ONTO-SEMIOTIC ANALYSIS OF THE EMERGENCE AND EVOLUTION OF FUNCTIONAL REASONING
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Keywords:
epistemology
history
function
mathematics education
onto-semiotic approach
Main Article Content
Issue:
Vol. 1 No. 1 (2024): junio
Section: Artículos de Educación Matemática
Abstract
Developing students’ adequate functional reasoning requires paying attention to the design and planning of teaching from the first educational levels. This implies considering and progressively articulating the diversity of meanings of the function, attending to the generality and formalization levels that emerged in its historical evolution. In this paper, we review historical and epistemological studies on function using theoretical tools of the Onto-semiotic Approach to characterize different levels of functional reasoning. We interpret meaning in terms of systems of operative and discursive practices related to solving types of problems. In line with previous research, we identify partial meanings of function (operative-tabular, operative-graphic, algebraic-geometric, analytic, arbitrary correspondence between numerical sets, and mapping between arbitrary sets) that should be part of the overall reference meaning in the planning and management of function teaching and learning processes. This study provides a complementary view of the multiple investigations that describe the phylogenesis of the concept of function in mathematics with a historical and epistemological approach.
Article Details
Godino, J. D., Burgos, M., & Wilhelmi, M. R. (2024). ONTO-SEMIOTIC ANALYSIS OF THE EMERGENCE AND EVOLUTION OF FUNCTIONAL REASONING. RIME, 1(1), 9-37. https://doi.org/10.32735/S2810-7187202400013181
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References
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25. Font V, Godino JD, Gallardo J. The Emergence of Objects from Math-ematical Practices. Educational Studies in Mathematics. 2013; 82, 97-124. http://dx.doi.org/10.1007/s10649-012-9411-0
26. Wittgenstein L. Philosophical investigations. London: Basil Blackwell Ltd, 1953.
27. Barrow-Green J, Gray J, Wilson R. The history of Mathematics: a Source-Based Approach. Volume 1. American Mathematical Society 2019.
28. Barrow-Green J, Gray J, Wilson R. The history of mathematics: a source-based approach. Volume 2. American Mathematical Society, 2022.
29. Bell ET. The Development of Mathematics. McGraw-Hill, 1945.
30. Bos HJM. Newton, Leibniz and the Leibnizian Tradition. In Grattan-Guinness I, editor, From the Calculus to Set Theory 1630-1910. An In-troductory History. (pp. 49-92). Princeton and Oxford: Princeton Uni-versity Press. 1980.
31. Bottazzini U. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag. 1986.
32. Boyer CB. The History of the Calculus andiIts Conceptual Develop-ment. Dover. 1949.
33. Boyer CB, Merzbach UC. A History of Mathematics (3rd ed). John Wiley & Sons. 1968/2011.
34. Edwards CH. The Historical Development of the Calculus. Cham: Springer-Verlag. 1979.
35. Grattan-Guinness I. From the Calculus to Set Theory 1630-1910. An Introductory History. Princeton University Press. 1980.
36. Kline M. Mathematical Thought from Ancient to Modern Times. Ox-ford University Press. 1972.
37. Rüthing D. Some Definitions of the Concept of Function from Joh. Bernoulli to N. Bourbaki. Mathematical Intelligencer. 1984; 6 (2), 72–77. https://link.springer.com/content/pdf/10.1007/BF03026743.pdf?pdf=core
38. Malik MA. Historical and Pedagogical Aspects of the Definition of Function. International Journal of Mathematics Education in Science and Technology. 1981; 11, 489-492. http://dx.doi.org/10.1080/0020739800110404
39. Markovits Z, Eylon BS, Bruckheimer M. Functions Today and Yester-day. For the learning of mathematics. 1986; 6(2), 18- 413.
40. Piaget J, Grize B, Szeminska A, Bang V. Epistemologie et Psychologie de la Fonction. PUF. 1968.
41. Ponte JP. The history of the Concept of Function and Some Education-al Implications. The Mathematics Educator. 1992; 3-8.
42. Thompson PW, Carlson MP. Variation, covariation, and functions: Foundational ways of thinking mathematically. In Cai J, editor, Com-pendium for Research in Mathematics Education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics. 2017.
