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Tipificación de errores en evaluaciones matemáticas de un primer cur= so universitario

 

Typification of errors in mathematical assessments in a first university course

 

= 1=

Ramón Antonio Abancin Ospina

https://orcid.org/0000-0002-2417-6671

 

 <= /span>

Escuela Superior Politécnica de Chimborazo (ESPOCH), Facultad de Ciencias, Carrer= a de Matemática, Grupo CIDED, Riobamba, Ecuador, Panamericana Sur, Km 11/2. Universidad Simón Bolívar (USB), Valle de Sartenejas, Venezuela.

ramon.abancin@espoch.edu.ec

= 2=

Zenaida Natividad Castillo Marrero<= /span>

https://orcid.org/0000-0002-4424-8652

 

 <= /span>

 Escuela Superior Politécnica de Chimbo= razo (ESPOCH), Facultad de Ciencias, Carrera de Matemática, Grupo CIDED, Rioba= mba, Ecuador, Panamericana Sur, Km 11/2. Universidad Central de Venezuela (UCV= ), Venezuela.

zenaida.castillo@espoch.edu.ec

&nbs= p;

&nbs= p;

&nbs= p;

<= o:p> 

 

Artículo de Investigación Científica y Tecnológi= ca

Enviado: 05/04/2022

Revisado: 21/05/2022

Aceptado: 23/06/2022

Publicado:05/07/2022

DOI: https://doi.org/10.33262/exploradordigital.v6i3.2196<= /span>    =     

 

<= o:p> 

 <= /o:p>

 

Cítese:

 

&= nbsp;

Abancin Ospina, R. A., & Castillo Marrero, Z. N. (2022). Tipificación de erro= res en evaluaciones matemáticas de un primer curso universitario . Explorador Digital, 6(3), 6-27. https://doi.org/10.33262/exploradordigital.v6i3.2196

&= nbsp;

 

 

EXPLORADOR DIGITAL, <= i>es una Revista electrónica, Trimestral, qu= e se publicará en soporte electrónico tiene como misión contribuir a la   formación de profesionales competentes con visión humanística y crítica que sean capaces de exponer sus resultados investigativos y científicos en la misma medida que se promueva mediante su intervención cambios positivos en la sociedad. 

La revista es editada por la Editorial Ciencia Digital (Editorial de prestigio registra= da en la Cámara Ecuatoriana de Libro con No de Afiliación 663) <= span style=3D'font-size:8.0pt;font-family:"Times New Roman",serif;mso-fareast-= font-family: "MS Mincho";color:#0563C1'>www.celibro.org.ec

 

 

 

Esta revista está protegida bajo una licencia Creative Commons AttributionNonCommercialNoDerivatives 4.0 International. Copia de la licencia: http://creativecommons.org/licenses/by-nc-nd/4= .0/.

Palabras claves:

matemáticas básicas, errores matemáticos, exámenes, identificación de errores, tipificación de errores.<= /o:p>

&= nbsp;

Resumen

Introd= ucción: los errores que se observan constantemente en las evaluaciones de matemática, en cursos universitarios o preuniversitarios, están latentes en las producciones escritas y orales, incidiendo negativamente en los procesos de enseñanza y aprendizaje, tanto de las ma= temáticas como de las otras áreas y disciplinas que las utilizan como herramienta auxiliar. Objetivo: en este sentido, el propósito de esta investigación es construir una tipificación de errores matemáticos derivada del análisis, identificación y clasificación= de errores cometidos por estudiantes en la resolución escrita de ejercicios = y/o problemas de matemáticas correspondientes a los exámenes regulares de un primer curso de matemática a nivel universitario. Metodología: se realiza la investigación bajo un enfoque cualitativo y con propósito descriptivo, usando una muestra de 141 exámenes parciales presentados por= un grupo de estudiantes de matemáticas I del Ciclo Básico de la Universidad Simón Bolívar. Como resultado, se identificaron diez categorías que recog= en la diversidad de errores en las producciones escritas de los estudiantes, obteniendo una tipificación en errores de tipo: simbólico, notación, aplicación de fórmulas, operaciones, manipulación, cálculos, redacción y escritura, graficación y problemas prácticos. Resultados: se concl= uye que los errores matemáticos entorpecen la adquisición de nuevos conocimientos, de allí, la importancia de identificarlos y analizarlos, p= ara diseñar e implementar estrategias puntuales que ayuden a suprimirlos o, p= or lo menos, minimizar su incidencia.

 <= /o:p>

Keywords: basic mathematics, mathematical mist= akes, exams, mistakes identification, mistakes classification.

 

Abstract<= /span>

Int= roduction: the errors that are constantly observed in mathematics evaluations, in university or pre-university courses, are latent in written and oral productions, negatively affecting the teaching and learning processes, bo= th in mathematics and in the other areas and disciplines that use them as an auxiliary tool. Objective: in this sense, the purpose of this research is= to construct a typification of mathematical errors derived from the analysis, identification and classification of errors committed by students in the written resolution of exercises and / or problems of mathematics corresponding to the regular exams of a first course of mathematics at the university level. Methodology: se conducts the research under a qualitati= ve approach and with a descriptive purpose, using a sample of 141 partial ex= ams presented by a group of students of mathematics I of the Basic Cycle of t= he Simón Bolívar University. As a result, ten categories were identified that collect the diversity of errors in the written productions of the student= s, obtaining a typification in errors of type: symbolic, notation, applicati= on of formulas, operations, manipulation, calculations, writing and writing, graphing, and practical problems. Results: se concludes that mathematical errors hinder the acquisition of new knowledge, hence the importance of identifying and analyzing them, to design and implement specific strategi= es that help suppress them or, at least, minimize their incidence.

