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This practice guide provides evidence-based practices that can help teachers tailor their instructional approaches and/or their mathematics intervention programs to meet the needs of their students.

This practice guide provides three recommendations for teaching algebra to students in middle school and high school. Each recommendation includes implementation steps and solutions for common roadblocks. The recommendations also summarize and rate supporting evidence. This guide is geared toward teachers, administrators, and other educators who want to improve their students’ algebra knowledge.

This practice guide presents five recommendations intended to help educators improve students’ understanding of fractions. Recommendations include strategies to develop young children’s understanding of early fraction concepts and ideas for helping older children understand the meaning of fractions and the computations involved. The guide also highlights ways to build on students’ existing strategies to solve problems involving ratios, rates, and proportions.

Taking early action may be key to helping students struggling with mathematics.

Pre-K Mathematics is a supplemental curriculum designed to develop informal mathematical knowledge and skills in preschool children. Specific mathematical concepts and skills from each unit are taught in the classroom through teacher-guided, small-group activities using concrete manipulatives. Take-home activities with materials that parallel the small-group classroom activities are designed to help caregivers support their children’s mathematical development at home.

Cognitive Tutor^®, published by Carnegie Learning, is a math curricula that combines textbooks and interactive software.

Pre-K Mathematics is a supplemental curriculum designed to develop informal mathematical knowledge and skills in preschool children. Specific mathematical concepts and skills from each unit are taught in the classroom through teacher-guided, small-group activities using concrete manipulatives. Take-home activities with materials that parallel the small-group classroom activities are designed to help caregivers support their children’s mathematical development at home.

Odyssey® Math is a web-based program developed by Compass Learning® for mathematics instruction in grades K–8. The online program includes a mathematics curriculum and formative assessments designed to support differentiated and data-driven instruction. Based on assessment results, the program generates an individualized sequence of mathematics topics and skills—a “learning path.” Odyssey® Math is often used as a prescriptive tool, where students can start by taking a diagnostic assessment aligned with local or state standards. Teachers can modify learning paths to match their lesson plans or to align them with district scopes and sequences.

Fraction Face-Off! is a supplemental math program developed to support fourth-grade students who need assistance solving fraction problems. Teachers use program materials with individual students or small groups to promote understanding of the magnitude of fractions, to compare two fractions, to put three fractions in order, and to place fractions on a number line.

I CAN Learn® is a computer-based math curriculum for students in middle school, high school, and college. It provides math instruction through a series of interactive lessons that students work on individually at their own computers. Students move at their own pace and must demonstrate mastery of each concept before progressing to the next one. Classroom teachers may provide individual, small-group, or whole-class instruction based on students’ performance on the software program.

University of Chicago School Mathematics Project (UCSMP) Algebra, designed to increase students’ skills in algebra, is appropriate for students in grades 7–10, depending on the students’ incoming knowledge. This 1-year course highlights applications, uses statistics and geometry to develop the algebra of linear equations and inequalities, and includes probability concepts in conjunction with algebraic fractions. The curriculum emphasizes graphing, delaying manipulation with rational algebraic expressions until later courses. This curriculum uses the UCSMP textbook.

UCSMP is a core mathematics curriculum that emphasizes problem solving, real-world applications, and the use of technology. The curriculum is based on a student-centered approach with a focus on active learning that incorporates reading and uses a flexible lesson organization.

Everyday Mathematics® is a core curriculum for students in prekindergarten through grade 6. At each grade level, the Everyday Mathematics® curriculum provides students with multiple opportunities to learn concepts and practice skills. Across grade levels, concepts are reviewed and extended in varying instructional contexts. The distinguishing features of Everyday Mathematics® are its focus on real-life problem solving, student communication of mathematical thinking, and appropriate use of technology. This curriculum also emphasizes balancing different types of instruction (including collaborative learning), using various methods for skills practice, and fostering parent involvement in student learning.

DreamBox Learning is a supplemental online mathematics program that provides adaptive instruction for students in grades K–5 and focuses on number and operations, place value, and number sense. The program aims to individualize instruction for each student using unique paths through the curriculum ihat match each student’s level of comprehension and learning style.

Core-Plus Mathematics is a 4-year curriculum that replaces the traditional sequence with courses that each feature interwoven strands of algebra and functions, statistics and probability, geometry and trigonometry, and discrete mathematics. The curriculum emphasizes mathematical modeling, using technology to emphasize reasoning with multiple representations (verbal, numerical, graphical, and symbolic) and to focus on goals in which mathematical thinking and problem solving are central. Instructional materials promote active learning and teaching centered around collaborative small-group investigations of problem situations, followed by teacher-led whole-class summarizing activities that lead to analysis, abstraction, and further application of underlying mathematical ideas.

The Expert Mathematician is designed to help middle school students develop the thinking processes for mathematical applications and communication. A 3-year program of instruction, The Expert Mathematician uses a software and print materials package with 196 lessons that teach the Logo programming language. Each lesson ranges from 40–120 minutes. A test of unit concepts is administered at the end of each instructional unit. The curriculum covers general mathematics, pre-algebra, and algebra I.

University of Chicago School Mathematics Project (UCSMP) is a core mathematics curriculum that includes materials and a routinized instructional approach with an option for teacher training. The curriculum uses an inquiry-based approach with a focus on active learning where students frequently engage in hands-on activities and small-group activities. Pre-Transition Mathematics teaches arithmetic, algebra, geometry, probability, and statistics. Transition Mathematics teaches more advanced arithmetic, algebra, and geometry, and connects these areas to measurement, probability, and statistics.

Math Expressions is a curriculum for students in prekindergarten through sixth grade that aims to build students’ conceptual understanding of mathematics and to develop fluency in mathematical problem solving and computation. The curriculum encourages student learning of mathematics through real-world situations, visual supports such as drawings and manipulatives, multiple approaches to solving problems, and opportunities for students to explain their mathematical thinking.

Accelerated Math®, published by Renaissance Learning, is a software tool that provides practice problems for students in grades K–12 and provides teachers with reports to monitor student progress. Accelerated Math® creates individualized student assignments, scores the assignments, and generates reports on student progress. The software is typically used with the math curriculum being used in the classroom to add practice for students and help teachers differentiate instruction through the program’s progress-monitoring data.

Accelerated Math®, published by Renaissance Learning, is a software tool that provides practice problems for students in grades K–12 and provides teachers with reports to monitor student progress. Accelerated Math® creates individualized student assignments, scores the assignments, and generates reports on student progress. The software is typically used with the math curriculum being used in the classroom to add practice for students and help teachers differentiate instruction through the program’s progress-monitoring data.

I CAN Learn® is a computer-based math curriculum for students in middle school, high school, and college. It provides math instruction through a series of interactive lessons that students work on individually at their own computers. Students move at their own pace and must demonstrate mastery of each concept before progressing to the next one. Classroom teachers may provide individual, small-group, or whole-class instruction based on students’ performance on the software program.

I CAN Learn® is a computer-based math curriculum for students in middle school, high school, and college. It provides math instruction through a series of interactive lessons that students work on individually at their own computers. Students move at their own pace and must demonstrate mastery of each concept before progressing to the next one. Classroom teachers may provide individual, small-group, or whole-class instruction based on students’ performance on the software program.

Saxon Math is a curriculum for students in grades K–12. The amount of new math content students receive each day is limited and students practice concepts every day. New concepts are developed, reviewed, and practiced cumulatively rather than in discrete chapters or units.

Connected Mathematics Project (CMP) is a math curriculum for students in grades 6–8. It uses interactive problems and everyday situations to explore mathematical ideas, with a goal of fostering a problem-centered, inquiry-based learning environment. At each grade level, the curriculum covers numbers, algebra, geometry/measurement, probability, and statistics.

Cognitive Tutor^®, published by Carnegie Learning, is a math curricula that combines textbooks and interactive software.

enVisionMATH is a core curriculum for students in kindergarten through grade 6. The program seeks to help students develop an understanding of math concepts through problem-based instruction, small-group interaction, and visual learning with a focus on reasoning and modeling. Differentiated instruction and ongoing assessment are used to meet the needs of students at all ability levels.

Cognitive Tutor^®, published by Carnegie Learning, is a math curricula that combines textbooks and interactive software.

Cognitive Tutor^®, published by Carnegie Learning, is a math curricula that combines textbooks and interactive software.

Cognitive Tutor^®, published by Carnegie Learning, is a math curricula that combines textbooks and interactive software.

Cognitive Tutor^®, published by Carnegie Learning, is a math curricula that combines textbooks and interactive software.

Saxon Math is a core curriculum for students in grades K–12 that uses an incremental approach to instruction and assessment. This approach limits the amount of new math content delivered to students each day and allows time for daily practice. New concepts are introduced gradually and integrated with previously introduced content so that concepts are developed, reviewed, and practiced over time rather than being taught during discrete periods of time, such as in chapters or units.

Saxon Math is a core curriculum for students in grades K–12 that uses an incremental approach to instruction and assessment. This approach limits the amount of new math content delivered to students each day and allows time for daily practice. New concepts are introduced gradually and integrated with previously introduced content so that concepts are developed, reviewed, and practiced over time rather than being taught during discrete periods of time, such as in chapters or units.

Saxon Math is a core curriculum for students in grades K–12 that uses an incremental approach to instruction and assessment. This approach limits the amount of new math content delivered to students each day and allows time for daily practice. New concepts are introduced gradually and integrated with previously introduced content so that concepts are developed, reviewed, and practiced over time rather than being taught during discrete periods of time, such as in chapters or units.

Saxon Math is a textbook series covering grades K–12 based on incremental development and continual review of mathematical concepts to give students time to learn and practice concepts throughout the year. The program is built on the premise that students learn best when instruction is incremental and explicit, previously learned concepts are continually reviewed, and assessment is frequent and cumulative. At each grade level, math concepts are introduced, reviewed, and practiced over time in order to move students from understanding to fluency.

Saxon Math is a textbook series covering grades K–12 based on incremental development and continual review of mathematical concepts to give students time to learn and practice concepts throughout the year. The program is built on the premise that students learn best when instruction is incremental and explicit, previously learned concepts are continually reviewed, and assessment is frequent and cumulative. At each grade level, math concepts are introduced, reviewed, and practiced over time in order to move students from understanding to fluency.

UCSMP is a core mathematics curriculum that emphasizes problem solving, real-world applications, and the use of technology. The curriculum is based on a student-centered approach with a focus on active learning that incorporates reading and uses a flexible lesson organization.

UCSMP is a core mathematics curriculum that emphasizes problem solving, real-world applications, and the use of technology. The curriculum is based on a student-centered approach with a focus on active learning that incorporates reading and uses a flexible lesson organization. 

UCSMP is a core mathematics curriculum that emphasizes problem solving, real-world applications, and the use of technology. The curriculum is based on a student-centered approach with a focus on active learning that incorporates reading and uses a flexible lesson organization. 

UCSMP is a core mathematics curriculum that emphasizes problem solving, real-world applications, and the use of technology. The curriculum is based on a student-centered approach with a focus on active learning that incorporates reading and uses a flexible lesson organization. 

Singapore Mathematics, produced in the United States under the names “Primary Mathematics” (grades 1-6), and “New Elementary Mathematics” (grades 7-12), is a year-long program designed for schools and homeschoolers alike. Based upon the rigorous mathematics curriculum used in Singapore from 1982 to 2001, it provides core curriculum and supplementary materials in the form of course books, manipulatives, teaching guides, and workbooks. The curriculum takes a “concrete-to-pictorial-to-abstract” approach which teaches mastery to each grade level in order to create a foundation for higher mathematical development. An alternative mathematics curriculum for low-performing students which focuses on the same math topics is available, and is taught at a slower pace with greater repetition by expert teachers.

Scott Foresman-Addison Wesley Elementary Mathematics is a core curriculum for students at all ability levels in kindergarten through grade 6. The program supports students’ understanding of key math concepts and skills, and it covers a range of mathematical content across grades. The curriculum focuses on questioning strategies, problem-solving skills, embedded assessment, and exercises tailored to students of different ability levels. The program provides explicit problem-solving instruction, hands-on activities, and opportunities to extend students’ mathematical understanding through reading and writing connections.

Investigations in Number, Data, and Space is an activity-based, K–5 mathematics curriculum designed to help students understand number and operations, geometry, data, measurement, and early algebra. Each instructional unit focuses on a particular content area and lasts for 2–5.5 weeks. The curriculum encourages students to develop their own strategies for solving problems and engage in discussion about their reasoning and ideas. The lessons are activity-based in order to facilitate increased comprehension of basic math fundamentals. The curriculum is presented through a series of resource books called ”curriculum units” that provide teachers with guidance on implementation. One or more of the units for each year has a software program associated with it. Other materials include manipulatives, flash cards, overheads, and textbooks.

Peer-Assisted Learning Strategies is a peer-tutoring program for grades K–6 that aims to improve student proficiency in several disciplines. During the 30-35 minute peer-tutoring sessions, students take turns acting at the tutor, coaching and correcting one another as they work through problems. The designation of tutoring pairs and skill assignment is based on teacher judgement of student needs and abilities, and teachers reassign tutoring pairs regularly.  

Repeated reading is an academic practice that aims to increase oral reading fluency. Repeated reading can be used with students who have developed initial word reading skills but demonstrate inadequate reading fluency for their grade level. During repeated reading, a student sits in a quiet location with a teacher and reads a passage aloud at least three times. Typically, the teacher selects a passage of about 50 to 200 words in length. If the student misreads a word or hesitates for longer than 5 seconds, the teacher reads the word aloud, and the student repeats the word correctly. If the student requests help with a word, the teacher reads the word aloud or provides the definition. The student rereads the passage until he or she achieves a satisfactory fluency level.

PLATO® Achieve Now is a software-based curriculum for the elementary and middle school grades that focuses on pre-algebraic concepts and includes content pertaining to rational numbers in related organizational patterns, proportion and percent, integers, probability, statistics, problem solving, geometry, measurement, and the foundational concepts of algebra I. Instructional content is delivered via the PlayStation Portable (PSP®) system, allowing students to access learning materials in various settings. Software-based assessments are used to customize individual instruction, allowing students to learn at their own pace with content appropriate for their skill level.

Developed by the Math Learning Center with support from the National Science Foundation, Bridges in Mathematics is a grades K–2 curriculum that emphasizes problem solving. It was designed to meet the National Council of Teachers of Mathematics (NCTM) standards and utilizes whole group, small group, and individual instruction. The concepts covered by this curriculum include numbers and operations, data analysis, geometry, probability, sorting, and measurement. Each grade level in the Bridges program provides one year’s worth of mathematics lessons as well as daily basic skills practices, textbooks, homework assignments, teacher guides, student assessments, manipulatives, and visual models.

Kumon, a supplemental after-school mathematics program for pre-K-12 grades, is geared towards students seeking to catch up or gain extra enrichment. Children are taught to master fundamental math skills with individualized programs that allow them to comfortably work to their full potential at their own pace. Once placement is determined through a series of tests, worksheets provide practice in the form of simple problems that gradually build up to more challenging lessons. Students attend a Kumon center twice a week and are encouraged to remain in the program for at least six months to notice results; worksheets are assigned for daily practice and take 20 minutes to complete. Before students may proceed to the next assignment, they must pass a timed test on their current subject and receive a perfect score.

A middle school mathematics curriculum for grades 5 through 8. Mathematics in Context was developed to align with the 1989 National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards. It is also based on the Dutch Realistic Mathematics Education approach of first engaging students in understanding real problems and then gradually moving to abstract concepts. Rather than focusing on one mathematical domain at a time, Mathematics in Context teaches students to explore the relationships among different domains of mathematics (such as algebra and geometry) and to develop strategies for reasoning through problems mathematically. Mathematics in Context also encourages students to collaborate on problem solving.

Mathematics Their Way presents an activity-centered, child-centered, manipulative math program in which the children learn through the five senses. The program is designed to meet the needs of kindergarten, first, and second grade students. The children follow a sequence of activities, varying between total group, small group, and independent work. The goal of the activities is to help children develop an understanding of and insight into the patterns of mathematics. The activities are designed to help young children see relationships and interconnections in mathematics and to enable them to deal flexibly with mathematical ideas and concepts.

New Century Mathematics provides a pre-K through eighth grade curriculum designed to meet the National Council of Teacher’s of Mathematics (NCTM) standards. In addition to teaching state curriculum objectives and basic math skills in a classroom format, teachers may utilize the New Century Integrated Instructional System. This computer-based assessment program designs independent individualized lessons for every student in order to meet their particular learning needs and remedy weaknesses. It monitors student progress, provides teachers with diagnostic student profiles, updates profiles to indicate gains in student achievement, and can print a variety of reports on particular lessons, individuals, or groups. New Century Mathematics incorporates mental arithmetic, audio instruction and assistance at all levels, and also supports students with limited English skills. Rulers, protractors, and technology (calculators, online tools) may be used to support lesson comprehension.

