Main Article Content
Abstract
Studies that address preservice teachers’ knowledge of area measurement emphasize their lack of knowledge and their tendency towards the use of formulas, without offering a body of knowledge that helps to address such difficulties. This study offers an approximation of the mathematical knowledge necessary for preservice teachers to solve area tasks. For this, the Mathematics Teacher's Specialized Knowledge model is used with emphasis on the subdomain of Knowledge of Topics and Knowledge of the Structure of Mathematics. Preservice teachers' resolutions and written justifications are analyzed using qualitative and quantitative tools. The results indicate that those resolutions that manage to mobilize mathematical knowledge are associated with the joint mobilization of different procedures, properties, and geometric principles. Results also indicate that the strategic coordination between different registers of representation allows Preservice Teachers to mobilize categories of specialized knowledge and establishing connections with other mathematical contents.
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References
- Caviedes Barrera, S., de Gamboa, G., & Badillo Jiménez, E. R. (2021). Mathematical objects that configure the partial area meanings mobilized in task-solving. International Journal of Mathematical Education in Science and Technology, 1-20. https://doi.org/10.1080/0020739X.2021.1991019
- Caviedes, S., de Gamboa, G., & Badillo, E. (2022a). Knowledge used by preservice teaching students when comparing areas of 2D figures. Uniciencia, 36(1), 639-658. https://doi.org/10.15359/ru.36-1.41
- Caviedes, S., De Gamboa, G., & Badillo, E. (2022b). Pre-service teacher`s specialised knowledge on area of flat figures. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 123-130). PME. http://hdl.handle.net/10045/126553
- Aguilar-González, Á., Muñoz-Catalán, M. C., & Carrillo, J. (2018). An example of connections between the mathematics teacher’s conceptions and specialised knowledge. EURASIA Journal of Mathematics, Science and Technology Education, 15(2), em1664. https://doi.org/10.29333/ejmste/101598
- Aguilera, S., & Flores, P. (1998). Comparación de áreas de figuras por estudiantes de primero de magisterio [PDF file]. https://www.ugr.es/~pflores/textos/aRTICULOS/Propuestas/Jaen_areas.pdf
- Baturo, A., & Nason, R. (1996). Student teachers' subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31(3), 235-268. https://doi.org/10.1007/BF00376322
- Bailey, K. (1994) Methods of Social Research. The free press.
- Carrillo-Yañez, J., Climent, N., Contreras, L., & Muñoz-Catalán, M. (2013). Determining specialised knowledge for mathematics teaching. In B. Ubuz, C. Hasery M. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 2985-2994). ERME. http://cerme8.metu.edu/wgpapers/WG17/Wg17_Climent.pdf
- Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, A., Ribeiro, M., & Muñoz-Catalán, M. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236-253. https://doi.org/10.1080/14794802.2018.1479981
- Carpenter, T., Fennema, E., Peterson, P., & Carey, D. (1988). Teachers' pedagogical content knowledge of students' problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19(5), 385-401. https://doi.org/10.5951/jresematheduc.19.5.0385
- Chamberlin, M., & Candelaria, M. (2018). Learning from teaching teachers: A lesson experiment in area and volume with prospective teachers. Mathematics Teacher Education and Development, 20(1), 86-111. https://mted.merga.net.au/index.php/mted/article/view/314
- Clements, D. H., Sarama, J., Van Dine, D., Barrett, J., Cullen, C., Hudyma, A., Dolgin, R., Cullen, A., & Eames, C. (2018). Evaluation of three interventions teaching area measurement as spatial structuring to young children. The Journal of Mathematical Behaviour, 50, 23-41. https://doi.org/10.1016/j.jmathb.2017.12.004
- Cohen, L., Manion, L. & Morrison, K. (2000). Research Methods in Education, 5th ed. Routledge Falmer. https://doi.org/10.4324/9780203224342
- Douady, R., & Perrin-Glorian, M. J. (1989). Un processus d’apprentissage du concept d’aire de surface plane. Educational Studies in Mathematics, 20(4), 387–424. https://doi.org/10.1007/BF00315608
- Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z
- Duval, R. (2017). Understanding the mathematical way of thinking – The registers of semiotic representations. Springer. https://doi.org/10.1007/978-3-319-56910-9
- Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Reidel.
