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• Presented by Daniel C. Hyde for Psychology 216

Why Study Number? Math Education

• Math is taught in most education levels, including elementary, high school, college and graduate school.
• Early number skills are predictors of life achievement, even better than literacy skills.
• Dyscalculia, a difficulty in learning/comprehending mathematics, affects 1-7% of the population (P.O.S.T., 2004).
• U.S. math education has room for improvement.
• For example, the U.S. ranks 35th in math and 27th in science education out of the top 64 countries (PISA, 2012).

Overview of Numerical Development Lecture

• What numerical abilities are present at birth, and in what sense are they numerical?
• Beginnings of learning about symbolic numbers will be covered.
• Discuss stages of counting development and core number's role in math development.
• How can the science of numerical development inform math education?
• Consider how core numerical abilities might be harnessed to enhance early mathematics education.

Approximate Number System (ANS)

• Exists before formal education.
• It is also known as "Number Sense".

Approximate Number System - Key features

• Facilitates rough estimates and number comparisons without counting.
• Precision is limited.
• Ratio between numbers decreases the closer they are.
• Number as an approximate and imprecise concept.

Approximate Number Comparison - Research

• Included research by Barth, Kanwisher, & Spelke, 2003.
• Participants were asked to estimate which of two dot sets had more dots.

Numerosity Discrimination by Adults

• Number discrimination relies upon the ratio between numbers being compared.
• Comparison for equal ratios are the same, for example: for example, 8 vs. 16; 16 vs. 32; 32 vs. 64; 50 vs. 100.

Numerical Comparisons & Properties

• Numerical comparisons are centered on NUMBER, not other sensory properties.
• Evidence comes from controls.
• Changing size of objects, spacing, density, or brightness does not significantly impair numerical comparison abilities.

Cross-Modal Comparisons

• Involves comparison of two different senses: visual and audio.

Cross-Modal Results

• Audio-visual comparison is nearly as accurate as a visual comparison.
• Comparisons stem from abstract notions rather than direct non-numerical sensory stimulus properties like amount of visual stimulation.
• ANS allows for rough approximation and comparison of number without counting.

Approximate Numbers

• Ratio with limited precision.
• Abstract mental representations are not tied to non-numerical sensory properties of individual objects.
• Can be used productively for arithmetic.

Ratio Dependency in Arithmetic

• Ratio dependency occurs between the real sum and the foil sum.

Specialized Brain Regions for Number Sense

• The intraparietal sulcus (IPS) region responds specifically to number {Dehaene et al., 2003}.
• Impaired IPS causes impaired numerical abilities.
• Permanent brain damage to the IPS impairs numerical processing.
• Transcranial Magnetic Stimulation (TMS) to the IPS temporarily impairs numerical processing (Cappelletti et al., 2009).
• The TMS method helps establish a causal relationship between IPS and numerical abilities.
• Specialized cortical regions of the parietal lobe play causal role in numerical abilities
• Approximate number system.

Number Sense in the Munduruku

• Even with limited numerical language and without formal number system, people still have an understanding of approximate number.
• ANS and basic arithmetic intuitions are universal in humans (Pica, et al., 2004; Piazza et al., 2011).

Infant Number Change Detection

• Research was conducted by Brannon Lab-UPenn.
• Number sense exhibit ratio limit in infants.
• 8 vs 16 dots had success at 6 months.
• 8 vs 12 dots had failure at 6 months.
• Infants can compare quantities up to a point.
• At 6 months, the ratio limit is 1:2.
• By 9 months, precision increases to 2:3.
• In adults, it is ~7:8 (Xu, Spelke, Lipton, and others).
• Number sense is present from birth.
• Newborns match the precise number of audio and visual items (Izard et al., 2009).
• Infants engage the IPS selectively for number, like adults (Izard et al., 2008/Hyde et al., 2010).

Approximate Number System

• Facilitates rough approximations and comparisons without needing to count.
• Ratio limited precision.
• Mental number representations are abstract.
• Can be applied productively for arithmetic.
• Present from birth and is continuous acorss the lifespan, changes in precision.
• Specialized cortical regions of the parietal lobe play roles in numerical abilities
• Present universally to humans

Brannon Lab-Duke

• Brannon Lab-Duke studies non-human primate addition and subtraction
• Non-human animals are known to represent approximate numerosities.
• Show same signatures to number as human infants & adults.
• There's single cell tuning to number in the monkey brain.
• Nieder and Dehaene, 2009 led to the finding.

