TECH presents this Postgraduate diploma as an opportunity to implement the most innovative concepts on Mathematics teaching for problem-solving in the early childhood classroom" 

Solving math problems is very complicated for many children, especially for the youngest ones when they are starting in this science. However, developing their logical thinking through this practice is fundamental because, as many specialists have determined, the use of this pedagogical strategy is highly beneficial to enhance their cognitive skills, because it allows them to face more effectively, not only in the subject but in everyday situations of life, in the family or social environment. For this reason, more and more education professionals wish to implement techniques related to problem solving in their curricula, so that students can understand the meaning of each of the operations, establish inferences and cause-effect relationships.  

And so that the graduate can know in detail the best teaching strategies for this, especially applicable to the first levels of education (3 to 6 years), TECH presents this complete Postgraduate diploma. This is a multidisciplinary academic experience through which the teacher will be able to delve into the teaching of Arithmetic, Algebra, Geometry and measurement, but in a dynamic and innovative way: through games. They will also be able to learn in detail the best techniques to promote Logical-Mathematical Thinking in Early Childhood Education, focusing their quarterly programming on effective problem solving.

To do so, you will have 540 hours of the best theoretical, practical and additional content, which has been compacted in a convenient and flexible 100% online format. Therefore, the graduate will be able to access the Postgraduate Certificate course from anywhere, anytime and from any device with an Internet connection: PC, Tablet or cell phone. Furthermore, they will be able to download all the material for consultation, even when they do not have coverage or when the academic experience has been completed. This way they will not have to worry about schedules or on-site classes when studying for this high-level Postgraduate diploma that adapts not only to their needs, but also to the demands of the modern educational sector.  

You will have 540 hours of the best theoretical, practical and additional content at your disposal to expand each of the sections of the syllabus according to your needs and goals"

This Postgraduate diploma in Problem Solving and Mental Arithmetic in the Early Childhood Classroom contains the most complete and up-to-date educational program on the market. The most important features include:

• The examination of practical cases presented by experts in Mathematics teaching 
• The graphic, schematic and practical contents of the book provide technical and practical information on those disciplines that are essential for professional practice
• Practical exercises where the self-assessment process can be carried out to improve learning
• Its special emphasis on innovative methodologies
• Theoretical lessons, questions to the expert, debate forums on controversial topics, and individual reflection work
• Content that is accessible from any fixed or portable device with an Internet connection

A Postgraduate diploma that will convert you into an expert, by building comprehensive knowledge of arithmetic, algebraic, geometric and measurement strategies for children"

The program’s teaching staff includes professionals from the field who contribute their work experience to this educational program, as well as renowned specialists from leading societies and prestigious universities.  

The multimedia content, developed with the latest educational technology, will provide the professional with situated and contextual learning, i.e., a simulated environment that will provide immersive education programmed to learn in real situations.  

This program is designed around Problem-Based Learning, whereby the professional must try to solve the different professional practice situations that arise during the course. For this purpose, students will be assisted by an innovative interactive video system created by renowned experts in the field of educational coaching with extensive experience.  

If you are looking for a program that enhances your skills for the practice of Logical-Mathematical Thinking in Early Childhood, this is the perfect opportunity for you"

You will have access to a state-of-the-art Virtual Campus where you will find all the material from the beginning of the course and which you can access from any device with an Internet connection"

For the design of the structure and, in general, the content of this program, TECH has taken into consideration the criteria of the teaching team, since, being developed by professionals in the field of Early Childhood Education, they know in detail the aspects that the graduate must master to improve their academic skills. Based on their research work, the 540 hours of theoretical, practical and additional material included in the Postgraduate diploma have been created, giving the student the opportunity to access it from any device with an Internet connection thanks to its convenient 100% online format.

