Cognitive and educational neuroscience of

math learning: implications for

interventions in learning disabilities

Vinod Menon

Department of Psychiatry & Behavioral Sciences

Department of Neurology & Neurological Sciences

Symbolic Systems Program

Program in Neuroscience

Stanford University

http://braindevelopment.stanford.edu

Acknowledgements

Miriam Rosenberg-Lee

Teresa Iuculano

Soohyun Cho

Sarit Ashkenazi

Shaozheng Qin

Kaustubh Supekar

Dietsje Jolles

Other lab members

Dave Geary

Lynn Fuchs

Stanford Brain Development Project

Research Questions

in Educational Neuroscience

How does brain development influence learning?

What brain systems and circuits support academic skill acquisition?

How does learning go awry in some children?

What can be done to successfully remediate poor skills in struggling students?

Math Ability and Life Success

• Children’s math ability is highly predictive of future academic and

professional success, more than reading

• Math Disability (MD): 20% of children (3-6% dyscalculia)

• MD affects education, employment, everyday life situations

Current Research Topics in

Mathematical Cognition & Dyscalculia

Cognitive

• Differentiating between various components of mathematical information processing (e.g. calculation and retrieval)

• Contributions of working memory, attention, visuospatial abilities, abstract reasoning

• Performance under pressure , Math anxiety, Stereotype threat

• Mindset, Motivation

Developmental

• Development of number sense and problem solving abilities in children

Educational

• Development of remediation programs for individuals with poor math performance

Neurobiological

• Relative contribution of specific brain regions vs. distributed system function

• Neural basis of performance differences across individuals

• Neural basis of development and dysfunction in mathematical reasoning

Outline

• Brain organization for mathematical cognition– Multiple cognitive processes

– Multiple brain circuits

– Multiple sources of deficits in learning disabilities

• Developmental changes underlying mathematical skill development– Protracted developmental trajectory

– Fine-grain developmental differences during critical stages of skill acquisition

• Development of memory-based knowledge – Learning to use efficient strategies

– Crucial role of associative memory system

– Core memory systems play an important role in knowledge acquisition

• Training and brain plasticity – Normalization of brain response with training in children with learning disabilities

– Changes in functional and structural circuits

– Brain-based predictors of learning

• Implications for educational neuroscience and interventions in learning disabilities

Levels of Mathematical Information Processing

• Basic Number Processing– Symbol mapping

– Magnitude Judgment

• Fact Retrieval – Calculation vs. Automatic Retrieval

• Complex Mathematical Computation– Involves many other cognitive functions

• Working Memory

• Attention

• Visuospatial Processing

Which is bigger?

3 5

6 x 6 = 36

17 - 8 = ?

Basic number processing and fluent fact retrieval are key bottlenecks in dyscalculia.

3

Canonical brain areas involved in numerical

problem solving

Results of meta-analysis - 44

studies of arithmetic in adults

(Neurosynth, Yarkoni et al.

2011)

Ashkenazi et al., 2012 JLD.

Neurobiological basis of math and

reading disabilities

Distributed systems involved in math cognition and

learning

Menon (2013) Handbook Math Cogn.

Fias et al. (2013) Trends in Neursci Ed.

Cannot assume that same brain systems are

similarly involved in different stages of learning

Mathematical Skill Development

Developmental of Arithmetic skills

8 10 12 14 16 18 20

0

1

2

3

4

Nu

mb

er

of su

bje

cts

AgeRivera (2005) Cerebral Cortex

• Aim: Examine neurodevelopmental changes in mental arithmetic (ages 8-19)

• Task: Single-digit addition and subtractiona + b = ca – b = c

• Analysis: Age-related increases and decreases in

brain activation

Developmental changes: Behavior

• Accuracy: asymptotes by age 9

• RT: changes continue through adolescence

8 10 12 14 16 18 20

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Accura

cy

Age

8 10 12 14 16 18 20

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

1.02

Accura

cy

Age

8 10 12 14 16 18 20

600

800

1000

1200

1400

1600

1800

2000

2200

2400

Reaction t

ime (

msec)

Age

8 10 12 14 16 18 20

600

800

1000

1200

1400

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1800

2000

2200

2400

Reaction t

ime (

msec)

