A Critique of the Structure of U.S. Elementary School Mathematics

Abstract

R esearch in mathematics education can be partitioned in many ways. If research in elementary mathematics education is partitioned into just two categories, content and method, this article may seem to belong only to the latter. It argues that consideration of the way in which the content of elementary mathematics is organized and presented is worthwhile for both U.S. and Chinese elementary mathematics educators. But, as illustrated in this article, organizing structure may affect the content that is presented. Two distinguishing features of organizing structures for elementary mathematics are categorizations of elementary mathematics content and the nature of the relationship among the categories. These features are illustrated by the two examples in Figure 1 below. Example A has a “core-subject structure”. The large gray cylinder in the center represents school arithmetic. Its solid outline indicates that it is a “self-contained subject”. (The next section of this article elaborates the meaning of this term.) School arithmetic consists of two parts: whole numbers (nonnegative integers) and fractions (nonnegative rational numbers). Knowledge of whole numbers is the foundation upon which knowledge of fractions is built. The smaller cylinders represent the four other components of elementary mathematics, shown according to the order in which they appear in instruction. These are: measurement (M); elementary geometry, simple equations (E); and simple statistics (S). (The last is similar to the U.S. “measurement and data” and includes tables, pie charts, line graphs, and bar graphs.) The dotted outlines and interiors of these components indicate that they are not self-contained subjects. The sizes of the five cylinders reflect their relative proportions within elementary mathematics, and their positions reflect their relationship with arithmetic: arithmetic is the main body of elementary mathematics, and the other components depend on it. Each nonarithmetic component occurs at a stage in the development of school arithmetic that allows the five components to interlock to form a unified whole. Example B has a “strands structure”. Its components are juxtaposed but not connected. Each of the ten cylinders represents one standard in Principles and Standards for School Mathematics. The content standards appear in the front, and the process standards appear in the back. No selfcontained subject is shown. This type of structure has existed in the U.S. for almost fifty years, since the beginning of the 1960s. Over the decades, the strands have been given different names (e.g., “strands”, “content areas”, or “standards”) and their number, form, and content have varied many times. In these two structural types, the main difference is that the core subject structure has a selfcontained subject that continues from beginning to end. In contrast, the strands structure does not, and all its components continue from beginning Liping Ma is an independent scholar of mathematics education. Her email address is maliping51@gmail.com.