Numerical Thinking Lecture Notes PDF

Numerical Thinking Daniel C. Hyde Psychology 216 Psyc 216  Why Study Number? Math Education Math taught in most levels of education – Elementary, high-school, college, graduate school Early number skills predict life achievement – Better predictor tha...

Numerical Thinking Daniel C. Hyde Psychology 216 Psyc 216  Why Study Number? Math Education Math taught in most levels of education – Elementary, high-school, college, graduate school Early number skills predict life achievement – Better predictor than literacy (Bynner & Parsons, 1997; Richie & Bates, 2013) Many struggle with numbers/math – Dyscalculia: difficulty learning/comprehending mathematics 1%-7% of the population (P.O.S.T., 2004) Room for improvement in U.S. math education – Example: U.S. ranks 35th in math and 27th in science education out of top 64 countries (PISA, 2012) Psyc 216  Overview of numerical development lecture What numerical abilities are we born with? – In what sense are they numerical? How do we begin to learn about symbolic numbers? – What are the stages of counting development? – What is the role of core number in math development? How might the science of numerical development inform math education – How might core numerical abilities be harnessed to learn or improve early mathematics education? Psyc 216  What numerical abilities do we have before education? Approximate Number System (ANS) “Number Sense” Psyc 216  How Many? Psyc 216  Approximate Number System Allows for rough approximation (not exact) and comparison of number without counting Limited precision (ratio signature) Psyc 216  100 80 8 v. 24 Accuracy 60 8 vs. 16 40 8 vs. 10 20 0 Less accurate as ratio between numbers decreases (numbers become more similar/closer) Shows that number is approximate and imprecise Psyc 216  Approximate number comparison (Barth, Kanwisher, & Spelke, 2003) 2 1 Which has more? Psyc 216  Numerosity Discrimination by Adults Chance (50%) 100 95 Ratio Signature: 90 Number discrimination is a 85 function of the ratio between 80 75 large (40-80) medium (20-40) the two numbers to be 70 small (10-20) compared 65 60 55 Comparison equal for equal 50 ratios: e.g. 8 vs. 16; 16 vs. 32; 2 1.5 1.25 1.15 1.1 32 vs. 64; 50 vs. 100 Psyc 216 set size ratio  Are numerical comparisons actually based on number or some other property? Yes, they are based on NUMBER, not other properties – Evidence from Controls: Change size of objects, spacing, density, brightness… Doesn’t significantly impair your ability to make numerical comparisons Psyc 216  Cross-modal comparisons 2 1 Which has more? Psyc 216. Cross-modal results Cross-modal (audio- visual) comparison is Accuracy nearly as accurate as 100 visual comparison 90 Suggests comparisons are % Correct 80 being made over an 76 abstract notion of 70 73 number rather some direct non-numerical 60 sensory property of the stimulus (e.g., amount of visual stimulation) 50 Visual Crossmodal Comparison Comparison Psyc 216  Approximate Number System Allows for rough approximation (not exact) and comparison of number without counting Ratio limited precision Abstract mental representations/not tied to non-numerical sensory properties of individual objects Psyc 216  Approximate Number System Allows for rough approximation and comparison without counting Ratio limited precision Abstract mental representations Can be used productively for arithmetic Psyc 216  Arithmetic? Psyc 216  LESS Psyc 216  Arithmetic? Psyc 216  LESS Psyc 216  Ratio dependency in arithmetic (between real sum and foil sum) Psyc 216 e.g., Jang & Hyde, 2020  Specialized brain regions for number sense: IPS Region of Intraparietal sulcus (IPS): responds specifically to number Psyc 216 Dehaene et al., 2003  Impaired IPS = Impaired Numerical Abilities Permanent brain damage to IPS impairs numerical processing Transcranial magnetic stimulation (TMS) to the IPS will temporarily impair numerical processing (Cappelletti et al., 2009) TMS method helps establish a causal relationship between IPS and numerical abilities Psyc 216 https://www.youtube.com/watch?v=3bca2X3xtwo  Approximate Number System Allows for rough approximation and comparison without counting Ratio-limited precision Mental representations of number are abstract Can be used productively for arithmetic Specialized cortical regions of the parietal lobe—Play causal role in numerical abilities Psyc 216  Number sense in the Munduruku Even with restricted numerical language and no formal number system, Munduruku have intuitions of approximate number Psyc 216  Approximate addition in the Munduruku Munduruku perform nearly as well as French control subjects in ANS and basic