Cómo motivar a los estudiantes de Economía para el aprendizaje de las matemáticasHow to motivate economics students to learn mathematicsAna Ida Vilier Cintra1 Edgar Prieres Vilier1Ana Karen Prieres Vilier21Universidad de Guantánamo. Cuba.2Delegación Provincial del MINTUR. Guantánamo. Cuba. ORCID: https://orcid.org/0000-0003-0106-8093 https://orcid.org/0000-0002-8986-4254 https://orcid.org/0000-0001-8238-0569Correo electrónico:aviliercintra@gmail.comprieres81@gmail.comakprieres@gmail.comRecibido: 14/02/2021Aceptado: 15/06/2021Resumen: Las dificultades que históricamente poseen los estudiantes en la modelación como antesala para resolver problemas, y en especial en la esfera económica, motivó la realización de este artículo donde a través de ejemplos se plantea como con la modelación de problemas de la práctica económica y utilizando herramientas como: funciones lineales, cálculo diferencial e integral de funciones reales de una variable, función que define el ingreso y cálculo de costos, el estudiante puede no solo motivarse por el aprendizaje de las matemáticas, sino además, interiorizar que estas pueden utilizarse para comprender y modificar la realidad.Palabras clave: Costo marginal; Costo promedio; Precio de equilibrio; Modelación de problemas.Abstract: The difficulties that students historically have in modeling as a prelude to solve problems, and especially in the economic sphere, motivated the realization of this article where, through examples, it is proposed as with the modeling of problems of economic practice and using tools such as: linear functions, differential and integral calculus of real functions of a variable, a function that defines income and calculation of costs, the student can not only be motivated by learning mathematics, but also internalize that these can be used to understand and modify reality Keywords: Marginal cost; Average cost; Equilibrium price; Problem modelingIntroducciónEn el proceso de asimilación de la actividad por el estudiante, lo primero que debe hacer el profesor es motivar, pues es esto lo que va a garantizar la efectividad del conocimiento y el desarrollo de las habilidades exigidas en el programa. Por otro lado cada maestro sabe, que si el estudiante no quiere estudiar no es posible que aprenda, quiere decir esto que en cada estudiante debe existir un motivo para estudiar y están los motivos internos y externos, estando dentro de los primeros, el interés cognoscitivo vinculado con un objeto dado. Exteriormente la actividad de todos los estudiantes es similar, ahora internamente es muy diversa y esta diversidad está dada ante todo por los motivos de la actividad. Son los motivos los que determinan el sentido que tiene la actividad para la persona que la realiza y cuando se habla de las vías para elevar la efectividad de la actividad docente el carácter de los motivos es totalmente decisivo. La cual es sustancialmente más elevada en presencia de los motivos internos. Alves (2009, p. 35), afirma: “Motivar es despertar el interés y la atención de los alumnos por los valores contenidos en la materia, excitando en ellos el interés de aprenderla, el gusto de estudiarla y la satisfacción de cumplir las tareas que exige”. Por otro lado, es conocido que el no saber de modo preciso para qué puede servir lo que se estudia, puede no resultar motivador incluso para aquellos estudiantes que buscan aprender o adquirir competencias, pues se considera mejor ser competente en algo que resulta útil, que en algo que no se sabe para qué sirve. Todas estas consideraciones constituyen un argumento para demostrar que a través de la modelación de problemas de la especialidad, puede motivarse a los estudiantes de economía para el aprendizaje de las matemáticas, siendo el objetivo de este trabajo. Uno de los medios que coadyuvan a la motivación cognoscitiva, es la enseñanza problémica. No debe olvidarse que la necesidad cognoscitiva es el motivo más efectivo que empuja a la persona al estudio y en este sentido, la modelación y resolución de problemas son consideradas como poderosas dentro de las matemáticas, por cuanto promueven a los estudiantes, a la colaboración, integración, motivación, interés, y estimulan el desarrollo de ciertas habilidades intelectuales tales como el razonamiento, y la creatividad en el aprendizaje.En el proceso de formación el docente