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Edwards Deming

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e-ISSN: 2576-0971

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Approved: May 14, 2023

Page 15-25

E-activities for the identification and use of

immediate integration formulas.

E-actividades para la identificación y uso de las fórmulas de

integración inmediata

Segundo Bienvenido Camatón Arízabal

Marco Vinicio Añazco Maldonado

Janine Noemí Cueva Palacios

Yefferson Ricardo Litardo Mieles

ABSTRACT

The present research work proposes the theory of

conceptual fields as a means to reach the use and

recognition of the formulas of immediate integration;

because through the development of the four essential

characteristics of the cognitive schemes, implicit in the e-

activities, it is sought to facilitate the teaching-learning

process in this part of Integral Calculus. In this way, a

research with a quasi-experimental design is carried out,

which recognized the veracity of the facts with

correlational hypotheses and a test directed to the

students; confirming that the theory of conceptual fields

does influence the understanding and association of the

recognition and use of the formulas of immediate

integration.

Keywords: Immediate integrals, conceptual fields,

schemes.

RESUMEN

El presente trabajo de investigación propone a la teoría de

campos conceptuales como un medio para llegar al uso y

el reconocimiento de las fórmulas de integración

* Magister en Enseñanza de la Matemática, Universidad de Guayaquil

segundo.camatona@ug.edu.ec https://orcid.org/0000-0001-8327-2869

Magíster en Investigación Operativa, Universidad de Guayaquil

marco.anazcom@ug.edu.ec https://orcid.org/0000-0003-1022-6487

Licenciada en Pedagogía de las Matemáticas y la Física, Unidad Educativa San

Antonio de Padua, janine.cuevap@ug.edu.ec, https://orcid.org/0000-0002-

9235-2489

Licenciado en Pedagogía de las Matemáticas y la Física Unidad Educativa

Comandante Antonio José de Sucre , yefferson.litardom@ug.edu.ec

https://orcid.org/0000-0003-4824-0794

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inmediata; pues a través del desarrollo de las cuatro

características esenciales de los esquemas cognitivos,

implícitas en las e-actividades, se busca facilitar el proceso

de enseñanza-aprendizaje en esta parte de Cálculo Integral.

De esta forma se efectúa una investigación con diseño

cuasi-experimental que reconoció la veracidad de los

hechos con hipótesis correlacionales y un test dirigido a

los estudiantes; confirmando que la teoría de campos

conceptuales si influye en la comprensión y asociación del

reconocimiento y uso de las fórmulas de integración

inmediata.

Palaras clave: Integrales inmediatas, campos

conceptuales, esquemas.

INTRODUCTION

Today's society has evolved in all possible areas, among them we can highlight the school,

where specifically the teaching-learning process has sought the necessary mechanisms

to reach perfection, especially through theories that propose the student as the main

actor of the educational system; thus, changes are observed in the different branches of

science. In spite of the fact that for Mathematics there are always the greatest difficulties,

here the Theory of Conceptual Fields has been highlighted, which pretends that students

come to understand the object of study as a whole, visualizing it through real situations.

This is stated by Pabón et al. (2020) who show that the use of conceptual fields oriented

with other tools and methodological strategies allows the assessment of the student's

formative processes, enabling them to generate a significant context in the analytical and

systematic capacity of understanding Integral Calculus (pp.179).

However, one part is the research, which leads us to listen or read a number of steps

that sound very good together, and a very different one is to propose it in the classroom,

because in the classroom this theory is very little applied, as Mateus-Nieves (2021) says,

(Scagnoli, 2006)(Scagnoli, 2006), who mentions that "the lack of a deep conceptual

teaching can be evidenced, due to the fact that the most used approach is the mechanical

resolution".

del Mastro & Monereo, (2014). All this has repercussions on the student's learning, who,

not internalizing what is being executed, will present difficulties during their studies to

reach the Third level, even more so in the moments when they must implement them

in everyday life situations.

Definitely, the object of study must be internalized regardless of the profession to be

exercised in the future, that is, it should not matter whether the trainee is going to

become an engineer or a teacher; since, if we speak specifically of the integration

formulas, both must have mastery of them, in the case of the engineer for its execution

during the calculation of irregular areas or a teacher to exemplify to his students how

to implement it.

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To this end (Romero, 2016), (Capraro et al., 2010), (Caballero-Montañez & Sime-Poma,

2006)The updating of knowledge on the part of the teacher is indispensable, since this

way he can be informed of new methods, activities, strategies, teaching techniques that

facilitate or allow the improvement of learning.

From this point of view, the problem that arises can be described from two perspectives:

on the one hand, the analysis of the theory of conceptual fields in the use and

identification of the formulas of immediate integration in the future mathematics

teachers of the pedagogy career of experimental sciences of Mathematics and Physics

belonging to the University of Guayaquil and on the other hand, the scarce relevant

information on the methodology of conceptual fields that motivates the use of such

methodology in the use and identification of formulas of immediate integrals, not only

as a concept but as a complex cognitive scheme (conceptual-procedural).

