sesiones en video. Este material fue analizado de forma independiente por tres investigadoras, quienes para cada uno de los problemas, identificaron los niveles y estrategias para abordar la resolución de un problema. Dado que nos centramos en la etapa Concreta y Representacional el análisis de datos realizado distingue entre:El tipo de estrategia seguida, según la tipología establecida por Mulligan (1992): reparto uno a uno, reparto por múltiplos, reparto por ensayo y error. Se interpreta que estas acciones pueden estar asociadas al uso de material concreto, al pictomaterial o a otro tipo de dibujos realizados por el estudiante.El tipo de representación utilizada, según la clasificación de Hegarty y Kozhevnikov (1999): pictórica o esquemática, de las cantidades de los problemas de la etapa Representacional.Posteriormente, se contrastaron los análisis realizados por las tres investigadoras y se discutieron los casos hasta alcanzar un 100% de acuerdo.En algunos problemas, el estudiante llevo a cabo varios intentos de resolución. En esos casos se recogió la información de cada uno de los intentos. Así mismo, la tutora registró los aspectos actitudinales y de comportamiento del estudiante antes de la sesión y durante la misma. Esta información se utilizó para anular algunas de las sesiones en las que se detectó una actitud negativa y falta de concentración por parte del estudiante que fueron repetidas en momentos de mayor receptividad.ResultadosSe presentan a continuación las características de las estrategias seguidas por el estudiante y el tipo de representación utilizada durante la resolución de problemas aritméticos verbales de división partitiva, atendiendo al formato en el que se le presenta al estudiante el problema: problemas sin dibujo, y con apoyo de material concreto o pictomaterial (etapa Concreta) y problemas sin apoyo de material (etapa Representacional).4.1 Características de las estrategias al resolver problemas con material concreto o pictomaterialEn los problemas presentados en este formato, se manifestaron las tres estrategias de reparto identificadas por Mulligan (1992): (a) reparto uno a uno,(b) reparto por múltiplos y (c) reparto por ensayo y error. La tabla 1 muestra la evolución de las estrategias de los problemas de la etapa Concreta. Se distingue las estrategias, con y sin éxito, en la resolución de los problemas planteados. Cada celda, corresponde a una sesión y a un tipo de estrategia, y aparece marcada con “X” cuando en dicha sesión el alumno ha manifestado dicha estrategia en algún momento de la resolución de un problema. Puede darse el caso que ante un mismo problema el alumno emplee varias estrategias diferentes, especialmente cuando no ha llegado a éxito y la instructora lo invita a resolverlo de nuevo.Tabla 1Evolución temporal de las estrategias problemas con material: reparto uno a uno (RUU), reparto por múltiplos (RM) y ensayo y error (EE)Sesion esSSSS1234SSSS5678SS1S1901S1S123S1S145M1-81143635114300Exito00ExitoRUUXXXXXXRMXXXXXXXXX EEXRUUX114363583185No00NoRMXXEEXXLa estrategia de reparto por múltiplos fue la preferida por el estudiante dentro de los problemas con material concreto. En las tres primeras sesiones tuvo algunas dificultades, pero posteriormente las superó con éxito. Se dio la circunstancia de que en ninguno de los casos hizo ajuste posterior, es decir, eligió directamente el cociente como grupo inicial. Se interpretó que hacía algún tipo de cálculo mental previo, de tipo sumativo, que le permitía obtener el cociente de forma sencilla cuando los números dados eran pequeños. Por ejemplo, en la Figura 3 se muestra cómo hace un reparto por múltiplos de 8 bolígrafos en 4 vasos, repartiendo directamente 2 bolígrafos en cada vaso.211963019050000Figura 3. Estrategia reparto por múltiplos sin ajuste posterior con respuesta correcta (división 8:4)En los casos en los que obtuvo resultados incorrectos con esta estrategia fue, bien porque eligió una cantidad inicial mayor que el resultado, dejando recipientes vacíos (ver Figura 4), o bien porque eligió una cantidad inicial menor que el resultado, sobrándole elementos que no repartió.285623018986500Figura 4. Ejemplo de estrategia de reparto por múltiplos con respuesta errónea (división 6:3)La siguiente estrategia más utilizada en contexto de material fue la de reparto uno a uno. Esta estrategia no surgió de forma espontánea, sino que fue introducida por la tutora para ayudarle a resolver los problemas sin material. Una vez que conoció esta estrategia, el estudiante comenzó a usarla también en los problemas con material. De hecho, el reparto uno a uno fue la acción que finalmente siguió utilizando con éxito de forma estable en el