es la herramienta esencial para el aprendizaje de sus estudiantes y lejos de ser un simple transmisor de conocimientos debe transformarse en un facilitador de estos. En definitiva, un docente es aquel individuo que está a disposición en cada momento del desarrollo cognoscitivo de sus estudiantes, busca las herramientas necesarias para que sea efectiva la adquisición de todo conocimiento nuevo. La familiarización de los estudiantes con la modelación de problemas matemáticos comienza desde los primeros grados y continua hasta el final de los estudios en la enseñanza superior, pero la metodología es aproximadamente la misma: los estudiantes leen el problema, analizan los datos de manera colectiva o individual, precisan lo que les pide el problema, llevan el texto del problema a un lenguaje formal y luego resuelven el mismo apoyándose en los modelos dados por el profesor. En este sentido y en correspondencia con las experiencias de los autores se conoce que de manera general la minoría de los estudiantes resuelven con independencia cognoscitiva el problema propuesto y en este trabajo se defiende una de las causas que propicia esta situación: la modelación. Las dificultades que históricamente poseen los estudiantes en la modelación como antesala para resolver problemas, y en especial en la esfera económica, además de la escasa información sobre el tema según la búsqueda realizada por los autores, motivó la realización de este trabajo dirigido a ilustrar a través de ejemplos ¿cómo motivar a los estudiantes de economía para el aprendizaje de las matemáticas?Se trata de que el estudiante de economía perciba que con las herramientas cognitivas de las matemáticas en la modelación, puedan dar mayor importancia a los temas de estas, ayudándoles a relacionar los conceptos matemáticos con el mundo real o sea mediante situaciones problémicas, además de interiorizar que estas pueden utilizarse para comprender y modificar la realidad. Pero se conoce también según Babanski (1982, p. 45). “que la eficiencia de los conocimientos se pone de manifiesto en la habilidad del individuo para utilizarlos en situaciones concretas”. De ahí la idea de demostrar que con la modelación de problemas de la especialidad, el estudiante es motivado para aprender matemáticas.Desarrollo Un modelo es una abstracción simplificada de la realidad más compleja, es el proceso de representación del mundo real, en términos matemáticos. Los modelos matemáticos que estudiamos son sistemas que evolucionan con el tiempo y desde el punto de vista matemático, modelar significa llevar una situación de la realidad objetiva al lenguaje de las matemáticas, claro para ello se necesita tener un sistema de conocimientos matemáticos, previos que en este caso constituyen las herramientas cognitivas que les permitan al que modela, traducir del lenguaje común al matemático, además de resolver el problema.Las cantidades en los modelos seleccionados se agrupan en tres categorías: la variable independiente, las variables dependientes y los parámetros. Por ejemplo: en ecuaciones diferenciales la variable independiente casi siempre es el tiempo. En tanto que las variables dependientes son cantidades que son funciones de la variable independiente. Por ejemplo en Física la posición es una función del tiempo. Puede preguntarse entonces si la variable dependiente aumenta o disminuye, o si oscila, o tiende a un límite.Un modelo nunca será una representación completamente exacta de una situación, en un buen modelo la realidad se simplifica lo suficiente para también reducir los cálculos matemáticos, pero incluso así, es bastante exacto para dar conclusiones valiosas. Las relaciones matemáticas formales entre los objetos del modelo, deben representar de alguna manera las relaciones reales existentes entre las diferentes entidades o aspectos del sistema u objeto real. Así una vez traducido o representado cierto problema en forma de modelo matemático, se pueden aplicar: el cálculo, el álgebra, y otras herramientas matemáticas para deducir el comportamiento del sistema bajo estudio.El modelo es ciencia porque está sustentado en un conjunto de procesos estructurados: análisis y detección de las relaciones entre los datos, estableciéndose las suposiciones y aproximaciones en la representación de los problemas desarrollados de algoritmos específicos de