Two Systems of Core Number

• According to Feigenson, Dehaene, & Spelke, 2004 that there are two evolutionarily ancient systems to calculate numerical computations.
• Approximate Number System (ANS) allows for comparisons using approximate numerical magnitudes.
• The object tracking system (OTS) allows one to represent, remember, distinguish between, and track individual objects.
• Allows comparisons using 1 to 1 correspondence.
• Subitizing is the abilitty to enumerate a limited number of items instantly and very accurately, usually about 3-4 items in adults.
• OTS enables simultaneous selection (subitizing), tracking, and remembering of a limited number of individual items.
• System is capacity limited
• In infants, OTS capacity is limited to 3.
• Both ANS and OTS are present from early in the development and persist over the lifetime.

Learning a Symbolic Number System: Core limitations

• There are limitations, since as core number (ANS and OTS) can not fully represent integers/natural number used in basic counting.
• It's typically thought of as positive integers or NATURAL NUMBER.
• Counting, Number words(seven, ten, etc.) and Arithmetic and other math operations are natural numbers.
• ANS can represent numerical magnitudes, but only in an approximate manner.
• OTS can represent individual objects up to 3-4.
• Exact numbers like 10, for example, are not represented by OTS or ANS.
• Natural number development can be difficult.
• Seems to require cultural access to natural number (i.e., a number system in language).
• Occurs between 2-5 years of age (in Western societies).
• From learning to recite the count list (one, two, three..) to acquiring integer/natural number concepts usually takes about 2-2.5 years

Stages of Counting Development

• Children can learn the count list though may not understand meaning.
• Includes One, two, and three-knowers, who all gradually use this knowledge
• Counting principle (CP)-knower emerge around "three” or "four”, generalize the counting principles to other numbers.

Universal Development

• Do natural numbers develop in all humans, or is it a cultural invention?
• Piraha people of the Brazilian Amazon are test case.
• Piraha as a group are hunter-gatherers residing in a remote region of the Amazon.
• They have no numerical terms beyond “one”, "two", and "many” in their language.
• Gordon (2004) used a variety of matching tasks.

Matching Task with Piraha

• Matching results show 2 core systems.
• Perfect performance on small numbers.
• No ability to match exact large cardinal values.
• Suggest natural number may not be innate/universal.
• Requires more than just core number abilities and/or brain development:
• Instruction.
• Number list/language/other cultural invention.
• Is core number related to the development symbolic number/mathematics learning?
• There are associations between symbolic number and ANS
• Symbolic number system is influenced by the ANS in adults and older children.
• There are distance effects when comparing symbolic numbers.
• Difficulty determined by distance between numbers.
• Just like with non-symbolic object arrays.

Brain Regions

• Number-sensitive brain regions (intraparietal sulcus, IPS) respond to both symbolic and non-symbolic comparisons.
• Individual differences in approximate numerical ability is associated with math achievement in adults and children (e.g., Halberda et al., 2008; Gilmore, McCarthy, & Spelke, 2010; Libertus et al., 2013).
• Impaired ANS ability is associated with developmental dyscalculia (e.g., Piazza et al., 2010).
• Training the ANS improves math in children and adults (e.g., Park and Brannon, 2013; Hyde et al., 2014). Evidence suggests an important number relationships.

Interventions

• Interventions may be able to enhance or remediate symbolic math abilities. E.g., Hyde et al., 2014 saw practicing an approximate number task improving arithmetic in 1st graders..
• Dillon et al., 2017 found that larger-scale intervention in preschools in India (~1500 students total, 4 months) improved abilities among pupils and children..
• Random assignment to intervention between social and non-symbolic.
• Children improved in abilities after non-symbolic number/magnitude.
• Compared to control (social intervention).

Intervention Summary

• Intervention studies suggest comparing numerical magnitudes can improves understanding of numbers and arithmeti Suggests a role of core number/magnitude in learning about symbolic number and mathematics
• Easy and cheaper interventions may work.
• Pairing magnitudes with number are common in mathematics and elementary school.

Conclusions

• Humans are born with two core cognitive systems of number.
• Each allows numerical abilities before education/instruction.
• Not the same as Natural Number/Symbolic Number System.
• Relationships are interesting for abilities and math, core system may be able to help development
• Building on core intuitions of number may be a way to facilitate early number and math learning.
• A curriculum that builds on core abilities to improve math/number learning may be useful.