The Partnership for Access to Higher Mathematics (PATH Mathematics) is a partnership among the Southwest Texas State University, the local telephone company, and a local school district. This mathematics and social curriculum is designed for low-track students. The pilot project targeted 9th-graders, but it has been used with 8th-graders. The goals of PATH Mathematics are to increase the achievement of low-track students and to enhance their aspirations to take further mathematics classes and eventually to attend college. According to the developers, the math curriculum adheres to standards proposed by the National Council of Teachers of Mathematics, emphasizes real-world applications, and integrates the use of manipulatives and technology to solve problems. The curriculum has six sequential topic areas: (1) patterns, integers, and solving equations, (2) measurement, probability, and rational numbers, (3) solving equations and graphing with proportional reasoning, (4) problem solving with geometry and irrational numbers; (5) linear and nonlinear relations; and (6) polynomials.

The SuccessMaker program is a set of computer-based courses used to supplement regular classroom reading instruction in grades K–8. Using adaptive lessons tailored to a student’s reading level, SuccessMaker aims to improve understanding in areas such as phonological awareness, phonics, fluency, vocabulary, comprehension, and concepts of print.

The Integrated Mathematics, Science, and Technology (IMaST) program is a 6th, 7th, and 8th grade curriculum that promotes both hands-on learning for students and teamwork among teachers from different disciplines. IMaST emphasizes learning based on constructivist theory and active student participation involving a hands-on approach. Coordination of a wide variety of activities helps students grasp the many natural interdisciplinary connections in the curriculum. A team of mathematics, science, and technology specialists, in collaboration with other field experts, did the research to create the curriculum, built on major themes that are presented in modules. Each theme develops the focus of all disciplines in relation to several key concepts that lead toward the same objective. IMaST has developed its activities using benchmarks, national standards, and state frameworks.

The Heath Mathematics Connections Series curriculum may be used with elementary and middle school grades as the primary classroom text. Teacher-led classroom instruction is supplemented with suggested classroom activities - such as practice problems, mathematics drills, and cooperative learning - to increase skill knowledge and determine underlying concepts. Classroom assessment is provided through warm-up problems, a problem of the day for each lesson, enrichment worksheets, chapter reviews and tests, standardized format tests, math logs and portfolios. The curriculum also provides educators with a classroom textbook, teaching editions, practice worksheets and copymasters, and family involvement activities. Some classroom and homework materials are also available in Spanish.

Core-Plus Mathematics is a 4-year curriculum that replaces the traditional sequence with courses that each feature interwoven strands of algebra and functions, statistics and probability, geometry and trigonometry, and discrete mathematics. The curriculum emphasizes mathematical modeling, using technology to emphasize reasoning with multiple representations (verbal, numerical, graphical, and symbolic) and to focus on goals in which mathematical thinking and problem solving are central. Instructional materials promote active learning and teaching centered around collaborative small-group investigations of problem situations, followed by teacher-led whole-class summarizing activities that lead to analysis, abstraction, and further application of underlying mathematical ideas.

Progress in Mathematics © 2006 is a core curriculum for students in kindergarten through grade 6. Progress in Mathematics © 2006 uses a sequence of systematic lesson plans to teach mathematical concepts and skills. It incorporates the following features at each grade level: explicit instruction of mathematics content; development of conceptual understanding through a three-step process that begins with hands-on activities (concrete thinking to visual thinking to symbol use); fluency in numerical computation; problem solving; development of mathematical vocabulary; practice and review; and different types of assessment. Student textbooks, student workbooks, and teacher’s editions are available for each grade level, as well as manipulatives and online practice exercises.

Houghton Mifflin Mathematics is a core mathematics curriculum for students at all ability levels in kindergarten through grade 6. At each grade level, the program focuses on basic skills development, problem solving, and vocabulary expansion to help students master key math concepts. Students practice daily math lessons through instructional software, enrichment worksheets, manipulatives, and workbooks, in addition to student textbooks. The program incorporates assessments—including lesson-level interventions to meet the needs of all learners—to monitor students’ progress.

Transition Mathematics aims to increase applied arithmetic, pre-algebra, and pre-geometry skills in students in grades 7–12 . This 1-year curriculum also addresses general application to different wordings of problems, types of numbers, and contexts for problems and aims to promote mathematical reading skills. The curriculum uses the University of Chicago School Mathematics Project (UCSMP) textbook. The sequence of the topics intends to assist the transition from arithmetic to algebra and geometry.

The current study investigated the effectiveness of three distinct educational technologies--two game-based applications (From Here to There and DragonBox 12+) and two modes of online problem sets in ASSISTments (an Immediate Feedback condition and an Active Control condition with no immediate feedback) on Grade 7 students' algebraic knowledge. More than 3,600 Grade 7 students across nine in-person and one virtual schools within the same district were randomly assigned to one of the four conditions. Students received nine 30-minute intervention sessions from September 2020 to March 2021. Hierarchical linear modeling analyses of the final analytic sample (N = 1,850) showed significantly higher posttest scores for students who used From Here to There and DragonBox 12+ compared to the Active Control condition. No significant difference was found for the Immediate Feedback condition. The findings have implications for understanding how game-based applications can affect algebraic understanding, even within pandemic pressures on learning.

The current study investigated the effectiveness of three distinct educational technologies--two game-based applications (From Here to There and DragonBox 12+) and two modes of online problem sets in ASSISTments (an Immediate Feedback condition and an Active Control condition with no immediate feedback) on Grade 7 students' algebraic knowledge. More than 3,600 Grade 7 students across nine in-person and one virtual schools within the same district were randomly assigned to one of the four conditions. Students received nine 30-minute intervention sessions from September 2020 to March 2021. Hierarchical linear modeling analyses of the final analytic sample (N = 1,850) showed significantly higher posttest scores for students who used From Here to There and DragonBox 12+ compared to the Active Control condition. No significant difference was found for the Immediate Feedback condition. The findings have implications for understanding how game-based applications can affect algebraic understanding, even within pandemic pressures on learning.

We report the results of a preregistered, cluster randomized controlled trial of a mathematics learning intervention known as interleaved practice. Whereas most mathematics assignments consist of a block of problems devoted to the same skill or concept, an interleaved assignment is arranged so that no 2 consecutive problems require the same strategy. Previous small-scale studies found that practice assignments with a greater proportion of interleaved practice produced higher test scores. In the present study, we assessed the efficacy and feasibility of interleaved practice in a naturalistic setting with a large, diverse sample. Each of 54 7th-grade mathematics classes periodically completed interleaved or blocked assignments over a period of 4 months, and then both groups completed an interleaved review assignment. One month later, students took an unannounced test, and the interleaved group outscored the blocked group, 61% versus 38%, d = 0.83. Teachers were able to implement the intervention without training, and they later expressed support for interleaved practice in an anonymous survey they completed before they knew the results of the study. Although important caveats remain, the results suggest that interleaved mathematics practice is effective and feasible. [For the corresponding grantee submission, see ED595322.]

Policymakers face dilemmas when choosing a policy, program, or practice to implement. Researchers in education, public health, and other fields have proposed a sequential approach to identifying interventions worthy of broader adoption, involving pilot, efficacy, effectiveness, and scale-up studies. In this paper, we examine a scale-up of an early math intervention to the state level, using a cluster randomized controlled trial. The intervention, "Pre-K Mathematics," has produced robust positive effects on children's math ability in prior pilot, efficacy, and effectiveness studies. In the current study, we ask if it remains effective at a larger scale in a heterogeneous collection of pre-K programs that plausibly represent all low-income families with a child of pre-K age who live in California. We find that "Pre-K Mathematics" remains effective at the state level, with positive and statistically significant effects (effect size = 0.30, p < 0.01). In addition, we develop a framework of the dimensions of scale-up to explain why effect sizes might decrease as scale increases. Using this framework, we compare the causal estimates from the present study to those from earlier, smaller studies. Consistent with our framework, we find that effect sizes have decreased over time. We conclude with a discussion of the implications of our study for how we think about the external validity of causal relationships. [This is the online version of an article published in "Evaluation Review."]

This study used a randomized controlled trial design to investigate the ROOTS curriculum, a 50-lesson kindergarten mathematics intervention. Ten ROOTS-eligible students per classroom (n = 60) were randomly assigned to one of three conditions: a ROOTS five-student group, a ROOTS two-student group, and a no-treatment control group. Two primary research questions were investigated as part of this study: What was the overall impact of the treatment (the ROOTS intervention) as compared with the control (business as usual)? Was there a differential impact on student outcomes between the two treatment conditions (two- vs. five-student group)? Initial analyses for the first research question indicated a significant impact on three outcomes and positive but nonsignificant impacts on three additional measures. Results for the second research question, comparing the two- and five-student groups, indicated negligible and nonsignificant differences. Implications for practice are discussed. [For the corresponding grantee submission, see ED578431.]

Two intervention approaches designed to address the multifaceted academic and cognitive difficulties of low-income children who enter pre-K with very low math knowledge were tested in a randomized experiment. Blocking on classroom, children who met screening criteria were assigned to a Math + Attention condition in which the Pre-Kindergarten Mathematics Tutorial (PKMT) intervention was implemented (4 days/week for 24 weeks) in addition to 16 adaptive attention training sessions, a Math-Only condition using the PKMT intervention, or a business-as-usual condition. Five hundred eighteen children were assessed at pretest and posttest. There was a significant effect of the PKMT intervention on a broad measure of informal mathematical knowledge and a small but significant effect on a measure of numerical knowledge. Attention training was associated with small effects on attention, but did not provide additional benefit for mathematics. A main effect of state on math outcomes was associated with a stronger, numeracy-focused Tier 1 mathematics curriculum in one state. Findings are discussed with respect to increasing intensity of math-specific and domain-general interventions for young children at risk for mathematical learning difficulties.

Many college students never take, or do not pass, required remedial mathematics courses theorized to increase college-level performance. Some colleges and states are therefore instituting policies allowing students to take college-level courses without first taking remedial courses. However, no experiments have compared the effectiveness of these approaches, and other data are mixed. We randomly assigned 907 students to (a) remedial elementary algebra, (b) that course with workshops, or (c) college-level statistics with workshops (corequisite remediation). Students assigned to statistics passed at a rate 16 percentage points higher than those assigned to algebra (p < 0.001), and subsequently accumulated more credits. A majority of enrolled statistics students passed. Policies allowing students to take college-level instead of remedial quantitative courses can increase student success.

In a randomized field trial with 2,850 seventh-grade mathematics students, we evaluated whether an educational technology intervention increased mathematics learning. Assigning homework is common yet sometimes controversial. Building on prior research on formative assessment and adaptive teaching, we predicted that combining an online homework tool with teacher training could increase learning. The online tool ASSISTments (a) provides timely feedback and hints to students as they do homework and (b) gives teachers timely, organized information about students' work. To test this prediction, we analyzed data from 43 schools that participated in a random assignment experiment in Maine, a state that provides every seventh-grade student with a laptop to take home. Results showed that the intervention significantly increased student scores on an end-of-the-year standardized mathematics assessment as compared with a control group that continued with existing homework practices. Students with low prior mathematics achievement benefited most. The intervention has potential for wider adoption. [For the corresponding grantee submission, see ED575159.]

This report presents results from the "Ready To Learn" Prekindergarten Transmedia Mathematics Study, a principal part of the summative evaluation of "Ready To Learn," which is a partnership between the US Department of Education, the Corporation for Public Broadcasting, and PBS. Researchers found that preschool children who experienced a PBS KIDS Transmedia Math Supplement developed essential early mathematics skills. The PBS KIDS Transmedia Math Supplement was centered around public media videos and digital games, played on a selected set of learning technologies (interactive whiteboards and laptop computers). The important skills measure--counting; subitizing; recognizing numerals; recognizing, composing, and representing shapes; and patterning--increased significantly for the study's four- and five-year-old children, who were from traditionally economically disadvantaged communities where children are often less prepared for kindergarten than are their more socially and economically advantaged peers. Also important, preschool teachers who enacted the PBS KIDS Transmedia Math Supplement reported significant changes in their confidence and comfort with early mathematics concepts and teaching with technology. [This report was co-produced by SRI's Center for Technology in Learning (CTL).]

This study employed a cluster randomized trial design to evaluate the effectiveness of a research-based intervention for improving the mathematics education of very young children. This intervention includes the "Building Blocks" mathematics curriculum, which is structured in research-based learning trajectories, and congruous professional development emphasizing teaching for understanding via learning trajectories and technology. A total of 42 schools serving low-resource communities were randomly selected and randomly assigned to 3 treatment groups using a randomized block design involving 1,375 preschoolers in 106 classrooms. Teachers implemented the intervention with adequate fidelity. Pre- to posttest scores revealed that the children in the Building Blocks group learned more mathematics than the children in the control group (effect size, g = 0.72). Specific components of a measure of the quantity and quality of classroom mathematics environments and teaching partially mediated the treatment effect. (Contains 5 tables and 1 footnote.)

This report presents findings from a randomized control trial designed to inform the decisions of policymakers who are considering using online courses to provide access to Algebra I in grade 8. It focuses on students judged by their schools to be ready to take Algebra I in grade 8 but who attend schools that do not offer the course. The study tested the impact of offering an online Algebra I course on students' algebra achievement at the end of grade 8 and their subsequent likelihood of participating in an advanced mathematics course sequence in high school. The study was designed to respond to both broad public interest in the deployment of online courses for K-12 students and to calls from policymakers to provide students with adequate pathways to advanced coursetaking sequences in mathematics (National Mathematics Advisory Panel 2008). This study is the first of its kind to rigorously evaluate the impact of offering an online version of Algebra I in schools that otherwise do not typically offer the course, even though they have students who are ready to take it. For educators and students facing similar challenges, the results of this study may be particularly informative and promising. Results showed that offering an online course to AR students is an effective way to broaden access to Algebra I in grade 8 and later, to more challenging mathematics course opportunities. The study demonstrates that an online course as implemented is more effective in promoting students' success in mathematics than existing practices in these schools. Appended are: (1) Study Design, Study Samples, and Statistical Precision; (2) Measures; (3) Intervention Features; (4) Estimation Methods and Hypothesis Testing; (5) Sensitivity Analyses; and (6) Missing Data and Multiple Imputation. (Contains 77 tables, 12 figures and 61 footnotes.)

The authors present three studies (two randomized controlled experiments and one embedded quasi-experiment) designed to evaluate the impact of replacement units targeting student learning of advanced middle school mathematics. The studies evaluated the SimCalc approach, which integrates an interactive representational technology, paper curriculum, and teacher professional development. Each study addressed both replicability of findings and robustness across Texas settings, with varied teacher characteristics (backgrounds, knowledge, attitudes) and student characteristics (demographics, levels of prior mathematics knowledge). Analyses revealed statistically significant main effects, with student-level effect sizes of 0.63, 0.50, and 0.56. These consistent gains support the conclusion that SimCalc is effective in enabling a wide variety of teachers in a diversity of settings to extend student learning to more advanced mathematics. (Contains 4 tables, 5 figures, and 1 note.)

This project entailed a three-year efficacy evaluation of the Computer and Team Assisted Mathematical Acceleration (CATAMA) Lab developed by the Center for Social Organization of Schools at Johns Hopkins University. The CATAMA Lab was proposed as an immediate and practical approach to addressing the different types of math deficits held by students at urban high-poverty schools. The Lab required only 1 teacher per school reducing staff and professional development requirements. It used multiple instructional techniques (including individualized computer instruction, direct instruction, pair and team learning, and individual instruction) to teach math concepts and skills. By taking the place of an elective it allowed students to continue with their on-grade math class. For a more detailed description of the Lab see Appendix 2. The original goal of the project was to establish the Lab at three urban schools serving high-poverty high-minority middle grade students (grades 5-8). Students underperforming in mathematics (as established by district standardized tests) were to take a trimester course of study in the Lab to increase their knowledge of math concepts and skills taught by a regular math teacher receiving extensive ongoing professional development. Students were to take the Lab as an elective course while continuing with their regular math class. From each school's pool of students eligible to participate, students were to be randomly assigned to take the Lab. An implementation analysis was to measure the teaching of the concepts and skills to be taught in the Lab. To evaluate the impact of the intervention, students' math achievement, as measured by standardized math tests, was to be compared to eligible students not assigned to the Lab. This report discusses the project in three sections: (1) A comparison of the actual project with the planned project; (2) The descriptive results from the project; (a) Description of the sample; (b) Description of implementation of the CATAMA Lab; and (3) The evaluative results from the project.

