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Unformatted text preview: usions), sensory experience cannot
be trusted to give us truth with certitude. The rigor, precision, and certainty of
mathematics, a purely deductive discipline, remains the rationalists’ prime example
in support of the power of reason. When they had to explain the origin of this power,
many rationalists ended up saying that certain lands of knowledge or concepts are
innate and that these unfold as a function of maturation.
Piaget saw elements of truth and untruth in both camps. As a scientist
trained in biology, he was convinced that the only way to answer epistemological questions was to study them scientifically rather than by continuing to argue
on the basis of speculation. With this conviction, he wanted to study humanity’s
construction of mathematics from its prehistoric beginning, because to under
stand human knowledge he believed that it was necessary to study its develop
ment rather than only the end product. However, the prehistoric and historical
evidence was no longer available to him, and this is why he decided that a good
way to study the evolution of empirical knowledge and reason was to study their
development in children. His study of children was thus a means to answer
epistemological questions scientifically.
While Piaget saw the importance of bodi sensory information and reason, his
sympathy lay on the rationalist side of the fence. His 60 years of research with chil
dren was motivated to a large extent by a desire to prove the inadequacy of empiri
cism. The three kinds of knowledge and the nature oflogico-mathematical knowl
edge, which are discussed next, should be understood in light of this background. THE NATURE OF LOGICO-MATHEMATICAL KNOWLEDGE
Three Kinds of Knowledge
Piaget (1967/1971, 1945/1951) distinguished three kinds of knowledge ac
cording to their ultimate sources and modes of structuring: physical knowledge,
social (conventional) knowledge, and logico-mathematical knowledge. How Do Children Acquire Number Concepts? 5 Physical knowledge is knowledge of objects in external reality. The color
and weight of counters or any other object are examples of physical knowledge.
The fact that counters do not roll away like marbles is also an example of physi
cal knowledge. The ultimate source of physical knowledge is thus partly in ob
jects, and physical knowledge can be acquired empirically through observation.
(Our reason for saying “partly” will be explained shortly.)
Examples of social knowledge are languages such as English and Spanish,
which were created by convention among people. Other examples of social
knowledge are holidays such as the Fourth of July, the rule of extending our
right hand to shake hands, and rules about when to say “Good morning.” The
ultimate source of social knowledge is thus partly in conventions made by people.
(Our reason for saying “partly” will also be clarified shortly.)
Logico-mathematical knowledge consists of mental relationships, and the
ultimate source of these relationships is in each individual. For instance, when
we are presented with a red counter and a blue one, we can think about them as
being different or similar. It is just as true to say that the counters are different
(because one is red and one is blue) as it is to say that they are similar (because
they are both round and made of plastic). The similarity and difference exist
neither in the red counter nor in the blue one, and if a person did not put the
objects into a relationship, these relationships would not exist for him or her.
Other examples of relationships the individual can create between the two
counters are the same in weight and two. From the point of view of weight, the
two counters are the same. If the individual wants to think about the same
counters numerically, the counters become “two.” The counters are observable,
but the “twoness” is not. Number is a relationship created mentally by each
We hasten to say that “two...
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This document was uploaded on 03/13/2014.
- Spring '14
- The Land