mgonzalezAnonymous202019-10-29T17:36:00Z2019-10-29T17:36:00Z2019-10-29T17:36:00Z17487126793223633160115

The transversal treatment of the calculation of combined operations in primary education

El tratamiento transversal del cálculo de operaciones combinadas en la educación primaria

Olga Lidia Torres-Gamboa

Academia de las Artes Vicentina de la Torre, Cuba

Karel Pérez-Ariza

Universidad Ignacio Agramonte Loynaz, Camagüey, Cuba

José Julián García-Muñoz

Universidad Central Marta Abreu, Villa Clara, CubaCorreo electrónico(s):

olga.torres@reduc.edu.cukarel.perez@reduc.edu.cu

josejulian@uclv.cuReceived: 14 de marzo de 2018Accepted: 5 de junio de 2018

Abstract: The treatment of the calculation of combined operations occupies a central place in primary education since it allows to systematize important conceptual and procedural contents of Mathematics. The analysis of documents and the products of the teaching activity reveal inadequacies in the learning of the operational order and in the development of calculation skills. Consequently, the purpose of the article is to offer theoretical-methodological considerations for the transversal treatment of the calculation of combined operations in the aforementioned subject. The use of theoretical analysis-synthesis and historical-logical methods was useful for research

EduSol

Vol. 19. Núm.6

ISSN: 1729-8091

Publicada en línea: octubre de 2019 (http://edusol.cug.co.cu)iVBORw0KGgoAAAANSUhEUgAAAboAAACOCAYAAABHa82KAAAAAXNSR0IArs4c6QAAAARnQU1BAACx jwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAAAAlwSFlz AAAh1QAAIdUBBJy0nQAAQr9JREFUeF7tfQmYXEXVdieZnpksoIBASHqSsGg0CuqHy4eKfKgooiDw IyibIPALIhi2DwRlEdmVRWRHUDYBk8lkAREEBGRHCWHf9zUQErbsk+99q6t7qm/Xvbfq9r29zZnn 6ad7uuvW8tby1jl1zqnc9Hz+6OtyuVVz8heKwFW53PBp+fwxN+dyowQmQUAQEAQEgRZDoC+ffwKL +Mt9nZ3brMjlhrRY9TOv7syOjo2Bz8PAZ/G1udzKmRcoBQgCgoAgIAikiwCJDov4Crz6saBPndLd PT7dElozN5Da6sDkQryWaXyE6FqzK6XWgoAgMNgRMIiOZLdiWmfnO72dnb+gum4wYnNfLpfv7ej4 GXCYqwlO4SIS3WAcDdJmQUAQaAsEgkRXWtwh3T2N1w6DRZ3JdqLtW2k1ZYnczHeR6NpixEsjBAFB YNAhEEZ0hjRzz/Surs3blfDYrmldXV9He28PSHBBshOiG3SzQxosCAgCbYGAA9GVFnwSwXeuzuWG tUPD2Q6oJ7fA67YYghPVZTt0uLRBEBAEBi8CHkRXOsN7AM/sPiOXG9GKqP09lxsJ9eSeILgHHAlO iK4VO1rqLAgIAoJACQFfojPO8N7As6fAcOUTrYAmyG393nz+NLwHjUxs53FyRtcKnSp1FAQEAUHA BYGkRGdIQ8vx+U6cc+3XN3z4GJcy65WGrhKQ3A5C/e7De7+nBCdndPXqKClHEBAEBIEsEQABHErT 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55Keywords: Arithmetic calculation; Calculation of combined operations; Transversality; Primary education; Mathematics teaching-learning.

Resumen

El tratamiento del cálculo de operaciones combinadas ocupa un lugar central en la educación primaria ya que permite sistematizar importantes contenidos conceptuales y procedimentales de la Matemática. El análisis de documentos y de los productos de la actividad docente revela insuficiencias en el aprendizaje del orden operacional y en el desarrollo de habilidades de cálculo. Consecuentemente, el propósito del artículo reside en ofrecer consideraciones teórico-metodológicas para el tratamiento transversal del cálculo de operaciones combinadas en la referida asignatura. Para la investigación fue de utilidad el empleo de los métodos teóricos análisis-síntesis e histórico-lógico.