Age

Experimental trials Control trials

Anterior to Posterior Shift in Arithmetic

Processing with Age

Rivera et al., 2005

Increases with age

Decreases with age

Addition & Subtraction

Combined

Age range: 8 – 19 years

Developmental changes: Children vs. Adults

PFC, MTL and basal ganglia are more

engaged in children

8 10 12 14 16 18 20

-3

-2

-1

0

1

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3

L S

FG

Age

8 10 12 14 16 18 20

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-2

-1

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L M

FG

Age

8 10 12 14 16 18 20

-3

-2

-1

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3

L I

FG

Age

8 10 12 14 16 18 20

-3

-2

-1

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1

2

3

L V

entr

al str

iatu

m

Age

8 10 12 14 16 18 20

-3

-2

-1

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3

L P

uta

me

n

Age

8 10 12 14 16 18 20

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-2

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G/P

HG

Age

8 10 12 14 16 18 20

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-2

-1

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3

L H

ipp

oca

mp

us

Age

But also - Basal ganglia,

hippocampus and

parahippocampal gyrus

Prefrontal cortex

Effortful processing Procedural and declarative memory systems

L Parietal and Lateral occipital cortex activation increases with age

8 10 12 14 16 18 20

-3

-2

-1

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1

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3

L S

upra

ma

rgin

al g

yru

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Age

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-2

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nf.

Occip

ita

l g

yru

s

Age

Parietal and LOC are more engaged in adults

Summary: Neurodevelopmental changes in

mental arithmetic

• Older individuals showed greater activation in the left parietal cortex, along the supramarginal gyrus and adjoining anterior intra-parietal sulcus

• By contrast, younger subjects showed greater activation in the prefrontal cortex, including dorsolateral and ventrolateral prefrontal cortex and the anterior cingulate cortex – Reflects comparatively more working memory and attentional resources to

achieve similar levels of mental arithmetic performance

• Increased functional specialization of the left parietal cortex in mental arithmetic, a process that is accompanied by decreased dependence on memory and attentional resources with development

• Younger subjects also showed greater activation of the hippocampus and dorsal basal ganglia – Reflects greater demands placed on both declarative and procedural

memory systems

How does a child solve 7 + 8 ? Decoding

brain activity patterns associated with

counting and retrieval strategies

Counting Fingers

Sum

Max

Min

Verbal Counting

Sum

Max

Min

Counting-based

Memory-based

Decomposition

Retrieval

Increasing maturity

Dec

reas

ing

reac

tion t

imes

and c

ogn

itiv

e re

sourc

es

Geary (1994)

How does a child solve 7 + 8 ? Decoding brain activity

patterns associated with counting and retrieval strategies

Strategy session fMRI session

Cho (2011) Developmental Science

Performance Matched

How does a child solve 7 + 8 ?:

Decoding Children’s Brain Activity Patterns

during Counting versus Retrieval

• Participants

2-3 grade children (7-10 years)

• Behavioral task

1) strategy assessment session (e.g., 3 + 4 = ?)

- % of trials retrieved, % of trials counted, etc.

2) fMRI session

- standard addition (both addends greater than 1; e.g., 3 + 4 = 8 ?)

- plus 1 addition control (one of the addend is 1; e.g., 7 + 1 = 8 ?)

• Functional image analysis

- univariate approach based on GLM

- multivariate approach using searchlight classification

Distinct Neural RepresentationsRetrievers (n = 19) vs. Counters (n = 17) MVPA

Functional reorganization and

Enhanced Cognitive control Activation differences Retrievers vs. Counters

L VLPFC

overlaps with the

MVPA map

Summary

Retrieval and counting strategies during early learning are

characterized by distinct patterns of activity in a distributed

network of brain regions involved in arithmetic problem

solving and controlled retrieval of arithmetic facts.

Reorganization and refinement of neural activity patterns in

multiple brain regions, including the hippocampus, plays a

dominant role in the transition to memory-based arithmetic

problem solving.

Multivariate approaches can provide novel insights into fine-

scale developmental changes in the brain.

Cho (2011) Developmental Science

Hippocampal-prefrontal engagement and

dynamic causal interactions in the

maturation of children’s fact retrieval

Cho (2012) J Cogn Neurosci.

Retrieval Fluency

Increased PFC-PPC-MTL responses with

Retrieval Fluency

Increased MTL connectivity with

Retrieval Fluency

Dynamic Causal PFC-MTL Interactions

Cho (2012) J Cogn Neurosci.

Summary

Higher retrieval fluency associated with increased recruitment of hippocampus and

prefrontal cortex.

Significant effective connectivity of the right hippocampus with bilateral PFC.

