University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function

Carlos Ledezma; Juan Pablo Vargas; Diana Hidalgo-Moncada; Vicenç Font

doi:10.29333/ejmste/13864

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Carlos Ledezma ¹ , Juan Pablo Vargas ¹ , Diana Hidalgo-Moncada ¹ , Vicenç Font ¹ ^*

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¹ Department of Linguistic, Scientific, & Mathematics Education, Faculty of Education, University of Barcelona, Barcelona, SPAIN^* Corresponding Author

Abstract

Literature shows a tendency to relegate the role of coefficient b to second place in the teaching of the quadratic function. We report an experience with Chilean university teachers, who designed a teaching and learning sequence with this function for construction engineering students. Our focus was on the didactic-mathematical knowledge about the effects of varying coefficient b on the graphical representation of this function that the participating teachers made evident. We constituted a focus group with 10 teachers and then qualitatively analyzed their dialogues using the mathematics teacher’s didactic-mathematical knowledge and competencies model. We highlight the following results: (a) the importance of mathematical knowledge and that of the epistemic facet to interpret the effect of coefficient b on the graphical representation of the quadratic function and (b) the proposal of an interpretation for the graphical behavior of coefficient b that contributes to the teaching of the quadratic function.

Keywords

coefficient b
didactic-mathematical knowledge
quadratic function
university teachers

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2023, Volume 19, Issue 12, Article No: em2370

