Problem Solving in Chilean Higher Education: Figurations Prior in Interpretations of Cartesian Graphs
A Cartesian graph, as a mathematical object, becomes a tool for configuration of change. Its best comprehension is done through everyday life problem-solving associated with its representation. Despite this, the current educational framework favors general graphs, without consideration of their argumentation. Students are required to find the mathematical function without associating it to the development of graphical language. This research describes the use made by students of configurations made prior to Cartesian graphs with regards to an everyday life problem related to a time and distance variation phenomenon. The theoretical framework describes the function conditions of study and their modeling. This is a qualitative, descriptive study involving six undergraduate case studies that were carried out during the first term in 2016 at University of Los Lagos. The research problem concerned the graphic modeling of a real person’s movement phenomenon, and two levels of analysis were identified. The first level aims to identify local and global graph interpretations; a second level describes the iconicity and referentiality degree of an image. According to the results, students were able to draw no figures before the Cartesian graph, highlighting the need for students to represent the context and the movement of which causes the phenomenon change. From this, they managed Cartesian graphs representing changes in position, therefore, achieved an overall view of the graph. However, the local view only indicates specific events in the problem situation, using graphic and verbal expressions to represent movement. This view does not enable us to identify what happens on the graph when the movement characteristics change based on possible paths in the person’s walking speed.
[1] R. Barthes, “The rhetoric of the image” in Image Music Text, S. Heath, ed. London: Fontana Press, 1964, pp. 32-52.
[2] L. Blanco, “La investigación en educación”, Educatio Siglo XXI, vol. 29, no. 1, pp. 109-128, 2011.
[3] G. Buendía, “El uso de las gráficas cartesianas. Un estudio con profesores”, Revista Educación Matemática, vol. 24, no. 2, pp. 9-35, 2012.
[4] G. Buendía, and F. Cordero, “Prediction and the periodical aspects generators of knowledge in social practice framework. A socioepistemological study”, Educational Studies in Mathematics, vol. 58, no. 3, pp. 299–333, 2005.
[5] C. Campos, La argumentación gráfica en la transformación de funciones cuadráticas. Una aproximación socioepistemológica, master's dissertation, Cinvestav-IPN, D.F., México, 2003.
[6] R. Cantoral, and R. Farfán, “Pensamiento y lenguaje variacional en la introducción al análisis”, Epsilon, vol. 42, no. 3, pp. 854–856, 1998.
[7] C. Cen, Los funcionamientos y formas de las gráficas en los libros de texto: una práctica institucional en el bachillerato, master's dissertation, Cinvestav-IPN, D.F., México, 2006.
[8] J. Clement, “The Concept of variation and misconceptions in cartesian graphing”, Focus on Learning Problems in Mathematics,vol. 11, no. 1-2, pp.77-87, 1989.
[9] C. Coll, “Las competencias en la educación escolar: algo más que una moda y mucho menos que un remedio”. Aula de Innovación Educativa, vol. 1, no. 161, pp.34-39, 2007.
[10] F. Cordero, and R. Flores, “El uso de las gráficas en el discurso matemático escolar. Un estudio socio epistemológico en el nivel básico a través de los libros de texto”, Revista Latinoamericana de Investigación en Matemática Educativa, vol. 10, no. 1, pp. 7-38. 2007.
[11] F. Cordero, “La modellazione e la rappresentazione grafica nell'insegnamento–apprendimento dellamatematica”. La Matematica e la suaDidattica, vol. 20, no. 1, pp.59–79, 2006.
[12] V. Díaz, and I. Pérez, “Uso de gráficas en una situación de modelación del movimiento en matemática en la enseñanza secundaria en Chile”, Revista Paradigma, vol. XXXVII, no. 1, pp.161-180,2016.
[13] V. Diaz, and A. Poblete, “Training in Mathematics Based on Competences: Successful Experiences in Chilean Schools”, in Proc. 8th Int. Conf. of Education, Research and Innovation, Seville, 2015, pp. 3059-3067.
[14] V.Diaz, and A. Poblete, “Resolución de problemas en matemática desde la transversalidad: educar en valores éticos”,Revista Paradigma, vol. 35, no. 2, pp.155-182, 2014.
[15] V. Diaz, and A. Poblete, “Resolución de problemas en matemática y su integración con la enseñanza de valores éticos: el caso de Chile”, Boletim de Educação Matemática, vol. 27, no. 45, pp. 117-141, 2013.
