Front Cover

The Mathematical Brain Across the Lifespan

Copyright

Contributors

Contents

Preface

Chapter 1: Neuronal foundations of human numerical representations

1. Introduction

2. A Core Numerical Representation in Parietal Cortex

2.1. Numerical Processing and the Coarse Scale Functional Neuroanatomy of Parietal Cortex

2.2. Fine-Scale Representation of Numerical Information: Findings From Macaque Neurophysiology

2.3. Fine-Scale Representation of Numerical Information: fMRI in Humans

3. The Extraction of Numerical Information: Format-Specific Contributions Within and Beyond Parietal Cortex

3.1. The Extraction of Numerosity from Concrete Sets of Objects

3.2. The Extraction of Number from Symbols

4. Concluding Remarks

References

Chapter 2: What counts in estimation? The nature of the preverbal system

1. The Preverbal System

1.1. Accumulator Model

1.2. Analogue Magnitude System

2. Neural Implementation of a Preverbal System and Verbal Counting Series

3. Our Aim

4. Binomial Accumulator

5. Poisson Accumulator

6. Doubly Stochastic Process

7. Implications of Linear Accumulator Models

8. Numerical Consequences of the AMS Hypothesis

9. Utility of AMS Hypothesis

10. AMS Integrator

11. Building an AMS Accumulator

12. AMS Accumulator or AMS Integrator?

12.1. Method

12.1.1. Participants

12.1.2. Stimuli

12.1.3. Analysis

12.2. Results and Discussion

13. Representations of Magnitude Orders: Stochastic Cascades

14. Log vs Linear: Is This an Issue (for Learning)?

15. Conclusions

Acknowledgments

References

Chapter 3: Core mathematical abilities in infants: Number and much more

1. Introduction

2. Two Cognitive Systems for Nonverbal Numerical Representation

3. Infants' Arithmetical Computations on Numerosities

4. Beyond Number: Other Quantitative Dimensions

5. Infants' Mappings Across Quantitative Dimensions

6. A Spatially Oriented Representation of Number in Infants

7. Conclusions

References

Chapter 4: Cognitive and brain systems underlying early mathematical development

1. Intuitions of Number

1.1. The OTS

1.2. The ANS

1.2.1. Brain systems

1.2.2. Development

2. Relations Between ANS Acuity and Mathematical Achievement

3. The Role of Domain-general Abilities

3.1. Mathematics Learning in Evolutionary Context

3.2. Executive Functions and Mathematics Learning

3.2.1. Executive functions and working memory

3.2.2. Potential evolutionary mechanisms

4. Conclusion

Acknowledgments

References

Chapter 5: Individual differences in children's mathematics achievement: The roles of symbolic numerical magnitude proces ...

1. Introduction

2. Neurocognitive Development of Arithmetic in Children

3. Symbolic Numerical Magnitude Processing

4. Working Memory

5. Phonological Processing

6. Conclusions and Future Directions

Acknowledgments

References

Chapter 6: Similarity interference in learning and retrieving arithmetic facts

1. Typical Development of Arithmetic Facts Network

1.1. Models of Arithmetic Facts Network

1.2. Similarity Interference Through Development

1.3. Similarity Interference in Arithmetic Facts: Brain-Imaging Evidence

2. Atypical Development of Arithmetic Facts Network

2.1. Explanation for Arithmetic Facts Learning/Retrieving Deficit

2.2. The Hypersensitivity-to-Interference in Memory Hypothesis

2.2.1. Single-case study

2.2.2. Group study

2.2.3. One specific profile of dyscalculia

3. Discussion and Conclusion

References

Chapter 7: Memory and cognitive control circuits in mathematical cognition and learning