https://doi.org/10.29333/ejmste/13864

Publication date: 01 Dec 2023

Online publication date: 05 Nov 2023

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Article Downloads: 385

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References

Arana-Pedraza, R. A., Ibarra, S. E., & Font, V. (2019, May 5–10). Conocimientos y competencias didáctico-matemáticas del profesor de matemáticas en ingeniería: Un primer acercamiento [Mathematics teacher’ didactic-mathematical knowledge and competencies in engineering: A first approach] [Paper presentation]. Inter-American Conference on Mathematics Education, Medellin, Colombia.
Araya, D., Pino-Fan, L. R., Medrano, I. G., & Castro, W. F. (2021). Epistemic criteria for designing limit tasks on a real variable function. Bolema: Boletim de Educação Matemática [Bulletin: Mathematics Education Bulletin], 35(69), 179-205. https://doi.org/10.1590/1980-4415v35n69a09
Breda, A., Font, V., & Pino-Fan, L. (2018). Criterios valorativos y normativos en la Didáctica de las Matemáticas: El caso del constructo idoneidad didáctica [Evaluative and normative criteria in Didactics of Mathematics: The case of didactic suitability construct]. BOLEMA: Boletim de Educação Matemática, 32(60), 255-278. https://doi.org/10.1590/1980-4415v32n60a13
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. https://doi.org/10.12973/eurasia.2017.01207a
Breda, A., Pochulu, M., Sánchez, A., & Font, V. (2021). Simulation of teacher interventions in a training course of mathematics teacher educators. Mathematics, 9(24), 3228. https://doi.org/10.3390/math9243228
Burns-Childers, A., & Vidakovic, D. (2018). Calculus students’ understanding of the vertex of the quadratic function in relation to the concept of derivative. International Journal of Mathematical Education in Science and Technology, 49(5), 660-679. https://doi.org/10.1080/0020739X.2017.1409367
Clement, L. L. (2001). What do students really know about functions? The Mathematics Teacher, 94(9), 745-748. https://doi.org/10.5951/MT.94.9.0745
Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education. Routledge. https://doi.org/10.4324/9781315456539
Davis, J. D. (2012). Investigating the effects of parameters on quadratic functions. The Mathematics Teacher, 106(1), 64-69. https://doi.org/10.5951/mathteacher.106.1.0064
Denscombe, M. (2014). The good research guide. Open University Press.
Díaz, V., Aravena, M., & Flores, G. (2020). Solving problem types contextualized to the quadratic function and error analysis: A case study. EURASIA Journal of Mathematics, Science and Technology Education, 16(11), em1896. https://doi.org/10.29333/ejmste/8547
Garcés, W. (2021). Análisis de las pautas que rigen la práctica del profesor en la enseñanza de derivadas en ciencias básicas en carreras de ingeniería [Analysis of the guidelines governing the teacher’s practice in the teaching of derivatives in basic sciences in engineering careers]. REDIMAT: Journal of Research in Mathematics Education, 10(3), 239-268. https://doi.org/10.17583/redimat.7957
García-García, J. (2019). Escenarios de exploración de conexiones matemáticas [Exploration scenarios of mathematical connections]. Números: Revista de Didáctica de las Matemáticas [Numbers: Magazine of Mathematics Didactics], 100, 129-133.
Gibbs, A. (2012). Focus groups and group interviews. In J. Arthur, M. Waring, R. Coe, & L. V. Hedges (Eds.), Research methods and methodologies in education (pp. 186-192). SAGE.
Godino, J. D. (2009). Categorías de análisis de los conocimientos del profesor de matemáticas [Analysis categories of the mathematics teacher’s knowledge]. UNIÓN: Revista Iberoamericana de Educación Matemática [UNION: Ibero-American Magazine of Mathematics Education], 20, 13-31.
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. https://doi.org/10.1007/s11858-006-0004-1
Graf, E. A., Fife, J. H., Howell, H., & Marquez, E. (2018). The development of a quadratic functions learning progression and associated task shells. ETS Research Report Series, 2018(1), ETS RR-18-47. https://doi.org/10.1002/ets2.12234
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372-400. https://doi.org/10.5951/jresematheduc.39.4.0372
Hydén, L.-C., & Bülow, P. H. (2003). Who’s talking: Drawing conclusions from focus groups–Some methodological considerations. International Journal of Social Research Methodology, 6(4), 305-321. https://doi.org/10.1080/13645570210124865
Kleiner, I. (1989). Evolution of the function concept: A brief survey. The College Mathematics Journal, 20(4), 282-300. https://doi.org/10.1080/07468342.1989.11973245
Kunter, M., Kleickmann, T., Klusmann, U., & Richter, D. (2013). The development of teachers’ professional competence. In M. Kunter, J. Baumert, W. Blum, U. Klusmann, S. Krauss, & M. Neubrand (Eds.), Cognitive activation in the mathematics classroom and professional competence of teachers: Results from the COACTIV project (pp. 63-77). Springer. https://doi.org/10.1007/978-1-4614-5149-5_4
Ledezma, C., Breda, A., & Font, V. (2023). Prospective teachers’ reflections on the inclusion of mathematical modelling during the transition period between the face-to-face and virtual teaching contexts. International Journal of Science and Mathematics Education. Advanced online publication. https://doi.org/10.1007/s10763-023-10412-8
Ledezma, C., Sol, T., Sala-Sebastià, G., & Font, V. (2022). Knowledge and beliefs on mathematical modelling inferred in the argumentation of a prospective teacher when reflecting on the incorporation of this process in his lessons. Mathematics, 10(18), 3339. https://doi.org/10.3390/math10183339
Mallart, A., Font, V., & Diez, J. (2018). Case study on mathematics pre-service teachers' difficulties in problem posing. EURASIA Journal of Mathematics, Science and Technology Education, 14(4), 1465-1481. https://doi.org/10.29333/ejmste/83682
Morgan, D. L. (1988). Focus groups as qualitative research. SAGE.
Neubrand, M. (2018). Conceptualizations of professional knowledge for teachers of mathematics. ZDM–Mathematics Education, 50(4), 601-612. https://doi.org/10.1007/s11858-017-0906-0
Ninow, V., & Kaiber, C. T. (2019). Affine function: An analysis from the perspective of the epistemic and cognitive suitability of the onto-semiotic approach. Acta Scientiae [Journal of Science], 21(6), 130-149. https://doi.org/10.17648/acta.scientiae.5506
Ozaltun Celik, A., & Bukova Guzel, E. (2017). Revealing Ozgur’s thoughts of a quadratic function with a clinical interview: Concepts and their underlying reasons. International Journal of Research in Education and Science, 3(1), 122-134.
Parra-Urrea, Y., & Pino-Fan, L. (2022). Proposal to systematize the reflection and assessment of the teacher’s practice on the teaching of functions. Mathematics, 10(18), 3330. https://doi.org/10.3390/math10183330
Petrou, M., & Goulding, M. (2011). Conceptualizing teachers’ mathematical knowledge in teaching. In T. Rowland, & K. Ruthven (Eds.), Mathematical knowledge in teaching (pp. 9-25). Springer. https://doi.org/10.1007/978-90-481-9766-8_2
Pino-Fan, L. R., Assis, A., & Castro, W. F. (2015). Towards a methodology for the characterization of teachers’ didactic-mathematical knowledge. EURASIA Journal of Mathematics, Science and Technology Education, 11(6), 1429-1456. https://doi.org/10.12973/eurasia.2015.1403a
Pino-Fan, L. R., Castro, W. F., & Font, V. (2023). A macro tool to characterize and develop key competencies for the mathematics teacher’s practice. International Journal of Science and Mathematics Education, 21(5), 1407-1432. https://doi.org/10.1007/s10763-022-10301-6
Pino-Fan, L., & Godino, J. D. (2015). Perspectiva ampliada del conocimiento didáctico-matemático del profesor [An expanded view of teachers’ didactic-mathematical knowledge]. Paradigma [Paradigm], 36(1), 87-109.
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. https://doi.org/10.1007/s10857-016-9349-8
Pino-Fan, L., & Parra-Urrea, Y. (2021). Criterios para orientar el diseño y la reflexión de clases sobre funciones: ¿Qué nos dice la literatura científica? [Criteria to guide the design and reflection on lessons about functions: What does scientific literature say?] Uno: Revista de Didáctica de las Matemáticas [One: Journal of Mathematics Didactics], 91, 45-54.
Pino-Fan, L., Parra-Urrea, Y., & Castro, W. F. (2019). Significados de la función pretendidos por el currículo de matemáticas chileno [Intended meanings of function in the Chilean mathematics curriculum]. Magis: Revista Internacional de Investigación en Educación [Magis: International Journal of Education Research], 11(23), 201-220. https://doi.org/10.11144/Javeriana.m11-23.sfpc
Ramírez, R., Cañadas, M. C., & Damián, A. (2022). Structures and representations used by 6^th graders when working with quadratic functions. ZDM–Mathematics Education, 54(6), 1393-1406. https://doi.org/10.1007/s11858-022-01423-w
Rowland, T. (2013). The knowledge quartet: The genesis and application of a framework for analyzing mathematics teaching and deepening teachers’ mathematics knowledge. Sisyphus–Journal of Education, 1(3), 15-43. https://doi.org/10.25749/sis.3705
Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: The knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281. https://doi.org/10.1007/s10857-005-0853-5
Schoenfeld, A. H., & Kilpatrick, J. (2008). Towards a theory of proficiency in teaching mathematics. In D. Tirosh, & T. Woods (Eds.), The international handbook of mathematics teacher education Vol. 2: Tools and processes in mathematics teacher education (pp. 321-354). Sense Publishers. https://doi.org/10.1163/9789087905460_016
Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-23. https://doi.org/10.17763/haer.57.1.j463w79r56455411

