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Cognitive, Semantic, Conceptual Space in Math Learning（初学者版）

2024-10-8 15:11

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Infant's Cognitive, Semantic, and Conceptual Spaces in Math Learning by DIKWP Semantic Mathematics

Yucong Duan

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation(DIKWP-SC)

World Artificial Consciousness CIC(WAC)

World Conference on Artificial Consciousness(WCAC)

(Email: duanyucong@hotmail.com)

Abstract

This document continues the comprehensive simulation of an infant's cognitive development by focusing on the math learning stage, emphasizing the state changes in the cognitive space, semantic space, and conceptual space within the Data-Information-Knowledge-Wisdom-Purpose (DIKWP) Semantic Mathematics framework proposed by Prof. Yucong Duan. By detailing each incremental change as the infant interacts with numerical and spatial stimuli, we illustrate how cognitive structures evolve during math learning. This step-by-step account provides a deep understanding of how exposure to mathematical concepts influences semantics and concepts, contributing significantly to the infant's overall cognitive development.

Table of Contents

Introduction

1.1 Overview
1.2 Objectives

Foundational Concepts

2.1.1 Cognitive Space
2.1.2 Semantic Space
2.1.3 Conceptual Space

2.1 Cognitive Spaces in DIKWP Semantic Mathematics
2.2 Mathematical Representation of Spaces

Simulated Scenario: Math Learning Stage

3.1 Setting and Characters
3.2 Overview of Developmental Timeline

Detailed State Changes in Math Learning

4.3.1 Cognitive State Expansion
4.3.2 State Changes per Interaction

4.2.1 Cognitive State Evolution
4.2.2 State Changes per Interaction

4.1.1 Initial Cognitive State
4.1.2 State Changes per Interaction

4.1 Stage 1: Early Numerical Awareness (6-9 Months)
4.2 Stage 2: Understanding Quantity and Space (10-12 Months)
4.3 Stage 3: Basic Mathematical Concepts (13-18 Months)

Mathematical Modeling of State Changes in Math Learning

5.1 Semantic Space Transformations
5.2 Conceptual Space Transformations
5.3 Cognitive Space Integration

Visualization of State Changes

6.1 Diagrams of Cognitive Spaces
6.2 Graphical Representation of Transformations

Discussion

7.1 Insights from Detailed State Changes
7.2 Implications for AI and Cognitive Science
7.3 Limitations and Future Directions

Conclusion
References

1. Introduction1.1 Overview

Mathematical cognition begins early in infancy, with the development of numerical and spatial awareness forming the foundation for later mathematical understanding. This document simulates the infant's cognitive development during the math learning stage, focusing on the state transitions within the cognitive, semantic, and conceptual spaces as modeled by the DIKWP Semantic Mathematics framework.

1.2 Objectives

Detail every state change in the infant's cognitive, semantic, and conceptual spaces during math learning.
Provide mathematical representations of these changes.
Illustrate the evolution of cognitive structures influenced by numerical and spatial experiences.
Enhance understanding of how math learning contributes to overall cognitive development.

2. Foundational Concepts2.1 Cognitive Spaces in DIKWP Semantic Mathematics2.1.1 Cognitive Space ( $C\mathcal{C}$ )

Definition: The overall mental space encompassing all cognitive processes, including those related to mathematical perception and reasoning.
Components: Includes the semantic and conceptual spaces specific to mathematical elements.

2.1.2 Semantic Space ( $S$ )

Definition: A multidimensional space representing the meanings associated with numerical and spatial inputs and experiences.
State Changes: Occur when new mathematical semantic units are formed or existing ones are modified.

2.1.3 Conceptual Space ( $C$ )

Definition: A structured space where mathematical concepts are formed by organizing semantic units related to numbers, quantities, and spatial relations.
State Changes: Happen when new mathematical concepts are created or existing concepts are updated.

2.2 Mathematical Representation of Spaces

Mathematical Semantic Units ( $s_i$ ): Represented as vectors in $S$ with attributes like quantity, size, shape, and spatial orientation.
Mathematical Concepts ( $c_k$ ): Formed by functions mapping sets of mathematical semantic units to vectors in $C$ .
State ( $σ\sigma$ ): At any time $t$ , the state of each space is $σS(t)\sigma_S^{(t)}$ and $σC(t)\sigma_C^{(t)}$ .

3. Simulated Scenario: Math Learning Stage3.1 Setting and Characters

Infant: Emma, from 6 to 18 months.
Parents: Alice and Bob, who provide opportunities for mathematical exploration.
Environment: Home setting with toys and objects facilitating numerical and spatial interactions (e.g., blocks, shape sorters).