Thanks to the dozens of hours of additional material that you will find in the Virtual Campus, you will be able to expand each section in a personalized way according to your interests and requirements”  

Module 1. Logical-Mathematical Thinking in Pre-School

1.1. Logical-Mathematical Thinking

1.1.1. What is Mathematical Logic?
1.1.2. How is Mathematical Knowledge Acquired?
1.1.3. The Formation of Logical-Mathematical Concepts at an Early Age
1.1.4. Mathematical Concepts
1.1.5. Characteristics of Logical-Mathematical Thinking

1.2. Training Skills Related to Logical-Mathematical Development

1.2.1. Cognitive Development (Piaget)
1.2.2. Evolutionary Stages
1.2.3. Division of Thought in Knowledge (Piaget)
1.2.4. Evolution of Logical-Mathematical Knowledge
1.2.5. Physical Knowledge vs. Logical-Mathematical Knowledge
1.2.6. Knowledge of Space and Time

1.3. Development of Logical-Mathematical Thinking

1.3.1. Introduction
1.3.2. Knowledge and Reality
1.3.3. Development of Mathematical Knowledge
1.3.4. Development of Logical Thinking by Age
1.3.5. Components of Logical Development
1.3.6. Mathematical Language
1.3.7. Logical-Mathematical Development and Core Curriculum

1.4. Psychopedagogical Foundations in the Construction of Mathematical Knowledge

1.4.1. Sensorimotor Intelligence
1.4.2. Formation of Objective Symbolic Thinking
1.4.3. Formation of Concrete-Logical Thinking
1.4.4. Reasoning and its Types
1.4.5. Bloom's Taxonomy in the Development of Logical-Mathematical Thinking

1.5. Logical-Mathematical Learning I

1.5.1. Introduction
1.5.2. Structuring of the Body Scheme

1.5.2.1. Body Concept
1.5.2.2. Body image
1.5.2.3. Postural Adjustment
1.5.2.4. Coordination

1.6. Notions of Order

1.6.1. Comparison
1.6.2. Correspondence
1.6.3. Quantifiers
1.6.4. Quantity Conservation
1.6.5. Sets or Groupings
1.6.6. Formation of Sets
1.6.7. Numerical Cardinality
1.6.8. The Number Concept
1.6.9. Comparison of Sets
1.6.10 Set Equivalence
1.6.11 Recognition of Natural Numbers
1.6.12 Ordinal Numbers
1.6.13 Mathematical Operations: Addition and Subtraction

1.7. Prenumerical Knowledge: Classification

1.7.1. What is Classification?
1.7.2. Processes
1.7.3. Types of Classification
1.7.4. Cross Classifications
1.7.5. Classification Games

1.8. Seriation Games

1.8.1. Importance of Making Series
1.8.2. Logical Operations in the Construction of Series
1.8.3. Types of Series
1.8.4. Seriation in Pre-school Education
1.8.5. Seriation Games

1.9. Prenumerical Knowledge: Enumeration

1.9.1. Conceptualization and Function of Enumeration
1.9.2. Logical Operations Involved in Enumeration
1.9.3. Enumeration in Pre-school Education Design of Activities
1.9.4. Design of Activities
1.9.5. Task-Based Achievements

1.10. Representation and Manipulative Mathematics

1.10.1. Development of Logical-Mathematical Thinking Through the Senses
1.10.2. Representation, Visualization and Reasoning
1.10.3. Design of Activities Supported by Representation
1.10.4. Manipulative Mathematics: Functions and Resources
1.10.5. Design of Activities that Rely on Manipulation

Module 2. Arithmetic, Algebra, Geometry and Measurement Games with Numbers

2.1. Initiation to Number

2.1.1. Number Concept
2.1.2. Construction of the Number Structure
2.1.3. Numerical Development: Counting

2.1.3.1. Phases in Learning the Numerical Sequence

2.1.3.1.1. Rope or String Level
2.1.3.1.2. Unbreakable Chain Level
2.1.3.1.3. Breakable Chain Level
2.1.3.1.4. Numerable Chain Level
2.1.3.1.5. Bidirectional Chain Level