- Gras, R., & Kuntz, P. (2008). An overview of the statistical implicative analysis (SIA) development. In R. Gras, E. Suzuki, F. Guillet & F. Spagnolo (Eds.), Statistical Implicative Analysis (pp. 11–40). Springer. https://doi.org/10.1007/978-3-540-78983-3_1
- Hong, D., & Runnalls, C. (2020). Examining preservice teachers' responses to area conservation tasks. School Science and Mathematics, 120(5), 262-272. https://doi.org/10.1111/ssm.12409
- Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. https://doi.org/10.3102/00028312042002371
- Krippendorff, K. (2004) Content Analysis: An Introduction to its Methodology. Sage.
- Liñan, M., Barrera, V. & Infante, J. (2014). Teacher trainees’ specialized knowledge: solving a problem with division of fractions. Escuela Abierta, 17(1), 41-63. https://doi.org/10.29257/EA17.2014.04
- Livy, S., Muir, T., & Maher, N. (2012). How do they measure up? Primary pre-service teachers' mathematical knowledge of area and perimeter. Mathematics Teacher Education and Development, 14(2), 91-112. https://eric.ed.gov/?id=EJ1018652
- Montes, M., Aguilar, A., Carrillo, J., & Muñoz-Catalán, M. (2013). MTSK: From Common and Horizon Knowledge to Knowledge of Topics and Structures. In B. Ubuz, C. Haser, & M. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 3185-3194). ERME. http://www.mathematik.tu-dortmund.de/~erme/doc/CERME8/CERME8_2013_Proceedings.pdf
- Murphy, C. (2012). The role of subject knowledge in primary prospective teachers’ approaches to teaching the topic of area. Journal of Mathematics Teacher Education, 15(3), 187-206. https://doi.org/10.1007/s10857-011-9194-8
- Palmas, S., Rojano, T., & Sutherland, R. (2021). Digital technologies as a means of accessing powerful mathematical ideas. A study of adults with low schooling in Mexico. Teaching Mathematics and Its Applications: An International Journal of the IMA, 40(1), 16-39. https://doi.org/10.1093/teamat/hraa004
- Policastro, M., Mellone, M., Ribeiro, M., & Fiorentini, D. (2019). Conceptualising tasks for teacher education: from a research methodology to teachers’ knowledge development. In Jankvist, U. T., Van den Heuvel-Panhuizen, M., & Veldhuis, M. (Eds.). Proceedings of the 11th Congress of the European Society for Research in Mathematics (No. 24). ERME. https://hal.archives-ouvertes.fr/hal-02430487ERME.
- Puig, L., & Guillén, G. (1983). Necesidad y experimentación de un nuevo modelo para el estudio de la geometría en la EGB y Escuelas de Magisterio. [Memoria de Investigación] Universitat de Valencia.
- Ribeiro, M., & Amaral, R. (2015). Early years prospective teachers' specialised knowledge on problem posing. In Beswick, K.., Muir, T., & Wells, J. (Eds.). Proceedings of 39th Psychology of Mathematics Education conference, (Vol. 4, pp. 81-88). PME.
