Hacia profesores artificiales en la resolución algebraica de problemas verbales

Presentamos los fundamentos del diseño de un sistema tutorial inteligente para la enseñanza y aprendizaje de la resolución de problemas verbales aritmético-algebraicos. El programa resultante admite resoluciones aritméticas y algebraicas, establece la validez de las acciones atendiendo a las restricciones del problema y puede generar ayudas a demanda, tras identificar la línea de resolución que se está siguiendo. El sistema es capaz de determinar parte de las características del estudiante como resolutor con respecto al modelo de competencia. Con este fin usamos la idea de esquema conceptual. De esta forma podemos clasificar las acciones de los estudiantes cuando resuelven problemas verbales de más de una etapa mediante un criterio que tiene en cuenta la tarea y las decisiones que se toman. Para finalizar, ofrecemos líneas futuras y preguntas de investigación que quedan abiertas

Personalization of Learning Materials for Mathematics Learning Using a Case-Based Reasoning Algorithm

Personalization is important to ensure that learning can cater to the needs of individual learners. The Intelligent Tutoring System (ITS) is a technology that can ease the personalization process; one of the most widely used algorithms in ITS is case-based reasoning (CBR). This study measures the ability of the CBR algorithm to give suggestions for the most suitable learning material based on specific information supplied by the user of the system. In order to test the ability of the application to recommend learning material, two versions of the application were created. The first version displayed the most suitable learning material, and the second version displayed the least preferable learning material. The results show that the first version of the application successfully assigns students to the most suitable learning material when compared with the second version

La influencia de proporcionar los nombres de las cantidades en la resolución aritmética de problemas verbales

Cuando un profesor supervisa la resolución de un problema que lleva a cabo un estudiante, debe valorar el potencial de las ayudas que proporciona. Presentamos parte de una investigación que tenía como objetivo estudiar el papel de las ayudas expresadas en lenguaje natural en la resolución aritmética de problemas. En el experimento participaron 32 estudiantes de quinto curso de primaria (10-11 años). En concreto, hemos analizado cómo influye, en el proceso de resolución de un problema, el hecho de proporcionar un conjunto de nombres (o etiquetas) que hacen referencia a un número suficiente de cantidades para resolverlo. Los nombres usados eran del tipo “kilos de naranjas”, “precio de las naranjas que se han comprado”, etc. El análisis de los resultados nos ha permitido elaborar un catálogo de actuaciones en el que se reflejan los procesos de gestión y las dificultades de los estudiantes para integrar estas etiquetas en el proceso de resolución

Integrating knowledge tracing and item response theory: A tale of two frameworks

Traditionally, the assessment and learning science commu-nities rely on different paradigms to model student performance. The assessment community uses Item Response Theory which allows modeling different student abilities and problem difficulties, while the learning science community uses Knowledge Tracing, which captures skill acquisition. These two paradigms are complementary - IRT cannot be used to model student learning, while Knowledge Tracing assumes all students and problems are the same. Recently, two highly related models based on a principled synthesis of IRT and Knowledge Tracing were introduced. However, these two models were evaluated on different data sets, using different evaluation metrics and with different ways of splitting the data into training and testing sets. In this paper we reconcile the models' results by presenting a unified view of the two models, and by evaluating the models under a common evaluation metric. We find that both models are equivalent and only differ in their training procedure. Our results show that the combined IRT and Knowledge Tracing models offer the best of assessment and learning sciences - high prediction accuracy like the IRT model, and the ability to model student learning like Knowledge Tracing

Investigation of the Misconceptions Related to the Concepts of Equivalence and Literal Symbols Held by Underprepared Community College Students