2.1.4. Counting Principles

2.1.4.1. One-to-one Correspondence Principle
2.1.4.2. Stable Order Principle
2.1.4.3. Cardinality Principle
2.1.4.4. Abstraction Principle
2.1.4.5. Irrelevance of Order Principle

2.1.5. Procedures used by the Child in Counting

2.1.5.1. Term to Term Correspondence
2.1.5.2. Subset to Subset Correspondence
2.1.5.3. Purely Visual Estimation
2.1.5.4. Subitization
2.1.5.5. Count the Elements of a Collection
2.1.5.6. Recount
2.1.5.7. Discount
2.1.5.8. Overcount
2.1.5.9. Calculation Procedures

2.1.6. Fundamental Cardinal and Ordinal Situations
2.1.7. The Importance of Zero
2.1.8. Strategies to Enhance the Concept and Use of Number

2.2. Number Acquisition Process

2.2.1. Introduction
2.2.2. Number Concept

2.2.2.1. Perception of General Quantities
2.2.2.2. Distinguishing and Comparing Quantities of Objects
2.2.2.3. Uniqueness Principle
2.2.2.4. Generalization
2.2.2.5. Summative Action
2.2.2.6. Capture of Named Quantities

2.2.2.6.1. Oral Numeric Series
2.2.2.6.2. Counting Objects
2.2.2.6.3. Cardinal Representation
2.2.2.6.4. Compare Magnitudes

2.2.2.7. Identification of the Name with its Representation
2.2.2.8. Invariance of Named Quantities

2.2.3. From Experimental Psychology

2.2.3.1. Distance Effect
2.2.3.2. Size Effect
2.2.3.3. Numerical Spatial Arrangement

2.2.4. From Developmental Psychology

2.2.4.1. Behavioral, Cognitive and Constructivist Theory

2.2.4.1.1. Exercise Law
2.2.4.1.2. Law of Effect

2.2.5. Theories on the Process of Number Acquisition
2.2.6. Piaget

2.2.6.1. Stages
2.2.6.2. Requirements for the Understanding of the Notion of Number

2.2.7. Dienes

2.2.7.1. Principles

2.2.7.1.1. Dynamic Principle
2.2.7.1.2. Constructive Principle
2.2.7.1.3. Economic Variability Principle
2.2.7.1.4. Constructive Variability Principle

2.2.7.2. Stages

2.2.7.2.1. Free Play
2.2.7.2.2. Game with Rules
2.2.7.2.3. Isomorphic Games
2.2.7.2.4. Representation
2.2.7.2.5. Description
2.2.7.2.6. Deduction

2.2.8. Mialaret

2.2.8.1. Stages

2.2.8.1.1. Action Itself
2.2.8.1.2. Action Accompanied by Language
2.2.8.1.3. Conduct of the Narrative
2.2.8.1.4. Application of the Story to Real Situations
2.2.8.1.5. Graphical Expression of the Actions already Reported and Represented
2.2.8.1.6. Symbolic Translation of the Studied Problem

2.2.9. Information Processing

2.2.9.1. Numerical Apprehension Model
2.2.9.2. Pre-Linguistic Numerical Skills

2.2.10 Counting Principles (Gelman and Gallistel)

2.2.10.1. Biunivocal Correspondence Principle
2.2.10.2. Stable Order Principle
2.2.10.3. Cardinality Principle
2.2.10.4. Abstraction Principle
2.2.10.5. Inconsequence of Order Principle

