1. •Designing educational experiences without an
understanding of the brain is like designing a glove
without an understanding of the human hand.
-attributed to Leslie Hart (1983)
1
Tokuhama-Espinosa Feb 2017

2. The Mathematical Brain:
What Teachers Need to Know
Tracey Tokuhama-Espinosa, Ph.D.
tracey.tokuhama@gmail.com
www.conexiones.com.ec
www.thelearningsciences.com
September 2019

3. Tokuhama-Espinosa 3
Conexiones: The Learning Sciences Platform
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Suggested citation format:
Tokuhama-Espinosa, T. (2019). The Mathematical Brain: What Teachers Need to Know [PowerPoint slides]. Retrieved from
http://thelearningsciences.com/site/portfolio-items/mathematical_brain/

4. Tracey Tokuhama-Espinosa
• Professor, Harvard University Extension School: Psych 1609 “The
Neuroscience of Learning: Introduction to Mind, Brain, Health and
Education science”
• OECD: Member of the expert panel on Teachers New Pedagogical
Knowledge based on contributions from Technology and Neuroscience
• Latin American Social Science Research Faculty, Ecuador: Educational
Researcher and Professor
• Interdisciplinary researcher in neuroscience, cognitive psychology and
education (cultural anthropology and linguistics).
• Associate Editor of the Nature Partner Journal Science of Learning
• Boston University: BA, BS, magna cum laude; Harvard University:
Master’s in International Educational Development; Capella University:
Ph.D. In Professional Studies in Education (Mind, Brain and Education
Science)
• Former Director of the Teaching and Learning Institute at the
Universidad San Francisco de Quito Ecuador
• Former Dean of Education at the Universidad de las Américas, Quito,
Ecuador
• Teacher at all levels of education (K-University, continuing education)
with more than 29 years of experience in 34 countries.
4

5. Invitation
Tokuhama-Espinosa 5
www.conexiones.com.ec
www.thelearningsciences.com
tracey.tokuhama@gmail.com

6. 3-2-1
• 3 things that are new (unknown before)
• 2 two things so interesting you will continue to research them or share with
someone else
• 1 thing you will change about your practice based on the information shared
today
Tokuhama-Espinosa 6

7. conexiones.com.ec

8. Today
1. Definitions
2. The Story
3. Ten (10) key ideas about Education
from Neuroscience Perspective
(Mind, Brain and Education)

9. Definitions
Tokuhama-Espinosa 9

10. 24 January 2021 Tokuhama-Espinosa 10

11. 24 January 2021 Tokuhama-Espinosa 11

12. 24 January 2021 Tokuhama-Espinosa 12

13. 24 January 2021 Tokuhama-Espinosa 13

14. 24 January 2021 Tokuhama-Espinosa 14

15. 24 January 2021 Tokuhama-Espinosa 15

16. 24 January 2021 Tokuhama-Espinosa 16

17. Heuristics and Bias
• Your brain is efficient and tries to save energy where ever it can.
• Heuristics are inevitable.
• Heuristics are useful, but at the same time, they can be
dangerous because they can lead to unconscious bias that
blocks learning.
Tokuhama-Espinosa 17

18. From “Education” to the “Learning Sciences”

19. EDUCATIONAL
NEUROSCIENCE
vs.
(NEURO EDUCATION)
THE LEARNING
SCIENCES
SUB- OR SECONDARY FIELD + PRIMARY FIELD
Educational Neuroscience
Neuro Education
Tokuhama-Espinosa 2017

20. EDUCATIONAL
NEUROSCIENCE
vs.
(NEURO EDUCATION)
THE LEARNING
SCIENCES
MIND, BRAIN, AND
EDUCATION SCIENCE
The focus is not on Learning or Teaching, but rather
“THE TEACHING-LEARNING DYNAMIC”
Tokuhama-Espinosa 2017

21. learning
teaching
teaching
teaching
teaching
teaching
learning
learning
learning
learning
learning
Tokuhama-Espinosa 2017
The teaching-learning dynamic
B
A

22. learning
teaching
teaching
teaching
teaching
teaching
learning
learning
learning
learning
learning
Tokuhama-Espinosa 2017
The teaching-learning dynamic
B
A
X

23. learning
teaching
teaching
teaching
teaching
teaching
learning
learning
learning
learning
learning
diagnosing
planning
context
reflection
application
feedback
motivation
interpretation
understanding
knowledge building
rehearsal
reinforcement
consolidation
transfer, new application, innovation
Tokuhama-Espinosa 2017

