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GeoGebra como recurso de enseńanza de matemática en primero de bachillerato

GeoGebra as a resource for teaching mathematics in the first year of high school

Inés Maribel Barros Tapia = https://orcid.org/0000-0001= -5673-9558

Universidad Católica de Cuenca, Azogues, Ecuador

ines.barros.85@est.ucacue.edu.ec

Roxana Auccahuallpa Fernández https://orcid.org/0000= -0002-5242-2083

Universidad Católica de Cuenca, Azogues, Ecuador

roxana.auccahuallpa@ucacue.edu.ec

Juan Carlos Era= zo Álvarez https://orcid.org/0000-0001-= 6480-2270

Universidad Católica de Cuenca, Azogues, Ecuador

jcerazo@ucacue.edu.ec

	Artículo de Investigación Científica y Tecnológica Enviado: 12/= 07/2022 Revisado: 15= /08/2022 Aceptado: 05= /09/2022 Publicado:06= /10/2022 DOI: https://doi= .org/10.33262/exploradordigital.v6i4.2346 <= /span>
= =	<= /o:p>
Cítese: <= /o:p>		&= nbsp; Barros Tapia, I. M., Auccahuallpa Fernández, R., & Erazo Álvarez, J. C. (202= 2). GeoGebra como recurso de enseńanza de matemática en primero de bachillera= to. Explorador Digital, 6(4), 42-59. https://doi= .org/10.33262/exploradordigital.v6i4.2346 &= nbsp;
<= /span>		EXPLORADOR DIGITAL, <= em>es una Revist= a electrónica, Trimestral, que se publicará en soporte electrónico tiene como misión contribuir a la formación de&nbs= p;profesionales competentes con visión humanística y crítica que sean capaces de exp= oner sus resultados investigativos y científicos en la misma medida que se promueva mediante su intervención cambios positivos en la sociedad. https://exploradordigital.org<= span class=3DMsoHyperlink> La revista es editada por la Editorial Ciencia Digital (Editorial de prestigio registrada en la Cámara Ecuatoriana de Li= bro con No de Afiliación 663) = www.celibro.org.ec
<= /span>		Esta revista está protegida bajo una licencia Creative Commons AttributionNonCommercialNoDerivatives 4.0 International. Copia de la licencia: http://crea= tivecommons.org/licenses/by-nc-nd/4.0/.
Palabras claves: Matemáticas y estadística, Geometría, programa, lógica matemática, tecnología educacional, método de enseńanza=	&n= bsp;	Resumen Uno de los problemas fundamentales en la matemát= ica, por parte de los estudiantes del BGU son las operaciones con vectores, por los diferentes métodos gráficos correspondientes y de allí su importancia= de estudio y búsqueda de un proceso a ser realizado, que posea creatividad, dinamismo, fácil de utilización y más ventajas a favor del educando, que proporciona el programa de GeoGebra. Para la metodología de la enseńanza = de la matemática, es necesario que el docente conjuntamente con el educando explore las diferentes posibilidades de resolución de los ejercicios y un= ir lo tradicional. Lo que se deduce que anteriormente no se incorporaba la tecnología, como una comparación entre métodos gráficos y analíticos de interacción dicotómica de la teoría y la práctica realizados en GeoGebra y aplicando las TIC, con los tradicionales realizados en la pizarra; como diferencias en el ahorro del tiempo, la precisión en los gráficos, la creatividad, etc., para que el estudiante sea el observador directo de sus cambios y éxitos de comprensión. Más aún cuando realizamos una investigac= ión de estudiantes del sector rural y sus deficiencias para el manejo de la tecnología.
Keywords: Mathematics and statistics, geometry, software, mathematical logic, educational technology, teaching metho= d		Abstract One of the fundamental problems in mathematic= s, on the part of the students of the BGU, is the operations with vectors, by the different corresponding graphic methods and hence its importance of s= tudy and search for a process to be carried out, that possesses creativity, dynamism, easy to use and more advantages in favor of the learner, provid= ed by the GeoGebra program. For the methodology of teaching mathematics, it = is necessary that the teacher together with the student explore the different possibilities of solving the exercises and join the traditional. What fol= lows is that technology was not previously incorporated, such as a comparison between graphical and analytical methods of dichotomous interaction of th= eory and practice carried out in GeoGebra and applying ICT, with the tradition= al ones carried out on the blackboard; such as differences in time savings, accuracy in graphics, creativity, etc., so that the student is the direct observer of their changes and successes in understanding. Even more so wh= en we carry out an investigation of students from the rural sector and their deficiencies for the management of technology.

Introduc= ción.

Uno de los problemas fundamentales en la matemática, por parte de los estudiantes del = BGU son las operaciones con vectores, por los diferentes métodos gráficos correspondientes y de allí su importancia de estudio y búsqueda de un proce= so a ser realizado, que posea creatividad, dinamismo, fácil de utilización y más ventajas a favor del educando, que proporciona el programa de GeoGebra.