solución. Es arte pues materializa una visión interpretación de la realidad objetiva, no siempre única.Los modelos matemáticos pueden clasificarse de la siguiente manera:Determinista. Se conoce de manera puntual la forma del resultado ya que no hay incertidumbre. Además, los datos utilizados para alimentar el modelo son completamente conocidos y determinados. Estocástico. Probabilístico, que no se conoce el resultado esperado, sino su probabilidad y existe por tanto incertidumbre. Además con respecto a la función del origen de la información utilizada para construirlos los modelos pueden clasificarse en heurísticos y empíricos.Modelos heurísticos. Son los que están basados en las explicaciones sobre las causas o mecanismos naturales que dan lugar al fenómeno estudiado. Modelos empíricos. Son los que utilizan las observaciones directas o los resultados de experimentos del fenómeno estudiado. Según su campo de aplicación los modelos pueden clasificarse en:Modelos conceptuales. Son los que reproducen mediante fórmulas y algoritmos matemáticos más o menos complejos los procesos físicos que se producen en la naturaleza Modelo matemático de optimización. Los modelos matemáticos de optimización son ampliamente utilizados en diversas ramas de la Ingeniería para resolver problemas que por su naturaleza son indeterminados, es decir presentan más de una solución posible. Etapas para desarrollar un modelo.Acondicionamiento de la información e interpretación del problema, Para su traducción del lenguaje común, al lenguaje matemático, en esta etapa los datos son vitales para la veracidad de las soluciones.Formulación matemática.Es muy importante la formulación y escritura mediante ecuaciones, matemáticas que describan el problema. Es este precisamente el problema fundamental que tradicionalmente poseen los estudiantes. El modelado del problema en cuestión. Resolución: Es donde se selecciona un algoritmo para su soluciónNo basta poseer los conocimientos y transmitirlos, sino además es necesario apoderarse de otros, comprenderlos, desarrollar habilidades y muy importante aplicarlos a otras áreas, de ahí que la modelación puede ser considerada un método para tales objetivos.Un conocimiento que debe quedar imperecedero en un economista es el de modelar procesos económicos mediante funciones matemáticas, es por ello que numerosos tipos de funciones son aplicables a la modelación y en el presente trabajo serán aplicadas algunas de ellas, siendo esto ilustrado mediante ejemplos de situaciones económicas que se prestan para modelar con tales funciones y simultáneamente, se ilustra la importancia de lo expresado anteriormente.El punto en que se cruzan las curvas de oferta y demanda, se llama punto de equilibrio del mercado. Cuando el precio del mercado coincide con el del punto de equilibrio, la cantidad ofrecida y la cantidad demandada del bien es la misma. El precio correspondiente a ese punto es llamado precio de equilibrio. La cantidad que se ofrece y se demanda, en otras palabras, la cantidad del bien que se intercambia, es llamada cantidad de equilibrio. En ese punto: Todo lo que se produce se vende y todo lo que se demanda se puede adquirir. En el contenido anterior son aplicables las funciones lineales y cuadráticas de una variable real, y en estos casos se habla de modelos lineales y cuadráticos respectivamente. Las funciones de oferta y demanda involucradas en el equilibrio se modelan también mediante estas funciones, en la búsqueda del punto y la cantidad de equilibrio.Para la obtención de un aprendizaje significativo no debe obviarse el asentamiento de los conocimientos previos necesarios para el desarrollo del nuevo contenido, en tal sentido para afrontar con éxito el equilibrio de mercado, es necesario que el estudiante haya profundizado en los contenidos matemáticos referentes a las funciones, representación gráfica y propiedades de cada una de ellas, además de la resolución de ecuaciones. A continuación se ilustra a través de un ejemplo, cómo motivar al estudiante de economía con el uso de las funciones:I. Un equipo de béisbol juega en un estadio con una capacidad de 55,000 espectadores. Con precios de los boletos de 10 dólares, la asistencia promedio fue de 27,000. Cuando el precio bajó hasta 8 dólares, la asistencia promedio subió hasta 33,000.a) Encuentre la función de demanda, suponiendo