intuitions of an approximate addition task arithmetic are universal in humans Psyc 216 (Pica, et al., 2004; Piazza et al., 2011)  Approximate Number System Allows for rough approximation and comparison without counting Ratio limited precision Mental representations of number are abstract Can be used productively for arithmetic Specialized cortical regions of the parietal lobe Universal to humans Psyc 216  Infant Number Change Detection (Brannon Lab-UPenn) Psyc 216  Number sense shows ratio limit in infants 8 16 Habituation (…) (…) Test Psyc 216  8 vs. 16 success at 6 months 35 30 25 20 habituation new number 15 old number 10 5 0 1 2 3 4 5 6 1 2 3 Habituation Trials Test trials Psyc 216  Number sense shows ratio limit in 8 infants 12 Habituation (…) (…) Test Psyc 216  8 vs. 12 failure at 6 months 35 30 25 20 habituation new number 15 old number 10 5 0 1 2 3 4 5 6 1 2 3 Habituation Trials Test trials Psyc 216  Further investigations into ratio limits in infants 6 month success 6 month failure 8 vs. 16 dots 8 vs. 12 dots 16 vs. 32 dots 16 vs. 24 dots 4 vs. 8 dots 4 vs. 6 dots Similar to adults, infants ability to compare quantities is ratio-limited Actual ratio limit depends on age – At 6 months ratio limit is 1:2 – By 9 months this increases in precision to 2:3 – In adults it is ~7:8 Xu, Spelke, Lipton, and others Psyc 216  Number sense is present from birth Newborns match number of auditory tones to correct number of visual items (Izard et al., 2009) Psyc 216  Infants engage IPS selectively for number (like adults) Psyc 216 Izard et al., 2008/Hyde et al., 2010  Approximate Number System Allows for rough approximation and comparison without counting Ratio limited precision Mental representations of number are abstract Can be used productively for arithmetic Specialized cortical regions of the parietal lobe Universal to humans Present from birth, continuous across the lifespan (only changes in precision) Psyc 216  Brannon Lab-Duke non-human primate addition and subtraction Psyc 216  Non-human animals have number sense Non-human animals represent approximate numerosities and show the same signatures as those of human infants & adults Psyc 216  Single-cell tuning to number in the monkey brain Nieder and Dehaene, 2009 Psyc 216  Approximate Number System Allows for rough approximation and comparison without counting Ratio limited precision Mental representations of number are abstract Can be used productively for arithmetic Specialized cortical regions of the parietal lobe Present from birth, continuous across the lifespan (only changes in precision) Shared with many non-human animals Psyc 216  Two Systems of Core Number (see Feigenson, Dehaene, & Spelke, 2004 reading) Two evolutionarily ancient systems that allow for numerical computations – Approximate number system (ANS) Allows comparison using approximate numerical magnitudes – Object tracking system (OTS) Ability to represent, remember, distinguish between, and track individual objects Allows comparison using 1 to 1 correspondence Psyc 216  3500 3000 2500 2000 1500 1000 500 0 verbal counting 1 2 3 4 5 6 7 8 9 “subitizing” Psyc 216  Subitizing (Jevons, 1871) Ability to enumerate a limited number of items instantaneously and very accurately – Usually limited to about 3-4 items in adults Indicative of a system for representing individual objects simultaneously – System is capacity-limited Psyc 216  same or different? Psyc 216 Psyc 216  same or different? Psyc 216  3-4 item capacity limit Psyc 216  Object tracking system (OTS) Allows for the simultaneous selection (subitizing), tracking, and remembering of a limited number of individual items – Enumeration – Tracking – Short-term memory Limited to ~3-4 total items in adults – Comparisons are not ratio dependent Psyc 216  Same limitations of OTS in infants (Feigenson et al., 2002) Succeed at 1 vs. 2 2 vs. 3 1 vs. 3 Fail at: 2 vs. 4 1 vs. 4 Capacity limit of 3: Only 3 objects can be tracked in any given location at one time Psyc 216  Object tracking system (OTS) Allows for the simultaneous selection (subitizing), tracking, and remembering of a limited number of individual items Limited to about 3-4 total items in adults – Comparisons are not ratio dependent Present in infants – Similar capacity limit/signature Psyc 216  Two Systems of Core Number (see Feigenson, Dehaene, & Spelke, 2004 reading) Two evolutionarily ancient systems that allow for numerical computations – Approximate number system (ANS) – Object tracking system (OTS) Both present from early in the development and persist over the lifetime Psyc 216  Learning a symbolic number system Psyc 216  Core number (i.e., ANS and OTS) can not represent