La enseñanza de la resolución algebraica de problemas verbales mediante un sistema tutorial inteligente

Technology has usually been considered (perhaps with frequency, without a sound theoretical base) an instrument which facilitates learning in practically every educational field. In this respect, intelligent tutoring systems (ITS) stood out for the great expectations generated in their birth, based on the didactic possibilities that the one-to-one teaching situations offer (a teacher working with a single student). The field of algebraic solving of word problems, on which the present work is focused, is not an exception. In fact, the use of computer environments for the teaching and learning in the solving of word problems has been a regular topic of research in educational mathematics. Some of these programs have tried to replace some teacher¿s tasks; others offered environments in which the solver could recourse to different systems of representation or could get rid of monotonous tasks such as the calculation of arithmetic operations. Numerous studies have described environments that had this intention and the consequences of its use with educational aims. Some examples would be: Word Problem Assistant (Thompson, 1989), ANIMATE (Nathan, 1990), HERON (Reusser, 1993), PAT (Koedinger, & Anderson, 1998), AnimalWatch (Beal & Arroyo, 2002) and MathCAL (Chang, Sung, & Lin, 2006). Arnau, Arevalillo-Hérraez and Puig (2011) pointed out that the interactive learning environments (ITS would be included here) for the solving of word problems designed to that date, had not achieved to combine flexibility when allowing the solver to follow any valid resolution path and being able to verify their actions. Among those which actually supervised the solving process, the lack of flexibility was reflected in: a problem was associated with a resolution; a quantity was associated with an expression; and each problem was associated with some error messages. These limitations of the existing systems could constitute an explanation to the fact that the educational impact of the ITS would not have been as high as it was expected at first. In view of this outlook, this work intended to give an answer to two questions: 1) How does the teaching of algebraic solving of word problems by using an intelligent tutoring system influence the proficiency of secondary school students in solving word problems with pencil and paper? 2) What are the performances of students when they solve word problems in an intelligent tutoring system after being instructed in algebraic solving of word problems using the system? In this study, the ITS Hypergraph Based Problem Solver (HBPS) was used (Arnau et al., 2011; Arnau, Arevalillo-Herráez, Puig, & González-Calero, 2013). This ITS is characterized by being able to supervise the solving of arithmetic-algebraic word problems. Unlike other systems, HBPS offers a greater flexibility with regard to the decision-making of the solver. Concretely, and in relation to the flexibility in the supervision of solver¿s actions, HBPS meets the following criteria: - Independence with respect to the solving method. The ITS lets students solve problems in an arithmetical or an algebraic way. - Independence with respect to the use of one or a system of equations. When a problem is solved in an algebraic way, the use of one or more letters is possible and, consequently, the use of one or more equations. - Independence between quantity and its representation. The program does not store a predetermined assignment between a quantity and its representation, but it checks the validity of an expression in accordance with the problem restrictions and the solver¿s decisions. - Independence in the ITS operation with respect to the problem. The verification of the mathematical expressions that are introduced and the error and (basic) help messages that the ITS provides are independent of the specific problem that is being solved. (Arnau et al., 2011, p. 259) In this study, we adopt the theoretical and methodological perspective of Local Theoretical Models (LTM) (Filloy, 1999; Filloy, Rojano, & Puig, 2008, Kieran, & Filloy, 1989). This framework is especially suitable for the study of learning phenomena, considering the essential elements of the whole process of teaching and learning. These elements are considered in terms of four interrelated component: a teaching model, a model for the cognitive processes, a model of formal competence and a model of communication. To be precise, and in relation to the model of formal competence, our study required to embrace two fundamental aspects: on the one hand, the algebraic solving of word problems with pencil and paper, and on the other hand, the algebraic solving of word problems using the ITS. For this reason, the study takes the Cartesian Method (CM) as reference for the algebraic solving of arithmetic-algebraic word problems. From this method, the model of competence was determined for the algebraic solving of word problems in HBPS, paying special attention to how the different elements of competence are affected when solving a word problem in this environment in comparison with the pencil and paper environment. With regard to the teaching model, it was required to build a teaching sequence on the algebraic solving of word problems in HBPS, comparable to a parallel sequence of teaching to be developed with pencil