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Developmental Theory

Piaget, as well as many others, did research on stages of development. Significant research has been done on Development Theory in Mathematics,

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:
col/cogsys/piaget.html. Quoting from the Website:

  1. Sensorimotor stage (Infancy). In this period (which has 6 stages), intelligence is demonstrated through motor activity without the use of symbols. Knowledge of the world is limited (but developing) because its based on physical interactions / experiences. Children acquire object permanence at about 7 months of age (memory). Physical development (mobility) allows the child to begin developing new intellectual abilities. Some symbolic (language) abilities are developed at the end of this stage.
  2. Pre-operational stage (Toddler and Early Childhood). In this period (which has two substages), intelligence is demonstrated through the use of symbols, language use matures, and memory and imagination are developed, but thinking is done in a nonlogical, nonreversable manner. Egocentric thinking predominates
  3. Concrete operational stage (Elementary and early adolescence). In this stage (characterized by 7 types of conservation: number, length, liquid, mass, weight, area, volume), intelligence is demonstrated through logical and systematic manipulation of symbols related to concrete objects. Operational thinking develops (mental actions that are reversible). Egocentric thought diminishes.
  4. Formal operational stage (Adolescence and adulthood). In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts [emphasis added by Moursund]. Early in the period there is a return to egocentric thought. Only 35% of high school graduates in industrialized countries obtain formal operations; many people do not think formally during adulthood.

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.


Huitt, W. and Hummel, J. (January 1998). Cognitive Development [Online]. Accessed 2/28/02:
col/cogsys/piaget.html. Quoting from the Website:

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: 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."

The result of the research of Hallas and Schoen is that they recommend that all high school students be taught a core curriculum of mathematics, including algebra, geometry, trigonometry, probability, statistics, and discrete mathematics. However, the slower learners, or non-college-bound students could be taught these subjects in a much more concrete manner, using examples from each discipline that would apply to real world experiences. This approach is supported by other curriculum recommendations, including the NCTM Curriculum and Evaluation Standards for School Mathematics. Topics in all mathematical areas could be presented in interesting but concrete ways.

van Hiele Model of Geometric Thought [Online]. Accessed 3/11/02:
Van_Hiele_Model_of_Geometr.html. Quoting from the Website:

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.

The van Hiele model asserts that the learner moves sequentially through five levels of understanding. Different numbering systems are found in the literature but the van Hieles spoke of levels 0 through 4.

Level 0 (Basic Level): Visualization Students recognize figures as total entities (triangles, squares), but do not recognize properties of these figures (right angles in a square).

Level 1: Analysis Students analyze component parts of the figures (opposite angles of parallelograms are congruent), but interrelationships between figures and properties cannot be explained.

Level 2: Informal Deduction Students can establish interrelationships of properties within figures (in a quadrilateral, opposite sides being parallel necessitates opposite angles being congruent) and among figures (a square is a rectangle because it has all the properties of a rectangle). Informal proofs can be followed but students do not see how the logical order could be altered nor do they see how to construct a proof starting from different or unfamiliar premises.

Level 3: Deduction At this level the significance of deduction as a way of establishing geometric theory within an axiom system is understood. The interrelationship and role of undefined terms, axioms, definitions, theorems and formal proof is seen. The possibility of developing a proof in more than one way is seen.

Level 4: Rigor Students at this level can compare different axiom systems (non-Euclidean geometry can be studied). Geometry is seen in the abstract with a high degree of rigor, even without concrete examples.

The majority of high school geometry courses is taught at Level 3. The van Hieles also identified some characteristics of their model, including the fact that a person must proceed through the levels in order, that the advancement from level to level depends more on content and mode of instruction than on age, and that each level has its own vocabulary and its own system of relations.The van Hieles proposed sequential phases of learning to help students move from one level to another.





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