1. Introduction

2. Parietal-Frontal Working Memory Systems

2.1. Core and Noncore Parietal Systems Overlap in the IPS

2.2. Multiple Parietal-Frontal Working Memory Circuits

2.3. Parietal-Frontal Working Memory Systems in Mathematical Cognition and Its Development

2.4. Parietal-Frontal Impairments in Children with Mathematical Disabilities

2.5. Hyperactive Parietal-Frontal Working Memory Circuits in Children with MD

3. Hippocampal-Frontal Declarative Memory System

3.1. The Medial Temporal Lobe: A System for Associative Learning

3.2. Hippocampal-Frontal Cortex Circuits

3.3. Hippocampal-Prefrontal Coactivation in Children's Mathematical Skill Development

3.4. Hippocampal-Frontal Circuits in Children's Mathematical Skill Development and Learning

4. Cognitive Control Systems in Mathematical Cognition

4.1. Flexible Hubs for Cognitive Control

4.2. Dynamic Parietal-Frontal Control Signals

4.3. Dynamic Hippocampal-Frontal Control Signals

5. Summary and Conclusions

References

Chapter 8: On the ordinality of numbers: A review of neural and behavioral studies

1. General Introduction

2. How Different Are Ordinality and Cardinality?

2.1. Ordinal and Cardinal Processing in the Brain

2.2. Distance Effects: Different Signatures of Ordinal and Cardinal Processing

2.3. Symbolic vs Nonsymbolic Ordinal Processing

2.4. Summary

3. Is Numerical Order Special?

3.1. Specificity of Numerical Order in the Brain

3.2. How Number Specific Are Canonical and Reverse Distance Effects?

3.3. Summary

4. Increasing Ordinal Complexity: From Nonhuman Animals to Development and Acquisition of Ordinality in Humans

4.1. Complex Ordinal Processing in Nonhuman Animals

4.2. Going Beyond Simple Item-Item Ordinal Associations in Human Development and Learning

4.3. Summary

5. Mechanisms that Support Numerical Ordinal Processing

5.1. Magnitude-Based Mechanisms

5.2. Serial-Order WM

5.3. Spatial Mechanisms

5.4. The Mechanisms Underlying Acquisition and Access of Ordinal Associations

5.5. Summary

6. Ordinality and Implications for More Complex Numerical Processing

6.1. Limitations

7. Conclusions

References

Chapter 9: On the instability and constraints of the interaction between number representation and spatial attention in h ...