[16] A. DiSessa, D. Hammer, B. Sherin, and T. Kolpakowski, “Inventing graphing: meta-representational expertise in children”,Journal of Mathematical Behavior,vol. 10, no. 2, pp. 117-160, 1991.
[17] C. Dolores, and I. Cuevas, “Lectura e interpretación de gráficas socialmente compartidas”, Revista Latinoamericana de Investigación en Matemática Educativa,vol. 10, no. 1, pp.69-96, 2007.
[18] I. Domínguez, La resignificación de lo asintótico en la aproximación socioepistemológica, master's dissertation, Cinvestav-IPN, D.F., México, 2003.
[19] L.F. Doorman, Modelling motion: From trace graphs to instantaneous change. Nederlands: CD-β Press, 2005.
[20] R. Duval, Semiosis y pensamiento humano. Cali: Universidad del Valle, 1999.
[21] R. Duval, “Graphiques et equations: L’articulation de deux registres”, Annales de Didactique et de Sciences Cognitives,vol. 1, pp. 235-253, 1998.
[22] R. Flores, Variaciones simultáneas de primer y segundo órdenes en una situación de graficación y modelación de movimiento, master's dissertation, 2007, Retrieved from http://www.matedu.cicata.ipn.mx/tesis/maestria/flores_2007.pdf Accessed 6/05/2015.
[23] Ministerio de Educación, Objetivos Fundamentales y Contenidos Mínimos Obligatorios de la Educación Básica y Media. Actualización 2009. Santiago: Impresos Universitaria, 2015.
[24] Ministerio de Educación, Programa de Estudio Matemática - 8° Básico, Marzo 2014. Santiago: Impresos Universitaria, 2014.
[25] Ministerio de Educación, Bases Curriculares 2013. Matemática 7° Básico a 2° Medio. Santiago: Impresos Universitaria, 2013.
[26] I. Miranda, L. Radford, and J. Guzmán, “One mathematical origin vs. two phenomenological origins: The meaning of the movement of objects with respect to the point (0,0)”, Journal of Research in Mathematics Education, vol. 2, no. 2, pp.183-208, 2013.
[27] I. Miranda, L. Radford, and J. Guzmán, “Interpretación de gráficas cartesianas sobre el movimiento desde el punto de vista de la teoría de la objetivación”, Revista Educación Matemática, vol. 19, no. 3, pp.5-30, 2007.
[28] R. Nemirovsky, C. Tierney, and W. Tracy, “Body and graphing”,Cognition and Instruction, vol. 16, no. 2, pp.119-172, 1998.
[29] R. Nemirovsky, “On ways of symbolizing: The case of Laura and the velocity sign”, Journal of Mathematical Behavior, vol. 13, no. 4, pp.389-422, 1994.
[30] OECD Organization for Economic Cooperation and Development, Computers and Learning: Making the Connection. PISA, OECD Publishing, Paris DOI: http://dx.doi.org/10.1787/9789264239555-en, 2015. Accessed 6/04/2016.
[31] OECD Organization for Economic Cooperation and Development, PISA 2015 Collaborative Problem Solving Framework. OECD Publishing, 2013.
[32] OECD Organization for Economic Cooperation and Development, PISA 2009 Results: Learning Trends: Changes in Student Performance Since 2000 (Volume V) http://dx.doi.org/10.1787/9789264091580, 2010. Accessed 11/12/2012.
[33] OECD Organization for Economic Cooperation and Development,The PISA 2003 Assessment Framework: Mathematics, Reading, Science and Problem Solving Knowledge and Skills. OECD Publishing, 2003.
[34] J. Pino, J. and L.J. Blanco, “Análisis de los problemas de los libros de texto de Matemáticas para alumnos de 12 a 14 años de edad de España y de Chile en relación con los contenidos de proporcionalidad”, Publicaciones, vol. 38, pp.63-88, 2008.
[35] L. Radford, “Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings”, Educational Studies in Mathematics, vol. 70, no. 2, pp. 11-26, 2009.
[36] L. Radford, S. Demers, J. Guzmán, and M. Cerulli, Calculators, graphs, gestures and the production of meaning. InProc.27th Conf. International Group for the Psychology of Mathematics Education,Hawaii, 2003, pp. 55-62.