Dynamic causal modeling analysis revealed strong bidirectional interactions

between the hippocampus and the PFC.

Causal influences from the left VLPFC to the hippocampus served as the main

“top–down” component, whereas causal influences from the hippocampus to the

left DLPFC served as the main “bottom–up” component of this retrieval network.

Hippocampal–prefrontal circuits are important in the early development of

arithmetic retrieval fluency.

Cho (2012) J Cogn Neurosci.

Hippocampus critical for learning

Hippocampal-neocortical functional

reorganization underlying children’s

cognitive development

• The ability to efficiently retrieve basic facts

from memory and bring them to bear in ever-

changing situations is a cardinal feature of

memory-based problem solving. (Siegler, 1996;

Geary, 2006)

• Little is known about how this ability

develops as the brain matures from childhood

through adolescence into adulthood?

Qin et al., Nature Neurosci. 2014

Overlapping waves theory

• Maturation of mathematical problem solving skills in children

Perc

enta

ge u

se

(Adapted from Chen & Siegler, 2000)Age & Experience

counting

Strategy 2 Strategy 3

Fact retrieval

• Over development, problem solving skills become less dependent on effortful

procedures i.e counting, and more gradually dependent on memory-based

strategies

Experimental designPerc

enta

ge u

se

Age & Experience

Strategy 1

Strategy 2 Strategy 3Fact retrieval

Exp-1: longitudinal fMRI study in children

Exp-2: cross-sectional in adolescents and adults

Experimental Design

Demographics

Experimental design and behavioral results

Developmental changes in accuracy and

reaction time during solving addition problems

Block design Event-related

Brain areas activated involved in solving

arithmetic problems in children

Longitudinal changes in hippocampal

engagement in children

Longitudinal decrease in prefrontal-parietal

engagement

Longitudinal changes in hippocampal-neocortical

functional coupling in relation to individual

improvements in children’s retrieval fluency

Developmental changes in medial temporal lobe

engagement from childhood through adolescence

into adulthood

Time-1 (a) Time-2 (b) Adolescents (c) Adults (d)

Longitudinal changes in hippocampal

engagement during childhood, and further

development through adolescence into adulthood

Trial-by-trial stability of representations in children at

Time-1 and Time-2 children, adolescents, and adults

Learning mechanisms in hippocampus

Summary

• Hippocampal system is pivotal to strategic transition from procedure-based to

memory-based problem-solving strategies.

• Transition to memory-based retrieval parallels increased hippocampal and

decreased prefrontal-parietal engagement during arithmetic problem solving.

• Longitudinal improvements in retrieval-strategy use are predicted by increased

hippocampal-neocortical functional connectivity.

• Beyond childhood, retrieval-strategy use continues to improve through

adolescence into adulthood and is associated with decreased activation but more

stable inter-problem representations in the hippocampus.

• Dynamic role of the hippocampus in the maturation of memory-based problem

solving and establish a critical link between hippocampal-neocortical

reorganization and children's cognitive development.

2. What are the neural mechanisms that drive

some children to acquire math skills faster than

others?

1. How do brain circuits change in

response to math learning in childhood?

Training and Neuroplasticity

• Growing interest in training-related changes of brain

structure and function associated with math and

dyscalculia

• Little is known about changes in brain circuits in

academic domains: especially math

Training and Neuroplasticity

Untrained vs. Trained Problems

Ischebek (2007, 2009).

Mathematical Disabilities

• Developmental math disability (dyscalculia) affects 5-6%

of school-age children (stable and persistent)

• The ability to quickly and accurately retrieve basic

arithmetic facts is one of the most consistent deficits in

children with mathematical disabilities

• Mathematical difficulties are common in school-age

children

• Much less studied than dyslexia

• Major consequences for academic success

Cognitive Learning: Study design

Training and Neuroplasticity

Normalization of aberrant functional brain responses in

dyscalculic children after 8 weeks of math tutoring

Normalization of aberrant functional brain

responses in dyscalculic children after 8 weeks of

math tutoring

Changes in MLD vs. TD groups

Brain plasticity and normalization

• 8 weeks of 1:1 cognitive tutoring not only remediates poor performance in

children with MLD, but also induces widespread changes in brain activity

• Neuroplasticity manifests as normalization of aberrant functional responses in a

distributed network of parietal, prefrontal and ventral temporal-occipital areas

that support successful numerical problem-solving

• Remarkably, machine learning algorithms show that brain activity patterns in

children with MLD are significantly discriminable from neurotypical peers

before, but not after, tutoring

• Multivariate measures reveal that children with MLD with greater

neuroplasticity also exhibit larger performance gains with tutoring

• Identifies functional brain mechanisms underlying effective intervention in

children with MLD and provides novel metrics to track remediation

Summary

2. How do intrinsic brain circuits

change in response to math learning in

childhood?