Palabras clave:

Cálculo aritmético; Cálculo de operaciones combinadas; Transversalidad; Educación primaria; Enseñanza-aprendizaje de la Matemática

Revista EduSol

Volumen 16. Número 57.

ISSN: 1729-8091

Publicada en línea:11 de enero de 2017 (http://edusol.cug.co.cu)

PAGE \* MERGEFORMAT2

Introduction

Mathematics is within the study program of the Cuban school, its content favors the development of the ability to calculate combined operations. Upon completion of primary education students must have developed skills in the calculation with natural and fractional numbers, safety and

speed in the calculation of any type of operations including combined operations in any form of representation.

There are several investigations around the teaching and learning of calculus in primary education. The ones related to oral or mental calculus stand out among them (Geissler, Sieber, Starke and Wolf, 1975; MINED, 1987; Bernabeu, 2005; Albarrán, 2007a); writing (Villalón, Peña, León, Garea, Bello and Varela et al, 2001; Rodríguez, 2005; Albarrán, 2007b and 2007c; Rojas, 2012); with magnitude data (GeisslerSieber, Starke and Wolf, 1975; Villalón, Marrero and Toulon, 1984; González, Santiesteban and Piñeiro, 2013); and with combined operations (García, 2000; León, Villalón, Peña, Bello, Rizo, García et al, 2001; Torres, 2010; Torres, Pérez and García, 2017).

Because of the importance of understanding the term combined operations in the article, we conceptualize it as those where several arithmetic operations are combined following the established operational order. Based on the ideas assumed about the ability to calculate and the concept of combined operations, we define the ability to calculate combined operations as the system of theoretical and practical actions aimed at consciously seeking a quantitative result from the use of the calculation operations involved in the exercise.

The calculation of exercises with combined operations is an effective way to systematize various arithmetic contents (mastery of concepts, laws, properties and procedures). In addition, it favors the formation and development of a scientific thought because it allows the use of heuristic procedures with a generalized character and their system approach; and establish part-all relationships, analogy, among others.

Through the solution of exercises, the development of combinatorial thinking continues and the work begins with the conjunctist language and symbology, which constitutes the preparatory phase of the work that will be carried out in this sense in later grades. The major weight must be assumed by the work for the development of calculation exercises with arithmetic operations with natural numbers. Particular attention is paid to the order in which these operations are performed when they appear combined.

When reviewing the textbooks of primary education, the insufficient amount, and little variety and complexity of the exercises with combined operations, are observed. For example, 40 exercises with combined operations appear in the sixth grade textbook, which is insufficient to develop this skill. On the other hand, none of the exercises combine all the calculation operations that are worked in that educational level, and do not adjust to the aspirated difficulty levels.

In the operations of the quality of education and the provincial groups of the subject, difficulties have been detected in the learning of calculus with combined operations in primary education. Among them are: a) a poor application of the operational order, and b) an insufficient systematization of calculation skills, with emphasis when the numbers are fractional and expressed in different notation systems. Consequently, theoretical-methodological considerations are offered for the transversal treatment of the calculation of combined operations in primary education.

Development

Transversality in the teaching-learning process

Transversality has been an aspect tackled strongly in the educational field during the last decades. According to Ruiz (2003), this has been conditioned, among other factors, by the following needs: a) to include in the curricula the results of the vertiginous development of science, b) to attend to the social and individual needs of the process of development of new generations, and c) avoid excess information in the curricula to avoid curriculum overload.

The analysis of specialized literature (Klingberg, 1978; Muñoz, 1997; Rico, Bonet, Castillo, García, Martín, Rizo et al, 2000; Ruiz, 2003; Redon, 2007 and Fiallo, 2012), reveals that studies on transversality and its conception in curricular designs point to two essential directions: a) mainstreaming at a particular level (subject), and b) mainstreaming at a general level (curriculum). From there the applicability of this idea can be inferred at the macro, meso and micro levels of concretion of the curricular design (Páez, Miranda, Addine and Fernández, 2013).