and paper. With the aim of giving an answer to the first research question, we conducted a group study. The population of this study consisted of a group of 56 students in the fourth year of secondary school (15-16 years old) in a Spanish public secondary school. The choice of the population responded to our intention to work with students previously instructed in the algebraic solving of word problems but who would not had reached full competence in the field. At this stage, students were randomly divided into three groups; each of them was instructed using different teaching sequences. The sequencing at this experimental stage was identical in the three groups: 1) pre-test, 2) teaching sequence and, 3) post-test. The problems in pre and post-test and those that were used through the teaching sequence were the same in the three groups. All the problems used were characterized by being usually solved in an algebraic way. Each problem used in the post-test was isomorph to one of the problems in the pre-test, i.e. they had the same structure of relations in their analytical readings. In addition, problems of pre-test and post-test were structurally different from the ones used in the teaching sequences. In the three groups, the teaching was practically limited for students to solve a set of problems without any human help. A group of students was entirely taught in a paper and pencil environment (group PP); another group worked with a complete version of HBPS (group CT); and a third group worked with a limited version of the system that did not provide explicit helps (group RT). Being concrete, the version with which group CT worked offered a set of strategies to support the resolution process and, when demanded by the user, provided aids that always let the student continue the solving process informing him about how to perform the next step. However, the version of group RT offered support strategies but the user was never told exactly what to do in case of error. In both versions, the system offered to the user a structuring of the resolution process using the CM as skeleton and validated the correctness of the user¿s actions at each moment. Hints on demand provided by the complete version of HBPS follow a three-level structure. At the first level, the system provided aids about the resolution method with no information related to the problem content. At the second and third levels, help messages were contextualized within the problem being solved. In fact, at the last level, help messages offer the user detailed instructions about how to continue in the solving process. Given the fact that the group study tried to determine the effect of the different types of teaching on the students¿ proficiency of solving word problems in an algebraic way, the number of problems that students were able to be solved correctly before being instructed (pre-test) and immediately after (post-test) was analysed. The results of this study show a statistically significant improvement in the proficiency of the algebraic solving of word problems with pencil and paper in group CT, compared to groups RT and PP. In spite of the existence of studies that point out that the systems offering too much help to the solver can generate limited learning (e.g. Baker, Corbett, Koedinger, & Wagner, 2004; Shute, Woltz, & Regian, 1989; Walonoski, & Heffernan, 2006). These results highlight that the higher level of help offered to group CT gave rise to a higher improvement of the proficiency in the algebraic solving of word problems. These results discard the possibility of the fact that the students from group CT used gaming strategies (e.g., Walonoski, & Heffernan, 2006) intensively throughout the teaching sequence. This possibility is a risk always associated to ITS that offer a high level of help and a fine supervision of the user¿s actions. In general, these results confirmed the thesis of Koedinger and Aleven (2007) that effective tutoring is more related to strategies that provide information rather than other methods that attempt to withhold it. At the same time, the group study did not show significant differences between groups PP and RT. This fact seems to emphasise that the mere structuring and sequencing of the ideal steps that a solver must undertake in order to solve a word problem and informing students about the correctness of each step are not sufficient to generate an improvement in the proficiency of the algebraic solving of this type of problems. The group study was also used with the intention of classifying the students who belong to the groups that worked with HBPS during the teaching sequence (groups CT and RT) and, therefore, being able to select the pairs that would take part in the following stage of the research: the case study. This stage intended to give an answer to the second of the research questions. Thus, the case study tried to describe and analyse the students¿ performances when they solved arithmetic-algebraic word problems using the tutor HBPS, considering not only correct resolution strategies but also incorrect ones. Seven pairs of students took part in this experimentation stage. Each of them solved a variable number of problems, according to the ones that they were able to solve within 35 minutes. Basically, the problems proposed at this stage were selected from the group study. In fact, students were grouped in pairs according to