2.2.11 Comparison of Counting Principles between Piaget's, Gelman's and Gallistel's Theory

2.3.  Informal Arithmetic I

2.3.1. Introduction
2.3.2. Towards an Informal and Intuitive Arithmetic in Pre-School Education

2.3.2.1. Recognize Quantities
2.3.2.2. Relate Quantities
2.3.2.3. Operate Quantities

2.3.3. Objectives
2.3.4. Early Arithmetic Skills

2.3.4.1. Preservation of Inequality

2.3.5. Arithmetic Skills and Chants

2.3.5.1. Preliminary Considerations

2.3.5.1.1. Socio-Cognitive Conflict
2.3.5.1.2. Role of the Language
2.3.5.1.3. Creation of Contexts

2.3.5.2. Procedures and Mastery of the Chants

2.4. Informal Arithmetic II

2.4.1. Memorization of Numerical Facts

2.4.1.1. Activities to Work on Memorization
2.4.1.2. Domino
2.4.1.3. Hopscotch

2.4.2. Didactic Situations for the Introduction of Addition

2.4.2.1. Dialed Number Game
2.4.2.2. Race to 10
2.4.2.3. Christmas Greeting

2.5. Basic Arithmetic Operations

2.5.1. Introduction
2.5.2. Additive Structure

2.5.2.1. Phases of Mialaret

2.5.2.1.1. Approach Through Manipulation
2.5.2.1.2. Action Accompanied by Language
2.5.2.1.3. Mental Work Supported by Verbalization
2.5.2.1.4. Purely Mental Work

2.5.2.2. Strategies to Add
2.5.2.3. Initiation to Subtraction
2.5.2.4. Addition and Subtraction

2.5.2.4.1. Direct and Object Modeling
2.5.2.4.2. Counting Sequences
2.5.2.4.3. Recalled Numeric Data
2.5.2.4.4. Strategies to Add
2.5.2.4.5. Subtraction Strategies

2.5.3. Multiplication and Division
2.5.4. Arithmetic Problem Solving

2.5.4.1. Addition and Subtraction
2.5.4.2. Multiplications and Divisions

2.6. Space and Geometry in Pre-school Education

2.6.1. Introduction
2.6.2. Objectives Proposed by the NCTM
2.6.3. Psychopedagogical Considerations
2.6.4. Recommendations for Teaching Geometry
2.6.5. Piaget and his Contribution to Geometry
2.6.6. Van Hiele Model

2.6.6.1. Levels

2.6.6.1.1. Visualization or Recognition
2.6.6.1.2. Analysis
2.6.6.1.3. Sorting and Classification
2.6.6.1.4. Rigor

2.6.6.2. Learning Phases

2.6.6.2.1. Phase 1: Consultancy
2.6.6.2.2. Phase 2: Directed Guidance
2.6.6.2.3. Phase 3: Explication
2.6.6.2.4. Phase 4: Guidance
2.6.6.2.5. Phase 5: Integration

2.6.7. Geometry Types

2.6.7.1. Topological
2.6.7.2. Projective
2.6.7.3. Metrics

2.6.8. Visualization and Reasoning

2.6.8.1. Spatial Orientation
2.6.8.2. Spatial Structuring
2.6.8.3. Gálvez y Brousseau

2.6.8.3.1. Microspace
2.6.8.3.2. Mesospace
2.6.8.3.3. Macrospace

2.7. Magnitudes and their Measurement

2.7.1. Introduction
2.7.2. Construction of the Notion of Magnitude in the Child

2.7.2.1. Piagetian Phases in the Construction of Magnitudes

2.7.2.1.1. Consideration and Perception of a Magnitude
2.7.2.1.2. Conservation of Magnitude
2.7.2.1.3. Ordering with Respect to Magnitude
2.7.2.1.4. Correspondence of Numbers to Quantities of Magnitude

2.7.2.2. Stages in the Construction of the Measure

2.7.2.2.1. Direct Perceptual Comparison
2.7.2.2.2. Displacement of Objects
2.7.2.2.3. Operability of the Transitive Property

2.7.2.3. Stages in the Teaching-Learning of Magnitudes

2.7.2.3.1. Sensory Stimulation
2.7.2.3.2. Direct Comparison
2.7.2.3.3. Indirect Comparison
2.7.2.3.4. Choice of Unit
2.7.2.3.5. Irregular Measurement System
2.7.2.3.6. Regular Measurement System

2.7.3. Measuring Magnitudes
2.7.4. Length Measurement
2.7.5. Length Measurement
2.7.6. Measurement of Capacity and Volume
2.7.7. Measurement of Time
2.7.8. Phases of the Different Magnitudes

