Criteria used by mathematics teachers to pose problems in the classroom
DOI:
https://doi.org/10.15359/ru.34-2.7Keywords:
Teacher education, mathematics, problem solving, design, mathematics education, mathematics teachersAbstract
The study of the problems posed by teachers to their students and the characteristics that should be considered to enhance mathematical competences is an issue that has gained interest in recent years. However, most of these studies have been approached from an epistemic, technological, or cognitive point of view, without effectively considering what math teachers usually do to pose problems in their classes. In this sense, the objective of this article is to explore the pedagogical conceptions and practices of a group of elementary and secondary teachers from southern Chile about the selection of mathematical problems for their classes, and the criteria they use for such selection. The interest of this study stems from assuming that, in the classroom, teachers are the articulating vehicle of educational policies. Another motivation for this paper is the results of research in Mathematics Education. For this purpose, a questionnaire was presented which showed that teachers choose any of the following three mathematical problems: i) problems taken from textbooks, internet, and other resources; ii) problems adapted from textbooks, internet, or other resources (own variations); and iii) problems created by the teacher. In any case, there is a growing interest in teachers to pose problems with contexts that students feel close to and arouse interest from an emotional point of view. In addition, it was observed that the criteria used by teachers for posing math problems in the classroom implicitly contemplate the didactical suitability criteria proposed within the framework of the Onto-Semiotic Approach to Mathematical Knowledge and Instruction (OSA).
References
Arcavi, A., & Friedlander, A. (2007). Curriculum developers and problem solving: the case of Israeli elementary school projects. ZDM Mathematics Education, 39(5-6), 355-364. doi: https://doi.org/10.1007/s11858-007-0050-3
Borasi, R. (1986). On the nature of problems. Educational Studies in Mathematics, 17(2), 125-141. doi: https://doi.org/10.1007/BF00311517
Breda, A.; Pino-Fan, L., & Font, V. (2017). Meta didactic-mathematical knowledge of teachers: criteria for the reflection and assessment on teaching practice. Eurasia Journal of Mathematics, Science and Technology Education, 13(6), 1893-1918. doi: https://doi.org/10.12973/eurasia.2017.01207a
Breda, A., Font, V., & Pino-Fan, L. (2018). Criterios valorativos y normativos en la didáctica de las matemáticas: El caso del constructo idoneidad didáctica. Bolema, Río Claro, 32(60), 255-278. doi: http://dx.doi.org/10.1590/1980-4415v32n60a13
Castro, W. F., Pino-Fan, L., & Velásquez-Echavarría, H. (2018). A proposal to enhance preservice teacher’s noticing. Eurasia Journal of Mathematics, Science and Technology Education, 14(11), 1-17. doi: https://doi.org/10.29333/ejmste/92017
Cohen, L., Manion, L., & Morrison, K. (2011). Research Methods in Education (7th ed.). London: Routledge.
Crespo, S. (2003). Learning to pose mathematical problems: exploring changes in preservice teachers’ practices. Educational Studies in Mathematics, 52(3), 243-270. doi: https://doi.org/10.1023/A:1024364304664
De Guzmán, M. (1993). Tendencias innovadoras en educación matemática. Madrid, España: Universidad Complutense. Recuperado de http://nautilus.fis.uc.pt/bspm/revistas/25/009-034.150.pdf
Felmer, P., Pehkonen, E., & Kilpatrick, J. (2016). Posing and solving mathematical problems. Suiza: Springer International Publishing. Recuperado de https://link.springer.com/book/10.1007%2F978-3-319-28023-3
Felmer, P., & Perdomo-Díaz, J. (2016). Novice Chilean Secondary Mathematics Teachers as Problem Solvers. In P. Felmer, E. Pehkonen, & J. Kilpatrick (2016), Posing and solving mathematical problems (pp. 287-308). Suiza: Springer International Publishing. doi: https://doi.org/10.1007/978-3-319-28023-3_17
Font, V., Planas, N., & Godino, J. D. (2010). Modelo para el análisis didáctico en educación matemática. Infancia y Aprendizaje, 33(1), 89-105. Recuperado de https://www.academia.edu/1252367/Modelo_para_el_an%C3%A1lisis_did%C3%A1ctico_en_educaci%C3%B3n_matem%C3%A1tica
Font, V. (2011). Competencias profesionales en la formación inicial de profesores de matemáticas de secundaria. Unión. Revista Iberoamericana de Educación Matemática, 26, 9-25. Recuperado de http://www.fisem.org/www/union/revistas/2011/26/archivo_5_de_volumen_26.pdf
Font, V., Godino, J. D., & Gallardo, J. (2013). The Emergence of Objects from Mathematical Practices. Educational Studies in Mathematics, 82(1), 97-124. doi: https://doi.org/10.1007/s10649-012-9411-0
Foster, C. & Inglis, M. (2017). Teachers’ appraisals of adjectives relating to mathematics tasks. Educational Studies in Mathematics, 95(3), 283–301. https://doi.org/10.1007/s10649-017-9750-y
Godino, J. D., Batanero, C., & Font, V. (2007). The onto-semiotic approach to research in mathematics education. ZDM Mathematics Education, 39(1-2), 127-135. doi: https://doi.org/10.1007/s11858-006-0004-1
Godino, J. D., Batanero, C., & Font, V. (2019). The onto-semiotic approach: implications for the prescriptive character of didactics. For the Learning of Mathematics, 39(1), 37- 42.
