This is a version of a workshop I have done for NCTM and other conferences.

1. YOUR BRAIN

2. ON NUMBER

3. HELLO
Dr. Sara Stetson
HELLO
My Name Is
@GoodCognitions
www.sstetson.wordpress.com
Dr. Sara Stetson

4. What is number sense?
Core Systems of number:
-Core system 1: Approximate Number
-Core System 2: Exact Number
-Object Attention System
Space and number
Integer representation
Your Brain on Math
- Dyscalculia
Agenda

5. Ramanujan
Turing
Tao
Gauss
Euler
Kovalevskaya
Noether
Mirzakhani
Great Mathematicians

6. Better Mathematicians

7. ACTIVITY
I give you one billion dollars
…But you have to count it first

8. Charles & Ray Eames Powers of Ten 1977

9. STUDENTS DAYS
1 0.20
2 3
7 7
1 14
1 15
1 17.5
1 31
1 4,380
1 6,935
1 7,300
1 10,000
1 694,449

10. Infants

11. Animals

12. Brain Studies

13. Indigenous People

14. Language shapes the
way we think, and
determines what we
think about--Benjamin Lee Whorf

15. Orwell: Newspeak
An extreme
Whorfian view

16. A Language Without Numbers

17. Recursion Recursion

18. Pirahã have no fixed numbers. The word that
he had long taken to mean “one” (hoi, on a
falling tone) is used by the Pirahã to refer,
more generally, to “a small size or amount,”
and the word for “two” (hoi, on a rising tone)
is often used to mean “a somewhat larger
size or amount.”
--John Colapinto The New Yorker

19. Approximate System

20. Fight Flight Foraging

21. McComb, Packer, & Pusey, 1994

22. The ANS at work

23. Number sense = ANS

24. (e.g., Brannon, 2001; Dehaene, et al., 2004; Lipton & Spelke, 2003; McCrink & Wynn, 2005; Wood & Spelke, 2005; Xu, 2002)
Cross-modality

25. Approximate Number System
Gross estimation
(e.g., Brannon, 2001; Xu, 2002; Lipton & Spelke, 2003; Wood & Spelke, 2005; McCrink & Wynn, 2005; deHevia& Spelke, 2008)
No tracking individual elements

26. Approximate Number System
(De Hevia& Spelke, 2008)
vs
DeHevia & Spelke 2010

27. 8 vs. 16
Ratio 1:2
Six Months
But Not
8 vs. 12

28. 16 vs. 24
Ratio 2:3
Nine Months
Adults
Ratio 9:10

29. Discrimination
(e.g., Brannon, 2001; Berger, et al., 2006; Libertus & Brannon, 2009 Lipton & Spelke, 2003; McCrink & Wynn, 2005; Wood & Spelke, 2005)
Always 1010 or 20
Rediculously
cute baby

30. What is Ratio-Dependence?
(Dehaene, 2003; Gallistel & Gelman, 2000; Huntley-Fenner, 2001; Temple & Posner, 1998)
2 4 8 16 32 64 128
5432 6 7 8

32. Fixation Sample Delay
TEST 1 TEST 2
(Nieder & Miller 2003,2004)

33. I Love You Nine!
Tudusciuc & Neider 2007

34. I Can’t Stand the Distance!

35. I Love You Both!

36. Fuzzy Math?
I’ll fire when you are near, but you’re sometimes not so clear...

37. Approximate Number System Precision
Target Number
(Neider, 2004; Neider & Miller, 2003)
Target Number
Representation of a number
Neider & Miller 2003

38. Target Number
(Neider, 2004; Neider & Miller, 2003)
Target Number
Low vs. high variability
Neider, 2004

39. Approximate Number System Precision
(Halberda, 2014)
Halberda, 2014; Dehaene, 1997

40. Weber’s Law

41. OK, Not So Much…

42. That’s Better
And the next Just Noticeable Difference goes to...

43. Me! Notice
Me!No! Over
here!
OUCH!

44. Weber’s Law
Discriminate 13 from 10 at 90%
Numerical distance of 3
Double the reference array to 20:
How far from 20 to reach 90% correct discrimination?
Double the target array to 26:
Numerical distance of 6
(From Dehaene, 2011)

45. Target Number
(Neider, 2004; Neider & Miller, 2003)
Target Number
Low Weber fraction = low
variability/high precision
High Weber fraction = high
variability and low precision
SD = n(W)
Halberda, 2014; Neider, 2004; Neider & Miller, 2003

46. (Pica, Izard, Lemer, & Dehaene, 2004)
Universal

47. Munduruku´ Math

48. Munduruku´ Math

49. 1
2
3
4
5
6
Munduruku´ American
7
8
9
Number of dots shown on screen
Pica, Izard, Lemer, &Dehaene, 2004

50. Look, here come
some blue dots!
Now they’re being
covered up!
Here come some
more blue dots
Barth et al., 2005

51. And here come
some red dots!
Are there more blue dots,
or more red dots?
Barth et al., 2005

52. (Gilmore, McCarthy, & Spelke, 2007)
Gilmore et al. (2007)
21 21 30 21 30 34
Sara has 21 candies She gets 30 more John has 34 candies.
Who has more?