2.7.8.1. Preparation Phase
2.7.8.2. Measurement Practice Phase
2.7.8.3. Consolidation Phase of Techniques and Concepts

2.8. Play in Pre-school Education

2.8.1. Introduction
2.8.2. Objectives
2.8.3. Playing Features
2.8.4. Evolution of the Game

2.8.4.1. Types of Games

2.8.4.1.1. Functional Game
2.8.4.1.2. Imitation or Symbolic Play
2.8.4.1.3. Game with Rules
2.8.4.1.4. Construction Game

2.8.5. Chance and Strategy
2.8.6. Competition in the Games
2.8.7. Didactic Considerations on the Game

2.9. Didactic Resources of the Game

2.9.1. Games and Logical Thinking

2.9.1.1. Three in a Row
2.9.1.2. Quarto
2.9.1.3. Portrait Games

2.9.2. Quantitative Games

2.9.2.1. Number to Compare

2.9.2.1.1. Home!

2.9.2.2. Number to Calculate

2.9.2.2.1. Couples
2.9.2.2.2. No more!
2.9.2.2.3. Cat and Mouse

2.9.3. Games and the Structure of Space

2.9.3.1. Puzzles

2.9.3.1.1. Two-Color Paintings
2.9.3.1.2. The Hex

2.10. Games in Different Spaces

2.10.1. Introduction
2.10.2. Games in the Classroom

2.10.2.1. The Butterfly Game
2.10.2.2. The Partitioning Game
2.10.2.3. Image Trains
2.10.2.4. The Newspaper
2.10.2.5. Flat Figures
2.10.2.6. The Containers

2.10.3. Games in Psychomotor Skills

2.10.3.1. Working with Sizes
2.10.3.2. Classify
2.10.3.3. We Play with the Hoops

2.10.4. Outdoor Games
2.10.5. Mathematical Games with ICT

2.10.5.1. Playing with the Turtle's Mind
2.10.5.2. Geometric Figures
2.10.5.3. For 3-Year-Old Students
2.10.5.4. Variety of Activities
2.10.5.5. Didactic Unit

Module 3. Problem Solving and Mental Arithmetic

3.1. Problem in Pre-School Education

3.1.1. Methodological Considerations
3.1.2. Psychopedagogical Considerations of the Initiation of Problem Idea Representation
3.1.3. What is a Problem?
3.1.4. How to Pose Problems in Pre-School?

3.2. Idea of a Problem to be Introduced in Pre-School Education

3.2.1.  Why do We Solve Problems?
3.2.2. Perspectives for the Inclusion of Comprehension and Problem Solving in Pre-school Education
3.2.3. Specific Didactic Contract for Problem Solving in Pre-School Education
3.2.4. Most Appropriate Models for Introducing the idea of Problem in Pre-School Education
3.2.5. Reading and Understanding Statements

3.2.5.1. Factors of Understanding Statements

3.2.6. Didactic Variables of the Statements

3.3. Towards a Didactic Approach to the Introduction to the idea of Problem in Pre-School Education

3.3.1. Factors to be Taken into Consideration in the Approach and Resolution of Problems in Pre-School
3.3.2. Learning Logical-Mathematical Concepts Through Problem Solving

3.3.2.1. Heuristic Strategies
3.3.2.2. Technique Most Commonly Used at These Ages for Problem Solving
3.3.2.3. Numerical Strategies

3.3.3. Various Situations for the Teaching of Proposition and Problem Solving
3.3.4. Problem Solving Constituent Elements of a Problem

3.3.4.1. Problems that Serve as Exercises to Practice the Problem Idea

3.3.5. Main Recommendations for Approaching the Problem Idea in Pre-school Education

3.4. Mathematical Value of Stories

3.4.1. Pre-school Learning and Mathematics
3.4.2. Stories and Mathematics
3.4.3. Examples of Stories and Mathematical Learning

