Developing Arithmetical Skills

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• Developing Arithmetical Skills

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Informal Arithmetic

• Piaget(1965) children did not have a conceptual
  understanding of basic arithmetic until the age
  of 7 or 8 years

Asked On which day they get to eat most of the
candies?
Day 1 4 4 candies Day 2 1 7 candies
Ans ___________________
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Can you see that the numbers of candies in the 2
rows are equal?
For 1st day afternoon
For 1st morning
Which day will you get to eat the cadies?
For 2nd day morning
For 2nd day afternoon
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2nd day, because the group of 7 is more.
The same.
7 gt4, but 4gt1
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Piaget A major conceptual knowledge of
arithmetic required an understanding that numbers
are composed of groups of smaller numbers and
that a variety of different combinations can
result in the same quantity.
Even 5-year-old cannot arrange 2 rows of beads (8
14) as equal numbers
Arithmetic conceptual knowledge develop at later
stage of development?
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Children have considerable informal knowledge
(Starkey et al)

• 24-month-olds child was able to represent and
  remember the number of balls (1,2,3) deposited
  and use this representation to guide her search
  (p.8)
• 36-month-olds able to represent and remember
  numerosities up to and including 4, ( not through
  counting).
• 18-month-old and nearly 2-year-olds, understand
  the addition increases the numerosity of a set
  substraction decreases a set.(p.9)

Humans have a fundamental sense of number and
quantity, addition and subtraction, independent
of language system.
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5 balls put into the box by the children
One ball taken out by the researcher
Children take out the balls, look for the 5th
ball.
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5 balls put into the box by the children
One ball put in by the researcher
Children look for the 6th ball?
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Childrens Early Arithmetic Knowledge

• Young Children, even infants, have a fundamental
  understanding that addition and subtraction
  influence quantity
• Early implicit knowledge limited to quantities up
  to and including three items and is accomplished
  nonverbally.
• By 4 or 5, most children use verbal counting in
  situations requiring and -.

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Arithmetic Knowledge

• Commutativity
• Base 10 Knowledge
• Fractions

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Commutativity ???

• What is commutativity
• Do Children understand commutativity? When? How
  (Taught or induced)

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Solving 136 and then 613

• 72 of the first grade and 83 of the second- and
  third-grade consistently (at least 3 or the
  trials) solve without counting
• first-grade children had not been explicitly
  taught commutativity
• 40 kindergarten children understand
  commutativity.
• Induced by noting the outcome of basic addition.

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Base-10 Knowledge

• Why is base-10 system important
• understand multi-digit numbers
• regrouping in addition
• carrying and borrowing
• How to teach Base-10 knowledge
• concrete representation versus rote-learning

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Fractions

• Why is the concept of fraction difficult to
  grasp?
• 1/5 1/6 -gt 2/11
• 1/3 gt 1/2
• Not encountered in daily life?
• Do not have intuitive understanding
• Internal knowledge of counting and addition
  required?

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Arithmetic Operations

• Addition
• Simple Addition
• Complex Addition
• Subtraction
• Simple Subtraction
• Complex Subtraction
• Multiplication
• Simple Multiplication
• Complex Multiplication
• Division

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Addition

• What strategies
• manipulative
• finger counting
• verbal counting without the use of manipulations
  (mentally)
• derived facts
• fact retrieval

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Manipulations

• What procedures involved?
• Sets of objects represent numbers to be counted
• pointing to the objects during counting

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Finger Counting

• How
• Advantage over Manipulatives
• What difficulties involved (abstraction)

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Verbal Counting

• Difficulties mentally keep tract of number
  counted and number not counted.
• Strategies
• counting all (sum)
• counting from the 1st number (first) - a short
  cut
• counting from the larger no. (min) -
  understanding the commutative law.
• Strategies seem to be self-discovered.

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Derived Facts (Decompositions)

• 67 -gt 661
• 85 -gt 23
• own constructed

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Fact Retrieval

• Remembered
• smaller numbers easier (why?)
• problem size effect
• found in , -, x
• the larger the number is in the problem, the
  slower and more error-prone the retrieval (why?)
• some addition facts, like n0n seems to be
  retrieved from the long-term memory.

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Errors

• Wild guess e.g., 4141
• Near Misses 1 or 2 higher than the correct sum,
  mirroring childs earlier counting error.
• Operation confusion correct answer of an
  analogous problem with a diff. Arith. Operations,
  e.g., 43 -gt 12
• table errors retrieving answers to related
  problems, e.g., 67 -gt 12 (57)

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Complex Addition

• Skills involved
• counting
• decomposition regrouping
• columnwise procedure
• trading (carrying)
• difficulties encountered
• mentally note a trade has been taken
• understanding of place value

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Errors in Complex Addition
(A) 46 58 ------------ 94
(B) 46 58 ------------ 914
(C) 22 64 ------------ 96
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Subtraction (see Table 2.2, page 72)

• Common Strategies
• manipulatives
• separating from
• addition on
• matching
• Counting fingers
• verbal counting
• counting up
• counting down
• Retrieval

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Complex Subtraction (see Table 2.2, page 72)

• Verbal counting
• Counting down
• Decompositions
• Down over the ten
• Take from the ten
• Delete 10s rule
• Columnar retrieval

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Multiplication Strategies

• Repeated addition 5x3 555
• counting by n 5 x3 counted as 5, 10, 15
• rule 5X0 0
• derived facts (decomposing) 5X6 5X55
• fact retrieval