43. Hacking I. ‘Style’ for historians and philosophers. Studies in History and Philosophy of Science. 1992; 23 (1), 1-20. http://dx.doi.org/10.1016/0039-3681(92)90024-Z
44. Toulmin S. Human Understanding. Oxford University Press. 1977.
45. Morin E. El método. Las Ideas. Su Hábitat, su Vida, sus Costumbres, su Organización. Cátedra. 1992.
46. White LA. The Locus of Mathematical reality: An Anthropological Footnote. Philosophy of Science. 1983; 14(4), 289–303.
47. Rheinberger H-J. On Historicizing Epistemology. An Essay. Stanford University Press. 2010.
48. Davis PJ, Hersch R, Marchisotto EA. The mathematical experience. Birkhäuser. 2012.
49. Awodey S. Category Theory. Carnegie Mellon University. 2010.
50. Godino JD. Ecology of Mathematical Knowledge: An Alternative Vi-sion of the Popularization of Mathematics. In Joseph A, Mignot F, Mu-rat F, Prum B, Rentschler R, editors, First European Congress of Math-ematics (vol. 3, pp. 150–156). Birkhauser. 1994.
51. Wilhelmi MR. Proporcionalidad en Educación Primaria y Secundaria. En J. M. Contreras y otros (Eds.), Actas del II Congreso International Virtual sobre el Enfoque Ontosemiótico del Conocimiento y la Instruc-ción Matemáticos. 2017. Available from: https://enfoqueontosemiotico.ugr.es/civeos.html
52. Wilhelmi MR, Godino JD, Lasa A. Significados Conflictivos de Ecua-ción y Función en Estudiantes de Profesorado de Secundaria. In Gon-zález MT, Codes M, Arnau D & Ortega T (Eds.), Investigación en Edu-cación Matemática XVIII (pp. 573-582). SEIEM. 2014.
53. Dubinsky E, Breidenbach D, Hawks J, Nichols F. Development of the Process Conception of Function. Educational Studies in Mathematics. 1992; 23, 247-285. https://link.springer.com/contnt/pdf/10.1007/BF02309532.pdf
54. Godino J D, Giacomone B, Batanero C, Font V. Enfoque Ontosemióti-co de los Conocimientos y Competencias del Profesor de Matemáticas. Bolema. 2017; 31 (57), 90-113. http://dx.doi.org/10.1590/1980-4415v31n57a05
2. Biehler R. Reconstruction of Meaning as a Didactical Task: The Concept of Function as an Example. In Kilpatrick J, Hoyles C, Skovsmose O, edi-tors. Meaning in Mathematics Education (pp. 61-81). Kluwer, 2005.
3. Kleiner I. Evolution of the Function Concept: A Brief Survey. The Col-lege Mathematics Journal. 1989; 20 (4), 282–300. https://doi.org/10.1080/07468342.1989.11973245
4. Kleiner I. Functions: Historical and Pedagogical Aspects. Science & Ed-ucation. 1993; 2, 183-209. http://dx.doi.org/10.1007/BF00592206
5. Kleiner I. Excursions in the History of Mathematics. Cham: Springer, 2012.
6. Sfard A. The Case of Function. In Harel G, Dubinsky E, editors. The concept of function. Aspects oi Epistemology and Pedagogy (p. 85-106). Mathematical Association of America, 1992.
7. Youschevitch AP. The Concept of Function up to the Middle of the 19th Century. Archive for History of Exact Sciences. 1976; 16, 36-85.
8. Dubinsky E, Harel G. The concept of function. Aspects of Epistemology and Pedagogy. USA: Mathematical Association of America (MMA), 1992.
9. Ruiz-Higueras L. Concepciones de los Alumnos de Secundaria sobre la Noción de Función. Análisis Epistemológico y Didáctico. Ph.D. Univer-sidad de Granada, 1994.
10. Sierpinska A. On understanding the notion of function. In Harel G, Dubinsky E, editors. The Concept of Function. Aspects of Epistemolo-gy and Pedagogy (p. 25-58). Mathematical Association of America, 1992.