MATERIALS AND METHODS

In order to verify if the Theory of Conceptual Fields helps students to use and implement

the formulas of immediate integration, a correlational research with quasi-experimental

design was carried out, for which we proceeded to form the research groups with the

students of the career of pedagogy of experimental sciences of Mathematics and Physics

at the University of Guayaquil Cycle I 2022-2023 and being defined the parallels 4-5 A2

as experimental group with a total of 19 members; while courses 4-6 C2 were

designated as the control group, with a total of 22 students.

Thus, it is defined that the experimental group will implement the instructional sequence

designed with e-activities, while the control group will only be taught a regular class, as

shown in Table 1.

Table 1. Variables Scheme

Experimental group

Control group

---

x: Instructional sequence with e-activities based on CBT on the immediate

integration formula with powers

: Successful identification and use of the formula.

: Success in identifying and using the formula, which to this group the

instructional sequence will not be applied.

Therefore, to collect the data, a test was applied to each research group, which was

composed of a questionnaire of 10 questions, valued at one point each; these questions

were based on what was taught through the e-activities and the regular class,

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respectively. The data collected were analyzed with the help of descriptive statistics,

thus giving the study a quantitative approach.

It is worth mentioning that the e-activities were planned and developed based on the

information found through the literature review; in addition, the structure and platforms

used were incorporated based on this research. Also, the relevant stages of the

Conceptual Fields Theory were defined.

The following is a description of the e-activities developed for the experimental group,

which allow students to manipulate the object of study and thus achieve the use and

application of the immediate integration formulas:

Subject

Integral Calculus Course

Unit(s)

Unit 1. Immediate Integrals

Topics covered

- Algebraic Immediate Integrals

Expected learning. Objective

Apply the algebraic immediate integration formula that

relates power to the properties of integrals to solve

specific exercises.

Duration of the

unit:

2 hours

Modality:

Online

Stage

1. Learning situation

Duration

20 minutes

Type

Individual

Display instruction

The frequent use of the calculus of a variable is based fundamentally on the problem of

calculating the area enclosed by the graph defined by a function.

It is often used in economics through statistical analyses, which are closely linked to

functions that have powers within their composition.

If it is stated that integration is the inverse operation of derivation, how could the

following integral be solved

∫

𝒙

𝒏

𝒅𝒙 ?

Describe at least 2 steps that you consider necessary to solve the integral with their

justification.

Cross-cutting artifacts

Type

Instructions for use

Padlet or Forum

Students should access the following link to access

the Padlet (an interaction wall)

https://padlet.com/jannyest30/sw5shf8k00ye20h4 , in

the case of teachers who have a virtual classroom,

they can do the same through a forum.

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Stage

2. Context of the problem

Duration:

25 minutes

Type

Group

Instruction screen

It is important to keep in mind that to solve this type of exercises we must not only

calculate its primitive, but also analyze the integral; which allows us to apply the most

appropriate method to solve. Therefore, we can analyze what type of integral it belongs

to in order to solve it, through the following steps:

1. Check if it is immediate integral

2. Check if it is almost immediate

3. Check that the exponent is a number and not the same variable

4. Check that the differential refers to the same variable of the function.

Cross-cutting artifacts

Type

Instructions for use

Google Sites

Celeriti Game

It is a blog created with the information that the

student needs, where they will have the availability of

a previous game to remember the previous concepts

seen in the previous section, this web site is available

at the following here

For the game, students should go to the following link

https://www.cerebriti.com/juegos-de-

matematicas/generalidades-de-calculo-integral and

complete the activity.

Stage

3. Operational Invariants

Duration:

25 minutes

Type

Individual

Display instruction

To start solving the exercise, it is vital to recognize the terrain first.

So we must list the elements, properties, concepts that we will use to solve it.

Therefore, we will exercise our knowledge trying to decipher which is the rule that the

algebraic immediate integrals with powers comply with, which will be done through the

following activity by accessing the following link https://es.educaplay.com/recursos-

educativos/12685180-integrales_inmediatas_algebraicas.html

Transversal artifacts

Type

Instructions for use

Educaplay

The student must enter the game with the link, which

will be provided by the teacher and the student will

enter to find the formula that allows us to solve

algebraic immediate integrals.

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Stage

4. Feedback and inferences

Duration

30 minutes

Type

Individual and Group

Instruction screen

Once the activity has been completed by the students, we now proceed to provide

feedback on the concepts we need and use to identify and use algebraic immediate

integrals with powers.

And at the same time, students exercise the activity a little by solving 5 simple exercises,

the activity can be found at the following link https://es.liveworksheets.com/pd3141121tu,

where it is important that the teacher guides the feedback and if doubts arise, they can

be solved at the same time.

Cross-cutting artifacts

Type

Instructions for Use

Liveworksheets worksheet

The activity will always be available which is

automatically qualified for the generation of

contributions, for teachers who need it just

enter the link and can perform the activity

without the need of an account.

Stage

5. Evaluation

Duration

20 minutes

Type

Individual

Instruction Screen

Once the whole learning process is finished, it is important to evaluate, so a google form

with four answer options will be made, where students identify and use the formulas of

immediate integration with powers, which are self-graded.

