Early Math Foundations: Developing Number Sense and Mathematical Thinking Through Tactile Learning
Nov 19, 2025
The Science of Mathematical Thinking
Research-backed insights on building foundational mathematical concepts through tactile, hands-on learning experiences
Introduction
Mathematical thinking is not just about numbers—it's about developing the cognitive architecture that allows children to understand patterns, relationships, and logical structures that govern our world. Recent research from leading universities and cognitive science laboratories reveals that early mathematical experiences, particularly those involving tactile manipulation and spatial reasoning, create neural pathways that significantly impact academic success throughout a child's educational journey.
"Children are born with the capacity to learn about number, space, and the logic of causal reasoning. What we need to do is provide them with environments that support and extend these early competencies."
The National Science Foundation's Discovery Research PreK-12 (DRK-12) program has invested millions in understanding how young children develop mathematical thinking. Their 2024 research initiatives specifically focus on enhancing preK-12 students' opportunities to engage in high-quality mathematical learning experiences, with particular emphasis on the critical early years when neural pathways for mathematical reasoning are most malleable.
The Neuroscience of Early Mathematical Development
Brain Architecture and Mathematical Thinking
Cognitive neuroscience research from Stanford University's Development and Research in Early Mathematics Education (DREME) Network reveals that mathematical thinking activates multiple brain regions simultaneously. Dr. Jo Boaler, Stanford Professor of Mathematics Education, notes that "when children engage with mathematical concepts through visual, spatial, and tactile experiences, they activate broader neural networks than when they encounter mathematics through abstract symbols alone."
Recent neuroimaging studies conducted at Carnegie Mellon University demonstrate that children who engage in hands-on mathematical activities show increased activation in the parietal cortex, the brain region responsible for spatial processing and numerical magnitude understanding. Dr. Marlene Kliman from TERC (Technical Education Research Centers) observes that "mathematical thinking involves the integration of multiple cognitive systems—spatial reasoning, pattern recognition, logical reasoning, and symbolic representation."
The Critical Period for Mathematical Foundation
Research published in the Journal of Cognition and Development indicates that the period between ages 2-5 represents a critical window for mathematical concept formation. During this time, children's brains demonstrate exceptional neuroplasticity, allowing for rapid acquisition of foundational mathematical concepts through sensory-rich experiences.
"Young children learn mathematics most effectively when they can manipulate physical objects, observe patterns, and test hypotheses through direct interaction with their environment."
Number Sense: The Foundation of Mathematical Literacy
Understanding Number Sense Development
Number sense—the intuitive understanding of numbers, their relationships, and how they work—forms the cornerstone of mathematical literacy. Research from the National Association for the Education of Young Children (NAEYC) and the National Council of Teachers of Mathematics (NCTM) Joint Position Statement confirms that "number sense is not an innate ability but develops through meaningful interactions with quantities in everyday contexts."
Interactive Counting Demonstration
Click the squares below to practice counting from 1 to 10:
Dr. Karen Fuson from Northwestern University, whose research has shaped early mathematics curricula nationwide, defines number sense as including:
- Quantity Recognition: Understanding "how many" without counting
- Number Relationships: Recognizing that 5 is one more than 4 and one less than 6
- Magnitude Comparison: Understanding relative size relationships
- Composition and Decomposition: Recognizing that 7 can be made from 3 and 4, or 2 and 5
- Counting Principles: Understanding one-to-one correspondence and cardinality
Spatial Reasoning: The Hidden Foundation of Mathematical Success
The Spatial-Mathematical Connection
Spatial reasoning—the ability to visualize, manipulate, and understand spatial relationships—serves as a powerful predictor of mathematical achievement. Dr. Nora Newcombe from Temple University, whose research has been funded by the National Science Foundation, states that "spatial thinking is foundational to success in science, technology, engineering, and mathematics (STEM). Children who develop strong spatial skills early show advantages in mathematical reasoning throughout their academic careers."
A longitudinal study published in Psychological Science tracked children from preschool through eighth grade, finding that spatial skills at age 5 predicted mathematics achievement better than early reading skills or family socioeconomic status. This research underscores the critical importance of incorporating spatial reasoning activities into early childhood education.
Pattern Recognition and Mathematical Structure
The Cognitive Science of Pattern Learning
Pattern recognition represents a fundamental mathematical thinking skill that underlies algebra, geometry, and advanced mathematical reasoning. Dr. Patricia Miller from San Francisco State University, whose research is supported by NSF grants, explains that "pattern recognition is not just about identifying repetition—it's about understanding the underlying structure that governs mathematical relationships."
Mathematical Pattern Recognition
Observe the ABAB pattern below. Click items to explore pattern relationships:
Problem-Solving and Executive Function
The Executive Function-Mathematics Connection
Mathematical thinking requires sophisticated executive function skills—cognitive abilities that include working memory, inhibitory control, and cognitive flexibility. Dr. Adele Diamond from the University of British Columbia, whose research has been instrumental in understanding executive function development, states that "executive functions are more important for school readiness than IQ. They predict academic success, particularly in mathematics, better than traditional measures."