 

 

Introducción

En el ámbito educativo, particularmente en el área de matemática, es común sub= estimar en la escritura matemática, tales como de: signos, operaciones básicas, ent= re otras; tanto por parte de los estudiantes a la hora de revisar su calificac= ión en una prueba, como quizás, algunos profesores al momento de calificar las producciones escritas de sus estudiantes. Es común que los estudiantes les manifiesten a sus profesores: “es solamente un error de signo”, “la respues= ta no es correcta, pero sé el procedimiento”, por ejemplo: palabras a las que, quizás, muchos profesores sucumben, y los llevan a reconsiderar (ignorar) l= os errores, y modificar (aumentar) la calificación.

La situación planteada tiene ramificaciones negativas que comienzan en la educ= ación media (Bachillerato) y repercuten significativamente en la Educación Superi= or, ya que tópicos como la factorización, radicación, trigonometría, etc., son = la base para la construcción de temas más avanzados en la disciplina. Estos co= ntenidos muchas veces no son desarrollados correctamente o con el suficiente tiempo = para su consolidación, y la recuperación por parte de los estudiantes no es inmediata, o podría contener perturbaciones que se materializan al momento = de resolver ejercicios y/o problemas en un primer curso a nivel universitario.=

Esta situación entorpece el desenvolvimiento académico de los estudiantes y repercute en el óptimo desarrollo de su carrera universitaria, causando baj= as calificaciones, estancamiento en algunas asignaturas, frustraciones, etc., eventos que pueden desencadenar en la deserción escolar voluntaria (abandon= o) u obligatoria (políticas de permanencia) en los programas académicos. De allí= la importancia del reconocimiento y tipificación de estos errores comunes que aparecen en las evaluaciones de los estudiantes.

En este sentido, los motivos que soportan este estudio se resumen en: a) los e= rrores matemáticos están latentes en las evaluaciones de los estudiantes, y entorp= ecen el aprendizaje; b) si los errores matemáticos básicos no son tratados a tie= mpo, el aprendizaje de  temas avanzados = no se logra de manera efectiva; y c) al analizar, identificar y clasificar los er= rores matemáticos que cometen los estudiantes comenzando la etapa universitaria, = proporcionan información importante para diseñar cursos de nivelación, y estrategias rem= ediales y/o preventivas para abordar esta problemática desde la universidad. <= /o:p>

En la literatura científica se encuentra una amplia gama de investigaciones relacionadas con el tema de errores producidos por la comunidad estudiantil cuando toman cursos de matemáticas en los diferentes niveles de educación, = resaltando el tradicional análisis que se realiza con el propósito de categorizar y/o clasificar. En este sentido, la revisión bibliográfica nos lleva a tres gra= ndes grupos no exhaustivos, ni excluyentes: a) categorización de errores emergen= tes del análisis de un conjunto de datos, b) clasificación de errores utilizando categorías existentes en la literatura y c) uso de los errores matemáticos = como fuente de información.

El proceso de tipificación de estos errores consiste en construir categorías, = en principio, empíricas a partir del análisis de las producciones (escritas y/u orales) de estudiantes, aunque en algunos casos, se construyen a priori a partir de la revisión bibliográfica. También se encuentran categorías que se estructuran a partir del estudio de estos errores desde otras perspectivas = o teorías (Socas, 1997). Respecto a las categorizaciones, se pueden encontrar investigaciones como las de Esteley & Villarreal (1990), Pochulu (2005), Abrate et al. (2006). Dentro de los trabajos, cuya intención fue clasificar errores se ha= llan autores como Del Puerto et al. (2006), Carazo & Brey (2012), Gamboa et = al. (2019), etc. Contemplando ambos enfoques se tienen estudios como los de Gan= dulfo et al. (2013), Ramírez et al. (2018), etc.

A modo de ilustración, en el tercer grupo se localizan trabajos como el de Fa= rías et al. (2021) identificando los errores originados por el uso incorrecto de= las propiedades de potenciación en las producciones de los estudiantes que inic= ian estudios universitarios. Después diseñaron y aplicaron actividades como estrategias para minimizar la incidencia de estos errores. La experiencia se llevó a cabo con 108 (77 experimental y 48 control) estudiantes del ciclo d= e iniciación universitaria de la Universidad Simón Bolívar (USB).

La persistencia de ciertos errores en las evaluaciones de cursos de la matemát= ica ha permitido a muchos investigadores su categorización, al respecto se puede decir que: a) no existe una categorización universal ni exhaustiva para esta clase de errores, esto se debe a que un error puede estar asociado a distin= tos aspectos y causas, y puede ser clasificado de diversas formas; b) la clasificación de estos errores depende de la categoría con la cual se obser= va o analiza ; y c) la categorización de los errores no es una tarea finiquitada, así que se puede seguir tipificando errores, en contextos particulares, y en términos a los avances educativos.  

Contextualizando este estudio en el ámbito de los estudiantes que ingresan al sistema de Edu= cación Superior pero focalizado en los contenidos de Bachillerato, el propósito es construir una tipificación de errores producto del análisis, identificación= y clasificación de errores cometidos por un grupo (muestra) de estudiantes en= la resolución escrita de ejercicios y/o problemas de matemáticas, correspondie= ntes a los exámenes parciales que se dan en un primer curso regular de matemátic= as a nivel universitario.

En este sentido, la investigación estuvo orienta a dar repuesta a: ¿cuáles son= las categorías de errores, construidas a partir de los errores que comenten los estudiantes en el curso de Matemáticas I (MAI) de la USB en la resolución d= e ejercicios y/o problemas de matemáticas en exámenes parciales?  Para aportar información a esta interrog= ante se propuso: analizar las respuestas y/o soluciones de los estudiantes a los= ejercicios y problemas de matemáticas de los exámenes parciales del curso MAI; identif= icando los errores matemáticos cometidos por los alumnos; y tipificando estos erro= res matemáticos después de una revisión detallada.

