The document discusses elementary computational models and processes for addition and subtraction. It describes external representations which display knowledge through facts and procedures. Internal representations are how we mentally represent the external world. Key addition and subtraction concepts children develop include the compare schema to make judgments, part-whole schema to understand how parts form a whole, and change-by-adding and change-by-subtracting schemas to solve word problems by manipulating quantities.

1. Elementary
computational
models and
processes – Addition
and Subtraction

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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.

9. 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?”

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Part-part whole model
Separating one part
from the whole leaves one
part; combining two parts
makes the whole.