The purpose of this paper is to show and validate a design for a deep understanding of the derivative from its local perspective. Based on a historical and epistemological study of the derivative, we performed an extension of the Sierpinska theoretical framework – Thinking Modes– of the domain of the derivative from its local perspective, and therefore identified three ways of thinking about the derivative, described as the Synthetic-Geometric-Convergent (SGC), Analytic-Operational (AO) and Analytic-Structural (AE, for its name in Spanish), as the components that, along with its articulators comprise a design for deep understanding of its local aspect.  Methodologically, hypothetically proposed articulators are compared to data obtained in case studies with two groups of university students, in addition to a semi-structured interview to mathematicians, researchers and scholars, through which the proposed articulators elements were validated and clarified. The result is a design for the understanding of the derivative from its local aspect as a viable tool to cause the rupture from a purely algebraic thinking about this topic, and that benefits its deep understanding, being this, the ability of a student to articulate these thinking modes.

• Artigue, M. (1995). La enseñanza de los principios del cálculo: problemas epistemológicos, cognitivos y didácticos. In P. Gómez (Ed.), Ingeniería Didáctica en Educación Matemática (pp.97-140). México: Grupo Editorial Iberoamericano.
• Bachelard, G. (2000). La formación del espíritu científico.  México DF: Siglo XXI.
• Badillo, E., Azcárate, C. y Font, V. (2011). Analysis of comprehension levels of objects f’(a) and f’(x) in Mathematics teachers. Enseñanza de las Ciencias, 29(2), 191-206.
• Boyer, C. (1959). The History of the Calculus and its Conceptual Development. New York: Dover Publications.
• D’Amore, B. and Godino, J. (2007) El enfoque Ontosemiótico como un desarrollo de la teoría Antropológica en Didáctica de la matemática. Revista Latinoamericana de Investigación en Matemática Educativa, 10(002), 191-218.
• Ferrini-Mundy, J. & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives and integrals. In J. Kaput & E. Dubinsky (Eds.), Research issues in undergraduate mathematics learning (pp. 31-45). Washington: MAA Notes.
• Grabiner, J. (1983). The changing concept of change: The derivative from Fermat to Weierstrass. Mathematics Magazine, 56(4), 195-206.
• Jaafar, R. and Lin, Y. (2017). Assessment for Learning in the Calculus Classroom: A Proactive Aproach to Engange Students in Active Learning. IEJME-Mathematics Education 12(5), 503-520.
• Kinley, M. (2016). Grade Twelve Students Establishing the Relationship Between Differentiation and Integration in Calculus Using graphs. IEJME-Mathematics Education, 11(9), 3371-3385.
• Lang, S. (1972). Introduction to Differential Manifold. Yale: Springer.
• Mena, A., Mena, J., Montoya, E, Morales, A. y Parraguez, M., (2015).  El obstáculo epistemológico del infinito actual. The epistemological obstacle of the current infinite. Revista Latinoamericana de Investigación en Matemática Educativa, 18(3), 329-358.
• Montoya Delgadillo, E. & Vivier, L. (2015). ETM de la noción de tangente en un ámbito gráfico - Cambios de dominios y de puntos de vista, Proceedings of CIAEM XIV, (pp.5-7), Chiapas: Tuxtla Gutiérrez.
• Park, J. (2015) Is the derivative a function? If so, how do we teach it? Educational Studies in Mathematics, 89(2), 233-250.
• Parraguez, M. (2012). Teoría los modos de pensamiento. Didáctica de la Matemática. Valparaíso: Ediciones Instituto de Matemática de la Pontificia Universidad Católica de Valparaíso.
• Pino-Fan, L., Godino, J. y Font, V. (2011). Faceta epistémica del conocimiento Didáctico-Matemático sobre la Derivada. Educação Matemática Pesquisa, 13(1), 141-178.
• Poole, D. (2014). Linear Algebra: A Modern Introduction. New York: Brooks/Cole.
• Redon, S. y Angulo, J. (2017). Investigación Cualitativa en Educación. Argentina: Miño y Dávila.
• Sánchez-Matamoros, García, G., García Blanco, M., and Llinares Ciscar, S. (2008). El desarrollo del esquema de derivada. Enseñanza de las ciencias. Revista de investigación y experiencias didácticas, 24(1), 85-98.
• 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.
• Sierpinska, A. (1985). Obstacles épistémologiques relatifs à la notion de limite. Recherches en Didactique des Mathématiques, 6(1), 5-67.
• Sierpinska, A. (2000). On some Aspects of Student´s thinking in Linear Algebra. In J. Dorier.(Ed.), The Teaching of Linear Algebra in Question (pp. 209-246). Netherlands: Kluwer Academic Publishers.
• Sierpinska, A. (2005) On Practical and Theoretical Thinking and Other False Dichotomies in Mathematics Education. In M.H. Hoffmann, J. Lenhard, F. Seeger. (Ed) Activity and Sign (pp.117-135). Boston: Springer.
• Sierpinska, A., Nnadozie, A. y Oktaç, A. (2002) A Study of relationships between theoretical thinking and high achievement in linear algebra. Concordia University: Montreal.
• Stake, R. (2010). Investigación con estudio de casos. Madrid: Morata.
• Steward, I. (2007). Historia de las matemáticas en los últimos 10.000 años. España: Drakontos.
• Tallman, M., Carlson, M. P., Bressoud, D., & Pearson, M. (2016). A characterization of calculus I final exams in U.S. colleges and universities. International Journal of Research in Undergraduate Mathematics Education, 2(1), 105-133.
• Vandebrouck, F. (2011). Perspectives et domaines de travail pour l’étude des fonctions. Annales de Didactique et de Sciences Cognitives, 16(1), 149-185.
• Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. Schoenfeld, J. Kaput, (Ed.), Research in collegiate mathematics education. IV. Issues in mathematics education (pp. 103-127). Providence, RI: American Mathematical Society.
• Zimmermann, W. y Cunningham, S. (1991). Visualisation in Teaching and Learning Mathematics. Washington DC, USA: Mathematical Association of America.

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