Many students struggle to learn mathematics in K-8 grades. Research has shown that lower grade students often misconceive equivalence as an operation rather than a relation, and that students also form various misconceptions of literal symbols. Many students arrive at college seriously underprepared in mathematics, but there is scant research on the difficulties and misconceptions of these college students. The purpose of this research was to learn if underprepared community college students harbor misconceptions of equivalence and of literal symbols similar to K-8 students. For this study, 191 underprepared college students were surveyed for misconceptions by a questionnaire of 43 items selected from the established suite of effective items. The items for each concept were further partitioned into the definition, properties, and applications of each concept. Many students (84%) were expert regarding the definition of equivalence. An additional 13% of the students also demonstrated knowledge of the concept, although they did not always take advantage of it. Similarly, over 40% of the students demonstrated expert understanding of the properties of equivalence, but an additional 53% demonstrated a restricted understanding of the concept. Only 5% of the students were considered expert with the fundamental applications of equivalence and less than 60% demonstrated a basic knowledge of the applications. Few students (33%) were knowledgeable of the definition of literal symbols, and fewer (\u3c 5%) demonstrated knowledge of the properties of the literal symbol. Consequent to their minimal knowledge of the concept, very few students were able to demonstrate knowledge of literal symbol applications. Community college students underprepared in mathematics are generally aware of the relational definition of equivalence, but many are not fluent in its use. Most attention needs to be directed to the applications of equivalence. The same students are generally not aware of the concept of literal symbols and much attention needs to be directed not only at the applications of literal symbols, but also at their definition and properties

A semi-automatic computer-aided assessment framework for primary mathematics

Assessment and feedback processes shape students behaviour, learning and skill development. Computer-aided assessments are increasingly being used to support problem-solving, marking and feedback activities. However, many computer-aided assessment environments only replicate traditional pencil-and-paper tasks. Attention is on grading and providing feedback on the final product of assessment tasks rather than the processes of problem solving. Focusing on steps and problem-solving processes can help teachers to diagnose strengths and weaknesses, discover problem-solving strategies, and to provide appropriate feedback to students. This thesis presents a semi-automatic framework for capturing and marking students solution steps in the context of elementary school mathematics. The first focus is on providing an interactive touch-based tool called MuTAT to facilitate interactive problem solving for students. The second focus is on providing a marking tool named Marking Assistant which utilises the case-based reasoning artificial intelligence methodology to carry out marking and feedback activities more efficiently and consistently. Results from studies carried out with students showed that the MuTAT prototype tool was usable, and performance scores on it were comparable to those obtained when paper-and-pencil was used. More importantly, the MuTAT provided more explicit information on the problem-solving process, showing the students thinking. The captured data allowed for the detection of arithmetic strategies used by the students. Exploratory studies conducted using the Marking Assistant prototype showed that 26% savings in marking time can be achieved compared to traditional paper-and-pencil marking and feedback. The broad feedback capabilities the research tools provided can enable teachers to evaluate whether intended learning outcomes are being achieved and so decide on required pedagogical interventions. The implications of these results are that innovative CAA environments can enable more direct and engaging assessments which can reduce staff workloads while improving the quality of assessment and feedback for students