24. learning
teaching
teaching
teaching
teaching
teaching
learning
learning
learning
learning
learning
Tokuhama-Espinosa 2017
(Global) Societal
Classroom
Molecular
Individual
Big data on multiple levels
EDUCATION
SOCIOLOGY
ANTROPOLOGY
PSICOLOGY
NEUROCIENCE

25. Challenges to Educaiton
• Do teachers know enough about the brain?
25
Tokuhama-Espinosa Aug 2016

26. Why are neuromyths problematic?
They do harm!
• Studies of teachers around the world by
Gleichgerrcht and colleagues (2015) and Howard-
Jones (2009, 2012, 2014) show that more than 50%
of teachers believe in myths.
• This misinformation can do harm to student
learning.
• Neuromyth beliefs in Latin America is higher than
those in Europe, Asia and the US.
• ¿Y usted? Sabe suficiente sobre el cerebro
y cómo aprenden los estudiantes?
26
Gleichgerrcht, E., Lira Luttges, B., Salvarezza, F., & Campos, A. L. (2015). Educational neuromyths among teachers in Latin America. Mind, Brain, and Education, 9(3), 170-178.

27. New OECD
publication (2017)
Tokuhama-Espinosa 2017
Available free on: http://www.keepeek.com/Digital-
Asset-Management/oecd/education/pedagogical-
knowledge-and-the-changing-nature-of-the-teaching-
profession_9789264270695-en#.WLMBlxIrIzU

28. Key idea 1 (write this down!)
• Teaching is the most important profession in
society.
• We must base our choice on evidence
• We should begin to professionalize teaching
by calling ourselves “learning scientists”.
Tokuhama-Espinosa 28

29. Big Idea 2 (write this down!)
• “Designing educational experiences without an
understanding of the brain is like designing a
glove without an understanding of the human
hand”-Leslie Hart (1983)
Tokuhama-Espinosa 29

30. Big Idea 3 (write this down!)
• Language and Mathematics are the cornerstones of all educational programs and are
vital to the success of each citizen and their countries related to competitiveness
(OECD, 2016).
• Domain skills, such as Language and Mathematics, are developed according to a (neuro)
constructivist trajectory.
• Students who start behind, stay behind throughout their studies (Isaacs, 2012;
Stanovich, 1986), unless a great teacher helps them fill in gaps of knowledge.

31. The story, part 1
Tokuhama-Espinosa 31

32. Costa Rica (2013)

33. World Education Research Association
(sept 2015)

34. March 2019
Tokuhama-Espinosa 34

35. Problem
Not all students complete first grade
successfully (learn to read and have basic
math skills)
• Many children (estimated to be
17% of the Costa Rican
government) enter school
(approximately 5-6 years of age)
with inadequate knowledge in
Mathematics and Language to be
successful in school.
Why?
• Inadequate early childhhod
experiences
• At home?
• In state pre-schools

36. ”We’ve tried everything else, so ….”
Tokuhama-Espinosa 36
Modelos
psicológicos

37. "... as a last resort, we want to know what
neuroscience can tell us."
http://www.brainfacts.org/In-the-Lab/Tools-and-Techniques/2014/Brain-Scans-Technologies-that-Peer-Inside-
Your-Head
37

38. Hypothesis (1 of 3):
• School failure is due, at least
in part, to the
underdevelopment of
important neural pathways in
the early years.

39. Hypothesis (2 of 3):
• More students would find
greater school success in
Math and Language if the
order of key concepts was
changed.

40. Hypothesis (3 of 3):
• The order (hierarchy) of
math and language concepts
in school does not always
correlate with order in the
brain.