tiempo, incluso en los casos en los que no era eficiente, por tener números grandes. Por ejemplo, en la Figura 5 se muestra la acción de reparto uno a uno que lleva a cabo el estudiante en un problema en el que ha de repartir 28 alubias en 4 platos.109601019177000300101019177000474535519177000Figura 5. Estrategia de reparto uno a uno con respuesta correcta (división 28:4)La estrategia de reparto uno a uno dio lugar a una solución correcta en casi todos los casos.Los casos en los que realizó un reparto por ensayo y error en problemas con material fueron muy escasos (3 de 24) en las primeras sesiones de la instrucción. Un ejemplo se muestra en el problema de las Figuras 6, en el que el estudiante hizo un primer reparto aleatorio, para posteriormente ajustarlo de manera correcta.11224761744524043553179913Figura 6. Estrategia de ensayo y error con respuesta correcta (división 15:3)El comportamiento del estudiante fue igual con material concreto y con pictomaterial en términos de las estrategias manifestadas. En algunas ocasiones en las que se le presentó un enunciado sin material en la etapa Representacional, este solicitó el pictomaterial para resolverlo.4.2. Características de las estrategias al resolver problemas sin materialesLos problemas presentados en este formato de texto sin material concreto ni pictomaterial se plantearon a partir de la cuarta sesión y en ellos Tom también manifestó las tres estrategias de reparto: (a) reparto uno a uno, (b) reparto por múltiplos y (c) reparto por ensayo y error. La Tabla 2 muestra la evolución de las estrategias de los problemas sin apoyo de material, con y sin éxito.Tabla 2Evolución temporal de las estrategias problemas sin material: reparto uno a uno (RUU), reparto por múltiplos (RM), ensayo y error (EE) y multiplicativa (M)S1S2S3S4S5S6S7S8S9S10S11S12S13S14S15M1-8ExitosaRU UXXXXXXRMXXXXXEEXXXRU UXXNoRMXXXXXXEEXAl comienzo de estas sesiones, el estudiante manifestó una estrategia errónea de reparto por múltiplos, con una acción de tipo multiplicativo, dibujando la cantidad total a repartir en cada uno de los recipientes, en lugar de separar el total en grupos. Esto se observa en el ejemplo de la Figura 7, cuando el estudiante, para dividir 12 piruletas en dos bolsas, dibuja 12 piruletas en cada bolsa.299593018986500Figura 7. Respuesta de reparto por múltiplos con respuesta errónea (división 12:2)Para superar esta dificultad, se propuso a Tom que, antes de comenzar a repartir, dibujase la cantidad total de objetos fuera de los recipientes y que después uniese cada objeto a un recipiente, mediante flechas (ver Figura 8).Figura 8. Estrategia reparto por múltiplos con dibujo previo del dividendo con respuesta correcta (división 12:4)Aunque esta ayuda sirvió para que abandonara la estrategia multiplicativa descrita en la Figura 7, resultaba poco operativa con cantidades elevadas, por la confusión que le producían las líneas conectoras. Por ello se le sugirió que en lugar de unir mediante conectores, tachara el objeto una vez que lo dibujaba en cada recipiente.239522018986500Figura 9. Estrategia por ensayo y error, con dibujo previo del dividendo con respuesta incorrecta (división 15:5)Un ejemplo es el de la Figura 9 en el que sigue una estrategia de ensayo y error errónea. Con el fin de ayudarle a superar estas dificultades, se le instruyó en la estrategia de reparto uno a uno, complementada con el dibujo previo de la cantidad a repartir. Desde un comienzo el estudiante asumió esta estrategia e incluso, como hemos indicado antes, la trasladó a los problemas con material. A partir de la sesión 12 siguió esta estrategia que le llevó a éxito en todos los problemas y la mantuvo hasta la última sesión. Una vez aprendida esta estrategia, el estudiante la ha mejorado, llegando a llevar la cuenta de los objetos repartidos mentalmente.En resumen, el alumno resuelve los problemas de división partitiva con rasgos del nivel de modelado directo con conteo, aspecto esperado debido a las características de la instrucción y a la forma de plantear los enunciados de los problemas. Como se aprecia en las Tablas 1 y 2, los problemas con materiales resultaron más exitosos que los de enunciado con dibujo. Lo cual indica que la secuencia seguida fue correcta para este estudiante con TEA, pues le ayudó a cambiar las estrategias incorrectas. En los problemas con material el estudiante se decanta primero por la estrategia de reparto por múltiplos, logrando éxito en la resolución y a continuación, adopta la estrategia de reparto uno a uno, tambiéncon éxito. Por otra parte, en los problemas con dibujo manifestó una mayor variedad de estrategias. Dado que presentó dificultades en algunos casos, se le presentó la