Information Age Education
   Issue Number 20
June, 2009   

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:
  1.  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.
  2. An increasing capacity in the logical, critical, creative reasoning, and thinking involved in understanding and solving problems and doing proofs.
  3. An increasing capability to communicate effectively in the language and ideas of mathematics.
  4. An increasing capacity to learn mathematics—to build upon one’s current mathematical knowledge and to take increasing personal responsibility for this learning.
  5. 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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