Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria
The overall objective of this research is to study the development of informal knowledge on base 10 grouping and place value. This is done through the study of the strategies used by children in solving arithmetic word problems and the analysis of representations of discrete quantities used in their...
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Universidad Complutense de Madrid (España)
2015
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Conocimientos informales resolución de problemas problemas aritméticos verbales decena y valor posicional caminos de aprendizaje Informal knowledge problems resolving arithmetic word problems ten and positional value learning paths |
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Conocimientos informales resolución de problemas problemas aritméticos verbales decena y valor posicional caminos de aprendizaje Informal knowledge problems resolving arithmetic word problems ten and positional value learning paths Ramírez García, Mónica Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
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The overall objective of this research is to study the development of informal knowledge on base 10 grouping and place value. This is done through the study of the strategies used by children in solving arithmetic word problems and the analysis of representations of discrete quantities used in their procedures, and the description of the evolution of strategies and representations along one academic year.
Fifty-four students have participated in the research. They were studying first grade of primary education in a public school in the northwest of Madrid (Spain). We have designed an arithmetic problem-solving workshop composed of 25 sessions, one per week, developed over a school year. The workshop posed problems of multiplicative structure, of equal groups, with 10 groups, of multiplication and division; other problems of equal groups, without groups of ten; and additive structure problems with two digit numbers. The problems were based on stories previously read in the classroom. We offered to the students various manipulatives (structured and unstructured), without instruction on its use, among which they could choose freely. In the workshops, there was a phase of individual work followed by sharing strategies, and writing a letter explaining the problem solving process.
Data collection has been done through individual interviews videotaped in the classroom. We have also taken notes on record sheets and photographs of the resolution process, while students used manipulatives. Finally, we have collected the students' worksheets and their written letters.
For the analysis of strategies, we start with a categorization from previous studies. Direct modeling strategies have been analyzed according to the representation of the quantities and the mode to carry out counting. This circumstance coupled with the freedom given in the selection and use of materials, has led to the detection of great diversity of modalities for implementing strategies, not described in previous studies. Some of them are transition strategies from direct modeling to counting strategies and other strategies involving the use of number facts, promoted by the use of rekenrek and hundred chart. It also shows, with greater detail than previous studies, the development of direct modeling strategies, since the lack of representation of numbers in groups of 10, to the representation of the amounts separated in tens and units, using non structured manipulatives as cartons of ten eggs and ten bars constructed by children with interlocking cubes. This has allowed the description of the evolution from informal direct modeling strategies to formal strategies, as well as developing an understanding of tens, for which we describe transitions between levels of understanding identified in previous studies.
After categorization of strategies, in a second analysis, we describe strategies as successions of capacities and represent them, in a diagram, as possible learning paths to solve a problem.
When analyzing representations of discrete quantities, we consider each quantity as composed by a number and an object type, and we identify each component as iconic, symbolic, or a mixture of both, which results in a classification scheme. Representations have been classified depending on the phase in which they are produced: the process of solving the problem, the moment of annotation of the solution, or the stage of writing the letter in which the process and the answer are communicated. Children have most often used iconic representations to solve problems. The frequency of symbolic representations has been increasing throughout the workshop, because of the use of hundred charts and the application of algorithms. At the stage of communication of the written solution, and in the elaboration of the letter, they use more formal representations, as writing the number and the name of the object in figures or words.
Among the conclusions of the investigation, we noted that first grade children applied preferably, throughout the entire course, direct modeling strategies that reflect an informal knowledge. This has occurred in a learning situation governed by rules that allowed free choice of strategies and manipulatives, and despite formal mathematical knowledge, which students were learning in their daily math classes.
As an implication for teaching, we propose to include tasks in the classroom that promote the use of informal knowledge, by providing experiences that enable children building ideas on mathematical concepts, prior to their formal teaching. For example, problems of multiplicative structure may be included in first grade of primary education. This supposes a change of teaching approach, in which we think about problem solving as a way of construction of mathematical content, to overcome an applicationist approach. The use of manipulatives also must undergo reflection, paying more attention to thinking that children develop using manipulatives of their own choice, that to the structure of the manipulatives.
