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Developmental Theory. Piaget, as well as many others, did
research on stages of development. Piaget, for example,
talks about a child beginning at the level of Sensory Motor,
moving to Preoperational, then Concrete Operations and
eventually reaching Formal Operations. Click
here to read about Piaget's theory of Cognitive
Constructivism. Or, read the following materials from Huitt,
W. and Hummel, J. (January 1998). Cognitive Development
[Online]. Accessed 2/28/02: http://chiron.valdosta.edu/whuitt/
Developmental Theory is certainly applicable to learning mathematics. If we attempt to teach a math topic to a student who is far from being developmentally ready for it, the child tends to have little recourse but to attempt to "get by" by memorizing and regurgitating. Secondary school math teachers see this all the time, perhaps most especially in geometry courses and more advanced courses. The "Algebra for all." movement is suspect partly because it appears to be pushing many students into classes for which they are not mathematically developmentally ready. Algebra and Formal Operations seem to be closely related. The table given below is from Huitt and Hummel (January 1998).
Notice that only 35% of students reach formal operations by the time they finish high school. This is suggestive that there is a substantial mismatch between our secondary school math curriculum and the developmental level of students. Arnold Arons wrote extensively about the developmental level of college students in his science courses. He noted that a significant percentage (and, an even higher percentage of education majors) were not at the Formal Operations level. Thus, they learned science by rote memorization. He noted that about 10% of students in the General Physics course at the University of Washington (this is the course for students who are seeking a serious course in Physics and who have a strong high school math background) were not at the Formal Operations level. He developed an intervention that could help bring students to Formal Oerations. It was many weeks in length. A Google search on 2/12/06, using the expression Arnold Arons University of Washington physics produced about 10,000 hits. ReferencesHuitt, W. and Hummel, J. (January 1998). Cognitive
Development [Online]. Accessed 2/28/02: http://chiron.valdosta.edu/whuitt/ Jean Piaget (1896-1980) was one of the most influential researchers in the area of developmental psychology during the 20th century. Piaget originally trained in the areas of biology and philosophy and considered himself a "genetic epistimologist." He was mainly interested in the biological influences on "how we come to know." He believed that what distinguishes human beings from other animals is our ability to do "abstract symbolic reasoning." Piaget's views are often compared with those of Lev Vygotsky (1896-1934), who looked more to social interaction as the primary source of cognition and behavior. Keyes, Cynthia (1997). A Review of Research on General Mathematics Research [Online]. Accessed 2/28/02: http://www.gsu.edu/~mstlls/res_ck.htm. Quoting from this brief student-written review: The van Hiele model of the development of geometric thought illustrates what might be taught based on a student's level of geometric maturity. The levels of the model, in order, are "visualization," "analysis," "informal deduction," "formal deduction," and "rigor." The van Hiele research asserts that students must move through the levels in order, and they must have understanding at each level before moving on to the next. The progress through these levels is very dependent on the type of instruction, so teachers must provide appropriate activities at each level. Many hands-on activities can be provided for students at all levels, particularly the lower levels. For specific ideas on activities to use in each of the van Hiele levels, a good source is Mary L. Crowley's "The van Hiele Model of the Development of Geometric Thought." van Hiele Model of Geometric Thought [Online].
Accessed 3/11/02: http://euler.slu.edu/teachmaterial/ Two Dutch educators, Dina and Pierre van Hiele, suggested that children may learn geometry along the lines of a structure for reasoning that they developed in the 1950s. Educators in the former Soviet Union learned of the van Hiele research and changed their geometry curriculum in the 1960s. During the 1980s there was interest in the United States in the van Hieles' contributions; the {Standards} of the National Council of Teachers of Mathematics (1989) brought the van Hiele model of learning closer to implementation by stressing the importance of sequential learning and an activity approach.
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