Because standardized tests in science are not given to PreK-3 students in Ohio, this report examined the longitudinal effects of learning from a teacher who had participated in the NURTURES professional development program. Specifically, it looked at the effects on students' mathematics and reading learning in grades 2-5 and science learning in 5th grade in 2017. Students who were in 5th grade at that time could have had a NURTURES-trained teacher at any time between kindergarten and 3rd grade. Thus, the study followed students up to 5 years after having a NURTURES teacher. The sample included the population of students enrolled in the 41 elementary schools in the Toledo Public School District. Students who never learned from a teacher who participated in NURTURES served as the control group. The data came from the 2017 administration of the Ohio Measure of Academic Progress (MAP) (NWEA, 2019) for mathematics and reading and the Ohio Achievement Test in Science for science (Ohio Achievement Assessment, 2015). The total number of students from these schools who took the May 2017 reading MAP was 6759 and the total number who took mathematics was 6703. The number of those students who had at least one NURTURES teacher was 2801 (41.4%) for reading and 2707 (41.6%) for mathematics. Analysis of the reading scores showed 2.14 advantage points for NURTURES students as compared to the average non-intervention student to an annual growth rate of 7.02 units (p < .001). The treatment effect size (Hedges' g) was 0.12. For mathematics there were 1.55 advantage points to an annual growth rate estimated to be 8.17 units (p < .001) as compared to the average non-intervention student. The treatment effect size (Hedges' g) was 0.09. Analysis of the 5th grade Ohio Achievement Science Subtest showed that students associated with at least one NURTURES-trained teacher was modeled to have a 5.86 advantage points as compared to the average non-intervention student. The treatment effect size (Hedges' g) was 0.08, which is to be interpreted as a treatment group having, on average, 0.08 higher scores in standard deviation units as compared to the scores of the control cohort. When compared with our earlier evaluation report (2016; revised in 2018), we see that students who had a NURTURES-trained teacher, on average, continued to show greater gains compared to students who did not. In addition, the achievement gaps between non-minority and minority students in reading and mathematics were reduced when the minority students had a NURTURES-trained teacher and the non-minority students did not. In science, the impact of the intervention roughly compensated for the attainment gap between boys and girls and partially ameliorated the gap between minority and non-minority children's scores associated with these demographic factors. [Published May 6, 2020 with minor revisions based in WWC inquiries published January 5, 2024.]

On average, Florida students earn only half of the points possible in the statistics content area of the state's annual mathematics assessment. Leaders in Broward County Public Schools, a large, diverse, urban school district, viewed changes to statistics curriculum and instruction as one way to address this issue. This study randomly assigned 40 middle schools in the district to either implement a replacement curriculum unit with four days of teacher professional development in probability and statistics or continue with their practice-as-usual instruction in probability and statistics. The replacement unit supported teaching and learning of all the probability and statistics standards in the grade 7 course description. The replacement unit with the associated professional development, called the Supporting Teacher Enactment of the Probability and Statistics Standards program, improved student understanding of statistics and statistics instruction. The magnitude of the effect on student understanding was 23 percent of 1 standard deviation, which is comparable to an increase of 9 percentile points for an average student. [For the appendixes, see ED610168. For the study snapshot, see ED610167.]

This paper discusses the implementation and evaluation of an innovative professional development program that supported 8th grade mathematics teachers in designing learning activities that assisted students in developing a conceptual understanding of mathematics. Funded by an Investing in Innovation (i3) grant from the U.S. Department of Education, the Collaboration Resulting in Educators Applying Technology Effectively (CREATE) Project assembled teams of teachers to work with their coaches to create dynamic and interactive learning aimed at helping students make sense of mathematics rather than simply practice procedures. As a result, student achievement was positively affected while participating teachers developed their capacity to address the learning needs of their students. Coaching was also examined in a parallel study to explore how the role of trust affected the work of the participating teachers.

This brief aims to help educators understand the implications of math curriculum choice in the early elementary grades by presenting new findings from a study that examined how four math curricula affect students' achievement across two years--from 1st through 2nd grades. The four curricula were (1) Investigations in Number, Data, and Space (Investigations); (2) Math Expressions; (3) Saxon Math (Saxon); and (4) Scott Foresman-Addison Wesley Mathematics (SFAW), which the developer revised and renamed enVision Math (enVision) during the study. These curricula are widely used and differ in their approaches to teaching and learning. Within districts, we randomly assigned one of the four curricula to each school that participated in the study. After one year (by the end of 1st grade), students taught with Math Expressions and Saxon made greater gains in achievement than students taught with Investigations and SFAW. After two years (by the end of 2nd grade), Investigations students continued to lag behind Math Expressions and Saxon students, while SFAW/enVision students caught up to Math Expressions and Saxon students. Therefore, Math Expressions, Saxon, and SFAW/enVision improved 1st-through-2nd-grade math achievement by similar amounts, and all three outperformed Investigations. Our findings also suggest that switching between some of the study's curricula does not harm student achievement and can even be beneficial. (Contains 24 endnotes, 3 figures, and 2 tables.) [For "After Two Years, Three Elementary Math Curricula Outperform a Fourth. NCEE Technical Appendix. NCEE 2013-4019", see ED544187.]

A socioeconomic status (SES) gap in mathematical knowledge emerges early and widens prior to school entry. To address this gap, a curricular intervention, "Pre-K Mathematics," was developed and found to be effective in prior efficacy research. In the present project, the next step was taken in evaluating this intervention. Specifically, an effectiveness study was conducted to determine the degree to which the intervention improves pre-kindergarten (4-year-old) children's mathematical knowledge when implemented by local program staff in multiple settings that serve a heterogeneous population of low-SES families. In contrast with the prior efficacy study, the effectiveness study required that all teachers, rather than volunteer teachers, in their public preschool programs be available for random selection and random assignment. It also used curriculum coaches who were either members of the participating school districts or Head Start programs' permanent training staff or independent contractors, depending on the way a program a routinely supported teacher learning for its in-service teaching staff. Participating programs included publicly funded Head Start and state preschool programs serving low-income, ethnically/racially diverse, urban families in California and low-income, predominantly White, rural families in Kentucky and Indiana. A trainer-of-trainers model was used (1) to train curriculum coaches to support teachers' implementation and (2) to train teachers to implement Pre-K Mathematics with adequate fidelity. A two-condition (treatment and control) RCT was conducted, with clusters of pre-kindergarten classrooms as the unit of randomization. Treatment teachers implemented Pre-K Mathematics and control teachers continued their usual classroom practices. Children were assessed at pretest, posttest, and kindergarten follow up using the Child Math Assessment (CMA) and the Test of Early Mathematics Ability, 3rd Edition (TEMA-3). Classroom observations were made to measure the nature and amount of math support provided by treatment and control teachers during the school year. Coaches supported implementation and teachers implemented with adequate to high levels of fidelity. Multi-level analyses revealed that treatment children made significantly greater gains in mathematical knowledge than control children during the pre-kindergarten year as measured by the CMA (ES = 0.83) and TEMA-3 (ES = 0.45). A multilevel mediation analysis found evidence that time spent in mathematically focused small-group activities had a significant indirect effect on children's math outcomes. Thus, this effectiveness study found that the Pre-K Mathematics intervention had a significant positive effect on low-SES children's mathematical knowledge. An implication of this finding is that early mathematics intervention is a promising educational strategy for reducing the SES gap in mathematical knowledge.

We used a cluster randomized trial to evaluate the effectiveness of a research-based model for scaling up educational interventions, focusing on the persistence of effects with and without a follow-through intervention. The instantiation of the Technology-enhanced, Research-based, Instruction, Assessment, and professional Development (TRIAD) model emphasized teaching early mathematics for understanding via learning trajectories and technology. The TRIAD implementation began in 42 schools in two city districts serving low-resource communities, randomly assigned to three conditions. In pre-kindergarten, the 2 experimental interventions were identical, but 1 included follow-through in the kindergarten year, including knowledge of the pre-K intervention and ways to build upon that knowledge using learning trajectories. Intent-to-treat analyses showed that students in both the follow-through condition (g = 0.33) and non-follow-through condition (g = 0.22) scored statistically significantly higher than children in the control condition. Both groups outperformed the control condition in treatment-on-the-treated analyses (g = 0.38, follow-through; g = 0.30 non-follow-through). Moderators and mediators were also analyzed. We conclude that the instantiation of the TRIAD model was successful and that follow through may contribute to the persistence of the effects of preschool interventions. (Contains 5 tables and 3 figures.)

National achievement data show that elementary school students in the United States, particularly those from low socioeconomic backgrounds, have weak math skills (National Center for Education Statistics 2009). In fact, data show that, even before they enter elementary school, children from disadvantaged backgrounds are behind their more advantaged peers in basic competencies such as number-line ordering and magnitude comparison (Rathburn and West 2004). Furthermore, after a year of kindergarten, disadvantaged students still have less extensive knowledge of mathematics than their more affluent peers (Denton and West 2002). This study examines whether some early elementary school math curricula are more effective than others at improving student math achievement in disadvantaged schools. A small number of curricula, which are based on different approaches for developing student math skills, dominate elementary math instruction--7 curricula make up 91 percent of those used by K-2 educators, according to a 2008 survey (Resnick et al. 2010). Little rigorous evidence exists to support one approach over another, however, which means that research does not provide educators with much useful information when choosing a math curriculum to use. The key findings in this report include the following: (1) Teachers used their assigned curriculum, and the instructional approaches of the four curriculum groups differed as expected; (2) Math instruction varied in other notable ways across the curriculum groups; (3) In terms of student math achievement, the curriculum used by the study schools mattered; and (4) The curriculum used in different contexts also mattered, and some of these findings are consistent with findings based on all students whereas others are not. Appendices include: (1) Data Collection and Response Rates; (2) Teacher-Reported Frequency of Implementing Other Curriculum-Specific Activities; (3) Glossary of Curriculum-Specific Terms; and (4) Constructing the Analyses Samples and Estimating Curriculum Effects. (Contains 82 tables, 7 figures and 97 footnotes.) [For the executive summary, see ED512553.]

We explored how a progress monitoring and instructional management system can be used to help educators differentiate instruction and meet the wide-ranging learning needs of their increasingly diverse classrooms. We compared classrooms in 24 states that used a curriculum-based progress monitoring and instructional management system, Accelerated Math, to same school control classrooms that did not use it. Among the major findings were the following: (1) At every grade level there were large differences in grade equivalent score and percentile gains for students in the experimental and control classrooms; (2) Gains were experienced across the achievement spectrum. An analysis of low-, middle-, and high-achieving students showed consistent rates of gain for each math objective mastered; (3) Intervention integrity had a significant effect on student achievement; (4) Teachers using the progress monitoring and instructional management system spent more time providing individual versus group instruction and felt better able to meet the individual needs of their students; and (5) Significantly more students who were in classrooms where teachers used the progress monitoring and instructional management system reported that they like math, help each other with math, and like math better this year than last year. Addition of a progress monitoring and instructional management system to ongoing mathematics instruction improves mathematics outcomes for students. The effects of the program clearly are a function of intervention integrity; when progress monitoring and instructional management practices are implemented with high fidelity or integrity, the mathematics performance of all students is significantly enhanced. Implications for practice are discussed.

The purpose of this study was to explore the paths by which word-problem intervention, with versus without embedded prealgebraic reasoning instruction, improved word-problem performance. Students with mathematics difficulty (MD; n = 304) were randomly assigned to a business-as-usual condition or 1 of 2 variants of word-problem intervention. The prealgebraic reasoning component targeted relational understanding of the equal sign as well as standard and nonstandard equation solving. Intervention occurred for 16 weeks, 3 times per week, 30 min per session. Sequential mediation models revealed main effects, in which each intervention condition significantly and substantially outperformed the business-as-usual condition, corroborating prior research on the efficacy of schema word-problem intervention. Yet despite comparable effects on word-problem outcomes between the two word-problem conditions, the process by which effects accrued differed: An indirect path via equal-sign understanding and then equation solving was significant only for the word-problem intervention condition with embedded prealgebraic reasoning instruction. Additionally, the effect of this condition on equal-sign reasoning was strong. Given the link between equal-sign reasoning for success with algebra and the importance of algebra for success with advanced mathematics, results suggest an advantage for embedding prealgebraic reasoning instruction within word-problem intervention.

In 2015, Bergen Community College (BCC) received a grant from the U.S. Department of Education First in the World Grant Program. The grant entitled Alternatives to Mathematics Education: An Unprecedented Program (AMP-UP), was awarded to conduct a randomized control trial on a corequisite approach to developmental math education. This study was conducted by researchers at the Education and Employment Research Center (EERC) at Rutgers, The State University of New Jersey. EERC investigated whether an accelerated delivery of developmental and college-level mathematics coursework would improve student retention, gateway course completion, credit accumulation, and degree completion over three years. The intervention group enrolled in accelerated developmental and college-level coursework; those in the group who placed into developmental arithmetic also participated in a self-paced Summer Bridge program. The comparison group followed the college's usual developmental mathematics sequence, generally enrolling in their first math course in the Fall term of their first year. The study found that both groups enrolled in a similar number of terms over three years. But in that period, intervention group students were 13 percentage points more likely to complete a developmental mathematics course and 30 percentage points more likely to complete a college-level mathematics course. The intervention group also earned 5.1 more credits and was 8 percentage points more likely to complete a degree in the study period. [This report was published by Rutgers' Education and Employment Research Center at the School of Management and Labor Relations.]

In 2016, Union County College began a four-year experiment with corequisite developmental mathematics as part of a grant from the U.S. Department of Education's First in the World (FITW) program. In this experiment, students assessed as needing to take developmental mathematics courses would be eligible to receive a waiver from their developmental requirements and instead proceed to college-level mathematics courses. Students selected to receive a waiver would also be required to participate weekly in tutoring services offered by the college. The Education and Employment Research Center at Rutgers University served as the external evaluator for the study. The evaluation focuses on three key outcomes: continuous enrollment, passing college-level mathematics, and degree completion. The outcomes assessment found that students assigned to the intervention group -- those who had the immediate opportunity to proceed to college-level mathematics with support -- benefitted primarily from the intervention itself. In other words, intervention group students were substantially more likely to have passed a college-level mathematics course within three years than their counterparts in the comparison group, who would have had to first complete a developmental mathematics sequence prior to enrolling in college-level math. Assignment to the treatment group did not, however, have a measurable impact on either student persistence at the college or on degree completion in the study period. [This report was produced by Rutgers' Education and Employment Research Center.]

The purposes of this study were to assess the effects of fractions intervention for students who are at-risk for poor outcomes and to examine whether a component that combines self-regulated learning with growth mindset instruction (SR-GM) provides added value for improving outcomes. At-risk students (N = 84) were randomly assigned to 3 conditions: fractions intervention, fractions intervention with embedded SR-GM, and a control group. Intervention was conducted 3 times per week for 35 min per session for 13 weeks. Multilevel models indicated both fractions intervention conditions produced strong effects, with no added value for SR-GM. Posttest fractions achievement gaps for both intervention conditions held steady, narrowed, or closed, while the control group's gaps remained sizeable or grew. Results suggest that intervention can address challenging mathematics standards for at-risk learners and that SR-GM instruction may not be necessary in the context of strong intervention. [This is the in press version of an article published in "Exceptional Children."]

The purpose of this study was to explore the efficacy of fractions intervention with and without an embedded self-regulation (SR) component for third-grade students at risk for mathematics disabilities. Fractions intervention focused on magnitude understanding and word problems. Embedded SR was designed to support a growth mindset (fostering belief that intellectual and academic abilities can be developed) along with SR processes in which students set goals, self-monitor, and use strategies to engage motivationally, metacognitively, and behaviorally through challenging tasks. Students (n = 69) were randomly assigned to business-as-usual control and the two versions of fractions intervention. Multilevel models, accounting for the nested structure of the data, identified a moderation effect on fraction word problems: For students receiving fractions intervention with embedded SR, response to intervention was robust across the continuum of students' pretest word problem skill; by contrast, without SR, response to fractions intervention depended on students' pretest word problem skill. On the remaining outcomes, results reflected stronger outcomes when fractions intervention embedded SR instruction without moderation. [This paper will be published in the "Journal of Learning Disabilities."]

Students in the elementary grades often experience difficulty setting up and solving word problems. Using an equation to represent the structure of the problem serves as an effective tool for solving word problems, but students may require specific pre-algebraic reasoning instruction about the equal sign as a relational symbol to set up and solve such equations successfully. We identified students with mathematics difficulty (n = 138) from a sample of 916 third-grade students. We randomly assigned students to a word-problem intervention with a pre-algebraic reasoning component, a word-problem intervention without pre-algebraic reasoning, or the business-as-usual. Students in the 2 active intervention conditions participated in 45 individual sessions and learned about 3 additive word-problem schemas. Students who received word-problem intervention with a pre-algebraic reasoning component demonstrated improved nonstandard equation solving, equal sign understanding, and word-problem solving compared to students in the other two conditions. [The paper will be published in "ZDM Mathematics Education."]

The effectiveness of an experimental middle school fraction intervention was evaluated. The intervention was centered on the number line and incorporated key principles from the science of learning. Sixth graders (N = 51) who struggled with fraction concepts were randomly assigned at the student level to the experimental intervention (n = 28) or to a business-as-usual control who received their school's intervention (n = 23). The experimental intervention occurred over 6 weeks (27 lessons). Fraction number line estimation, magnitude comparisons, concepts, and arithmetic were assessed at pretest, posttest, and delayed posttest. The experimental group demonstrated significantly more learning than the control group from pretest to posttest, with meaningful effect sizes on measures of fraction concepts (g = 1.09), number line estimation as measured by percent absolute error (g = -0.85), and magnitude comparisons (g = 0.82). These improvements held at delayed posttest 7 weeks later. Exploratory analyses showed a significant interaction between classroom attentive behavior and intervention group on fraction concepts at posttest, suggesting a buffering effect of the experimental intervention on the normally negative impact of low attentive behavior on learning. A number line-centered approach to teaching fractions that also incorporates research-based learning strategies helps struggling learners to make durable gains in their conceptual understanding of fractions. [This paper will be published in the "Journal of Educational Psychology."]