Si revisamos la historia, el educador era quien guiaba a sus estudiantes en las respectivas actividades que requerían las gráficas de las operaciones con vectores (sum= as, restas, ángulos, etc.) que se las hacía en la pizarra e inclusive hasta aho= ra y en sus cuadernos de materia; pero se necesitaba precisión, una escala adecu= ada a cada ejercicio planteado y un tiempo no limitado dentro de la hora clase;= y así era el docente quien al final realizaba el mayor esfuerzo.

En el Ecuador = si hacemos un análisis de la educación actualmente, encontramos diferencias marcadas entre las regiones de la Sierra, Costa, Oriente e Insular. Especialmente las instituciones educativas alejadas al sector urbano y es a= sí que en ciertas partes del Azuay específicamente en Cumbe; el conocimiento y= uso de tecnología por los educandos es limitada y la gran necesidad de fortalec= er en matemática la aplicabilidad del programa GeoGebra; a más de otras necesidades como: la infraestructura, internet adecuado, etc. para una educación de calidad y funcional.

Respondiendo a= sí a las necesidades de una buena tecnología que se pueda aplicar a innumerables temas matemáticos y en esta investigación en particular a vectores como propuesta de mejora, y al indicado= r de evaluación de matemática planteado por el (Ministerio de Educación, 2016) “I.M.5.6.2. Realiza operaciones en el e= spacio vectorial R2; calcula la distancia entre dos puntos, el módulo y la direcci= ón de un vector; reconoce cuando dos vectores son ortogonales; y aplica este conocimiento en problemas físicos, apoyado en las TIC.” <= /p>

Como dice Duval (1995= ) enseńar la programación matemática dentro de funciones lineales que representaría la función objeto con su respectiva restricción. En el caso de vectores cuyo propósito e= s; que los educandos aprendan los métodos analíticos y gráficos (polígono y paralelogramo) en las operaciones con los mismos, como segmentos orientados= a la aplicabilidad de problemas del ámbito común; y utilizando ecuaciones en = las operaciones que involucren restricciones y función objeto estaría insertánd= ose también en el campo de la programación lineal entre otros campos. Sin embar= go, contestando las interrogaciones de:

· = żLos estudiantes de primero de BGU alcanzan mejores resultados con el uso del software GeoGebra en las operaciones básicas de vectores?

· = żSe visualiza fácilmente la diferencia de operaciones bás= icas de vectores, con el uso del programa GeoGebra; en los diferentes métodos de resolución analíticos y gráficos?

Al utilizar el GeoGebra en las gráficas de las operaciones con vectores como recurso tecnológicas con los estudiantes = de primero de bachillerato paralelo B de la Unidad Educativa Cumbe, para que l= os educandos sean los creadores de sus propios aprendizajes y posean una herramienta eficaz, precisa, dinámica y clara de dichas operaciones, que refleje claramente mejores aprendizajes significativos y un ahorro de tiempo considerable; para la posterior aplicabilidad de problemas reales de diversos ámbitos y campos como: científico, ingeniería, física, deportivo; etc.

En el futuro es indispensable que las unidades educativas cuenten con internet en cada aula= y se pueda aplicar esta herramienta básica que es el GeoGebra en los diversos temas que requieran gráficas y en particular las operaciones con vectores.<= o:p>

La matemática = en el transcurso de éstos últimos 14 ańos aproximadamente, desde la primera dé= cada del 2000 se ha tenido unos cambios significativos para poder ingresar a la Universidad, primero con el examen “ser bachiller”, luego el examen EAES y ahora el examen Transformar y es así igualmente que los típicos ejercicios = de vectores empiezan a aplicarse en problemas de la vida cotidiana e inclusive en otras asignaturas como la física, problemas sobre: aviones, trenes, automóviles, torres ,personas etc. Es aho= ra un tema realmente necesario y precedente en la aplicabilidad directa de situaciones reales o hipotéticas.

Aplicar GeoGeb= ra, en el aprendizaje de operaciones básicas con vectores en los estudiantes de primeros de bachillerato paralelo B de la Unidad Educativa Cumbe, para que = los educandos sean los creadores de sus propios aprendizajes.=

Como dice Díaz-= Barriga (2011) cuando los docentes se encuentran laborando normalmente, observan en su constante práctica, la necesidad de innovar una pedagogía prevista en el tiempo y espacio de una visión tecnológica, ante la abstracta concisión en el aula; lo que genera cambiar = de estrategia adaptada a la didáctica con enfoques de aprendizajes dinámicos, interactivos agradables con reflex= ión y raciocinio ; cuyo objetivo principal para los maestros es enseńar, modifica= r el sistema a uno que utilice recursos que involucren variedad de tipos o secuencias didácticas innovadoras y tecnológicas.