3.2 Overview of Developmental Timeline

Stage 1 (6-9 Months): Early numerical awareness; formation of basic mathematical semantic units.
Stage 2 (10-12 Months): Understanding of quantity and space; initial mathematical concept formation.
Stage 3 (13-18 Months): Development of basic mathematical concepts; refinement of numerical and spatial reasoning.

4. Detailed State Changes in Math Learning4.1 Stage 1: Early Numerical Awareness (6-9 Months)4.1.1 Initial Cognitive State

Time $t_0$ :

Cognitive Space ( $C(t0)\mathcal{C}^{(t_0)}$ ): Initial structures for mathematical elements are minimal.
Semantic Space ( $σS(t0)\sigma_S^{(t_0)}$ ): Lacks mathematical semantic units.
Conceptual Space ( $σC(t0)\sigma_C^{(t_0)}$ ): No mathematical concepts formed.

4.1.2 State Changes per Interaction

Interaction 1: Observing Object Quantities

Scenario: Emma sees two toys placed in front of her.
Sensory Input: Visual perception of two objects ( $dtwo_objectsd_{\text{two\_objects}}$ ).
State Change in Semantic Space ( $ΔσS(t1)\Delta \sigma_S^{(t_1)}$ ):

$σS(t1)=σS(t0)∪{squantity(1)}\sigma_S^{(t_1)} = \sigma_S^{(t_0)} \cup \{ s_{\text{quantity}}^{(1)} \}$ .
$squantity(1)s_{\text{quantity}}^{(1)}$ : Perception of "two-ness."
Formation of mathematical semantic units:
Update:

Interaction 2: Hearing Counting Rhymes

Sensory Input: Parents recite "One, Two, Buckle My Shoe" ( $dcounting_rhymed_{\text{counting\_rhyme}}$ ).
State Change in Semantic Space ( $ΔσS(t2)\Delta \sigma_S^{(t_2)}$ ):

$σS(t2)=σS(t1)∪{snumerical_sounds(1)}\sigma_S^{(t_2)} = \sigma_S^{(t_1)} \cup \{ s_{\text{numerical\_sounds}}^{(1)} \}$ .
$snumerical_sounds(1)s_{\text{numerical\_sounds}}^{(1)}$ : Sounds associated with counting.
Formation of:
Update:

Conceptual Space Changes:

At this stage, no significant mathematical concepts are formed; $σC(t2)=σC(t0)\sigma_C^{(t_2)} = \sigma_C^{(t_0)}$ .

Cognitive Space Integration:

The cognitive space now includes basic mathematical semantic units related to quantity and numerical sounds.

4.2 Stage 2: Understanding Quantity and Space (10-12 Months)4.2.1 Cognitive State Evolution

Accumulated Mathematical Semantic Units:

$squantity(2)s_{\text{quantity}}^{(2)}$ : Understanding "more" vs. "less."
$sspatial_relation(1)s_{\text{spatial\_relation}}^{(1)}$ : Notions of "in" and "out."

State of Semantic Space ( $σS(t3)\sigma_S^{(t_3)}$ ):

$σS(t3)=σS(t2)∪{squantity(2),sspatial_relation(1)}\sigma_S^{(t_3)} = \sigma_S^{(t_2)} \cup \{ s_{\text{quantity}}^{(2)}, s_{\text{spatial\_relation}}^{(1)} \}$ .

4.2.2 State Changes per Interaction

Interaction 3: Playing with Blocks

Scenario: Emma stacks blocks and knocks them down.
Sensory Input: Visual and tactile experience of stacking ( $dstacking_blocksd_{\text{stacking\_blocks}}$ ).
State Change in Semantic Space:

Formation of $sspatial_relation(2)s_{\text{spatial\_relation}}^{(2)}$ : Understanding "up" and "down."

Concept Formation:

$cspatial_concepts(1)=fC({sspatial_relation(1),sspatial_relation(2)})c_{\text{spatial\_concepts}}^{(1)} = f_C(\{ s_{\text{spatial\_relation}}^{(1)}, s_{\text{spatial\_relation}}^{(2)} \})$ .

State Change in Conceptual Space ( $ΔσC(t4)\Delta \sigma_C^{(t_4)}$ ):

$σC(t4)=σC(t3)∪{cspatial_concepts(1)}\sigma_C^{(t_4)} = \sigma_C^{(t_3)} \cup \{ c_{\text{spatial\_concepts}}^{(1)} \}$ .