Transversality at a particular level has been revealed from the consideration of the so-called “guidelines” in the curriculum. They were introduced in the country from the 70s, mainly due to

the contributions of the German didact Lothar Klingberg. However, it is worthy to recognize that in the Mathematics programs of measured education (basic and higher), it has been where with more strength and systematicity it has been treated, highlighting in that sense the contributions of: Ballester, Santana, Hernández, Cruz, Arango, García et al.

(1992); Ballester, Quintana, Fernández, Báez, Santana and Rodríguez (2002); Álvarez, Almeida and Villegas (2014).

In its general sense, it has been tackled - at least with more force - from the transformations that occurred in Cuban education at the end of the twentieth century and the beginning of the twenty-first. At this stage, the so-called transverse axes are conceived as part of the Generic Model of the Cuban School, which are considered as general objectives that are achieved, with the required depth and mastery, only from the content of the different subjects (Horruitiner, 2006; Fiallo, 2012). Hence, most of them respond to the characteristics of the so-called educational or training content, which are aimed primarily at the formation of feelings, attitudes, rules of conduct, etc. (Chávez and Pérez, 2015). In higher education these transversal axes are known as curricular strategies (Horruitiner, 2006).

Among the most common cross-cutting axes currently at all Cuban educational levels stand out: patriotic and citizen education; education for health, sexuality, and family life; environmental education; economic education

.The exposed so far allows to reveal the value of the transversal conception of the curriculum in its different levels of concretion. Therefore, for the direction of the teaching - learning process at any educational level, those axes or themes that penetrate the different levels of organization of the curriculum must be considered: subject, discipline, year or grade, cycle and level or career. Taking into account the hierarchical relationship of the different levels referred to, it is necessary to clarify that the axes identified in the upper levels must cross the curriculum of the lower ones, but in the lower ones new axes can be identified - which we call secondary -, which do not necessarily have to cross The curriculum of the superiors.

For example:

Environmental education, as a transversal axis in the education of primary school children, must go through the curriculum of all grades, developmental moments, and level.

The treatment of mathematical problems (in a broad sense), as a transversal axis of the Mathematical subject, does not have to go through the curriculum of the degree, moment of development, and level.

In this way, the transversality of the curriculum is projected and executed in a bidirectional way, surpassing the unidirectional conception that currently prevails in the conception of the curricula and the direction of the teaching-learning process. This, in turn, would allow the transversal conception of the curriculum to favor the transit from intradisciplinarity to transdisciplinarity of the contents taught and the systematization of knowledge in a more effective way.

The treatment of the calculation of combined operations

The arithmetic calculation belongs to the Mathematics knowledge system, so it implies the development of specific skills (Simeón, León, Rodríguez, Pulgarón, Brito, Bermúdez et al., 1991; León, Villalón, Peña, Bello, Rizo, García et al., 2001). Its learning process has a significant character since it prepares the subject to face the minimum demands of daily life that arise as a result of the accelerated development of science and technology. On the other hand, it allows us to develop ways of thinking by systematizing conceptual and procedural contents of the subject, becoming not only an exercise in the application of knowledge but also in a mathematical skill of vital importance.

To calculate implies to carry out a set of calculation operations to characterize the object of study (García, 2000; Rico et al., 2001). Torres (2010) states that calculating is an existential form of an algorithm that can be carried out manually, verbally (orally or in writing), mentally and through the use of tables, calculators or computers. That author herself maintains that in order to develop that ability, it is necessary to master the following operations: a) identification of the calculation objects, b) characterization of the calculation objects, c) determination of the operation or operations that are established between the objects of the calculation, d) establishment of the algorithm representing the operation or operations, e) check the result of the algorithm.

To achieve the sufficient development of the calculation skill it is necessary that from the first grade it is systematically addressed. In the same way it is advisable to use the potential offered by the different subject complexes for their treatment, since it is generally done only from the

calculation units or the exercise and consolidation of contents. Here are some considerations in this regard.

In the first grade, the teacher tells the students that in everyday life we do everything in an order: we get up to go to school, we arrive, we return from it, we bathe, so on. That is a good opportunity to explain that in the Mathematics subject they learned to add and subtract, and now they will learn to combine these two calculation operations.

Example of exercise:

Calculate. Write the numeral.

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Thus, the difficulties are gradually graduating because, although it is not a priority objective of the degree, the foundations are created from the first degree for the development of the referred ability.