How to cite this article

APA

Ledezma, C., Vargas, J. P., Hidalgo-Moncada, D., & Font, V. (2023). University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function. Eurasia Journal of Mathematics, Science and Technology Education, 19(12), em2370. https://doi.org/10.29333/ejmste/13864

Vancouver

Ledezma C, Vargas JP, Hidalgo-Moncada D, Font V. University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function. EURASIA J Math Sci Tech Ed. 2023;19(12):em2370. https://doi.org/10.29333/ejmste/13864

AMA

Ledezma C, Vargas JP, Hidalgo-Moncada D, Font V. University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function. EURASIA J Math Sci Tech Ed. 2023;19(12), em2370. https://doi.org/10.29333/ejmste/13864

Chicago

Ledezma, Carlos, Juan Pablo Vargas, Diana Hidalgo-Moncada, and Vicenç Font. "University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function". Eurasia Journal of Mathematics, Science and Technology Education 2023 19 no. 12 (2023): em2370. https://doi.org/10.29333/ejmste/13864

Harvard

Ledezma, C., Vargas, J. P., Hidalgo-Moncada, D., and Font, V. (2023). University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function. Eurasia Journal of Mathematics, Science and Technology Education, 19(12), em2370. https://doi.org/10.29333/ejmste/13864

MLA

Ledezma, Carlos et al. "University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient b on the quadratic function". Eurasia Journal of Mathematics, Science and Technology Education, vol. 19, no. 12, 2023, em2370. https://doi.org/10.29333/ejmste/13864