que es lineal.Solución.Con los pares ordenados (27000, 10) y (33000, 8), es obtenida la pendiente de la recta.y con esta y uno de los pares ordenados, entonces se busca a n, obteniéndose finalmente la ecuación que define a la función de demanda, la cual sería: a) la función de venta (o función de ingreso), está dada por: Observación: En el anterior ejemplo se ha modelado el problema dado a), mediante la aplicación de una función lineal.En economía, es muy importante además conocer la velocidad de variación de las cantidades con el tiempo, por ejemplo predecir el crecimiento de una determinada población, para estimar una demanda futura, para ello se necesita información sobre la tasa de variación y para describir a esta desde el punto de vista matemático, el concepto que se utilizará es el de derivada. Por tanto, es vital para un estudiante de economía conocer del Cálculo Diferencial de funciones de una variable real, pues necesitan de la modelación de problemas que involucran, las funciones totales, medias y marginales, además de la maximización de la ganancia y la minimización del costo, en estos últimos casos, sin dejar de considerar la optimización clásica -extremos libres y condicionados-, porque es innegable que el futuro egresado para ser objetivo, necesita ser un optimizador económico, lo cual significa: maximizar los beneficios y minimizar los costos económicos, pero se sabe que para ello necesita de las herramientas matemáticas, por ello le es necesario el cálculo diferencial e integral para funciones de una y varias variables reales, lo que combinado con los conocimientos económicos lo arma para su desempeño como profesional. Por ejemplo, si en el problema anterior se formulara la pregunta siguiente:b) ¿A qué precio deben fijarse los boletos para maximizar el ingreso? Se estaría precisado a utilizar algunas cuestiones de la teoría anterior, en este caso: la función de ingreso total -o función de ventas-, dada por el producto de la función de precio -o función de demanda-, por la cantidad de unidades vendidas x, o sea: y la función de ingreso marginal, dada por: esta sería igual a cero, para x = 28500, este valor de x da un máximo absoluto, y el precio correspondiente sería:, luego para maximizar los ingresos, el precio de los boletos debe ser fijado a $9,50Esto presupone que, el estudiante debe poseer habilidades para modelar y resolver problemas relativos a la optimización de funciones económicas, pero para ello debe dominar lógicamente algunos elementos del cálculo diferencial previamente, tales como: la tasa instantánea de variación de en a denotada por , La tasa proporcional de variación de en a, dada por , también llamada tasa relativa de variación, y normalmente se dan en tanto por ciento, ó cuando el tiempo es la variable independiente, en porcentajes anuales. Se conoce que:C(x) costo de producción de x unidadesR(x) ingreso por venta de x unidades Beneficio de producción (y venta) de x unidades, como normalmente una empresa produce muchas unidades de x, entonces se puede considerar que es un número cercano a cero y es obtenida la aproximación: De esta manera, el coste marginal es aproximadamente igual al incremento de coste que es el coste adicional de producir una unidad más de x.Véase un ejemplo, que ilustra cómo aplicar algunos de estos elementos de la teoría a la economía. II. Una compañía estima que el costo (en dólares) para producir x artículos es C(x) = 25000 + 120x + 0.1 x2a) Encuentre el costo, costo promedio y el costo marginal para producir 1000 artículos.b) ¿A qué nivel de producción el costo promedio será el más bajo?c) ¿Cuál es el costo promedio mínimo?Solución:La función de costo promedio será: C'(x) coste marginal en x; R’(x) ingreso marginal; beneficio marginal.La función de costo marginal será:c (1000) = 25 + 120 + 100 = $245C (1000) = 25000 + 120 (1000) + 0.1 (1000)2 = $245000La siguiente tabla contiene el Costo, Costo promedio y Costo marginal (en dólares).Tabla 1: Costo, Costo promedio y Costo marginalb) Para minimizar el costo promedio debe tenerse en cuenta que:Costo marginal = Costo promedio, entonces:c) El costo promedio mínimo sería: Como se había hecho referencia anteriormente, son numerosos los problemas económicos que se auxilian de los extremos libres y condicionados. Ilústrese su utilización. Por ejemplo.III. Halle la cantidad de los factores de producción y del producto para que la ganancia sea máxima si x representa la cantidad de productos, y y z las cantidades de los factores de producción. Además, los precios unitarios son: y la función de producción es Para solucionarlo, debe hallarse la función que define a la ganancia, la cual se debe maximizar (sería la función objetivo).