New analysis of US elementary school mathematics finds half-century of problematic 'strands' structure

In her article, Ma notes that, in many countries where students do well in mathematics, elementary mathematics has school arithmetic as its main organizing structure. School arithmetic is developed as a self-contained subject consisting of whole numbers and fractions, with the whole numbers forming the basis for understanding of fractions. Other components of elementary mathematics, such as measurement or geometry, are not presented as self-contained subjects but are taught in relation to the main subject of school arithmetic.

In the U.S., by contrast, the organizing structure for elementary mathematics has no self-contained structure at its core but rather consists of several strands that are juxtaposed but not explicitly connected. Over the decades, the strands have been given different names -- such as "strands," "content areas," or "standards" -- and their number, form, and content have varied many times.

Examining developments in U.S. mathematics education going back to the 19th century, Ma notes that although U.S. scholars made significant contributions to school arithmetic, the U.S. never had, as some other countries do, a well-developed school arithmetic. Nevertheless, arithmetic was the core of elementary mathematics in the U.S. for almost one hundred years. Ma describes how this began to change during the 20th century with the advent of the "New Math" of the 1960s and the NCTM Standards of the 1990s. Among the effects of the strands structure are instability of curricular content, discontinuity in instruction, and incoherence in concepts.

In the United States, "the potential of school arithmetic to unify elementary mathematics is not sufficiently known," Ma argues. "This is a blind spot for current U.S. elementary mathematics." Too often school arithmetic is equated with basic computational skills that require only inferior cognitive activity such as rote learning. Although many people in mathematics education view arithmetic as an ugly duckling -- that is, a collection of algorithms to be learned by rote -- Ma notes that "in the eyes of mathematicians it is often a swan" because of the mathematical structure mathematicians see in arithmetic.

Liping Ma became well known among mathematicians for her 1999 book Knowing and Teaching Elementary Mathematics (the book was reviewed by Roger Howe in the Notices; see http://www.ams.org/notices/199908/rev-howe.pdf). In this book, Ma studied the understanding of mathematics possessed by school teachers and described a quality called "profound understanding of fundamental mathematics." Ma found that about 10 percent of very experienced teachers in China have this quality. Their profound understanding was acquired not by studying advanced mathematics, but by studying and teaching school mathematics with arithmetic as its core. In an ironic twist, she finds that today China seems to be moving toward adopting a more strand-like structure for its school mathematics standards.