Development of the cardinality principle, an understanding that the last number-word recited in counting a collection of items specifies the number of items in that collection, is a critical milestone in developing a concept of number. Researchers in early number development have endeavored to theorize its development. Here we critique two widely…

The purpose of this study was to investigate how teachers understand one specific aspect of proportional reasoning. We were interested in understanding the extent to which practicing teachers were able to make sense of reasoning that involved the fixed number of variable-sized parts perspective. We used two items, drawn from a larger dataset, that…

Two-hour long developmental teaching interviews were conducted with each of 14 sixth grade students, ages 11-12. The purposes of the interviews were to investigate how students solved arrangement problems (APs), and how their solutions of these problems differed from their solution of Cartesian product problems (CPPs). The 14 students represented…

To test the hypothesis that a higher tendency to "s"pontaneously "f"ocus "o"n "m"ultiplicative "r"elations (SFOR) leads to improvements in rational number knowledge via more exact estimation of fractional quantities, we presented sixth graders (n = 112) with fraction number line estimations and a…

Three iterative, 18-episode design experiments were conducted after school with groups of 6-9 middle school students to understand how to differentiate mathematics instruction. Prior research on differentiating instruction (DI) and hypothetical learning trajectories guided the instruction. As the experiments proceeded, this definition of DI…

Students often show difficulties in understanding rational numbers. Often, these are related to the natural number bias, that is, the tendency to apply the properties of natural numbers to rational number tasks. Although this phenomenon has received a lot of research interest over the last two decades, research on the existence of qualitatively…

In this study, we analyzed the imprecise (i.e., less mathematically precise) responses that 148 third- to fifth-grade Chinese students made on selected-response problems that were part of a spontaneous mathematical focusing task, the Quantitative Relations Test for Chinese Children (QRTC[superscript 2]). The purpose for this analysis was to…