The efficacy of a research-based fraction sense intervention for sixth graders with or at risk for mathematics difficulties (N = 52) was examined. The intervention aimed to build understanding of fraction magnitudes on the number line. Key concepts were taught with a narrow range of denominators to develop deep understanding. The intervention was centered on a visual number line in the meaningful context of a color run race. Students were randomly assigned to the fraction sense intervention (n = 25) or a business-as-usual control group (n = 27). Students in the intervention condition received 21 lessons in small groups (45 min each) during their regular mathematics intervention period. Students in the intervention group performed significantly better than those in the control group on a measure of fraction number line estimation and a more general measure of fraction concepts, both at immediate posttest and delayed posttest, with large effect sizes; lesser effects were shown for fraction arithmetic. [This paper will be published in "Remedial and Special Education."]

The purpose of this study was to investigate the effects of an early numeracy intervention delivered by kindergarten teachers to students identified as having mathematics difficulties. A multigroup growth-modeling-with-random-assignment-to-intervention-condition design was employed. Thirty-two teachers were randomly assigned to the treatment or comparison condition. A total of 71 students participated in the study, 47 in the treatment group and 24 in the comparison group. Results indicated that the treatment condition students outperformed comparison students (g* = 0.99) and demonstrated statistically significantly higher scores on all proximal measures of early numeracy. Also, about 80% to 100% of the variance was accounted for at the student level. Performance on distal measures was less impressive, with no significant differences between groups; the effect size was 0.44. Teachers rated components of the intervention highly, reflecting strong teacher satisfaction.

This study evaluated the effects of a mathematics software program, the Building Blocks software suite, on young children's mathematics performance. Participants included 247 Kindergartners from 37 classrooms in 9 schools located in low-income communities. Children within classrooms were randomly assigned to receive 21 weeks of computer-assisted instruction (CAI) in mathematics with Building Blocks or in literacy with Earobics Step 1. Children in the Building Blocks condition evidenced higher posttest scores on tests of numeracy and Applied Problems after controlling for beginning-of-year numeracy scores and classroom nesting. These findings, together with a review of earlier CAI, provide guidance for future work on CAI aiming to improve mathematics performance of children from low-income backgrounds. [This paper was published in "Journal for Research in Mathematics Education" (EJ1100307).]

The purpose of this closely aligned conceptual replication study was to investigate the efficacy of a Tier 2 kindergarten mathematics intervention. The replication study differed from the initial randomized controlled trial on three important elements: geographical region, timing of the intervention, and instructional context of the counterfactual. Similar to the original investigation, however, the current study tested the same intervention, used the same outcome measures and statistical analyses, and involved the same population of learners. A total of 319 kindergarten students with mathematics difficulties from 36 kindergarten classrooms participated in the study. Students who were randomly assigned to the treatment condition received the intervention in small-group formats, with 2 or 5 students per group. Control students participated in a no-treatment control condition. Significant effects on proximal and distal measures of mathematics achievement were found. Effect sizes obtained for all measures fell within or exceeded the upper bound of the effects reported in the initial study. Implications for systematically situating replication studies in larger frameworks of intervention research and reporting rates of treatment response across replication studies are discussed.

The purpose of this closely aligned conceptual replication study was to investigate the efficacy of a Tier 2 kindergarten mathematics intervention. The replication study differed from the initial randomized controlled trial on three important elements: geographical region, timing of the intervention, and instructional context of the counterfactual. Similar to the original investigation, however, the current study tested the same intervention, used the same outcome measures and statistical analyses, and involved the same population of learners. A total of 319 kindergarten students with mathematics difficulties from 36 kindergarten classrooms participated in the study. Students who were randomly assigned to the treatment condition received the intervention in small-group formats, with 2 or 5 students per group. Control students participated in a no-treatment control condition. Significant effects on proximal and distal measures of mathematics achievement were found. Effect sizes obtained for all measures fell within or exceeded the upper bound of the effects reported in the initial study. Implications for systematically situating replication studies in larger frameworks of intervention research and reporting rates of treatment response across replication studies are discussed. [This paper was published in "Exceptional Children" (EJ1116305).]

The goal of the current research was to better understand when and why feedback has positive effects on learning and to identify features of feedback that may improve its efficacy. In a randomized experiment, second-grade children (N = 75) received instruction on a correct problem-solving strategy and then solved a set of relevant problems. Children were assigned to receive no feedback, immediate feedback, or summative feedback from the computer. On a posttest the following day, feedback resulted in higher scores relative to no feedback for children who started with low prior knowledge. Immediate feedback was particularly effective, facilitating mastery of the material for children with both low and high prior knowledge. Results suggest that minimal computer-generated feedback can be a powerful form of guidance during problem solving.

The goal of the current research was to better understand when and why feedback has positive effects on learning and to identify features of feedback that may improve its efficacy. In a randomized experiment, second-grade children (N = 75) received instruction on a correct problem-solving strategy and then solved a set of relevant problems. Children were assigned to receive no feedback, immediate feedback, or summative feedback from the computer. On a posttest the following day, feedback resulted in higher scores relative to no feedback for children who started with low prior knowledge. Immediate feedback was particularly effective, facilitating mastery of the material for children with both low and high prior knowledge. Results suggest that minimal computer-generated feedback can be a powerful form of guidance during problem solving.

The purpose of this study was to test the promise of the NumberShire Level 1 Gaming Intervention (NS1) to accelerate math learning for first-grade students with or at risk for math difficulties. The NS1 intervention was developed through the Institute of Education Sciences, Small Business Innovation Research Program (Gause, Fien, Baker, & Clarke, 2011 Gause, M., Fien, H., Baker, S. K., & Clarke, B. (2011). Project NumberShire I: A game-based integrated learning and assessment system to target whole number concepts. This study used a randomized controlled trial design to test the promise of the NS1 intervention. In total, 250 first-grade students were randomly assigned within classrooms to the treatment condition or a control condition. Results indicate significant effects favoring the treatment group on proximal measures of whole-number concepts and skills. Intervention effects were not statistically significant for distal outcome measures. Treatment effects were not moderated by special education or English learner status; however, the condition by initial skill level interaction approached significance. Additionally, there was no relationship between dosage variables and students' response to the intervention. Limitations and future directions for research are discussed. [This paper was published in "Journal of Research on Educational Effectiveness" (EJ1115268).]

The focus of the present study was enhancing word problem and calculation achievement in ways that support prealgebraic thinking among second-grade students at risk for mathematics difficulty. Intervention relied on a multitier support system (i.e., responsiveness to intervention, or RTI) in which at-risk students participate in general classroom instruction and receive supplementary small-group tutoring. Participants were 265 students in 110 classrooms in 25 schools. Teachers were randomly assigned to one of three conditions: calculation RTI, word problem RTI, or business-as-usual control. Intervention lasted 17 weeks. Multilevel modeling indicated that calculation RTI improved calculation but not word problem outcomes, word problem RTI enhanced proximal word problem outcomes as well as performance on some calculation outcomes, and word problem RTI provided a stronger route than calculation RTI to prealgebraic knowledge.

The focus of the present study was enhancing word problem and calculation achievement in ways that support prealgebraic thinking among second-grade students at risk for mathematics difficulty. Intervention relied on a multitier support system (i.e., responsiveness to intervention, or RTI) in which at-risk students participate in general classroom instruction and receive supplementary small-group tutoring. Participants were 265 students in 110 classrooms in 25 schools. Teachers were randomly assigned to one of three conditions: calculation RTI, word problem RTI, or business-as-usual control. Intervention lasted 17 weeks. Multilevel modeling indicated that calculation RTI improved calculation but not word problem outcomes, word problem RTI enhanced proximal word problem outcomes as well as performance on some calculation outcomes, and word problem RTI provided a stronger route than calculation RTI to prealgebraic knowledge.

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation solving and equal-sign understanding of students with mathematics difficulty (MD). In the present study, second-grade students with MD (n = 51) were randomly assigned to standard equations tutoring, combined tutoring (standard and nonstandard equations), and no-tutoring control. Combined tutoring students demonstrated greater gains on equation-solving assessments and equal-sign tasks compared to the other two conditions. Standard tutoring students demonstrated improved skill on equation solving over control students, but combined tutoring students' performance gains were significantly larger. Results indicate that exposure to and practice with nonstandard equations positively influence student understanding of the equal sign.

Students often misinterpret the equal sign (=) as operational instead of relational. Research indicates misinterpretation of the equal sign occurs because students receive relatively little exposure to equations that promote relational understanding of the equal sign. No study, however, has examined effects of nonstandard equations on the equation solving and equal-sign understanding of students with mathematics difficulty (MD). In the present study, second-grade students with MD (n = 51) were randomly assigned to standard equations tutoring, combined tutoring (standard and nonstandard equations), and no-tutoring control. Combined tutoring students demonstrated greater gains on equation-solving assessments and equal-sign tasks compared to the other two conditions. Standard tutoring students demonstrated improved skill on equation solving over control students, but combined tutoring students' performance gains were significantly larger. Results indicate that exposure to and practice with nonstandard equations positively influence student understanding of the equal sign.

A typical mathematics assignment consists primarily of practice problems requiring the strategy introduced in the immediately preceding lesson (e.g., a dozen problems that are solved by using the Pythagorean Theorem). This means that students know which strategy is needed to solve each problem before they read the problem. In an alternative approach known as "interleaved practice," problems from the course are rearranged so that a portion of each assignment includes different kinds of problems in an interleaved order. Interleaved practice requires students to choose a strategy on the basis of the problem itself, as they must do when they encounter a problem during a comprehensive examination or subsequent course. In the experiment reported here, 126 seventh-grade students received the same practice problems over a three-month period, but the problems were arranged so that skills were learned by interleaved practice or by the usual blocked approach. The practice phase concluded with a review session, followed 1 or 30 days later by an unannounced test. Compared to blocked practice, interleaved practice produced higher scores on both the immediate and delayed tests (Cohen's d = 0.42 and 0.79, respectively). Two appendices include: (1) Serial Position of Each Graph and Slope Problem in the Assignments (table); and (2) Frequency of Responses of Three Teachers to Statements About Interleaved Practice (table). [Note: This article was "in press" at the time of submission. No citation information is available at this time.]

Most mathematics assignments consist of a group of problems requiring the same strategy. For example, a lesson on the quadratic formula is typically followed by a block of problems requiring students to use the quadratic formula, which means that students know the appropriate strategy before they read each problem. In an alternative approach, different kinds of problems appear in an interleaved order, which requires students to choose the strategy on the basis of the problem itself. In the classroom-based experiment reported here, grade seven students (n = 140) received blocked or interleaved practice over a nine-week period, followed two weeks later by an unannounced test. Mean test scores were greater for material learned by interleaved practice rather than by blocked practice (72% vs. 38%, d = 1.05). This interleaving effect was observed even though the different kinds of problems were superficially dissimilar from each other, whereas previous interleaved mathematics studies required students to learn nearly identical kinds of problems. We conclude that interleaving improves mathematics learning not only by improving discrimination between different kinds of problems but also by strengthening the association between each kind of problem and its corresponding strategy. [This article was published in: "Psychonomic Bulletin & Review" v21 n5 p1323-1330 Oct 2014; http://dx.doi.org/ 10.3758/s13423-014-0588-3.]

This pilot study examined the efficacy of a Tier 2 first-grade mathematics intervention program targeting whole-number understanding for students at risk in mathematics. The study used a randomized block design. Students (N = 89) were randomly assigned to treatment (Fusion) or control (standard district practice) conditions. Measures of mathematics achievement were collected at pretest and posttest. Treatment and control students did not differ on mathematics assessments at pretest. A series of random-effects models were estimated to compare gains between treatment and control conditions. Gain scores of intervention students were significantly greater than those of control peers on a proximal measure of mathematics achievement. The role of a strong theory-of-change model in the development and evaluation of mathematics interventions is articulated. Implications for researchers and educators designing and delivering instruction for at-risk students in a response-to-intervention model are discussed.

This intervention study compared the efficacy of small-group tutoring on the mathematics learning of third-grade students at risk for mathematics difficulty using either a school-provided standards-based curriculum (SBC) or a schema-based instruction (SBI) curriculum. The SBI curriculum placed particular emphasis on the underlying mathematical structure of additive problems to represent and solve word problems. At-risk students (N = 136) from 35 classrooms scoring below a proficiency level on their district accountability assessment were assigned randomly to treatment groups. Results indicated interaction effects on the word problem-solving (WPS) posttest and retention tests such that SBI students with higher incoming (pretest) WPS scores outperformed SBC students with higher pretest scores, whereas SBC students with lower pretest scores outperformed SBI students with lower pretest scores. No effects were found on number combinations automaticity, and mathematics and reading achievement. Implications to improve the problem-solving performance of at-risk students are discussed. (Contains 4 tables and 2 figures.)

This study compared the effects of delivering a supplemental, small-group tutoring intervention on the mathematics outcomes of third-grade students at risk for mathematics difficulties (MD) who were randomly assigned to either a schema-based instruction (SBI) or control group. SBI emphasized the underlying mathematical structure of additive problems. All students at risk for MD identified through screening received a mathematics intervention in groups of 2-4 for 12 weeks across the school year. Results revealed that students in the SBI group outperformed students in the control group on a word problem solving (WPS) posttest ("g" = 0.46). The effect of SBI proved to be equivalent for students in both high and low at-risk subgroups. On a district-administered mathematics achievement test, SBI students scored significantly higher than control students (g = 0.34); however, there were no significant effects on the WPS retention test (8 weeks later). (Contains 3 tables and 1 figure.)

The purpose of this quantitative study was to investigate the effectiveness of a system-wide Response to Intervention (RTI) program on the mathematical achievement of seventh and eighth grade students. The study consisted of five district schools with a total of 502 participants. The students were identified as belonging to one of two tiers, which differed in regard to amount of intervention. The first tier (Tier 1) of students only received the regular classroom instruction while the second tier (Tier 2/3) received an additional thirty minutes of intervention strategies. The students receiving interventions, the Tier 2/3 students, were divided into two groups. One group received primarily teacher-directed instruction (TDI) as an intervention while the other group received computer-assisted instruction (CAI) as an intervention. For the purpose of this study, the CAI intervention involved the use of the commercial program, Odyssey Math. The students were benchmark tested at the beginning and end of the 2010-2011 school year using the STAR Math assessment program and also progress monitored on a regular basis. In an attempt to determine the effectiveness of the RTI program, a gain score ANOVA was conducted using the scaled scores of the two tiers from the beginning and the end of the school year. The analysis indicated that Tier 2/3 students did demonstrate greater growth than the students in Tier 1. The gain scores of the two groups of Tier 2/3 students were also used in a gain score ANOVA to measure differences in growth. An additional analysis of their mean scores was also conducted using ANCOVA. Both analyses indicated that the CAI group demonstrated greater gains. A third analysis was conducted in order to determine how accurately the STAR Math assessment program could predict student success (reaching either a Proficient or Advanced level) on the state assessment. While the STAR Math program did not accurately predict the students' level in every case, the logistic regression analysis did indicate that the program was successful in identifying struggling students. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]

Background: A key learning outcome in problem-solving domains is the development of procedural flexibility, where learners know multiple procedures and use them appropriately to solve a range of problems (e.g., Verschaffel, Luwel, Torbeyns, & Van Dooren, 2009). However, students often fail to become flexible problem solvers in mathematics. To support flexibility, teaching standards in many countries recommend that students be exposed to multiple procedures early in instruction and be encouraged to compare them. Aims: We experimentally evaluated this recommended instructional practice for supporting procedural flexibility during a classroom lesson, relative to two alternative conditions. The alternatives reflected the common instructional practice of delayed exposure to multiple procedures, either with or without comparison of procedures. Sample: Grade 8 students from two public schools (N= 198) were randomly assigned to condition. Students had not received prior instruction on multi-step equation solving, which was the topic of our lessons. Method: Students learned about multi-step equation solving under one of three conditions in math class for about 3 hr. They also completed a pre-test, post-test, and 1-month-retention test on their procedural knowledge, procedural flexibility, and conceptual knowledge of equation solving. Results: Novices who compared procedures immediately were more flexible problem solvers than those who did not, even on a 1-month retention test. Although condition had limited direct impact on conceptual and procedural knowledge, greater flexibility was associated with greater knowledge of both types. Conclusions: Comparing procedures can support flexibility in novices and early introduction to multiple procedures may be one important reason. (Contains 5 tables and 2 figures.)

We compared the learning from playing a linear number board game of preschoolers from middle-income backgrounds to the learning of preschoolers from low-income backgrounds. Playing this game produced greater learning by both groups than engaging in other numerical activities for the same amount of time. The benefits were present on number line estimation, magnitude comparison, numeral identification, and arithmetic learning. Children with less initial knowledge generally learned more, and children from low-income backgrounds learned at least as much, and on several measures more, than preschoolers from middle-income backgrounds with comparable initial knowledge. The findings suggest a class of intervention that might be especially effective for reducing the gap between low-income and middle-income children's knowledge when they enter school.

The purpose of this study was to determine the effects of an early numeracy preventative Tier 2 intervention on the mathematics performance of first-grade students with mathematics difficulties. Researchers used a pretest-posttest control group design with randomized assignment of 139 students to the Tier 2 treatment condition and 65 students to the comparison condition. Systematic instruction, visual representations of mathematical concepts, purposeful and meaningful practice opportunities, and frequent progress monitoring were used to develop understanding in early numeracy skills and concepts. Researchers used progress-monitoring measures and a standardized assessment measure to test the effects of the intervention. Findings showed that students in the treatment group outperformed students in the comparison group on the progress-monitoring measures of mathematics performance and the measures that focused on whole-number computation. There were no differences between groups on the problem-solving measures. (Contains 5 tables.)