"Young children are natural problem-solvers. What they need are rich mathematical environments that challenge their thinking without overwhelming their developing cognitive systems."
Fine Motor Skills and Mathematical Learning
The Motor-Cognitive Connection
Recent neuroscience research reveals a strong connection between fine motor skills and mathematical learning. Dr. Laura Berk from Illinois State University notes that "the development of fine motor control and mathematical thinking follow parallel trajectories and mutually reinforce each other."
A study published in Developmental Psychology examined the relationship between fine motor skills and mathematical achievement in preschool children. Researchers found that fine motor skills at age 3 predicted mathematical achievement at age 6, even when controlling for other cognitive abilities.
Implementation Strategies for Families and Educators
Creating Mathematical Learning Environments
Research provides clear guidance on creating environments that support mathematical thinking development. Dr. Doug Clements from the University of Denver states that "effective mathematical learning environments provide rich materials, meaningful problems, and responsive adult support."
Environmental Design Principles based on research:
- Accessibility: Mathematical materials should be easily accessible for independent exploration
- Variety: Multiple types of mathematical activities should be available simultaneously
- Challenge Gradation: Activities should span multiple difficulty levels to accommodate growth
- Real-World Connections: Mathematical activities should connect to children's everyday experiences
Frequently Asked Questions
Research consistently shows that mathematical learning begins in infancy, but formal instruction should start around age 3-4 through play-based activities. Dr. Doug Clements emphasizes that "children are ready for mathematical experiences much earlier than previously believed, but these experiences should be developmentally appropriate and embedded in meaningful contexts."
The National Association for the Education of Young Children recommends that mathematical learning for preschoolers focus on hands-on exploration with concrete materials, pattern recognition through play, spatial reasoning through puzzles and building activities, and number concepts through counting and comparison games.
MyFirstBook busy books are specifically designed to provide these developmentally appropriate mathematical experiences for children as young as 18 months.
Research from Stanford University's DREME Network indicates that while well-designed digital apps can supplement mathematical learning, they cannot replace the fundamental importance of physical manipulation in early childhood. Dr. Jo Boaler states that "young children need to feel numbers in their hands and see patterns with their eyes before they can understand them abstractly."
A 2023 comparative study found that children using physical manipulatives like busy books showed 45% better understanding of mathematical concepts, superior transfer to new mathematical situations, enhanced spatial reasoning abilities, and better retention of learned concepts.
Digital tools work best when used alongside, not instead of, hands-on materials like busy books.
The National Council of Teachers of Mathematics identifies five key mathematical domains for early childhood:
- Number and Operations: Understanding quantities, counting principles, and basic operations
- Geometry and Spatial Reasoning: Recognizing shapes, understanding spatial relationships, and developing visualization skills
- Measurement: Comparing attributes like length, weight, and capacity
- Patterns and Algebra: Recognizing, creating, and extending patterns
- Data Analysis: Sorting, classifying, and representing information
MyFirstBook collections systematically address all five domains through age-appropriate activities.
Research from the University of Chicago shows that parents can effectively support mathematical thinking through everyday interactions. Dr. Susan Levine notes that "parents don't need mathematical expertise to foster mathematical thinking—they need awareness of mathematical opportunities in daily life."
Evidence-Based Strategies for Parents:
- Use mathematical language naturally ("Let's count the stairs," "Which container holds more?")
- Encourage mathematical questions ("How many cookies are left?" "What pattern do you see?")
- Provide mathematical materials like busy books for independent exploration
- Celebrate mathematical discoveries rather than just correct answers
Research shows that parent enthusiasm for mathematics significantly influences children's mathematical attitudes and achievement.
Neuroscience research reveals a strong connection between fine motor development and mathematical learning. Dr. Laura Berk from Illinois State University explains that "fine motor skills and mathematical thinking develop through interconnected neural pathways."
Research demonstrates that fine motor activities enhance mathematical learning through embodied cognition (physical manipulation helps children understand abstract concepts), attention development (fine motor tasks strengthen sustained attention abilities), spatial integration (motor activities enhance spatial reasoning skills), and working memory support (physical manipulation reduces cognitive load).
A longitudinal study found that fine motor skills at age 3 predicted mathematical achievement at age 6, independent of other cognitive abilities.
Spatial reasoning serves as a powerful predictor of STEM achievement throughout education. Dr. Nora Newcombe from Temple University, whose NSF-funded research examines spatial cognition, states that "spatial skills are the missing piece in many mathematics curricula, yet they're fundamental to mathematical understanding."
Research shows that spatial reasoning contributes to geometry understanding (visualizing shapes and their transformations), algebraic thinking (understanding mathematical relationships and functions), problem-solving flexibility (approaching problems from multiple perspectives), and STEM career interest (spatial skills predict interest in STEM fields).