Al respecto Analytics (2022) está muy claro la diferencia entre docentes nativos digitales y los inmigra= ntes digitales; sin embargo ambos grupos distinguen las ventajas amplias de un software como el GeoGebra para la aplicabilidad en el proceso de aprender y enseńar la matemática y materias afines. Ante esta situación se visualiza u= na realidad de la no constante utilización de la herramienta tecnológica efica= z y precisa para gráficas, cálculos, estadística y más necesidades en todos los= niveles educativos. Primero el temor de ciertos docentes inmigrantes digitales de equivocarse en la aplicabilidad de la herramienta tecnológica funcional, por falta de buena preparación profesional y segundo por el escaso equipamiento= de tecnología que ofrecen las instituciones educativas que no están a la vanguardia de las necesidades actuales de hoy.

Característica= s y ventajas del GeoGebra que permiten su utilización en el proceso de enseńanz= a - aprendizaje de la matemática en la secundaria básica. GeoGebra como programa nació en Austria, en la Universidad de Salzburgo, del resultado del trabajo= de tesis de Markus Hohenwarter en el ańo 2002; a manera de una combinación de programas de geometría dinámica y cálculo simbólico (Markus-Hohenwarter, 2009). El autor tuvo esta brillante idea al darse cuenta de lo complicado de los sistemas del cálculo simbólico que representaba por su compleja conexión y = para los docentes no resultaba agradable y no así los programas de geometría dinámica que surgían de una buena conexión física y funcional entre dos apa= ratos (interfaz); se evidenciaba entre la persona y la máquina una comunicación factible que facilitaba su uso. Es así que GeoGebra se convertiría con los ańos en la mejor herramienta tecnológica p= ara aprender matemática de una forma dinámica, funcional, interactiva; traducid= a a decenas de idiomas y creando institutos locales y el Instituto Internaciona= l de GeoGebra con nuevas funciones y comunidades de usuarios investigadores para tales fines.

Al decir que GeoGebra es uno de los mejores programas que relaciona la participación, comunicación, reciprocidad; decimos que es un software interactivo para la aplicación de docentes y estudiantes en el proceso enseńanza-aprendizaje de la matemática. Que se utiliza para el Cálculo, geometría y el álgebra; de forma dinámica y libre adaptado a cualquier plataforma.

Markus-Hohenwarter (2009) el programa GeoGebra y sus múltiples funciones se representa de tres maneras diferentes= y principales de cada objeto matemático: una vista numérica, una vi= sta gráfica, vista algebraica, como también, una vista de hoja de cálculo. Esta variedad es lo que con lleva a apreciar los objetos matemátic= os en tres representaciones diferentes: gráfica (como en el caso de puntos, gráficos de funciones, vectores, etc.), algebraica (como coordenadas de pun= tos, ecuaciones), y en celdas de una hoja de cálculo. <= /p>

Estas tres p= erspectivas se unifican en un solo proceso automático de interacción, de manera muy dinámica y motivante para quien utilice el programa GeoGebra; Así mismo es fácil observar los cambios que generan en cualquiera de los campos.

<= /o:p>

Figura 1

Interfaz de GeoGebra

Fuente: Elab= oración propia<= /span>

Junquera (2008= ) indica que un vector es un segmento orientado = que queda determinado por dos puntos A y B, y el orden de estos. El primer punt= o se llama origen (A) y el segundo se llama extremo (B). El vector se nombra con= las dos letras mayúsculas de su origen y extremo o con una letra minúscula:

Vectores, elementos y componentes. (Katz, 2013) un vector es un segmento orientado que queda determinado por dos puntos A y B, y el orden de estos. El primer punto se l= lama origen (A) y el segundo se llama extremo (B). El vector se nombra con las d= os letras mayúsculas de su origen y extremo o con una letra minúscula:<= !--[if gte msEquation 12]>AB o .

Figura 2

Dos vectores de fuerza

<= /span>

Fuente: Simulador Phet.

<= /o:p>

Figura 3

Coordenada de un vector

Fu= ente: Elaboración propia

C= omo indica Castro-Gordón (2013)<= span lang=3DES> la operaciones con vectores requieren de dos procesos denominados analítico y gráfico, como por ejemplo Juan camina varios kilómetros hasta el punto (3, 2) y luego vuelve a caminar varios kilómetros hasta el punto (2, 5). żEncontrar las coordenadas del vector desplazamiento a la que se encuentra Juan desde el punto de partida? (origen de coordenadas). Resolució= n: para determinar las coordenadas del vector desplazamiento desde el punto de partida, sumamos los dos vectores anteriores componente a componente.

Software GeoGebra como herramienta en enseńanza y aprendizaje de la Geometría.