In second grade, after they have acquired the contents on the addition and subtraction of natural numbers from one place and two places, without and with excess, the teacher develops in his student’s skills in the combined operations, even if it is not objective of the degree. Then the calculation operations are increased, according to the program, the multiplication and division by three, four, five, one, zero, six, seven, eight and nine. Once the four basic operations are known by schoolchildren, exercises with combined operations - first with two and then with more operations - should be inserted for the consolidation of the arithmetic calculation. For this it is essential to explain the procedure for calculating this type of exercise, based on the operational order; In addition to instilling the habit of working with cleanliness and organization.

The third grader must have a notion about the exercises with simple combined operations, even if it is not the objective of the grade. Students know in this grade the natural numbers up to 10,000 and the written calculation procedures with these numbers, therefore, they are able to apply this content in the calculation of combined operations.

In fourth grade, students expand their knowledge about natural numbers and perform written calculation exercises with greater complexities, which favors the possibilities for working with combined operations. In this degree the empowerment and the radication are not worked, but the powers of 10 and their expression in exponential notation are learned. This makes it possible to calculate exercises such as the following:

Calculate. Consider the operational order.

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Students have not worked with empowerment, but when they decompose as a sum of power multiple of 10, they know that 10

is 10. 10. Likewise, it happens with any power of multiple 10. In this grade the following operational order must be generalized: 1) Operations within parentheses, 2) Multiplication and division in the order that appears, 3) Addition and subtraction in the order that appears; but if the teacher has worked the decomposition of sum as a multiple of powers of 10, it is explained that after parentheses, empowerment corresponds and that in the fifth grade it will deepen it.

In fifth grade, work continues with known calculation operations, expanding the level of complexity in the multiplication and division of natural numbers. The calculation of powers and roots is increased, and the work with fractional numbers expressed in decimal notation or as a fraction. In this grade the operational order is extended to calculate combined operations, being the following: 1) Operations within parentheses, 2) Empowerment and filing in the order that they appear, 3) Multiplication and division in the order in which they appear, and 4) Addition and subtraction in the order in which they appear.

Once the work with the natural numbers is finished, unit 2: Numerical fractions is introduced. Calculations with fractions. This work with fractions will help to extend the calculation with natural numbers, since emphasis will be placed on the decimal expressions and the rules for calculating with them, which are the same for the natural ones but taking into account the comma. Then they work with common operations and with decimal fractions that are: a) Addition and subtraction with fractions of equal and different denominator; b) Addition, subtraction and

multiplication of decimal expressions; and c) Multiplication and division by the unit followed by zero. Subsequently, the combined operations are worked.

We consider that one aspect that affects the teaching process - learning the calculation of exercises with combined operations is the outdated current textbooks. The exercises that appear in them have a lower degree of difficulty than those that are required for evaluation purposes in the national educational quality operations and other evaluation systems used in Cuba. From there arises the need for the elaboration, by the teacher, of more complex exercises, which complement those contained in the textbooks.

Proposal of activities for sixth grade

1- Mathematical riddle

Objective. - Calculate combined exercises with mental speed.

The activity is oriented to the students to ensure the preconditions or for the motivation of the class. The teacher explains that when they express each order they have to execute until they reach the end of the exercise. When the student concludes, each student will obtain the result, at that moment the individual activity is controlled.

Synopsis: this activity is used to ensure preconditions or motivate schoolchildren in classes dedicated to the calculation of combined operations.

Think of a number that is less than 10 and that is not zero.

Add 29.

Remove the first decimal order from the sum.

Multiplythenumberby 10.

Add four to the product.

Multiply the obtained by three.

Subtract two from the product.

If they did not make calculation or procedural errors the number is __

2- Objective. - Memorize the operational order.

The teacher orients that the student has to memorize the order, then apply it in the order according to the operational order. The student executes and the teacher controls when they mark correct alternatives in the notebooks

Activity. - Sort according to the order in which they are made when combined.

__ Potency and roots according to the order in which they appear.

__ Operations within parentheses.

__ The products and the quotients in the order that they appear.

__ The sum and difference in the order in which they appear.

Mark the right alternative

____ BADC 3- ____ BDAC

____ CDAB 4- ____ ABCD

3- Title: Game “The Heel”

Objective. - Calculate combined exercises through the game The heel, demonstrating speed and safety in the calculation.