Godino, J. D., Giacomone, B., Font, V., & Pino-Fan, L. (2018). Conocimientos profesionales en el diseño y gestión de una clase sobre semejanza de triángulos. Análisis con herramientas del modelo CCDM. Avances de Investigación en Educación Matemática, 13, 63-83. Recuperado de https://www.aiem.es/index.php/aiem/article/view/224
Halmos, P. R. (1980). The heart of mathematics. The American Mathematical Monthly, 87(7), 519-524. Recuperado de https://www.jstor.org/stable/2321415
Kaiber, C. T., Lemos, A., & Pino-Fan, L. (2017). Enfoque ntossemiótico do Conhecimento e da Instrução Matemática (EOS): um panorama das pesquisas na América Latina. Perspectivas da Educação Matemática, 10(23), 531 –552. Recuperado de https://periodicos.ufms.br/index.php/pedmat/article/download/5056/4091.
Kilpatrick, J. (1985). A Retrospective Account of the Twenty-five Years of Research on Teaching Mathematical Problem Solving. En E. A. Silver (Ed.), Teaching and learning mathematical problem solving: multiple research perspective (pp.1-15). Hillsdale, USA: Lawrence Erlbaum.
Klinshtern, M., Koichu, B., & Berman, A. (2015). What do high school teachers mean by saying “I pose my own problems”. In F. M., Singer, N., Ellerton & J., Cai. (2015). Mathematical Problem Posing. From research to effective practice (pp. 449-467). New York: Springer. doi: https://doi.org/10.1007/978-1-4614-6258-3_22
Liljedahl, P., Santos-Trigo, M., Malaspina, U., & Bruder, R. (2016). Problem solving in Mathematics Education. Hamburg, Germany: ICME-13 Topical Surveys, SpringerOpen. doi: https://doi.org/10.1007/978-3-319-40730-2_1
Love, E., & Pimm, D. (1996). ‘This is so’: a text on texts. En A. Bishop, K. Clements, C. Keitel, J. Kilpatrick, y C. Laborde (Eds.), International handbook of mathematics education (pp. 371– 409). Dordrecht, Holanda: Kluwer. Recuperado de https://link.springer.com/chapter/10.1007/978-94-009-1465-0_11
Malaspina, U., Mallart, A., & Font, V. (2015). Development of teachers' mathematical and didactic competencies by means of problem posing. En K. Krainer y N. Vondrová (Eds.), Proceedings of the Ninth Congress of the European Society for Research in Mathematics Education (pp. 2861-2866). Prague, Czech Republic: CERME. Recuperado de https://hal.archives-ouvertes.fr/hal-01289630/document
Malaspina, U. (2016). Creación de problemas: Sus potencialidades en la enseñanza y aprendizaje de las matemáticas. Cuadernos de Investigación y Formación en Educación Matemática, 11(15), 321-331. Recuperado de https://revistas.ucr.ac.cr/index.php/cifem/article/view/23946/24101
Malaspina, U. (2017). La creación de problemas como medio para potenciar la articulación de competencias y conocimientos del profesor de matemáticas. En J. M. Contreras, P. Arteaga, G. R. Cañadas, M. M. Gea, B. Giacomone y M. M. López-Martín (Eds.), Actas del Segundo Congreso International Virtual sobre el Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos. Pontificia Universidad Católica del Perú, Perú. Recuperado de http://enfoqueontosemiotico.ugr.es/civeos/malaspina.pdf
Morales-López, Y. & Font, V. (2019). Evaluation by a teacher of the suitability of her mathematics class. Educação e Pesquisa, 45, 1-19. doi: https://dx.doi.org/10.1590/s1678-4634201945189468
NCTM. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, Virginia, USA: NCTM. Recuperado de http://csmc.missouri.edu/PDFS/CCM/summaries/standards_summary.pdf
NCTM. (2016). De los principios a la acción. Para garantizar el éxito matemático para todos. Reston, Virginia, USA: NCTM.