Alvarez et al. (2019) real= izó el posterior análisis a todo el trabajo investigativo realizado en la utilizac= ión del Software GeoGebra con los educandos; se observa primero el logro del ma= nejo de las diversas funciones del GeoGebra, en donde se fusionan el método tradicional de realizar en cuadernos y a mano con el nuevo método tecnológi= co del GeoGebra. Los estudiantes des= cubren las funciones y aplicaciones básicas del GeoGebra en la geometría dinámica = que este programa presenta. Segundo el cumplimiento de todas las facetas o dimensiones del conocimiento didáctico-matemático (Godino, 2011) o de= la idoneidad didáctica presentadas en este artículo que son: faceta interaccio= nal, cognitiva, epistémica, ecológica, afectiva, mediacional. =

En la primera faceta la interaccional se dio en cuanto los estudiantes estuvieron constantemente utilizando el programa GeoGebra de manera interactiva y dinámica. En la seg= unda faceta, cognitiva se dio en cuanto se enseńó a los estudiantes el marco teó= rico completo del uso del GeoGebra, se respondió a las dudas del manejo de la mi= sma y la relación con la geometría dinámica en el contexto realizado. En la ter= cera faceta, epistémica se dio en cuanto se estableció dos elementos de dos dimensiones: la dimensión matemática en los diversos cálculos realizados po= r el estudiante y la dimensión didáctica en el conocimiento del proceso de apren= der y enseńar; para esto se evidenció la teoría de las situaciones didácticas y= a- didácticas; por cuanto se dio el aprendizaje de un conocimiento (matemático= )en las teorías didácticas y se dio también el desarrollo de conocimientos en l= os educandos, como: la manera de expresarse, escuchar ,argumentar, una interac= ción dinámica en la teoría a-didáctica.

En la cuarta faceta, la ecoló= gica se dio en el trabajo de campo y los estudiantes formaron parte de la poblac= ión de estudio en la investigación realizada, la adaptación de los educandos se realizó tomando en cuenta las necesidades e intereses de la población. En la quinta Faceta, afectiva; mostraron los estudiantes gran interés, motivación= y carińo por realizar las actividades utilizando el GeoGebra y así aprender haciendo. finalmente, en la faceta de mediaciones, se dio todo momento, ya = que se integró didácticamente las TIC en el uso del software del GeoGebra, en d= onde se realizó tablas, gráficos, cálculos y más en base al acceso de la tecnolo= gía proporcionada. El GeoGebra: una herramienta tecnológi= ca para aprender matemática en la secundaria básica haciendo matemática.=

Arteaga-Valdés et al. (2019) mediante ejemplos ilustrados y desarrollados se evidencia que el Software GeoGebra no solo es una herramienta que resuelve todo tipo de ejercicios co= mo problemas de matemática; es también un recurso didáctico que funciona eficientemente en el desarrollo de la creatividad del estudiante y lo hace = más dinámico, mejor motivado por el sin número de funciones que tiene el progra= ma y el reto de descubrir haciendo.

Marisol et al. (2007) indica que el discente aprende a utilizar el programa de cálculo simbólico (CAS) software Maple, en la resolución de problemas matemáticos a

diferentes campos como la Ingeniería civil entre otros; de una manera que ejerce un desequilibrio cognitivo el cual mejora el pensamiento y obliga al razonamie= nto, luego de la interacción y construcción del conocimiento de la disciplina fundamental y compleja que es la matemática. Por tal razón se evidencia la importancia y mejoramiento de la matemática desde nińo, utilizando tecnolog= ías como software libre; entre otras aplicaciones el Geogebra hasta programas m= as complejos como Maple. (Miguel, Francisco, Jesús, & Luis, 2= 008)Manifiestan que= la nueva generación actual y en toda su enseńanza, presenta a la tecnología y = sus programas informáticos utilizados en aulas diseńadas pedagógicamente y en t= odos los niveles de la enseńanza-aprendizaje, inclusive en entornos de enseńanza bimodal (presencial y online),y que fortalece el trabajo en grupo y aprendi= zajes que valoran el error y la comparación , como también la investigación en la simplificación de aprendizajes y flexibilidad en la adaptación. Es así como= en países como Reyno Unido aplican modelos pedagógicos que requieren utilizar eficazmente ordenadores en cada aula.

Almada et al. (2018) menciona la necesidad = de integrar en la Educación Superior las TIC, mediante infraestructuras acorde= s a la tecnología, capacitaciones a los docentes y el cumplimiento operativo de= los educandos en el uso adecuado de las misma; por lo que se refleja la certeza= de la precisión de software libres en el futuro, como herramientas imprescindi= bles de la Educación en General y más aún la Educación Superior.

Izasa et al. (2019) indica que todo lo referido anteriormente se llega al punto central de la investigación, que son los materiales multimedia en el uso del software educativo, para agregar códigos de información como imagen, texto, animació= n y sonido; ya que se requiere anexar un entorno visual favorable y sencillo qu= e se manipule todo tipo de gráficas y representaciones, en problemas de vectores, matrices, problemas de álgebra en general, cálculo, etc.; mediante programas libres de acceso gratuito para tod= as las personas, programes como GeoGebra, Maple, Derive versión (6.1), todos unidos con un mismo fin que es mejorar el aprendizaje en la matemática.