In this paper the augmentation of worked examples with animations for teaching problem-solving skills in mathematics is advocated as an effective instructional method. First, in a cognitive task analysis different knowledge prerequisites are identified for solving mathematical word problems. Second, it is argued that so called hybrid animations would be most effective for acquiring these prerequisites, because they show the continuous transition from a concrete, but superficial problem representation to a more abstract, mathematical problem model that forms a basis for solving a problem. An experiment was conducted, where N = 32 pupils from a German high school studied either only text-based worked examples explaining different problem categories from the domain of algebra or worked examples augmented with hybrid animations. Learners with hybrid animations showed superior problem-solving performance for problems of different transfer distance relative to those in the text-only condition.

Student achievement in mathematics has been a focal concern in the United States for many years. The National Research Council's 2001 report and the recent report of the National Mathematics Advisory Panel (2008) both called attention to student achievement in mathematics, and both called for all students to learn algebra by the end of eighth grade. Reports have argued, further, that achieving this goal requires that students first successfully learn several topics in rational numbers--fractions, decimals, ratio, rate, proportion, and percent. These topics are typically covered in grades 4 through 7, yet many students continue to struggle with them beyond the seventh grade. The National Mathematics Advisory Panel wrote that--difficulty with fractions (including decimals and percent) is pervasive and is a major obstacle to further progress in mathematics, including algebra. The panel also specified that by the end of seventh grade, students should be able to solve problems involving percent, ratio, and rate, and extend this work to proportionality. The U.S. Department of Education's National Center for Educational Evaluation and Regional Assistance (NCEE)--within the Institute of Education Sciences--initiated the Middle School Mathematics Professional Development Impact Study to test the impact of a professional development (PD) program for teachers that was designed to address the problem of low student achievement in topics in rational numbers. The study focuses on seventh grade, the culminating year for teaching those topics and has three central research questions: (1) What impact did the PD program provided in this study have on teacher knowledge of rational number topics? (2) What impact did the PD program provided in this study have on teacher instructional practices? and (3) What impact did the PD program provided in this study have on student achievement in rational number topics? The study produced the following results: (1) The study's PD program was implemented as intended; (2) The PD program did not produce a statistically significant impact on teacher knowledge of rational numbers (effect size = 0.19, p-value = 0.15); (3) The PD program had a statistically significant impact on the frequency with which teachers engaged in activities that elicited student thinking, one of the three measures of instructional practice used in the study (effect size = 0.48); and (4) The PD program did not produce a statistically significant impact on student achievement (effect size = 0.04, p-value = 0.37). This report presents the study's findings after 1 year of implementing the PD in the treatment schools. A subsequent report will present findings after 2 years of implementing the PD. Chapter 1 presents an overview of the study. Chapter 2 describes the study design and its realization, including a description of the sample and tests of baseline equivalence of the treatment and control groups on observed characteristics. Chapter 3 describes the design and implementation of the PD program and the extent of service contrast between the treatment and control groups. Chapter 4 addresses the impact of the PD program on teacher knowledge, instructional practice, and student mathematics achievement. Chapter 5 provides several nonexperimental analyses that explore additional questions related to the impact findings. Appended are: (1) Data Collection; (2) Details of the Study Samples and Analytic Approaches; (3) Supplemental Information on the Design and Implementation of the PD Program; (4) Supporting Tables and Figures for Impact Analyses; and (5) Exploratory Analyses: Approaches and Additional Results. (Contains 9 exhibits, 9 figures, and 90 tables.)

The primary purpose of this study was to assess the effects of strategic counting instruction, with and without deliberate practice with those counting strategies, on number combination (NC) skill among students with mathematics difficulties (MD). Students (n = 150) were stratified on MD status (i.e., MD alone versus MD with reading difficulty) and site (proximal versus distal to the intervention developer) and then randomly assigned to control (no tutoring) or 1 of 2 variants of NC remediation. Both remediations were embedded in the same validated word-problem tutoring protocol (i.e., Pirate Math). In 1 variant, the focus on NCs was limited to a single lesson that taught strategic counting. In the other variant, 4-6 min of practice per session was added to the other variant. Tutoring occurred for 16 weeks, 3 sessions per week for 20-30 min per session. Strategic counting without deliberate practice produced superior NC fluency compared to control; however, strategic counting with deliberate practice effected superior NC fluency and transfer to procedural calculations compared with both competing conditions. Also, the efficacy of Pirate Math word-problem tutoring was replicated. (Contains 6 tables.)

The primary purpose of this study was to assess the effects of strategic counting instruction, with and without deliberate practice with those counting strategies, on number combination (NC) skill among students with mathematics difficulties (MD). Students (n = 150) were stratified on MD status (i.e., MD alone versus MD with reading difficulty) and site (proximal versus distal to the intervention developer) and then randomly assigned to control (no tutoring) or 1 of 2 variants of NC remediation. Both remediations were embedded in the same validated word-problem tutoring protocol (i.e., Pirate Math). In 1 variant, the focus on NCs was limited to a single lesson that taught strategic counting. In the other variant, 4-6 min of practice per session was added to the other variant. Tutoring occurred for 16 weeks, 3 sessions per week for 20-30 min per session. Strategic counting without deliberate practice produced superior NC fluency compared to control; however, strategic counting with deliberate practice effected superior NC fluency and transfer to procedural calculations compared with both competing conditions. Also, the efficacy of Pirate Math word-problem tutoring was replicated. (Contains 6 tables.)

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students' math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20-30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students' mathematics difficulty status. The tutoring protocols proved transportable across sites. (Contains 5 tables and 8 footnotes.)

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students' math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20-30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students' mathematics difficulty status. The tutoring protocols proved transportable across sites. (Contains 5 tables and 8 footnotes.)

Researchers in both cognitive science and mathematics education emphasize the importance of comparison for learning and transfer. However, surprisingly little is known about the advantages and disadvantages of what types of things are being compared. In this experimental study, 162 seventh- and eighth-grade students learned to solve equations (a) by comparing equivalent problems solved with the same solution method, (b) by comparing different problem types solved with the same solution method, or (c) by comparing different solution methods to the same problem. Students' conceptual knowledge and procedural flexibility were best supported by comparing solution methods and to a lesser extent by comparing problem types. The benefits of comparison are augmented when examples differ on relevant features, and contrasting methods may be particularly useful in mathematics learning. (Contains 3 figures, 8 tables, and 1 footnote.)

In the No Child Left Behind Act (NCLB), Congress called for the U.S. Department of Education (ED) to conduct a rigorous study of the conditions and practices under which educational technology is effective in increasing student academic achievement. A 2007 report presenting study findings for the 2004-2005 school year, indicated that, after one school year, differences in student test scores were not statistically significant between classrooms that were randomly assigned to use software products and those that were randomly assigned not to use products. School and teacher characteristics generally were not related to whether products were effective. The second year of the study examined whether an additional year of teaching experience using the software products increased the estimated effects of software products on student test scores. The evidence for this hypothesis is mixed. For reading, there were no statistically significant differences between the effects that products had on standardized student test scores in the first year and the second year. For sixth grade math, product effects on student test scores were statistically significantly lower (more negative) in the second year than in the first year, and for algebra I, effects on student test scores were statistically significantly higher in the second year than in the first year. The study also tested whether using any of the 10 software products increased student test scores. One product had a positive and statistically significant effect. Nine did not have statistically significant effects on test scores. Five of the insignificant effects were negative and four were positive. Study findings should be interpreted in the context of design and objectives. The study examined a range of reading and math software products in a range of diverse school districts and schools. But it did not study many forms of educational technology and it did not include many types of software products. How much information the findings provide about the effectiveness of products that are not in the study is an open question. Products in the study also were implemented in a specific set of districts and schools, and other districts and schools may have different experiences with the products. The findings should be viewed as one element within a larger set of research studies that have explored the effectiveness of software products. Three appendixes are included: (1) Second-Year Data Collection and Response Rates; (2) Description of Sample for the 10 Products; and (3) Details of Estimation Methods. (Contains 29 footnotes, 4 figures and 24 tables.

Research indicates that a socioeconomic status-related gap in mathematical knowledge appears early and widens during early childhood. Young children from economically disadvantaged families receive less support for mathematical development both at home and in preschool. Consequently, children from different socioeconomic backgrounds enter elementary school at different levels of readiness to learn a standards-based mathematics curriculum. One approach to closing this gap is the development and implementation of effective mathematics curricula for public preschool programs enrolling economically disadvantaged children. A randomized controlled trial was conducted in 40 Head Start and state preschool classrooms, with 278 children, to determine whether a pre-kindergarten mathematics intervention was effective. Intervention teachers received training that enabled them to implement with fidelity, and a large majority of parents regularly used math activities teachers sent home. Intervention and control groups did not differ on math assessments at pretest; however, gain scores of intervention children were significantly greater than those of control children at posttest. Thus, the intervention reduced the gap in children's early mathematical knowledge. (Contains 3 tables and 1 figure.)

A randomized-trials design was used to evaluate the effectiveness of a preschool mathematics program based on a comprehensive model of research-based curricula development. Thirty-six preschool classrooms were assigned to experimental (Building Blocks), comparison (a different preschool mathematics curriculum), or control conditions. Children were individually pre- and posttested, participating in 26 weeks of instruction in between. Observational measures indicated that the curricula were implemented with fidelity, and the experimental condition had significant positive effects on classrooms' mathematics environment and teaching. The experimental group score increased significantly more than the comparison group score (effect size = 0.47) and the control group score (effect size = 1.07). Early interventions can increase the quality of the mathematics environment and help preschoolers develop a foundation of mathematics knowledge. (Contains 7 tables, 1 figure, and 1 note.)

The major purposes of this study were to assess the efficacy of tutoring to remediate 3rd-grade computational deficits and to explore whether remediation is differentially efficacious depending on whether students experience mathematics difficulty alone or concomitantly with reading difficulty. At 2 sites, 127 students were stratified on mathematics difficulty status and randomly assigned to 4 conditions: word recognition (control) tutoring or 1 of 3 computation tutoring conditions: fact retrieval, procedural computation and computational estimation, and combined (fact retrieval + procedural computation and computational estimation). Results revealed that fact retrieval tutoring enhanced fact retrieval skill, and procedural computation and computational estimation tutoring (whether in isolation or combined with fact retrieval tutoring) enhanced computational estimation skill. Remediation was not differentially efficacious as a function of students' mathematics difficulty status. (Contains 4 tables and 1 footnote.)

A key learning outcome in problem-solving domains is the development of flexible knowledge, where learners know multiple strategies and adaptively choose efficient strategies. Two interventions hypothesized to improve flexibility in problem solving were experimentally evaluated: prompts to discover multiple strategies and direct instruction on multiple strategies. Participants were 132 sixth-grade students who solved linear equations for three hours. Both interventions improved students' flexibility in problem solving and did not replace, nor interfere with, one another. Overall, the study provides causal evidence that exposure to multiple strategies leads to improved flexibility in problem solving and that discovery learning and direct instruction are compatible instructional approaches. (Contains 6 tables.)

Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventh-grade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom.

This study evaluated the efficacy of a preschool mathematics program based on a comprehensive model of developing research-based software and print curricula. Building Blocks, funded by the National Science Foundation, is a curriculum development project focused on creating research-based, technology-enhanced mathematics materials for pre-K through grade 2. In this article, we describe the underlying principles, development, and initial summative evaluation of the first set of resulting materials as they were used in classrooms with children at risk for later school failure. Experimental and comparison classrooms included two principal types of public preschool programs serving low-income families: state funded and Head Start prekindergarten programs. The experimental treatment group score increased significantly more than the comparison group score; achievement gains of the experimental group approached the sought-after 2-sigma effect of individual tutoring. This study contributes to research showing that focused early mathematical interventions help young children develop a foundation of informal mathematics knowledge, especially for children at risk for later school failure.

This study investigated the differential effects of two problem-solving instructional approaches--schema-based instruction (SBI) and general strategy instruction (GSI)--on the mathematical word problem-solving performance of 22 middle school students who had learning disabilities or were at risk for mathematics failure. Results indicated that the SBI group significantly outperformed the GSI group on immediate and delayed posttests as well as the transfer test. Implications of the study are discussed within the context of the new IDEA amendment and access to the general education curriculum.

Compared students who had four years of the University of Chicago School Mathematics Project (UCSMP) secondary curriculum to two distinct groups of comparable students at three different sites (one urban and two suburban). UCSMP students at all three sites achieved higher but registered little difference in attitude. (Contains 48 references.) (MDH)

The purpose of this research was to design and evaluate a computer based adaptive strategy for teaching mathematics. Results indicated preference for materials with personalized examples for motivation and meaningful learning of problem solving procedures. Practical advantages of the computer-based model for adapting instruction are also considered. (Author/JAZ)

An explicit strategy method developed from a composite of basal arithmetic texts was used for mathematics problem-solving instruction for a group of fourth graders. Posttest results were positive. Implications for teaching mathematics problem solving to low performing students are discussed. (Author/DF)

Scholars and practitioners have called for personalized and widely accessible professional development (PD) for teachers. Yet, a long-standing tension between customizing support and increasing access to such support has hindered the scale-up of high-quality PD for individual teachers. This study addresses this challenge by developing a computerized program for middle school mathematics teachers that provides frequent opportunities for teachers to interact with and obtain personalized and real-time feedback from a virtual facilitator based on natural language processing. Based on the data collected from 1727 middle school students in an experiment in which the teachers of these students were randomly assigned to the program or the business-as-usual condition (i.e., the control group), we found that the program had a statistically significant impact on students' mathematics performance. These results demonstrate the potential of incorporating an automated, interactive feedback tool supported by artificial intelligence to create effective, scalable teacher PD.

The Math Curriculum Impact Study was a large-scale randomized controlled trial (RCT) to test the efficacy of a digital core curriculum for Grade 5 mathematics. Reasoning Mind's Grade 5 Common Core Curriculum was a comprehensive, adaptive, blended learning approach that treated schools implemented for an entire school year. The study was completed in 46 schools throughout West Virginia, resulting in achievement data from 1,919 students. It also included exploratory investigations of teacher practice and student engagement. The main experimental finding was a null result; achievement was similar in both experimental groups. The exploratory investigations help to clarify interpretation of this result. As educational leaders throughout the United States adopt digital mathematics curricula and adaptive, blended approaches, our findings provide a relevant caution. However, our findings are not generalizable to all digital offerings, and there is a continuing need for refined theory, study of implementation, and rigorous experimentation to advise schools.

The current study investigated the effectiveness of three distinct educational technologies--two game-based applications (From Here to There and DragonBox 12+) and two modes of online problem sets in ASSISTments (an Immediate Feedback condition and an Active Control condition with no immediate feedback) on Grade 7 students' algebraic knowledge. More than 3,600 Grade 7 students across nine in-person and one virtual schools within the same district were randomly assigned to one of the four conditions. Students received nine 30-minute intervention sessions from September 2020 to March 2021. Hierarchical linear modeling analyses of the final analytic sample (N = 1,850) showed significantly higher posttest scores for students who used From Here to There and DragonBox 12+ compared to the Active Control condition. No significant difference was found for the Immediate Feedback condition. The findings have implications for understanding how game-based applications can affect algebraic understanding, even within pandemic pressures on learning.

The purpose of this study was to explore the paths by which word-problem intervention, with versus without embedded prealgebraic reasoning instruction, improved word-problem performance. Students with mathematics difficulty (MD; n = 304) were randomly assigned to a business-as-usual condition or 1 of 2 variants of word-problem intervention. The prealgebraic reasoning component targeted relational understanding of the equal sign as well as standard and nonstandard equation solving. Intervention occurred for 16 weeks, 3 times per week, 30 min per session. Sequential mediation models revealed main effects, in which each intervention condition significantly and substantially outperformed the business-as-usual condition, corroborating prior research on the efficacy of schema word-problem intervention. Yet despite comparable effects on word-problem outcomes between the two word-problem conditions, the process by which effects accrued differed: An indirect path via equal-sign understanding and then equation solving was significant only for the word-problem intervention condition with embedded prealgebraic reasoning instruction. Additionally, the effect of this condition on equal-sign reasoning was strong. Given the link between equal-sign reasoning for success with algebra and the importance of algebra for success with advanced mathematics, results suggest an advantage for embedding prealgebraic reasoning instruction within word-problem intervention.

Preschoolers' repeating patterning knowledge is predictive of their concurrent and later math and numeracy knowledge, but strong experimental evidence is needed to determine if these relations are causal. The purpose of the current Study was to examine the causal effects of repeating patterning and numeracy tutoring on repeating patterning, numeracy, and general mathematics knowledge in the year before kindergarten (i.e., pre-K). Children in pre-K (N = 211) were randomly assigned to receive five sessions of researcher-delivered tutoring (a) on repeating patterns and numeracy or (b) on numeracy (and literacy as an active control), or received no tutoring and business as usual classroom instruction(control). Children who received tutoring in repeating patterning and numeracy improved in their repeating patterning knowledge the most. However, children's general math and numeracy knowledge improved similarly across conditions, and a specific aspect of numeracy emphasized during the tutoring did not improve. Children's repeating patterning knowledge is malleable, but this initial attempt to demonstrate causal links between repeating patterning and math knowledge was not successful. Results parallel mixed success in research training other skills, such as working memory or spatial skills, for improving mathematics knowledge. Findings are discussed in terms of the relations between patterning, numeracy, and general math knowledge in preschoolers. [This paper will be published in "Journal of Educational Psychology."]

This randomized controlled trial examined effects of the MyTeachingPartner-Math/Science intervention on the quality and quantity of teachers' mathematics and science instruction, and children's mathematics and science outcomes in 140 pre-kindergarten classrooms. Teachers participated in the intervention for two years with consecutive cohorts of children. Results from Year 1 are considered experimental, however due to high levels of attrition, results from Year 2 are considered quasi-experimental. Across both years, intervention teachers exhibited higher quality and quantity of instruction. In Year 1, there were no significant effects of the intervention on children's outcomes. In Year 2, children in intervention classrooms made greater gains in teachers' ratings of mathematics and science skills and performed better on a spring assessment of science skills. These results have implications for designing and evaluating professional development aimed at supporting children's mathematics and science knowledge and skills.

Dual-credit courses expose high school students to college-level content and provide the opportunity to earn college credits, in part to smooth the transition to college. With the Tennessee Department of Education, we conduct the first randomized controlled trial of the effects of dual-credit math coursework on a range of high school and college outcomes. We find that the dual-credit advanced algebra course alters students' subsequent high school math course-taking, reducing enrollment in remedial math and boosting enrollment in precalculus and Advanced Placement math courses. We fail to detect an effect of the dual-credit math course on overall rates of college enrollment. However, the course induces some students to choose four-year universities instead of two-year colleges, particularly for those in the middle of the math achievement distribution and those first exposed to the opportunity to take the course in eleventh rather than twelfth grade. We see limited evidence of improvements in early math performance during college.

The purpose of this project evaluation was to assess the impact of data-based individualization (DBI) on the mathematics achievement of students with intensive mathematics learning needs, including students with disabilities. The evaluation study used a cluster randomized trial in which elementary schools were randomly assigned to treatment using a delayed-intervention design. Since this was a development project, the evaluation delineated between the primary, confirmatory impact question and exploratory research questions. The confirmatory question included students in Grades 1-2 and was concerned with the relationship of one year of DBI implementation support in comparison with a business-as-usual, delayed intervention group. Because of the developmental, iterative nature of the project, exploratory questions were concerned with cumulative longitudinal relations between years of DBI implementation support between two cohorts of elementary schools. In addition, project staff supported DBI implementation pilot in two middle schools and tracked student progress in those sites. Analytic results provided preliminary evidence to suggest that there may be contextual factors that govern the likelihood a student will profit from DBI. In addition, schools may require significant ongoing support to sustain implementation.

This study used a randomized controlled trial design to investigate the ROOTS curriculum, a 50-lesson kindergarten mathematics intervention. Ten ROOTS-eligible students per classroom (n = 60) were randomly assigned to one of three conditions: a ROOTS five-student group, a ROOTS two-student group, and a no-treatment control group. Two primary research questions were investigated as part of this study: What was the overall impact of the treatment (the ROOTS intervention) as compared with the control (business as usual)? Was there a differential impact on student outcomes between the two treatment conditions (two- vs. five-student group)? Initial analyses for the first research question indicated a significant impact on three outcomes and positive but nonsignificant impacts on three additional measures. Results for the second research question, comparing the two- and five-student groups, indicated negligible and nonsignificant differences. Implications for practice are discussed. [For the corresponding grantee submission, see ED578431.]

Middle school algebra students (N = 125) randomly assigned within classroom to a Problem-solving control group, a Correct worked examples control group, or an Incorrect worked examples group, completed an experimental classroom study to assess the differential effects of incorrect examples versus the two control groups on students' algebra learning, competence expectancy, and sense of belonging to math class. The study also explored whether prior knowledge impacted the effectiveness of the intervention. A greater sense of belonging and competence expectancy predicted greater learning overall. Students' sense of belonging to math and competence expectancies were high at the start of the study and did not increase as a result of the intervention. A significant interaction between prior knowledge and incorrect worked examples on post-test scores revealed that students with low prior knowledge who struggle with learning math benefit most from reflecting on highlighted errors within an incorrect worked examples intervention. The unique contributions of these findings as well as educational implications are discussed. [This article was published in "Learning and Individual Differences," v48 p36-44 May 2016.]

Middle school algebra students (N = 125) randomly assigned within classroom to a Problem-solving control group, a Correct worked examples control group, or an Incorrect worked examples group, completed an experimental classroom study to assess the differential effects of incorrect examples versus the two control groups on students' algebra learning, competence expectancy, and sense of belonging to math class. The study also explored whether prior knowledge impacted the effectiveness of the intervention. A greater sense of belonging and competence expectancy predicted greater learning overall. Students' sense of belonging to math and competence expectancies were high at the start of the study and did not increase as a result of the intervention. A significant interaction between prior knowledge and incorrect worked examples on post-test scores revealed that students with low prior knowledge who struggle with learning math benefit most from reflecting on highlighted errors within an incorrect worked examples intervention. The unique contributions of these findings as well as educational implications are discussed. [This article was published in "Learning and Individual Differences," v48 p36-44 May 2016.]

Thirty-two students enrolled in one of four sixth-grade classrooms across two elementary schools participated in this study. Students receiving supplemental and intensive instruction in math and those with math-related disabilities were participants. A treatment and control, pre/postexperimental design was used to examine the effect of preteaching using a gradual instructional sequence on students' accuracy in solving fraction computations. Prior to each unit, students were pretaught three essential prerequisite skills related to the upcoming general education core math unit. Findings indicate that the combination of preteaching using the concrete-representational-abstract instructional sequence can be effective at improving the overall fraction computations of students with or at risk for disabilities.

Homework is transforming at a rapid rate with continuous advances in educational technology. Computer-based homework, in particular, is gaining popularity across a range of schools, with little empirical evidence on how to optimize student learning. The current aim was to test the effects of different types of feedback on computer-based homework. In the study, middle school students completed a computer-based pretest, homework assignment, and posttest containing challenging algebraic problems. On the homework assignment, students were assigned to different feedback conditions. In Experiment 1 (N = 103), students received no feedback or correct-answer feedback after each problem. In Experiment 2 (N = 143), students received (1) no feedback, (2) correct-answer feedback, (3) try-again feedback, or (4) explanation feedback after each problem. For students with low prior knowledge, feedback resulted in better posttest performance than no feedback. However, students with high prior knowledge learned just as much whether they received feedback or not. Results suggest the provision of basic feedback on computer-based homework can benefit novice students' mathematics learning.

This five-year evaluation examined the effectiveness of a promising middle-school mathematics intervention funded through an Investing in Innovation (i3) development grant. Evaluation objectives were to: (1) study the impact of an intervention aimed at increasing the academic achievement of students in Algebra I--a gate-keeping course--as measured by student performance on an end-of-year state test in mathematics; and (2) better understand the relationship between intervention impact and implementation fidelity, as measured by levels of compliance by teachers with the study protocol. The intervention was piloted in Year 2 of the grant (2011-12 school year) that was followed by a two-year [randomized control trial] RCT in grant years 3 (2012-13 school year) and 4 (2013-14 school year). Data collected in the RCT years were focused on impact and exploratory analyses, respectively. For the RCT component, 70 Grade 8 Algebra I teachers were recruited from 15 school districts across California. Randomization, conducted by WestEd in spring 2012, was conducted at the teacher level. Students were assigned to classrooms without knowledge of the group membership of teachers (treatment vs. control), using each district's routine placement policies. Fidelity of implementation study was monitored by collecting systematically information from teachers assigned to the treatment condition throughout the course of the study. The contrast of interest was performance on a standardized Algebra I test by students assigned to classrooms taught by treatment teachers compared to performance by students assigned to classrooms taught by control teachers. The final analytic sample for the 2012-13 cohort included 1,384 students assigned to 28 treatment teachers and 1,088 students assigned to 27 control teachers. None of the contrasts showed a statistically significant difference at the 0.05 level. Students who were assigned to classrooms taught by treatment teachers did not perform differently in relation to those assigned to classrooms taught by control teachers. Overall findings from the implementation study indicated that great variability emerged in the ways in which teachers implemented the intervention. The threshold for fidelity was reached with only one component (Instructional Unit #1) of the four studied (three instructional units and professional coaching). The following appendices are included: (1) Logic Model: SLOPE (DEV11) v.13; (2) Teacher Background Survey; (3) Interpreting Intervention Impact through the Lens of Implementation Fidelity: Findings from a Federally Funded Evaluation--Paper Presented at the Annual Meeting of the American Educational Research Association (Chicago, Illinois, April 19, 2015); (4) Implementation Survey for Air Traffic Control; (5) 2012-2013 Measuring Fidelity of Implementation for Algebra I Drop-in Units: DEV11 (SLOPE); (6) Teacher-Level Participation in i3 SLOPE Evaluation (2011-2014); and (7) Findings from Evaluator Study of Implementation: Implementation Year 1.

Educators often use manipulatives when teaching mathematics because manipulatives are assumed to promote learning. However, research indicates that instructional variables impact the effectiveness of manipulatives. In this article, the authors consider the relations between two instructional characteristics: (a) level of instructional guidance and (b) perceptual qualities of manipulatives. Results from the randomized experiment with preschoolers (N = 72) suggest that learning is improved when instruction is conducted with high levels of instructional guidance and is impacted by the perceptual qualities of manipulatives. Perceptually rich manipulatives decreased learner performance on outcomes associated with conceptual knowledge and improved performance on transfer of learning. In addition, transfer was positively affected by perceptually rich manipulatives when low levels of instructional guidance were present.

Researchers and practitioners indicate students require explicit instruction on mathematics vocabulary terms, yet no study has examined the effects of an embedded vocabulary component within mathematics tutoring for early elementary students. First-grade students with mathematics difficulty (MD; n = 98) were randomly assigned to addition tutoring with an embedded vocabulary component, addition tutoring without the embedded vocabulary component, or business-as-usual control. At posttest, students who received addition tutoring without vocabulary demonstrated greater gains than control students on addition fluency. On a measure of mathematics vocabulary, students in the active tutoring conditions demonstrated improved performance on mathematics vocabulary over control students. Results indicate that exposure to addition tutoring with or without an embedded vocabulary component positively improves mathematics vocabulary performance.

Researchers and practitioners indicate students require explicit instruction on mathematics vocabulary terms, yet no study has examined the effects of an embedded vocabulary component within mathematics tutoring for early elementary students. First-grade students with mathematics difficulty (MD; n = 98) were randomly assigned to addition tutoring with an embedded vocabulary component, addition tutoring without the embedded vocabulary component, or business-as-usual control. At posttest, students who received addition tutoring without vocabulary demonstrated greater gains than control students on addition fluency. On a measure of mathematics vocabulary, students in the active tutoring conditions demonstrated improved performance on mathematics vocabulary over control students. Results indicate that exposure to addition tutoring with or without an embedded vocabulary component positively improves mathematics vocabulary performance.

Researchers and practitioners indicate students require explicit instruction on mathematics vocabulary terms, yet no study has examined the effects of an embedded vocabulary component within mathematics tutoring for early elementary students. First-grade students with mathematics difficulty (MD; n = 98) were randomly assigned to addition tutoring with an embedded vocabulary component, addition tutoring without the embedded vocabulary component, or business-as-usual control. At posttest, students who received addition tutoring without vocabulary demonstrated greater gains than control students on addition fluency. On a measure of mathematics vocabulary, students in the active tutoring conditions demonstrated improved performance on mathematics vocabulary over control students. Results indicate that exposure to addition tutoring with or without an embedded vocabulary component positively improves mathematics vocabulary performance.

"MyTeachingPartner--Math/Science" ("MTP-MS") is a system of two curricula (math and science) plus teacher supports designed to improve the quality of instructional interactions in pre-kindergarten classrooms and to scaffold children's development in mathematics and science. The program includes year-long curricula in these domains, and a teacher support system (web-based supports and in-person workshops) designed to foster high-quality curricular implementation. This study examined the impacts of the intervention on the development of mathematics and science skills of 444 children during pre-kindergarten, via school-level random assignment to two intervention conditions ("Basic: MTP-M/S" mathematics and science curricula, and "Plus: MTP-M/S" mathematics and science curricula plus related teacher support system) and a Business-as-Usual control condition ("BaU"). There were intervention effects for children's knowledge and skills in geometry and measurement as well as number sense and place value: Children in "Plus" classrooms made greater gains in geometry and measurement, compared with those in "BaU" classrooms. Children in "Plus" classrooms also performed better on the number sense and place value assessment than did those in "Basic" or "BaU" classrooms. We describe the implications of these results for supporting the development of children's knowledge and skills in early childhood and for developing and providing teachers with professional development to support these outcomes.

"MyTeachingPartner--Math/Science" ("MTP-MS") is a system of two curricula (math and science) plus teacher supports designed to improve the quality of instructional interactions in pre-kindergarten classrooms and to scaffold children's development in mathematics and science. The program includes year-long curricula in these domains, and a teacher support system (web-based supports and in-person workshops) designed to foster high-quality curricular implementation. This study examined the impacts of the intervention on the development of mathematics and science skills of 444 children during pre-kindergarten, via school-level random assignment to two intervention conditions ("Basic: MTP-M/S" mathematics and science curricula, and "Plus: MTP-M/S" mathematics and science curricula plus related teacher support system) and a Business-as-Usual control condition ("BaU"). There were intervention effects for children's knowledge and skills in geometry and measurement as well as number sense and place value: Children in "Plus" classrooms made greater gains in geometry and measurement, compared with those in "BaU" classrooms. Children in "Plus" classrooms also performed better on the number sense and place value assessment than did those in "Basic" or "BaU" classrooms. We describe the implications of these results for supporting the development of children's knowledge and skills in early childhood and for developing and providing teachers with professional development to support these outcomes.

This addendum to previously published results presents alternative analyses of data from large-scale effectiveness studies of Cognitive Tutor Algebra I in middle schools and high schools. These alternative analyses produce results that are substantively the same as previously reported. We find a significant positive effect of 0.21 standard deviation units for high school students in the second year of the study. An appendix containing additional tables is included. [See the study: "Effectiveness of Cognitive Tutor Algebra I at Scale," "Educational Evaluation and Policy Analysis," v36 n2 p127-144 Jun 2014 at EJ1024233.]

This article examines the effectiveness of a technology-based algebra curriculum in a wide variety of middle schools and high schools in seven states. Participating schools were matched into similar pairs and randomly assigned to either continue with the current algebra curriculum for 2 years or to adopt Cognitive Tutor Algebra I (CTAI), which uses a personalized, mastery-learning, blended-learning approach. Schools assigned to implement CTAI did so under conditions similar to schools that independently adopt it. Analysis of posttest outcomes on an algebra proficiency exam finds no effects in the first year of implementation, but finds evidence in support of positive effects in the second year. The estimated effect is statistically significant for high schools but not for middle schools; in both cases, the magnitude is sufficient to improve the median student's performance by approximately eight percentile points.

This study investigated the role of strategy instruction on solution accuracy in children with and without serious math difficulties (MD) in problem solving. Children's posttest solution accuracy was compared on standardized and experimental measures as a function of strategy conditions. Strategy conditions included curriculum materials that gradually increased the number of irrelevant propositions within word problems. Children in Grade 3 (N = 193) were randomly assigned to one of five conditions: materials + verbal strategies (e.g., underlining the question), materials + verbal + visual strategies, materials + visual strategies (e.g., correctly placing numbers in diagrams), materials only-no overt strategies, and an untreated control. Compared to children with MD in the control condition, posttest outcomes for children with MD on standardized measures improved significantly under verbal + visual conditions, whereas posttest scores on the experimental problem-solving measures improved under the materials-only condition. Those strategy conditions found least effective made substantial demands on children's working memory capacity. The authors discuss benefits and limitations of strategy instruction.

The authors' purpose was to determine the effects of computer-based practice and conceptual interventions on computational fluency and word-problem solving of fourth- and fifth-grade students with mathematics difficulties. A randomized pretest-posttest control group design found that students assigned to the computer-based practice intervention group outperformed students in the comparison group on the retention measure. Students assigned to the conceptual intervention did not outperform the comparison group on any of the outcome variables. Implications for instruction and interventions are discussed.

The authors' purpose was to determine the effects of computer-based practice and conceptual interventions on computational fluency and word-problem solving of fourth- and fifth-grade students with mathematics difficulties. A randomized pretest-posttest control group design found that students assigned to the computer-based practice intervention group outperformed students in the comparison group on the retention measure. Students assigned to the conceptual intervention did not outperform the comparison group on any of the outcome variables. Implications for instruction and interventions are discussed.

The authors' purpose was to determine the effects of computer-based practice and conceptual interventions on computational fluency and word-problem solving of fourth- and fifth-grade students with mathematics difficulties. A randomized pretest-posttest control group design found that students assigned to the computer-based practice intervention group outperformed students in the comparison group on the retention measure. Students assigned to the conceptual intervention did not outperform the comparison group on any of the outcome variables. Implications for instruction and interventions are discussed.

