🔢 Chapter 10: Numerical Thinking and Arithmetic Skills
(from Children’s Thinking: Cognitive Development and Individual Differences)
🧠 OVERVIEW
This chapter explores how children develop number concepts, understand quantities, and
build arithmetic skills. It includes foundational theories like Siegler and Lortie-Forgues’s
Integrative Theory of Numerical Development, details of counting strategies, and how
culture, disability, and gender affect math learning.
🧮 INTEGRATIVE THEORY OF NUMERICAL
DEVELOPMENT (Siegler & Lortie-Forgues, 2014)
This theory describes four major milestones of numerical development:
1. Representing nonsymbolic numbers with increasing precision
2. Linking nonsymbolic and symbolic number representations
3. Extending numerical representation to larger ranges
4. Representing non-whole numbers (fractions, decimals, negatives)
● Based on the Approximate Number System (ANS): an intuitive, early-developing
system for estimating quantities
● ANS precision improves with age:
○ 2:1 ratio (6 months)
○ 3:2 (9 months)
○ 4:3 (3 years)
○ 6:5 (6 years)
, ○ 11:10 (some adults)9781544361338-childrens…
🔡 CONCEPTUAL VS. PROCEDURAL MATHEMATICAL
KNOWLEDGE
📘 Conceptual Knowledge
● Understanding principles behind math (e.g., "Each object gets one number").
● Examples:
○ Knowing what counting means
○ Recognizing that fractions = part of a whole
🔧 Procedural Knowledge
● Knowing how to perform operations (e.g., counting, adding).
● Includes strategies like:
○ Sum strategy: count all
○ Min strategy: start with the bigger number and count up
○ Fact retrieval: recall math facts from memory
○ Decomposition: break down a problem into easier
parts9781544361338-childrens…
🔄 DEVELOPMENTAL PROGRESSION
● Early arithmetic is vocal and often done with fingers.
● Over time, children move from counting strategies to memory-based strategies.
(from Children’s Thinking: Cognitive Development and Individual Differences)
🧠 OVERVIEW
This chapter explores how children develop number concepts, understand quantities, and
build arithmetic skills. It includes foundational theories like Siegler and Lortie-Forgues’s
Integrative Theory of Numerical Development, details of counting strategies, and how
culture, disability, and gender affect math learning.
🧮 INTEGRATIVE THEORY OF NUMERICAL
DEVELOPMENT (Siegler & Lortie-Forgues, 2014)
This theory describes four major milestones of numerical development:
1. Representing nonsymbolic numbers with increasing precision
2. Linking nonsymbolic and symbolic number representations
3. Extending numerical representation to larger ranges
4. Representing non-whole numbers (fractions, decimals, negatives)
● Based on the Approximate Number System (ANS): an intuitive, early-developing
system for estimating quantities
● ANS precision improves with age:
○ 2:1 ratio (6 months)
○ 3:2 (9 months)
○ 4:3 (3 years)
○ 6:5 (6 years)
, ○ 11:10 (some adults)9781544361338-childrens…
🔡 CONCEPTUAL VS. PROCEDURAL MATHEMATICAL
KNOWLEDGE
📘 Conceptual Knowledge
● Understanding principles behind math (e.g., "Each object gets one number").
● Examples:
○ Knowing what counting means
○ Recognizing that fractions = part of a whole
🔧 Procedural Knowledge
● Knowing how to perform operations (e.g., counting, adding).
● Includes strategies like:
○ Sum strategy: count all
○ Min strategy: start with the bigger number and count up
○ Fact retrieval: recall math facts from memory
○ Decomposition: break down a problem into easier
parts9781544361338-childrens…
🔄 DEVELOPMENTAL PROGRESSION
● Early arithmetic is vocal and often done with fingers.
● Over time, children move from counting strategies to memory-based strategies.
