"Assumptions" play a fundamental role in disciplinary mathematical practice, especially concerning the relativity of truth. However, much is still unclear about ways to help students recognize key aspects of this role. In this paper, we propose a set of principles for task design to introduce students to the role of assumptions in…

Difficulties with rational numbers have been explained by a natural number bias, where concepts of natural numbers are inappropriately applied to rational numbers. Overcoming this difficulty may require a radical restructuring of previous knowledge. In order to capture this development, we examined third- to fifth-grade students' understanding of…

This study examines presentations of the distributive property (DP) in two widely used U.S. elementary text series and one main Chinese text series along three dimensions: problem contexts, typical problem types within each problem context, and variability in using the DP. In general, the two U.S. texts were found to resemble each other but to…

We present an empirical study that investigated seventh-, ninth-, and eleventh-grade students' understanding of the infinity of numbers in an interval. The participants (n = 549) were asked how many (i.e., a finite or infinite number of numbers) and what type of numbers (i.e., decimals, fractions, or any type) lie between two rational numbers. The…

This study reports findings from comparative samples of sixth-grade Chinese and U.S. students' interpretations of the equal sign. Ninety-eight percent of the Chinese sample correctly answered 4 items indicating conceptions of equality and provided conceptually accurate explanations. In contrast, only 28% of the U.S. sample performed at this level.…

Documented differences in frequency and type of mathematical explanations during lessons in U.S., Taiwanese, and Japanese first- and fifth-grade classrooms. Found that explanations occurred more frequently in Japanese and Taiwanese classrooms than in U.S. classrooms. Typical explanations in Asian classrooms were more substantive than in U.S.…