41. Terminology
(A) Urban Child Institute http://www.urbanchildinstitute.org/why-0-3/baby-and-brain
(B) National Geographic https://www.youtube.com/watch?v=nvXuq9jRWKE
(C)Soon-Beom HongAndrew ZaleskyLuca CocchiAlex FornitoEun-Jung ChoiHo-Hyun KimJeong-Eun SuhChang-Dai KimJae-Won KimSoon-Hyung Yi -
http://www.plosone.org/article/info%3Adoi%2F10.1371%2Fjournal.pone.0057831;
(D) Human Connectome Project http://www.humanconnectomeproject.org/
Synapsis
Pathways
Networks
A
B
C
D
Learning depends on
multiple well-functioning
networks.
Math is not in a single
part of the bran but
rather in dozens of
neural networks
throughout the brain.
Nodes

42. Schematic breakdown
Pathways
• Magnitude recognition
• Addition
• Subtraction
• Multiplication
• Division
• …
Sub domains
• Symbols
• Size
• Whole numbers
• Communal properties
• Associative properties
• Counting
• Mental number line
• …
Cognitive ability (“academic
domains”)
 Mathematics
Observable behavior
 “understand the notion of
quantity, the relationships of
order, addition, and
subtraction, using concrete
materials to resolve problems
in daily settings.
 …
Educación General Básica Preparatorio (2016), Currículo Integrador, 4. Relaciones lógico matemáticas; O.M.1.2., EGB, p.60

43. Where is ”math” in the brain?

44. Studies on the brain and mathematics
Dehaene, Piazza, Pinel, & Cohen, 2003 (http://www.unicog.org/pm/pmwiki.php/Main/Arithmetics)
Arithmetic
Triple Code:
• “3”
• “three”
• “***”

45. Studies on the mathematical brain
Mental addition
and subtraction
Price, Mazzocco & Ansari, 2013 (http://www.jneurosci.org/content/33/1/156.full.pdf)

46. Studies on the mathematical brain
Jacob, Vallentin & Nieder, 2012 (http://www.cell.com/cms/attachment/2002984313/2011335179/gr3.jpg)
Magnitude
estimation

47. Studies on the mathematical brain
Moeller, Willmes & Klein, 2015 (http://journal.frontiersin.org/researchtopic/1363/abstract-mathematical-cognition)
Numerical
cognition

48. Studies on the mathematical brain
Grabner & Stern, n/d [2010?],(http://www.ifvll.ethz.ch/research/Learning)
Multiplication vs.
Spatial roation
tasks

49. Studies on the mathematical brain
Pletzer, Kronbichler, Nuerk & Kerschbaum, 2015 (http://journal.frontiersin.org/article/10.3389/fnhum.2015.00202/full)
Math Anxiety

50. Big Idea 5
(write this down!)
There are (at least)
16 networks related
to Mathematics
• There are four broad categories of learning networks
influencing mathematics learning:
• Physiological (sensory perception) (n=3)
• Cultural-Social-Emotional (n=4)
• Cognitive (general) (n=3) and
• Cognitive (domain specific) (n=6)
which are sub-divided into 16 neural networks:
I. Hearing
II. Vision
III. Touch and Graphomotor
IV. Social cognition
V. Relations with caregivers
VI. Self-esteem
VII. Motivation and self-regulation
VIII. Memory
IX. Attention
X. Executive Functions
XI. “Language Sense” Receptive Language
XII. Symbols
XIII. Order
XIV. Patterns
XV. Categories
XVI. Relationships

51. The 16 neural networks are sub-divided into
106 neural pathways:
• Physiological
II. Vision
9. Color
10. Luminance
11. Motion
12. Size, Monocular-Binocular
13. Proximity
14. Perception versus Action
15. Spatio-temporal contrast
16. Faces versus other objects
17. Search and Saliency
18. Visual crowding
19. Spatial frequency
Physiology
(All learning)

52. The 16 neural networks are sub-divided
into 106 neural pathways:
• Physiological
III. Motor
20. Visual-Motor Integration
(graphomotor)
21. Tactile recognition of shapes
(graphomotor)
22. Writing/Scribbling (graphomotor)
23. Drawing/Tracing/Copying
(graphomotor)
24. Variant Expressions Drawing
(graphomotor)
25. Speech
Physiology
(All learning)

53. Cultural-Social-
Emotional
(All learning)
The 16 neural networks are sub-divided into
106 neural pathways:
• Cultural-Social-Emotional
IV. Cultural influence on learning
26. Cultural neuroscience (domain
variation)
27. Theory of Mind (class position)
V. Student-Teacher (Caregiver) relations
28. Cognitive-Affective dimension
29. Modeling for resiliency
VI. Self esteem
30. Self-belief, -efficacy, -worth
31. Math anxiety
VII. Motivation
32. Intrinsic motivation
33. Self-regulation