Interim assessments are increasingly common in U.S. schools. We use high-quality data from a large-scale school-level cluster randomized experiment to examine the impact of two well-known commercial interim assessment programs on mathematics and reading achievement in Indiana. Results indicate that the treatment effects are positive but not consistently significant. The treatment effects are smaller in lower grades (i.e., kindergarten to second grade) and larger in upper grades (i.e., third to eighth grade). Significant treatment effects are detected in Grades 3 to 8, especially in third- and fourth-grade reading and in fifth- and sixth-grade mathematics.

In a series of two in vivo experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly assigned to complete their unit on solving two-step linear equations with the traditional Tutor program (control) or one of three versions which incorporated examples; results indicate that explaining worked examples during guided practice leads to improved conceptual understanding compared with guided practice alone. In Experiment 2, a more comprehensive battery of conceptual and procedural tests was used to determine which type of example is most beneficial for improving different facets of student learning. Results suggest that incorrect examples, either alone or in combination with correct examples, may be especially beneficial for fostering conceptual understanding. (Contains 3 tables, 4 figures, and 1 footnote.) [A version of this paper was published in "Learning and Instruction," v25 p24-34 Jun 2013.]

This study highlights the connections between two facets of teachers' skills--those supporting teachers' mathematical instructional interactions and those underlying social interactions within the classroom. The impact of the Responsive Classroom (RC) approach and use of RC practices on the use of standards-based mathematics teaching practices was investigated in third-grade classrooms. Eighty-eight third-grade teachers from 24 elementary schools in a large suburban district were selected from a sample of teachers participating in a larger randomized-control study. Results showed that teachers at schools assigned randomly to receive training in the RC approach showed higher use of standards-based mathematics teaching practices than teachers in control schools. These findings were supported by analyses using fidelity of implementation: greater adherence to the intervention predicted the use of more standards-based mathematics teaching practices. Findings support the use of the RC approach for creating classroom social environments that facilitate standards-based mathematical practices. (Contains 3 tables and 1 note.)

Early number sense is a strong predictor of later success in school mathematics. A disproportionate number of children from low-income families come to first grade with weak number competencies, leaving them at risk for a cycle of failure. The present study examined the effects of an 8-week number sense intervention to develop number competencies of low-income kindergartners ("N" = 121). The intervention purposefully targeted whole number concepts related to counting, comparing, and manipulating sets. Children were randomly assigned to either a number sense intervention or a business as usual contrast group. The intervention was carried out in small-group, 30-min sessions, 3 days per week, for a total of 24 sessions. Controlling for number sense at pretest, the intervention group made meaningful gains relative to the control group at immediate as well delayed posttest on a measure of early numeracy. Intervention children also performed better than controls on a standardized test of mathematics calculation at immediate posttest. (Contains 5 tables, 2 figures, and 1 note.)

In this article, we present the results from a longitudinal examination of the impact of a "Standards"-based or reform mathematics curriculum (called CMP) and traditional mathematics curricula (called non-CMP) on students' learning of algebra using various outcome measures. Findings include the following: (1) students did not sacrifice basic mathematical skills if they are taught using a "Standards"-based or reform mathematics curriculum like CMP; (2) African American students experienced greater gain in symbol manipulation when they used a traditional curriculum; (3) the use of either the CMP or a non-CMP curriculum improved the mathematics achievement of all students, including students of color; (4) the use of CMP contributed to significantly higher problem-solving growth for all ethnic groups; and (5) a high level of conceptual emphasis in a classroom improved the students' ability to represent problem situations. (However, the level of conceptual emphasis bears no relation to students' problem solving or symbol manipulation skills.) (Contains 12 tables and 3 figures.)

This is the second and final report of the Middle School Mathematics Professional Development Impact Study, which examines the impact of providing a professional development (PD) program in rational number topics to seventh-grade mathematics teachers. An interim report (Garet et al. 2010) described the findings after one year of PD. The current report documents the impact after providing a second year of PD in a subset of the original participating districts and includes supplemental analyses that use data from both years of the study. The study produced the following core second-year results: (1) The study's PD program was implemented as intended, but teacher turnover limited the average dosage received; (2) At the end of the second year of implementation, the PD program did not have a statistically significant impact on teacher knowledge; and (3) At the end of the second year of implementation, the PD program did not have a statistically significant impact on average student achievement in rational numbers. Appended are: (1) Details of the Study Samples; (2) Details of Data Collection and Analytical Approaches; (3) Supplemental Information on the Design and Implementation of the PD Program; (4) Supporting Tables and Figures for Impact Analyses; and (5) Exploratory Analyses: Approaches and Additional Results. (Contains 6 exhibits, 6 figures, 81 tables and 124 footnotes.) [For "Middle School Mathematics Professional Development Impact Study: Findings after the Second Year of Implementation. Executive Summary. NCEE 2011-4025," see ED519923. For "Middle School Mathematics Professional Development Impact Study: Findings After the First Year of Implementation. NCEE 2010-4009," see ED509306.]

This study was conducted by the Central Region Educational Laboratory (REL Central) administered by Mid-continent Research for Education and Learning to provide educators and policymakers with rigorous evidence about the potential of Classroom Assessment for Student Learning (CASL) to improve student achievement. CASL is a widely used professional development program in classroom and formative assessment published by the Assessment Training Institute of Pearson Education. Schools were recruited from across Colorado to participate in the study. Colorado was chosen as the target state primarily because it has one of the largest populations in the Central Region from which to recruit schools and because its statewide achievement test is vertically scaled. This cluster randomized trial of the CASL professional development program had sufficient statistical power to detect an impact of at least 0.25 standard deviation on student achievement. An intent-to-treat analysis was conducted to estimate the impact of CASL on student achievement; all schools were included in the analysis and were analyzed as randomized regardless of the level of implementation fidelity. Analysis did not reveal a statistically significant impact of CASL on the school-level average mathematics achievement of grade 4 and grade 5 students. Results from sensitivity analyses revealed that the impact estimates on student achievement were robust to decisions regarding the inclusion of covariates, estimation method, and the treatment of missing data. In other words, design and analysis decisions made by the research team did not change whether the impact results would have been statistically significant. Appendices include: (1) Power analysis; (2) Response rates by data collection wave, instrument, and experimental group; (3) Data collection instruments; (4) Development, reliability, and validity of teacher outcomes; (5) Teacher Assessment Work Sample; (6) Impact analysis models; (7) Calculation of effect sizes; (8) Treatment of missing data; (9) Variance components estimates and intraclass correlations; (10) Raw means and standard deviations; and (11) Complete mixed model results. (Contains 1 box, 4 figures and 56 tables.

Analyzing mathematics and reading achievement outcomes from a district-level random assignment study fielded in over 500 schools within 59 school districts and seven states, the authors estimate the 1-year impacts of a data-driven reform initiative implemented by the Johns Hopkins Center for Data-Driven Reform in Education (CDDRE). CDDRE consultants work with districts to implement quarterly student benchmark assessments and provide district and school leaders with extensive training on interpreting and using the data to guide reform. Relative to a control condition, in which districts operated as usual without CDDRE services, the data-driven reform initiative caused statistically significant districtwide improvements in student mathematics achievement. The CDDRE intervention also had a positive effect on reading achievement, but the estimates fell short of conventional levels of statistical significance. (Contains 1 figure, 3 tables, and 16 notes.)

This study examined the efficacy of a core kindergarten mathematics program, ELM, a 120-lesson comprehensive curriculum providing instruction in (a) number operations, (b) geometry, (c) measurement, and (d) vocabulary. ELM is designed to address the learning needs of all students, including at-risk students in the general education or Tier I classroom setting. The study utilized a randomized block design, with 64 classrooms randomly assigned within schools to treatment (ELM) or control (standard district practices) conditions. Measures of achievement were collected at pretest and posttest to measure student achievement. Students did not differ on mathematics assessments at pretest. Gain scores of at-risk treatment students were significantly greater than control peers, and the gains of at-risk treatment students were greater than the gains of peers not at risk, effectively reducing the achievement gap. Implications for Tier I instruction in a Response to Intervention (RTI) model are discussed. (Contains 4 tables.)

This article reports on the development and field trial of an integrated Head Start curriculum (Evidence-Based Program for Integrated Curricula [EPIC]) that focuses on comprehensive mathematics, language, and literacy skills. Seventy Head Start classrooms (N = 1,415 children) were randomly assigned to one of two curriculum programs: EPIC or the Developmental Learning Materials Early Childhood Express, with curricula implemented as stand-alone programs. EPIC included instruction in mathematics, language, literacy, and approaches to learning skills; formative assessment; and a learning community for teachers. Multilevel growth modeling through four direct assessments revealed significant main effects and growth rates in mathematics and listening comprehension favoring EPIC, controlling for demographics and special needs and language status. Both programs produced significant growth rates in literacy. (Contains 2 figures, 2 tables, and 2 notes.)

This randomized, controlled field trial estimated the causal impact of a technology-based geometry curriculum on students' geometry achievement, as well as their attitudes toward mathematics and technology. The curriculum combines learner-centered classroom pedagogy with individualized, computer-based student instruction. Conducted over a 3-year period in eight high schools within an urban fringe district, the study found that students assigned to the treatment curriculum scored 19% of a standard deviation lower on the geometry posttest than their counterparts assigned to the district's standard curriculum, but found no statistically significant impact on students' attitudes toward mathematics and technology. Researchers also collected observation and interview data on teachers' instructional practices. These data suggest that many teachers had difficulty implementing the treatment curriculum's learner-centered pedagogy. In fact, observed levels of learner-centered practices were only modestly higher in treatment classes than in control classes. In both treatment and control classes, however, higher levels of learner-centered pedagogy were associated with higher student achievement in geometry. (Contains 4 figures, 10 footnotes, and 5 tables.)

The primary purpose of this study was to assess the effects of strategic counting instruction, with and without deliberate practice with those counting strategies, on number combination (NC) skill among students with mathematics difficulties (MD). Students (n = 150) were stratified on MD status (i.e., MD alone versus MD with reading difficulty) and site (proximal versus distal to the intervention developer) and then randomly assigned to control (no tutoring) or 1 of 2 variants of NC remediation. Both remediations were embedded in the same validated word-problem tutoring protocol (i.e., Pirate Math). In 1 variant, the focus on NCs was limited to a single lesson that taught strategic counting. In the other variant, 4-6 min of practice per session was added to the other variant. Tutoring occurred for 16 weeks, 3 sessions per week for 20-30 min per session. Strategic counting without deliberate practice produced superior NC fluency compared to control; however, strategic counting with deliberate practice effected superior NC fluency and transfer to procedural calculations compared with both competing conditions. Also, the efficacy of Pirate Math word-problem tutoring was replicated. (Contains 6 tables.)

In the No Child Left Behind Act (NCLB), Congress called for the U.S. Department of Education (ED) to conduct a rigorous study of the conditions and practices under which educational technology is effective in increasing student academic achievement. A 2007 report presenting study findings for the 2004-2005 school year, indicated that, after one school year, differences in student test scores were not statistically significant between classrooms that were randomly assigned to use software products and those that were randomly assigned not to use products. School and teacher characteristics generally were not related to whether products were effective. The second year of the study examined whether an additional year of teaching experience using the software products increased the estimated effects of software products on student test scores. The evidence for this hypothesis is mixed. For reading, there were no statistically significant differences between the effects that products had on standardized student test scores in the first year and the second year. For sixth grade math, product effects on student test scores were statistically significantly lower (more negative) in the second year than in the first year, and for algebra I, effects on student test scores were statistically significantly higher in the second year than in the first year. The study also tested whether using any of the 10 software products increased student test scores. One product had a positive and statistically significant effect. Nine did not have statistically significant effects on test scores. Five of the insignificant effects were negative and four were positive. Study findings should be interpreted in the context of design and objectives. The study examined a range of reading and math software products in a range of diverse school districts and schools. But it did not study many forms of educational technology and it did not include many types of software products. How much information the findings provide about the effectiveness of products that are not in the study is an open question. Products in the study also were implemented in a specific set of districts and schools, and other districts and schools may have different experiences with the products. The findings should be viewed as one element within a larger set of research studies that have explored the effectiveness of software products. Three appendixes are included: (1) Second-Year Data Collection and Response Rates; (2) Description of Sample for the 10 Products; and (3) Details of Estimation Methods. (Contains 29 footnotes, 4 figures and 24 tables.

In the No Child Left Behind Act (NCLB), Congress called for the U.S. Department of Education (ED) to conduct a rigorous study of the conditions and practices under which educational technology is effective in increasing student academic achievement. A 2007 report presenting study findings for the 2004-2005 school year, indicated that, after one school year, differences in student test scores were not statistically significant between classrooms that were randomly assigned to use software products and those that were randomly assigned not to use products. School and teacher characteristics generally were not related to whether products were effective. The second year of the study examined whether an additional year of teaching experience using the software products increased the estimated effects of software products on student test scores. The evidence for this hypothesis is mixed. For reading, there were no statistically significant differences between the effects that products had on standardized student test scores in the first year and the second year. For sixth grade math, product effects on student test scores were statistically significantly lower (more negative) in the second year than in the first year, and for algebra I, effects on student test scores were statistically significantly higher in the second year than in the first year. The study also tested whether using any of the 10 software products increased student test scores. One product had a positive and statistically significant effect. Nine did not have statistically significant effects on test scores. Five of the insignificant effects were negative and four were positive. Study findings should be interpreted in the context of design and objectives. The study examined a range of reading and math software products in a range of diverse school districts and schools. But it did not study many forms of educational technology and it did not include many types of software products. How much information the findings provide about the effectiveness of products that are not in the study is an open question. Products in the study also were implemented in a specific set of districts and schools, and other districts and schools may have different experiences with the products. The findings should be viewed as one element within a larger set of research studies that have explored the effectiveness of software products. Three appendixes are included: (1) Second-Year Data Collection and Response Rates; (2) Description of Sample for the 10 Products; and (3) Details of Estimation Methods. (Contains 29 footnotes, 4 figures and 24 tables.

In the No Child Left Behind Act (NCLB), Congress called for the U.S. Department of Education (ED) to conduct a rigorous study of the conditions and practices under which educational technology is effective in increasing student academic achievement. A 2007 report presenting study findings for the 2004-2005 school year, indicated that, after one school year, differences in student test scores were not statistically significant between classrooms that were randomly assigned to use software products and those that were randomly assigned not to use products. School and teacher characteristics generally were not related to whether products were effective. The second year of the study examined whether an additional year of teaching experience using the software products increased the estimated effects of software products on student test scores. The evidence for this hypothesis is mixed. For reading, there were no statistically significant differences between the effects that products had on standardized student test scores in the first year and the second year. For sixth grade math, product effects on student test scores were statistically significantly lower (more negative) in the second year than in the first year, and for algebra I, effects on student test scores were statistically significantly higher in the second year than in the first year. The study also tested whether using any of the 10 software products increased student test scores. One product had a positive and statistically significant effect. Nine did not have statistically significant effects on test scores. Five of the insignificant effects were negative and four were positive. Study findings should be interpreted in the context of design and objectives. The study examined a range of reading and math software products in a range of diverse school districts and schools. But it did not study many forms of educational technology and it did not include many types of software products. How much information the findings provide about the effectiveness of products that are not in the study is an open question. Products in the study also were implemented in a specific set of districts and schools, and other districts and schools may have different experiences with the products. The findings should be viewed as one element within a larger set of research studies that have explored the effectiveness of software products. Three appendixes are included: (1) Second-Year Data Collection and Response Rates; (2) Description of Sample for the 10 Products; and (3) Details of Estimation Methods. (Contains 29 footnotes, 4 figures and 24 tables.

In the No Child Left Behind Act (NCLB), Congress called for the U.S. Department of Education (ED) to conduct a rigorous study of the conditions and practices under which educational technology is effective in increasing student academic achievement. A 2007 report presenting study findings for the 2004-2005 school year, indicated that, after one school year, differences in student test scores were not statistically significant between classrooms that were randomly assigned to use software products and those that were randomly assigned not to use products. School and teacher characteristics generally were not related to whether products were effective. The second year of the study examined whether an additional year of teaching experience using the software products increased the estimated effects of software products on student test scores. The evidence for this hypothesis is mixed. For reading, there were no statistically significant differences between the effects that products had on standardized student test scores in the first year and the second year. For sixth grade math, product effects on student test scores were statistically significantly lower (more negative) in the second year than in the first year, and for algebra I, effects on student test scores were statistically significantly higher in the second year than in the first year. The study also tested whether using any of the 10 software products increased student test scores. One product had a positive and statistically significant effect. Nine did not have statistically significant effects on test scores. Five of the insignificant effects were negative and four were positive. Study findings should be interpreted in the context of design and objectives. The study examined a range of reading and math software products in a range of diverse school districts and schools. But it did not study many forms of educational technology and it did not include many types of software products. How much information the findings provide about the effectiveness of products that are not in the study is an open question. Products in the study also were implemented in a specific set of districts and schools, and other districts and schools may have different experiences with the products. The findings should be viewed as one element within a larger set of research studies that have explored the effectiveness of software products. Three appendixes are included: (1) Second-Year Data Collection and Response Rates; (2) Description of Sample for the 10 Products; and (3) Details of Estimation Methods. (Contains 29 footnotes, 4 figures and 24 tables.