[37] L. Radford, S. Demers, J. Guzmán, and M. Cerulli, The sensual and the conceptual: artefact-mediated kinesthetic actions and semiotic activity, in Proc.28th Conf. International Group for the Psychology of Mathematics Education, Norway, pp. 73-80, 2004.
[38] P. Rosado, Una resignificación de la derivada. El caso de la linealidad del polinomio en la aproximación socioepistemológica, master's dissertation,2004, Retrieved from http://cimate.uagro.mx/ivanlopez/seminario/archivos/pilar.pdf Accessed 3/04/2015.
[39] M. Santos, La Resolución de Problemas Matemáticos. Fundamentos Cognitivos. México: Trillas, 2007.
[40] B. Sherin, “How students invent representations of motion: A genetic account”, Journal of Mathematical Behavior, vol. 19, no. 4, pp.399-44, 2000.
[41] A. H. Schoenfeld, “Problem Solving in The United States, 1970-2008: Research and theory, practice and politics”, Zentralblatt für Didaktik der Mathematik, vol 9, nos. 5-6, pp. 537-551, 2008.
[42] L. Suárez, and F. Cordero, “Modelación–graficación, una categoría para la matemática escolar. Resultados de un estudio socioepistemológico”. Revista Latinoamericana de Investigación en Matemática Educativa, vol. 13, nos. 4-II, pp.319-333, 2010.
[43] L. Suárez, and F. Cordero, “Elementos teóricos para estudiar el uso de las gráficas en la modelación del cambio y de la variación en un ambiente tecnológico”, Revista Electrónica de Investigación en Educación en Ciencias, vol. 3, no. 1, pp. 51-58, 2008.
[44] A. Torres, La modelación y las gráficas en situaciones de movimiento con tecnología, master's dissertation, CICATA-IPN., D.F., México, 2004.
[45] C. Yoon, M. Thomas, and T. Dreyfus,“Grounded blends and mathematical gesture spaces: Developing mathematical understandings via gestures”, Educational Studies in Mathematics, vol. 7, no. 3, pp. 271-303, 2011.
[1] R. Barthes, “The rhetoric of the image” in Image Music Text, S. Heath, ed. London: Fontana Press, 1964, pp. 32-52.
[2] L. Blanco, “La investigación en educación”, Educatio Siglo XXI, vol. 29, no. 1, pp. 109-128, 2011.
[3] G. Buendía, “El uso de las gráficas cartesianas. Un estudio con profesores”, Revista Educación Matemática, vol. 24, no. 2, pp. 9-35, 2012.
[4] G. Buendía, and F. Cordero, “Prediction and the periodical aspects generators of knowledge in social practice framework. A socioepistemological study”, Educational Studies in Mathematics, vol. 58, no. 3, pp. 299–333, 2005.
[5] C. Campos, La argumentación gráfica en la transformación de funciones cuadráticas. Una aproximación socioepistemológica, master's dissertation, Cinvestav-IPN, D.F., México, 2003.
[6] R. Cantoral, and R. Farfán, “Pensamiento y lenguaje variacional en la introducción al análisis”, Epsilon, vol. 42, no. 3, pp. 854–856, 1998.
[7] C. Cen, Los funcionamientos y formas de las gráficas en los libros de texto: una práctica institucional en el bachillerato, master's dissertation, Cinvestav-IPN, D.F., México, 2006.
[8] J. Clement, “The Concept of variation and misconceptions in cartesian graphing”, Focus on Learning Problems in Mathematics,vol. 11, no. 1-2, pp.77-87, 1989.
[9] C. Coll, “Las competencias en la educación escolar: algo más que una moda y mucho menos que un remedio”. Aula de Innovación Educativa, vol. 1, no. 161, pp.34-39, 2007.
[10] F. Cordero, and R. Flores, “El uso de las gráficas en el discurso matemático escolar. Un estudio socio epistemológico en el nivel básico a través de los libros de texto”, Revista Latinoamericana de Investigación en Matemática Educativa, vol. 10, no. 1, pp. 7-38. 2007.
[11] F. Cordero, “La modellazione e la rappresentazione grafica nell'insegnamento–apprendimento dellamatematica”. La Matematica e la suaDidattica, vol. 20, no. 1, pp.59–79, 2006.