Perdomo-Díaz, J., Felmer, P., Randolph, V., González, G. (2017). Problem Solving as a Professional Development Strategy for Teachers: a Case Study with Fractions. Eurasia Journal of Mathematics, Science and Technology Education, 13(3), 987-999. doi: https://doi.org/10.12973/eurasia.2017.00653a
Pino-Fan, L., Assis, A., & Castro, W. F. (2015). Towards a methodology for the characterization of teachers’ didactic-mathematical knowledge. Eurasia Journal of Mathematics, Science & Technology Education, 11(6), 1429-1456. Recuperado de https://pdfs.semanticscholar.org/dea9/70789e49dbe1d8206a04b19f4096fd186b89.pdf
Pino-Fan, L., & Godino, J. D. (2015). Perspectiva ampliada del conocimiento didáctico-matemático del profesor. PARADIGMA, 36(1), 87-109. Recuperado de https://pdfs.semanticscholar.org/bb35/fef73c0071dd63b94156d8757a9cd7b6f6bd.pdf?_ga=2.187459496.1402721529.1572471938-923436786.1565796163
Pino-Fan, L., Font, V., & Breda, A. (2017). Mathematics teachers’ knowledge and competences model based on the onto-semiotic approach. In B. Kaur, W. K. Ho, T. L. Toh & B. H. Choy (Eds.), Proceedings of the 41st Conference of the International Group for the Psychology of Mathematics Education (pp. 33-40). Singapure, Asia: PME. Recuperado de https://www.dropbox.com/s/6t1yfkuzy0ynbb4/Pino-Fan,Font %26 Breda_CCDM_PME.pdf?dl=0
Pino-Fan, L. Godino, J. D., & Font, V. (2018). Assessing key epistemic features of didactic-mathematical knowledge of prospective teachers: the case of the derivative. Journal of Mathematics Teacher Education, 21(1), 63-94. doi: http://dx.doi.org/10.1007/s10857-016-9349-8
Pino-Fan, L., Guzmán, I., Larraín, M., & Vargas, C. (2018). La formación inicial de profesores en Chile: 'voces' de la comunidad chilena de investigación en educación matemática. Uniciencia, 32(1), 68-88. doi: http://dx.doi.org/10.15359/ru.32-1.5
Polya, G. (1986). How to Solve it. [Cómo plantear y resolver problemas]. New Jersey, USA: Princeton University Press.
Presmeg, N. (2014). Visualization and Learning in Mathematics Education. In S. Lerman (Ed.), Encyclopedia of Mathematics Education. Dordrecht, Holanda: Springer. doi: https://doi.org/10.1007/978-94-007-4978-8
Ron, G., Zaslavsky, O. & Zodik, I. (2013). Engaging teachers in the web of considerations underlying the design of tasks that foster the need for new mathematical concept tools. In C. Margolinas (Ed.), Task design in mathematics education; Proceedings of the International Commission on Mathematical Instruction Study 22, (pp. 641–647), Oxford, UK. Recuperado de http://hal.archives-ouvertes.fr/hal-00834054
Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academic Press. doi: https://doi.org/10.1016/C2013-0-05012-8
Singer, F.M., Ellerton, N., & Cai, J. (2015). Mathematical Problem Posing. New York: Springer. doi: 10.1007/978-1-4614-6258-3
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
1. Authors guarantee the journal the right to be the first publication of the work as licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgment of the work's authorship and initial publication in this journal.
2. Authors can set separate additional agreements for non-exclusive distribution of the version of the work published in the journal (eg, place it in an institutional repository or publish it in a book), with an acknowledgment of its initial publication in this journal.
3. The authors have declared to hold all permissions to use the resources they provided in the paper (images, tables, among others) and assume full responsibility for damages to third parties.
4. The opinions expressed in the paper are the exclusive responsibility of the authors and do not necessarily represent the opinion of the editors or the Universidad Nacional.
Uniciencia Journal and all its productions are under Creative Commons Atribución-NoComercial-SinDerivadas 4.0 Unported.
There is neither fee for access nor Article Processing Charge (APC)