53. 0
0.2
0.4
0.6
0.8
1.0
1.2
0 2 4 6 8 10 12 14 16 18 20 22
W
Age
Typical Weber Development

54. Exact System

55. Monkey Math

56. Target Number
(Neider, 2004; Neider & Miller, 2003)
Number of Items
1000
2000
3000
4000
5000
1 2 3 4 5 6
Adapted from: Smith-Chant et al., 2005; Piazza et al., 2002
K
1
2

57. Newborns do Math!
tu– tu- tu- tu- tu- tu- tu- tu- tu- tu- tu -tu
tuuuuu= tuuuuu- tuuuuu- tuuuuu
ra- ra- ra- ra- ra- ra- ra- ra- ra- ra- ra- ra
raaaaa- raaaaa- raaaaa- raaaaa
Iizard et al., 2008

58. Newborns do Math!
Izard, et al., 2008
30
10
20
40
4 vs 12 6 vs 18 4 vs 8
look
time
congruent
incongruent

59. (Wynn, 1992)
Object permanence?

60. Berger, et al 2006

61. (Rugani, Regolin, & Vallortigara, 2010)
Rugani, et al., 2013
Don’t Count Your Chickens

62. Mama Mia!

64. Rugani, et al., 2013
Or Your Chickens Count You

65. Bolek, Wittlinger, & Wolf, 2002

66. ANT
FOOD
Bolek, Wittlinger & Wolf, 2002
Amputee Ants?

67. (Feigenson& Carey, 2003; Feigenson, Carey & Hauser, 2002; VanMarle, 2005)
Search

68. (Feigenson& Carey, 2003; Feigenson, Carey & Hauser, 2002; VanMarle, 2005)
Persistence

69. Limits
0 10 20 30 40 50 60 70 80 90
1 vs 2
2 vs 3
3 vs 4
2 vs 4
1 vs 4
Percent choosing more
Choiceratio
12 month olds- Manual Search
(Adapted from Hauser, et al., 2000)
Adapted from Carey, 2014

70. Linear Number

71. The log to linear shift is hard

72. Why?

73. 2 yrs. ______________________________________________________5 yrs.
Cardinality Principle
Spelke & Snedecker, 2012

74. (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
1:1 Principle

75. (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
Abstraction Principle

76. (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
Order Irrelevance Principle

77. (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
Stable Order Principle

78. (Butterworth, 2006; Gelman & Gallistel, 1978; Wynn, 1990)
N + 1Successor Function Principle

79. Remember This?
(De Hevia& Spelke, 2008)
vs
DeHevia & Spelke 2010

80. Number and Space
Smaller Larger
65

81. Number and Space
Smaller Larger
65
60 70504030 80 90 100
Dehaene, Bossini, & Giraux 1993

83. Number vs Number

84. Space: Magnitude

85. Number vs Relative Size

86. Space: Relative Magnitude
1 3 6

87. 16 vs. 24
Ratio 2:3
Remember This?
Adults
Ratio 9:10
Nine Months

88. Number vs Position

89. Space: Position
(Cantlon & Brannon, 2006; DeHevia 2013; Lyons & Beilock, 2011; Picozi, et al., 2010)
1 3 6

91. Merritt, et al., 2011

92. Number vs. Extent

95. Space: Continuous Extent
(Brannon, Lutz, & Cordes, 2006; Cordes & Brannon, 2009; De Hevia, 2013; De Hevia & Spelke, 2010)
1 3 6

96. Remember This?
(De Hevia& Spelke, 2008)
vs
DeHevia & Spelke 2010

97. An Age-Old Problem: Number vs.
Extent
Which row has more?

98. An Age-Old Problem
Now we both have the same, right?

99. Feigensen, Carey, & Hauser, 2002

100. Language vs. Space

101. Counting Rules are Preverbal
Skill Example Studies Visual Representation
1:1 with objects and
amounts
Potter & Levy, 1968; Sarnecka
& Gelman, 2004
± one = A new count
word
Lipton & Spelke, 2006;
Sarnecka & Carey, 2008
Stirring ǂ A new
count word
Lipton & Spelke, 2006
Recognize counting
errors
Gelman & Meck, 1983