Metodologia.<= /p>

La presente investigación es con diseńo de tipo cuasiexperimental, pues se la realizó en el aula de clases, tomando en cuenta la metodología descriptiva = del estudio y explicación de las variables (independiente – GeoGebra, dependien= te - Aprendizaje de vectores) que se correlacionan con el contexto. Con un corte longitudinal en dos tiempo= s; por cuanto la población lo constituyen todos los 30 estudiantes del primero B de bachillerato especialidad contabilidad, de la Unidad Educativa Cumbe en el = ańo lectivo 2021-2022; quienes fueron seleccionados al estar dentro de la carga horaria del investigador y al ser de primeros de bachillerato, cuyo currícu= lo posee el tema de Vectores. =

E= sta investigación por tanto se la realizó correctamente en los dos tiempos establecidos y con la técnica de la encuesta. Primero se ejecutó el pretest= con 14 preguntas cerradas (escala de Likert) el 10 de mayo del 2022 acerca de la utilización o no del GeoGebra en cualquier tema en ańos anteriores; cabe destacar que dicha escala presenta un paradigma cuantitativo y cualitativo = en tanto que refleja los resultados alfanuméricos; es importante mencionar tam= bién el poco uso de la herramienta antes mencionada.

E= n el segundo tiempo se realizó el postest con fecha 31 de mayo; como un trabajo = de aula, luego de aplicar en varias clases en el Laboratorio de Computación de= la Unidad Educativa Cumbe, el GeoGebra clásico, que fue instalado con anterior= idad en todas las computadoras para este trabajo investigativo en las operaciones con vectores. Se realizó y se hizo firmar adjuntamente la carta de autoriza= ción a los representantes de los estudiantes para participar en la investigación, cumpliendo así con los fundamentos éticos requeridos.

D= entro del análisis estadístico en esta investigación descriptiva- inferencial se utilizó el programa Editor Estadístico de Datos (SPSS) para la creación de = las tablas, gráficas, y análisis de datos. Por lo que se refiere a la aplicación del instrumento con 14 pregunt= as, mismo que fue validado a través del coeficiente de fiabilidad alfa de Cronb= ach con un valor de 0,739.

Resultados:

Se escogió la encuesta como instrumento para recolección de información, por cuanto facil= ita el desarrollo del trabajo investigativo escogido y la utilización de la esc= ala de calificación (Likert) que mide cuantitativa y cualitativamente los datos cuando se utiliza las TIC.

Al realizar un análisis de normalidad a través de la prueba Shapiro-Wilk se denota que las= 14 variables analizadas son paramétricas. Se evidenció que GeoGebra aplicado por los estudiantes; lograron demostrar capacidad para ser los propios creadores de sus aprendizajes en operaciones básicas de vectores.

Tabla 1

Ge= oGebra un programa sencillo y fácil

	Frecuencia	Porcentaje	Porcentaje válido	Porcentaje acumulado
Válidos	Casi nunca	1	3,3	3,3	3,3
A veces	7	23,3	23,3	26,7
Casi siempre	5	16,7	16,7	43,3
Siempre	17	56,7	56,7	100,0
Total	30	100,0	100,0

Fuente: Elaboración propia =

En la tabla 1 = se evidencia que 56,7% de los estudiantes encuestados mencionan que GeoGebra e= s un programa sencillo y fácil de utilizar, porcentaje que ratifica que la mayor= ía de la población están de acuerdo que este software es sencillo y fácil de utilizar; por lo que es necesario aplicar GeoGebra para la enseńanza de vectores en matemática, como se muestra en la siguiente tabla 2.=

Tabla 2

Te gustó GeoGebra para aprender vectores

	Frecuencia	Porcentaje	Porcentaje válido	Porcentaje acumulado
Válidos	A veces	7	23,3	23,3	23,3
Casi siempre=	13	43,3	43,3	66,7
Siempre	10	33,3	33,3	100,0
Total<= /p>	30	100,0	100,0

Fuente: Elaboración propia

En la tabla 2 = se evidencia que el 66,7% de los estudiantes encuestados confirman un gusto pa= ra aprender vectores mediante GeoGebra, dato que corrobora un porcentaje basta= nte aceptable de la población estudiantil por el gusto de la práctica de vector= es con GeoGebra; por ende, hay que aprender a realizar operaciones con vectore= s en el programa matemático GeoGebra.

Figura 4

Aprendió a realizar gráficos con vectores en GeoGebra

Fuente: Elaboración propia

En la figura 4, hay la certeza que 56%= de los educandos encuestados aprendieron por lo general a realizar operaciones= con vectores, mediante el programa GeoGebra; dato que aclara que se puede apren= der vectores por los estudiantes como creadores de sus propios aprendizajes, sin dificultades en un programa matemático dinámico como el GeoGebra; en tanto = que es necesario que todas las Instituciones Educativas apliquen este programa dentro del currículo.