Marco teórico referencial

Este apartado expone y discute los principales elementos teóricos en donde se enmarca la investigación y dan sustento al análisis del problema planteado.=

Errores en las evaluaciones de matemática

Una estrategia común para aprender matemáticas se basa en la ejercitación de las herramientas que cada tema despliega. Al respecto Pochulu (2005), afirma qu= e en muchos casos los alumnos simplemente desean saber el algoritmo que permite resolver un ejercicio, sin preocuparse por los conceptos subyacentes del te= ma. Esto puede resultar en primera instancia, pero a medida que se van añadiendo nue= vos contenidos, el estudiante podría tener problemas para la recuperación, y utilización de conocimientos previos y sus aplicaciones.

La segunda visión es aceptar que “en todo proceso educativo y de construcción = de conocimiento el error es posible” (Gamboa et al., 2019, p.6), y considerar = la producción de errores como parte inseparable del proceso de aprendizaje (Del Puerto et al., 2006). En este sentido, los errores matemáticos “forman part= e de las producciones de la mayoría de los alumnos, y constituyen, generalmente,= un elemento estable en los procesos de enseñanza y aprendizaje de las matemáti= cas en todos los niveles del sistema educativo (Pochulu, 2005).

El presente estudio se circunscribe en la primera línea de investigación, y se= plantea que los errores que cometen los alumnos en los cursos de matemática se deben principalmente a deficiencias en conocimientos, originados por un aprendiza= je incompleto y/o no consolidado de las herramientas matemáticas. Se sustenta = que estos errores se materializan en aplicaciones de estrategias, métodos, regl= as y/o técnicas personales de los estudiantes, muchas veces informales pero originales y cuyos razonamientos son válidos para los resolutores frente a algunos ejercicios y/o problemas de matemáticas propuestos, pero que son no admitidas por los conocedores dentro de la comunidad matemática.=

Además, las oportunidades de los estudiantes para aprender matemática dependen del entorno, del tipo de tareas, del discurso en el que participan, y de cómo se implican en estas actividades, y algunos autores como Vera & Volta (2009), Gandulfo et al. (2013), Gamboa et al. (2019), coinciden en que los errores matemáticos pueden estar influenciados por diferentes variables, tales como: profesores, currículos, instituciones educativas, contexto sociocultural, e= tc.

Específicamente, los errores matemáticos son desequilibrios producidos por deficiencias y de= sconocimientos de los contenidos matemáticos que se materializan en aplicaciones de estrategias, métodos, reglas y/o técnicas personales inventadas, muchas vec= es informales pero originales y cuyo razonamientos son válidos para los resolutores frente a algunos ejercicios y/o problemas de matemáticas propuestos, pero que son no admitidas tanto por los conocedores como por la comunidad matemática, dificultando la enseñanza y aprendizaje de las matemáticas, en especial en la construcción de nuevos conocimientos.

Clasificaciones de errores en las evaluaciones de cursos matemático= s

En la tabla 1 se presentan algunas clasificaciones a partir de l= as categorizaciones propuestas por algunos autores mencionados en el apartado anterior, y que se utilizarán como insumo en el estudio propuesto.

Tabla 1

Algunos trabajos de clasificaciones de er= rores que usaron categorías de otros autores

Autor(es) y año

Contexto(s)

Situación y temas

Autores de las categorías<= /o:p>

Resultados

Esteley & Villarreal (1996)

Educación Superior: Alumnos del primer año de ciencias agropecuaria

Resolver problemas o ejercicios sobre función, límite y continuidad de funciones reales de una variable.

Esteley & Villarreal (1990)

Los errores estaban incluidos en tres categorías= : B, C y E.

 

 

Tabla 1

Algunos trabajos de clasificaciones de er= rores que usaron categorías de otros autores (continuación)=

Autor(es) y año

Contexto(s)

Situación y temas

Autores de las categorías<= /o:p>

Resultados

Del Puerto et al. (2006)<= o:p>

Bachillerato y Educación Superior: alumnos del f= inal del

nivel medio y comienzos de los niveles terciario= y universitario.

 

Cuestionarios con contenidos de álgebra y teoría básica de funciones.

Radatz (1979)

Los errores matemáticos están distribuidos en to= das las categorías.

Vera & Volta (2009)

 

 

Educación Superior: examen de suficiencia para ingreso.

Resolución de situaciones problemáticas. Álgebra, geometría analítica, ecuaciones lineales y cuadráticas, desigualdades, trigonometría y gráficas de funciones.

Radatz (1979)

Las dificultades se clasificaron en las categorí= as: A, C, D y E.

Carazo & Brey (2012)

Educación Superior: alumnos universitarios de un= curso de matemáticas financieras.

Hojas de problemas y exámenes de alumnos.

Mosvshovitz-Hadar et al. (1987)

Apoyados en la categoría previa encontraron dos bloques de errores: Errores transversales y Errores de origen e incidencia específicos.

 

Gandulfo et al. (2013)

Educación Superior: Exámenes finales de ingreso.=

Conjuntos numéricos. Operaciones y propiedades.<= o:p>

Gandulfo et al. (2013)

La clasificación se encontró en las categorías: = ,  y .

Ramírez et al. (2018)

 

Bachillerato: proyecto de estimulación del talento matemático.<= o:p>

Resolver tareas geométricas durante 3 sesiones de enriquecimiento curricular focalizado en técnicas de argumentación.=

Ramírez et al. (2018)

Los errores observados pertenecieron a alguna de= las categorías delimitadas a priori: .

 

Tabla 1

Algunos trabajos de clasificaciones de er= rores que usaron categorías de otros autores (continuación)=

Autor(es) y año

Contexto(s)

Situación y temas

Autores de las categorías

Resulta= dos

Gamboa et al. (2019)

 

 

Educación Superior: pruebas diagnósticas de estudiantes de primer ingreso

Ejercicios sobre contenidos básicos (funciones, trigonometría, factorización, fórmulas notables y potenciación, ecuacione= s, etc.) para el desempeño universitario.

 

Abrate et al. (2006)

Los resultados presentar= on todas las categorías. Además, encontraron errores de identificación de la prioridad de las operaciones, uso de paréntesis, concepto de valor absolu= to, operaciones con polinomios, fórmulas notables, leyes de potencias, factorización, etc.

&n= bsp;

 

Se resalta que existen otros tipos de clasifica= ción para los errores utilizando otras perspectivas, enfocados en sus orígenes (= Socas, 1997), o la propuesta de Ruano et a= l. (2008) quienes se focalizan en tres procesos específicos del lenguaje algebraico: sustitución formal, generalización y modelización. <= /span>

Metodología

Método y muestra

En esta investigación se emplea la metodología cualitativa con propósito descriptivo. Se describen y explican aspectos relacionados con los errores en evaluaciones matemáticas, a través del análisis, identificación y categorización de los errores cometidos por un g= rupo de estudiantes en sus producciones escritas (exámenes) al resolver ejercici= os y/o problemas en un primer curso básico de matemáticas universitarias.=

Los participantes del estudio constituyeron una muestra intencional (selección de escenario), escogida usando varios criter= ios: C1) Niv= el educativo: todos los participantes debían ser estudiantes regulares de un primer curso de matemáticas a nivel universitario, para garantizar que cada= uno de ellos tuviera algún tipo de interés por cursar la asignatura; C2) Perfil: edades, ubicadas a partir de los 16 años= , y de distintos géneros; y C3) Factibilidad de recolectar el material escrito (exámenes) producido por los estudiantes.

La muestra estuvo constituida inicialmente por 52 alumnos regulares, inscritos en el curso de MAI del trimestre enero-marzo 2= 019 del ciclo básico de la USB, ubicada en Caracas, Venezuela. Se destaca que, debido a la deserción escolar o la posibilidad de un retiro voluntario de la asignatura, después de conocer las calificaciones del primer y segundo exam= en, la cantidad de alumnos iniciales bajo a 50, en el 2do parcial, y= a 39 en el 3er parcial.

Recopilación de información y estrategias de análisis =

Siguiendo lo establecido por McMillan & Schuma= cher (2005), la investigación se organizó por las fases que se describen a continuación, donde  denota la fase del método cualitativo y <= !--[if gte msEquation 12]>Pi,j=  corresponderá al paso  de la fase  del mismo método:

F1 ) Planificació= n: para identificar y tipificar los errores que aparecieron en las soluciones escritas a los exámenes parciales del grupo de alumnos, se procedió de la siguiente manera:

P1,1= ) Datos: para = la recolección de información se utilizaron los tres exámenes parciales contemplados en el curso de MAI durante enero-marzo 2019. Cada examen analizado constaba de 5, 7 y 6 ejercicios y/o problemas de desarrollo, respectivamente; que los estudiantes debían resolver en dos hor= as. Los exámenes parciales fueron elaborados por el profesorado del Departament= o de Matemáticas Puras y Aplicadas de la USB.

P1,2 ) Foco de la revisión: se enfatizó en el análisis de los errores relacionados con los co= nocimientos matemáticos previos y básicos abordados en Bachillerato, donde el uso incorrecto (o la omisión) de las herramientas matemáticas derivadas de estos contenidos, entorpecieron de alguna forma la solución presentada.

) Criterios pa= ra el análisis, identificación y clasificación de errores: no se tomó como base ninguna categorización particular de errores previamente establecida en la literatura, debido a que la idea fue que la tipificación de errores matemát= icos resultante emergiera del análisis de los datos.

) Periodo de recopilación de datos: en cada parcial se esperó por la aplicación y calificación de los exámenes parciales por parte del profesor de la asignat= ura, además, de la respectiva entrega y discusión de resultados con los estudian= tes. Luego, se procedió a un análisis estructurado por pasos: =

) Inicio: se analizaron en forma detallada= 52 exámenes parciales, a través de la resolución escrita de ejercicios y/o problemas por parte de los estudiantes; identificando errores (según lo descrito en el paso ). Se analizar= on los errores emergentes sin hacer referencia al tema particular de los ejercicios y/o problemas; y los errores repetidos correspondientes a un mis= mo examen fueron considerados como un único error.

) Intermedio: esta porción estuvo formada por 50 exámenes parciales. Una vez identificados los errores y su frecuencia se compararon con los grupos ya creados, con el fin de revisar si se podían incluir en alguno de estos o si era necesario crear nuevos grupos.

) Cierre: se c= erró el ciclo de recolección con 39 exámenes parciales y siguiendo las estrategi= as aplicadas en los pasos anteriores.

) Finalización= : se dio paso al análisis y articu= lación de la información recolectada apoyada en la elaboración de tablas y diagram= as de frecuencia, siguiendo:

) Temas: se le= yó la información recolectada y pre-procesada derivada de la revisión de los exámenes parciales, y se definen temas naturales del conjunto de datos.

) Categorías: = se hizo una clasificación más detallada de la obtenida en los conjuntos preliminares para facilitar la tipificación.

) Modelos: basándose en las categorías encontradas se formularon los modelos.

El cierre de esta última fase dio paso a la presentación y discusión de los resultados.

Resultados y discusión

Resultados

Por motivos de espacio, solo se ilustran en detalle la primera categoría emerge= nte del análisis se la información recolectada a través de la tabla 2 y la figu= ra 1, donde se presentan los elementos envueltos con los errores, así como, sus respectivas frecuencias. Las demás categorías siguieron el mismo tratamient= o.

De un total de 141 exámenes parciales revisados, el 100% presentó por lo menos= un error matemático. Los errores identificados y analizados se distribuyeron y agruparon para construir una tipificación basada e= n categorías que contemplan los siguientes tipos:

) Errores Tipo Símbolos (ETS): son aquellos donde se incurre en la omisión y/o uso incorrecto de los símbolos matem= áticos clásicos. Los símbolos matemáticos tienen un significado explícito dentro de un contexto del discurso, y se utilizan para abreviar, con un sen= tido intrínseco (entonces, etc.), representar conceptos (ángulos, etc.,), indicar operaciones (suma, unión, etc.) y relaciones (igualdad, menor que, etc.) en= tre números (o valores).

 

 

 

Tabla 2

Símbolos y frecuencias de errores tipo símbolos matemáticos

Error

 

 

Significado

Frecuencia

ETS

Etiqueta

Referencia

 

Omisión

Uso incorrecto

Símbolos

matemáticos

igualdad

135

57

dos puntos

21

25

implicación o entonces

19

71

agrupación

133

25

relación de orden

8

37

noción de infinito

3

13

conjunto vacío

6

16

∈ ,  ∉

pertenencia o no a un conjunto=

2

14

noción de derecha e izquierda<= /p>

6

33

 ,   

intersección y unión de conjuntos

11

31

aproximado

4

6

valor absoluto

7

4

) Errores Tipo Notación (ETN= ): involucrados en la omisión y/o uso incorrecto de la notación matemá= tica clásica. Aquí, la notación matemática es el lenguaje formal y abreviado que= se establece durante la introducción y desarrollo de un tema con un significado explícito y de utilidad.

) Errores <= /b>Tipo Uso de aspectos fundamentales (ETU): consiste en las interpretaciones y concepciones inadecuadas de definiciones y resultados matemáticos, tales co= mo propiedades, teoremas, etc., usados de forma incorrecta; incluyendo la omis= ión de la previa verificación de las hipótesis donde sea necesario, para hacer = uso de las correspondientes conclusiones. 

) Errores <= /b>Tipo Aplicación de fórmulas (ETA): contemplan tanto la recuperación incorrecta de las fórmulas matemáticas básicas como sus aplicaciones. Aquí, la aplicación= de fórmulas matemáticas consiste en la utilización de un tipo especial de ecuaciones que relacionan algunas diferentes variables, de las cuales se co= noce el valor de algunas y se desea hallar el valor de otras. =

) Errores Tipo Operaciones (ETO): relacionados con las realizaciones incorrectas de las operaciones matemáticas básicas, tales como: suma, multiplicación, división, potenciación, radicación, trigonometría, composición de funciones, desigualdades, ángulos, etc.

) Error= es Tipo Manipulación algebraica (ETM): consisten en el incorrecto manej= o de las expresiones matemáticas, a través de técnicas tales como despejes, simplificación, sustitución, productos notables, uso de identidades, etc.

) Errores Tipo Cálculos (ETC): relacionados con la omisión y/o cálculos matemáticos incorrectos, apoyados en un conjunto de operaciones matemáticas para averig= uar el resultado, valor, medida o solución del problema o ejercicio planteado, tales como, dominios, conjuntos soluciones, etc.

) Errores Tipo Graficación (ETG): involucra las gráficas o dibujos omitidos y/o trazados de forma incorrecta, solicitados en forma directa, o como requisito para obten= er la respuesta a un ejercicio y/o problema planteado.  

) Errores Tipo Redacción y escritura (ETR): abordan la omisión o incorrecta representación= por escrito del lenguaje matemático de cálculos, ideas, gráficos, etc., apoyado= s en letras, palabras, números o símbolos, con el propósito de comunicar (durant= e y después) el proceso de construcción de soluciones, obtención de resultados y conclusiones a los ejercicios y/o problemas planteados.

) Errores Tipo= Problemas prácticos (ETP): envuelve aquellos ejercicios contextualizados resue= ltos de forma incorrecta a causa de deficiencias en el uso de herramientas matemáticas básicas o, simplemente, fueron omitidos de las propuestas de soluciones. En este caso, se analizan los resultados de ejercicios contextualizados en la vida cotidiana con el propósito de poner en práctica= los contenidos matemáticos aprendidos durante el estudio de un tema. Estos problemas fueron planteados en un lenguaje natural para que el resolutor, primero, busque su traducción al leguaje matemático y luego lo resuelva apo= yado en los contenidos aprendidos. En este sentido, se resalta que sólo los dos últimos exámenes parciales tenían problemas prácticos, para un total de 89. Para el análisis y organización de los errores, se estructuro la solución d= el problema en distintos pasos basados en las respuestas dadas por los alumnos, para luego analizar los errores.

Finalmente, como resultado de los pasos descritos, en la figura 2 se puede apreciar los tres modelos derivados del estudio luego de un nivel más de abstracción en el proceso de análisis de l= os datos.

Figura 2

Modelos y categ= orías para la tipificación errores en evaluaciones matemáticas

Discusión=

En este apartado se analizan, y describen, en forma general, los principales resultados del procesamiento de la información recolectada a través del análisis de las soluciones de los exámenes parciales de los estudiantes. Ad= icionalmente, se discuten algunos casos representativos para ilustrar los errores cometid= os por los estudiantes de la muestra.

