Unformatted text preview: the initial equality: “But remem
ber how you put a red counter in front of each blue one before? An
other child said there are just as many red ones as blue ones now be
cause all I did was move them. Who do you think is right, you or the
The Three Levels Found Level 1. At Level 1, the child cannot make a set that has the same number.
Some children put out all the red counters as shown in Figure 1.2a.
They stop putting counters out only because there are no more left.
Figure 1.2b shows a more advanced response within Level 1. The chil
dren who do this do not put out the same number of red counters as
blue ones but carefully use the spatial frontiers of the rows as the cri
terion for deciding the “sameness” of the two quantities. (When chil
dren have not yet built the logic of number, they use the best crite
rion they can think of to judge the quantitative “sameness.” In this case,
the criterion is the spatial frontiers, which they can see.)
Level 2. At Level 2 children can make a set that has the same number by
using one-to-one correspondence, but they cannot conserve this equal
ity. When asked the conservation question, they reply, for example,
“There are more red ones because the red line is longer.”
Level 3. At Level 3 children are conservers. They give correct answers to
all the questions, are not swayed by countersuggestions, and give one
of the following three arguments to explain why they think the two
rows have the same quantity:
• “There’s just as many blue ones as red ones because you didn’t add
anything or take anything away” (the identity argument).
• “We could put all the red ones back to the way they were before,
and you’ll see that there’s the same number” (the reversibility
argument). Theoretical Foundation 8 Figure 1.2. Two sublevels within Level 1. • “The red row is longer, but there’s more space in between. So the
number is still the same” (the compensation argument).
Intermediate level. Conservation is not achieved overnight, and there is an
intermediate level between Levels 2 and 3. Intermediate-level chil
dren hesitate and/or keep changing their minds (“There are more blue
ones . . . , no, red ones . . . , no, they’re the same_ ”). When children
give the correct answer but cannot justify it, they are also categorized
at the intermediate level.
The conservation task is a test of children’s logico-mathematical knowledge.
Counters are cultural objects (social knowledge), and knowing that counters stay
on the table without melting like ice cubes is physical knowledge. However,
physical knowledge is not enough to deduce that the quantity in the two rows
stays the same when their empirical appearance changes. Only when children
can make numerical relationships among the chips can they deduce, with the
force of logical necessity, that the two rows have the same number. This state
ment will be clarified by the following discussion of two lands of abstraction
Piaget distinguished. How Do Children Acquire Number Concepts? 9 Empirical and Constructive Abstraction
In empirical abstraction, we focus on a certain property of the object and
ignore the others. For example, when we abstract the color of an object, we sim
ply ignore the other properties such as weight and the material with which the
object is made (plastic or glass, for instance).
Constructive abstraction involves making mental relationships between and
among objects, such as “the same,” “similar,” “different,” and “two.” As stated
earlier, these relationships do not have an existence in external reality. The simi
larity or difference between one counter and another is constructed, or men
tally made, by each individual by constructive abstraction.
Constructive abstraction is also known as “reflective” or “reflecting” abstrac
tion. The French term Piaget usually used was abstractio...
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- Spring '14
- The Land, number concepts