Network View of Math Cognition

• Math cognition is accomplished via the dynamic assembly of brain regions.

• Connectivity between brain regions changes with development.

• Learning and remediation involve strengthening connections between key brain regions.

1. Select a seed: e.g. Left IPS

4. Functional connectivity of Left IPS

y= -48 z = 44 x = -46

2. Extract time course from seed

3. Correlated with time courses from other areas

Functional Connectivity Analyses

Math tutoring strengthens functional

connectivity between parietal cortex and

hippocampus

p < .01 height, 128 voxels

5

0t-sc

ore

0

0.02

0.04

0.06

beta

x = -22

Jolles et al, under review

Left Intra-Parietal

Sulcus (IPS)

Hippocampus

Hippocampus

z = 6

p < .01 height, 128 voxels

5

0t-sc

ore

Pre Post

0

0.05

0.1

0.15

Beta

L IFG R IFG

0

0.02

0.04

0.06

beta

x = -22

Jolles et al, under review

Left Intra Parietal

Sulcus (IPS)

Hippocampus

Bilateral IFG

Hippocampus

Math tutoring strengthens functional

connectivity

Performance gains are associated with

increased connectivity

Parietal-Hippocampal connectivity changes

Per

form

ance

chan

ges

• No changes in control group

0

0.5

1

1.5

2

-0.05 0 0.05 0.1 0.15

Brain-behavior correlations:

Jolles et al, under review

Cognitive Training Reduces Math Anxiety

Cognitive Training Reduces Amygdala

Reactivity

Supekar et al, under review

Cognitive Training Reduces Amygdala

Connectivity

Changes in Amygdala Reactivity Predicts

Reduction in a Child’s Math Anxiety

Remediation of aberrant amygdala reactivity

predicts tutoring-induced reductions in math anxiety

Neural predictors of individual differences in response to

math tutoring in primary-grade school children

Quantitative measures of brain structure and intrinsic

brain organization can provide a more sensitive marker of

skill acquisition than behavioral measures.

Supekar et al. PNAS 2013

Anatomical and functional predictors of

learning in children

Supekar et al. PNAS 2013

8-week Math Training

Neuropsychological assessments

(fMRI scan and math tasks)

fMRI scan and math tasks

Participants:

3rd grade children (N = 22)

Wide range of math abilities: 5th– 96th percentile

One-on-One Math tutoring

Math tutoring improves arithmetic performance

©2013 by National Academy of Sciences

Strategy shift

p = .067

Gray matter volume in hippocampus correlates with

improvement in arithmetic performance in response to

8 wk of one-to-one math tutoring

Functional connectivity of the hippocampus predicts

arithmetic performance improvements in response to

tutoring

©2013 by National Academy of Sciences

Functional connectivity of the hippocampus shows

highest correlation with arithmetic performance

improvement

©2013 by National Academy of Sciences

Summary – Training & Plasticity

• Tutoring normalizes widespread hyper-activation in children

with learning disabilities

• Multivariate measures reveal that children with MLD who

demonstrate greater neuroplasticity also exhibit larger

performance gains with tutoring

• Quantitative measures of brain structure and intrinsic brain

organization can provide a more sensitive marker of skill

acquisition than behavioral measures

• In typically developing children, training strengthens intrinsic

functional circuits linking numerical and mnemonic processes,

resulting in more efficient performance

• Cognitive tutoring can reduce anxiety

Conclusions

• Brain organization for mathematical cognition– Multiple cognitive processes

– Multiple brain circuits

– Multiple sources of deficits in learning disabilities

• Developmental changes underlying mathematical skill development– Protracted developmental trajectory

– Fine-grain developmental differences during critical stages of skill acquisition

• Development of memory-based knowledge – Learning to use efficient strategies

– Crucial role of associative memory system

– Core memory systems play an important role in knowledge acquisition

• Training and brain plasticity – Normalization of brain response with training in children with learning disabilities

– Changes in functional and structural circuits

– Brain-based predictors of learning

• Implications for educational neuroscience and interventions in learning disabilities

Brain systems involved in math learning change

dynamically with age and skill

Menon (2013) Handbook Math Cogn.