Tabla 3

Estadísticos= de grupo

	Test<= /p>	N	Media	Desviación típ.	Error típ. de la media
GeoGebra un programa sencillo y fácil	Pretest	30	4,27	0,944	0,172
Postest	30	4,50	0,572	0,104
Te gusto GeoGebra para aprender vectores	Pretest	30	4,10	0,759	0,139
Postest	30	4,90	0,305	0,056
Aprendió a realizar gráficos con vectores en GeoGebra	Pretest	30	3,57	0,935	0,171
Postest	30	4,53	0,681	0,124

Prueba de muestra independientes<= /o:p>

	GeoGebra un programa sencillo y fácil	Te gustó GeoGebra para aprender vectores	Aprendió a realizar gráficos con vectores en GeoGebra
Se han asumido varianzas iguales	No se han asumido varia= nzas iguales	Se han asumido varianzas iguales	No se han asumido varia= nzas iguales	Se han asumido varianzas iguales	No se han asumido varia= nzas iguales
Prueba de Levene para la igualdad de varianzas	F	12,624		20,119		2,172=
Sig.	,001<= /p>		,000<= /p>		,146<= /p>
Prueba T para la igualdad de medias	T	-1,157	-1,157	-5,357	-5,357	-4,575	-4,575
Gl	58	47,770	58	38,138	58	53,021
Sig. (bilateral)	,252<= /p>	,253<= /p>	,000<= /p>	,000<= /p>	,000<= /p>	,000<= /p>
Diferencia de medias	-,233=	-,233=	-,800=	-,800=	-,967=	-,967=
Error típ. de la diferencia	,202<= /p>	,202<= /p>	,149<= /p>	,149<= /p>	,211<= /p>	,211<= /p>
95% Intervalo de confianza para la diferencia	Inferior	-,637=	-,639=	-1,099	-1,102	-1,390	-1,390
= Superior	,170	,172	-,501	-,498	-,544	-,543

Fuente: Elaboración propia

En la tabla 3,= estadísticos de grupo; se evidencia que la variable GeoGebra un programa sencillo y fácil, tiene un incremento de la media de 0= .33 y en la Prueba de muestras independientes una sig. de 0.001 que corrobora p= or tanto que si están relacionadas las variables del antes y del después y que= son datos significativos. Estadísticos= de grupo; se transparenta que la variable Te gustó GeoGebra para aprender vectores, tiene un incremento de la media de 0.80 y en la prueba de muestras independientes una sig. de 0.000 que ratifica por tanto que si están relacionadas las variables del antes y del después y que son datos significativos. Estadísticos de grupo; se aclara que la variable Aprendió a realizar gráficos con vectores en GeoGebra tiene un incremento de la media = de 0,96 y en la Prueba de muestras independientes una sig. de 0.146 que ratifi= ca por tanto que no hay relación de las variables del Pretest y Postest y que = son datos no significativos. Por lo qu= e es menester mayor prácticas en la realización de gráficos con vectores en GeoGebra.

Discusión.

Al realizar la comparación Alvarez et al. (2019) que concluye que la fusión del mét= odo tradicional con el método tecnológico del GeoGebra, incrementó los aprendizajes significati= vos; ya que se evidenció una transforma= ción positiva de las actitudes relacionadas con la matemática de la mayoría de l= os estudiantes, gracias al trabajo con GeoGebra. De manera semejante en esta investigación se concluye que el programa GeoGebra aplicado a los estudiantes del sector rural lograron demostrar capacidad para ser los propios creadores de sus aprendizajes en operaciones básicas con vectores; es así también que se verificó en las tres variables realizadas tabla 1 GeoGebra un programa sencill= o y fácil; tabla 2 te gustó GeoGebra para aprender vectores; tabla 3 Aprendió a realizar gráficos con vectores en GeoGebra, en un porcentaje ≥ al 56 % que corrobora resultados positivos de la investigación.<= o:p>

En la siguiente comparación Arteaga-Valdés et al. (2019) el autor concluye que se forma también la trilogía alumno– GeoGebra–conteni= do en los procesos enseńanza-aprendizaje de la matemática básica de la secunda= ria y por ende en la Geometría, y es el educando el creador de sus propios aprendizajes mediante el software de GeoGebra. De igual forma en la presente investigación los resultados en la tabla 1 GeoG= ebra un programa sencillo y fácil; son claros en confirmar que el programa GeoGebra es sencillo y fácil de utilizar por los estudiantes en un 56,7%, p= or ende, los contenidos de Geometría y específicamente en este caso de vectore= s el educando es un manipulador y entonces el creador de los aprendizajes propio= s.

Propuesta.<= /o:p>

Utilizar el GeoGebra como herramienta tecnológica eficaz y precisa todo el ańo lectivo = en las operaciones con vectores y demás temas que aplique la geometría dinámic= a, para gráficas, cálculos, estadística y más necesidades en todos los niveles educativos; como una metodología activa dentro del aula, contemplada en el currículo y los estándares de calidad. En donde el educando sea el principal agente de la educación, como un constructor dinámico, activo y creativo de = los aprendizajes, mientras que el docente necesariamente se capacitará continuamente en las actualizaciones de las TIC y por ende del GeoGebra.