their level of proficiency and to the fact that the students had shown similar difficulties in the same problems of the post-test. The case study allows us to create a catalogue of student performances. From this catalogue we should highlight the following: 1) the difficulty in interpreting the names of the quantities that HBPS proposes; 2) the reinterpretation of names provided by HBPS over the resolution process; 3) the tendency to isolate propositions from the statement in which relations between quantities are described and to translate these isolated propositions into algebraic language without interpreting the relation in the context of the situation described by the statement; 4) the tendency to assign a letter to the quantity demanded in the statement and how it leads to generate difficulties in the step 4 of CM; 5) the tendency to make direct translations from the natural language to the algebraic language, causing reversal errors; 6) the tendency to build equations in the form x = f (x); by expressing the equation in this way, some students considered that it was possible to calculate the value of the isolated quantity; and, 7) the interpretation of equations as a representation of the association among related quantities. As the group study indicates that the HBPS can be an useful instrument in the teaching of the algebraic solving of word problems, the case study also reveals the appearance of undesirable performances when secondary school students solve problems in such environment. Some of the performances could have their origin in the design characteristics of HBPS and they should be taken into consideration when developing new versions of the ITS. The results of this research suggest that the evolution of this ITS should be orientated towards a greater level of adaptability to the individual characteristics of each student.La tecnología se ha considerado habitualmente (quizá con frecuencia, sin una base teórica sólida) un instrumento facilitador del aprendizaje en prácticamente cualquier ámbito educativo. En este sentido, los sistemas tutoriales inteligentes (en adelante, STI) destacan por las grandes expectativas que generaron en su nacimiento, basadas en las posibilidades didácticas que ofrecen la situaciones de enseñanza uno a uno (un profesor para un estudiante). El campo de la enseñanza de la resolución algebraica de problemas verbales, en el que se centra el presente trabajo, no es una excepción. De hecho, el uso de entornos informáticos para la enseñanza y aprendizaje de la resolución de problemas verbales ha sido un tema habitual de investigación en la Matemática Educativa. Algunos de estos programas han pretendido sustituir el papel del profesor; otros ofrecían entornos en los que el resolutor podía recurrir a distintos sistemas de representación o podía liberarse de tareas rutinarias como el cálculo de operaciones aritméticas. Numerosos estudios han descrito entornos que tenían este propósito y las consecuencias de su uso con intenciones educativas. Algunos ejemplos serían: Word Problem Assistant (Thompson, 1989), ANIMATE (Nathan, 1990), HERON (Reusser, 1993), PAT (Koedinger y Anderson, 1998), AnimalWatch (Beal & Arroyo, 2002) y MathCAL (Chang, Sung y Lin, 2006). En Arnau, Arevalillo-Hérraez y Puig (2011) se señalaba que los entornos interactivos de aprendizaje (entre los que incluirían los STI) para la resolución de problemas verbales diseñados hasta esa fecha, no habían conseguido conjugar flexibilidad a la hora de permitir que el resolutor pueda decidir seguir caminos distintos y poder verificar sus acciones. Entre aquellos que tutorizaban realmente el proceso de resolución, la falta de flexibilidad se reflejaba en: un problema se asociaba a una solución; una cantidad se asociaba a una expresión; y cada problema se asociaba a unos mensajes de error. Estas limitaciones de los sistemas existentes podrían constituir una explicación al hecho de que el impacto educativo de los STI no hubiera sido tan elevado como se prevía en un principio. Ante este panorama, nuestra investigación pretendía dar respuesta a dos preguntas: 1) ¿Cómo influye la enseñanza de la resolución algebraica de problemas verbales mediante un sistema tutorial inteligente en la competencia de estudiantes de secundaria cuando resuelven problemas verbales en lápiz y papel? 2) ¿Cuáles son las actuaciones de los estudiantes cuando resuelven problemas verbales en un sistema tutorial inteligente tras haber sido instruidos previamente en la resolución algebraica de problemas verbales mediante dicho sistema? En la investigación se utilizó el STI Hypergraph Based Problem Solver (en adelante, HBPS) (Arnau et al., 2011; Arnau, Arevalillo-Herráez, Puig y González-Calero, 2013). Este STI se caracteriza por ser capaz de supervisar la resolución de problemas verbales aritmético-algebraicos. A diferencia de otros sistemas, HBPS ofrece una mayor flexibilidad respecto a la toma de decisiones del resolutor. En concreto, en relación con la flexibilidad frente a las acciones del resolutor, HBPS satisface los siguientes criterios: Independencia respecto al método de resolución. El problema permite resolver problemas de manera aritmética y de manera algebraica. Independencia respecto al uso de una o más ecuaciones. Cuando se resuelve de manera algebraica, es posible el uso de una o más letras y el consiguiente recurso a una o más ecuaciones. Independencia entre