How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study

José María Gavilán-Izquierdo; Inés Gallego-Sánchez

doi:10.29333/iejme/10892

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José María Gavilán-Izquierdo ¹ , Inés Gallego-Sánchez ¹ ^*

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¹ Universidad de Sevilla, SPAIN^* Corresponding Author

Abstract

This paper analyzes the discursive activity of an upper secondary school teacher on the topic of the derivative. Specifically, we focus on the discursive activity when introducing the three essential concepts related to this topic: the average rate of change, the derivative at a point, and the derivative function. The theoretical framework used is the theory of commognition (Sfard, 2008). Using this framework, we identify different types of visual mediators (symbols, graphs, gestures) and routines (mainly construction and substantiation ones, with various subtypes). The analysis of these visual mediators and routines allows us to examine some resources students will have in their transition to university discourse. We also draw some other conclusions and present implications for teaching in this stage.

Keywords

teacher’s discursive activity
commognition
routines
visual mediators
transition
derivative

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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

INT ELECT J MATH ED, 2021, Volume 16, Issue 2, Article No: em0634

https://doi.org/10.29333/iejme/10892

Publication date: 07 May 2021

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Declaration of Conflict of Interest: No conflict of interest is declared by author(s).

Data sharing statement: Data supporting the findings and conclusions are available upon request from the corresponding author(s).