- Rocco, T., Bliss, L., Gallagher, S., Pérez, A., & Prado, P. (2003). Taking the next step: Mixed methods taking the next step: Mixed methods research in organizational systems research in organizational systems. Information Technology, Learning, and Performance Journal, 21(1), 19-29. https://www.proquest.com/docview/219816553
- Runnalls, C., & Hong, D. (2020). “Well, they understand the concept of area”: Pre-service teachers’ responses to student area misconceptions. Mathematics Education Research Journal, 32(4), 629-651. https://doi.org/10.1007/s13394-019-00274-1
- Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
- Simon, M., & Blume, G. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25(5), 472-494. https://doi.org/10.5951/jresematheduc.25.5.0472
- Sarama, J., & Clements, D. H. (2009). Early Childhood Mathematics Education Research: Learning Trajectories for Young Children. Routledge. https://doi.org/10.4324/9780203883785
- Trigueros, M. & Escandón, C. (2008). Los conceptos relevantes en el aprendizaje de la graficación. Revista Mexicana de Investigación Educativa, 13(36), 59-85. https://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-66662008000100004
References
Caviedes Barrera, S., de Gamboa, G., & Badillo Jiménez, E. R. (2021). Mathematical objects that configure the partial area meanings mobilized in task-solving. International Journal of Mathematical Education in Science and Technology, 1-20. https://doi.org/10.1080/0020739X.2021.1991019
Caviedes, S., de Gamboa, G., & Badillo, E. (2022a). Knowledge used by preservice teaching students when comparing areas of 2D figures. Uniciencia, 36(1), 639-658. https://doi.org/10.15359/ru.36-1.41
Caviedes, S., De Gamboa, G., & Badillo, E. (2022b). Pre-service teacher`s specialised knowledge on area of flat figures. In C. Fernández, S. Llinares, A. Gutiérrez, & N. Planas (Eds.), Proceedings of the 45th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 123-130). PME. http://hdl.handle.net/10045/126553
Aguilar-González, Á., Muñoz-Catalán, M. C., & Carrillo, J. (2018). An example of connections between the mathematics teacher’s conceptions and specialised knowledge. EURASIA Journal of Mathematics, Science and Technology Education, 15(2), em1664. https://doi.org/10.29333/ejmste/101598
Aguilera, S., & Flores, P. (1998). Comparación de áreas de figuras por estudiantes de primero de magisterio [PDF file]. https://www.ugr.es/~pflores/textos/aRTICULOS/Propuestas/Jaen_areas.pdf
Baturo, A., & Nason, R. (1996). Student teachers' subject matter knowledge within the domain of area measurement. Educational Studies in Mathematics, 31(3), 235-268. https://doi.org/10.1007/BF00376322
Bailey, K. (1994) Methods of Social Research. The free press.
Carrillo-Yañez, J., Climent, N., Contreras, L., & Muñoz-Catalán, M. (2013). Determining specialised knowledge for mathematics teaching. In B. Ubuz, C. Hasery M. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 2985-2994). ERME. http://cerme8.metu.edu/wgpapers/WG17/Wg17_Climent.pdf
Carrillo-Yañez, J., Climent, N., Montes, M., Contreras, L., Flores-Medrano, E., Escudero-Ávila, D., Vasco, D., Rojas, N., Flores, P., Aguilar-González, A., Ribeiro, M., & Muñoz-Catalán, M. (2018). The mathematics teacher’s specialised knowledge (MTSK) model. Research in Mathematics Education, 20(3), 236-253. https://doi.org/10.1080/14794802.2018.1479981
Carpenter, T., Fennema, E., Peterson, P., & Carey, D. (1988). Teachers' pedagogical content knowledge of students' problem solving in elementary arithmetic. Journal for Research in Mathematics Education, 19(5), 385-401. https://doi.org/10.5951/jresematheduc.19.5.0385
Chamberlin, M., & Candelaria, M. (2018). Learning from teaching teachers: A lesson experiment in area and volume with prospective teachers. Mathematics Teacher Education and Development, 20(1), 86-111. https://mted.merga.net.au/index.php/mted/article/view/314
Clements, D. H., Sarama, J., Van Dine, D., Barrett, J., Cullen, C., Hudyma, A., Dolgin, R., Cullen, A., & Eames, C. (2018). Evaluation of three interventions teaching area measurement as spatial structuring to young children. The Journal of Mathematical Behaviour, 50, 23-41. https://doi.org/10.1016/j.jmathb.2017.12.004
Cohen, L., Manion, L. & Morrison, K. (2000). Research Methods in Education, 5th ed. Routledge Falmer. https://doi.org/10.4324/9780203224342
Douady, R., & Perrin-Glorian, M. J. (1989). Un processus d’apprentissage du concept d’aire de surface plane. Educational Studies in Mathematics, 20(4), 387–424. https://doi.org/10.1007/BF00315608
Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131. https://doi.org/10.1007/s10649-006-0400-z
Duval, R. (2017). Understanding the mathematical way of thinking – The registers of semiotic representations. Springer. https://doi.org/10.1007/978-3-319-56910-9
Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Reidel.