Análisis general

La formación en matemáticas de los estudiantes antes de cursar MAI puede ser heterogénea, debido a que provienen de distintas instituciones educativas (oficiales o privadas, prestigiosas, especializadas, o simplemente estándar= es, etc.), por lo que el conocimiento previo es variado, lo que pudo influir en= la abundancia o escases de algunos de los errores matemáticos identificados, p= or eso la importancia de documentarlos todos. A pesar de que el curso MAI en su primera parte, repasa, amplia y profundiza los contenidos matemáticos elementales abordados en el Bachillerato, los errores detectados, relaciona= dos con estos temas, aún persisten en las producciones de los estudiantes.

Los ETS resaltan que los alumnos no respetan el correcto uso de algunos símbolos matemáticos, incluyendo la omisión en ciertos casos donde son estrictamente necesarios. En este último, sobresalió el olvido de los símbolos de agrupac= ión e igualdad. Además, fue común que reemplazaran la igualdad por una implicac= ión y viceversa. Esto muestra que los estudiantes no están consiente del significado de igualdad (cuándo dos cosas o más son iguales) y de la implic= ación (cuando la verdad del antecedente asegura la verdad del consecuente). Los E= TN manifiestan el desacato hacia los acuerdos de notación matemática básica establecidos para cada tema, escribiéndola erróneamente, omitiendo parte de= su estructura o confundiéndola entre contextos diferentes.

Los ETU muestran el uso deficiente de aspectos fundamentales de la matemática, además, de la ausencia de verificación de hipótesis para cotejar si los resultados son aplicables en alguna situación en cuestión. Los ETA reflejan= la dificultad para recuperar correctamente las fórmulas matemáticas clásicas, además, de presentar dificultades en el momento de aplicarlas, así como el = uso inconsciente de estas, aun cuando no es estrictamente necesario. Mientras q= ue, los ETO reflejan poco dominio de manipulación y uso de las operaciones matemáticas elementales por parte de los estudiantes.

Los ETM resaltan la precariedad existente en el uso de técnicas básicas de manipulación de expresiones en la resolución de ejercicios y/o problemas de matemáticas. En cuanto a los ETC, el cálculo de dominios es el error más destacable en este tipo, lo que podría explicar parte de los otros errores matemáticos. Además, calcular conjuntos soluciones cuando se combinan desigualdades con funciones, reflejó un grado de dificultad superior.<= /o:p>

Entre los ETG se encontraron que algunos alumnos no respetaron las escalas de sus propios gráficos lo que conlleva a confusión a la hora de interpretarlos. También, corrompieron los gráficos clásicos para los intervalos, combinándo= los entre sí para formar sus propios dibujos, que para un lector conocedor son inapropiados porque rompen los acuerdos establecidos en la matemática. Hubo= un notorio desconocimiento sobre la construcción de gráficos, algunos optaron = por omitirlos a pesar de que se les pedía realizarlos explícitamente, y también= se encontraron gráficas de funciones parabólicas trazadas como líneas rectas.<= o:p>

En los ETR se encontró que en general, los alumnos no respetan la correcta colocación del signo igual que señala la línea principal de una fracción. U= na parte considerable de los estudiantes se equivocó al transcribir el enuncia= do o los datos de los ejercicios y/o problemas en el cuadernillo, o cuando necesitaban continuar la resolución en otra página. Otros omitieron el cier= re o no expresaban correctamente las conclusiones. Aquellos estudiantes que se animaron a construir las soluciones a los problemas prácticos, algunos mostraron ETP debido a deficiencias para definir las variables y sus relaciones, plantear correctamente la situación y/o usar las herramientas matemáticas básicas, y finalmente dar respuesta a la pregunta planteada.   

Análisis particular: casos significativos

Esta sección complementa los resultados con algunos casos representativos de err= ores matemáticos para la descripción y ejemplificación de la tipificación constr= uida a partir del estudio.

Ejemplos

Se transcriben los errores detectados en lugar de utilizar una copia escaneada= de la situación original del error, para respetar la confidencialidad de los participantes.   

Tabla 3

 Algunos ejemplos de <= /i>errores matemáticos en evaluaciones<= /o:p>

Error=

Conte= xto

Repue= stas frecuentes

Descr= ipción

 

 

ETS

 

Resolver

 

  

  

El uso inadecuado del símbolo ‘implica’ rompe con el significado de la cadena de igualdades y soluciones de una ecuación.

 

=  

=  

ETN

=  

=  

=  

=  

=  

=  

Dominio de

 

Dom=3D(-∞,-1]            b)

Domf(x)=3D(-∞,-1]    d)

e)

Dom(f)=3D-x-1≥= ;0

 

Se encontró una divers= idad de notaciones incorrectas, basadas en combinaciones tales como las condiciones para el hallar dominios de ciertas funciones, evaluaciones puntuales, omisión, etc.

 

 

 

Tabla 3

 Algunos ejemplos de <= /i>errores matemáticos en evaluaciones (continuación)

Error=

Conte= xto

Repuestas frecuentes

Descr= ipción

=  

 

ETU

=  

=  

Resolver

=

 

a)

b) <= /span> ,

 

c)

Intentos fallidos al tratar de aplicar propiedades donde no es permisible. 

=  

=  

ETU

=  

 

 

Propiedades=

a)

b) , para todo <= ![if !msEquation]>  

c)

Aplican propiedades por inercia sin tomarse un tiempo de reflexión para revisar si lo obtenido ti= ene sentido matemático.  <= /span>

=  

=  

=  

=  

=  

ETA

=  

=  

=  

=  

 

Si  y , entonces hallar

 

dP1,P2=3D (-1-4)<= /i>2+(1+5)2

dP1,P2=3D (-1-4)<= /i>2+(1-5)2

Se observó dificultades para reemplazar números negativos en fórmulas que involucran menos. 

 

 

Diferentes situaciones con trigonometría<= /o:p>

sen2x=3Dsen2sen(x)    

sen2x=3D2sen(x)

<= m:rPr>cos= x+<= /span>π3= =3D<= m:rPr>cosx+<= m:rPr>cos= π3=                                   =       

1-4= <= m:rPr>cos= 2x=3D4<= m:rPr>sen= 2(x)= .

Asumen propiedades que= no son permisibles, como linealidad, producto, etc. Aplican fórmulas que ya consideran la manipulación de los signos y, aun así, lo toman en cuenta.<= o:p>

=  

=  

=  

=  

ETO

 

 

Diferentes situaciones con operaciones básicas=

2x-2=3Dx       

1-2= <= m:rPr>cosx=3D-cos⁡(x)

x2+12=3D= 2x+2= 2      d= )

50=3D0                 f) 1-= xx=3D1-0=3D1

g)

h)   

 

 

Suman términos solo tomando en cuenta los coeficientes. Aplican los algoritmos de la suma de fracciones erróneamente. Confunden los productos con sumas y, las divisio= nes con restas.

 

 

Tabla 3

 Algunos ejemplos de <= /i>errores matemáticos en evaluaciones (continuación)

Error=

Conte= xto

Repue= stas frecuentes

Descr= ipción

=  

=  

=  

ETO

 

 

Resolver

a)

3= x+1<2 ,  |2x-3|<2

3x+12-2x-32<22

3x+1≥0, -3x-1<0<= ![endif]--> , ,

Se encontraron intentos fallidos por eliminar el valor absoluto de forma fácil y rápida, sin percatarse que no son técnicas válidas en esta situación.

 

 

=  

=  

=  

=  

=  

=  

ETM

 

 

 

 

 

Diferentes situaciones

-x<6 ⟹x>6        b)  

c)      d)

f)

g)

h)  

3x+1-2x-32=3D<= /span>3x<= m:r>+12

                                              =          

En situaciones de pote= ncia de una suma aplican potencia de un producto.

 

 

ETM<= /p>

 

 

Factorización

a) <= ![if !msEquation]>

b)  

2x-2=3D(x-1)= (x+2)

Se refleja la carencia= en el manejo de las técnicas de factorización rápida.

=  

=  

ETC

 

Diferentes situaciones

a)

x2=3D25⟹x= =3D5          d)          f)

Domg=3D0= ,∞

Domg=3Dx2-xx+3≥0

Reconocen que existe u= na condición mayor o igual a cero para una raíz cuadrada, pero no saben plan= tear y concretar los cálculos.  <= /o:p>

=  

=  

=  

 

=  

=  

ETG

 

 

 

 

 

 

Situaciones de graficación <= /p>

 

 

 

a)

b) 

Fijan el infinito en un punto de la recta real, lo que le quita su esencia. Combinan los dibujos estándar para graficar intervalos. Dibujan las parábolas como si estuviera formada por segmentos de rectas gráficos y no como una curva suave. No puntean las gráficas para indicar que están se extienden a lo largo de su dominio.

=  

ETR

Dadas

    y

 

 

Fue común encontrar que no se respeta la línea principal de una fracción.

=  

=  

=  

=  

ETP

Una hoja de papel de forma rectangular debe contener 18 cm2 de texto impreso. Los márgenes superior e infe= rior han de ser de 2 cm y los laterales de 1 cm. Calcular las dimensiones de la hoja de tal manera que el área de esta sea mínima.

 

 

Se encontró que algunos estudiantes tratan de resolver el problema empíricamente, sin hacer uso de las herramientas matemáticas.

 

 

 

 

 

 

Conclusiones<= /b>

·&nb= sp;        En cuanto a los objetivos sobre analizar, identifi= car y tipificar los errores que cometieron un grupo de estudiantes en la resolu= ción de ejercicios y/o problemas de matemáticas en los exámenes parciales correspondientes a MAI de la USB, se pudieron construir diez categorías: ET= S, ETN, ETU, ETA, ETO, ETM, ETC, ETG, ETR y ETP; formando la tipificación de errores matemáticos derivada del estudio.

·&nb= sp;        En general, la investigación confirmo que: a) los = errores matemáticos están latentes en las producciones de los estudiantes, debido a= que todos los exámenes parciales analizados mostraron algún tipo de error; b) l= os alumnos poseen deficiencias en los aspectos fundamentales de las matemáticas básicas, especialmente como herramientas y aplicabilidad; c) los errores originados por deficiencia en conocimientos previos entorpecen la adquisici= ón de nuevos contenidos; d) la omisión, o resistencia para aplicar las herrami= entas matemáticas durante la resolución ejercicios y/o problemas de matemáticas, = no solo revela la carencia de conocimientos previos sino que pudiera apuntar también al miedo a fracasar cometiendo errores, u otras razones objeto de estudio.  Al respecto Gómez (1998) = señala que cuando el profesor resuelve un ejercicio o cuando el libro lo hace a tr= avés de un ejemplo, la solución se presenta en limpio, sin que haya la menor indicación del proceso de borrador por medio del cual se llegó a la solució= n; y e) para los lectores conocedores y usuarios de las matemáticas, estas dificultades entorpecen la lectura de textos especializados o publicaciones= de interés, dificultando la adquisición de conocimientos.

