This article investigates the interplay of time words with how children position hands on an analog clock. Using a mathematics discourse framework (Sfard, 2008), we analyzed how students interpreted precise (e.g., 2:30) and relative (e.g., half past 11) times, finding that particular words are dynamically interwoven with activity. Interviews with…

This paper outlines a framework to explain the early development of place-value understanding based on an analysis of data from 84 five- to seven-year-old children from diverse cultural and linguistic backgrounds. The children were assessed individually on number knowledge tasks (recalled facts, subitizing, counting, place-value understanding) and…

Research findings have established that students often struggle with mathematical symbols including common misconceptions for literal symbolic representations of variables but provide little evidence of when or how these misconceptions arise. This article reports findings from a study of grade 4-6 students' conception(s) for various…

Proportional reasoning is important to the field of mathematics education because it lies at the crossroads of additive reasoning in the elementary school and multiplicative reasoning needed for more advanced mathematics. This research reports on the representations used by pre-service teachers (PSTs) as they responded to tasks involving…

In contrast to previous studies on Spontaneous Focusing on Quantitative Relations (SFOR), the present study investigated not only the "extent" to which children focus on (multiplicative) quantitative relations, but also the "nature" of children's quantitative focus (i.e., the types of quantitative relations that children focus…

The present study investigated the development of fraction comparison strategies through a longitudinal analysis of children's responses to a fraction comparison task in 4th through 6th grades (N = 394). Participants were asked to choose the larger value for 24 fraction pairs blocked by fraction type. Latent class analysis of performance over item…

Do concrete learning materials promote strong learning outcomes, or do they simply make learning tasks more initially accessible? Although concrete materials may offer an intuitive foothold on a topic, research on desirable difficulties suggests that more challenging tasks facilitate greater retention and transfer. In the approach introduced here,…

The current study examined the development of two effects that have been found in single-digit multiplication errors: relatedness and distance. Typically achieving (TA) second, fourth, and sixth graders and adults, and sixth and eighth graders with a mathematics learning disability (MLD) performed a verification task. Relatedness was defined by a…

In this paper the authors present three design principles they use to develop preservice teachers' mathematical content knowledge for teaching in their mathematics content and/or methods courses: (1) building on currently held conceptions, (2) modeling teaching for understanding, (3) focusing on connections between content knowledge and other…

This report describes reflections from two cycles of developmental research that involved creating and refining a series of computer-based applets for reasoning about the relative magnitude of fractional quantities. The applet sequence stemmed from a cognitively demanding task used in face-to-face teacher education settings that involved placing…

A hierarchy for learning to solve different types of addition with fractions problems was hypothesized on the basis of both content analysis and psychological considerations. Problem types were defined according to the relationship of the two denominators to each other (e.g., equal, prime, etc.) Students in grades 4 through 8 were each given 45…

The purpose of the two studies reported was to develop and validate clinical assessment instruments designed to identify both the progress of learners through a hierarchy of mathematics content and the mode of presentation through which learners can best handle the content. In Study 1, 73 second graders were given one of two forms of a 15-item…

A 20-task learning hierarchy for subtraction of fractions was deductively derived using Gagne's task analysis. To test this analysis empirically, composite items were written for each level and administered to students in grades 3-6. Test results were analyzed by the Walbesser Technique and Pattern Analysis; the acceptance levels developed by…

The mathematics curriculum provides a framework of instruction for exceptional child education in grades K-12. Content areas include: numeration, whole numbers, rational numbers, real/complex numbers, calculator literacy, measurement, geometry, statistics, functions/relations, computer literacy, and pre-algebra. The guide is organized by content…