A major source of errors in rational number tasks is the inappropriate application of natural number rules. We hypothesized that this is an instance of intuitive reasoning and thus can persist in adults, even when they respond correctly. This was tested by means of a reaction time method, relying on a dual process perspective that differentiates…

We investigated whether specific input helps 3-1/2-year-olds discover that the last word in a count represents its cardinal value (i.e., the cardinal word principle). In Study 1, we contrasted four training approaches. The only approach to yield significant improvement was to label a set's cardinality and then immediately count it. This training…

Counting complete subgraphs of three vertices in complete graphs, yields combinatorial arguments for identities for sums of squares of integers, odd integers, even integers and sums of the triangular numbers.

How do children learn the meanings of number words like "one," "two," and "three"? Whereas many words that children learn in early acquisition denote individual things and their properties (e.g., cats, colors, shapes), numerals, like quantifiers, denote the properties of sets. Unlike quantifiers such as "several" and "many," numerals denote…

It is of deep interest to both linguists and psychologists alike to account for how young children acquire an understanding of number words. In their commentaries, Barner and Butterworth both point out that an important question highlighted by the work of Syrett, Musolino, and Gelman, and one that remains highly controversial, is where number…

The ability to count has traditionally been considered an important milestone in children's development of number sense. However, using counting (e.g., counting on, counting all) strategies to solve addition problems is not the best way for children to achieve their full mathematical potential and to prepare them to develop more complex and…

The goal of this study was to investigate understanding of inservice elementary school teachers in Taiwan about number sense, teaching strategies of number sense and the development of number sense of students; and the profile of integrating number sense into mathematical instruction , and teaching practice. Data was gathered through interviews of…

Students around the world have difficulties in learning about fractions. In many countries, the average student never gains a conceptual knowledge of fractions. This research guide provides suggestions for teachers and administrators looking to improve fraction instruction in their classrooms or schools. The recommendations are based on a…

Exploring number systems of other cultures can be an enjoyable learning experience that enriches students' knowledge of numbers and number systems in important ways. It helps students deepen mental computation fluency, knowledge of place value, and equivalent representations for numbers. This article describes how the author designed her…

Research on probability judgment has traditionally emphasized that people are susceptible to biases because they rely on "variable substitution": the assessment of normative variables is replaced by assessment of heuristic, subjective variables. A recent proposal is that many of these biases may rather derive from constraints on cognitive…

Numbers are fundamental entities in mathematics, but their cognitive bases are unclear. Abundant research points to linear space as a natural grounding for number representation. But, is number representation fundamentally spatial? We disentangle number representation from standard number-to-line reporting methods, and compare numerical…

Integers are sometimes used in physics problems to simplify the mathematics so the arithmetic does not distract students from the physics concepts. This is particularly important in exams where students should not have to spend a lot of time using their calculators. Common uses of integers in physics problems include integer solutions to…

Several recent behavioral studies have shown that the enumeration of a small number of items (a process termed "subitizing") depends on the availability of attentional resources and is not a preattentive process as previously thought. Here we studied the neural correlates of visual enumeration under different attentional loads in a dual-task…

The commentary by Treccani and Umilta (2011) on our recent paper in this journal (Fischer et al., 2010) usefully broadens the perspective on numerical cognition for a general readership and clarifies (or raises) some fundamental issues of cognitive representation. Our response to this commentary is organized into three main sections that focus…

The present study involved examining whether a storybook reading intervention targeting mathematics vocabulary, such as "equal," "more," and "less," and associated number concepts would increase at-risk children's vocabulary knowledge and number competencies. Children with early numeracy difficulties (N = 124) were…