Kyle, a kindergartner from a low-income family, is shown a set of three black dots on a white mat. His teacher then hides the dots with a small box lid and lays down an additional set of two dots. She pushes the two dots under the cover, one at a time. Kyle must now choose the number of dots "hiding" under the box from a set of four…

Despite convincing research and strong public support, education budget cuts have forced schools and districts to limit or even eliminate arts instruction. Author Robin Ward argues that educators need to address the critical need to preserve and, in those cases, re-integrate the arts back into the school curriculum--mostly because the arts are…

Rational number interpretations can include part-whole, measure, ratio, quotient, and operator. These are all subconstructs of partitioning (Barnett-Clarke et al. 2010; Behr et al. 1980; Clarke, Roche, and Mitchell 2008; Flores, Samson, and Yanik 2006). Each of these subconstructs uses different cognitive skills (Driscoll 1984), so it is important…

This article reports on a study in which researchers asked children to "make up as story and a picture about marbles for this number sentence: 3 x 5 = 15." Students in this study came from pre - dominantly low- to average-income families living in three distinct geographical areas within the United States. A similar division task was…

In the study of music from a mathematical perspective, several types of counting problems naturally arise. For example, how many different rhythms of a specified length (in beats) can be written if we restrict ourselves to only quarter notes (one beat) and half notes (two beats)? What if we allow whole notes, dotted half notes, etc.? Or, what if…

The use of enabling and extending prompts allows tasks to be both accessible and challenging within a classroom. This article provides an example of how to use enabling and extending prompts effectively when employing a challenging task in Year 2.

Reciprocal triangular numbers have appeared in series since the very first infinite series were summed. Here we attack a number of subseries of the reciprocal triangular numbers by methodically expressing them as integrals.

This unlikely cast of characters, by working collaboratively in a trusting learning community, was able to identify an approach to teaching rational numbers through measuring from the everyday practices of Yup'ik Eskimo and other elders. "The beginning of everything," as named by a Yup'ik elder, provided deep insights into how practical…

The simple process of iteration can produce complex and beautiful figures. In this article, Robin O'Dell presents a set of tasks requiring students to use the geometric interpretation of complex number multiplication to construct linear iteration rules. When the outputs are plotted in the complex plane, the graphs trace pleasing designs…

In general, units coordination refers to the relationships that students can maintain between various units when working within a numerical situation. It is critical that middle school students learn to coordinate three levels of units not only because of their importance in understanding fractions but also because of their implications for…

The collection of studies in this special issue make an important contribution to our understanding and measurement of the core cognitive and noncognitive factors that influence children's emerging quantitative competencies. The studies also illustrate how the field has matured, from a time when the quantitative competencies of infants and young…

Almost everyone loves chocolate. However, the same cannot be said about fractions, which are loved by markedly fewer. Middle school students tend to view them with wary respect, but little affection. The authors attempt to sweeten the subject by describing a type of game involving division of chocolate bars. The activity they describe provides a…

An analysis of one-step equations from a cognitive load theory perspective uncovers variation within one-step equations. The complexity of one-step equations arises from the element interactivity across the operational and relational lines. The higher the number of operational and relational lines, the greater the complexity of the equations.…

How can educators encourage and better prepare students to pursue science, technology, engineering, and mathematics (STEM)-based fields? To start, students are more likely to pursue these fields if they enjoy and perceive themselves to be good at them. This means introducing relevant concepts and skills at an early age and embedding them…

Skilled instructors observe consistent errors that young children may make in particular situations and seek ways to help children reform their thinking through conceptual understanding. Such teacher observations as those presented at the beginning of this article, combined with similar observations by teachers in other preschool and kindergarten…