(A) G = I – C, donde I = 12x (Ingreso) y C= 6y + 4z sustituyendo en (A) se obtiene: (B) (costo), sustituyendo A y B en la función de producción se obtendría:Determínense las primeras derivadas parciales de la función ganancia: , luego determínense los puntos críticos, a través de la resolución del sistema siguiente: Obteniéndose que: Por otro lado: No debe olvidarse que y representan volúmenes de producción, luego deben desecharse los valores negativos.IV. Hallar las cantidades de los factores de producción y del producto para que la ganancia sea máxima, sabiendo que x representa la cantidad del producto y, z, las cantidades de los factores de producción y además la función de producción es y los precios unitarios SoluciónLa función ganancia está dada por G = I – C pero I= 2x Luego: Pero como la función de producción essustituyendo en la ganancia se obtiene que: (A). Sustituyendo en A la función de producción se obtiene: Función a maximizar; Y Con vista a encontrar los puntos críticos será resuelto el siguiente sistema de ecuaciones: Sustituyendo en la función de producción se obtiene: Luego ; No debe olvidarse que y representan volúmenes de producción, luego deben desecharse los valores negativos, obteniéndose como punto crítico a , observe que en este problema no se hace necesario que sea aplicado criterio alguno, para decidir si P es un máximo por la propia naturaleza del problema, dado que la función G es de ganancia.Para hallar la cantidad del producto x, sustituirá los valores de y = 2 y z = 3 en la función de producción, obteniéndose , finalmente para la obtención de la máxima ganancia sustituimos los valores de en (B), siendo Los problemas de optimización restringida que son tratados en economía necesitan usualmente de más de dos variables, de ahí que no puede pasarse por alto la aplicación de los extremos condicionados a problemas económicos, por lo que de manera análoga a como se ha procedido con los extremos libres, se ilustrará con un ejemplo puramente de la práctica económica. Previamente es necesario precisar que, a diferencia de los extremos libres, los condicionados son identificados porque las variables de las cuales depende la función a optimizar, no son independientes. El método de los multiplicadores de Lagrange (Sydsaeter y Hammond, 2003) muy utilizado por los economistas por sus importantes interpretaciones económicas, será el aplicado para resolver estos tipos de problemas.Interpretación económica del multiplicador de Lagrange Considérese el problema sujeta a Sean los valores de que resuelven este problema. En general, dependen de c. Supóngase que e son funciones diferenciables de c. Entonces es también función de c. A se le llama función valor óptimo para el problema. Cuando es utilizado el método Lagrangiano, el valor correspondiente del multiplicador de Lagrange también depende de c. Si se satisfacen ciertas condiciones de regularidad es obtenido resultado de que: El multiplicador de Lagrange es la tasa de variación del valor óptimo de la función objetivo cuando la constante de restricción c cambia.Observación:En aplicaciones económicas, c designa a menudo el Stock disponible de un cierto recurso y designa a la utilidad o beneficio. Entonces la medida aproximada del aumento de utilidad o beneficio que se puede obtener de unidades más del recurso escon > 0. Los economistas llaman aun precio sombra del recurso.A continuación es explicado ¿cómo aplicar el método para los extremos condicionados?:i. Se construye la función auxiliar F, como la suma de la función objetivo más el multiplicador de Lagrange por la condición (que debe estar igualada a cero).ii. Son determinados los puntos críticos a través de la solución del sistema de derivadas parciales, igualadas a cero y la condición de enlace, también igualada a cero. iii Son calculados los determinantes y aplicado el criterio correspondiente. Véase un ejemplo:V. Una fábrica produce tres productos en cantidades A, B, y C de los cuales se puede obtener