The Numerate Brain

What is Number?

• Number can be a symbol, a unit of measurement, a quantity, a code, or a magnitude.

How do we study number ability?

• Relative vs. Absolute judgments.

• Symbolic vs. Non-symbolic representations.

• Subitizing as a special case.

Is our sense of number innate and unique?

• Where there is ability, there is a biological process.

  • Number in the brain.

  • Spatial representations.

Models of Number Processing

• Cognitive models and how they deal with different representations and/or transcoding (switching forms).

Visual Number Perception

• Factors that influence judgments of visual quantity/magnitude.

  • 2D vs. 3D.

Visual Models

• Is number coded or is it the density of items in an area?

  • The primacy of our number sense.

  • Dakin’s model for coding number per area (i.e., coding number as density of items).

The Meaning of Numbers

• The meaning of a number is a quantity, magnitude, or size of a collection (numerosity).

• A telephone number is a code that uses number symbols but doesn’t convey numerical meaning.

• Meaning is abstract.

  • It is independent of the symbol denoting it (3, three, trois, 3 fingers) and the object being enumerated (3 ideas, 3 elephants, 3 computers).

Non-Symbolic Number Cognition: Quantities

• Tasks often involve judging the size of an array.

  • Which has more? (relative).

  • Are there 8? (absolute).

• Relative judgments of array size depend on:

  • Number of items (size effect) e.g. 20:10 harder than 10:5.

  • Difference in set size (distance effect) e.g. 6:5 harder than 7:5.

• Performance correlates with school-based maths measures (Halberda et al. 2008).

  • Impaired in developmental dyscalculia (Piazza et al. 2010).

Quantities

• Absolute judgments of array size (exactly how many?) depend on two mechanisms:

  • Subitizing (up to 3-4 items at a glance, highly accurate, parallel processing).

  • Counting (4+ items, conversion to symbolic code, serial processing, error-prone).

  • Approximating- we don’t/can’t always count.

Symbolic Number Cognition

• Deciding which of two number symbols is larger depends on:

  • The distance between the numbers (distance effect; e.g. "5 4" slower than "5 3").

  • The size of the numbers themselves (size effect; e.g. "5 3" slower that "4 2").

Interactions Between Symbolic & Non-Symbolic Number Codes

• Congruent Examples:

  • Which is physically larger?

    • 2 (423 msec) vs. 9 (532 ms).

  • Which is numerically larger?

    • 2 vs. 9.

• Incongruent Examples:

  • Which is physically larger?

    • 9 vs. 2.

  • Which is numerically larger?

    • 2 (450 msec) vs. 9 (619 msec).

• Number size influences physical size judgment.

• Physical size influences number size judgment.

• Girelli, Lucangeli and Butterworth (2000).

Interactions Between Symbolic & Non-Symbolic Number Codes

• Koechlin et al. (1998).

• Priming study:

  • Judge whether a stimulus is greater than 5.

  • Stimuli consist of names (SEVEN), digit (7) or dot arrays.

  • Before each trial, participants are shown a brief subliminal prime that is either on the same side of 5 as the target, or the opposite side.

  • Interference is found when the prime and target are on opposite sides of 5.

  • Influences extend across representations.

  • Suggests semantic priming.

Number Sense

• Ability to sense, compare, and estimate numbers.

• How common is this?

• How innate is this?

• How does the brain do it?

Innate Versus Cultural Aspects of Number

• Names within the subitizing range seem to be a general ability across cultures (Amazonian Munduruku number naming).

• Implies a common underlying mechanism or set of mechanisms.

• Other aspects of numerical cognition (such as written symbols, number names, equations, algebra etc.) may be cultural inventions.

• An appreciation of quantity (how much?) and numerosity (how many?) may be a basic skill of almost all humans and adults.

• Infants habituate if shown the same number of items but dishabituate if the number changes.

Number Neurons?

• Animal work: physiology measuring action potentials (Nieder, 2005).

• Some respond to sequential presentation of dots as well as simultaneous ones.

• Some respond to the number of sounds too (i.e. multi-sensory coding of number).

• Training monkeys to associate digits to set sizes reveals neurons (in PFC, not IPS) that respond to symbolic and non-symbolic number.

• Human children show frontal-to-parietal development shift in processing number (fMRI); suggesting symbols become linked to ‘core’ number code of meaning in IPS over time.

A Neural Region For Number Meaning?

• Converging evidence suggests the importance of a region in the intraparietal sulcus (IPS).

• More active in calculation than number reading.

• Sensitive to subliminal priming when prime and target differ in number.

• Shows a distance effect for both digits and number words.

• Habituation to repeated numbers of items in a dot array.

A Separate Region for Visual Recognition of Number Symbols?

• Left hemisphere.

  • Preference for shapes.

  • Biased connectivity to language.

• Right hemisphere.

  • Biased connectivity to numerosity.

• T&F, NFA, VWFA.

Left and Right Brain in Number Meaning

• Studies of acquired/developmental dyscalculia emphasize the importance of the left hemisphere, but functional imaging studies show bilateral activity. Why?

• One suggestion is that the left hemisphere deals with exact quantities (e.g., 5), whereas the right hemisphere deals with approximate quantities (e.g., between 4 and 6).

