NUMERACY

NUMERACY

1. numbers represent: magnitude, amount, order

   • development. Rats can tell between 2 and 4 light flashes (church and meck) preverbal number skills in infants. Cooper (1984) 10m detect equality, 14m detect less than

2. WINN (1992) violation of expectation paradigm 5m

   • Look longer when two teddies, violates expectation - suprising
     • criticism Simon et al (1995)
       • Knowledge of physical objects not numerical understanding. Changed object to a clown. Infants recognise change in number but not object, supports WYNN, simple additive reasoning. Only true up to 3 'subitising'

3. Xu Spelke and Goddard, infants could discriminate between 8 and 16 but not 1 and 2.

   • Larger numbers are dealt with differently, analogue magnitude representation. - internal mental representation of continuous quantities.
   • Webers law - perceptual descrimination depends on the similarity of the stimulus intensity. It is ratio sensitive.

4. AMR

   • Van Oeffen and Vos
     • Less precise at judging 12 vs. 10 and 11, than 12 vs. 4 and 20. Symbolic distance effect, is a marker of analogue coding

5. Counting and number sense

   • count to 5 by 3;0. counting does not mean an understanding of number. Piaget - dont have understanding until they have equivalence of sets.
   • Frydman and Bryant, children aged 4 try to count rather than draw inferences

6. Counting principles

   • how to count
     • One- one correspondence, each item gets one tag - need physical and mental tracking, tagging, recognition that names are abstract and unrelated
       • Problems - rhythm determining speed. innacurate finger pointing.
     • Stable order (ordinality)
       • same order across situations, Memorise number order, rhythm and intonation help. Problems - child hasn't memorised order
     • Cardinality
       • final number is size of set. (Fuson and Hall) 1. recite last number without knowing it means quantity, but know its right. 2. understand last number - quantity. 3. Understand progressive nature of cardinality. problem - havent memorised order
   • applying counting
     • Abstraction
       • real and imagined can be counted, need to understand that you can count imagined, problems if only conceive physical, and if counting affected by colour.
     • Order irrelevence
       • skills - all of the others. name tags are temporary. items are not numbers. order irrelevence does not affect cardinality.

7. Nativist (dehaene, 1992)

   • two innate abilities give rise to number sense. children have access to approximate non-symbolic analogue systems from birth. Make appropriate judgements about numerical quantity, innate knowledge of counting principles. evidence of innate ability from animals and young infants. But not clear how this relates to precise system in mathematics

8. Empiricist - Carey (2004)

   • 3 ways children (0-3) represent number. 1. Analogue approximate system. 2. parallel individuation system. 3. set based quantification
   • criticism - emphasis on language, children with poor language can still do maths. bootstrapping hypothesis is circular. Depends on number knowledge which is what its meant to explain

9. Interactionist Piaget

   • Understanding relations between quatity. gained through schemes of action. representation of actions that can be applied to any object. child as scientist, predicting outcomes based on experience. 3 core insights, equivalance, order and class inclusion (whole - sum of parts)
   • prompted most experimental discovery, most successful of theories RUnes and Bryant.

10. Our number system, recursive, base 10 , need additive and multiplicative principles.

    • oksamin tribe - non recursive uses body. Most european irregular up to 100. chinese base 10 spoken as written equivalent

11. If language effects counting, should tell us whether counting gives sense of ordinality.

    • Miura, kim et al (1988) one to one, canonical, non canonical. Blocks. Us china japan korea, US did one to one and couldnt think of another way to do it.
      • But, saxton and howse (1998) Taught the children - behaved the same, Vasilyeva did it on kindergarteners, no difference. Experience >language
        • language is important when learning to count. Counting is important for symoblic non- analogue. arithmatc. Piagetia reversibility and inversion 4+5 = 9