1. Introduction

1.1. Introspective Number Forms: The Mental Number Line

1.2. The SNARC Effect

1.3. The Attentional SNARC Effect

1.4. The Present Study: The Influence of Task Demands and the Set-Size of Numerical Cues on the Attentional SNARC Effect

2. Experiment 1: Attentional SNARC

2.1. Experiment 1A: Four-Digit Cues (1, 2, 8, and 9)

2.1.1. Method

2.1.2. Results

2.2. Experiment 1B: Eight-Digit Cues (1, 2, 3, 4, 6, 7, 8, and 9)

2.2.1. Method

2.2.2. Results

2.2.3. Comparison of the strength of the Att-SNARC between the four- and eight-digit cues conditions

3. Experiment 2: Imagery Attentional SNARC

3.1. Experiment 2A: Four-Digit Cues (1, 2, 8, and 9)

3.1.1. Method

3.1.2. Results

3.2. Experiment 2B: Eight-Digit Cues (1, 2, 3, 4, 6, 7, 8, and 9)

3.2.1. Method

3.2.2. Results

3.2.3. Comparison of the strength of the Att-SNARC between the four- and eight-digit cues conditions

4. Experiment 3: Spatial Attentional SNARC

4.1. Experiment 3A: Four-Digit Cues (1, 2, 8, and 9)

4.1.1. Method

4.1.2. Results

4.2. Experiment 3B: Eight-Digit Cues (1, 2, 3, 4, 6, 7, 8, and 9)

4.2.1. Method

4.2.2. Results

4.2.3. Comparison of the strength of the Att-SNARC between the four- and eight-digit cues conditions

5. Experiment 4: Magnitude Attentional SNARC

5.1. Experiment 4A: Four-Digit Cues (1, 2, 8, and 9)

5.1.1. Method

5.1.2. Results

5.2. Experiment 4B: Eight-Digit Cues (1, 2, 3, 4, 6, 7, 8, and 9)

5.2.1. Method

5.2.2. Results

5.2.3. Comparison of the strength of the Att-SNARC between the four- and eight-digit cues conditions

6. Comparing the Strength of the Att-SNARC Among Experiments 1-4

7. Discussion

8. Conclusions

References

Chapter 10: Age-related changes in strategic variations during arithmetic problem solving: The role of executive control

1. Age-related Differences During Arithmetic Problem Solving

1.1. Aging Effects on Arithmetic Performance

1.2. Strategic Variations with Age in Arithmetic

2. The Role of Executive Processes in Strategic Variations with Age in Arithmetic

2.1. Aging, Executive Control Processes, and Arithmetic Strategy Use

2.2. Aging, Executive Control Processes, and Arithmetic Strategy Selection

2.2.1. Correlational data

2.2.2. Experimental data

2.2.3. Brain imaging data

2.3. Aging, Executive Control Processes, and Arithmetic Strategy Execution

2.3.1. Strategy switch costs

2.3.2. Strategy sequential difficulty effects

2.3.3. Sequential modulations of poorer strategy effects

3. Conclusions and Future Directions

References

Chapter 11: Subtypes and comorbidity in mathematical learning disabilities: Multidimensional study of verbal and visual m ...

1. MLD and DD

2. WM Models in MLD Research

3. Verbal and Visual Memory Deficits in MLD

4. Analysis of Study Data

4.1. Coverage of Memory Domains and Power

5. Effect Sizes from Studies

6. Matching Reading and IQ in MLD and Control Groups

6.1. Developmental Pathways

6.2. Fractionating Subtypes of Visual Memory

6.3. Fractionating EFs

6.4. Studies with Ability-Matched Young Controls and Intervention

7. Processing Networks and the Impact of General Task Difficulty

8. MLD Subtypes, Network Coordination, and Individual Variability

8.1. Overall Conclusions

Acknowledgment

Appendix

References

Chapter 12: Neurocognitive accounts of developmental dyscalculia and its remediation

1. Introduction

2. Multiple Cognitive Factors Involved in DD

2.1. Number Sense Deficits

2.2. Memory Deficits

2.3. Ordinality and Other Numerical Mapping Deficits

2.4. Other Domain-General Processing Deficits

2.5. Math Anxiety

3. Multiple Neurocognitive Systems Involved in DD

3.1. Dorsal and Ventral Streams' Deficits

3.2. Frontoparietal Deficits

3.3. Medial Temporal Lobe Deficits

3.4. Network-Level Deficits

4. Remediating DD

4.1. Pedagogical and Cognitive Studies

4.2. Neuroimaging Studies

4.3. Individual Differences in Intervention Outcomes

4.4. Remediation of Persistent DD

4.5. Emergent Approaches: Embodied Intervention

5. Conclusions and Future Directions

References

Chapter 13: Approximate numerical abilities and mathematics: Insight from correlational and experimental training studies

1. Cognitive Foundations for Mathematical Abilities

1.1. Approximate Number System

1.2. Associations Between Approximate Numerical Magnitudes and Symbolic Numbers

1.3. Correlations Between Approximate Numerical Abilities and Mathematics Achievement

1.4. Experimental Training Studies on the Relationship Between the ANS and Mathematics

1.5. Alternative Explanations

2. Emerging Ideas from Empirical Work

3. Conclusions

References

Chapter 14: Brain stimulation, mathematical, and numerical training: Contribution of core and noncore skills

1. Introduction

2. Core and Noncore Skills

3. Brain Stimulation

4. Brain Stimulation and Mathematical Training

5. Conclusions and Future Directions

Acknowledgments

References

Index

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