la ganancia . Halle los valores que hacen máxima la ganancia, si la producción está sujeta a la condición: SoluciónCondición de enlace Las derivadas parciales de G con respecto a A, B, y C respectivamente serán: Formando el sistema de ecuaciones se tendría: Despejando a en I, II y III respectivamente se obtiene: y Sustituyendo A, B, y C en la ecuación de enlace se tiene que:Como se trata de ganancia (magnitud positiva), resulta que: A =10, B= 20 y C= 30Pero además, la derivada en la economía, tiene otras aplicaciones importantes.En la propensión marginal al consumo -es la derivada de la función de consumo respecto al ingreso), en el producto marginal del trabajo ó productividad marginal del trabajo(es la derivada de la función de producción respecto al trabajo-, entre otros. Es necesario para el economista, tener claro el conocimiento referente a la integración de funciones de una variable real, especialmente el de integral definida, pues este es aplicable también al proceso de modelación en la teoría de los excedentes -del consumidor y del productor respectivamente, llamados también el beneficio de consumidor y del productor respectivamente-, claro a nivel de mercado, es decir, el beneficio total de consumidores y empresas. Un principio común en la economía consiste en que cuando una persona deja de ganar dinero está perdiendo, y cuando deja de pagar dinero, está ganando.Sin embargo, las ganancias y pérdidas señaladas no son tangibles porque no forman un flujo de dinero como es el caso, por ejemplo, de un flujo de caja en una empresa o una ganancia de una persona en un negocio.Con la utilización de la integral definida, puede determinarse cada uno de los excedentes mencionados. Ambos excedentes serán la suma de las áreas formadas entre la curva de la demanda y la curva de la oferta. Estos excedentes se forman cuando la demanda y la oferta, ambas, se expanden hasta que el mercado se equilibra. En el transcurso, los beneficios se van formando, y en el equilibrio, tanto consumidores como productores se benefician. Los primeros se benefician porque pagan un precio menor al que estaban dispuestos a pagar, y los segundos, porque el precio de su producto aumentó más allá de los que esperaban dada la expansión de la demanda.La economía precisa también de la integración para el modelado. Se ilustra a continuación.VI. La demanda de un producto -en dólares- es: p = 450/(x + 8). Encuentre el superávit del consumidor cuando el precio de venta es de 10 dólares.Como el precio es 10 dólares, entonces: 10 = 450/(x + 8)ConclusionesEl presente trabajo contiene algunas problemáticas de la práctica económica que exigen ser modeladas y resueltas mediante la aplicación de conocimientos matemáticos, entre ellas: las necesidades de modelar y resolver problemas de su esfera, los motiva, de manera especial fijan los conceptos matemáticos que van adquiriendo en la asignatura, además de motivarse para resolver problemas económicos de su preferencia, con ejemplos de la práctica económica y a través de la modelación, los estudiantes de economía se motivan por el aprendizaje de las matemáticas, porque se responde a: para qué les sirve la asignatura, dónde y cómo emplearla, pero simultáneamente se convierten en artífices conscientes de esta ciencia, tan importante para la especialidad referida y para la vida en general.Referencias BibliográficasAlves, M. L. (2009). Compendio de Didáctica General. Editorial Kapelusz. http://www.educa.aragob.es/aplicadi/didac/dida38.html Babanski, Y. K. (1982). Optimización del proceso de enseñanza. Pueblo y Educación.Sydsaeter, K. y Hammond, P. (2003). Matemáticas para el Análisis Económico. Félix Varela.EduSolVol. 21. Núm. Esp.ISSN: 1729-8091Publicada en línea: noviembre de 2021 (http://edusol.cug.co.cu)PAGE \* MERGEFORMAT442EduSol Vol.21 Núm. Esp. Andrés Ferrer Castillo; Raúl Hernández Heredia; Pedro Planes Cumbá: La cultura organizacional en el … PAGE \* MERGEFORMAT441Ana Ida Vilier Cintra; Edgar Prieres Vilier; Ana Karen Prieres Vilier: Cómo motivar a los estudiantes de Economía…0M8R4KGxGuEAAAAAAAAAAAAAAAAAAAAAPgADAP7/CQAGAAAAAAAAAAAAAAABAAAAAQAAAAAAAAAA EAAAAgAAAAEAAAD+////AAAAAAAAAAD///////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ///////////////////////////////////////////////////////////////////////////9 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