estrategia de reparto uno a uno, la cual consolidó, pues le daba seguridad en el procedimiento y éxito en sus respuestas. Esto le llevó a no volver a usar la estrategia de reparto por múltiplos. Este resultado difiere del alumnado sin discapacidad para quienes la estrategia de reparto por múltiplos resulta más sencilla (Carpenter, y otros, 1993, Mulligan, 1992).4.3 Características de las representacionesEn la etapa Representacional (recuérdese que comenzó en la sesión 4) cobró especial relevancia las representaciones gráficas realizadas por el estudiante. Por ello, se analizaron todas las representaciones realizadas por el estudiante a lo largo de las sesiones y se identificaron dos tipos: pictóricas y esquemáticas (ver Tabla 3). Téngase en cuenta que el estudiante, en ocasiones, realizó varias representaciones en el mismo problema, y se han analizado todas ellas.Tabla 3Tipo de representaciones manifestadas en los problemas sin materialSesionesRepresentaciónS4S5S6S7S8S9S10S11S12S13S14S15M1- 8TotalPictórica031212421111827Esquemática201115331221325Como se aprecia en la Tabla 3, se manifestaron casi con la misma frecuencia las representaciones pictóricas y esquemáticas. Ambas aparecen a lo largo de toda la instrucción y no se aprecia una evolución temporal desde los dibujos concretos hacia abstractos, como podría haberse esperado.El análisis de las representaciones pictóricas en relación a los objetos a repartir indicados en el enunciado de los problemas, refleja en algunos casos la realización de dibujos muy detallados por parte del estudiante. Este fue el caso del ejemplo de la Figura 10, en el que los objetos a representar eran flores, las cuales dibujó con alto nivel de detalle.217043019050000Figura 10. Representación pictórica (división 15:3)Otros objetos (coches, piruletas, bolígrafos) fueron representados con distinto nivel de detalle. A modo de ejemplo, se muestran distintas representaciones de bolígrafos: de representación pictórica a esquemática (Figura 11).Figura 11. Representaciones pictórica y esquemática (División 8:4)Algo similar ocurrió en los ejemplos de las Figuras 12, a la hora de dibujar coches, que el estudiante representó con distinto nivel de detalle, en este caso durante dos intentos del mismo problema.184023019050000Figura 12. Representaciones pictóricas (División 15:3)Otro ejemplo, de la tendencia a dibujar los detalles se observan en la Figura 13 de las piruetas, las cuales representa mediante un círculo y una raya, y en ocasiones, dibujó los palos con relieve.224663019050000Figura 13. Representación pictórica (División 24:4)En el caso de la representación de las piruletas, en ocasiones el niño realizaba una representación esquemática (dibujando círculos para representar las piruletas) y al terminar de resolver el problema completaba el dibujo (añadiendo los palos). Esto parece indicar que la representación pictórica no está relacionada con la resolución del problema, sino con un deseo de perfeccionamiento del dibujo y que realmente no la necesita ni utiliza para resolver el problema.Por otro lado, algunos objetos (gominolas, fichas y balones) fueron siempre representados con círculos, lo que agilizaba considerablemente la resolución y facilitaba que el estudiante se centrara en las relaciones entre las cantidades.Como se ha mencionado anteriormente, los niños de desarrollo típico se suelen apoyar en representación pictóricas en una etapa inicial de la resolución de problemas, y a medida que avanzan en el aprendizaje van abandonándolas para ir adquiriendo otras más esquemáticas (Hughes, 1987). En el caso de Tom, se ha observado que realiza numerosas representaciones pictóricas, combinadas con las esquemáticas a lo largo de toda la instrucción. El uso de las primeras no parece formar parte de la resolución de los problemas, sino de su gusto por el dibujo detallado. Esta tendencia a dibujar con detalle ha ralentizadoen ocasiones la resolución, y, como afirma Presmeg (1986) puede ser un motivo de distracción que aleje de los elementos matemáticos del problema.ConclusionesHemos presentado cómo un estudiante con TEA se inicia en el aprendizaje de la división, mediante problemas de división partitiva. Se ha observado que el alumno se sitúa en el nivel de modelado directo con conteo, y manifiesta las mismas estrategias que las identificadas en estudios con alumnos de desarrollo típico, pero se han apreciado diferencias significativas.El estudiante comenzó utilizando estrategias basadas en reparto por múltiplos en los problemas con material tangible, que resolvió con éxito en la mayoría de los casos. Sin embargo, necesitó apoyo específico para transferir las acciones que realizaba con objetos concretos