knower, there are the complementary dimensions of the subconstructs -- at least quotient, measure, operator, and ratio numbers.
The first of the complementarities described earlier is related logically to the existence of the 1/n fractions for any nonzero integer n. Thus, the unit fractions along with the quotient and ratio nature form a mathematical base for rational numbers. This base also is seen in the mathematical actions of young persons. Considering rational numbers as a humanly knowable activity means taking into account the properties indicated and the many others that distinguish rational from natural numbers.
Seeing humans as capable of knowing rational numbers brings into play ideas from two models. The first argues for the efficacy of seeing rational number knowing as one of four types: ethnomathematical, intuitive, technical symbolic, and axiomatic deductive. The challenge is to find the interrelationships among these knowing types and their appropriate places in the curriculum. The second model, a more general model of the growth of mathematical understanding, portrays an interweaving, nonlinear leveled structure. Each level is characterized by a complementarity of process and form. Because of this, outer-level knowing can derive from either the processes or the forms of inner levels. This model also suggests four bases for mathematical knowing: action, image, form, and structure. These are not to be confused with traditional concrete, pictoral, and symbolic modes. Under this model, understanding rational numbers is characterized as a dynamic whole, of knowing rational numbers at many levels at once.
What are the rational number knowledge structures and the methods of organization that characterize an understanding of rationals toward which one might guide young learners? To know and understand rational numbers is to know numbers that are at once quotients and ratios and to know them, simultaneously, in their many forms. This knowing is organized in many embedded levels that are transcendent but recursive, and in which folding back to go ahead characterizes the interweaving of intuitive and formal understanding.
Behr, M. J., Lesh, R., Post, T., & Silver, E. A. ( 1983). "A mathematical and curricular analysis of rational number concepts". In R. Lesh & M. Landau (Eds.), The acquisition of mathematics concepts and processes (pp. 92-98). New York: Academic.
Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. ( 1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15( 5), 323-341.
Bergeron, J., & Herscovics, N. ( 1987). "Unit fractions of a continuous whole". In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Psychology of mathematics education, PME-XI (Vol. 1, pp. 357-365). Montreal: PME.