Number Sense, Measurement, and Geometry in Early Childhood

Posted on Dec 15, 2024 in Mathematics

Building Number Sense in Early Childhood

Stages of Number Development

Third Stage: Cardinality Rule

  • Children aged 5 to 10 can recognize that the last number counted represents the total (cardinality) of the set.
  • They do not connect the order of counting to the relative size of numbers.
  • Children do exercises with numbers 1 to 9.
  • They identify characteristics and qualities of the set to be counted. If sorting, they understand the desired size and can represent it in words or pictures.

Fourth Stage: Relative Size of Numbers

Children can recognize the relative size of two numbers less than or equal to 10 (comparisons).

Building Number and Operation Notions

  • Observe quantitative aspects of their environment, considering cultural value and pre-existing knowledge.
  • Experience quantitative aspects through their own body.
  • Use stories, songs, and other resources like fingers or popular sayings involving quantitative elements. Dramatize situations to encourage the display of quantities.
  • Handle and manipulate objects (both specific and non-specific materials) to create knowledge schemes related to numbers and operations.
  • Relate (compare, classify, sort) different amounts of elements perceptively, overcoming the primacy of perception.
  • Play, considering it is a crucial part of this developmental stage.
  • Use songs like “The 5 Apples” and “When the Geese Were in the Field.”
  • Use other technical supports, like computers, to simulate quantities (after ensuring experimentation and manipulation, especially from 3 to 6 years).
  • In all situations, promote activities that encourage estimation of quantities without the material present.

Activities for Number Development

  • Identify, define, and recognize amounts:

    • Recognize numbers (using different graphics for the same quantities).
    • Group elements by quantitative criteria (using quantifiers, recognizing numbers, reading number symbols, recognizing basic ordinal numbers, introducing number spellings).

  • Match amounts (target):

    • Link sets when comparing amounts of elements: quantitative criteria for classification and sorting.
    • Match quantities relating elements of different groups: couples and series.

  • Operating with amounts:

    • The notion of adding, associated with actions like joining, grouping, etc.
    • The notion of subtracting, associated with actions like separating, collecting, hiding, etc.
    • Composition and decomposition of numbers.
    • Introduction to mental arithmetic.

Understanding Measurement in Early Childhood

When we refer to measurement, we are assigning a value to certain magnitudes (measurable objects) in relation to or compared with units. Magnitude is the object that is to be measured.

Treatment Sequence for Measurement

  • Recognize the need for effective measurement, using different units and instruments.
  • Understand that measurement is approximate and requires using the appropriate unit for the desired degree of approximation.
  • Strengthen estimation as a useful practical skill.
  • Introduce specific units of measurement.

Methodological Proposal: Phases of Oak

1. Perception of Quality:

Identify the attribute or property to be measured. Help children grasp the measurable qualities of their environment.

  • 1.1. Length:

    • As an occupied space (length of an object).
    • As a gap (distance between objects).

  • 1.2. Mass/Weight:

    • Mass: amount of matter in a body.
    • Weight: force with which a body is attracted to the Earth.
    • At the elementary level, these are often confused.

  • 1.3. Time:

    • Achieved through daily routines.
    • More complex for children to understand.
    • Differentiate between reading time and having a notion of time.
    • Raise awareness of rhythms, repetitions, and ways of recording time.

2. Comparison of Quality:

Once the quality is perceived, it can be compared. Use terms like: more than, less than, as much as, high-low, long-short, close-far.

  • 1.1. How is the comparison made?

    • Directly: when objects can be placed side by side.
    • Indirectly: when a proxy is needed for comparison.

3. Measurement and Estimation of Quality:

Relate the object to a unit to determine a measure.

4. Activities and Their Temporalization (Alsina):

  • 1.1. Activities to Identify, Define, and/or Recognize Factors:

    • 3-4 years: Recognize basic notions of length (short, long), volume (big, small), mass (light, heavy), capacity (full, empty), and time (day, night, morning, afternoon). Group 3-4 elements by criteria.
    • 4-5 years: Recognize basic notions of length, volume, mass, capacity, and time. Group 6-7 elements by criteria.
    • 5-6 years: Recognize basic notions of length, volume, mass, capacity, and time. Group up to 9 items by measurement criteria. Begin using units to express measure.

  • 1.2. Activities Related to Continuous Magnitude:

    • 3-4 years: Sort and classify groups of items based on simple criteria for length, volume, mass, capacity, and time. Make pairs of objects based on measurement criteria (short-long, etc.).
    • 4-5 years: Sort and classify groups of elements based on more complex criteria for length, volume, mass, capacity, and time. Make pairs of objects based on more complex criteria.
    • 5-6 years: Order and classify groups of items based on more complex criteria for length, volume, mass, capacity, and time. Make pairs of objects based on more complex criteria.