Students’ performance in the workshop has evidenced characteristics of learning with understanding, as the connection between informal and formal strategies, children's knowledge of the applicability of the algorithms of addition and subtraction, or the use of different strategies for the same problem. The introduction, in primary education, of a teaching approach as described in the workshop, inspired in cognitively guided instruction, can promote learning with understanding.
As implication for theory, I propose to use the teaching-learning trajectories and the learning paths for a task, as complementary tools for curriculum design and classroom planning. Strategies can subdivided into sequences of capacities, which constitutes the highest level of concretion of student learning expectations and, from a broader perspective, this analysis identifies the necessary capacities to move from a level of understanding to the following. These tools serve also to articulate informal and formal knowledge, establishing connections between them. |
| author2 |
Castro Hernández, Carlos de (Universidad Complutense de Madrid) |
| author_facet |
Castro Hernández, Carlos de (Universidad Complutense de Madrid) Ramírez García, Mónica |
| format |
text (thesis) |
| author |
Ramírez García, Mónica |
| author_sort |
Ramírez García, Mónica |
| title |
Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
| title_short |
Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
| title_full |
Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
| title_fullStr |
Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
| title_full_unstemmed |
Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
| title_sort |
desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria |
| publisher |
Universidad Complutense de Madrid (España) |
| publishDate |
2015 |
| url |
https://dialnet.unirioja.es/servlet/oaites?codigo=47140 |
| work_keys_str_mv |
AT ramirezgarciamonica desarrollodeconocimientosmatematicosinformalesatravesderesolucionesdeproblemasaritmeticosverbalesenprimercursodeeducacionprimaria |
| _version_ |
1718346642185256960 |
| spelling |
oai-TES00000094752021-05-04Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primariaRamírez García, MónicaConocimientos informalesresolución de problemasproblemas aritméticos verbalesdecena y valor posicionalcaminos de aprendizajeInformal knowledgeproblems resolvingarithmetic word problemsten and positional valuelearning pathsThe overall objective of this research is to study the development of informal knowledge on base 10 grouping and place value. This is done through the study of the strategies used by children in solving arithmetic word problems and the analysis of representations of discrete quantities used in their procedures, and the description of the evolution of strategies and representations along one academic year. Fifty-four students have participated in the research. They were studying first grade of primary education in a public school in the northwest of Madrid (Spain). We have designed an arithmetic problem-solving workshop composed of 25 sessions, one per week, developed over a school year. The workshop posed problems of multiplicative structure, of equal groups, with 10 groups, of multiplication and division; other problems of equal groups, without groups of ten; and additive structure problems with two digit numbers. The problems were based on stories previously read in the classroom. We offered to the students various manipulatives (structured and unstructured), without instruction on its use, among which they could choose freely. In the workshops, there was a phase of individual work followed by sharing strategies, and writing a letter explaining the problem solving process. Data collection has been done through individual interviews videotaped in the classroom. We have also taken notes on record sheets and photographs of the resolution process, while students used manipulatives. Finally, we have collected the students' worksheets and their written letters. For the analysis of strategies, we start with a categorization from previous studies. Direct modeling strategies have been analyzed according to the representation of the quantities and the mode to carry out counting. This circumstance coupled with the freedom given in the selection and use of materials, has led to the detection of great diversity of modalities for implementing strategies, not described in previous studies. Some of them are transition strategies from direct modeling to counting strategies and other strategies involving the use of number facts, promoted by the use of rekenrek and hundred chart. It also shows, with greater detail than previous studies, the development of direct modeling strategies, since the lack of representation of numbers in groups of 10, to the representation of the amounts separated in tens and units, using non structured manipulatives as cartons of ten eggs and ten bars constructed by children with interlocking cubes. This has allowed the description of the evolution from informal direct modeling strategies to formal strategies, as well as developing an understanding of tens, for which we describe transitions