Interaction 4: Requesting More Snacks

Scenario: Emma gestures for more snacks after finishing.
State Change in Semantic Space:

Reinforcement of $squantity(2)s_{\text{quantity}}^{(2)}$ .

Concept Formation:

$cquantity_concepts(1)=fC({squantity(1),squantity(2)})c_{\text{quantity\_concepts}}^{(1)} = f_C(\{ s_{\text{quantity}}^{(1)}, s_{\text{quantity}}^{(2)} \})$ .

Update:

$σC(t5)=σC(t4)∪{cquantity_concepts(1)}\sigma_C^{(t_5)} = \sigma_C^{(t_4)} \cup \{ c_{\text{quantity\_concepts}}^{(1)} \}$ .

Cognitive Space Integration:

Cognitive space now includes basic mathematical concepts related to spatial relations and quantity.

4.3 Stage 3: Basic Mathematical Concepts (13-18 Months)4.3.1 Cognitive State Expansion

New Mathematical Semantic Units:

$sshape(1)s_{\text{shape}}^{(1)}$ : Recognizing circles and squares.
$snumerical_words(1)s_{\text{numerical\_words}}^{(1)}$ : Saying "one" and "two."

State of Semantic Space ( $σS(t6)\sigma_S^{(t_6)}$ ):

( \sigma_S^{(t_6)} = \sigma_S^{(t_5)} \cup { s_{\text{shape}}^{(1)}, s_{\text{numerical_words}}^{(1)} } .

4.3.2 State Changes per Interaction

Interaction 5: Shape Sorting

Scenario: Emma plays with a shape sorter, matching shapes to corresponding holes.
State Change in Semantic Space:

Formation of $sshape(2)s_{\text{shape}}^{(2)}$ : Understanding shape properties.

Concept Formation:

$cshape_concepts=fC({sshape(1),sshape(2)})c_{\text{shape\_concepts}} = f_C(\{ s_{\text{shape}}^{(1)}, s_{\text{shape}}^{(2)} \})$ .

Conceptual Space Update:

$σC(t7)=σC(t6)∪{cshape_concepts}\sigma_C^{(t_7)} = \sigma_C^{(t_6)} \cup \{ c_{\text{shape\_concepts}} \}$ .

Interaction 6: Counting Objects

Scenario: Parents count toys with Emma: "One, two, three."
State Change in Semantic Space:

Reinforcement of $snumerical_words(1)s_{\text{numerical\_words}}^{(1)}$ .
Formation of $squantity(3)s_{\text{quantity}}^{(3)}$ : Associating numerical words with quantities.

Concept Formation:

$cnumber_concepts=fC({snumerical_words(1),squantity(3)})c_{\text{number\_concepts}} = f_C(\{ s_{\text{numerical\_words}}^{(1)}, s_{\text{quantity}}^{(3)} \})$ .

Update:

$σC(t8)=σC(t7)∪{cnumber_concepts}\sigma_C^{(t_8)} = \sigma_C^{(t_7)} \cup \{ c_{\text{number\_concepts}} \}$ .

Cognitive Space Integration:

Cognitive space now includes more complex mathematical concepts, integrating numerical understanding with language.

5. Mathematical Modeling of State Changes in Math Learning5.1 Semantic Space Transformations

Addition of Mathematical Semantic Units:

$σS(t+1)=σS(t)∪{snew}\sigma_S^{(t+1)} = \sigma_S^{(t)} \cup \{ s_{\text{new}} \}$ .

Strengthening Units:

$si(t+1)=si(t)+Δss_i^{(t+1)} = s_i^{(t)} + \Delta s$ , where $Δs\Delta s$ is proportional to the reinforcement.

Example:

$snumerical_words(1)(t+1)=snumerical_words(1)(t)+Δss_{\text{numerical\_words}}^{(1)(t+1)} = s_{\text{numerical\_words}}^{(1)(t)} + \Delta s$ .

5.2 Conceptual Space Transformations

Formation of Mathematical Concepts:

$c_k = f_C(\{ s_{i_1}, s_{i_2}, ..., s_{i_n} \})$ .

Updating Concepts:

$ck(t+1)=ck(t)+γ∑(si−ck(t))c_k^{(t+1)} = c_k^{(t)} + \gamma \sum (s_i - c_k^{(t)})$ .

Associations Between Concepts:

Establishing links, e.g., $cnumber_concepts↔cquantity_concepts(1)c_{\text{number\_concepts}} \leftrightarrow c_{\text{quantity\_concepts}}^{(1)}$ .