References

Alcock, L., & Simpson, A. (2002). Definitions: Dealing with categories mathematically. For the learning of mathematics, 22(2), 28-34.
Alibali, M. W., & Nathan, M. J. (2012). Embodiment in mathematics teaching and learning: Evidence from learners’ and teachers’ gestures. Journal of the Learning Sciences, 21(2), 247-286. https://doi.org/10.1080/10508406.2011.611446
Artigue, M. (1991). Analysis. In D. Tall (Ed.), Advanced mathematical thinking (pp. 167-198). Kluwer.
Badillo, E., Azcarate, C., & Font, V. (2011). Análisis de los niveles de comprensión de los objetos f’(a) y f’(x) en profesores de matemáticas [Analysis of mathematics teachers’ level of understanding of the objects f’(a) and f’(x)]. Enseñanza de las Ciencias, 29(2), 191-206. https://doi.org/10.5565/rev/ec/v29n2.546
Bayda, N. I., & Sutliff, G. (2020). Comparing extracted and stipulated definitions in Algebra 1 textbooks and Khan Academy. International Electronic Journal of Mathematics Education, 15(2), 1-8. https://doi.org/10.29333/iejme/7601
Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh (Ed.), Acquisition of mathematical concepts and processes (pp. 91-126). New Academic Press.
Biza, I., Giraldo, V., Hochmuth, R., Khakbaz, A. S., & Rasmussen, C. (2016). Research on teaching and learning mathematics at the tertiary level: State-of-the-art and looking ahead. Springer. https://doi.org/10.1007/978-3-319-41814-8
Borji, V., & Sánchez, A. (2019). An exploratory analysis of the representations of functions in the university entrance exam in Spain and Iran. Eurasia Journal of Mathematics, Science and Technology Education, 15(8), em1727. https://doi.org/10.29333/ejmste/106258
Borji, V., Font, V., Alamolhodaei, H., & Sánchez, A. (2018). Application of the complementarities of two theories, APOS and OSA, for the analysis of the university students’ understanding on the graph of the function and its derivative. Eurasia Journal of Mathematics, Science and Technology Education, 14(6), 2301-2315. https://doi.org/10.29333/ejmste/89514
Burn, R. (2002). The genesis of mathematical structures. In P. Kahn & J. Kyle (Eds.), Effective learning and teaching in mathematics and its applications (pp. 20-33). Kogan Page.
Clark, M., & Lovric, M. (2009). Understanding secondary-tertiary transition in mathematics. International Journal of Mathematical Education in Science and Technology, 40(6), 755-776. https://doi.org/10.1080/00207390902912878
Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Kluwer.
Corriveau, C., & Bednarz, N. (2016). The secondary-tertiary transition viewed as a change in mathematical cultures: An exploration concerning symbolism and its use. Educational Studies in Mathematics, 95(1), 1-19 (2017). https://doi.org/10.1007/s10649-016-9738-z
Dahl, B. (2017). First-year non-STEM majors’ use of definitions to solve calculus tasks: Benefits of using concept image over concept definition? International Journal of Science and Mathematics Education, 15(7), 1303-1322. https://doi.org/10.1007/s10763-016-9751-9
De Guzman, M., Hodgson, B., Robert, A., & Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. In G. Fischer & U. Rehmann (Eds.), Proceedings of the International Congress of Mathematicians (Vol III, pp. 747-762). Fakultät für Mathematik, Universität Bielefeld.
De Villiers, M. (1998). To teach definitions in geometry or teach to define? In A. Olivier & K. Newstead (Eds.), Proceedings of the 22nd conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 248-255). University of Stellenbosch.
Dreher, A., Lindmeier, A., Heinze, A., & Niemand, C. (2018). What kind of content knowledge do secondary mathematics teachers need? Journal für Mathematik-Didaktik, 39(2), 319-341. https://doi.org/10.1007/s13138-018-0127-2
Edwards, B., & Ward, M. (2008). The role of mathematical definitions in mathematics and in undergraduate mathematics courses. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 221-230). Mathematical Association of America.
Edwards, L. D. (2009). Gestures and conceptual integration in mathematical talk. Educational Studies in Mathematics, 70(2), 127-141. https://doi.org/10.1007/s10649-008-9124-6
Emre-Akdoğan, E., Güçler, B., & Argün, Z. (2018). The development of two high school students’ discourses on geometric translation in relation to the teacher’s discourse in the classroom. Eurasia Journal of Mathematics, Science and Technology Education, 14(5), 1605-1619. https://doi.org/10.29333/ejmste/84885
Farsani, D., Breda, A., & Sala, G. (2020). ¿Cómo los gestos de los maestros afectan a la atención visual de las estudiantes durante el discurso matemático? [How do teachers’ gestures affect students’ visual attention during mathematical discourse?]. REDIMAT – Journal of Research in Mathematics Education, 9(3), 220-242. https://doi.org/10.17583/redimat.2020.5185
Freudenthal, H. (1973). Mathematics as an educational task. Reidel.
Fuentealba, C., Badillo, E., & Sánchez-Matamoros, G. (2019). Identificación y caracterización de los subniveles de desarrollo del esquema de derivada [Identification and characterization of the development sublevels of the derivative schema]. Enseñanza de las Ciencias, 37(2), 63–84. https://doi.org/10.5565/rev/ensciencias.2518
Fuentealba, C., Badillo, E., Sánchez-Matamoros, G., & Cárcamo, A. (2019). The understanding of the derivative concept in higher education. Eurasia Journal of Mathematics, Science and Technology Education, 15(2), em1662. https://doi.org/10.29333/ejmste/100640
García, M., Gavilán, J. M., & Llinares, S. (2012). Perspectiva de la práctica del profesor de matemáticas de secundaria sobre la enseñanza de la derivada. Relaciones entre la práctica y la perspectiva del profesor [Perspective of the secondary mathematics teacher’s practice on the teaching of the derivative. Relations between the teacher’s perspective and practice]. Enseñanza de las Ciencias, 30(3), 219-235. https://doi.org/10.5565/rev/ec/v30n3.684
Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116-140. https://doi.org/10.2307/749505
Güçler, B. (2013). Examining the discourse on the limit concept in a beginning-level calculus classroom. Educational Studies in Mathematics, 82(3), 439-453. https://doi.org/10.1007/s10649-012-9438-2
Gueudet, G. (2008). Investigating the secondary-tertiary transition. Educational Studies in Mathematics, 67(3), 237-254. https://doi.org/10.1007/s10649-007-9100-6
Kaminski, J. A., Sloutsky, V. M., & Heckler, A. F. (2008). The advantage of abstract examples in learning math. Science, 320(5875), 454-455. https://doi.org/10.1126/science.1154659
Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://doi.org/10.2307/4149959
Kobiela, M., & Lehrer, R. (2015). The codevelopment of mathematical concepts and the practice of defining. Journal for Research in Mathematics Education, 46(4), 423-454. https://doi.org/10.5951/jresematheduc.46.4.0423
Lakatos, I. (1976). Proofs and refutations. Cambridge University Press.
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
Landau, S. I. (2001). Dictionaries: The art and craft of lexicography (2nd ed.). Cambridge University Press.
Lavie, I., Steiner, A., & Sfard, A. (2019). Routines we live by: From ritual to exploration. Educational Studies in Mathematics, 101(2), 153-176. https://doi.org/10.1007/s10649-018-9817-4
Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics. Educational Studies in Mathematics, 46(1-3), 87-113. https://doi.org/10.1023/A:1014031004832