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

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 CT en comparación con los grupos RT y PP. A pesar de la existencia de estudios que señalan que los sistemas que ofrecen demasiadas ayudas al resolutor, pueden generar escaso aprendizaje (p.ej. Baker, Corbett, Koedinger and Wagner, 2004; Shute, Woltz and Regian, 1989; Walonoski and Heffernan, 2006), los resultados del estudio de grupo apuntan a que el mayor nivel de ayudas ofrecido al grupo CT, produjo un incremento mayor de la competencia en la resolución algebraica de problemas verbales. Estos resultados descartan la posibilidad de que los estudiantes del grupo CT emplearan intensivamente estrategias de gaming a lo largo de la secuencia de enseñanza. Esta posibilidad es un riesgo siempre asociado a los STI que ofrecen alto nivel de ayudas y una tutorización fina de las acciones del usuario. En líneas generales, estos resultados confirman la tesis de Koedinger y Aleven (2007) sobre que la tutorización efectiva está más ligada a estrategias que facilitan información que a aquellos métodos que intentan restringir las ayudas. A su vez, el estudio de grupos no mostró diferencias significativas entre los grupos PP y RT. Este hecho parece señalar que la mera estructuración y secuenciación de los pasos ideales que ha de acometer un resolutor para resolver un problema verbal, no es por sí suficiente para generar una mejora en la competencia de la resolución algebraica de este tipo de problemas. El estudio de grupos también se utilizó con el objeto de clasificar a los estudiantes pertenecientes a los grupos que trabajaron con HBPS durante la secuencia de enseñanza (grupos CT y RT) y, así, poder seleccionar las parejas que participarían en la siguiente fase de la investigación: el estudio de casos. Esta fase tenía el propósito de dar respuesta a la segunda de las preguntas de investigación. Así, el estudio de casos pretendía describir y analizar las actuaciones de los estudiantes cuando resolvían problemas verbales aritmético-algebraicos usando el tutor HBPS, considerando tanto las estrategias de resolución correctas como incorrectas. Siete de parejas de estudiantes participaron en esta fase de la experimentación. Cada una de ellas abordó un número variable de problemas, en función de los que fueran capaces de resolver en el tiempo asignado a cada pareja para el estudio de casos (aprox. 35 minutos). Básicamente los problemas propuestos en esta fase fueron seleccionados del estudio de grupo. De hecho, uno de los criterios utilizados para la formación de las parejas fue agrupar a estudiantes que hubieran tenido dificultades en los mismos problemas del cuestionario final del estudio de grupo. Entre el catálogo de actuaciones de los estudiantes construido a raíz del estudio de casos, destacamos las siguientes: 1) la dificultad para interpretar los nombres de las cantidades que propone HBPS; 2) el nombrado inapropiado de cantidades en el transcurso del proceso de resolución; 3) la dificultad para interpretar globalmente la red de relaciones descrita en un enunciado; 4) la tendencia a designar mediante una letra la cantidad por la que se pregunta en el enunciado y cómo esto se puede traducir en dificultades en el paso cuarto del método cartesiano; 5) la tendencia a recurrir a procesos de traducción directa desde fragmentos del enunciado en lenguaje natural al lenguaje algebraico, favoreciéndose la comisión de errores de inversión; 6) la tendencia a construir ecuaciones en la forma x = f (x), en ocasiones bajo la interpretación errónea de que una ecuación constituye una fórmula para el cálculo directo de la incógnita; y, 7) la interpretación de la ecuación como una asociación entre cantidades. Así como el estudio de grupo apunta a que HBPS puede ser instrumento útil en la enseñanza de la resolución algebraica de los problemas verbales, el estudio de casos también pone de manifiesto la aparición de actuaciones indeseadas cuando estudiantes de secundaria resuelven problemas en dicho entorno. Algunas de estas actuaciones podrían tener su origen en las características de diseño de HBPS y deben ser tomados en consideración a la hora de desarrollar nuevas versiones del STI. A su vez, los resultados de esta investigación sugieren que la evolución de este sistema debe orientarse hacia un mayor nivel de adaptación a las características individuales de cada estudiante.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

Proceedings of the tenth international conference Models in developing mathematics education: September 11 - 17, 2009, Dresden, Saxony, Germany

This volume contains the papers presented at the International Conference on “Models in Developing Mathematics Education” held from September 11-17, 2009 at The University of Applied Sciences, Dresden, Germany. The Conference was organized jointly by The University of Applied Sciences and The Mathematics Education into the 21st Century Project - a non-commercial international educational project founded in 1986. The Mathematics Education into the 21st Century Project is dedicated to the improvement of mathematics education world-wide through the publication and dissemination of innovative ideas. Many prominent mathematics educators have supported and contributed to the project, including the late Hans Freudental, Andrejs Dunkels and Hilary Shuard, as well as Bruce Meserve and Marilyn Suydam, Alan Osborne and Margaret Kasten, Mogens Niss, Tibor Nemetz, Ubi D’Ambrosio, Brian Wilson, Tatsuro Miwa, Henry Pollack, Werner Blum, Roberto Baldino, Waclaw Zawadowski, and many others throughout the world. Information on our project and its future work can be found on Our Project Home Page http://math.unipa.it/~grim/21project.htm It has been our pleasure to edit all of the papers for these Proceedings. Not all papers are about research in mathematics education, a number of them report on innovative experiences in the classroom and on new technology. We believe that “mathematics education” is fundamentally a “practicum” and in order to be “successful” all new materials, new ideas and new research must be tested and implemented in the classroom, the real “chalk face” of our discipline, and of our profession as mathematics educators. These Proceedings begin with a Plenary Paper and then the contributions of the Principal Authors in alphabetical name order. We sincerely thank all of the contributors for their time and creative effort. It is clear from the variety and quality of the papers that the conference has attracted many innovative mathematics educators from around the world. These Proceedings will therefore be useful in reviewing past work and looking ahead to the future

The Generalized Distributive Law as Tacit Knowledge in Algebra.