3.4.3.1. Logical Development
3.4.3.2. Numerical Development
3.4.3.3. Development of Magnitudes and their Measurement
3.4.3.4. Development of Geometric Thinking
3.4.3.5. Problem Solving

3.5. Logical Basis of Mental Arithmetic in Pre-school Education

3.5.1. Logical Operations

3.5.1.1. Classifications
3.5.1.2. Relationships of Order

3.5.2. Mental Arithmetic, Written Arithmetic and Estimated Arithmetic
3.5.3. Counting Process
3.5.4.  Phases for Learning the Counting Activity

3.6. Informal Arithmetic

3.6.1. Arithmetic Strategy
3.6.2. Comparison and Equivalence
3.6.3. Composition and Decomposition
3.6.4. Initiation to Operational Activity: Adding, Subtracting, Folding and Distributing

3.7. Mental Arithmetic in Pre-school Education

3.7.1. Calculation Examples for Pre-school Education
3.7.2. Perform Calculation by Manipulating Material
3.7.3. Calculation Without Material Handling
3.7.4. Proposal for Mental Arithmetic in Pre-School Education

3.7.4.1. Guessing Game
3.7.4.2. It Is Learned by Heart

3.7.5. Mechanics Acquired at the End of Pre-School Education
3.7.6. Resources to Achieve Apprenticeships
3.7.7. Practical Issues

3.8. Resource Bank for Calculation in Pre-School Education

3.8.1.  Abacus

3.8.1.1. Description
3.8.1.2. Possibilities for Didactic Use
3.8.1.3. Classroom Didactic Situations

3.8.2. Multibase Blocks

3.8.2.1. Description
3.8.2.2. Possibilities for Didactic Use
3.8.2.3. Classroom Didactic Situations

3.8.3. Cuisenaire Strips

3.8.3.1. Description
3.8.3.2. Possibilities for Didactic Use
3.8.3.3. Classroom Didactic Situations

3.8.4. Domino

3.8.4.1. Description
3.8.4.2. Possibilities for Didactic Use
3.8.4.3. Classroom Didactic Situations

3.8.5. Battle Game

3.8.5.1. Description
3.8.5.2. Possibilities for Didactic Use
3.8.5.3. Classroom Didactic Situations

3.9. Open Calculation Method based on ABN Numbers

3.9.1. What is the ABN Algorithm Method?

3.9.1.1. Quantity and Cardinality of Sets
3.9.1.2. Number Structure and Set Comparison

3.9.1.2.1. Figurative Representation
3.9.1.2.2. Symbolic Representation
3.9.1.2.3. Symbol-Sign Representation
3.9.1.2.4. Representation by Signs

3.9.1.3. Counting Well Over Ten
3.9.1.4. Number Transformations First Operations

3.9.2. Background of the ABN Method
3.9.3. Intuitive Approach vs. Traditional Approach

3.10. ABN Method Activities Proposal

3.10.1. Block 1: Numerical and Cardinal

3.10.1.1. Search for Equivalent Sets
3.10.1.2. Establishment of a Physical Pattern
3.10.1.3. Pattern Sorting
3.10.1.4. Numeric String Start of Counting
3.10.1.5. Subitization
3.10.1.6. Estimate

3.10.2. Block 2: Number Structure and Comparison

3.10.2.1. Introduction to the Ten
3.10.2.2. Ordering, but not Counting
3.10.2.3. Arrangement of Disordered Sets
3.10.2.4. Interaction of Missing Elements
3.10.2.5. Arrangement with Non-Manipulable Material
3.10.2.6. Comparison of Real Objects
3.10.2.7. Comparison of Figurative Elements

3.10.3. Block 3: Number Transformations

3.10.3.1. Number Transformations
3.10.3.2. Addition with the Number Line
3.10.3.3. Subtraction with Toothpicks
3.10.3.4. Finding the Double with Grid
3.10.3.5. Finding Half with the Number Line

3.10.4. Assessment

The teaching materials of this program, elaborated by these specialists, have contents that are completely applicable to your professional experiences"