a situaciones en las que no disponía de dicho material. Por este motivo, se le enseñó la estrategia básica de reparto uno a uno mediante el dibujo del total y el tachado de los objetos a medida que los iba repartiendo. El estudiante mantuvo esta estrategia de reparto uno a uno de forma estable, e incluso la trasladó a los problemas con material en los que había utilizado estrategias más sofisticadas con anterioridad. Esto supone una diferencia en el orden que suelen manifestarse las estrategias en estudiantes de desarrollo típico quienes suelen utilizar estrategias de reparto uno a uno al inicio del aprendizaje de la división para desarrollar posteriormente estrategias más eficientes, basadas en reparto por múltiplos (Carpenter, y otros, 1993, Mulligan, 1992). Esta predilección del estudiante sujeto del estudio por una estrategia más rudimentaria podría estar relacionada con la preferencia de los alumnos del espectro autista por métodos mecánicos que les aportan más seguridad (Bae, 2013).En relación con las representaciones realizadas por el alumno, se ha observado una propensión por el dibujo detallado realista en muchas resoluciones. Este gusto por el detalle en los dibujos, frecuente en personas del espectro autista (Booth, y otros, 2003), ha supuesto en algunos casos para Tom una distracción, por desviar su atención de los datos y preguntas del problema, entorpeciendo en ocasiones la resolución. A la vista de estos resultados, parece necesario aprovechar la fortaleza que ciertas personas con TEA muestran por el dibujo, pero llevándolos a formas de utilizarlo de manera efectiva, sin que un detalle excesivo aleje la atención de los elementos matemáticos del problema. Con este fin, pensamos que la implicación de objetos sencillos en los enunciados de los problemas, cuya representación concreta no difiera en exceso de una representación esquemática (como balones o fichas en el caso del sujeto del estudio), resulta beneficiosa.La incorporación del pictomaterial como intermediario entre el material concreto y el pictograma ha proporcionado de forma exitosa una familiarización con la división partitiva y ha servido para trabajar la superación de algunos de los errores manifestados por el alumno. Al igual que se mostró beneficioso el uso de esquemas asociados a las estructuras de los enunciados de los problemas aditivos con estudiantes con TEA (Rockwell, y otros, 2011), nuestro trabajo sugiere la posibilidad de utilizar el pictomaterial a modo de lenguaje aumentativo para trabajar también problemas de multiplicación y división.Los resultados expuestos arrojan información sobre el proceso de aprendizaje de la división en estudiantes con TEA y sugieren pautasmetodológicas que ayudan al estudiante a resolver con éxito problemas de división partitiva. Se trata de un primer avance imprescindible para una posterior comprensión de los hechos numéricos y el algoritmo de la división. Como en muchos estudios con estudiantes que tienen dificultades de aprendizaje en matemáticas, se ha seguido una metodología de estudio de casos que no permite generalizar, pero que puede servir de base para posteriores investigaciones con personas de este colectivo.Agradecimientos: Este trabajo ha sido financiado por el Proyecto de Investigación “Resolución de problemas y competencia matemática en la educación primaria y secundaria y en la formación de profesores”. EDU2017- 84276-R. MINECO. Madrid.Referencias bibliografícasAmerican Psychiatric Association. (2000). Diagnostic and statistical manual of mental disorders (4th ed., text rev.). Washington, DC: Author.Bae, Y. S. (2013). Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development (Ph.D. Dissertation). Columbia University. ProQuest.Bryant, P. (1997). Mathematical understanding in the nursery school years. In T. Nunes y P. Bryant (Eds.), Learning and teaching mathematics: An international perspective (pp. 53-67). Hove: Psychology Press.Booth, R., Charlton, R., Hughes, C., y Happé, F. (2003). Disentangling weak coherence and executive dysfunction: Planning drawing in Autism and ADHD. Philosophical Transactions of the Royal Society, 358, 387–392.Carpenter, T., Ansell, E., Franke, M., Fennema, E., y Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children’s problem- solving processes. Journal for Research in Mathematics Education, 24, 428-441.Castro, E., Rico, L., y Castro, E. (1995). Estructuras aritméticas elementales y su modelización. México D. F.: Grupo Editorial Iberoamérica.Cihak, D. F., y Foust, J. L. (2008). Comparing number lines and touch points to teach addition facts to students with autism. Focus on Autism and Other Developmental Disabilities, 23, 131-137.Correa, J., Nunes, T., y