Developing Cognitive
Thinking
The goal of the Structural Arithmetic Programme is to
develop cognitive thinking and an appreciation for the
exactness and clarity of mathematics. Arithmetic, a
branch of mathematics, can and should be taught from
the beginning so as to enable children to think and
to reason things out for themselves. For example, in
the 10-box children fit together pairs of blocks that
total 10 before number names have been assigned.
Experiments with the Structural Arithmetic materials
form the core of the programme. The materials, which
are designed to make the structure of the number system
visible, enable pupils to discover number concepts and
to gain insight into the meaning of each operation:
addition, subtraction, multiplication, and division.
Teaching Number Facts in Structurally
Related Groups
Children who are taught number facts by rote and in
isolation are denied the joy of using their minds. Such
an approach prevents pupils from developing the ability
to think and to reason. Children often work out an addition
fact by counting beads or using rods. Since they learn
it in isolation, they miss an opportunity to reason.
The Structural Arithmetic Programme introduces facts
in groups, giving children insight into the characteristic
structures common to these facts. This procedure allows
children to develop the ability to think mathematically
as they work out for themselves the relationships between
the number facts and express the generalisation in their
own words.
As an example of the Structural Arithmetic approach,
let us look at the single fact, 3 + 7 = 10. Children
discover all the combinations that make 10 by fitting
combinations of blocks into the 10-box. They reason
that if 9 needs 1 to make 10, then 8 needs 2, and 7
needs 3 to make 10. By switching the blocks around,
they discover that the order of the addends can be changed
without changing the sum. Thus, they grasp the interchangeable
nature of addition and can put it to use. This enables
them to reason that if 7 + 3 = 10, then 3 + 7 = 10.
This fact has not been learned in isolation, but has
been studied in a context where its relation to the
other facts can be seen.
Developing Concepts by Measuring,
Not by Counting
When children see an example such as 5 + 4 = and don't
know the answer, they often respond by counting "6,
7, 8, 9." Teachers may assume that encouraging
children to count, will one day result in their stopping
counting and saying, '9'. In actuality, each time they
see + 4 (as in 6 + 4 or 9 + 4), they automatically practise
counting. The numbers themselves don't have meaning.
For the counting child, 5 plus 4 does not equal 9; it
makes 9 by counting. No picture in the child's mind
makes the number fact 5 + 4 = 9 unforgettable. Furthermore,
if children count the total incorrectly as 10, they
have no certain way to check that result except by another
uncertain counting procedure. On the other hand, in
Structural Arithmetic, the two addends 5 and 4 actually
measure 9 in the Number Track.* It becomes obvious,
then, that counting is a senseless rote procedure that
prevents children from learning to think or reason.
Developing Spatial Thinking
and Reasoning
In Structural Arithmetic, children add together two
quantities and measure their total; for instance, the
7-block plus the 3-block measures 10. Working with materials
in this way allows children to experiment with ideas
and to work out other relationships. They might take
one of the blocks from the total and leave the other
one, thus discovering the related subtraction fact 10-3
= 7. Whether they are measuring with blocks or working
with patterns of cubes, they are using spatial thinking
to help them reason. Each experiment leaves a mental
picture that the children can turn around in their minds
to explore new relationships.
Progressing from Thinking to
Writing Equations
As soon as children want to communicate what they have
been thinking spatially, they must put their ideas into
words. Teachers should encourage them to use their own
phrasing, such as '7 needs 3 to be as big as 10,' and
'When you take 3 away from 10, 7 is left.' The final
step, then, is easy - to set down these thoughts in
equation form: 7 + 3 = 10 and 10 - 3 = 7. Experimenting
with the materials has enabled children to comprehend
and use the symbols and signs of mathematics to form
any equation. They have reached the stage of mastery;
they fully understand the operations and can give the
answers with confidence.
Economy of Learning - The Result
of Transfer
By making the structure of our number system visual,
the Structural Arithmetic materials make it possible
for children to transfer a newly learned fact to other
areas. If they know that 5 + 2 = 7, they can discover
that this fact holds true in any decade by measuring
in the Number Track,* 15 + 2 = 17, 25 + 2 = 27, or 65
+ 2 = 67. By working with cubes and 10-blocks in the
Dual Board* the children can find that what is true
for 'ones' is also true for 'tens', 50 + 20 = 70. The
result is an immense economy in the number of facts
that have to be learned.
The Structural Arithmetic Programme is designed to
help children develop number concepts and arrive at
generalisations essential to the understanding of mathematics.
To the degree in which they develop their ability to
think with numbers, their work will be creative and
fulfilling.
*The Number Track and Dual Board are part of Kit
B and are used in Books 3. 4 and 5.
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