This free Information Age Education Newsletter is written by David
Moursund and produced by Ken Loge. For more information, see the end of
this newsletter.
"An individual understands a concept, skill,
theory, or domain of knowledge to the extent that he or she can apply
it appropriately in a new situation." (Howard Gardner, The Disciplined
Mind: What All Students Should Understand.)
The capabilities of
a human brain depend both on nature (genetics) and nurture (informal
and formal education, diet, exercise, and so on). As the capabilities
of one’s brain increase, one is better able to “understand,” as
indicated in the quote from Howard Gardner.
During the past
couple of years, I have gotten quite interested in the topics of
cognitive development (CD) and Intelligence quotient (IQ) as they apply
to students gaining an increasing level of understanding, expertise,
and maturity in various academic disciplines.
Jean Piaget was
interested both in discipline-independent and in discipline specific
aspects of CD. Howard Gardner is interested in multiple
intelligences—that is, different kinds of intelligence that a person
has.
Piaget had a special interest in math CD, and
logical/mathematical is one of the eight areas of intelligence
identified by Gardner. Many people use the term math maturity as they
talk about a student’s progress in learning math. This newsletter
summarizes how math-oriented CD, IQ, and other aspects of math learning
and understanding have been combined into the concept of math maturity.
Many of the ideas given here are applicable in any academic discipline.
Intelligence Quotient
There is substantial research to support the contention that
students of higher IQ learn faster and better than students of lower
IQ. A teacher in a typical elementary school classroom may have one or
two students who can learn twice as fast (and better) than the average
students in the class, and one or two who learn half as fast (and not
as well) as compared to the average students in the class.
Have
you ever wondered why an average person has an IQ of about 100, and
that this does not change much over the years? Surely an average person
develops quite a bit mentally as he or she grows from infancy to
adulthood and learns a great deal during this time through informal and
formal education and through life experiences.
The explanation
to this situation lies in the way that intelligence is measured and
reported. Measures of intelligence are usually normed in a manner that
makes one's IQ a relatively stable number over the years. That is, as
child gets older, the child’s intelligence has to steadily increase in
order to maintain a particular IQ level. One’s intelligence must
increase faster than people with a similar IQ to produce an increasing
IQ.
Cognitive Development
Cognitive development is measured and studied in terms of a stage
theory. Piaget is well known for the initial four-level stage theory
that he developed. According to Piaget’s theory, a child moves from the
Sensory Motor Stage to the Pre Operational Stage to the Concrete
Operations Stage to the Formal Operations Stage.
A modern version of Piaget’s stage theory contains 15 levels (see
http://www.tiac.net/~commons/Commons&Richards04282004.htm).
This large number of stages comes from dividing the Piaget stages into
sub stages and adding several stages above formal operations. In brief
summary:
- A person moves up through the stages, passing through each stage in succession.
- Proper environments (schooling, etc.) and higher IQ help a person move through a stage faster than average.
- Many people peak out at much below the higher levels of formal
operations. However, with proper environments (schooling, etc.) a
person may well progress one or two stages higher than without this
extra help.
Evidence is beginning to mount to support the
contention that the rate that students are progressing upward through
the Piagetian-type stages is declining. A recent study in England found
11 to 12 year old students lagging two to three years below the
cognitive development levels of students 15 or so years earlier. See
http://www.guardian.co.uk/education/2006/jan/24/schools.uk.
See also
http://newsroom.ucla.edu/portal/ucla/is-technology-producing-a-decline-79127.aspx, a study pointing to a decline in critical thinking and analysis, and
http://news.bbc.co.uk/2/hi/uk_news/education/7692843.stm,
a study of a decline in higher-order problem solving skills in British
14 year olds. The authors of these studies suggest that the declines
may come from children’s extensive use of computer games and other
computer-based entertainment systems.
Math Maturity
The following list contains some components of math maturity. An increasing level of math maturity is demonstrated by:
- An increasing capacity to move beyond rote memorization in recognizing,
posing, representing, and solving math problems. This includes transfer
of learning of one’s math knowledge and skills to problems in many
different disciplines.
- An increasing capacity in the logical,
critical, creative reasoning, and thinking involved in understanding
and solving problems and doing proofs.
- An increasing capability to communicate effectively in the language and ideas of mathematics.
- An increasing capacity to learn mathematics—to build upon one’s current
mathematical knowledge and to take increasing personal responsibility
for this learning.
- Improvements in other factors affecting
math maturity such as attitude, interest, intrinsic motivation, focused
attention, perseverance and delayed gratification, having math-oriented
habits of mind, and acceptance of and fitting into the “culture” of the
discipline of mathematics.
Thus, math maturity grows as one’s
intelligence and cognitive development grow, and as one learns math in
an understanding, problem-solving, critical thinking manner.
Final Remarks
IQ researchers have found that average IQ (as measured by a wide
variety of tests) in increasing. This is called the Flynn effect. See http://en.wikipedia.org/wiki/Flynn_effect.
However, the studies cited in the previous section suggest that CD is
declining. This presents a major challenge in the teaching of
higher-order thinking and problem solving in all disciplines. In math
education, for example, this may help to explain why many students seem
to lack the math maturity to learn for understanding in the math
courses they take starting in middle school.
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