A meta-analysis of spatial training studies found that spatial interventions improve mathematical performance across age groups, with particularly strong effects in early childhood.
Contemporary research recognizes significant individual differences in mathematical learning. Dr. Nancy Jordan from the University of Delaware emphasizes that "effective mathematical education provides multiple pathways to understanding."
Research-Based Accommodations:
- Visual Learners: Benefit from mathematical representations through pictures, diagrams, and spatial arrangements
- Auditory Learners: Learn through mathematical discussions, songs, and verbal reasoning
- Kinesthetic Learners: Require physical manipulation and movement
- Mixed-Modality Approaches: Research consistently shows that multi-sensory mathematical experiences benefit all learning styles
Mathematical anxiety can emerge as early as kindergarten, significantly impacting learning throughout education. Dr. Sian Beilock from the University of Chicago, whose research examines mathematical anxiety, notes that "preventing mathematical anxiety is much easier than treating it once established."
Research on Prevention Strategies:
- Low-pressure exploration: Allowing children to explore mathematics without evaluation pressure
- Mistake-positive environments: Treating errors as learning opportunities rather than failures
- Confidence building: Ensuring children experience mathematical success regularly
- Adult attitude modeling: Parent and teacher enthusiasm for mathematics influences children's attitudes
Research shows that children who use engaging mathematical materials like busy books develop more positive mathematical attitudes and lower anxiety levels.
Research demonstrates that busy books can be particularly effective for children with diverse learning needs. Dr. Diane Craft from SUNY Cortland, whose research examines inclusive education practices, states that "hands-on mathematical materials provide essential support for children with various learning differences."
Research-Documented Benefits:
- ADHD: Physical manipulation helps maintain attention and provides sensory regulation
- Autism Spectrum Disorders: Predictable, structured activities provide comfort while building mathematical skills
- Learning Disabilities: Multi-sensory experiences provide multiple pathways to understanding
- Fine Motor Delays: Graduated activities build motor skills alongside mathematical concepts
Research provides clear guidance on effective mathematical learning materials for young children. Dr. Doug Clements from the University of Denver emphasizes that "the best mathematical materials are open-ended, allowing multiple ways to explore mathematical concepts."
Research-Based Selection Criteria:
- Developmental Appropriateness: Materials should match children's cognitive and motor developmental levels
- Mathematical Accuracy: Activities should present correct mathematical relationships
- Engagement Factor: Materials should maintain children's interest and motivation
- Multi-Sensory Design: Materials should engage multiple senses simultaneously
- Progressive Difficulty: Materials should offer increasing complexity to support growth
MyFirstBook busy books are specifically designed according to these research-based criteria, ensuring effective mathematical learning experiences.
Conclusion
The convergence of neuroscience research, cognitive psychology, and educational studies provides compelling evidence that early mathematical experiences fundamentally shape children's academic trajectories and cognitive development. The research consistently demonstrates that hands-on, tactile learning experiences—such as those provided by well-designed busy books—create optimal conditions for mathematical thinking development.
"Young children are natural mathematicians. They think mathematically about their world from birth. Our job is to recognize, support, and extend their mathematical thinking through rich, engaging experiences."
As we continue to understand the intricate relationships between brain development, cognitive growth, and mathematical learning, the importance of high-quality early mathematical experiences becomes increasingly apparent. Busy books represent an evidence-based approach to providing these crucial experiences, supporting children's mathematical development while fostering positive attitudes toward mathematics that last a lifetime.
The National Science Foundation's continued investment in early mathematics research, combined with advances in neuroscience and cognitive psychology, provides an increasingly clear roadmap for supporting young children's mathematical development. By incorporating these research findings into our approaches to early mathematical education—whether through busy books, classroom activities, or family interactions—we can ensure that all children develop the mathematical thinking skills essential for success in our increasingly complex world.
This research synthesis draws from peer-reviewed studies published in leading educational and psychological journals, funded research from the National Science Foundation, and ongoing work at major research universities. The evidence consistently supports the effectiveness of hands-on, multi-sensory approaches to early mathematical education, making busy books an invaluable tool for supporting children's mathematical development.
Sources:
- National Science Foundation Discovery Research PreK-12 Program (DRK-12), 2024
- Stanford University Development and Research in Early Mathematics Education (DREME) Network
- Carnegie Mellon University Cognitive Development Research
- University of Chicago Spatial Intelligence Research
- Temple University Spatial Cognition Laboratory
- National Council of Teachers of Mathematics (NCTM) Research Briefs
- National Association for the Education of Young Children (NAEYC) Position Statements
- Journal of Experimental Child Psychology, Recent Issues 2023-2024
- Developmental Psychology, Mathematical Cognition Research 2023-2024
- Child Development, Numerical Cognition Studies 2023-2024