[12] V. Díaz, and I. Pérez, “Uso de gráficas en una situación de modelación del movimiento en matemática en la enseñanza secundaria en Chile”, Revista Paradigma, vol. XXXVII, no. 1, pp.161-180,2016.
[13] V. Diaz, and A. Poblete, “Training in Mathematics Based on Competences: Successful Experiences in Chilean Schools”, in Proc. 8th Int. Conf. of Education, Research and Innovation, Seville, 2015, pp. 3059-3067.
[14] V.Diaz, and A. Poblete, “Resolución de problemas en matemática desde la transversalidad: educar en valores éticos”,Revista Paradigma, vol. 35, no. 2, pp.155-182, 2014.
[15] V. Diaz, and A. Poblete, “Resolución de problemas en matemática y su integración con la enseñanza de valores éticos: el caso de Chile”, Boletim de Educação Matemática, vol. 27, no. 45, pp. 117-141, 2013.
[16] A. DiSessa, D. Hammer, B. Sherin, and T. Kolpakowski, “Inventing graphing: meta-representational expertise in children”,Journal of Mathematical Behavior,vol. 10, no. 2, pp. 117-160, 1991.
[17] C. Dolores, and I. Cuevas, “Lectura e interpretación de gráficas socialmente compartidas”, Revista Latinoamericana de Investigación en Matemática Educativa,vol. 10, no. 1, pp.69-96, 2007.
[18] I. Domínguez, La resignificación de lo asintótico en la aproximación socioepistemológica, master's dissertation, Cinvestav-IPN, D.F., México, 2003.
[19] L.F. Doorman, Modelling motion: From trace graphs to instantaneous change. Nederlands: CD-β Press, 2005.
[20] R. Duval, Semiosis y pensamiento humano. Cali: Universidad del Valle, 1999.
[21] R. Duval, “Graphiques et equations: L’articulation de deux registres”, Annales de Didactique et de Sciences Cognitives,vol. 1, pp. 235-253, 1998.
[22] R. Flores, Variaciones simultáneas de primer y segundo órdenes en una situación de graficación y modelación de movimiento, master's dissertation, 2007, Retrieved from http://www.matedu.cicata.ipn.mx/tesis/maestria/flores_2007.pdf Accessed 6/05/2015.
[23] Ministerio de Educación, Objetivos Fundamentales y Contenidos Mínimos Obligatorios de la Educación Básica y Media. Actualización 2009. Santiago: Impresos Universitaria, 2015.
[24] Ministerio de Educación, Programa de Estudio Matemática - 8° Básico, Marzo 2014. Santiago: Impresos Universitaria, 2014.
[25] Ministerio de Educación, Bases Curriculares 2013. Matemática 7° Básico a 2° Medio. Santiago: Impresos Universitaria, 2013.
[26] I. Miranda, L. Radford, and J. Guzmán, “One mathematical origin vs. two phenomenological origins: The meaning of the movement of objects with respect to the point (0,0)”, Journal of Research in Mathematics Education, vol. 2, no. 2, pp.183-208, 2013.
[27] I. Miranda, L. Radford, and J. Guzmán, “Interpretación de gráficas cartesianas sobre el movimiento desde el punto de vista de la teoría de la objetivación”, Revista Educación Matemática, vol. 19, no. 3, pp.5-30, 2007.
[28] R. Nemirovsky, C. Tierney, and W. Tracy, “Body and graphing”,Cognition and Instruction, vol. 16, no. 2, pp.119-172, 1998.
[29] R. Nemirovsky, “On ways of symbolizing: The case of Laura and the velocity sign”, Journal of Mathematical Behavior, vol. 13, no. 4, pp.389-422, 1994.
[30] OECD Organization for Economic Cooperation and Development, Computers and Learning: Making the Connection. PISA, OECD Publishing, Paris DOI: http://dx.doi.org/10.1787/9789264239555-en, 2015. Accessed 6/04/2016.
[31] OECD Organization for Economic Cooperation and Development, PISA 2015 Collaborative Problem Solving Framework. OECD Publishing, 2013.
[32] OECD Organization for Economic Cooperation and Development, PISA 2009 Results: Learning Trends: Changes in Student Performance Since 2000 (Volume V) http://dx.doi.org/10.1787/9789264091580, 2010. Accessed 11/12/2012.
[33] OECD Organization for Economic Cooperation and Development,The PISA 2003 Assessment Framework: Mathematics, Reading, Science and Problem Solving Knowledge and Skills. OECD Publishing, 2003.