cantidad y su representación. El programa no supone una asignación predeterminada entre una cantidad y su representación, sino que comprueba la validez de una expresión atendiendo a las restricciones del problema y a las decisiones del resolutor. Independencia de funcionamiento del STI respecto del problema. La verificación de la validez de las expresiones matemáticas que se introducen y los mensajes (básicos) de error y ayuda que proporcionará el STI son independientes del problema concreto que se está resolviendo. (Arnau et al., 2011, p. 259) La investigación se organizó alrededor del marco teórico y metodológico que ofrecen los Modelos Teóricos Locales (Filloy, 1999; Filloy, Rojano y Puig, 2008, Kieran y Filloy, 1989). Este marco es especialmente adecuado para el estudio de fenómenos de aprendizaje al tomar en consideración los elementos esenciales de todo proceso de enseñanza y aprendizaje. Estos elementos son considerados al construirse: un modelo de enseñanza, un modelo de actuación, un modelo de competencia formal y un modelo de comunicación. En concreto, en relación con el modelo de competencia, nuestro estudio requería recoger dos aspectos fundamentales: por un lado, la resolución algebraica de problemas verbales en lápiz y papel, y por otro, la resolución algebraica de problemas verbales en el STI. Para ello, el estudio toma como modelo de referencia en la resolución algebraica de problemas verbales aritmético-algebraicos, el método cartesiano. A partir de este método, se determinó el modelo de competencia para la resolución algebraica de problemas verbales en HBPS, prestando especial a cómo se ven afectados los diferentes elementos de competencia a la hora de resolver un problema verbal en el entorno de resolución de HBPS en comparación con las resoluciones en lápiz y papel. En cuanto al modelo de enseñanza, se requería construir una secuencia de enseñanza sobre la resolución algebraica de problemas verbales en HBPS, comparable con una secuencia de enseñanza paralela a desarrollar en lápiz y papel. La población de esta investigación estaba formada por un grupo de 56 estudiantes de cuarto curso de Educación Secundaria Obligatoria de un centro público de Castilla-La Mancha. Los estudiantes tenían una edad de entre 15 y 16 años. La elección de la población y del momento de observación respondió a nuestro propósito de trabajar con estudiantes previamente instruidos en la resolución algebraica de problemas verbales pero que aún no hubieran alcanzado plena competencia en el campo. Con el propósito de dar respuesta a la primera pregunta de investigación, realizamos un estudio de grupo. En esta fase la población fue dividida en tres grupos de trabajo, los cuales fueron instruidos mediante secuencias de enseñanza diferentes. La secuenciación de esta fase experimental fue idéntica en los tres grupos: 1) realización de un cuestionario inicial, 2) secuencia de enseñanza y, 3) realización de un cuestionario final. En los tres grupos se emplearon los mismos problemas durante la secuencia de enseñanza. De la misma manera, los cuestionarios inicial y final fueron los mismos en los diferentes grupos. Los problemas de los cuestionarios inicial y final eran isomorfos entre sí y estructuralmente distintos a los empleados en las secuencias de enseñanza. La secuencia de enseñanza fue el único elemento que difirió para cada uno de los grupos. En los tres grupos, la enseñanza estuvo prácticamente limitada a los estudiantes resolvieran una colección de problemas sin ayuda humana. Un grupo de estudiantes fue instruido íntegramente en lápiz y papel (grupo PP); otro grupo trabajó con una versión con todas las funcionalidades de HBPS (grupo CT); y un tercer grupo trabajó con una versión limitada del sistema la cual no proporcionaban ayudas explícitas (grupo RT). En concreto, la versión con la que trabajó el grupo CT se caracterizaba porque el tutor ofrecía un conjunto de estrategias que asistían el proceso de resolución y, cuando el usuario lo socilitaba, estas ayudas permitían siempre completar el proceso al recibir información explícita sobre cómo completar el siguiente paso. En cambio, la versión del grupo RT ofrecía estrategias de ayuda pero no hasta el grado de informar al usuario de qué acción debía acometer. En ambas versiones, el sistema, apoyándose en el método cartesiano, ofrecía al usuario una estructuración del proceso de resolución y validaba en cada momento la corrección de las acciones del usuario. Sin embargo, sólo en el grupo CT, HBPS ofrecía ayudas bajo solicitud. Estas pistas siguen una estructura de tres niveles. En el primer nivel, el sistema provee ayudas para asistir al usuario con el método de resolución pero sin aportar información relacionada con el contenido del problema. En los niveles segundo y tercero, los mensajes de ayuda están relacionados directamente con el contenido del problema. Dado que el estudio de grupo pretendía determinar el efecto de los distintos tipos de enseñanza en la competencia de los estudiantes para resolver problemas verbales de manera algebraica, se analizó el número de problemas que eran capaces de resolver correctamente antes de ser instruidos mediante la secuencia de enseñanza correspondiente e inmediatamente después. Los resultados del estudio de grupo muestran una mejora estadísticamente significativa en la competencia en la resolución algebraica de problemas verbales en lápiz y papel en el grupo