Please enable JavaScript.
Coggle requires JavaScript to display documents.
EP1:Conceptual Development of Whole Numbers and Operations - Coggle Diagram
EP1:Conceptual Development of Whole Numbers and Operations
Conceptual Development of Whole Numbers
Children’s Development of Whole Number Concepts
1）Base-Ten System Complexity:
Value of a digit is both its value and its position within the numeral.
Children must grasp that a digit's value depends on both its numerical value and its position within a numeral, introducing complexity to the base-ten system.
2）Counting Stage
Children initially don't differentiate between single-digit and multi-digit numbers.
Numbers seen as representations of quantities, not digit compositions.
3）Unitary vs. Multiunit Conceptual Structure
Unitary structure: Counting by ones or tens, where ten serves as a shortcut.
Place value, unitizing, cardinality, one-to-one correspondence, and hierarchical inclusion.
4）Developmental Stages of "Ten" Concept
Numerical composite: Ten as a collection, not a unit.
Abstract composite unit: Recognizing ten as 10 ones but struggling with counting by tens.
Iterable unit: Viewing ten as both a single unit and a collection of 10 single objects.
5）Important Developmental Notions
Place value, unitizing, cardinality, one-to-one correspondence, and hierarchical inclusion.
Piaget's Theory
Four Stages of Cognitive Development
Piaget's theory outlines sensory-motor, pre-operational, concrete operational, and formal operational stages, each building on the previous one.
Sensory-motor stage (birth to age 2)
Pre-operational stage (ages 2 to 7)
Concrete operational stage (ages 7 to 11)
Formal operations stage (ages 12 to 15)
Factors Influencing Development
Nervous maturation, encounters with experience, social transmission, and equilibration.
Development of Number Concepts
Children progress from global comparisons to intuitive understanding and, finally, concrete operational understanding of numbers.
Global comparison stage (up to age 6)
Intuitive stage (ages 6 to 7)
Concrete operations stage (from age 7)
Operations Necessary for Number Understanding
Essential operations include one-to-one correspondence, understanding transitive relations (greater than/less than), ordering, and seriating.
Conceptual Development of Number Operations
Emphasizing the importance of conceptual understanding.
Linking conceptual understanding with procedural fluency.
The role of place value, addition, subtraction, and multiplication in developing proficiency.
Addition Strategies
Counting-on
Counting on from a larger addend.
Near Doubling
Utilizing "double plus 1" or "double minus 1" approach.
Adding to Ten
Decomposing numbers to make a sum of 10.
Adding One-Less-Than-Ten
Efficient strategy for adding nine.
Subtraction Strategies
Front-end Subtraction
Subtracting tens first.
Compensation
Adjusting to make the bottom number a multiple of ten.
Taking Extra and Adding Back
Changing one number, then adjusting.
Using Place Value Understanding:
Subtracting by equal additions, understanding place value.
Multiplication Strategies
Repeated Addition
Using repeated addition for solving multiplication.
Skip Counting
Counting groups using multiples.
Using Known Number Facts
Leveraging known multiplication facts.
Products with Nine as One Factor
Utilizing a "double minus 1" approach.
Division Strategies
Repeated Subtraction
Using repeated subtraction for division
Skip Counting
Counting groups for division.
Using Known Number Facts
Applying known facts for division.
Division with No Regrouping
Combining reasoning strategies for division without regrouping.