Gras, R., & Kuntz, P. (2008). An overview of the statistical implicative analysis (SIA) development. In R. Gras, E. Suzuki, F. Guillet & F. Spagnolo (Eds.), Statistical Implicative Analysis (pp. 11–40). Springer. https://doi.org/10.1007/978-3-540-78983-3_1
Hong, D., & Runnalls, C. (2020). Examining preservice teachers' responses to area conservation tasks. School Science and Mathematics, 120(5), 262-272. https://doi.org/10.1111/ssm.12409
Hill, H., Rowan, B., & Ball, D. L. (2005). Effects of teachers’ mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2), 371-406. https://doi.org/10.3102/00028312042002371
Krippendorff, K. (2004) Content Analysis: An Introduction to its Methodology. Sage.
Liñan, M., Barrera, V. & Infante, J. (2014). Teacher trainees’ specialized knowledge: solving a problem with division of fractions. Escuela Abierta, 17(1), 41-63. https://doi.org/10.29257/EA17.2014.04
Livy, S., Muir, T., & Maher, N. (2012). How do they measure up? Primary pre-service teachers' mathematical knowledge of area and perimeter. Mathematics Teacher Education and Development, 14(2), 91-112. https://eric.ed.gov/?id=EJ1018652
Montes, M., Aguilar, A., Carrillo, J., & Muñoz-Catalán, M. (2013). MTSK: From Common and Horizon Knowledge to Knowledge of Topics and Structures. In B. Ubuz, C. Haser, & M. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 3185-3194). ERME. http://www.mathematik.tu-dortmund.de/~erme/doc/CERME8/CERME8_2013_Proceedings.pdf
Murphy, C. (2012). The role of subject knowledge in primary prospective teachers’ approaches to teaching the topic of area. Journal of Mathematics Teacher Education, 15(3), 187-206. https://doi.org/10.1007/s10857-011-9194-8
Palmas, S., Rojano, T., & Sutherland, R. (2021). Digital technologies as a means of accessing powerful mathematical ideas. A study of adults with low schooling in Mexico. Teaching Mathematics and Its Applications: An International Journal of the IMA, 40(1), 16-39. https://doi.org/10.1093/teamat/hraa004
Policastro, M., Mellone, M., Ribeiro, M., & Fiorentini, D. (2019). Conceptualising tasks for teacher education: from a research methodology to teachers’ knowledge development. In Jankvist, U. T., Van den Heuvel-Panhuizen, M., & Veldhuis, M. (Eds.). Proceedings of the 11th Congress of the European Society for Research in Mathematics (No. 24). ERME. https://hal.archives-ouvertes.fr/hal-02430487ERME.
Puig, L., & Guillén, G. (1983). Necesidad y experimentación de un nuevo modelo para el estudio de la geometría en la EGB y Escuelas de Magisterio. [Memoria de Investigación] Universitat de Valencia.
Ribeiro, M., & Amaral, R. (2015). Early years prospective teachers' specialised knowledge on problem posing. In Beswick, K.., Muir, T., & Wells, J. (Eds.). Proceedings of 39th Psychology of Mathematics Education conference, (Vol. 4, pp. 81-88). PME.
Rocco, T., Bliss, L., Gallagher, S., Pérez, A., & Prado, P. (2003). Taking the next step: Mixed methods taking the next step: Mixed methods research in organizational systems research in organizational systems. Information Technology, Learning, and Performance Journal, 21(1), 19-29. https://www.proquest.com/docview/219816553
Runnalls, C., & Hong, D. (2020). “Well, they understand the concept of area”: Pre-service teachers’ responses to student area misconceptions. Mathematics Education Research Journal, 32(4), 629-651. https://doi.org/10.1007/s13394-019-00274-1
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
Simon, M., & Blume, G. (1994). Building and understanding multiplicative relationships: A study of prospective elementary teachers. Journal for Research in Mathematics Education, 25(5), 472-494. https://doi.org/10.5951/jresematheduc.25.5.0472
Sarama, J., & Clements, D. H. (2009). Early Childhood Mathematics Education Research: Learning Trajectories for Young Children. Routledge. https://doi.org/10.4324/9780203883785
Trigueros, M. & Escandón, C. (2008). Los conceptos relevantes en el aprendizaje de la graficación. Revista Mexicana de Investigación Educativa, 13(36), 59-85. https://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-66662008000100004