54. Cultural-Social-
Emotional
(All learning)
The 16 neural networks are sub-divided into
106 neural pathways:
• Cultural-Social-Emotional
IV. Cultural influence on learning
26. Cultural neuroscience (domain
variation)
27. Theory of Mind (class position)
V. Student-Teacher (Caregiver) relations
28. Cognitive-Affective dimension
29. Modeling for resiliency
VI. Self esteem
30. Self-belief, -efficacy, -worth
31. Math anxiety
VII.Motivation
32. Intrinsic motivation
33. Self-regulation

55. Cultural-
Social-
Emotional
(All learning)
The 16 neural networks are sub-divided into
106 neural pathways:
• Cultural-Social-Emotional
IV. Cultural influence on learning
26. Cultural neuroscience (domain
variation)
27. Theory of Mind (class position)
V. Student-Teacher (Caregiver) relations
28. Cognitive-Affective dimension
29. Modeling for resiliency
VI. Self esteem
30. Self-belief, -efficacy, -worth
31. Math anxiety
VII. Motivation
32. Intrinsic motivation
33. Self-regulation

56. General Cognitive
(All learning)
The 16 neural networks are sub-divided into
106 neural pathways:
• Cognitive (general)
VIII.Memory
34. Short-term
35. Working
36. Long-term (procedural)
37. Long-term (fact retrieval, semantic)
38. Long-term (autobiographical)
39. Long-term (episodic)
IX. Attention
40. Sustained
41. Alerting
42. Orienting
X. Executive Functions
43. Cognitive flexibility
44. Working memory
45. Inhibitory control

57. General Cognitive
(All learning)
The 16 neural networks are sub-divided into
106 neural pathways:
• Cognitive (general)
VIII.Memory
34. Short-term
35. Working
36. Long-term (procedural)
37. Long-term (fact retrieval, semantic)
38. Long-term (autobiographical)
39. Long-term (episodic)
IX. Attention
40. Sustained
41. Alerting
42. Orienting
X. Executive Functions
43. Cognitive flexibility
44. Working memory
45. Inhibitory control

58. General Cognitive
(All learning)
The 16 neural networks are sub-divided into
106 neural pathways:
• Cognitive (general)
VIII.Memory
34. Short-term
35. Working
36. Long-term (procedural)
37. Long-term (fact retrieval, semantic)
38. Long-term (autobiographical)
39. Long-term (episodic)
IX. Attention
40. Sustained
41. Alerting
42. Orienting
X. Executive Functions
43. Cognitive flexibility
44. Working memory
45. Inhibitory control

59. Mathematisa:
Five Pillars
• There are four broad categories of learning networks
influencing mathematics learning:
• Physiological (sensory perception) (n=3)
• Cultural-Social-Emotional (n=4)
• Cognitive (general) (n=3) and
• Cognitive (domain specific) (n=6)
which are sub-divided into 16 neural networks:
I. Hearing
II. Vision
III. Touch and Graphomotor
IV. Social cognition
V. Relations with caregivers
VI. Self-esteem
VII. Motivation and self-regulation
VIII. Memory
IX. Attention
X. Executive Functions
XI. “Language Sense” Receptive Language
XII. Symbols
XIII. Order
XIV. Patterns
XV. Categories
XVI. Relationships

60. The 16 neural networks are sub-divided
into 106 neural pathways:
• Cgnition (domain specific)
XI. The Number Sense
46. Initial: Non-verbal grouping
(categories and patterns)
47. Initial: Non-verbal comparative value
and magnitude estimation
48. Secondary: Verbalized grouping and
comparative magnitude estimation
Domain specific
(Mathematical)

61. The 16 neural networks are sub-divided
into 106 neural pathways:
• Cognition (domain specific)
XII. Symbols
49. Coding (abstract to tangible symbols)
50. Analogic symbolic systems
51. Coding (whole numbers [Arabic])
52. Triple code
53. Symbolic vs. Non-symbolic magnitude
(identification/labeling)
54. Symbolic vs. Non-symbolic magnitude (production)
55. Symbolic vs. Non-symbolic magnitude (arithmetic)
56. Coding (non numerical symbols in math)
Domain specific
(Mathematical)

62. The 16 neural networks are sub-divided into
106 neural pathways:
IV. Cognition (domain specific)
IV. Symbols
64. Form/Shape/Geometry (space and rotation)
65. Form/Shape/Geometry (geometric description)
66. Form/Shape/Geometry(identify and name angles)
67. Form/Shape/Geometry (alignment and identification of
advanced geometric forms)
68. Form/Shape/Geometry (shape replication)
69. Form/Shape/Geometry (description of advanced
geometric figures)
Domain specific
(Mathematical)