·&nb= sp;        Se resalta que a pesar de que el estudio está enmarcado en un contexto específico y con una muestra limitada, los resulta= dos exhiben un acercamiento a los errores que cometen los estudiantes que cursa= n un primer curso de matemáticas en la mayoría de las universidades. Concretamen= te, los resultados del estudio permitieron articular una tipificación de errores matemáticos como principal contribución a la literatura, aportando informac= ión para ampliar el conocimiento acerca de la problemática planteada. Además, de contribuir a contextualizar las deficiencias de los conocimientos previos (abordados en Bachillerato) relacionados con la ocurrencia de errores matemáticos, la tipificación puede ser un referente que permita una mejor articulación de su ejercicio docente, y puede ser utilizada para diseñar materiales y/o actividades de nivelación, remediales y/o preventivas que ay= uden a los estudiantes a superar las dificultades.

·&nb= sp;        Finalmente, la categorización de errores sigue sie= ndo una labor sin finiquitar. Esto sucede, a pesar de que se pueden encontrar t= ipos de errores comunes al comparar las categorías existentes o con futuras categorizaciones, siempre existe la posibilidad de introducir nuevos tipos = de errores matemáticos, o simplemente, analizarlos desde diferentes perspectivas, criterios o causas.

 

Referencias bibliográficas=

Abrate, R., Pochulu, M. & Vargas, J. (2006). Errores y dificu= ltades en matemática: análisis de causas y sugerencias de trabajo. Buenos Aire= s, Argentina: Universidad Nacional de Villa María.

Ca= razo, A. & Brey, R. (2012). Errores en el aprendizaje de las matemáticas financieras. Enseñanza de las ciencias, 30(2), 73-92.

Del Puerto, S., Minnaard, C. = & Seminara, S. (2006). Análisis de los errores: una valiosa fuente de informa= ción acerca del aprendizaje de las Matemáticas. Revista Iberoamericana de Educación, 38(4),1-12. https://dialnet.unirioja.es/servlet/artic= ulo? Código=3D 1704266

Esteley, C. & Villarreal,= M. (1990). Categorización de errores en Matemática. XIII Reunión de Educación Matemática. San Luis.

Esteley, C. & Villarreal,= M. (1996). Análisis y categorización de errores en matemática. Revista de Educación Matemática, 11(1), 16-35.

Farías, D., Abancin, R., & Pérez, J. (2021). Potenciación en el aula de clases en estudiantes que inic= ian estudios superiores. PARADIGMA, 42<= /i>(2), 110-129. https://doi.org/10.37618/PARADIGMA.1011-2251.2021.p110-129.id958

Gamboa, R., Castillo, M. & Hidalgo, R. (2019). Errores matemáticos de estudiantes que ingresan a la universidad. Revista Electrónica Actualidades Investigativas en Educación, 19(1), 1-31. DOI: 10.15517/aie.v19i1.35278= .

Gandulfo, M., Benítez, I., Ra= mírez, R., Bandolín, J., Gemignani, M., De Zan, M., Musto, D. & Gimenez, L. (2013). El aprendizaje de la matemá= tica a partir de los errores. VII Congreso Iberoamericano de Educación Matemát= ica. https://semur.edu.uy/cibem.org/7/ actas/pdfs/982.pdf.

Gómez, P. (= 1998). Profesor: no entiendo. Reflexiones alrededor de una experiencia en docencia de las matemáticas. Bogotá, Colombia: Una empresa docente.

McMillan, J. & Schumacher, S. (2005). Investigación educativa: una introducción conceptual. 5a Edición. Madrid, España: Pearson Educación.

Movshovitz-Hadar, N., Zaslavs= ky, O. & Inbar, S. (1987). An empirical classification model for errors in high school mathematics. Research in Mathematics Education, 18(1), 3-14.

Pochulu, M. (2005). Análisis y categorización de errores en el aprendizaje de la matemática en alumnos que ingresan a la universidad. Revista Iberoamericana de Educación, 35(4), 1-15.

Radatz, H. (1979). Error anal= ysis in the mathematics. Journal for research in math= ematics education, 9, 163-172. http://www.ugr.es/local/jgodino/edumat-maestros/

Ramírez, R., Flores, P. & Ramírez, I. (2018). Análisis de los errores en tareas geométricas de argumentación visual por estudiantes con talento matemático.= Revista Latinoamericana de Investigaci= ón en Matemática Educativa, 21(1), 29-56. DOI: 10.12802/relime.18.2112.<= /o:p>

Ruano, R., = Socas, M. & Palarea, M. (2008). Análisis y clasificación de errores cometidos = por alumnos de secundaria en los procesos de sustitución formal, generalización= y modelización en álgebra. PNA, 2(2), 61-74.

Socas, M. (= 1997). Dificultades, obstáculos y errores en el aprendizaje de las matemáticas en = la educación secundaria. En L. Rico (Coord.). La educación matemática en la enseñanza secundaria, (pp. 125-154). Barcelo= na: Horsori.

Ve= ra, O. & Volta, L. (2009). Análisis de errores en evaluaciones de suficienc= ia. Ingreso a la universidad nacional de Quilmes. Eje lógico matemático departa= mento de ciencia y tecnología. II Jornadas de Enseñanza e Investigación Educativa= en el campo de las Ciencias Exactas y Naturales, La Plata, Argentina. http://www.memoria.fahce.unlp.edu. ar/trab_eventos/ev.611/ev.611.pdf.<= /o:p>

 

 

Conflicto de intereses

Los autores deben declarar si existe o no conflict= o de intereses en relación con el artículo presentado.

 

 

El artículo que se publica es de exclu= siva responsabilidad de los autores y no necesariamente reflejan el pensamiento = de la Revista Explorador Digital.

 


 

 

 

El artículo queda en propiedad d= e la revista y, por tanto, su publicación parcial y/o total en otro medio tiene = que ser autorizado por el director de la Revista Explorador Digital.

 

&nb= sp;


 

 

 

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ISSN: 2661-6831

Vol. 6 No. 3, pp. 6 – 27, julio – septiembre 2022
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