Synopsis: to develop the activity, an area corresponding to the number of schoolchildren is sought. Ten rectangles are drawn in advance and in each one a number is written (from one to ten). Listed cards are placed, in correspondence with the numbers of the rectangles, on a table with the activities to be carried out. The realization of the game and its rules are oriented. A dice that must fall into the first rectangle is rolled, then the number one card is read. If you answer correctly, you will have the opportunity to start jumping in the rectangles and so on with all the cards.

The formation of the teams will be done in correspondence with the diagnosis of the schoolchildren, appointing a boss for each one. It is important to clarify that each card of the game depends on the corresponding actions of each according to the rules of the game, then the game is developed. In this activity, actions such as the application of the procedure required by

the game must be carried out with emphasis on the solution of each individual activity presented on each card.

The teacher must provide during the course of this stage a different attention to the students taking into account their potential. The first four activities will be done in written form, which is why different forms of control are used (oral, review of writings, and self-control). During the activity, the relations between schoolchildren are stressed and encouraged, creating a climate of trust and security.

Tarjeta número 1: ¿Te gustaría calcular operaciones combinadas jugando? ¿Por qué?

Card number 2: True or false. If it is false, make it true.

___ Add and rest in the first operational order.

___ Parenthesis is the first one made.

___ I multiply and divide in the order they appear, as a third party.

___ Roots and power in the order they appear, as second.

Card number 3: Calculate. Consider the operational order.

15 × 24 ÷ 15 + 856 – 92

Card number 4: Calculate. Leave the calculations in writing.

312 – (16 + 8) × 15 ÷ 12

Card number 5: Calculate. Remember the operational order.

7, 52 + (65 – 38) ÷ 100

Card number 6: Calculate. Leave all calculations in writing.

74 + 44 × (3/2 + 1/4) – 132

Card number 7: Calculate. Then explain the procedure used.

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Card number 8: Calculate.

Place the sign (+, -, x, ÷) punctuation and parenthesis as appropriate, the result of which is 46. 192; 4; 0,4; 3,6; 2/7

Card number 9: Calculate. Verbalize the procedure.

4/8 + 2/10 ÷ 25 × 103 – 9,5

Card number 10: Calculate. Explain what you did.

4- Title: Who calculates faster?

Objective. - Calculate exercises with combined operations demonstrating speed and safety in the calculation.

The teacher guides the activity and insists that when calculating exercises with combined operations they must apply the operational system. Once the doubts are clarified, the student executes the different activities. It is controlled individually and collectively.

Synopsis: the teacher divides the detachment into teams (must not exceed six schoolchildren) in the area of physical education. There, the hare and the turtle are waiting for you, who are discussing which of the two is more agile. Each team will have a path in which equal distances will be marked and each player will be able to advance one of the delimited sections (they will be determined in correspondence with the distance of the land where it is played, but they must not exceed the amount of three) for each exercise that calculates correctly. The first of the teams that will get all its members to finish the race will win.

Calculate as quickly as possible.

a) 6941 – 899 + 57590 ÷ 26 × 15

b) 8576 – 54 + 964488b ÷ 12 × 13

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d) 45% de 900 + 83420 ÷ 86

5- I know how to calculate

Objective, - Calculate combined exercises with the help of family members and the community.

The teacher guides an Olympiad with exercises with combined operations through a worksheet where parents, family members and people from the community participate. The activity is executed when the student, together with their family members and people in the community, begin to calculate it. Then the worksheets are collected, and controlled individually and collectively.

Synopsis: calculation Olympiad with exercises with combined operations. The teacher guides a worksheet with exercises to solve with the help of parents, family members and people in the community. The student who has had more participation and responded correctly will be the winner.

a) 5678 + 437 × 205 ÷ 50 – 9

b) 2/5 – 5/16 × 4/49 + 12

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d) 15% de 640 + 9

÷ 3/4

e) 70dm + 5

cm – 7cm × 5cm

6- Calculate

Objective. - Calculate combined operations exercises quickly and safely.