Figura 4

Esquema de la propuesta del uso del GeoGebra en el aula

Fuente: Elaboración propia

A continuación= , se describe sus principales componentes:

Prueba diagnóstica: se inicia el ańo lectivo en la segunda semana con una prueba de diagnóstico sobre el uso de = las TIC en la aplicación de GeoGebra; de esta manera el docente verificará el avance de la tecnología en ańos anteriores y de donde se debe partir para el ańo lectivo en curso.

Planificaciones curriculares: como metodología activa y permanente se realiza la planificación curricular para todo el ańo lectivo; con la utilización del programa GeoGebra incluido en d= icha planificación; para lo cual se propone utilizar una hora pedagógica a la se= mana dentro de la carga horaria obligatoria para cada docente.=

Capacitación a docentes: para poder realizar exitosamente una hora a la semana el programa GeoGebra en la aplicación de los diferentes temas de geometría dinámica; se requiere docen= tes capacitados en la actualización y constante uso del GeoGebra, como herramie= nta activa y dinámica.

Recursos tecnológicos: incrementar en = las instituciones educativas computadoras actualizadas e instaladas el programa GeoGebra Clásico; también un constante mantenimiento con aulas grandes con adecuada ventilación e iluminación, para el uso correcto de los recursos tecnológicos.

Conclusiones.

· = En las prácticas desarrolladas por los educandos investig= ados se ha podido concluir que GeoGebra como software interactivo resultó del ag= rado completamente de los estudiantes, quienes demostraron interés, comunicación= y capacidad para el manejo en las operaciones con vectores y así se comprobó = que GeoGebra se interrelaciona perfectamente con la población motivo de estudio= ; en el cálculo, geometría, álgebra de una manera muy dinámica y libre.

· = En el proceso investigativo se observó la libertad, espontaneidad ,naturalidad y la inclinación de los practicantes en todo el proceso que duró la investigación en el pretest y sobre todo en el postest (escala de Likert); por lo que las variables aplicadas (independiente – GeoGebra clásico, dependiente - Aprendizaje de vectores) se correlacionaron bien con el contexto= ; el programa Editor Estadístico de Datos con sus respectivas tablas, gráficas y análisis de datos y su validación a través del coeficiente de fiabilidad al= fa de Cronbach con valor de 0,739 demuestran resultados favorables . Lo que implica que la metodología aplicada a la investigación con el corte longitudinal en los dos tiempos realizados y a pesar del poco tiempo fue adecuada y por tanto la correcta.

· = Al analizar los resultados de la variable de la tabla 1 <= i>GeoGebra un programa sencillo y fácil, se concluye que un porcentaje muy representativo de estudiantes consideran que GeoGebra es un programa sencillo y de fácil aplicabilidad en un56,7%; sin embargo, se refleja que el porcentaje restante 43,3% todavía tiene dificult= ades en este programa. Hay que hacer notar también que el sector rural donde se llevó a cabo la investigación son estudiantes que en su mayoría no cuentan = con computadoras y no trabajaron antes adecuadamente con el programa y por tant= o el tiempo de práctica de GeoGebra aún no es el adecuado en esta comunidad educativa investigada.

· = Los resultados del trabajo investigativo también reflejan= y confirman el deseo verdadero de aprender con el uso de la tecnología TIC y = la aplicación de vectores utilizando GeoGebra por parte de los la comunidad educativa investigada y por ende una tendencia a los demás temas geométricos dinámicos que convierten a los estudiantes en descubridores de procesos y algoritmos matemáticos divertidos, agradables y exactos de gráficas y cálcu= los.

· = Y por último los resultados indican que la variable del gráfico 1 Aprendió a realizar gráficos con vectores en GeoGebra fue = muy favorable (en el corto plazo) con el 56,7% de estudiantes y un incremento notorio en la media. Pero así mismo se concluye que se requiere mayor aplicabilidad (práctica) para que los resultados sean mayores representativ= os. Entonces se evidenció que GeoGebra aplicado por los estudiantes; lograron demostrar capacidad para ser los propios creadores de sus aprendizajes en operaciones básicas de vectores y verificaron de una manera clara y fácil de reemplazar gráficas complejas (a mano) en gráficas sencillas de realizar en= el software mencionado.

· = Según comparaciones en el grupo focal de la discusión (Es= tado del arte), los dos autores conjuntamente con la presente investigación coinciden en determinar que el educando, el programa GeoGebra como herramie= nta en la enseńanza y aprendizaje de la Geometría y los contenidos matemáticos;= son aprendizajes significativos o se incrementan cuando el estudiante se convie= rte en el principal descubridor y creador de sus propios aprendizajes en la trilogía educativa que forman, ante un programa dinámico, agradable y fácil= de aplicar.