63. The 16 neural networks are sub-divided
into 106 neural pathways:
IV. Cognition (domain specific)
XIII. Order
70. Ordinality (range [including before and after;
first-last])
71. Ordinality (fixed)
72. Ordinality (unique)
73. Ordinality (inverse)
74. Ordinality (counting aloud)
75. Ordinality (vs. cardinality; 3 vs. 3rd)
76. Ordinality (relative order)
77. Sequence order
78. Positional value sequence
Domain specific
(Mathematical)

64. The 16 neural networks are sub-divided
into 106 neural pathways:
IV. Cognition (domain specific)
XIV. Patterns
79. Repetition and Regularity
80. Identification of variety (in
multiple formats)
81. Formula processing (arithmetic)
Domain specific
(Mathematical)

65. The 16 neural networks are sub-
divided into 106 neural pathways:
IV. Cognition (domain specific)
XV. Categories
82. Initial Number Sense (pre-verbal
categorization)
83. Classification (characteristics)
84. Classification (relations)
Domain specific
(Mathematical)

66. The 16 neural networks are sub-divided
into 106 neural pathways:
IV. Cognition (domain specific)
XVI.Relations
85. Numerosity (sets)
86. Numerosity (order abstraction and relevancy)
87. Approximations and Estimations
88. Decomposition (conservation)
89. Decomposition (arithmetic)
90. Decomposition (shapes)
91. Decomposition (equivalencies)
92. Composition of forms
93. Equivalencies (like sets)
94. Equivalencies (rreproduction)
95. Equivalencies (division)
96. Equivalencies (fractions)
Domain specific
(Mathematical)

67. The story, part 2
Tokuhama-Espinosa 67

68. Plotted Math 0-24 years
Manuscript sent to 30
neuroscientists who
study mathematics and
the brain
Tokuhama-Espinosa 68

69. Not only Math and Language
• Architecture
• Gender studies
• Museology
• Design
• Peace studies
• Administration
• Economic
• Sciences
• International trade
• Communications
• Technology
• Artificial intelligence
• Gardening
• Shop keepers
• News paper journalists
• Nannies
• Bankers….
Tokuhama-Espinosa 69

70. Big Idea 6 (write this down!)
• Everything humans learn is a symbol, a pattern, an expression of
order a category and/or a relationship.
Tokuhama-Espinosa 70

71. Ages vs. Stages vs. Prior Experience
• The order in which information has been presented over the lifespan of the
student is more important than the age or cognitive stage of a person.
Tokuhama-Espinosa, 2014

72. Big Idea 7 (write this down!)
• Education is complicated.
• The brain is compex.
• Solutions to problems in education cannot
be simplistic.
Tokuhama-Espinosa 72

73. Big Idea 8-9 (write this down!)
• Different classroom activities stimulate
different neural networks.
• All of the networks must be linked before a
student can demonstrate behavior related to
the domain area.
Tokuhama-Espinosa 73

74. Big Idea10 (write this down!)
• The first rule of teaching is the same as
doctors: Do No Harm.
Tokuhama-Espinosa 74

75. Answers to the research questions
• Current knowledge of neuroscientific education is in its infancy.
• We know more than ever, but we know very little.
• We know that there are at least 16 related neural networks for
both early math and pre-literacy learning in the brain.
• "Typical" teaching practices apparently do not stimulate all
networks equally well. Likely that early school failure is at least
partly due to this lack of adequate stimulation.

76. Symbols
Tokuhama-Espinosa 76

77. Patterns
Tokuhama-Espinosa 77

78. Order
Tokuhama-Espinosa 78

79. Categories
Tokuhama-Espinosa 79

80. Relations
Tokuhama-Espinosa 80

81. 3-2-1
• 3: Three things you didn’t know before
• 2: Two things you will continue to research or talk about
• 1: One thing you will change in your personal or professional life based
on the information that was shared
Tokuhama-Espinosa 81

82. Contact:
Tracey Tokuhama-Espinosa, Ph.D.
Quito, Ecuador
www.conexiones.com.ec
tracey.tokuhama@gmail.com
82
-Attributed to John Cotton Dana