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Elementary
computational
models and
processes – Addition
and Subtraction
www.company.com
Forming connections among internal
and external representations
• External representation
•A material display of
knowledge and information
which may include facts,
and also procedures and
structures
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Each representation comprises
these four key components
Design
Way in which the representation is
structured
Format
refers to the lay-out or arrangement of
the external representation
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Actions
Designating how the materials
or symbols are manipulated
Explanation
verbalization that accompanies
the implementation of other
components
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Internal representations
It is created when we re-
present the outside world to
ourselves internally. We take
information in through our
senses and we make internal
representations.
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Conceptual model of whole numbers
Addition and Subtraction
Schwartz ( 1988 ) defined
addition and subtraction as
“referent preserving
composition”.
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Compare schema
The compare schema
allows the child to make
comparative judgment of given
amounts, initially in a
perceptual way, then later using
numerical comparison.
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Part/whole schema
Children can reason
about ways in which familiar
materials come together and
break apart.
CHANGE-BY-ADDING
In this process, the children
must switch from their
accumulator model to a
counting-sequence model.
They must count as they add
the required number of items
to their existing set of items.
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The solution to the problem will be
gained by counting all of the items
in the set. Then the accumulator
model would be invoked as the set
is counted, representing a switch
from the counting-sequence model
back to the accumulator model.
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Combine –
Additive problems of the combine
type involve similar processes for the
child, although they entail actions or
implied actions on two distinct quantities.
These problems are effectively modeled
using a part-part-whole approach
wherein the whole is represented by the
parts taken together.
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Embedded Integration
It is a conceptual development
associated with latter procedure for
both addends. The two parts and
the whole can be considered
simultaneously by embedding the
parts within the whole.
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This means that the child is able to
abstract the cardinal value of the
one parts and does not need to re-
count its items in determining the
whole.
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Change by subtracting
Take away process
The take-away model of
subtraction is the model that is
most familiar from elementary
school.
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“If Harry had 30 marbles
and 10 were taken away
by his friend Ron, how
many were left?”
www.company.com
Part-part whole model
Separating one part
from the whole leaves one
part; combining two parts
makes the whole.

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Elementary computational models and processes – addition and