· = Cotejando a los autores (Marisol, Edie, Luisa, &a= mp; Zulma, 2007) con= (Miguel, Francisco, Jesús, & Luis, 2008) se = llaga a una conclusión sencilla pero muy importante como es, que tanto el Software = de Geogebra y el programa de cálculo simbólico(CAS), como Maple entre otros ; = son programas libres muy utilizados por las personas de todas las edades en las intituciones educativas y superiores inclusive. También mencionan la necesidad de tener acceso en cada aula de e= stos nuevos enfoques educativos y tecnológicos gratuitos y efectivos, con gran manipulación y flexibilidad de utilización, e interrelación con la educación superior de carreras sobre todo que requieren cálculos simbólicos e investigaciones matemáticas.

· = Como resultado del proceso de investigación, realizado en= el centro educativo Cumbe, de la Provincia del Azuay, a los estudiantes del primero de Bachil= lerato B ,especialidad Contabilidad, se concluye que GeoGebra funciona perfectamen= te en los temas de la asignatura de matemática, específicamente en todo lo que concierne a vectores y operaciones con vectores, observando gráficas perfec= tas y cálculos analíticos comprobados; y al ser una herramienta tecnológica eficaz y precisa que debe ser usada durante todo el ańo lectivo en las operaciones matemáticas que requieran gráficos , tablas ,cálculos, estadística y demás temas que aplique la geometría dinámi= ca, y en todos los niveles educativos; como una metodología activa dentro del a= ula, que contemple el currículo y los estándares de calidad. Se finaliza mencion= ando entonces que la investigación de campo fue positiva en todos los ángulos descritos anteriormente y que abre la puerta a posteriores investigaciones = que involucren las TIC mediante la aplicación del GeoGebra.

Referencias bibliográficas.

Almada, A. M., Castro, S. F., Navarro, J. A., & Contreras, M. L. (2018). Apr= endizaje Formación. Obtenido de https://n9.cl/wsv3n

Alvarez, C., Cordero, J., Gonzalez, J., & Sepulveda, = O. (2019). Software GeoGebra como herramienta en enseńanza y. Educación y Ciencia, 22, 387–402. https://n9.cl/1sv0e

Analytics, S. (n.d.). El Geogebra: una herramienta tecnológica para aprender Matemática en la Secundaria Básica haciendo matemática. Retrieved May 7, 2022, from https://n9.cl/heou3<= /span>

Arteaga-Valdés, E., Medina Mendieta, J. F., & del Sol Martínez, J. L. (2019). El Geogebra: una herramienta tecnológica para apren= der Matemática en la Secundaria Básica haciendo matemática. Conrado, 15(70), 102–108. https://n9.cl/t0kd7

Castro-Gordón, L. (2013). Mate= mática.

Díaz-Barriga, Á. (2011). <= span lang=3DES style=3D'font-size:12.0pt;line-height:115%;font-family:"Times New= Roman",serif; mso-no-proof:yes'>Competencias en educación corrientes de pensamiento. R= IES Revista Iberoamericana de Educación Superior, II, 3–24. https://n9.cl/10wdb

Duval, R. (1995). Un análisis de los problemas cogniti= vos de comprension en el aprendizaje de las matemáticas. https://n9.cl/x68e= f%0A

Edie, D. C., Luisa, C. C., & Zulma, A. V. (2007). Rev= ista Electrónica "Actualidades. Obtenido de https://n9.cl/v27yu<= /span>

Godino, J. (2011). Indicadores de idoneidad didáctica de procesos de enseńanza y aprendizaje de las matemáticas. XIII Conferęncia Interamericana de Educaçăo Matemática (ciaem-iacme), 1–20. https://n9.cl/7h8ds

Izasa, W. P., Súarez, C. H., & Castro, W. A. (2019). Proposta didática para a aprendiz= agem da derivada com Derive. Obtenido de https://n9.cl/ynem9

Junquera, J. (2008). Vectores. https://n9.cl/qj7mm=

Katz, R. D. (2013). Raúl D. Katz 2013. Vectores. https://n9.cl/tao72

Marisol, C. Á., Edie, D. C., Luisa, C. C., & Zulma, A. V. (2007). Revista Electrónica "Actualidades. Obtenido de https://n9.cl/v27yu

Markus-Hohenwarter, J= .-H. (2009). Documento de Ayuda de GeoGebra. https://n9.cl/44gbr

Ministerio de educación. (2016). Currículo de los nive= les de educación obligatoria. https://n9.cl/u00c2

Miguel, A., Francisco, B., Jesús, E., &a= mp; Luis, T. (2008). La Gaceta= . Obtenido de https://n9.cl/e7nwh4

El artículo que se publica es de exclusiva 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. 4, pp. 42 – 59, octubre – diciembre 2022

www= .exploradordigital.org

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