11. Trujillo M, Atarés L, Canet MJ, Pérez-Pascual MA. Learning Difficul-ties with the Concept of Function in Maths: A Literature Review. Edu-cation Sciences. 2023; 13 (5), 495. https://doi.org/10.3390/educsci13050495
12. Vinner S, Dreyfus T. Images and Definitions for the Concept of Func-tion. Journal for Research in Mathematics Education. 1989; 20(4), 356–366. http://dx.doi.org/10.5951/jresematheduc.20.4.0356
13. Pino-Fan LR, Parra-Urrea YE, Castro WF. Significados de la Función Pretendidos por el Currículo de Matemáticas Chileno. Magis. 2019; 11 (23), 201-220. https://doi.org/10.11144/Javeriana.m11-23.sfpc
14. Parra-Urrea Y E. Conocimiento Didáctico-Matemático de Futuros Pro-fesores Chilenos de Enseñanza Media sobre la Noción de Función: Una Experiencia en Contextos de Microenseñanza. PH. D. Universidad de Los Lagos, Chile, 2021.
15. Godino JD. Onto-Semiotic Approach to the Philosophy of Educational Mathematics. Revista Paradigma Edição Temática: EOS. Questões e Métodos. 2023; 44(4), 7-33. https://doi.org/10.37618/PARADIGMA.1011-2251.2023.p07-33.id1377
16. Godino JD, Batanero C. Significado Institucional y Personal de los Ob-jetos Matemáticos. Recherches en Didactique des Mathématiques. 1994; 14(3), 325-355.
17. Godino JD, Batanero C, Font V. The Onto-Semiotic Approach to Re-search in Mathematics Education. ZDM. 2007; 39 (1-2), 127-135. http://dx.doi.org/10.1007/s11858-006-0004-1
18. Godino J D, Neto T, Wilhelmi MR, Aké L, Etchegaray S, Lasa A. Nive-les de Algebrización de las Prácticas Matemáticas Escolares. Articula-ción de las Perspectivas Ontosemiótica y Antropológica. Avances de Investigación en educación Matemática. 2015; 8, 117-142. https://aiem.es/article/view/3970
19. Godino J D, Burgos M, Gea M. Analysing Theories of Meaning in Mathematics Education from the Onto-Semiotic Approach. Internation-al Journal of Mathematical Education in Science and Technology. 2021. https://doi.org/10.1080/0020739X.2021.1896042
20. Batanero C. Significados de la Probabilidad en la Educación Secunda-ria. Relime. 2005; 8(3), 247-264.
21. Wilhelmi M R, Godino JD, Lacasta E. Configuraciones Epistémicas Asociadas a la Noción de Igualdad de Números Reales. Recherches en Didactique des Mathématiques. 2007; 27 (1), 77-120. https://revue-rdm.com/2007/configuraciones-epistemicas/
22. Godino JD, Font V, Wilhelmi MR, Lurduy O. Why Is the Learning of Elementary Arithmetic Concepts Difficult? Semiotic Tools for Under-standing the Nature of Mathematical Objects. Educational Studies in Mathematics. 2021; 77 (2), 247-265. http://dx.doi.org/10.1007/s10649-010-9278-x
23. Burgos M, Godino JD. Modelo Ontosemiótico de Referencia de la Pro-porcionalidad. Implicaciones para la Planificación Curricular en Prima-ria y Secundaria. AIEM. 2020; 18, 1-20. http://dx.doi.org/10.35763/aiem.v0i18.255
24. Godino JD, Aké L, Gonzato M, Wilhelmi MR. Niveles de Algebriza-ción de la Actividad Matemática Escolar. Implicaciones para la Forma-ción de Maestros. Enseñanza de las Ciencias. 2014; 32(1), 199-219. https://raco.cat/index.php/Ensenanza/article/view/287515
25. Font V, Godino JD, Gallardo J. The Emergence of Objects from Math-ematical Practices. Educational Studies in Mathematics. 2013; 82, 97-124. http://dx.doi.org/10.1007/s10649-012-9411-0
26. Wittgenstein L. Philosophical investigations. London: Basil Blackwell Ltd, 1953.
27. Barrow-Green J, Gray J, Wilson R. The history of Mathematics: a Source-Based Approach. Volume 1. American Mathematical Society 2019.
28. Barrow-Green J, Gray J, Wilson R. The history of mathematics: a source-based approach. Volume 2. American Mathematical Society, 2022.
29. Bell ET. The Development of Mathematics. McGraw-Hill, 1945.
30. Bos HJM. Newton, Leibniz and the Leibnizian Tradition. In Grattan-Guinness I, editor, From the Calculus to Set Theory 1630-1910. An In-troductory History. (pp. 49-92). Princeton and Oxford: Princeton Uni-versity Press. 1980.