Martín-Molina, V., González-Regaña, A. J., & Gavilán-Izquierdo, J. M. (2018) Researching how professional mathematicians construct new mathematical definitions: A case study. International Journal of Mathematical Education in Science and Technology, 49(7), 1069-1082. https://doi.org/10.1080/0020739X.2018.1426795
Martín-Molina, V., González-Regaña, A. J., Toscano, R., & Gavilán-Izquierdo, J. M. (2020). Differences between how undergraduate students define geometric solids and what their lecturers expect from them through the lens of the theory of commognition. Eurasia Journal of Mathematics, Science and Technology Education, 16(12), em1917. https://doi.org/10.29333/ejmste/9159
Mason, J., & Pimm, D. (1984). Generic examples: Seeing the general in the particular. Educational Studies in Mathematics, 15(3), 277-289. https://doi.org/10.1007/BF00312078
McNeill, D. (1992). Hand and mind: What gestures reveal about thought. University of Chicago Press.
Morgan, C. (2005). Word, definitions and concepts in discourses of mathematics, teaching and learning. Language and Education, 19(2), 102-116. https://doi.org/10.1080/09500780508668666
Nardi, E., Ryve, A., Stadler, E., & Viirman, O. (2014). Commognitive analyses of the learning and teaching of mathematics at university level: The case of discursive shifts in the study of Calculus. Research in Mathematics Education, 16(2), 182-198. https://doi.org/10.1080/14794802.2014.918338
Niss, M. (2003). Mathematical competencies and the learning of mathematics: The Danish KOM project. In A. Gagatsis & S. Papastavridis (Eds.), Proceedings of the 3rd Mediterranean Conference on Mathematics Education (pp. 115-124). Hellenic MS.
Panero, M., Arzarello, F., & Sabena, C. (2016). The mathematical work with the derivative of a function: Teachers’ practices with the idea of “generic”. Bolema, Rio Claro (SP), 54(30), 265-286. https://doi.org/10.1590/1980-4415v30n54a13
Park, J. (2011). Calculus instructors’ and students’ discourses on the derivative (Unpublished doctoral dissertation). Michigan State University, Michigan.
Park, J. (2015). Is the derivative a function? If so, how do we teach it? Educational Studies in Mathematics, 90(2), 233-250. https://doi.org/10.1007/s10649-015-9630-2
Pinto-Rojas, I. E., & Parraguez, M. (2017). Articulators for thinking modes of the derivative from a local perspective. International Electronic Journal of Mathematics Education, 12(3), 873-898.
Planas, N., & Civil, M. (2010). Discourse processes in critical mathematics education. In H. Alrø, O. Ravn, & P. Valero (Eds.), Critical mathematics education: Past, present and future (pp. 145-160). Sense Publishers.
Rach, S., & Heinze, A. (2017). The transition from school to university in mathematics: Which influence do school-related variables have? International Journal of Science and Mathematics Education, 15(7), 1343-1363. https://doi.org/10.1007/s10763-016-9744-8
Rach, S., & Ufer, S. (2020). Which prior mathematical knowledge is necessary for study success in the university study entrance phase? Results on a new model of knowledge levels based on a reanalysis of data from existing studies. International Journal of Research in Undergraduate Mathematics Education, 6(3), 375-403. https://doi.org/10.1007/s40753-020-00112-x
Rasmussen, C., Zandieh, M, King, K., & Teppo, A. (2005). Advancing mathematical activity: A practice-oriented view of advanced mathematical thinking. Mathematical Thinking and Learning, 7(1), 51-73. https://doi.org/10.1207/s15327833mtl0701_4
Rooch, A., Junker, P., Härterich, J., & Hackl, K. (2016). Linking mathematics with engineering applications at an early stage – Implementation, experimental set-up and evaluation of a pilot project. European Journal of Engineering Education, 41(2), 172-191. https://doi.org/10.1080/03043797.2015.1056095
Schüler-Meyer, A. (2018). Defining as discursive practice in transition – Upper secondary students reinvent the formal definition of convergent sequences. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N. M. Hogstad (Eds.), Proceedings of INDRUM 2018 (pp. 255-264). University of Agder.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36. https://doi.org/10.1007/BF00302715
Sfard, A. (2008). Thinking as communicating human development, the growth of discourses, and mathematizing. Cambridge University Press.
Sfard, A. (2009). What’s all the fuss about gestures? A commentary. Educational Studies in Mathematics, 70(2), 191-200. https://doi.org/10.1007/s10649-008-9161-1
Sfard, A. (2014). University mathematics as a discourse – Why, how, and what for? Research in Mathematics Education, 16(2), 199-203. https://doi.org/10.1080/14794802.2014.918339
Sfard, A. (2007). When the rules of discourse change, but nobody tells you: Making sense of mathematics learning from a commognitive standpoint. Journal of the Learning Sciences, 16(4), 565-613. https://doi.org/10.1080/10508400701525253
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. https://doi.org/10.3102/0013189X015002004
Tabach, M., & Nachlieli, T. (2016). Communicational perspectives on learning and teaching mathematics: Prologue. Educational Studies in Mathematics, 91(3), 299-306. https://doi.org/10.1007/s10649-015-9638-7
Tall, D. O. (1991). The psychology of advanced mathematical thinking. In D. O. Tall (Ed.), Advanced mathematical thinking (pp. 1-24). Kluwer.
Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169. https://doi.org/10.1007/BF00305619
Tasara, I. (2018). Commognitive analysis of a teacher’s mathematical discourse on the derivative. In F. Curtis (Ed.), BSRLM Proceedings, 2017: British Society for Research into Learning Mathematics Conference. British Society for Research into Learning Mathematics.
Thoma, A., & Nardi, E. (2018). Transition from school to university mathematics: Manifestations of unresolved commognitive conflict in first year students’ examination scripts. International Journal of Research in Undergraduate Mathematics Education, 4(1), 161-180. https://doi.org/10.1007/s40753-017-0064-3
Thomas, M. O. J., de Freitas Druck, I., Huillet, D., Ju, M. K., Nardi, E., Rasmussen, C., & Xie, J. (2015). Key mathematical concepts in the transition from secondary school to university. In S. J. Cho (Ed.), The proceedings of the 12th international congress on mathematical education (pp. 265-284). Springer.
Viirman, O. (2014). The functions of function discourse – University mathematics teaching from a commognitive standpoint. International Journal of Mathematical Education in Science and Technology, 45(4), 512-527. https://doi.org/10.1080/0020739X.2013.855328
Viirman, O. (2015). Explanation, motivation and question posing routines in university mathematics teachers’ pedagogical discourse: A commognitive analysis. International Journal of Mathematical Education in Science and Technology, 46(8), 1165-1181. https://doi.org/10.1080/0020739X.2015.1034206
Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Kluwer.
Weber, K. (2004). Traditional instruction in advanced mathematics courses: A case study of one professor’s lectures and proofs in an introductory real analysis course. Journal of Mathematical Behavior, 23(2), 115-133. https://doi.org/10.1016/j.jmathb.2004.03.001
Yin, R. K. (2003). Case study research: Design and methods (3rd ed. ). Sage Publications.
Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. CBMS Issues in Mathematics Education, 8, 103-122.
Zandieh, M., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. Journal of Mathematical Behavior, 25(1), 1-17. https://doi.org/10.1016/j.jmathb.2005.11.002
Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317-346.