Cross-cutting artifacts

Type

Instructions for use

Google Forms

The student enters the link to the evaluation

and performs it, he/she must choose the

correct answer in each item. If the teacher

cannot access the google form, he/she can

take the questions and run it on another

platform known to him/her.

Based on the objective of this quasi-experimental study, it is estimated that:

● Null hypothesis (H0): The application of an instructional sequence with e-activities

based on CBT does not influence the successful identification and use of the

immediate integration formula with powers.

Alternative hypothesis (H1): The application of an instructional sequence with e-

activities based on CBT does influence the successful identification and use of the

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immediate integration formula with powers.

RESULTS

The results of correct answers are detailed, in percentages, according to the Group in

which the test was applied.

Graph 1. Correct Answers

It can be observed that the Experimental Group (EG) outperforms the Control Group

(CG) in most of the questions, i.e., the students who made up the EG got to know the

steps to develop the power integrals, as well as to assimilate that only constants and not

variables can be added at the time of solving and especially to identify in which exercises

this immediate formula can be applied. Although, the CG was able to memorize the

formula; the GE was highlighted during its application in proposed activities. Since the

test has been designed for the first time, it is necessary to submit it to a statistical test

to verify its viability, so it was determined through the calculation of Cronbach's Alpha,

giving a result of 0.77 and with this we can affirm that the evaluation instrument is

Excellently reliable according to the table referring to this coefficient.

Analysis of the test results by means of the T-test

In order to perform the Student's t-test, the normality and the existence of equal

variances in the data must be verified, which are detailed below:

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Normality of the data Experimental and Control Group

Graph 2. Normality of the data Experimental Group

Graph N° 3. Normality of data Control Group

Once the Ryan-Joiner test was performed, the results collected were found to be

normal, since both groups obtained a value greater than 0.1.

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Equal Variances

Table 2. Equality of Variances Test

Null hypothesis

H₀:%σ₁%/%σ₂%= 1

Alternative hypothesis

H₁:%σ₁%/%σ₂%≠ 1

Significance level

α = 0.05

Method

Test statistics

GL1

GL2

p-value

0.96

0.131

Consequently, the F test is run where the p value = 0.131, and since it corresponds to

the premise of p≥0.05, the null hypothesis, which states that the variances of both groups

are statistically equal, can be accepted.

T-test

Finally, once the normality of the data and the equality of variances were confirmed, the

T-test was run, for which the arithmetic means of both groups were required, by means

of which a gain of 20% can be observed in the experimental group, which obtained 8.37

in the average. Table 3.

Table 3. Arithmetic Mean of the Evaluated Groups

Sample

Media

Std. dev.

Standard

error

of the

mean

6,55

2,86

0,61

8,37

1,54

0,35

Thus, with the results obtained and observed in Table 4, the null hypothesis is rejected

and it is affirmed that there is a significant difference between the group that received

the treatment and the group that had a normal intervention.

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Table 4. Student's t-test

Null hypothesis

H₀:%µ₁%- µ₂%= 0.

Alternative

hypothesis

H₁:%µ₁%- µ₂%≠ 0

T-value

p-value

2,59

0,007

DISCUSSION

After conducting a bibliographic and documentary research on the theory of conceptual

fields, its elements and possible applications in the field of mathematics, it has been

decided that the use of cognitive schemes to strengthen mathematical skills such as the

recognition and use of properties and formulas in the area of integral calculus is optimal

because it allows the teacher to promote a learning environment where the student

actively participates in the change of schemes that allow him to execute the algorithms

of resolution for various exercises through the linking of their previous knowledge with

those that eventually adapts in class.

Subsequently, for the application of the conceptual fields theory, this research work was

based on four essential characteristics of the cognitive schemas: goals, subgoals and

anticipations; rules of action, information gathering and control; operative invariants and

inferences, characteristics that allowed the student to identify the mathematical

concepts about the immediate integral of the power, which allow not only to memorize

the characteristics of the exercise but also to infer where the mathematical property to

be used in the algorithm for solving the integral comes from.

At the end of the application of different statistical tests on the data obtained, it is

necessary to mention that the evaluation test went through a validation process through

the statistical test called Cronbach's Alpha, where it obtained a value of 0.77 giving the

instrument a level of reliability "Excellent". In addition, it was established that the group

to which the treatment of the instructional sequence with e-activities based on the

theory of conceptual fields was applied had a higher mean than the control group, which

had a regular class on the subject.

Analyzing the data through the Ryan-Joiner statistic, the value p>0.1 could confirm the

normality of these, therefore it was necessary to calculate the equality of variances

through the F test where p=0.131 and with which the equality of variances can be

established. Once these results were obtained, it was possible to calculate the Student's

t-test for two independent samples taking p=0.007, which being lower than the reference

value, the alternative hypothesis of the research is accepted. Concluding that the theory

of conceptual fields does influence the understanding and association of the recognition

and use of immediate integration formulas.

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