The purposes of this study were to investigate theories that explain why common errors of the type (a \pm b)\sp{c} = a\sp{c} \pm b\sp{c} and \root c \of {a \pm b} = \root c \of {a} \pm \root c \of {b} occur in algebra problem solving by novices; and to develop and assess techniques for remediating these errors. The meaning theory of learning (ML), procedural learning theory (PL), and implicit structure learning theory (ISL) are alternative frameworks for the explanation of the errors. The ML theory hypothesizes that experts have rich semantic connections to the procedures and symbols of algebra, but novices lack such connections (Ausubel, Novak & Hanesian, 1978; Brownell, 1947; Wearne & Hiebert, 1985). The PL theory hypothesizes that adept problem solvers have technical proficiency in memorizing and applying mechanical rules (Anderson, 1983; Lewis, Milson, & Anderson, 1987; Matz, 1980). The ISL theory hypothesizes that students enter the classroom with nascent abstract rule structures on which to build a more mature grammar of algebra through inductive processes (Bolio, 1989; Drouhard, 1988; Kirshner, 1987). In order to obtain some measure of the relative efficacy of these theories for remedial purposes, three brief educational treatments have been designed to reflect the three frameworks for learning. An analysis of variance for repeated measures was used to assess the effectiveness of the treatments in reducing the occurrences of the (a \pm b)\sp{c} = a\sp{c} \pm b\sp{c} and \root c\of {a \pm b} = \root c \of {a} \pm \root c\of {b} errors. Forty students participated in the study. They were enrolled in four intact developmental intermediate algebra classes at Southern University in Baton Rouge. The study used a pretest-posttest-retention test, control group design with three treatments--ML, PL, and ISL--and one control (C) which receives no special instruction concerning the errors. Results indicate that no significant difference was found in the number of errors between the groups on the post and retention tests. However, there was a significant difference between the mean scores of the pretest and the posttest. These results do not provide support for one theory over another in reducing the error types mentioned above, but do indicate a small decrease in the error rate for distributivity overgeneralization for all treatment groups

Analysis of Equation and Diagram Construction in Applied Calculus Problem Solving

The purpose of this study was to assess algebra and geometric prerequisites skills as incorporated into the Applied Calculus Optimization Problem (ACOP) solution. The difficulties that students encounter in applying algebraic and geometric prerequisites at the early stages of the ACOP solution were identified. The study analyzes errors related to variables and equations (i.e. algebraic symbol/transformation skills), drawing of geometric diagrams (visualization skills) and those associated with application of basic differentiation concepts into ACOP solution process. The study’s goals were addressed as seven specific research questions further subdivided into three main parts: the first four research questions investigated prerequisite algebraic and geometric skills, while question five examined the ability to use some or all of the prerequisite skills to obtain the required ACOP model. Question six is concerned with how some prerequisite (differentiation) skills are use in ACOP solution process. Finally, question seven looked into students’ ability to fully bring into play all the prerequisite skills into ACOP solution process. Furthermore, each of the seven research questions was split into quantitative and qualitative parts. The quantitative data were collected using a test instrument; and a follow up interview was conducted to collect qualitative data. These qualitative data were used to supplement, support and illuminate results from the quantitative components. The target sample is freshmen students taking calculus I in the department of mathematics, Louisiana State University, Baton Rouge. Overall, the study has revealed that students have achieved a very low success rate on ACOP, immediately following instruction on ACOP solving in their calculus I class. In general, they failed to integrate the basic competences required in ACOP solution. Qualitative evidence from students’ test performance indicated that failure to visualize geometric diagrams from word problems tendered to preclude getting the required formula. More generally, failure in at least one competence lead to collapse in another, and hence the whole breakdown of the ACOP solution process. The overall finding of the research was that students generally failed in integrating the independent algebraic and geometric competences; in cases where integration occurred, students face structural and procedural setbacks that ultimately led to a weakening of the ACOP solution process