between levels of understanding identified in previous studies. After categorization of strategies, in a second analysis, we describe strategies as successions of capacities and represent them, in a diagram, as possible learning paths to solve a problem. When analyzing representations of discrete quantities, we consider each quantity as composed by a number and an object type, and we identify each component as iconic, symbolic, or a mixture of both, which results in a classification scheme. Representations have been classified depending on the phase in which they are produced: the process of solving the problem, the moment of annotation of the solution, or the stage of writing the letter in which the process and the answer are communicated. Children have most often used iconic representations to solve problems. The frequency of symbolic representations has been increasing throughout the workshop, because of the use of hundred charts and the application of algorithms. At the stage of communication of the written solution, and in the elaboration of the letter, they use more formal representations, as writing the number and the name of the object in figures or words. Among the conclusions of the investigation, we noted that first grade children applied preferably, throughout the entire course, direct modeling strategies that reflect an informal knowledge. This has occurred in a learning situation governed by rules that allowed free choice of strategies and manipulatives, and despite formal mathematical knowledge, which students were learning in their daily math classes. As an implication for teaching, we propose to include tasks in the classroom that promote the use of informal knowledge, by providing experiences that enable children building ideas on mathematical concepts, prior to their formal teaching. For example, problems of multiplicative structure may be included in first grade of primary education. This supposes a change of teaching approach, in which we think about problem solving as a way of construction of mathematical content, to overcome an applicationist approach. The use of manipulatives also must undergo reflection, paying more attention to thinking that children develop using manipulatives of their own choice, that to the structure of the manipulatives. Students’ performance in the workshop has evidenced characteristics of learning with understanding, as the connection between informal and formal strategies, children's knowledge of the applicability of the algorithms of addition and subtraction, or the use of different strategies for the same problem. The introduction, in primary education, of a teaching approach as described in the workshop, inspired in cognitively guided instruction, can promote learning with understanding. As implication for theory, I propose to use the teaching-learning trajectories and the learning paths for a task, as complementary tools for curriculum design and classroom planning. Strategies can subdivided into sequences of capacities, which constitutes the highest level of concretion of student learning expectations and, from a broader perspective, this analysis identifies the necessary capacities to move from a level of understanding to the following. These tools serve also to articulate informal and formal knowledge, establishing connections between them.El objetivo general de esta investigación es estudiar el desarrollo de los conocimientos informales sobre la agrupación de base 10 y los conocimientos del valor posicional, a través del estudio de las estrategias utilizadas por los niños en la resolución de problemas aritméticos verbales, así como el análisis de las representaciones de cantidades discretas utilizadas en sus procedimientos, describiendo además, la evolución de las estrategias y representaciones a lo largo de un curso. En la investigación, han participado 54 alumnos de primer curso de educación primaria de un centro público de la zona noroeste de Madrid. Se ha diseñado un taller de resolución de problemas compuesto por 25 sesiones, una por semana, desarrollado a lo largo de un curso escolar. En el taller se han planteado problemas de estructura multiplicativa, de grupos iguales, con agrupamientos de 10, de multiplicación y división; otros de grupos iguales, sin grupos de diez; y problemas de estructura aditiva con números de dos cifras. Los problemas estaban basados en cuentos leídos en el aula. A los alumnos se les ofrecían diversos materiales manipulativos (estructurados y no estructurados), sin instrucción sobre su uso, entre los cuales podían elegir libremente. En los talleres había una fase de trabajo individual, seguida de una puesta en común, y la escritura de una carta con la explicación de proceso de resolución del problema. La recogida de datos se realiza a través de entrevistas individuales, realizadas dentro del aula, grabadas en video o anotadas en hojas de registro. Se han tomado fotografías del proceso de resolución cuando los alumnos utilizaban materiales manipulativos. Finalmente, se han recogido las hojas de trabajo de los alumnos y las cartas escritas. Para analizar las estrategias, se parte de una categorización proveniente de estudios previos. Las