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Unformatted text preview: Behaviorism can explain changes in animals’ surface behaviors, and
associationism can explain children’s learning of bits of knowledge (such as
nonsense syllables and sums). However, the deep and general logic underly
ing children’s construction of number concepts can be explained only by
Piaget’s constructivism. It is true that human beings can be conditioned, but
there is much more to human knowledge than what animals and young chil
dren can learn.
As can be seen in Figure 1.5b, the relationship between behaviorism and
Piaget’s constructivism is analogous to the one between the geocentric and he
liocentric theories of the universe. The geocentric theory existed first and was
based on common sense. The heliocentric theory went beyond the primitive
theory by encompassing the old one.
An interesting phenomenon in a scientific revolution is that while the new
theory makes the old one obsolete, the old theory remains true within a limited
scope. The geocentric theory became untrue when people stopped believing
that the sun went around the earth. However, from the limited perspective of
earth, it is still true today that the sun rises and sets. This “truth” is reported
daily in the news. It is likewise still true, from the limited perspective of surface
behavior, that drill and reinforcement “work.” From a deeper and longer-range
perspective, however, we no longer think that human beings acquire knowledge
by internalization, reinforcement, and conditioning.
Figure 1.5c shows a similar relationship between Euclidean and nonEuclidean geometry. When non-Euclidean geometry was accepted, it became un- How Do Children Acquire Number Concepts? 17 Figure 1.5. The relationships between (a) behaviorism and Piaget's
constructivism, (b) the geocentric and heliocentric theories, and
(c) Euclidean and non-Euclidean geometry. true that the shortest distance between two points is a straight line. Within the
limited perspective of Euclidean geometry, however, it is still true that the shortest
distance between two points is a straight line. Many other examples of scientific
revolutions can be given to show that a more adequate, later theory goes beyond
a primitive theory by encompassing it. The relationship between Newtonian physics
and quantum physics is another example of a scientific revolution.
All that a scientific theory does is to describe and explain phenomena, and
the practical application of an explanatory theory to an applied field like medi
cine, architecture, or education is not the business of science itself. However, a
scientific theory can be enormously useful in an applied field because it enables
us to change the focus of the debate from “this method of teaching versus that
method of teaching” to how children acquire number concepts (or any other kind
As stated at the beginning of this chapter, there is no disagreement in
medicine about the fact that the cause of cancer has not been explained scien
tifically. Disagreement about how to treat cancer always begins with agreement
about what is known and unknown scientifically about the cause(s) of cancer. 18 TheoreticalFoundation In education, by contrast, debates about how to teach arithmetic rage on with
out even asking how children acquire number concepts. Debates in education
are often based on unproven assumptions, just as people in medicine used to
argue in favor of bloodletting and the use of leeches, citrus fruit, and herbs.
Once we agree, scientifically, on how children acquire number concepts,
we can debate at a higher level how best to foster children’s process of learning.
A growing minority of educators have recognized the superiority of Piaget’s
constructivism and have drastically changed their way of teaching. Just as con
serves cannot go back to nonconservation, and humanity cannot go back to the
geocentric theory after accepting the heliocentric theory, teachers who know
how children acquire number concepts cannot go back to empiricist teaching.
It took 150 years for the heliocentric theory to become universally accepted
(Taylor, 1949). We hope it will not take 150 years for Piaget’s constructivism to
be accepted by educators....
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