  • 1.3. Mathematical Language Activities for Continuous Operations:

    • 3-4 years: Put, join, group, separate, hide.
    • 4-5 years: Put, join, group, separate, hide, add, join, remain.
    • 5-6 years: Put, join, group, separate, hide, add, join, subtract, break, compose.

Introduction to Geometry in Early Childhood

Geometry helps us understand our environment and study different properties. It provides theoretical models to explain nature.

Skills Developed Through Geometry

  • Visual perception: seeing things from different perspectives.
  • Verbal expression.
  • Graphic representation.
  • Spatial reasoning (location).
  • Logical reasoning: classifications, analysis.
  • Application of results.

Geometric Properties According to Piaget

1.1. Topological Properties:

  • The child’s body is the main reference point.
  • Spatial relationships are expressed with simple words (top, bottom, front, back).
  • At 3 years, a child may not distinguish a circle from a square.
  • The distance between two objects seems smaller if another object is placed in between.
  • A path can be understood starting at the end but not in the same direction.

1.2. Projective Properties:

  • Around 6 years, topological concepts transform into projective concepts.
  • Projective transformations allow children to see how angles and lengths change in the representation of an object.

1.3. Euclidean Properties:

  • Children perceive objects as moving, not static.
  • They can draw the path of a car.
  • A body that undergoes translation or symmetry maintains properties like length, angles, areas, and volumes.

General Geometric Dimensions of Education (Alsina)

  • Visual Geometry: Teach students to see the environment with a mathematical lens.
  • Constructed Geometry: Manipulation is important for children to touch and see object features.
  • Drawn Geometry: Geometric patterns give meaning to what they do.
  • Measured Geometry: Geometry involves finding techniques to measure.

Van Hiele Levels of Geometric Reasoning

Developed by Dutch teachers, these levels describe the progression of geometric understanding.

  • Level 1: Basic (Recognition and Visualization):

    • Children see geometric figures as a whole, globally.
    • They can generalize features to other figures in the same class.
    • They use phrases like “it seems like…” or “it’s shaped like…”.

  • Level 2: Analysis:

    • Children realize figures are made up of parts with mathematical properties.
    • They can say, “It has 4 sides, 2 and 2, they are not equal, it has 4 vertices…”.
    • They see properties as independent, unrelated to each other.
    • They cannot make logical classifications based on elements or properties.

  • Level 3: Classification:

    • Children begin to reason.
    • They recognize that some properties are deduced from others.
    • They can understand a demonstration but may not be able to do it themselves.
    • They can make mathematically correct definitions.

  • Level 4: Formal Deduction and Rigor:

    • Children can understand and carry out formal logical reasoning.
    • They understand the axiomatic structure of mathematics, theorems, etc.

Teaching Strategies Based on Van Hiele

  • Information: Students become familiar with the structure of the material presented by the teacher.
  • Guided Orientation: Students research the material, guided by the teacher’s questions.
  • Explicitness: Students learn to express their understanding using proper language and create their own definitions.
  • Free Orientation: Students apply their language to new research on the material, based on findings or other knowledge.
  • Integration: Students gain an overview of the material and integrate it into their knowledge base.

Introduction to Statistics and Probability in Early Childhood

It is important to introduce these concepts in kindergarten to familiarize children with the language and features.

Statistics

Science that studies methods to collect, organize, summarize, and analyze data to draw valid conclusions and make reasonable decisions.

  • Classifications: Descriptive, inductive inferential, and comparative statistics.
  • Basic Concepts: Population, individual, size, sample, qualitative and quantitative characteristics.
  • Numerical Description: Parameters of centralization (average, mode) and dispersion.
  • Graphic Description: Important for children to understand simple graphs.

Probability

Measures the degree of occurrence of an event associated with a random experiment.

  • Deterministic experiments: No need to experiment to see the outcome.
  • Randomized experiments: We know the possible outcomes but not the specific result.
  • Intuitive Probability: The degree of probability of occurrence in an experiment. With children, use terms like “more likely” or “less likely.”

Considerations for Teaching Statistics and Probability

  • Introduce between 3 and 6 years (ideally at 4).
  • Activities should be related to observing the immediate environment.
  • Activities should be based on motor and/or sensory education.
  • Children should create their own schedules, logs, etc.
  • Activities should be closely linked to oral language.
  • Introduce simple graphical activities.
  • Work on data collection and representation.
  • Distribute tasks and avoid repeating questions to children.