The Theory of School Arithmetic: Whole Numbers

Abstract

In 1999, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States presented readers with a puzzle. Four of its six chapters concerned interviews of US and Chinese primary teachers. (Note that the two groups of teachers were not chosen to be representative or similar in status.) The teachers described how they would respond to four classroom situations: teaching subtraction with regrouping; addressing a mistake in multi-digit multiplication; creating a word problem for division by fractions; and addressing an incorrect student conjecture about area and perimeter. The puzzle: Why did the US teachers, all of whom had tertiary degrees and were enrolled in intensive post-graduate programs, show less knowledge of school mathematics than the Chinese teachers, who had so much less education? An important part of the answer may be that school arithmetic in the two countries was profoundly different. (We write ‘was’ because expectations have changed in both countries. For example, influenced by the 1989 US Curriculum and Evaluation Standards, significant changes in school arithmetic appear in the 2001 Chinese Mathematics Curriculum Standards.) The knowledge displayed by the Chinese teachers was supported by the solid substance of school arithmetic in China. But, such knowledge was not readily available to their US counterparts. In this article, we give an account of this ‘solid substance’ for whole numbers and end with some examples of its connection with the teachers’ responses.