estrategias de modelización directa han sido analizadas prestando especial atención a la representación de las cantidades y su conteo. Esta circunstancia, unida a la libertad que se ha dado en la selección y uso de materiales, ha dado lugar a la detección de gran diversidad de modalidades de aplicación de las estrategias no descritas en estudios previos. Algunas de ellas son estrategias de transición de modelización directa a estrategias de conteo y a otras que suponen el uso de hechos numéricos, facilitadas por el uso del rekenrek y la Tabla 100. Otras muestran, con más detalle que los estudios previos, la evolución de las estrategias de modelización directa, desde la ausencia de representación de las cantidades en grupos de 10, a la representación de las cantidades separadas en decenas y unidades con ayuda de materiales no estructurados como los cartones de decenas de huevos y barras de 10 formadas con cubos encajables. Todo esto ha permitido describir la evolución, desde las estrategias informales de modelización a estrategias formales, así como el desarrollo de la comprensión de la decena, para el que se describen transiciones entre niveles de comprensión señalados en estudios previos. Tras la categorización de las estrategias, en un segundo análisis, estas se describen como sucesión de capacidades y se representan, dentro de un diagrama, como posibles caminos de aprendizaje para la resolución de un problema. Al analizar las representaciones de cantidades discretas, cada cantidad se considera compuesta por un número y un tipo de objeto, identificando cada componente como icónica, simbólica, o ambas, lo que da lugar a un esquema de clasificación. Las representaciones se han estudiado en función del momento en que se producen: el proceso de resolución del problema, la anotación de la respuesta, o la carta en que se comunican el proceso y la respuesta. Los niños han utilizado con más frecuencia representaciones icónicas para resolver problemas, aumentando las representaciones simbólicas a lo largo del taller, por el uso de la Tabla 100 y los algoritmos. Para comunicar la solución por escrito o elaborar la carta, utilizan representaciones más formales como la escritura del número y el tipo de objeto en cifras o palabras. Entre las conclusiones de la investigación, los niños de primero de primaria han utilizado preferentemente, a lo largo de todo el curso, estrategias de modelización directa que reflejan el uso de conocimientos informales. Esto se ha producido en una situación de aprendizaje gobernada por normas que permitían elegir libremente estrategias y materiales, y a pesar de los conocimientos formales que se introducían en las clases ordinarias de matemáticas. Como implicación para el aula, se propone incluir en la enseñanza tareas que fomenten el uso de conocimientos informales, proporcionando experiencias en que los niños puedan construir ideas sobre los conceptos antes de su enseñanza formal. Por ejemplo, se pueden incluir problemas de estructura multiplicativa en primer curso de educación primaria. Esto supone un cambio de enfoque, pasando a ver la resolución de problemas como vía de construcción de contenidos matemáticos, superando un enfoque aplicacionista. Asimismo, el uso de materiales manipulativos debe someterse a reflexión, otorgando más importancia al pensamiento que desarrollan los niños con ayuda de materiales de su elección, que a la propia estructura del material. La actuación de los alumnos en el taller ha evidenciado características propias de un aprendizaje con comprensión, como la conexión entre estrategias informales y formales, el conocimiento infantil de la aplicabilidad de los algoritmos de adición y sustracción, o el uso de diferentes estrategias para un mismo problema. La introducción en la educación primaria de una metodología como la descrita en el taller, inspirada en la Instrucción Cognitivamente Guiada, puede favorecer el aprendizaje con comprensión. Como implicación para la teoría, propongo utilizar las trayectorias de enseñanza-aprendizaje y los caminos de aprendizaje de una tarea, como instrumentos complementarios para el diseño curricular y la planificación de aula. Las estrategias pueden desglosarse en secuencias de capacidades, que constituyen el nivel máximo de concreción de las expectativas de aprendizaje y, desde una perspectiva más amplia, este análisis permite identificar las capacidades necesarias para pasar de unos niveles de comprensión a otros superiores. Estas herramientas sirven para articular conocimientos informales y formales, estableciendo conexiones entre ambos.Universidad Complutense de Madrid (España)Castro Hernández, Carlos de (Universidad Complutense de Madrid)Bueno Álvarez, José Antonio (Universidad Complutense de Madrid)2015text (thesis)application/pdfhttps://dialnet.unirioja.es/servlet/oaites?codigo=47140spaLICENCIA DE USO: Los documentos a texto completo incluidos en Dialnet son de acceso libre y propiedad de sus autores y/o editores. Por tanto, cualquier acto de reproducción, distribución, comunicación pública y/o transformación total o parcial requiere el consentimiento expreso y escrito de aquéllos. 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