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APA

Gavilán-Izquierdo, J. M., & Gallego-Sánchez, I. (2021). How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study. International Electronic Journal of Mathematics Education, 16(2), em0634. https://doi.org/10.29333/iejme/10892

Vancouver

Gavilán-Izquierdo JM, Gallego-Sánchez I. How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study. INT ELECT J MATH ED. 2021;16(2):em0634. https://doi.org/10.29333/iejme/10892

AMA

Gavilán-Izquierdo JM, Gallego-Sánchez I. How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study. INT ELECT J MATH ED. 2021;16(2), em0634. https://doi.org/10.29333/iejme/10892

Chicago

Gavilán-Izquierdo, José María, and Inés Gallego-Sánchez. "How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study". International Electronic Journal of Mathematics Education 2021 16 no. 2 (2021): em0634. https://doi.org/10.29333/iejme/10892

Harvard

Gavilán-Izquierdo, J. M., and Gallego-Sánchez, I. (2021). How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study. International Electronic Journal of Mathematics Education, 16(2), em0634. https://doi.org/10.29333/iejme/10892

MLA

Gavilán-Izquierdo, José María et al. "How an Upper Secondary School Teacher Provides Resources for the Transition to University: A Case Study". International Electronic Journal of Mathematics Education, vol. 16, no. 2, 2021, em0634. https://doi.org/10.29333/iejme/10892