5.3 Cognitive Space Integration

Integration Function:

$C(t+1)=C(t)+ΔσS(t+1)+ΔσC(t+1)\mathcal{C}^{(t+1)} = \mathcal{C}^{(t)} + \Delta \sigma_S^{(t+1)} + \Delta \sigma_C^{(t+1)}$ .

State Changes:

Reflect cumulative changes influenced by mathematical experiences.

6. Visualization of State Changes6.1 Diagrams of Cognitive Spaces

Semantic Space Maps:

Nodes represent mathematical semantic units (e.g., shapes, quantities).
Edges indicate relationships like similarity or co-occurrence.

Conceptual Space Maps:

Clusters represent mathematical concepts.
Links show associations between concepts (e.g., numbers and quantities).

6.2 Graphical Representation of Transformations

Time-Series Graphs:

Display the development of semantic units and concepts over time.

State Transition Diagrams:

Highlight key interactions leading to state changes.

7. Discussion7.1 Insights from Detailed State Changes

Early Numerical Cognition:

Infants develop an awareness of quantity and number before they can speak.

Integration of Language and Math:

The association of numerical words with quantities marks a significant cognitive milestone.

Spatial Reasoning Development:

Interactions with physical objects facilitate understanding of spatial relations and geometry.

7.2 Implications for AI and Cognitive Science

Modeling Mathematical Learning:

AI systems can simulate human-like mathematical learning by integrating sensory inputs with symbolic representations.

Embodied Cognition:

Physical interactions with the environment are crucial for cognitive development, suggesting AI should incorporate embodied experiences.

Educational Strategies:

Understanding these developmental stages can inform early childhood education approaches.

7.3 Limitations and Future Directions

Complexity of Mathematical Concepts:

Higher-level mathematical thinking involves abstract reasoning not covered in this early stage.

Individual Variability:

Infants may develop mathematical understanding at different rates.

Integration with Other Domains:

Future models can explore how mathematical learning interacts with language and emotional development.

8. Conclusion

This detailed simulation demonstrates the significant role of math learning in shaping an infant's cognitive, semantic, and conceptual spaces. By modeling the state changes using the DIKWP Semantic Mathematics framework, we observe how mathematical experiences lead to the formation and reinforcement of semantic units and concepts, contributing to overall cognitive development. This approach enhances our understanding of human cognition and offers valuable insights for developing AI systems capable of human-like learning.

9. References

International Standardization Committee of Networked DIKWP for Artificial Intelligence Evaluation (DIKWP-SC),World Association of Artificial Consciousness(WAC),World Conference on Artificial Consciousness(WCAC). Standardization of DIKWP Semantic Mathematics of International Test and Evaluation Standards for Artificial Intelligence based on Networked Data-Information-Knowledge-Wisdom-Purpose (DIKWP ) Model. October 2024 DOI: 10.13140/RG.2.2.26233.89445 . https://www.researchgate.net/publication/384637381_Standardization_of_DIKWP_Semantic_Mathematics_of_International_Test_and_Evaluation_Standards_for_Artificial_Intelligence_based_on_Networked_Data-Information-Knowledge-Wisdom-Purpose_DIKWP_Model
Duan, Y. (2023). The Paradox of Mathematics in AI Semantics. Proposed by Prof. Yucong Duan:" As Prof. Yucong Duan proposed the Paradox of Mathematics as that current mathematics will not reach the goal of supporting real AI development since it goes with the routine of based on abstraction of real semantics but want to reach the reality of semantics. ".
Wynn, K. (1992). Addition and Subtraction by Human Infants. Nature, 358(6389), 749-750.
Starkey, P., & Cooper, R. G. (1980). Perception of Numbers by Human Infants. Science, 210(4473), 1033-1035.
Piaget, J. (1952). The Child's Conception of Number. Routledge & Kegan Paul.
Gibson, E. J., & Pick, A. D. (2000). An Ecological Approach to Perceptual Learning and Development. Oxford University Press.
Gelman, R., & Gallistel, C. R. (1978). The Child's Understanding of Number. Harvard University Press.
Gärdenfors, P. (2000). Conceptual Spaces: The Geometry of Thought. MIT Press.
Russell, S., & Norvig, P. (2021). Artificial Intelligence: A Modern Approach (4th ed.). Pearson.

Keywords: DIKWP Semantic Mathematics, Cognitive State Changes, Semantic Space, Conceptual Space, Infant Math Learning, Cognitive Development, Prof. Yucong Duan, Cognitive Modeling, Artificial Intelligence, Mathematical Cognition, Semantic Integration, Concept Formation, Numerical Awareness.

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