Bryant, P. (1998). Young children’s understanding of division: the relationship between division terms in a noncomputational task. Journal of Educational Psychology, 90, 321-329.Downton, A. (2008). Links between children’s understanding of multiplication and solution strategies for division. In M. Goos y K. Makar (Eds.), Proceedings of the 31st annual conference of the Mathematics Education Research Group of Australasia (pp. 171–178). Sydney, Australia: Mathematics Education Research Group of Australasia.Estes, A., Rivera, V., Bryan, M., Cali, P., y Dawson, G. (2011). Discrepancies between Academic Achievement and Intellectual Ability in Higher- Functioning School-Aged Children with Autism Spectrum Disorder. Journal of Autism and Developmental Disorders, 41(8), 1044-1052.Hegarty, M., y Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684.Hughes, M. (1987). Los niños y los números. Barcelona: Planeta.Levingston H. B., Neef, N. A, y Cihon, T. M. (2009). The effects of teaching precurrent behaviors on children's solution of multiplication and division word problems. Journal of Applied Behavior Analysis, 42, 361-367.Llorca, M., Plasencia, I. C. y Rodríguez, P. (2009). Diagramas para la comprensión matemática. Estudio de caso en personas con trastorno del espectro autista. Revista Educación Inclusiva, 2(1), 79-90.Miller, S.P., y Mercer, C.D. (1993). Using data to learn about concrete- semiconcrete-abstract instruction for students with math disabilities. Learning Disabilities Research and Practice, 8, 89-96.Mirenda, P. (2003). Toward Functional Augmentative and Alternative Communication for Students with Autism: Manual Signs, Graphic Symbols, and Voice Output Communication Aids. Language, Speech, and Hearing Services in Schools, 34, 203-216.Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.Mulligan, J. (1992). Children's solutions to partition problems. In B. Southwell, R. Perry, y K. Owens (Eds.), Proceedings of the 15th Annual Conference of the Mathematics Education Research Group of Australasia (pp. 410-420). Sydney: MERGA.Polo-Blanco, I., González, M.J., Bruno, A., Olivera, B. (2017). Estrategias informales de división de un estudiante con autismo. En J.M. Muñoz- Escolano, A. Arnal-Bailera, P. Beltrán-Pellicer, M.L- Callejo, y J. carrillo (Eds.), Investigación en Educación Matemática XXI (p. 569). Zaragoza: SEIEM.Presmeg, N.C. (1986). Visualization in high school mathematics. For Learning of Mathematics, 63, 42-46.Puig, L., y Cerdán, F. (1988). Problemas aritméticos escolares. Madrid: Síntesis. Rockwell, S.B., Griffin, C.C., y Jones, H.A. (2011). Schema-Based Strategy Instruction in Mathematics and the Word Problem-Solving Performance ofa Student with Autism. Focus on Autism and Other Developmental Disabilities, 26, 87-95.Stroizer, S., Hinton, V., Flores, M., y LaTonya, T. (2015). An investigation of the effects of CRA instruction and students with autism spectrum disorder. Education and Training in Autism and Developmental Disabilities, 50, 223- 236.Whitby, P.J.S. (2012). The effects of “Solve It!” on the mathematical word problem solving ability of adolescents with autism spectrum disorders. Focus on Autism and Other Developmental Disabilities, 28, 78-88.Xin, Y.P., and Jitendra, A.K. (1999). The effects of instruction in solving mathematical word problems for students with learning problems: A meta- analysis. The Journal of Special Education, 32, 40-78.---------------------------------------------Sobre las autoras:Dra. Irene Polo BlancoRealizó el doctorado en Ciencias Matemáticas en la Universidad de Groningen (Holanda). Desde 2011 disfruta de una plaza de contratado doctor en el área de Didáctica de las Matemáticas en la Universidad de Cantabria.Dra. Alicia BrunoDoctora en Ciencias Matemáticas por la Universidad de La Laguna. Desde 2001 es profesora Titular de Universidad en el área de Didáctica de las Matemáticas de la Universidad de la Laguna.Dra. María José GonzálezDoctora en Ciencias Matemáticas por la Universidad de Cantabria. En la actualidad es profesora Titular de Universidad en el área de Didáctica de las Matemática en la Universidad de Cantabria.Dña. Beatriz OliveraLicenciada en psicología por la Universidad Autónoma de Madrid. En 2016 realizó el Máster en Formación del Profesorado de Educación Secundaria en la Universidad de Cantabria.364998010088245 PAGE 18000 PAGE 180364998010088245 PAGE 17000 PAGE 1701086485441960Revista Nacional e Internacional de Educación InclusivaISSN: 1889-4208.; e-ISSN 1989-4643. Volumen 11, Número 2, Diciembre 201800Revista Nacional e Internacional de Educación InclusivaISSN: 1889-4208.; e-ISSN 1989-4643. 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