[34] J. Pino, J. and L.J. Blanco, “Análisis de los problemas de los libros de texto de Matemáticas para alumnos de 12 a 14 años de edad de España y de Chile en relación con los contenidos de proporcionalidad”, Publicaciones, vol. 38, pp.63-88, 2008.
[35] L. Radford, “Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings”, Educational Studies in Mathematics, vol. 70, no. 2, pp. 11-26, 2009.
[36] L. Radford, S. Demers, J. Guzmán, and M. Cerulli, Calculators, graphs, gestures and the production of meaning. InProc.27th Conf. International Group for the Psychology of Mathematics Education,Hawaii, 2003, pp. 55-62.
[37] L. Radford, S. Demers, J. Guzmán, and M. Cerulli, The sensual and the conceptual: artefact-mediated kinesthetic actions and semiotic activity, in Proc.28th Conf. International Group for the Psychology of Mathematics Education, Norway, pp. 73-80, 2004.
[38] P. Rosado, Una resignificación de la derivada. El caso de la linealidad del polinomio en la aproximación socioepistemológica, master's dissertation,2004, Retrieved from http://cimate.uagro.mx/ivanlopez/seminario/archivos/pilar.pdf Accessed 3/04/2015.
[39] M. Santos, La Resolución de Problemas Matemáticos. Fundamentos Cognitivos. México: Trillas, 2007.
[40] B. Sherin, “How students invent representations of motion: A genetic account”, Journal of Mathematical Behavior, vol. 19, no. 4, pp.399-44, 2000.
[41] A. H. Schoenfeld, “Problem Solving in The United States, 1970-2008: Research and theory, practice and politics”, Zentralblatt für Didaktik der Mathematik, vol 9, nos. 5-6, pp. 537-551, 2008.
[42] L. Suárez, and F. Cordero, “Modelación–graficación, una categoría para la matemática escolar. Resultados de un estudio socioepistemológico”. Revista Latinoamericana de Investigación en Matemática Educativa, vol. 13, nos. 4-II, pp.319-333, 2010.
[43] L. Suárez, and F. Cordero, “Elementos teóricos para estudiar el uso de las gráficas en la modelación del cambio y de la variación en un ambiente tecnológico”, Revista Electrónica de Investigación en Educación en Ciencias, vol. 3, no. 1, pp. 51-58, 2008.
[44] A. Torres, La modelación y las gráficas en situaciones de movimiento con tecnología, master's dissertation, CICATA-IPN., D.F., México, 2004.
[45] C. Yoon, M. Thomas, and T. Dreyfus,“Grounded blends and mathematical gesture spaces: Developing mathematical understandings via gestures”, Educational Studies in Mathematics, vol. 7, no. 3, pp. 271-303, 2011.
@article{"International Journal of Business, Human and Social Sciences:74816", author = "Verónica Díaz", title = "Problem Solving in Chilean Higher Education: Figurations Prior in Interpretations of Cartesian Graphs", abstract = "A Cartesian graph, as a mathematical object, becomes a tool for configuration of change. Its best comprehension is done through everyday life problem-solving associated with its representation. Despite this, the current educational framework favors general graphs, without consideration of their argumentation. Students are required to find the mathematical function without associating it to the development of graphical language. This research describes the use made by students of configurations made prior to Cartesian graphs with regards to an everyday life problem related to a time and distance variation phenomenon. The theoretical framework describes the function conditions of study and their modeling. This is a qualitative, descriptive study involving six undergraduate case studies that were carried out during the first term in 2016 at University of Los Lagos. The research problem concerned the graphic modeling of a real person’s movement phenomenon, and two levels of analysis were identified. The first level aims to identify local and global graph interpretations; a second level describes the iconicity and referentiality degree of an image. According to the results, students were able to draw no figures before the Cartesian graph, highlighting the need for students to represent the context and the movement of which causes the phenomenon change. From this, they managed Cartesian graphs representing changes in position, therefore, achieved an overall view of the graph. However, the local view only indicates specific events in the problem situation, using graphic and verbal expressions to represent movement. This view does not enable us to identify what happens on the graph when the movement characteristics change based on possible paths in the person’s walking speed.
", keywords = "Cartesian graphs, higher education, movement modeling, problem solving.", volume = "11", number = "2", pages = "384-9", }