The teacher guides the activity and insists that when calculating exercises with combined operations, the operational system must be applied. Once the doubts are clarified, the student executes the different activities. It is controlled individually and collectively.

Synopsis: consists of performing exercises in a time of 30 minutes. The student who has done more exercises will be the winner.

a) 6

(215 + 9003) ÷ 4

b) 20% de 25 + 4

/9 ÷ 8/3 - 5

c) 18hL + 210cL × 5kL

d) Be A = 8705 – 497 B = 562 ÷ 24 C = 78,9 – 3, 16

- Calculate A + B + C

- The average of A + C es:

____ 417 ___ 8207 ___ 1 ___ You cannot know.

Form the combined operation and calculate.

e) Find the quotient of 57 590 and 26. Then find the product of 201 and then the cube root of 729.

7- Decide which one will be correct. Argues.

Objective. - Calculate combined exercises demonstrating speed and safety.

The teacher guides the activity and remembers that when calculating exercises with combined operations they must apply the operational system. Once the doubts are clarified, the student executes the different activities. It is controlled individually and collectively, and notebooks are exchanged.

Synopsis: the student must develop the exercise to know which the correct alternative is

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9103 – 2,4 + 2,6 × 6

9103 – 2,4 + 156

9068 + 156

= 922,4

A) __922,4 B) __ 909,82 C) __9098,2 D) __ 990,28

8- I create and combine.

Objective. - Prepare a combined exercise where you use all calculation operations: sum, empowerment, root and parentheses.

The teacher orients that an exercise with combined operations will be developed complying with the operational system. Once the doubts are clarified, the student executes the calculation. It is controlled individually and collectively.

Synopsis: the student must prepare an exercise in compliance with the order and calculate it.

Make a combination where you have all known calculation operations.

9- Calculate.

Objective. - Calculate different exercises combined from exercises with texts and problems.

The teacher guides exercises with texts and problems where the student has to think to form the exercise with combined operations. It is calculated and controlled on the board since it is a complete exercise.

1-Render the product of 54 tenths and the square root of 36 to 4384/100.

2-Find the quotient of 31758 and 158. Set 3/4 and then, the square root of 36.

3. My neighbor earned 15 pesos a day for 24 work days. Then he spent 32 pesos on the purchase of products, and then earned 12 pesos on another work day. Calculate the amount of money my neighbor has.

Conclusions

The development of the ability to calculate combined operations is an essential objective in the learning of Mathematics at the primary level, so it is vitally important to treat them from the first grades and maximize the potential of the different subject complexes for their systematization by schoolchildren.

The transversal treatment of the calculation of combined operations in the Mathematical subject favors the mastery, by schoolchildren, of the functional invariants of the aforementioned ability, and solidity in learning those contents.

The proposed activities constitute an effective tool for the development of the ability to calculate combined operations since they are varied and respond to the functional invariants of the referred ability.

Bibliographic references

Albarrán, J. (2007a).

¿Cómo realizar el tratamiento del cálculo mental?

La Habana: Pueblo y Educación.

Albarrán, J. (2007b).

¿Cómo realizar el tratamiento de los procedimientos escritos de adición, sustracción y multiplicación con números naturales?

La Habana: Pueblo y Educación.

Albarrán, J. (2007c).

¿Cómo realizar el tratamiento del procedimiento escrito de la división con números naturales?

La Habana: Pueblo y Educación.

Álvarez, M., Almeida, B., y Villegas, E. (2014).

El proceso de enseñanza-aprendizaje de la Matemática. Documentos metodológicos.

La Habana: Pueblo y Educación.

Ballester, S., Santana, H., Hernández, S., Cruz, I., Arango, C., García, M. et al. (1992).

Metodología de la enseñanza de la Matemática.

Parte 1

La Habana: Pueblo y Educación.

Ballester, S. Quintana, A., Fernández, J., Báez, L., Santana, H., Rodríguez, M. et al. (2002).

El transcurso de las líneas directrices en los programas de Matemática y la planificación de la enseñanza.

La Habana: Pueblo y Educación.

Bernabeu, M. (2005).

Una concepción didáctica para el aprendizaje del cálculo aritmético en el primer ciclo.

(Tesis de doctorado). Instituto Central de Ciencias Pedagógicas, La Habana.