31. Bottazzini U. The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Springer-Verlag. 1986.
32. Boyer CB. The History of the Calculus andiIts Conceptual Develop-ment. Dover. 1949.
33. Boyer CB, Merzbach UC. A History of Mathematics (3rd ed). John Wiley & Sons. 1968/2011.
34. Edwards CH. The Historical Development of the Calculus. Cham: Springer-Verlag. 1979.
35. Grattan-Guinness I. From the Calculus to Set Theory 1630-1910. An Introductory History. Princeton University Press. 1980.
36. Kline M. Mathematical Thought from Ancient to Modern Times. Ox-ford University Press. 1972.
37. Rüthing D. Some Definitions of the Concept of Function from Joh. Bernoulli to N. Bourbaki. Mathematical Intelligencer. 1984; 6 (2), 72–77. https://link.springer.com/content/pdf/10.1007/BF03026743.pdf?pdf=core
38. Malik MA. Historical and Pedagogical Aspects of the Definition of Function. International Journal of Mathematics Education in Science and Technology. 1981; 11, 489-492. http://dx.doi.org/10.1080/0020739800110404
39. Markovits Z, Eylon BS, Bruckheimer M. Functions Today and Yester-day. For the learning of mathematics. 1986; 6(2), 18- 413.
40. Piaget J, Grize B, Szeminska A, Bang V. Epistemologie et Psychologie de la Fonction. PUF. 1968.
41. Ponte JP. The history of the Concept of Function and Some Education-al Implications. The Mathematics Educator. 1992; 3-8.
42. Thompson PW, Carlson MP. Variation, covariation, and functions: Foundational ways of thinking mathematically. In Cai J, editor, Com-pendium for Research in Mathematics Education (pp. 421-456). Reston, VA: National Council of Teachers of Mathematics. 2017.
43. Hacking I. ‘Style’ for historians and philosophers. Studies in History and Philosophy of Science. 1992; 23 (1), 1-20. http://dx.doi.org/10.1016/0039-3681(92)90024-Z
44. Toulmin S. Human Understanding. Oxford University Press. 1977.
45. Morin E. El método. Las Ideas. Su Hábitat, su Vida, sus Costumbres, su Organización. Cátedra. 1992.
46. White LA. The Locus of Mathematical reality: An Anthropological Footnote. Philosophy of Science. 1983; 14(4), 289–303.
47. Rheinberger H-J. On Historicizing Epistemology. An Essay. Stanford University Press. 2010.
48. Davis PJ, Hersch R, Marchisotto EA. The mathematical experience. Birkhäuser. 2012.
49. Awodey S. Category Theory. Carnegie Mellon University. 2010.
50. Godino JD. Ecology of Mathematical Knowledge: An Alternative Vi-sion of the Popularization of Mathematics. In Joseph A, Mignot F, Mu-rat F, Prum B, Rentschler R, editors, First European Congress of Math-ematics (vol. 3, pp. 150–156). Birkhauser. 1994.
51. Wilhelmi MR. Proporcionalidad en Educación Primaria y Secundaria. En J. M. Contreras y otros (Eds.), Actas del II Congreso International Virtual sobre el Enfoque Ontosemiótico del Conocimiento y la Instruc-ción Matemáticos. 2017. Available from: https://enfoqueontosemiotico.ugr.es/civeos.html
52. Wilhelmi MR, Godino JD, Lasa A. Significados Conflictivos de Ecua-ción y Función en Estudiantes de Profesorado de Secundaria. In Gon-zález MT, Codes M, Arnau D & Ortega T (Eds.), Investigación en Edu-cación Matemática XVIII (pp. 573-582). SEIEM. 2014.
53. Dubinsky E, Breidenbach D, Hawks J, Nichols F. Development of the Process Conception of Function. Educational Studies in Mathematics. 1992; 23, 247-285. https://link.springer.com/contnt/pdf/10.1007/BF02309532.pdf
54. Godino J D, Giacomone B, Batanero C, Font V. Enfoque Ontosemióti-co de los Conocimientos y Competencias del Profesor de Matemáticas. Bolema. 2017; 31 (57), 90-113. http://dx.doi.org/10.1590/1980-4415v31n57a05