the integers/natural number used in basic counting “Number”: Typically think of the positive integers or NATURAL NUMBER – Counting – Number words(seven, ten, etc.) – Arithmetic and other math operations ANS can represent numerical magnitudes, but only in an approximate manner OTS can represent individual objects up to 3-4 Exactly 10, for example, can not be represented by either OTS or ANS Psyc 216  How does natural number develop? Difficult developmental process 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 Psyc 216  Stages of counting development Child learn the count list (one, two, three…) but don’t understand the meaning of the words One-knower: Learn the meaning of “one” (but no other number word meanings) Two-knower: Know “one” and learn the meaning of “two” (but no other number word meanings) Three-knower: Know “one” and “two” and learn the meaning of “three” (but no other number word meanings) Counting principle (CP)-knower: Somewhere around “three” or “four”, generalize the counting principles to other numbers. Psyc 216  Universal development? Does natural number develop in all humans or is it a cultural invention (e.g., reading/writing)? Test cases: Piraha of the Brazilian Amazon (Gordon, 2004; also see Pica et al., 2004) Psyc 216  Use of number in the Piraha Does natural number develop in all humans or is it a cultural invention (e.g. reading/writing)? Test cases: Piraha of the Brazilian Amazon (Gordon, 2004) – Hunter-gatherer group in remote region of the Amazon – No numerical terms beyond “one”, “two”, and “many” in their language Peter Gordon (2004) used a variety of matching tasks Psyc 216  Matching Task with Piraha Matching results provide evidence for two core systems – Perfect performance on small numbers – No ability to match exact large cardinal values Continually worse performance (approximate) as number increases for larger numbers Suggest natural number may not be innate/universal Clus te r-Line M atch 1 Proportion Correct Requires more than just core 0.75 number abilities and/or brain 0.5 development – Instruction 0.25 – Number list/language/other 0 cultural invention Targe t 2 3 4 5 6 7 8 9 10 Psyc 216  Is core number related to the development symbolic number/mathematics learning? Psyc 216  Associations between symbolic number and ANS Symbolic number system is influenced by the ANS in adults and older children – Distance effects when comparing symbolic numbers Difficulty determined by distance between numbers – E.g. 60 vs. 65 is harder than 31 vs. 65 – Just like with non-symbolic object arrays Psyc 216  Overlap in brain regions for core and symbolic number – Number-sensitive brain regions (intraparietal sulcus, IPS) respond to both symbolic and non- symbolic comparisons Psyc 216 Individual differences in ANS and mathematics – Individual differences in approximate numerical ability 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 in 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) Psyc 216  Evidence provided suggests that there is an important relationship between the ANS and mathematics learning If ANS forms a basis for learning about symbolic number/math… – Can curricular interventions based on training the ANS actually enhance or remediate symbolic math abilities? Psyc 216  E.g., Hyde et al., 2014 194 + 71 ___________ ? Practicing an approximate number task improves arithmetic in 1st graders Psyc 216  Dillon et al., 2017 Larger-scale intervention in preschools in India (~1500 students total, 4 months) Randomly assigned (RCT) to either social or non-symbolic magnitude intervention Children improved in math abilities after non-symbolic number/magnitude intervention – Compared to control (social intervention) Psyc 216  Intervention Summary Intervention studies suggest comparing numerical magnitudes can improve understanding of numbers and arithmetic Suggests a role of core number/magnitude in learning about symbolic number and mathematics Also, relatively easy/low-cost interventions *Pairing of magnitudes with number is common in preschool and elementary school mathematics curricula, in part, due to this sort of work. Psyc 216  Conclusions Humans are born with at least two core cognitive systems of number – Each allows numerical abilities before education/instruction – NOT the same as Natural Number/Symbolic Number System Interesting relationships between core number and symbolic number/mathematical abilities – Suggest a role of core systems in mathematics development Building on core intuitions of number may be a way to facilitate early number and math learning – Curriculum that builds on core numerical abilities may improve math/number learning Psyc 216