Chávez, J., y Pérez, L. (2015).

Fundamentos de Pedagogía General.

Parte 1

La Habana: Pueblo y Educación.

Fiallo, J. (2012).

¿Cómo formar un pensamiento interdisciplinario desde la escuela?

La Habana: Pueblo y Educación.

García, J. J. (2000).

El cálculo de operaciones combinadas en la educación primaria.

(Tesis de doctorado). Instituto Superior Pedagógico “Félix Varela”, Villa Clara.

Geissler, E., Sieber, J., Starke, H., y Wolf, A. (1975).

Metodología de la enseñanza de la Matemática de 1ro. a 4to. grado

. Parte 3

La Habana: Pueblo y Educación.

González, R., Santiesteban, E., y Piñeiro, E. (2013). El cálculo con cantidades de magnitud en el aprendizaje desarrollador en los escolares primarios.

IPLAC5

, 219 – 229.

Horruitiner, P. (2006).

La universidad cubana: el modelo de formación.

La Habana: Félix Varela.

Klingberg, L. (1978).

Introducción a la Didáctica General.

La Habana: Pueblo y Educación.

León, N., Villalón, M., Peña, R. L., Bello, M., Rizo, C., García, I. et al. (2001).

Orientaciones Metodológicas. Cuarto Grado. Matemática.

La Habana: Pueblo y Educación.

Ministerio de Educación (1987).

Indicaciones a los maestros para lograr habilidades de cálculo.

La Habana: Empresa Gráfica del Ministerio de Educación.

Muñoz, A. (1997). Los temas transversales del currículo educativo actual.

Revista Complutense de Educación, 8

(2), 161 – 173.

Páez, V., Miranda, T., Addine, F., y Fernández, A. (2013).

Currículo y contexto educativo.

La Habana: Pueblo y Educación.

Redon, S. (2007). Significados de la transversalidad en el currículum: Estudio de Caso.

Revista Iberoamericana de Educación, 43

(2), 1 – 14.

Rico, P., Bonet, M., Castillo, S., García, M., Martín, V., Rizo, C. et al. (2000).

Hacia el perfeccionamiento de la escuela primaria.

La Habana: Pueblo y Educación.

Rodríguez, E. (2005). Propuesta didáctica para el tratamiento de la división con números naturales. En J. Albarrán, C. Suárez, D. González, M. Bernabeu, E. Villegas, E. Rodríguez et. al. (Comps.),

Didáctica de la Matemática en la escuela primaria

. (pp. 188 – 208). La Habana: Pueblo y Educación.

Rojas, D. A. (2012). Una aproximación al método de búsqueda independiente para la obtención de los procedimientos de cálculo independiente.

Revista IPLAC, 1

, 105 – 111.

Ruiz, A. (2003).

Teoría y práctica curricular.

La Habana: Pueblo y Educación.

Simeón, O., León, T., Rodríguez, V., Pulgarón, J., Brito, T., Bermúdez, J. et al. (1991).

Metodología de la enseñanza de la Matemática en la escuela primaria.

Parte1. La Habana: Pueblo y Educación.

Torres, O. L. (2010).

Sistema de actividades para el desarrollo de la habilidad calcular operaciones combinadas en escolares de sexto grado.

Tesis de Maestría. Universidad de Ciencias Pedagógicas “José Martí”, Camagüey.

Torres, O. L., Pérez, K., y García, J. (2017).

Propuesta de ejercicios para el desarrollo de la habilidad calcular operaciones combinadas en escolares de la educación primaria.

XIV Conferencia Internacional de Ciencias de la Educación. Universidad de Camag

ey, Cuba.

Villalón, M., Marrero, R., y Tolón, R. (1984). Orientaciones metodológicas complementarias de Matemática. Primero y segundo grados. La Habana: Pueblo y Educación.

Villalón, M., Peña, R. L., León, N., Garea, L., Bello, M., Varela, L. et al. (2001). Orientaciones Metodológicas de Matemática para tercer grado. Parte 2. La Habana: Pueblo y Educación.

Olga Lidia Torres-Gamboa; Karel Pérez-Ariza; José Julián García-Muñoz: The transversal treatment of the calculation of…

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