There is currently considerable discussion about the relative influences of evolutionary and cultural factors in the development of early numerical skills. In particular, there has been substantial debate and study of the relationship between approximate, nonverbal (approximate magnitude system [AMS]) and exact, symbolic (symbolic number system…

This paper is concerned with developing a new class of generalized numbers. The main advantage of this class is that it generalizes the two classes of generalized Fibonacci numbers and generalized Pell numbers. Some new identities involving these generalized numbers are obtained. In addition, the two well-known identities of Sury and Marques which…

Magnitude understanding is critical for students to develop a deep understanding of fractions and more advanced mathematics curriculum. The research reports in this special issue underscore magnitude understanding for fractions and emphasize number lines as both an assessment and an instructional tool. In this commentary, we discuss how number…

The natural numbers may be our simplest, most useful, and best-studied abstract concepts, but their origins are debated. I consider this debate in the context of the proposal, by Gallistel and Gelman, that natural number system is a product of cognitive evolution and the proposal, by Carey, that it is a product of human cultural history. I offer a…

We review research literature concerning "number sense" from several related fields. Whereas other authors have pointed to difficulty defining "number sense" or to some degree of inconsistency in the literature, we argue instead that this is a case of polysemy: There are 3 different constructs that go by the same name. In this…

In this commentary, I examine some of the distinctive, foundational difficulties in learning fractions and other types of rational numbers encountered by students with a mathematical learning disability and how these differ from the struggles experienced by students classified as low achieving in math. I discuss evidence indicating that students…

We determined the relative importance of the so-called approximate number system (ANS), symbolic number comparison (SNC) and verbal and spatial short-term and working memory (WM) capacity for mathematics achievement in 1,254 Grade 2, 4 and 6 children. The large sample size assured high power and low false report probability and allowed us to…

Use young children's quick attention to numerosity to evaluate their grasp of number while they engage in game play.

Humans and other animals are able to make rough estimations of quantities using what has been termed the "approximate number system" (ANS). Much evidence suggests that sensitivity to numerosity correlates with symbolic math capacity, leading to the suggestion that the ANS may serve as a start-up tool to develop symbolic math. Many…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

Thomas Kuhn (1962/2012) introduced the term "paradigm shift" to the scientific literature to describe how knowledge in science develops. The aims of this article are to identify paradigm shifts, or revolutions, that have occurred in mathematics, and to discuss their relevance to teaching mathematics in schools. The authors argue that…

While conceptual understanding of properties, operations, and the base-ten number system is certainly associated with the ability to access math facts fluently, the role of math fact memorization to promote conceptual understanding remains contested. In order to gain insight into this question, this study looks at the results when one of three…

Recent research has supported and extended earlier research on how and for how long Indigenous people of Australasia have been counting. This history values the long history of Indigenous knowledge and re-writes the limited and sometimes false history that many Australian teachers accept and teach about number systems. The current views on the…

To increase opportunities for students to take more advanced math courses in high school, many school districts enroll grade 8 students in Algebra I, a gateway course for advanced math. But students who take Algebra I in grade 8 and skip other math courses, such as grade 8 general math, might miss opportunities to develop the foundational…

Learning fractions is difficult for children in general and especially difficult for children with mathematics difficulties (MD). Recent research on developmental and individual differences in fraction knowledge of children with MD and typically achieving (TA) children has demonstrated that U.S. children with MD start middle school behind their TA…