With regard to transfer, is it better to provide pupils with ready-made representations or is it more effective to scaffold pupils' thinking in the process of generating their own representations with the help of peers and under the guidance of a teacher in a process of guided co-construction? The sample comprises 10 classes and 239 Grade 5 primary school students, age 10-11 years. A pretest-posttest control group research design was used. In the experimental condition, pupils were taught to construct representations collaboratively as a tool in the learning of percentages and graphs. Children in the experimental condition outperformed control children on the posttest and transfer test. Both high- and low-achieving pupils profited from the intervention. This study shows that children who learn to design are in a better position to understand pictures, graphs, and models. They are more successful in solving new, complex mathematical problems. (Contains 4 figures and 3 tables.)

Comparing and contrasting examples is a core cognitive process that supports learning in children and adults across a variety of topics. In this experimental study, we evaluated the benefits of supporting comparison in a classroom context for children learning about computational estimation. Fifth- and sixth-grade students (N = 157) learned about estimation either by comparing alternative solution strategies or by reflecting on the strategies one at a time. At posttest and retention test, students who compared were more flexible problem solvers on a variety of measures. Comparison also supported greater conceptual knowledge, but only for students who already knew some estimation strategies. These findings indicate that comparison is an effective learning and instructional practice in a domain with multiple acceptable answers. (Contains 2 figures and 6 tables.)

Background: Knowledge of concepts and procedures seems to develop in an iterative fashion, with increases in one type of knowledge leading to increases in the other type of knowledge. This suggests that iterating between lessons on concepts and procedures may improve learning. Aims: The purpose of the current study was to evaluate the instructional benefits of an iterative lesson sequence compared to a concepts-before-procedures sequence for students learning decimal place-value concepts and arithmetic procedures. Samples: In two classroom experiments, sixth-grade students from two schools participated (N = 77 and 26). Method: Students completed six decimal lessons on an intelligent-tutoring systems. In the iterative condition, lessons cycled between concept and procedure lessons. In the concepts-first condition, all concept lessons were presented before introducing the procedure lessons. Results: In both experiments, students in the iterative condition gained more knowledge of arithmetic procedures, including ability to transfer the procedures to problems with novel features. Knowledge of concepts was fairly comparable across conditions. Finally, pre-test knowledge of one type predicted gains in knowledge of the other type across experiments. Conclusions: An iterative sequencing of lessons seems to facilitate learning and transfer, particularly of mathematical procedures. The findings support an iterative perspective for the development of knowledge of concepts and procedures.

This study examined whether the quality of first graders' (mean age = 7.2 years) numerical magnitude representations is correlated with, predictive of, and causally related to their arithmetic learning. The children's pretest numerical magnitude representations were found to be correlated with their pretest arithmetic knowledge and to be predictive of their learning of answers to unfamiliar arithmetic problems. The relation to learning of unfamiliar problems remained after controlling for prior arithmetic knowledge, short-term memory for numbers, and math achievement test scores. Moreover, presenting randomly chosen children with accurate visual representations of the magnitudes of addends and sums improved their learning of the answers to the problems. Thus, representations of numerical magnitude are both correlationally and causally related to arithmetic learning.

This study used a randomized field trial design to evaluate the efficacy of a research-based model for scaling up an intervention focused on preschool mathematics. Although the successes of research-based educational practices have been documented, equally well known is the paucity of successful efforts to bring these practices to scale. The same research corpus provides guidelines to scale up successful interventions. We designed an intervention model based on that research, including mathematics curricula with an emphasis on teaching for understanding following developmental guidelines, or learning trajectories, and using technology at multiple levels. We then implemented that model and evaluated the implementation with a limited scale up study. Within a design involving 25 classrooms serving children at risk for later school failure, we examined the impact of the model, using measures of fidelity of implementation, classroom observations of mathematics environment and teaching, and child outcomes. High levels of fidelity of implementation resulted in consistently higher scores in the intervention, compared to control, classes on the observation instrument and significantly and substantially greater gains in children's mathematics achievement in the intervention, compared to the control, children (effect size = 0.62). (Contains 5 tables.)

The major purposes of this study were to assess the efficacy of tutoring to remediate 3rd-grade computational deficits and to explore whether remediation is differentially efficacious depending on whether students experience mathematics difficulty alone or concomitantly with reading difficulty. At 2 sites, 127 students were stratified on mathematics difficulty status and randomly assigned to 4 conditions: word recognition (control) tutoring or 1 of 3 computation tutoring conditions: fact retrieval, procedural computation and computational estimation, and combined (fact retrieval + procedural computation and computational estimation). Results revealed that fact retrieval tutoring enhanced fact retrieval skill, and procedural computation and computational estimation tutoring (whether in isolation or combined with fact retrieval tutoring) enhanced computational estimation skill. Remediation was not differentially efficacious as a function of students' mathematics difficulty status. (Contains 4 tables and 1 footnote.)

The major purposes of this study were to assess the efficacy of tutoring to remediate 3rd-grade computational deficits and to explore whether remediation is differentially efficacious depending on whether students experience mathematics difficulty alone or concomitantly with reading difficulty. At 2 sites, 127 students were stratified on mathematics difficulty status and randomly assigned to 4 conditions: word recognition (control) tutoring or 1 of 3 computation tutoring conditions: fact retrieval, procedural computation and computational estimation, and combined (fact retrieval + procedural computation and computational estimation). Results revealed that fact retrieval tutoring enhanced fact retrieval skill, and procedural computation and computational estimation tutoring (whether in isolation or combined with fact retrieval tutoring) enhanced computational estimation skill. Remediation was not differentially efficacious as a function of students' mathematics difficulty status. (Contains 4 tables and 1 footnote.)

Under the "Math Science Partnership Grant," the Maui Hawaii Educational Consortium sought scientifically based evidence for the effectiveness of Carnegie Learning's "Cognitive Tutor[R]" (CT) program as part of the adoption process for pre-Algebra program. During the 2006-2007 school year, the researchers conducted a follow-on study to a previous randomized experiment in the Maui School District of the effectiveness of "CT" in Algebra I. In this second year, the focus was on the newly developed "Bridge to Algebra" program for pre-Algebra. The question being addressed specifically by the research is whether students in classes that use "CT" materials achieve higher scores on the standardized math assessment, as measured by the Northwest Evaluation Association (NWEA) General Math Test, than they would if they had been in a control classroom using the pre-Algebra curricula the Maui schools currently have in place. The researchers found that most students in both "CT" and control groups improved overall on the NWEA General Math Test. They did not find a difference in student performance in math between groups. Their analysis of the Algebraic Operations sub-strand revealed that many students in both groups did not demonstrate the growth in this scale, again with no discernible group differences. However, for Algebraic Operations outcomes, the researchers found a significant interaction between the pre-test and "CT": student scoring low before participating in "CT" got more benefit from the program's algebraic operations instruction than students with high initial scores. (Contains 8 figures, 33 tables, and 14 footnotes.) [For "Comparative Effectiveness of Carnegie Learning's "Cognitive Tutor Bridge to Algebra" Curriculum: A Report of a Randomized Experiment in the Maui School District. Research Summary," see ED538962.]

Because a significant portion of U.S. students lacks critical mathematic skills, schools across the country are investing heavily in computerized curriculums as a way to enhance education output, even though there is surprisingly little evidence that they actually improve student achievement. In this paper we present results from a randomized study in three urban school districts of a well-defined use of computers in schools: a popular instructional computer program which is designed to teach pre-algebra and algebra. We assess the impact of the program using statewide tests that cover a range of math skills and tests designed specifically to target pre-algebra and algebra skills. We find that students randomly assigned to computer-aided instruction score at least 0.17 of a standard deviation higher on a pre-algebra/algebra test than students randomly assigned to traditional instruction. We hypothesize that the effectiveness arises from increased individualized instruction as the effects appear larger for students in larger classes and those in classes in which students are frequently absent. (Detailed data information is appended. Contains 40 footnotes and 17 tables.)

This study evaluated the effects of a personalized computer-assisted mathematics problem-solving program on the performance and attitude of Taiwanese fourth grade students. The purpose of this study was to determine whether the personalized computer-assisted program improved student performance and attitude over the nonpersonalized program. One-hundred-sixty-five (165) Taiwanese fourth-grade students participated in the study. The research used the results to determine if the computer-assisted program was effective between the two groups. The results of the study showed that the personalized computer-assisted program on mathematics improved student performance and attitude. The achievement of students in the personalized group was significantly higher than those in the nonpersonalized group. The posttest score of the personalized group was significantly higher than the posttest score of the nonpersonalized group. The attitude of the personalized group was significantly more positive than that of the nonpersonalized group. (Contains 2 figures and 5 tables.)

A yearlong experimental study showed positive effects of a professional development project that involved 19 urban elementary schools, 180 teachers, and 3735 students from one of the lowest performing school districts in California. Algebraic reasoning as generalized arithmetic and the study of relations was used as the centerpiece for work with teachers in Grades 1-5. Participating teachers generated a wider variety of student strategies, including more strategies that reflected the use of relational thinking, than did nonparticipating teachers. Students in participating classes showed significantly better understanding of the equal sign and used significantly more strategies reflecting relational thinking during interviews than did students in classes of nonparticipating teachers. (Contains 8 tables and 9 footnotes.)

Students who have particular difficulty in mathematics are a growing concern for educators. Graphic organizers have been shown to improve reading comprehension and may be applied to upper level secondary mathematics content. In two systematic replications, one randomly assigned group was taught to solve systems of linear equations through direct instruction and strategy instruction. The other group was taught with the same methods with the addition of a graphic organizer. Students who received instruction with the graphic organizers outperformed those who received instruction without the organizers. They also better understood the related concepts as measured by immediate posttests in both replications. The difference in understanding concepts was maintained on a 2-3 week posttest.

This study evaluated the efficacy of a preschool mathematics program based on a comprehensive model of developing research-based software and print curricula. Building Blocks, funded by the National Science Foundation, is a curriculum development project focused on creating research-based, technology-enhanced mathematics materials for pre-K through grade 2. In this article, we describe the underlying principles, development, and initial summative evaluation of the first set of resulting materials as they were used in classrooms with children at risk for later school failure. Experimental and comparison classrooms included two principal types of public preschool programs serving low-income families: state funded and Head Start prekindergarten programs. The experimental treatment group score increased significantly more than the comparison group score; achievement gains of the experimental group approached the sought-after 2-sigma effect of individual tutoring. This study contributes to research showing that focused early mathematical interventions help young children develop a foundation of informal mathematics knowledge, especially for children at risk for later school failure.

The importance of algebra instruction has increased in the United States in the past few years. Thus, in most states, middle school students are required to take Algebra 1. Middle school students with math difficulties in inclusion algebra settings may require a different instructional approach. The purpose of this research was to compare student achievement in solving linear algebraic functions across two procedural approaches: a multisensory algebra model using a concrete-to-representational-to-abstract sequence of instruction (CRA) and a repeated abstract explicit instruction model. Out of 231 students who participated, the students who learned through the CRA model scored significantly higher on the post- and follow-up test. The success of the CRA model was consistent for students with a history of low, medium, and high math achievement. Implications of this model and possibilities for future research are discussed. (Contains 3 tables and 2 figures.)

While mathematics instruction for very young children needs to be age-appropriate in format and content, it also needs to prepare children conceptually for the kinds of mathematics learning that will be expected of them in future years. This perspective, informed by the work of Russian psychologists and educators on a measurement-based approach to early mathematics instruction [e.g., V.V. Davydov, Children's Capacity for Learning Mathematics, National Council of Teachers of Mathematics, Reston, VA, pp. 109-205], was the basis for an experimental mathematics curriculum which focused on the concept of unit as it applies to enumeration, measurement, and the identification of relations among geometric shapes. The curriculum particularly emphasized two ideas about units that derive from a measurement perspective: first, that the numerical result we obtain from counting or other measurement operations will depend on our choice of a unit; and second, that units of one kind can be combined to form higher-order units or taken apart to form lower-order ones. The curriculum included a weekly project activity conducted by the Head Start teachers, suggestions for supplementary activities, and a weekly home activity for a parent or other family member to carry out with the child. It was implemented with children in three Head Start centers (N=46; age range 2 years, 9 months-4 years, 7 months at the beginning of the program). To evaluate the curriculum, two assessment instruments (the mathematics subscale of the DSC and a supplementary instrument constructed especially for this study) were administered, at the beginning and again at the end of the school year, to these children and to two comparison groups. One comparison group (N=48; age range 2 years, 6 months-4 years, 7 months) received a literacy intervention rather than a mathematics one; the other (N=29; age range 2 years, 8 months-4 years, 7 months) did not receive any experimental intervention. Results showed significant, albeit modest, positive effects of the intervention. The importance of reexamining current beliefs about what is possible--and desirable--within a preschool mathematics curriculum is emphasized.

Sixth- and seventh-grade students (n=68) with learning disabilities in mathematics received either concrete-to-representational-to-abstract (CRA) or traditional instruction in algebraic transformation equations. Students receiving the CRA instruction outperformed peers receiving traditional instruction on both post-instruction and follow-up tests and performed fewer procedural errors when solving for variables. (Contains references.) (Author/DB)

This study compared effectiveness of either a concrete-representational-abstract (CRA) or a representational-abstract (RA) instructional sequence in teaching fraction concepts to 50 middle school students with mathematics disabilities. On all achievement measures, students in the CRA group had overall higher mean scores than did students in the RA group. (Contains references.) (Author/DB)

Investigates the effects of personalized instruction on achievement and attitudes of Taiwanese elementary students on two-step mathematics and word problems. Students initially completed a survey on which they choice their favorite foods, sports, stores, classmates, and other selections. The most popular items were then used to create personalized math word problems for the pretest, personalized instructional program, and posttest. (Author/AEF)

Presents evidence of the influence of professional development and curriculum on upper elementary students' understanding of fractions. Subjects included three groups of teachers and their students. Indicates that the benefits of reform curriculum for students may depend upon integrated professional development. (Contains 60 references.) (Author/ASK)

Proposes that conceptual and procedural knowledge develop in an iterative fashion and improved problem representation is one mechanism underlying the relations between them. Two experiments were conducted with 5th and 6th grade students learning about decimal fractions. Results indicate conceptual and procedural knowledge do develop, iteratively, with improved problem representation as part of the process. (BF)

Investigates the effects of group personalization of instruction on the mathematics achievement and attitudes of 72 fifth-grade Taiwanese students. Personalization was accomplished by incorporating personal information and preferences provided by students into their mathematics word problems. Students were blocked by ability level, then randomly assigned to a personalized or nonpersonalized version of an instructional program. (Contains 30 references.) (Author/AEF)

Reports a study that investigated the effects of metacognitive instruction in mathematics on low achieving, predominantly Hispanic, elementary school students (grades 3-8). Students were randomly assigned to either experimental (metacognitive) or traditional instruction. Results on learning outcomes showed significant effects favoring students who received metacognitive instruction, independent of grade level. (SM)

Examines the effects of classwide peer-assisted learning strategies (PALS) in mathematics incorporating the use of curriculum-based measurement on acquisition and transfer learning of three types of students (average-achieving student, low-achieving students, and students with a learning disability). Analysis of variance indicates superior mathematics growth for students in PALS condition. (Author/JDM)

In 2 experiments with 74 high school students, subjects studied worked examples while learning how to translate English expressions into algebraic equations. Those using worked examples outperformed the control group on posttests, made fewer errors, completed work more rapidly, and required less assistance from the teacher. (SLD)

Twenty primary-aged students with learning disabilities were tutored by nondisabled students in grades 3-6. Tutored students' performance on single-digit addition facts improved compared to a no-treatment control group. There were no significant differences between two tutoring procedures: a counting-on approach and a rote-memorization approach. (Author/JDD)

Twenty primary-aged students with learning disabilities were tutored by nondisabled students in grades 3-6. Tutored students' performance on single-digit addition facts improved compared to a no-treatment control group. There were no significant differences between two tutoring procedures: a counting-on approach and a rote-memorization approach. (Author/JDD)

The effects of on-going curriculum-based measurement of student growth and expert system instructional consultation on teacher planning and student achievement in mathematics operations were studied for 33 elementary teachers assigned to one control or 2 experimental groups. Each teacher worked with two learning disabled or emotionally disturbed students. (SLD)

Describes a study of the effect of teachers' verbal feedback on the ability of bilingual sixth graders to solve mathematical word problems. Treatment was based on Mayer's model of problem solving. Results indicated that theory-based feedback targeted to bilingual students' individual and class needs improved student mathematics performance. (GH)

This study compared the effectiveness of a diagrammatic and a keyword method of teaching mathematics problem solving to 70 learning-disabled junior high school students. Results indicated no significant difference between problem-solving performance by students taught with different methods. Potential factors explaining inconsistencies between this and previous investigations are discussed. (Author/PB)

Twenty first-grade teachers' use of knowledge from research about mathematical thinking of children was compared with that of 20 teachers who did not participate in an analysis of research. Implications for teacher education and practice are discussed. (SLD)