Number: Pt 2.

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Number Pt 2.

• Early Years Lecture 13

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Q1. Is counting sufficient? No!

• Number 1-1 (item-to-item) correspondence
• (Piaget, 1952)

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Piagets concept of reversibility

• Conservation of number

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Piagets concept of reversibility

• E As many green as blue?
• C Yes
• E. lengthens one row
• Now are there as many green as blue?
• C No - more blue
• Under 6 non-conserving
• Key reversibility

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Piagets concept of reversibility

• Key reversibility of operations
• Ignore perceptually salient cues.
• Change must be mentally represented
• Recognize that appearances can deceive
• Realize that spreading-out operation can be
  reversed by a closing-up operation.
• the victory of operation over intuition
• (Piaget, 1952, p. 149)

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Piagets concept of number

• Conservation PLUS
• Class inclusion (Part-whole relationships)
• - distinction between classes / sub-classes
• (are there more cows or more brown cows?)
• Transitivity (e.g., If A gt B B gt C then A gt C)
• - seriation along particular dimension.
• logical roots of number concept

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Q1. Is counting sufficient? Yes!

• Number 1-1 (word-to-item) correspondence
• (Gelman Gallistel, 1978)

1 2 3 4 5 6 7
Q. What other rules, or principles govern
counting?
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Counting principles

• 1-1 correspondence one word-per-item.
• stable-order same words in same order.
• cardinality last word summary of set.
• order-irrelevance start at A or B or C etc.
• abstraction anything can be counted.
• (Gelman Gallistel, 1978)
• Q. If children obey these principles... is that
  all they need to learn?

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Counting principles

• By 5-years most obey the principles.
• ... but do they connect counting with number?
• Fuson (1988)
• Sets 2/3 1-1 stable order before cardinality
• Sets gt 16 cardinality before 1-1 / stable order
• cardinality or last-word-rule?

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Counting principles

• Wynn (1992)
• E. Give me 5 toys
• C - hands over fifth toy she counted
• child fails to make cardinal-to-count transition
• cardinal tag principle?
• cardinal term 5 ? ordinal term 5th

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When to count?

• Fuson (1988) Children have individual skills....
• Give me 4 blue chips
• 82 counted
• Give me as many red chips as there are blue
  chips
• 73 used matching technique
• ....and use them selectively?

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Q2. How does the ability to judge relative number
words develop?

• How do children go beyond rote application of
  early-learned procedures?
• What type of interaction leads to conceptual
  insight?
• How might children progress in the classroom?
• Identification of educational intervention

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Models of counting development

• 2 dominant theories question of when
• 1 principles-before-skill
• (e.g., Gelman Meck, 1983)
• 2 skills-before-principles
• (e.g., Briars Siegler, 1984)

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Principles-before-skill

• principles-first/before
• Nativist
• Idiosyncratic list (e.g., 1, 2, 4, 6!)
• Rarely non-number terms
• Builds up (i.e., 1, then 2, then 3, etc.)
• Error-detection evidence....

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Principles-after-skill

• Children detecting errors on large sets can count
  small sets!
• Fuson (1988) set-size effect on adherence.
• Briars Siegler (1984) 3-yr-olds less likely
  than 4-yr-olds to catch 1-1/stable-order errors.
• Frye et al. (1989) counting proficiency precedes
  error detection.
• understanding of principles develops with
  age/experience

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Non-verbal gt verbal enumeration A link?

• mapping theory (e.g., Gelman, 1991)
• (ie., words on to accumulator representations)
• Counting easy
• Fractions difficult
• Wynn (1992)
• start to count _at_2
• grasp cardinality _at_3/4

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Spatial conflict

• Focus on length B gt G (e.g., Michie, 1984)
• Emphasise 1-1 correspondence? (Cowan, 1984)
• Count both rows? (Cowan, 1987)
• Quantité vs Quotité (Greco, 1962)

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Frog-boats (Sophian et al., 1995)
Party goin on....
.. boats are moored outside
Frogs go by boat into house...
Q1. How many boats are there outside? Q2. How
many frogs are at the party?
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Frog-boats (Muldoon et al., 2004)
Q1. How many frogs are at the party? Q2. Can you
get the boats ready so the frogs can go
home...remember, get just the right amount!
Frogs go into house
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Principles vs skills A dead-end?

• What predicts use of strategic counting?
• Counting proficiency? - No.
• individual skills have to be understood in
  relation to one another
• counting vs set comparison/set creation
• gt understanding anothers counting and
  miscounting what Freeman et al.(2000) call a
  theory of error.

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Q3. Is there a role for psychological studies of
number in the classroom?

• "We don't know what they understand. I don't know
  what a 4-year-old child understands. I think the
  way we teach numeracy is rote counting and we
  hope that the understanding comes later. It's
  monkeys jumping through hoops is all it is.".
• (interview with reception-class teacher)

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Learning about counting and number in the
classroom

• What? - identify predictors of target skill
• - ability to detect and reason about anothers
  miscounts
• When? - identify window of change
• - consistent pattern during 1st school year (4
  - 6)
• How? - identify context for learning
• - self-explanations - error driven reasoning
• gtST gains
• Whats gained?
• Statistical modelling of change identifies
  predictors of development.

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Mathematical concepts
2 - 3 yrs
4 - 6 yrs
Number/ Cardinality
Detect procedural errors
Error driven reasoning
Procedural mastery
Generate appropriate problem-solving strategies
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Applied Modelling

• Identify precursors and catalysts of arithmetical
  development
• Identify factors that impede numeracy
• Promote appropriate teacher-pupil interaction
• Tailor maths instruction to childrens abilities
• Identify building blocks of number through
  transition
• Go beyond rote learning - no more monkeys!

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Reading (from Science Direct)

• Sarnecka, B. W. Gelman, S. A. (2004). Six does
  not just mean a lot preschoolers see number
  words as specific. Cognition, 92, 329-352.
• Sophian, C. (2004). Mathematics for the future
  developing a Head Start curriculum to support
  mathematics learning. Early Childhood Research
  Quarterly, 19, 59-81.
• Zur, O. Gelman, R. (2004). Young children can
  add and subtract by predicting and checking.
  Early Childhood Research Quarterly, 19,
  121-137.
• available from Science Direct