102. Spatial Properties are Preverbal
Skill Example Studies Visual Representation
Animals and babies
represent ordinal
numerical
relationships
Brannon & Terrace,
1998; 2000; Cassia et al.,
2012
Animals and babies
represent relative
magnitude through
quantity comparison
Nieder & Miller, 2003,
2004; Halberda, et al.,
2008
Animals and babies
map number to
continuous extent
Tudusciuc & Nieder,
2007; De Hevia & Spelke,
2010

103. Approximate vs. Exact

104. Exact Number System
OBJECT
FILE
Blue
Circle
Attention
Circle
Treisman & Gelade 1980

105. Bootstrapping
OBJECT
OBJECT
OBJECTOBJECT
FILE
One or 1
Two or 2
(Carey, 2004)

106. Treisman & Gelade, 1980

107. Approximate Number System
OR

108. How do We Measure Number Sense?

109. Quantity Comparison: Output

110. Quantity Comparison: Output

111. How do We Measure Number Sense?
(Halberda et al., 2008; Siegler & Booth, 2004)
0 100
Where is 76?

112. Number-to-Position: Output
(Siegler & Booth, 2004)

113. Measuring Number Sense
(Halberda et al., 2008)
7 9 7 9
Congruent Incongruent

114. Your Brain on Math

115. VS

116. 2 + 3
2 VS 3

117. Visuo-spatial Memory
Chinello & Piazza, 2013
Grasping
Objects
Quantity
comparison
Faces
Gnosis
Ventral
Dorsal
Learning calculation makes use of
cortical real estate essential for fine
motor coordination and visuo-
spatial processing

118. Penner-Wigler, et al

119. So What?

120. 64 teens 14
yrs. old
Weber Fraction and Math
Halberda, Mazzocco & Feigenson (2008))
Record of
math
achievement
scores
Number
Sense
acuity

121. Math Performance
TEMA- 2 Score at 8 yrs (high = good)
Weber fraction at 14 yrs (low = good)
Halberda, Mazzocco, & Feigenson, 2008

122. ANS Predicts Dyscalculia
Longitudinal
studies
- 71 teenagers 9th
grade
- Assessments of
discrimination
and enumeration
Mazzocco, Feigenson, & Halberda, 2011

123. Target Number
(Neider, 2004; Neider & Miller, 2003)
Low Weber fraction = low
variability/high precision
0
1
2
3
4
5
6
7
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Distribution of estimates
Adults
10 Yr Olds
5 Yr. Olds
10 Yr. Olds
With dyscalculia
(Piazza et al., 2010)

124. Dyscalculia
(Butterworth, 2002; Shalev & Gross-Tsur, 2001; Geary, 1993, 2004)
5 to 6%

125. Counting strategies
Prolonged use of finger counting
Decomposing numbers
Place value
Multi-step procedures
Observable Deficits
Wilson, 2008

126. Dyscalculia
Skill Example Studies Visual Representation
Poor discrimination Landerl, et al., 2004;2013;
Mazzocco, et al., 2011;
Piazza, et al., 2010
Slow dot
enumeration
Landerl, 2013; Mazzocco &
Halberda, 2011
Mental numberline
slow to develop
Butterworth, 2005; Landerl,
2013 1________________________10
Core system (ANS)
deficit
Butterworth, 1999; Wilson
& Dehaene, 2007
Number symbols
not automatic
Rousselle & Noel, 2007
Where is 7?
7 9 7 9
= 5

127. Subtypes

128. Spatial
Wilson & Dehaene (2007)

129. Verbal

130. Executive

131. EF and Math
If inside shape matches outside shape, name the color.
If inside shape is different, then name the outside shape
But only use the rules for designed that don’t have an arrow.
If there is a backwards arrow, do the rule backwards.

132. Depth of Knowledge

133. Continuous Extent

134. Ramani & Siegler, 2008
1 2 3 4 5 6 7 8 9 10
THE GREAT RACE

135. 7
8
9
10
50
60
70
80
90
100
SET COMPARISON COUNTING
Pretest Posttest Follow-up
Number Game
Color Game

136. Number Bonds
Ways to Make 3

137. Numberline Tap Numberline Frog My Numberline
There’s an App for That
Estimation Line

138. (Kucian, et al., 2011)
Kucian, et al., 2011
Rescue Calcularis: Demo

140. HELLO
Dr. Sara Stetson
GOODBYE
My Name Is
@GoodCognitions
www.sstetson.wordpress.com
Dr. Sara Stetson