• Warrington (1982): dyscalculic patient with LH lesion able to do approximate sums (e.g. $5 + 7$ = “roughly 13”).

• Dehaene and Cohen (1996): digits presented to LH and RH of a split-brain patient; RH gives approximate answers 5 "six" but LH gives exact answers.

Representing Number

• Spatial Code for Numbers.

  • SNARC = Spatial numerical association of response codes.

  • The SNARC effect = faster response by left hand to small numbers, and right hand to large numbers.

• Counting explicitly using body parts: The “Yupno” in Papua New Guinea and Torres Straight Islanders (Adapted from Ifrah (1985)).

Modeling Number Ability

• Conceptualize number representation through models:

  • Cognitive models.

  • Visual models.

Models of Numerical Cognition

• McCloskey Model.

  • Cognitive model.

  • Number size is represented as base-10 units (divisible into 10s, 100s, 1000s, etc.).

  • Separate routines or stores for arithmetical operations ($+,-,/,\times$).

  • Abstract (semantic) representations are used for all calculations.

  • Transcoding is performed semantically.

• Dehaene's Triple-Code Model.

  • Cognitive and neuroanatomical model.

  • Number size is represented in a logarithmically compressed form (larger numbers harder to discriminate).

  • No separate routines or stores for arithmetical operations ($+,-,/,\times$).

  • Some calculations are independent of number semantics (e.g., multiplication is verbal fact retrieval).

  • Transcoding may be performed without semantics.

How do interactions occur: Transcoding

• Transcoding = translating one type of symbol (e.g., 5) to another (five); reading and spelling are types of transcoding.

• Evidence suggests some non-semantic transcoding (favoring Dehaene’s model).

• Cipolotti and Butterworth (1995) case study:

  • Asked to write "seventy thousand" he wrote 17,000 but when asked to do 56,748 + 13,252 he wrote 70,000.

Models of Numerical Cognition: A Hybrid Account

• An adaptation of the McCloskey model by explicitly adding transcoding stages (From Butterworth (1999)).

How is number being encoded: Base-10 or Log-Linear

• Some evidence for both theories.

• McCloskey assumes separate representations for units, tens, hundreds, etc.

• Evidence for this comes from e.g., deciding which digit is larger (e.g., 51 vs. 65), performance is affected by both tens (6>5) and unit magnitude (5>1) relative to when there is a mismatch.

• Dehaene assumes a "mental number line" because number comparison shows the same law as comparing the length of two lines.

Models of numerosity perception

• Visual models: how do we estimate non-symbolic quantities?

  • The previous models assume simple counting, but often we estimate numbers too rapidly to count (e.g., crowds, sheep).

  • When we do, we are reasonably accurate too (10-20% JND).
    Professor John Ross did considerable work in this field.

Models of numerosity perception: Approximating number as density

• We can estimate large numbers relatively accurately and fast without counting- but how?

• Dakin et al, PANAS, 2011, argue that number is affected by area:

  • They showed that larger patches look more numerous (E), implying our sense of number is tied to the area being referenced.

  • Dots per area is saying we encode density rather than number…

But how to code: Approximating number as density

• Larger patches look more numerous.

• The Dakin model assumes that fine detail (High spatial frequency) processes the number of items.

• By contrast, coarse detail (low spatial frequency) processes the relevant area.

Response: Number Sensed Directly

• John Ross supported the Dahaene view that we sense number directly, i.e. number neuron/, logarithmic numberline.

• John hated the number = density argument, and there were several exchanges in the published literature…

• Durgin's suggestion, that number is extracted indirectly from a texture representation obtained from the statistical kurtosis of a scene evaluated over various scales, seems reminiscent of the legendary Australian stockman who, when asked to explain his uncanny ability to judge the number of cattle in a herd, replied that he counted the legs and divided by four.

• In the end, as often happens, the answer seems to be that we use, or sense number for smaller N and use a density metric for larger N (e.g. crowds).

• Anobile et al., Psychological Science, 2014.

• Consistent with number neurons for smaller magnitudes.

Some of my work in this area: 2D or 3D

• I’ve worked on the coding of number, testing Dakin’s model and whether it is right to be 2D or needs to incorporate 3D information.

2D or 3D

• I replicated an area bias for density but NOT for number (argues against Dakin model).

• I showed that density biases did not occur for 3D volumes.

How Primary is Our Sense of Number?

• I’ve also looked at what types of visual cues interact with our sense of number: Apthorp & Bell, Current Biology 2015.

• Symmetrical patterns appear less numerous (compare top row in A).

• We argue this is because of redundancy. We need only attend and process half the pattern.

• This biases our estimate of number down.

• Implies symmetry is processed before number.

Summary

• Number is a complex concept: symbols, magnitudes, codes, quantities.

• There are several commonalities across these that point to a central abstract concept of number.

• This concept seems to map to space (SNARC).

• Number ability seems somewhat innate, crosses cultures.

• Number involves several brain regions, visual, verbal, abstract number concepts.

• These concepts relate to models of number.

• The primacy of number has given rise to models.

• Cognitive and visual.