The purpose of this article is to describe the Taxonomy of Intervention Intensity, which articulates 7 dimensions for evaluating and building intervention intensity. We explain the Taxonomy's dimensions of intensity. In explaining the Taxonomy, we rely on a case study to illustrate how the Taxonomy can systematize the process by which special…

The purpose of this study was to identify cognitive and linguistic predictors of word problems with versus without irrelevant information. The sample was 701 2nd-grade students who received no specialized intervention on word problems. In the fall, they were assessed on initial arithmetic and word-problem skill as well as language ability, working…

Development of fraction number line estimation was assessed longitudinally over five time points between fourth and sixth grades. Although students showed positive linear growth overall, latent class growth analyses revealed three distinct growth trajectory classes: Students who were highly accurate from the start and became even more accurate (n…

The purposes of this study were to (a) investigate the efficacy of a core fraction intervention program on understanding and calculation skill and (b) isolate the effects of different forms of fraction word-problem (WP) intervention delivered as part of the larger program. At-risk 4th graders (n = 213) were randomly assigned at the individual…

The purpose of this article was to summarize results from 5 randomized control trials assessing the effects of intervention to improve the fraction performance of 4th-grade students at-risk for difficulty in learning about fractions. We begin by explaining the importance of competence with fractions and why an instructional focus on fractions…

The purpose of this study was to assess the added value of dynamic assessment (DA) beyond more conventional static measures for predicting individual differences in year-end 1st-grade calculation (CA) and word-problem (WP) performance, as a function of limited English proficiency (LEP) status. At the start of 1st grade, students (129 LEP; 163…

There is a strong research base on the underlying concomitants of early developing math skills. Fewer studies have focused on later developing skills. Here, we focused on direct and indirect contributions of cognitive measures (e.g., language, spatial skills, working memory) and numerosity measures, as well as arithmetic proficiency, on key…

The purpose of this study was to examine child-level pathways in development of prealgebraic knowledge versus word-problem solving, while evaluating the contribution of calculation accuracy and fluency as mediators of foundational skills/processes. Children (n = 962; mean 7.60 years) were assessed on general cognitive processes and early…

The main purposes of this study were to test the effects of teaching at-risk 4th graders to provide explanations for their mathematics work and examine whether those effects occur by compensating for limitations in cognitive processes. We randomly assigned 212 children to 3 conditions: a control group and 2 variants of a multi-component fraction…

Students who demonstrate persistent mathematics difficulties and whose performance is severely below grade level require "intensive intervention". Intensive intervention is an individualized approach to instruction that is more demanding and concentrated than Tier 2 intervention efforts. We present the elements of intensive intervention…

The purpose of this analysis was to examine achievement gaps on fractions for very-low-performing students as a function of whether they receive inclusive fraction instruction or specialized fraction intervention and with the shift to Common Core State Standards (CCSS). In three randomized control trials conducted in 3 consecutive years, 203…

The purpose of this study was to examine the cognitive predictors of calculations and number line estimation with whole numbers and fractions. At-risk 4th-grade students (N = 139) were assessed on 7 domain-general abilities (i.e., working memory, processing speed, concept formation, language, attentive behavior, and nonverbal reasoning) and…

The 3 purposes of this study were to: (a) describe fraction ordering errors among at-risk 4th-grade students; (b) assess the effect of part-whole understanding and accuracy of fraction magnitude estimation on the probability of committing errors; and (c) examine the effect of students' ability to explain comparing problems on the probability of…

The purpose of this study was to assess the relative contribution of teacher and tutor ratings of inattentive behavior in two different instructional settings in predicting students' performance on fraction concepts and whole-number calculations. Classroom teachers rated each student's attentive behavior in a whole-class setting and tutors rated…

This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation…