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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 five recommendations for improving students’ mathematical problem solving in grades 4 through 8. This guide is geared toward teachers, math coaches, other educators, and curriculum developers who want to improve the mathematical problem solving of students.

This practice guide provides five recommendations for teaching math to children in preschool, prekindergarten, and kindergarten. 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 build a strong foundation for later math learning.

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

The objective of this guide is to provide teachers with specific recommendations that can be carried out in the classroom without requiring systemic change.

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.

Great Explorations in Math and Science^® (GEMS^®) Space Science Sequence is an instructional curriculum for grades 3–5 that covers fundamental concepts, including planetary sizes and distance, the earth’s shape and movement, gravity, and moon phases and eclipses. Part of the GEMS^® core curriculum, GEMS^® Space Science Sequence uses the solar system as the focal point for learning. The sequence uses models, hands-on investigations, peer-to-peer discussions, reflection, and informational student readings. Students complete four units, each lasting between four and nine sessions. Each unit builds upon knowledge from previous units and can be used independently or in conjunction with one another for an overall learning progression.

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.

Building Blocks for Math is a supplemental mathematics curriculum designed to develop preschool children’s early mathematical knowledge through various individual and small- and large-group activities. It uses Building Blocks for Math PreK software, manipulatives, and print material. Building Blocks for Math embeds mathematical learning in children’s daily activities, ranging from designated math activities to circle and story time, with the goal of helping children relate their informal math knowledge to more formal mathematical concepts.

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.

Vmath®, distributed by Voyager Learning, is a supplemental mathematics curriculum for students in grades 2–8 who are struggling with math. The program aims to improve understanding of math concepts and performance on high-stakes assessments. A distinguishing feature of Vmath® is that it provides teachers with a specific, detailed script for each lesson. In-class instruction is supplemented with VmathLive®, a web-based program that allows students to practice their math skills outside of school.

Developed for grades K-5 by the School District of Hillsborough County, FL, in accordance with state and district standards, Voyages uses a compact curriculum to accelerate math instruction and is comprised of two main components to meet diverse learning needs. Excursion Lessons provide teacher-led, interactive, contextual approaches to solving real-life problems through lessons which may take two to three days. Anchor Lessons focus on developing basic math skills and algebraic concepts through the use of visual and abstract representations. Textbooks and teaching guides are provided for lessons, as well as manipulatives kits and assessment tools.

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.  

Great Explorations in Math and Science® (GEMS®) The Real Reasons for Seasons is a curriculum unit for grades 6–8 that focuses on the connections between the Sun and the Earth to teach students the scientific concepts behind the seasons. The unit utilizes models, hands-on investigations, peer-to-peer discussions, reflection, and informational student readings to help students understand science content and develop scientific investigation skills.

Project SEED is a supplemental mathematics program for low-achieving students in grades 3 through 8 and is intended to prepare students to be successful in high school and college math. Based on the Socratic method, instruction is delivered through a series of questions to the class. In addition to individual responses, the instructor solicits group feedback through silent hand signals, chorus responses, and quick surveys of written work. The program is intended to encourage active student learning, develop critical thinking, and strengthen articulation skills. Student learning is assessed regularly, and instructors adapt the lessons to accommodate different ability levels. The curriculum, taught by mathematics specialists, includes topics from advanced mathematics, such as advanced algebra, pre-calculus, group theory, number theory, calculus, and geometry. Project SEED instruction is provided in addition to regular math instruction four times a week for 14 to 16 weeks. The program also provides professional development for classroom teachers through modeling, coaching, and workshops.

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.

Destination MATH is a computer curriculum that is designed to demonstrate how mathematical issues arise in real-life situations. Courses IV and V are designed for students in grades 6–8. Each course contains a scope and sequence section that shows the skills to be learned in that section of the course. Course IV, Basic Mathematics, focuses on arithmetic, numbers, and operations included in the middle school math curriculum. The course also investigates properties of fractions, decimals, percents, and integers as well as the rules that govern their operations. Course V, Pre-Algebra, focuses on ratios, proportions, and irrational numbers. Additionally, the course includes the fundamental skills and concepts found in algebra, geometry, statistics, and probability. The program includes a Learning Management and Assessment System designed to help teachers relate assignments with district objectives. Teachers can monitor their students through online reports and can create students tests designed for various levels of difficulty. The course can be obtained by subscribing online or by purchasing a CD.

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.

MathThematics is a mathematics curriculum for grades 6 through 8 that combines activity-based, discovery learning with direct instruction. The textbook for each grade level has eight instructional modules, with each module focused on a theme. Open-ended questions and projects are utilized throughout the curriculum to assess problem-solving skills and the ability to communicate mathematically.

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.

MathSteps is a research-based supplementary program for elementary and middle school grades which focuses on providing students with the skills and tactics they need to meet grade-level standards. It was designed to supply teachers with the resources necessary to assess student performance and apply appropriate materials to correct perceived weaknesses in computational ability. The program lists four key elements: Skills Inventories tests (to assess student skill levels), Sequential Skills Outlines, Correlation Guidelines, and Skill and Practice sheets. MathSteps also incorporates built-in staff development, and utilizes four components: student workbooks, teacher resource copymasters, teacher editions, and computational skills tutorial kits. Student progress is monitored on a daily basis through ongoing assessment.

MathWings is a K-6 program which emphasizes cooperative learning and is designed to meet the National Council of Teachers of Mathematics (NCTM) standards. In each class students, regardless of skill level, participate in a whole class exploration of skills and real-world problem solving activities intended to extend their mathematical knowledge and experience. Individualized units are developed and allow students to accelerate or refine their skills and complete exercises at their own pace. The program requires some faculty training with an on-site facilitator, and utilizes teaching manuals, transparencies, blackline masters, manipulatives, and literature kits.

Moving with Math is a manipulatives-based mathematics curriculum designed for grades K–8, and follows standards established by the National Council of Teachers of Mathematics (NCTM). Essential math concepts and problem solving skills are taught by integrating manipulatives into each lesson in order to illustrate ideas and make real world connections. Student activity books allow students to become actively involved in the learning process through the methodology of “explore, discover the pattern, and communicate the approach.” Lessons combine manipulatives and worksheets, while assessment tools, such as diagnostic tests and reviews, are provided in teaching manuals.

LOGO is a computer programming language designed as a learning tool for assisting students of all levels in becoming more aware of their own mathematical capabilities and build skills. Programs in LOGO are usually formed through collections of small procedures, which are created by writing commands in LOGO’s text editor. The results of the programs are visually demonstrated on a computer screen by the LOGO turtle, which may ‘walk’ to form any number of geometric shapes or other mathematical outputs. Programming in turtle graphics creates an environment for problem solving, and forms a foundation of concepts to build upon and progress to higher levels of mathematical comprehension.

The Math Trailblazers curriculum for grades K-5 emphasizes active participation in developing problem-solving strategies which are applicable to real-world contexts. Through a balance of class work, small group learning, and individual instruction, students participate in longer (3-5 days) activities as well as shorter problem-solving tasks which integrate science, math, and language arts into hourly lessons. The curriculum provides different ways of looking at math problems to suit diverse learning styles, and incorporates textbooks, transparencies, manipulatives, and computer technology into each unit. The curriculum is designed to meet National Council of Teachers of Mathematics (NCTM) standards.

The Rational Number Project was a cooperative multi-university research and development project funded by the National Science Foundation (NSF) from 1979-2002 (with the exception of 1983-84). One of the results of this project is a curriculum of lessons designed to introduce students in grades 4-8 to fractions, decimals, ratios, and measurement and meet the standards of the National Council of Teachers of Mathematics (NCTM). Lessons are conducted in large and small group settings, with daily hands-on activities involving manipulative kits. Along with a spiral-bound lesson book, colored tiles and pie piece shapes are among the manipulatives used to assist students in fraction operation and comprehension.

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.

M Monrad: The Digi-Block Learning System is a hands-on math manipulative program designed to supplement and improve the teaching and understanding of the base-ten number system for grades K-4. Each classroom set of Digi-Blocks consists of blocks and empty rectangular holders which hold exactly small ten blocks; each holder can be packed into larger blocks, enabling students to visualize the concept of hundreds and thousands. Through physical interaction with Digi-Block materials, students learn number sense and place value while building a foundation for later comprehension of higher math concepts in upper grades.

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.

The Adventures of Jasper Woodbury Series, a curriculum-based program available for grades 5 and up, consists of 12 videodisc-based adventures that are formulated to meet the National Council of Teachers of Mathematics (NCTM) standards. Each adventure ends in a challenge designed to bridge the gap between natural and school learning environments through problem solving, reasoning, communicating, and making connections to other areas such as science, social studies, literature, and history. The episodes include all the data needed to solve the challenge, and may be re-visited by students as necessary.

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 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.

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.

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.

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.

As a promising tool for improving math education and closing the achievement gap, the use of educational technology has dramatically expanded in K-12 education in recent years, accelerated by the COVID-19 pandemic. ASSISTments was one of the few digital learning programs recommended for use in response to the COVID-19 pandemic (Sahni et al., 2021). Developed by researchers at Worcester Polytechnic Institute, ASSISTments is a free online formative assessment platform that applies innovative technology in education. This study measured the possible long-term impact of ASSISTments on student learning. Findings from the study can help schools, districts, and states make informed decisions regarding resource allocation and adoption of technology-based programs, as well as the specific skills and knowledge that educators need to integrate technology into their schools effectively--potentially leading to improved student performance while saving time and money in the long run. By prioritizing evidence-based practices, schools and districts can track the impact of the technologies they adopt and hold themselves accountable for meeting their goals. Key findings include: (1) ASSISTments had a significant positive impact on student learning in the long term; (2) ASSISTments helped close opportunity gaps; and (3) ASSISTments benefited more those students whose schools had a higher percentage of students from economically disadvantaged backgrounds or a lower percentage of White students. [Collaborators for this report include WestEd, North Carolina State University, SRI International, LFC Research, Digital Promise, and Worcester Polytechnic Institute.]

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.]

Evidence-based interventions exist for improving multiple fundamental math competencies, but delivering interventions with fidelity and within a data-driven, tiered framework is a practical challenge faced by most schools. The current study evaluated a math intervention program delivered with community-based resources via AmeriCorps. At the beginning of the school year, students in Grades 4-8 (n = 550) were randomly assigned to receive math support via the program or to a waitlist control group. Outcomes were measured with a broad-based assessment of math achievement in the winter. Results from intent-to-treat analyses showed a significant and positive effect (d = 0.17) for the program that increased slightly under optimal dosage conditions (d = 0.24). The observed results extend existing literature on math interventions in schools by illustrating the potential for partnerships between community-based organizations and schools to improve outcomes for at-risk students.

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.]

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.]

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.

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.

A large-scale randomized controlled trial tested the effects of researcher-developed learning games on a transfer measure of fractions knowledge. The measure contained items similar to standardized assessments. Thirty treatment and 29 control classrooms (~1500 students, 9 districts, 26 schools) participated in the study. Students in treatment classrooms played fractions games and students in the control classrooms played solving equations games. Multilevel multidimensional item response theory modeling of the outcome measure produced scaled scores that were more sensitive to the instructional treatment than standard measurement approaches. Hierarchical linear modeling of the scaled scores showed that the treatment condition performed significantly higher on the outcome measure than the control condition. The effect (d = 0.58) was medium to large (Cohen, 1992). Two appendices are included: (1) Descriptive Statistics of Pretest and Posttest Scores by Schools and Conditions; and (2) Summary of Efficacy Trial Procedures.

Teach For America (TFA) and the Teaching Fellows programs are an important and growing source of teachers of hard-to-staff subjects in high-poverty schools, but comprehensive evidence of their effectiveness has been limited. This report presents findings from the first large-scale random assignment study of secondary math teachers from these programs. The study separately examined the effectiveness of TFA and Teaching Fellows teachers, comparing secondary math teachers from each program with other secondary math teachers teaching the same math courses in the same schools. The study focused on secondary math because this is a subject in which schools face particular staffing difficulties.The study had two main findings, one for each program studied: (1) TFA teachers were more effective than the teachers with whom they were compared. On average, students assigned to TFA teachers scored 0.07 standard deviations higher on end-of-year math assessments than students assigned to comparison teachers, a statistically significant difference. This impact is equivalent to an additional 2.6 months of school for the average student nationwide; and (2) Teaching Fellows were neither more nor less effective than the teachers with whom they were compared. On average, students of Teaching Fellows and students of comparison teachers had similar scores on end-of-year math assessments. By providing rigorous evidence on the effectiveness of secondary math teachers from TFA and the Teaching Fellows programs, the study can shed light on potential approaches for improving teacher effectiveness in hard-to-staff schools and subjects. The study findings can provide guidance to school principals faced with the choice of hiring teachers who have entered the profession via different routes to certification. The findings can also aid policymakers and funders of teacher preparation programs by providing information on the effectiveness of teachers from various routes to certification that use different methods to identify, attract, train, and support their teachers. Seven appendixes present: (1) Supplementary Technical Information on Study Design and Data Collection; (2) Supplementary Information on Analytic Methods; (3) Supplementary Information on Teach For America and Teaching Fellows Programs; (4) Teach For America and Teaching Fellows Teachers Compared with Comparison Teachers by Entry Route (Alternative or Traditional); (5) Supplementary Information on Teach For America and Teaching Fellows Teachers Compared with Comparison Teachers; (6) Supplementary Analyses of the Impacts of Teach For America and Teaching Fellows Teachers; and (7) Supplementary Findings on Factors Associated with Teacher Effectiveness. (Contains 96 tables, 21 figures, and 30 footnotes.)

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).]

The purposes of this study were to investigate the effects of an intervention designed to improve at-risk 4th graders' understanding of fractions and to examine the processes by which effects occurred. The intervention focused more on the measurement interpretation of fractions; the control condition focused more on the part-whole interpretation of fractions and on procedures. Intervention was also designed to compensate for at-risk students' limitations in the domain-general abilities associated with fraction learning. At-risk students (n = 259) were randomly assigned to intervention and control. Whole-number calculation skill, domain-general abilities (working memory, attentive behavior, processing speed, listening comprehension), and fraction proficiency were pretested. Intervention occurred for 12 weeks, 3 times per week, 30 min per session, and then fraction performance was reassessed. On each conceptual and procedural fraction outcome, effects favored intervention over control (effect sizes = 0.29 to 2.50), and the gap between at-risk and low-risk students narrowed for the intervention group but not the control group. Improvement in the accuracy of children's measurement interpretation of fractions mediated intervention effects. Also, intervention effects were moderated by domain-general abilities, but not whole-number calculation skill.

The purpose of this study was to investigate the effects of 1st-grade number knowledge tutoring with contrasting forms of practice. Tutoring occurred 3 times per week for 16 weeks. In each 30-min session, the major emphasis (25 min) was number knowledge; the other 5 min provided practice in 1 of 2 forms. Nonspeeded practice reinforced relations and principles addressed in number knowledge tutoring. Speeded practice promoted quick responding and use of efficient counting procedures to generate many correct responses. At-risk students were randomly assigned to number knowledge tutoring with speeded practice (n = 195), number knowledge tutoring with nonspeeded practice (n = 190), and control (no tutoring, n = 206). Each tutoring condition produced stronger learning than control on all 4 mathematics outcomes. Speeded practice produced stronger learning than nonspeeded practice on arithmetic and 2-digit calculations, but effects were comparable on number knowledge and word problems. Effects of both practice conditions on arithmetic were partially mediated by increased reliance on retrieval, but only speeded practice helped at-risk children compensate for weak reasoning ability. (Contains 7 tables, 2 figures and 6 footnotes.)

The purpose of this study was to investigate the effects of 1st-grade number knowledge tutoring with contrasting forms of practice. Tutoring occurred 3 times per week for 16 weeks. In each 30-min session, the major emphasis (25 min) was number knowledge; the other 5 min provided practice in 1 of 2 forms. Nonspeeded practice reinforced relations and principles addressed in number knowledge tutoring. Speeded practice promoted quick responding and use of efficient counting procedures to generate many correct responses. At-risk students were randomly assigned to number knowledge tutoring with speeded practice (n = 195), number knowledge tutoring with nonspeeded practice (n = 190), and control (no tutoring, n = 206). Each tutoring condition produced stronger learning than control on all 4 mathematics outcomes. Speeded practice produced stronger learning than nonspeeded practice on arithmetic and 2-digit calculations, but effects were comparable on number knowledge and word problems. Effects of both practice conditions on arithmetic were partially mediated by increased reliance on retrieval, but only speeded practice helped at-risk children compensate for weak reasoning ability. (Contains 7 tables, 2 figures and 6 footnotes.)

The purpose of this study was to investigate the effects of 1st-grade number knowledge tutoring with contrasting forms of practice. Tutoring occurred 3 times per week for 16 weeks. In each 30-min session, the major emphasis (25 min) was number knowledge; the other 5 min provided practice in 1 of 2 forms. Nonspeeded practice reinforced relations and principles addressed in number knowledge tutoring. Speeded practice promoted quick responding and use of efficient counting procedures to generate many correct responses. At-risk students were randomly assigned to number knowledge tutoring with speeded practice (n = 195), number knowledge tutoring with nonspeeded practice (n = 190), and control (no tutoring, n = 206). Each tutoring condition produced stronger learning than control on all 4 mathematics outcomes. Speeded practice produced stronger learning than nonspeeded practice on arithmetic and 2-digit calculations, but effects were comparable on number knowledge and word problems. Effects of both practice conditions on arithmetic were partially mediated by increased reliance on retrieval, but only speeded practice helped at-risk children compensate for weak reasoning ability. (Contains 7 tables, 2 figures and 6 footnotes.)

This report presents the results of an experiment conducted in Alabama beginning in the 2006/07 school year, to determine the effectiveness of the Alabama Math, Science, and Technology Initiative (AMSTI), which aims to improve mathematics and science achievement in the state's K-12 schools. This study is the first randomized controlled trial testing the effectiveness of AMSTI in improving mathematics problem solving and science achievement in upper-elementary and middle schools. AMSTI is an initiative specific to Alabama and was developed and supported through state resources. An important finding is the positive and statistically significant effect of AMSTI on mathematics achievement as measured by the SAT 10 mathematics problem solving assessment administered by the state to students in grades 4-8. After one year in the program, student mathematics scores were higher than those of a control group that did not receive AMSTI by 0.05 standard deviation, equivalent to 2 percentile points. Nine of the 10 sensitivity analyses yielded effect estimates that were statistically significant at the 0.025 level, consistent with the main finding. The estimated effect of AMSTI on science achievement measured after one year was not statistically significant. Based on the SAT 10 science test administered by the state to students in grades 5 and 7, no difference between AMSTI and control schools could be discerned after one year. Changes in classroom instructional strategies, especially an emphasis on more active-learning strategies, are important to the AMSTI theory of action. Therefore, a secondary investigation of classroom practices was conducted, based on data from survey responses from teachers. For both mathematics and science, statistically significant differences were found between AMSTI and control teachers in the average reported time spent using the strategies. The effect of AMSTI on these instructional strategies was 0.47 standard deviation in mathematics and 0.32 standard deviation in science. Two years of AMSTI appeared to have a positive and statistically significant effect on achievement in mathematics problem solving, compared to no AMSTI. Two years of AMSTI appeared to have a positive and statistically significant effect on achievement in science. AMSTI appeared to have a positive and statistically significant effect on reading achievement as measured by the SAT 10 test of reading administered by the state to students in grades 4-8. AMSTI did not appear to have a statistically significant effect on teacher-reported content knowledge in mathematics or science after one year. AMSTI did not appear to have statistically significant differential effects on student achievement in mathematics problem solving or science based on racial/ethnic minority status, enrollment in the free or reduced-price lunch program, gender, or pretest level. Appended are: (1) Explanation of primary and secondary confirmatory outcome measures; (2) Explanation of exploratory research questions; (3) Selection and random assignment of schools; (4) Statistical power analysis; (5) Data collection procedures and timeline; (6) Description of program implementation data collected but not used in report; (7) Alabama Math, Science, and Technology Initiative (AMSTI) teacher survey #3; (8) Data cleaning and data file construction; (9) Attrition through study stages for samples used in the confirmatory analysis; (10) Description of degree rank; (11) Equivalence of Year 1 baseline and analyzed samples for confirmatory student-level and classroom practice outcomes; (12) Internal consistency and validity of active learning measures; (13) Number of students and teachers in schools in analytic samples used to analyze Year 1 confirmatory questions; (14) Attrition through study stages for samples used in Year 1 exploratory analysis; (15) Tests of equivalence for baseline and analytic samples for Year 1 exploratory outcomes; (16) Statistical power analyses for moderator analyses; (17) Derivation and motivation of the Bell-Bradley estimator when measuring estimated two-year effect of the Alabama Math, Science, and Technology Initiative (AMSTI); (18) Attrition through study stages for samples contributing to estimation of two-year effects; (19) Examination of equivalence in baseline and analytic samples used in the estimation of two-year effects; (20) Estimation model for two-year effects of the Alabama Math, Science, and Technology Initiative (AMSTI); (21) Topics and instructional methods used at the Alabama Math, Science, and Technology Initiative (AMSTI) summer institute; (22) Parameter estimates on probability scale for odds-ratio tests of differences between Alabama Math, Science, and Technology Initiative (AMSTI) and control conditions in Year 1 (associated with summer professional development and in-school support outcomes); (23) Descriptive statistics for variables that change to a binary scale used in the Alabama Math, Science, and Technology Initiative (AMSTI) and control conditions in Year 1; (24) Comparison of assumed parameter values and observed sample statistics for statistical power analysis after one year; (25) Parameter estimates for Stanford Achievement Test Tenth Edition (SAT 10) mathematics problem solving after one year; (26) Parameter estimates for Stanford Achievement Test Tenth Edition (SAT 10) science after one year; (27) Parameter estimates for active learning in mathematics after one year; (28) Parameter estimates for active learning in science after one year; (29) Sensitivity analyses of effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on Stanford Achievement Test Tenth Edition (SAT 10) mathematics problem solving achievement after one year; (30) Sensitivity analyses of effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on Stanford Achievement Test Tenth Edition (SAT 10) science achievement after one year; (31) Sensitivity analyses of effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on active learning instructional strategies in mathematics classrooms after one year; (32) Sensitivity analyses of effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on active learning instructional strategies in science classrooms after one year; (33) Tests for violations of factors associated with assumption of equal first year effects on students in Alabama Math, Science, and Technology Initiative (AMSTI) and control schools; (34) Post hoc adjustment to standard error for estimate of two-year effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on mathematics achievement after two years; (35) Parameter estimates for effect of the Alabama Math, Science, and Technology Initiative (AMSTI) after two years; (36) Parameter estimates for effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on student reading achievement after one year; (37) Parameter estimates for teacher content and student engagement after one year; (38) Estimates of effects for terms involving the indicator of treatment status in the analysis of the moderating effect of the three-level pretest variable; (39) Parameter estimates for the analysis of the moderating effect of racial/ethnic minority status on the impact of the Alabama Math, Science, and Technology Initiative (AMSTI) on reading after one year; (40) Parameter estimates for analysis of average effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on reading by racial/ethnic minority students after one year; and (41) Parameter estimates for effect of the Alabama Math, Science, and Technology Initiative (AMSTI) on reading for White students after one year. (Contains 26 figures, 136 tables, 1 box and 130 footnotes.)

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.)

This study examined the effect of linguistic modification on middle school students' ability to show what they know and can do on math assessments. REL West's study on middle school math assessment accommodations found that simplifying the language--or linguistic modification--on standardized math test items made it easier for English Language learners to focus on and grasp math concepts, and thus was a more accurate assessment of their math skills. The results contribute to the body of knowledge informing assessment practices and accommodations appropriate for English language learner students. The study examined students' performance on two sets of math items--both the originally worded items and those that had been modified. Researchers analyzed results from three subgroups of students--English learners (EL), non-English language arts proficient (NEP), and English language arts proficient (EP) students. Key results include: (1) Linguistically modifying the language of mathematics test items did not change the math knowledge being assessed; (2) The effect of linguistic modification on students' math performance varied between the three student subgroups. The results also varied depending on how scores were calculated for each student; and (3) For each of the four scoring approaches analyzed, the effect of linguistic modification was greatest for EL students, followed by NEP and EP students. The report is structured as follows. Following an Executive Summary and a Study Overview, Chapter 2 describes the study design, sample selection and recruitment, item set development processes, and standardized administration procedures. Chapter 3 describes the implementation of the accommodation (linguistic modification), including discussion of considerations and methods for data analysis. Chapter 4 presents findings from data analyses. Chapter 5 summarizes and interprets key findings, describes study challenges, comments on implications of the findings, and offers recommendations for future research. Appendices include: (1) Power analysis for primary research questions; (2) Operational test administration manual; (3) Student Language Background Survey; (4) Guide for developing a linguistically modified assessment; (5) Workgroup training materials; (6) Overview and protocol for cognitive interviews; (7) Item parameter estimates for IRT models; (8) Descriptive statistics from four scoring approaches; (9) ANOVA findings across four scoring approaches; (10) Cross-approach comparisons; (11) Results of the classical item-level analyses; (12) Summary of differential item functioning findings; (13) Exploratory factor analysis results; (14) Operational item set--original; and (15) Operational item set--linguistically modified. (Contains 31 tables, 10 figures, and 45 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.

We investigated the impact in ninth-grade mathematics classrooms of Proficiency-based Assessment and Reassessment of Learning Outcomes (PARLO), a standards-based grading system. Key components of PARLO are basing student final grades on the number of learning outcomes on which the student is high-performance or proficient, providing students with formative feedback, and encouraging students to reassess for full credit after further study. Our mixed-methods study employed a cluster randomized control trial with 2,736 participating ninth graders at 14 Treatment and 15 Control schools. Data included student achievement tests, interviews with 35 teachers, and student surveys. The program improved student performance on end-of-course algebra and geometry tests by a statistically significant 0.33 SD but did not impact students' value of or expectancies for success in mathematics. However, treatment effects on mathematics performance were moderated by these psychological antecedents of motivation, such that students with higher math expectancies and value benefited more from the treatment. Furthermore, teacher interviews suggested that PARLO may have also had positive effects on growth mindsets, mastery goals, autonomy, and relatedness.

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.]

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.

The National Math + Science Initiative's (NMSI's) College Readiness Program (CRP) is an established program whose goal is to promote science, technology, engineering, and mathematics education in high schools to improve students' readiness for college. It provides teacher, student, and school supports to promote high school students' success in mathematics, science, and English Advanced Placement (AP) courses, with a focus on students who are traditionally underrepresented in the targeted AP courses. Through a federal Investing in Innovation Fund (i3) validation grant awarded to NMSI in 2011, CRP was implemented in a total of 58 high schools in two states--Colorado and Indiana--beginning in the 2012-13 school year. American Institutes for Research (AIR) conducted an independent evaluation of the impacts of CRP on students' AP outcomes in these schools for the three cohorts of schools that adopted the program in sequential years, using a comparative interrupted time series (CITS) design that matched comparison schools to program schools in the two states. Overall, schools implementing CRP demonstrated significantly larger increases in the share of students taking and passing AP tests in targeted areas relative to comparison schools in each of the three cohorts of schools, and the gains in CRP schools were sustained over time. Fidelity of program implementation was evaluated using a fidelity matrix approach required as part of the National Evaluation of the i3 program, which showed that not all elements of the program were implemented with high fidelity. Teachers and students were not always able to attend all meetings, and schools did not always meet negotiated enrollment targets. Teacher survey data indicated that teachers found the professional development activities provided by CRP to be the most helpful support they received under CRP, and students reported that the tutoring and special study sessions were the most helpful. Although the program provided financial incentives to both teachers and students that were tied to student performance on AP tests, these incentives were considered the least important element of the program by both teachers and students

Replication studies are extremely rare in education. This randomized controlled trial (RCT) is a scale-up replication of Fuchs et al., which in a sample of 139 found a statistically significant positive impact for Number Rockets, a small-group intervention for at-risk first graders that focused on building understanding of number operations. The study was relatively small scale (one site) and highly controlled. This replication was implemented at a much larger scale--in 76 schools in four urban districts; 994 at-risk students participated. Intervention students participated in approximately 30 hours of small-group work in addition to classroom instruction; control students received typical instruction and whatever assistance the teacher would normally provide. Intervention students showed significantly superior performance on a broad measure of mathematics proficiency.

Mathematics Recovery (MR) is designed to identify first graders who are struggling in mathematics and provide them with intensive one-to-one tutoring. We report findings from a 2-year evaluation of MR conducted in 20 elementary schools across five districts in two states. The design allowed for the estimation of the counterfactual growth trajectory based on those students randomly assigned either to a tutoring cohort with a delayed start or to a wait list. Results demonstrate strong end of first grade effects on a diagnostic measure developed by MR and weak to moderate effects (effect size, 0.15-0.30) on measures administered by external evaluators. By the end of second grade, no significant effects were found on any measures. Practical and research implications are discussed. (Contains 7 tables, 3 figures, and 5 notes.)

This report summarizes the results of a scale-up project funded by the Institute of Education Sciences in 2006. "Scaling up TRIAD: Teaching Early Mathematics for Understanding with Trajectories and Technologies" was a project that took a preschool mathematics intervention to scale across three sites, following children from their Prekindergarten year--in which the intervention was delivered--through first grade. This study was designed to explore the following research questions: (1) What are the immediate and long-term effects of the intervention on children's math skills?; (2) How much variation was there in effects across sites?; (3) Were curricular effects different for different subgroups of children?; and (4) What are the effects of the math environment and the fidelity of implementation on children's immediate and long term math gains, overall and across sites? [This research was conducted in partnership with Doug Clements and Julie Sarama at the University at Buffalo SUNY (recently moved to the University of Colorado).]

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.)

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.

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.

A pretest-training-posttest design assessed whether training to improve spatial skills also improved mathematics performance in elementary-aged children. First grade students (mean age = 7 years, n = 134) and sixth grade students (mean age = 12 years, n = 124) completed training in 1 of 2 spatial skills--spatial visualization or form perception/VSWM--or in a nonspatial control condition that featured language arts training. Spatial training led to better overall mathematics performance in both grades, and the gains were significantly greater than for language arts training. The same effects were found regardless of spatial training type, or the type of mathematics tested. [For the corresponding grantee submission, see ED606395.]

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."]

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 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."]

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 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. [For the corresponding grantee submission, see ED595063.]

The purpose of this study was to assess whether intervention with an integrated focus on fraction and decimal magnitude provides added value in improving rational number performance over intervention focused exclusively on fractions. We randomly assigned 4th graders with poor whole-number performance to 3 conditions: a business-as-usual control group and 2 variants of a validated fraction magnitude (FM) intervention. One variant of FM intervention included an integrated component on fraction-decimal magnitude (FM+DM); the other included a fraction applications component (FM+FAPP) to more closely mirror the validated FM intervention and to control for intervention time. Cross-classified partially-nested analyses (N=225) provided the basis for 3 conclusions. First, FM intervention improves 4th-graders' fraction understanding and applications. Second, effects of FM intervention, even without a focus on decimals, transfer to decimal number line performance. Third, an intervention component integrating fraction-decimal magnitude does not provide added value over FM intervention on fraction or decimal performance, except on decimal tasks paralleling intervention tasks. [This is the in press version of an article published in "Contemporary Educational Psychology."]

The goal of this randomized controlled trial (RCT) was to evaluate the effectiveness of a small-group intervention in fractions for fifth-graders who are performing below grade level in mathematics. The impact of the fractions intervention was assessed on fifth-grade at-risk students' understanding of foundational fractions concepts and procedural competence with fractions. For the fractions intervention, lessons from the "TransMath"® curriculum (Level 2; Woodward & Stroh, 2015) were modified to create 52 thirty-five-minute lessons focused only on fourth- and fifth-grade level fractions content that could be used in a small-group setting. In this rigorous large-scale RCT, a sample of 1,123 students from three school districts across two U.S. states were screened at the beginning of the school year using a fractions measure developed by the research team. Two hundred and five students who scored between the 15th and the 37th percentile on the screening measure and received parental consent to participate were randomly assigned to the intervention (n = 102) and comparison (n = 103) conditions. Students in the fractions intervention condition received 35-minute tutoring sessions 3-4 times a week. Fraction instruction using the 52 "TransMath" lessons was provided by trained tutors. The comparison condition (n = 103) was business as usual instruction (i.e., core classroom fractions instruction, including intervention or support traditionally provided by the school). Results from the final analytic sample of 186 students (87 in intervention, 99 in comparison) showed that the intervention group significantly outperformed the comparison group on all outcome measures, which included an array of assessments used to measure both student understanding of foundational fractions concepts as well as procedural competence with grade-level fraction material (Hedges' g = 0.66 to 1.08; p < 0.0001).

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 acquisition of early mathematical knowledge is critical for successful long-term academic development. Mathematical language is one of the strongest predictors of children's early mathematical success. Findings from previous studies have provided correlational evidence supporting the importance of mathematical language to the development of children's mathematics skills, but there is limited causal evidence supporting this link. To address this research gap, 47 Head Start children were randomly assigned to a mathematical language intervention group or a business-as-usual group. Over the course of eight weeks, interventionists implemented a dialogic reading intervention focused on quantitative and spatial mathematical language. At posttest, students in the intervention group significantly outperformed the students in the comparison group not only on a mathematical language assessment, but on a mathematical knowledge assessment as well. These findings indicate that increasing children's exposure to mathematical language can positively affect their general mathematics skills. This study is an important first step in providing causal evidence of the importance of early mathematical language for children's general mathematical knowledge and the potential for mathematical language interventions to increase children's overall mathematics abilities.

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 purposes of this study were to (a) investigate the efficacy of a core fraction intervention program on understanding and calculation skill and (b) isolate the effects of different forms of fraction word-problem (WP) intervention delivered as part of the larger program. At-risk 4th graders (n = 213) were randomly assigned at the individual level to receive the school's business-as-usual program or 1 of 2 variants of the core fraction intervention (each 12 weeks, 3 sessions/week). In each session of the 2 variants, 28 min were identical, focused mainly on the measurement interpretation of fractions. The other 7 min addressed fraction WPs: multiplicative WPs versus additive WPs. Children were pre- and posttested on fraction understanding, calculations, and WPs. On understanding and calculations, both intervention conditions outperformed the control group, and the effect of intervention versus control on released fraction items from the National Assessment of Education Progress were mediated by children's improvement in the measurement interpretation of fractions. On multiplicative WPs, multiplicative WP intervention was superior to the other 2 conditions, but additive WP intervention and the control group performed comparably. By contrast, on additive WPs, there was a step-down effect in which additive WP intervention was superior to multiplicative WP intervention, which was superior to control. [This paper was published in "Elementary School Journal," v116 n4 p625-661 Jun 2016 (EJ1103953).]

The purposes of this study were to (a) investigate the efficacy of a core fraction intervention program on understanding and calculation skill and (b) isolate the effects of different forms of fraction word-problem (WP) intervention. At-risk fourth graders (n = 213) were randomly assigned to the school's business-as-usual program, or one of two variants of the core fraction intervention (each 12 weeks, 36 sessions). In each session of the two variants, 28 minutes were identical, focused mainly on the measurement interpretation of fractions. The other 7 minutes addressed multiplicative WPs versus additive WPs. Children were pre-/posttested on fraction understanding, calculations, and WPs. On understanding and calculations, both intervention conditions outperformed the control group. The effect of intervention versus control on released fraction items from the National Assessment of Education Progress was mediated by children's improvement in the measurement interpretation of fractions. On multiplicative WPs, multiplicative WP intervention was superior to the other conditions, but additive WP intervention and the control group performed comparably. On additive WPs, additive WP intervention was superior to multiplicative WP intervention, which was superior to control.

The main purposes of this study were to test the effects of teaching at-risk 4th graders to provide explanations for their mathematics work and examine whether those effects occur by compensating for limitations in cognitive processes. We randomly assigned 212 children to 3 conditions: a control group and 2 variants of a multicomponent fraction intervention. Both intervention conditions included 36 sessions, each lasting 35 min. All but 7 min of each session were identical. In the 7-min component, students were taught to provide high quality explanations when comparing fraction magnitudes or to solve fraction word problems. Children were pretested on cognitive variables and pre/posttested on fraction knowledge. On accuracy of magnitude comparisons and quality of explanations, children who received the explaining intervention outperformed those in the word-problem condition. On word problems, children who received the word-problem intervention outperformed those in the explaining condition. Moderator analyses indicated that the explaining intervention was more effective for students with weaker working memory, while the word-problem intervention was more effective for students with stronger reasoning ability.

The main purposes of this study were to test the effects of teaching at-risk 4th graders to provide explanations for their mathematics work and examine whether those effects occur by compensating for limitations in cognitive processes. We randomly assigned 212 children to 3 conditions: a control group and 2 variants of a multicomponent fraction intervention. Both intervention conditions included 36 sessions, each lasting 35 min. All but 7 min of each session were identical. In the 7-min component, students were taught to provide high quality explanations when comparing fraction magnitudes or to solve fraction word problems. Children were pretested on cognitive variables and pre/posttested on fraction knowledge. On accuracy of magnitude comparisons and quality of explanations, children who received the explaining intervention outperformed those in the word-problem condition. On word problems, children who received the word-problem intervention outperformed those in the explaining condition. Moderator analyses indicated that the explaining intervention was more effective for students with weaker working memory, while the word-problem intervention was more effective for students with stronger reasoning ability.

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).]

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.

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 efficacy of a research-based number-sense intervention for low-achieving kindergartners was examined. Children (N = 126) were randomly assigned to 1 of 3 conditions: a number-sense intervention followed by a number-fact practice session, an identical number-sense intervention followed by a number-list practice session, or a business-as-usual control group. The number-fact practice condition not only gave children an additional advantage over the number-list practice condition on the outcomes at delayed posttest 8 weeks later but also was especially effective for producing gains in English learners.

The efficacy of a research-based number-sense intervention for low-achieving kindergartners was examined. Children (N = 126) were randomly assigned to 1 of 3 conditions: a number-sense intervention followed by a number-fact practice session, an identical number-sense intervention followed by a number-list practice session, or a business-as-usual control group. The number-fact practice condition not only gave children an additional advantage over the number-list practice condition on the outcomes at delayed posttest 8 weeks later but also was especially effective for producing gains in English learners.

The efficacy of a research-based number-sense intervention for low-achieving kindergartners was examined. Children (N = 126) were randomly assigned to 1 of 3 conditions: a number-sense intervention followed by a number-fact practice session, an identical number-sense intervention followed by a number-list practice session, or a business-as-usual control group. The number-fact practice condition not only gave children an additional advantage over the number-list practice condition on the outcomes at delayed posttest 8 weeks later but also was especially effective for producing gains in English learners.

The Common Core State Standards for Mathematics will place more pressure on special education and math teachers to raise the skill levels of all students, especially those with disabilities in math (MD). The purpose of this study was to assess the effects of enhanced anchored instruction (EAI) on students with and without MD in co-taught general education classrooms. Results showed that students in the EAI condition improved their performance on math skills contained in several of the standards. Effect sizes were especially large for students with MD when the special education teacher more actively participated in the instructional activities with the math teacher. Classroom observations provided examples of how teachers can work together to benefit students in inclusive math settings.

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 study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation of fractions mediated those effects. At-risk students (n = 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude-treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects.

This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation of fractions mediated those effects. At-risk students (n = 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude-treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects.

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 investigated the role of strategy instruction and cognitive abilities on word problem solving accuracy in children with math difficulties (MD). Elementary school children (N = 120) with and without MD were randomly assigned to 1 of 4 conditions: general-heuristic (e.g., underline question sentence), visual-schematic presentation (diagrams), general-heuristic + visual-schematic, and an untreated control. When compared to the control condition that included children with MD, an advantage at posttest was found for children with MD for the visual-schematic-alone condition on measures of problem solving and calculation accuracy, whereas all strategy conditions facilitated posttest performance in correctly identifying problem solving components. The results also suggested that strategy conditions drew upon different cognitive resources. The General-heuristic condition drew primarily upon the executive component of working memory (WM), Visual-schematic condition drew upon the visual component of WM and the combined strategies condition drew upon number processing skills.

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.)

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.]

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.]

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.

This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional "operations on left side" format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving traditional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children's understanding of fundamental concepts.

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.)

The purpose of this study was to assess the efficacy of fact retrieval tutoring as a function of math difficulty (MD) subtype, that is, whether students have MD alone (MD-only) or have concurrent difficulty with math and reading (MDRD). Third graders (n = 139) at two sites were randomly assigned, blocking by site and MD subtype, to four tutoring conditions: fact retrieval practice, conceptual fact retrieval instruction with practice, procedural computation/estimation instruction, and control (no tutoring). Tutoring occurred for 45 sessions over 15 weeks for 15-25 minutes per session. Results provided evidence of an interaction between tutoring condition and MD subtype status for assessment of fact retrieval. For MD-only students, students in both fact retrieval conditions achieved comparably and outperformed MD-only students in the control group as well as those in the procedural computation/estimation instruction group. By contrast, for MDRD students, there were no significant differences among intervention conditions.

The purpose of this study was to assess the efficacy of fact retrieval tutoring as a function of math difficulty (MD) subtype, that is, whether students have MD alone (MD-only) or have concurrent difficulty with math and reading (MDRD). Third graders (n = 139) at two sites were randomly assigned, blocking by site and MD subtype, to four tutoring conditions: fact retrieval practice, conceptual fact retrieval instruction with practice, procedural computation/estimation instruction, and control (no tutoring). Tutoring occurred for 45 sessions over 15 weeks for 15-25 minutes per session. Results provided evidence of an interaction between tutoring condition and MD subtype status for assessment of fact retrieval. For MD-only students, students in both fact retrieval conditions achieved comparably and outperformed MD-only students in the control group as well as those in the procedural computation/estimation instruction group. By contrast, for MDRD students, there were no significant differences among intervention conditions.

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.

This study assessed the effects of preventative tutoring on the math problem solving of third-grade students with math and reading difficulties. Students (n = 35) were assigned randomly to continue in their general education math program or to receive secondary preventative tutoring 3 times per week, 30 min per session, for 12 weeks. Schema-broadening tutoring taught students to (a) focus on the mathematical structure of 3 problem types; (b) recognize problems as belonging to those 3 problem-type schemas; (c) solve the 3 word-problem types; and (d) transfer solution methods to problems that include irrelevant information, 2-digit operands, missing information in the first or second positions in the algebraic equation, or relevant information in charts, graphs, and pictures. Also, students were taught to perform the calculation and algebraic skills foundational for problem solving. Analyses of variance revealed statistically significant effects on a wide range of word problems, with large effect sizes. Findings support the efficacy of the tutoring protocol for preventing word-problem deficits among third-grade students with math and reading deficits. (Contains 3 tables and 1 figure.)

This study assessed the effects of small-group tutoring with and without validated classroom instruction on at-risk students' math problem solving. Stratifying within schools, 119 3rd-grade classes were randomly assigned to conventional or validated problem-solving instruction (Hot Math, schema-broadening instruction). Students identified as at risk (n = 243) were randomly assigned, within classroom conditions, to receive or not receive Hot Math tutoring. Students were tested on problem-solving and math applications measures before and after 16 weeks of intervention. Analyses of variance, which accounted for the nested structure of the data, revealed that the tutored students who received validated classroom instruction achieved better than the tutored students who received conventional classroom instruction (effect size = 1.34). However, the advantage for tutoring over no tutoring was similar whether students received validated or conventional classroom instruction (effect sizes = 1.18 and 1.13). Tutoring, not validated classroom instruction, reduced the prevalence of math difficulty. Implications for responsiveness-to-intervention prevention models and for enhancing math problem-solving instruction are discussed. (Contains 5 tables, 1 figure and 1 footnote.)

This study assessed the effects of small-group tutoring with and without validated classroom instruction on at-risk students' math problem solving. Stratifying within schools, 119 3rd-grade classes were randomly assigned to conventional or validated problem-solving instruction (Hot Math, schema-broadening instruction). Students identified as at risk (n = 243) were randomly assigned, within classroom conditions, to receive or not receive Hot Math tutoring. Students were tested on problem-solving and math applications measures before and after 16 weeks of intervention. Analyses of variance, which accounted for the nested structure of the data, revealed that the tutored students who received validated classroom instruction achieved better than the tutored students who received conventional classroom instruction (effect size = 1.34). However, the advantage for tutoring over no tutoring was similar whether students received validated or conventional classroom instruction (effect sizes = 1.18 and 1.13). Tutoring, not validated classroom instruction, reduced the prevalence of math difficulty. Implications for responsiveness-to-intervention prevention models and for enhancing math problem-solving instruction are discussed. (Contains 5 tables, 1 figure 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.)

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.)

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 examined the efficacy of preventive 1st-grade tutoring in mathematics, estimated the prevalence and severity of mathematics disability, and explored pretreatment cognitive characteristics associated with mathematics development. Participants were 564 first graders, 127 of whom were designated at risk (AR) for mathematics difficulty and randomly assigned to tutoring or control conditions. Before treatment, all participants were assessed on cognitive and academic measures. Tutoring occurred 3 times weekly for 16 weeks; treatment fidelity was documented; and math outcomes were assessed. Tutoring efficacy was supported on computation and concepts/applications, but not on fact fluency. Tutoring decreased the prevalence of math disability, with prevalence and severity varying as a function of identification method and math domain. Attention accounted for unique variance in predicting each aspect of end-of-year math performance. Other predictors, depending on the aspect of math performance, were nonverbal problem solving, working memory, and phonological processing.

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.

This study compared instruction in addition using either a minimum addend strategy or drill and practice with 84 students either with or without learning disabilities (LD). Students with LD improved significantly only in the strategy condition, whereas general education students improved in both the strategy and the drill-and-practice conditions. In a transfer task, students from both groups improved only in the strategy conditions. (Contains references.) (Author/DB)

This study compared instruction in addition using either a minimum addend strategy or drill and practice with 84 students either with or without learning disabilities (LD). Students with LD improved significantly only in the strategy condition, whereas general education students improved in both the strategy and the drill-and-practice conditions. In a transfer task, students from both groups improved only in the strategy conditions. (Contains references.) (Author/DB)

Examined the differential effects of two instructional strategies (explicit schema-based and traditional-based) on the acquisition, maintenance, and generalization of mathematical word problem solving among elementary students at risk or with mild disabilities. Pretesting and posttesting indicated that both groups' performance increased from pretest to posttest, though students in the schema-based increased more significantly. (Author/SM)

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)

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)

The paper evaluates math performance at four high-need middle schools during a four-year intervention, which was designed to help math teachers diagnose students' areas of need and to design lesson plans responsive to those needs. Before the intervention began, the researchers pre-selected four comparison schools by matching based on achievement and also on demographics. A difference-in-difference analysis finds a significant increase of about 0.11 standard deviation in test scores per year for students in the program schools. Supplementary event study and synthetic control analyses to detect year-by-year effects lack precision but are weakly suggestive of a smaller impact in year 1 than later years. A cost analysis considers the affordability of extending similar programs.

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.

Using a cluster-randomized controlled trial research design, this study investigates the effect of the first year of a three-year teacher professional development program on grades 1 and 2 student achievement in mathematics. Although the findings reported in the present report do not meet the standard cutoff for statistical significance (e.g., 95% confidence), effect-size estimates for student achievement in the first year of implementation were positive for the tests that focused on problem solving, applications of mathematics, and algebraic thinking, and they were negative for the computation-focused tests. Those effects varied by grade level. Implications and suggestions for next steps are discussed. [This report was prepared by the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM), Learning Systems Institute, Florida State University.]

This study tested the effects of implementing a narrative computer-based educational game within a middle-school math class. Gameplay consisted of navigating through a virtual spaceship and completing missions by periodically engaging in learning-by-teaching activities that involved helping an avatar solve math problems. In a pretest/posttest matched-groups design, 58 middle-school students either played the game for 10 hours over 4 days in place of their typical math instruction (game group), or they received conventional math instruction that consisted of a matched set of practice problems (control group). Contrary to our hypotheses, results from posttest measures indicated no significant differences in learning outcomes or motivation between the two groups. Importantly, supplementary observational data indicated that students in the game group spent much of their time during gameplay engaging in activities unrelated to the educational content of the game (e.g., navigating the virtual world) and only 20% of their time engaging in learning-by-teaching activities. These results highlight the importance of designing educational games that effectively balance features intended to entertain learners and features intended to promote learning. Implications for implementing educational games into classroom instruction are discussed.

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.

Math skills are critical for children's future success in school, as school-entry math knowledge is the strongest predictor of later academic achievement. Although there is a recent increase of literature on math with young children, there is a scarcity of research related to young children with disabilities. This quasi-experimental study with 50 preschool aged children with disabilities examined the effects of an intervention that integrated mathematics and literature on early numeracy skills. The intervention was conducted 3 days per week for 6 weeks, and consisted of an interactive shared storybook reading including mathematical content through scripted questioning and discussions and story-related mathematical activities after the reading of the story. Children who received the intervention scored significantly higher than the comparison group in total math ability, quantity comparison, one-to-one correspondence counting, and oral counting as measured by scores on the Test of Early Mathematics Ability, Third Edition (TEMA-3) and the Individual Growth & Development Indicators Early Numeracy (IGDIS-EN).

This experimental study was intended to examine whether the integration of game characteristics in the OpenSimulator-supported virtual reality (VR) learning environment can improve mathematical achievement for elementary school students. In this pre- and posttest experimental comparison study, data were collected from 132 fourth graders through an achievement test. The same math problem-solving tasks were provided to the experimental and control groups in the VR learning setting. Tasks for the experimental group involved the game characteristics of challenges, a storyline, immediate rewards, and the integration of game-play in the learning content. Analysis of covariance with the achievement test results indicated a significant effect of game-based learning (GBL) in the VR environment, in comparison with non-GBL in the VR environment, in promoting math knowledge test performance.

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."]

Since the 2000s, states have experimented with reforms to improve success among underprepared students traditionally assigned to developmental education (DE). Florida's reform under Senate Bill 1720 has been among the most comprehensive and wide-reaching. Recent public high school graduates and military personnel became exempt from DE, but nearly onethird of students, including those without a Florida standard high school diploma, were still required to take a placement test and enroll in DE if they scored below college-ready. The legislation also required colleges to offer accelerated instructional strategies for students remaining in DE, and provide enhanced advising and support services. Focusing specifically on nonexempt students, we use statewide data to conduct a difference-in-regression discontinuity analysis to examine differences in first-year math coursetaking outcomes for students on the margins of college readiness before and after the reform. While students narrowly assigned to DE tend to have a lower likelihood of taking and passing college-level courses relative to their college-ready peers, these students experienced larger gains after the reform when DE courses were offered in accelerated formats accompanied by support services. The reform also improved outcomes for students scoring above college-ready, which suggests that nonexempt students benefited from enhanced advising and support services too. [This article was published in "Journal of Higher Education" (EJ1281792).]

The purpose of this study was to examine the effects of algebraic equation solving intervention for sixth graders with mathematics learning difficulties (MD). A total of 48 students with MD were randomly assigned to either the algebraic equation solving intervention, "Mystery Math" (n = 24) or control condition (n = 24). The multicomponent intervention was based on the principles of explicit instruction and focused on improving conceptual and procedural knowledge of algebraic equation solving using concrete manipulatives. Students in the intervention group received instruction in pairs, 30 min per session, 3 sessions per week for 5 weeks (i.e., 15 sessions). The results indicated that the main effect of intervention was significant for 2 proximal measures of mathematics vocabulary, and conceptual and procedural understanding of algebraic equation solving with large effect sizes. However, the main effect of intervention was not significant for distal measures of comprehensive pre-algebra skills and whole-number computations. The findings demonstrate that grade-level standards can be successfully taught to students with MD. Implications for practice are discussed.

The Rural Math Innovation Network (RMIN) is a 4-year project that launched in January 2017 after receiving a $2.9 million Investing in Innovation (i3) development grant from the U.S. Department of Education (ED) and matching funds from the private sector. Virginia Ed Strategies and rural local education agencies (LEAs) in Virginia are implementing a project using a networked improvement community (NIC) of Pre-Algebra and Algebra 1 teachers to incorporate social-emotional learning (SEL) factors of academic self-efficacy and growth mindset into lesson plans for teaching career readiness math competencies. During Year 1, the project established Memos of Understanding with 18 school divisions in southwest and southside Virginia, which enabled math teachers within these divisions to submit applications to participate in the project. At the end of Year 1, December 2017, the project had a 38-member teacher cohort across 25 schools. By the end of Year 2 (January 1 - December 31, 2018), the cohort included 30 teachers (19 middle school teachers and 11 high school teachers) across 20 schools (12 middle schools and 8 high schools) within 16 participating divisions (several teachers dropped out of the project in Year 2 and a few teachers were added). During Year 3 (January 1 - December 31, 2019), several more teachers dropped out of the project, resulting in 26 teachers (17 middle school and 9 high school) across 18 schools within 15 participating divisions. Sixteen of the 26 teachers are located in the southside region, with the remaining 10 teachers located in the southwest region of Virginia One of the i3 requirements is to have an external evaluation conducted of the project; development grants must include both an implementation study and an impact study. To fulfill this requirement, Virginia Ed Strategies hired ICF to conduct an independent evaluation of the RMIN project throughout the 4-year period. The evaluation includes three components: a formative study to provide ongoing feedback about participants' reactions, learning, behaviors, and results; an implementation study focusing on how well the structural and programmatic aspects of the RMIN project are implemented, as well as facilitating or impeding factors; and an impact study to determine the extent to which the project impacts high-need students' math achievement. Previously in Year 2, the evaluation team recruited 10 comparison teachers across rural Virginia school divisions who were also teaching either Pre-Algebra or Algebra 1 (no school has both a participating and comparison teacher). The purpose of this report is to summarize key findings from the impact study. The primary audience is the RMIN project staff at Virginia Ed Strategies; secondary audiences include ED and other interested stakeholders. Although the impact study was designed originally to include students from the 2018-19 and 2019-20 school years, the COVID-19 pandemic in spring 2020 led to school closings and no administration of the statewide Standards of Learning (SOL) math assessment. Therefore, the impact study is based solely on the one year of SOL data. Findings are presented for the impact data and framed by the evaluation questions. Conclusions are presented below. Impact on student achievement. The one-year program impact on students' SOL scores was estimated and the results did not find evidence that the RMIN program significantly improved students' SOL performance. The program impact from the Pre-Algebra sample was negative, but it was not statistically significant, and the effect size was small. The program impact from the Algebra I analysis was positive but not statistically significant, and the effect size was small.

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.

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.

Community colleges are a large sector of postsecondary education. In 2016-2017, the United States had nearly 1,000 public 2-year postsecondary institutions (community colleges), serving almost nine million students, representing 39% of all undergraduates. The majority of entering community college students require developmental (or remedial) math. Success rates in the developmental math course sequence and college more broadly are discouragingly low. Policymakers, practitioners, and researchers alike are eagerly searching for reforms to improve success rates, but there is a dearth of causal evidence on the effectiveness of most proposed reforms. We sought to answer the following question: what effect does a modularized, computer-assisted, self-paced approach to developmental math (compared with a more "traditional" direct-instruction course alternative) have on students' likelihood of completing the developmental math course sequence? Findings from a randomized controlled trial (n=1,403) are presented. The program was well implemented; however, we did not find evidence that this approach was superior to the "traditional" math class. Although these results are disappointing, they are important because modularization and self-paced computer-assisted instruction are popular reforms. [This article was published in "Journal of Research on Educational Effectiveness" (EJ1229042).]

The National Math + Science Initiative's (NMSI's) College Readiness Program (CRP) is an established program whose goal is to promote science, technology, engineering, and mathematics education in high schools to improve students' readiness for college. The program provides teacher, student, and school supports to promote high school students' success in mathematics, science, and English Advanced Placement (AP) courses, with a focus on students who are traditionally underrepresented in the targeted AP courses. Through a scale-up grant awarded to NMSI by the Investing in Innovation (i3) program, the CRP was implemented in 28 schools in the 2016-2017 school year. CRESST conducted an independent evaluation of the impact of the CRP on students' AP outcomes using a randomized cluster trial with 28 CRP schools and 24 control schools in 10 states. The evaluation of the CRP consisted of two parts: (1) assessment of the program's impact on selected student AP exam outcomes and (2) assessment of the fidelity of implementation of the CRP. Program impact was evaluated using a 2-level hierarchical generalized linear model (HGLM) with students nested within schools The descriptive statistics showed that a higher percetange of students in the treatment schools took at least one AP course (30.7%) compared to those in the control schools (26.4%) by approximately 4.3%, however the difference was not statistically significant. In addition, students in the treatment schools were not more likely to achieve a score of 3 or higher, when compared to the delayed treatment schools. We further examined the effectiveness of the CRP using the prior year's school-level performance on the AP exam as a covariate. As with the above findings, the results indicated the probability of a student taking at least one AP course or scoring 3 or higher on at least one AP exam is not statistically different between students in the treatment schools and those in the control treatment schools. Fidelity of implementation was evaluated using a fidelity matrix approach (required as part of the evaluation of the i3 program), which showed that not all elements of the program were implemented with high fidelity. Overall results, however, indicated that 23 schools out of 28 treatment schools (82.1%) achieved 80% or better implementation fidelity, for an average fidelity score of 89.5%. Seven schools achieved a perfect 100% fidelity score. Looking at the different indicator groups (school, teacher and student), we found that all school support measures across all schools were implemented with fidelity. In over 80% of schools, not all teachers fulfilled their requirements for attending all training sessions, and so this component was not implemented with fidelity. Stipends and teacher awards were paid as expected as were student award payments. Teacher survey data indicated that teachers found the training and professional development activities provided by the CRP to be the most beneficial program supports relating to helping increase student achievement in AP courses. Teacher incentives were chosen as the least important program component relating to increasing student performance by 16% of teachers and student incentives by 12% of teachers. Teachers did, however, view the student incentives as an important program component to encourage enrollment in AP courses. Likewise, students rated the financial incentives on average as somewhat important in encouraging them to participate in AP courses.

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.]

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.

This "Math Snacks" intervention measured 741 fifth grade students' gains in conceptual understanding of core math concepts after game-based learning activities. Teachers integrated four "Math Snacks" games and related activities into instruction on ratios, coordinate plane, number systems, fractions and decimals. Using a randomized, controlled, quasi-experimental design, classrooms were randomly assigned to one of two conditions in a delayed treatment model. First Group A, and then Group B, received 5 weeks of "Math Snacks" intervention in addition to the regular district mathematics curriculum. Using a carefully refined test containing multiple choice and open-ended items, both groups were assessed three times: prior to any interventions, at the end of Phase 1 (when only Group A had received the intervention), and at the end of Phase 2 (when both groups had received the intervention). Students' mean gains over 5 weeks were significantly higher while receiving the "Math Snacks" intervention as compared to the group not receiving the intervention.

This is the final external evaluation report prepared by SRI International for the Rural Math Excel Partnership (RMEP) project, an investing in innovation (i3) development project funded by the U.S. Department of Education. Operated by Virginia Advanced Study Strategies, Inc. (VASS), the RMEP project included six rural school districts (LEAs) in five Virginia counties as partners. The project goal was to develop and implement a model of shared responsibility among families, math teachers, and communities in rural areas to prepare students enrolled in Algebra I, Geometry, Algebra II, and Algebra Functions and Data Analysis (AFDA) courses for success in advanced high school and postsecondary STEM studies. The long term outcome was for students to leave school ready, at a minimum, to enroll in postsecondary programs focused on technician-level careers in STEM-related fields considered important to the regional rural economy. Due to low levels of implementation by some teachers in project years 1 and 2, in fall 2015 (year 3) the RMEP team focused their supports and services on a group of 24 high-implementation teachers in the seven middle and seven high schools. Key implementation and impact findings were the RMEP team completed five of the six core implementation activities meeting the standard of performance set by the evaluation team; student and family access to technology did not meet performance standards. It was difficult and time consuming to locate students and families in need of tablets and broadband access at the 14 schools and then to provide these individuals with the necessary services in their homes. Furthermore, district firewalls, teacher comfort level with technology, and registration requirements for the MARi online video platform created significant delays for the RMEP project throughout Year 2. By the end of the project's third year, however, RMEP provided technology access to all the students and their families of the 24 teachers. Willingness of individual teachers to perform their role in the model of shared responsibility varied, especially in the number of videos that teachers assigned to students and their efforts to hold Family Math Nights. Although full implementation of the model was restricted to a single semester, there was evidence that this higher level of support was beginning to have positive impacts on the teachers in terms of video assignments and student completion of these assignments. Evaluations from families and students showed that participating in RMEP-related events were useful and worth their time, though attendance was lower than expected for these events. Teachers and community members reported that organizing these events required a large time commitment and that they needed more help in identifying ways to increase attendance. SRI evaluators found that the RMEP project had no impact on students' achievement or attitudes by the end of 2015. Possible reasons included differences between the content knowledge that the Virginia Standards of Learning (SOL) exams assess and the content emphasized by the RMEP project. Evaluators were not able to limit the sample to only those students whose teachers implemented the intervention in the 2015-16 school year (the high-implementing teachers). The small sample size of students may not have been large enough to detect a very small effect. Key model components were not fully implemented until fall 2015, an insufficient time period for the intervention to have a significant impact on the targeted outcomes. Nevertheless, the RMEP project can serve as an illustrative example for other such initiatives, and suggests that similar projects should consider level of participant buy-in, anticipate and be able to troubleshoot technology access issues, and provide enough time as well as staff support for full implementation.

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.

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.

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.

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.

This study investigated whether individual differences in working memory (WM) moderate effects of 2 variations of intervention designed to improve at-risk 4th graders' fraction knowledge. We also examined the effects of each intervention condition against a business-as-usual control group and assessed whether children's measurement interpretation of fractions mediated those effects. At-risk students (n = 243) were randomly assigned to control and 2 intervention conditions. The interventions each lasted 12 weeks, with three 30-min sessions per week. The major focus of both intervention conditions was the measurement interpretation of fractions. Across the 2 conditions, only 5 min of each 30-min session differed. One condition completed activities to build fluency with 4 measurement interpretation topics; in the other, activities were completed to consolidate understanding on the same 4 topics. Results revealed a significant aptitude-treatment interaction, in which students with very weak WM learned better with conceptual activities but children with more adequate (but still low) WM learned better with fluency activities. Both intervention conditions outperformed the control group on all outcomes, and improvement in the measurement interpretation of fractions mediated those effects.

Cognitive strategies are important tools for children with math difficulties (MD) in learning to solve word problems. The effectiveness of strategy training, however, depends on working memory capacity (WMC). Thus, children with MD but with relatively higher WMC are more likely to benefit from strategy training, whereas children with lower WMC may have their resources overtaxed. Children in Grade 3 (N = 147) were randomly assigned to 1 of 4 conditions: (a) verbal strategies (e.g., underlining question sentence), (b) visual strategies (e.g., correctly placing numbers in diagrams), (c) verbal plus visual strategies, or (d) an untreated control. In line with the predictions, children with MD and higher WMC benefited from verbal or visual strategies relative to those in the control condition on posttest measures of problem solving, calculation, and operation span. In contrast, cognitive strategies decreased problem-solving accuracy in children with low WMC. Thus, improvement in problem solving and related measures, as well as the impairment in learning outcomes, was moderated by WMC.

Cognitive strategies are important tools for children with math difficulties (MD) in learning to solve word problems. The effectiveness of strategy training, however, depends on working memory capacity (WMC). Thus, children with MD but with relatively higher WMC are more likely to benefit from strategy training, whereas children with lower WMC may have their resources overtaxed. Children in Grade 3 (N = 147) were randomly assigned to 1 of 4 conditions: (a) verbal strategies (e.g., underlining question sentence), (b) visual strategies (e.g., correctly placing numbers in diagrams), (c) verbal plus visual strategies, or (d) an untreated control. In line with the predictions, children with MD and higher WMC benefited from verbal or visual strategies relative to those in the control condition on posttest measures of problem solving, calculation, and operation span. In contrast, cognitive strategies decreased problem-solving accuracy in children with low WMC. Thus, improvement in problem solving and related measures, as well as the impairment in learning outcomes, was moderated by WMC.

The purpose of this study was to investigate the effectiveness of explicit, direct, and generative strategy training and working memory capacity (WMC) on mathematical word problem-solving accuracy in elementary schoolchildren. In this study, children in third grade ("N" = 82) identified as at risk for math difficulties (MD) were randomly assigned (within classrooms) to one of three treatment conditions that explicitly directed students' attention to different propositions within word problems--paraphrase question propositions (Restate), paraphrase relevant propositions (Relevant), and paraphrase all propositions (Complete)--or an untreated control condition. A significant treatment by covariate design indicated that generative strategy outcomes were conditional on the level of pretest WMC. A clear advantage in posttest problem-solving accuracy and solution planning was found for the complete generative condition relative to the control condition, but this advantage was conditional on setting WMC to a high level. Although no significant treatment advantages were found for solution accuracy when WMC was set to a low level, treatment advantages relative to the control condition were found for measures of schema activation. The results indicated that the effectiveness of generative strategies among children at risk for MD was directly dependent on the level of WMC.

The purpose of this study was to investigate the effectiveness of explicit, direct, and generative strategy training and working memory capacity (WMC) on mathematical word problem-solving accuracy in elementary schoolchildren. In this study, children in third grade ("N" = 82) identified as at risk for math difficulties (MD) were randomly assigned (within classrooms) to one of three treatment conditions that explicitly directed students' attention to different propositions within word problems--paraphrase question propositions (Restate), paraphrase relevant propositions (Relevant), and paraphrase all propositions (Complete)--or an untreated control condition. A significant treatment by covariate design indicated that generative strategy outcomes were conditional on the level of pretest WMC. A clear advantage in posttest problem-solving accuracy and solution planning was found for the complete generative condition relative to the control condition, but this advantage was conditional on setting WMC to a high level. Although no significant treatment advantages were found for solution accuracy when WMC was set to a low level, treatment advantages relative to the control condition were found for measures of schema activation. The results indicated that the effectiveness of generative strategies among children at risk for MD was directly dependent on the level of WMC.

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 effects of a mathematical problem-solving intervention on students' problem-solving performance and math achievement were measured in a randomized control trial with 1,059 7th-grade students. The intervention, "Solve It!," is a research-based cognitive strategy instructional intervention that was shown to improve the problem-solving performance of 8th-grade students with and without learning disabilities (LD). The purpose of the present study was to determine whether the effectiveness of the intervention could be replicated with younger students. Forty middle schools in a large urban school district were included in the study, with one 7th-grade math teacher participating at each school (after attrition, n = 34). "Solve It!" was implemented by the teachers in their inclusive math classrooms. Problem-solving performance was assessed using curriculum-based math problem-solving measures, which were administered as a pretest and then monthly over the course of the 8-month intervention. Students who received the intervention (n = 644) embedded in the district curriculum showed a significantly greater rate of growth on the curriculum-based measures than students in the comparison group (n = 415) who received the district curriculum only. Results of the Bayesian analyses indicated that the intervention effect was somewhat stronger for low-achieving students than for average-achieving students. Overall, findings from the present study as well as the previous study with 8th-grade students indicate that the intervention was effective across ability groups and is an appropriate program to use in inclusive classrooms with students of varying math ability.

"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.

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.

This study investigated the role of generative strategies and working memory capacity on word problem solving accuracy in children with math difficulties (MD). Within classrooms, children in Grade 3 with MD ("n" = 69) were randomly assigned to one of three treatment conditions: paraphrase question propositions (Restate), paraphrase relevant propositions (Relevant), and paraphrase all propositions (Complete), or to an untreated control. An additional control group included children without MD ("n" = 22). Mixed regression modeling showed that generative strategies significantly improved posttest scores for children with MD compared with the control condition, but outcomes were related to the type of dependent measures. The Relevant and Complete treatment conditions improved problem-solving accuracy, the Complete condition improved problem component identification, and the Restate and Relevant conditions improved operation span performance when compared with the control conditions. Only the Relevant and Complete generative learning treatments allowed children with MD to catch up to children without MD, but the results were moderated by working memory capacity.

This study investigated the role of generative strategies and working memory capacity on word problem solving accuracy in children with math difficulties (MD). Within classrooms, children in Grade 3 with MD ("n" = 69) were randomly assigned to one of three treatment conditions: paraphrase question propositions (Restate), paraphrase relevant propositions (Relevant), and paraphrase all propositions (Complete), or to an untreated control. An additional control group included children without MD ("n" = 22). Mixed regression modeling showed that generative strategies significantly improved posttest scores for children with MD compared with the control condition, but outcomes were related to the type of dependent measures. The Relevant and Complete treatment conditions improved problem-solving accuracy, the Complete condition improved problem component identification, and the Restate and Relevant conditions improved operation span performance when compared with the control conditions. Only the Relevant and Complete generative learning treatments allowed children with MD to catch up to children without MD, but the results were moderated by working memory capacity.

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.)

Stanford University's Education Program for Gifted Youth (EPGY) conducted a randomized-treatment experiment during the 2006-2007 school year to test the efficacy, for Title I students, of the technological and individualized EPGY Kindergarten through Grade 5 Mathematics Course Sequence, modified for the Title I schools. Restricting attention to students who were in the top half of the distribution of correct first-exercise attempts (a measure of work and engagement), we found substantial and statistically significant improvements in the 2007 California Standard Math Tests (CST07) scores compared to those of matched control students. Gains in second grade were larger than those in Grades 3 to 5. Less able students, as measured by their 2006 CST mathematics scores, also had, on average, larger gains.

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.]

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.)

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.]

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 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.)

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 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.

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.)

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.

In an effort to identify instructional methods that might improve mathematics learning at the grade 4 level when used in a variety of educational settings under typical conditions, the REL Mid-Atlantic research team looked for promising, replicable practices that were being used broadly by teachers in U.S. schools, for which research showed promising results but had not been conducted using methodologies that can establish causal relationships. CompassLearning's Odyssey[R] Math product met all of these criteria. Odyssey Math is a computer-based math curriculum developed by CompassLearning, Inc., to improve math learning for K-12 students. The software consists of a web-accessed series of learning activities, assessments, and math tools. These components constitute the basic framework of the software. CompassLearning professional development trainers presented the learning activities, math tools, and assessments as available options to intervention teachers during the summer professional development session. This study was the first randomized controlled trial to assess the impact of Odyssey Math on student achievement. The study had the statistical power needed to detect a 0.20 effect size and was well designed in that comparable groups were created at baseline and maintained through posttesting. Implementation during the school year was documented and shown to be consistent with typical implementation of the Odyssey Math software. The results from the multilevel model with pretest covariates also indicate that Odyssey Math did not yield a statistically significant impact on end-of-year student achievement. This study generated a statistically unbiased estimate of the effect of Odyssey Math on student achievement when implemented in typical school settings with typical teacher and student use. However, the findings apply only to participating schools, teachers, and students because the study used a volunteer sample. Twelve appendices include: (1) Detailed Professional Development Agenda Sessions; (2) Statistical Power Analysis; (3) Probability of Assignment to Study Conditions; (4) Sample Size from Random Assignment to Data Analysis; (5) Teacher Survey, Fall 2007; (6) Observation Protocols; (7) Odyssey Math Sample Screens; (8) Fidelity Observation Comparisons; (9) Model Variance and Intraclass Correlations; (10) Complete Multilevel Model Results for Research Question 1; (11) Comparison of Assumed Population Parameters for Statistical Power (During Planning Phase) with Corresponding Sample Statistics (During Analysis Phase); and (12) Equations for Multilevel Model Analyses. (Contains 3 figures, 22 tables, 10 exhibits, and 23 footnotes.) [This report was prepared for the National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences under contract with Regional Educational Laboratory Mid-Atlantic administered by Pennsylvania State University.]

The purpose of this study was to assess the efficacy of fact retrieval tutoring as a function of math difficulty (MD) subtype, that is, whether students have MD alone (MD-only) or have concurrent difficulty with math and reading (MDRD). Third graders (n = 139) at two sites were randomly assigned, blocking by site and MD subtype, to four tutoring conditions: fact retrieval practice, conceptual fact retrieval instruction with practice, procedural computation/estimation instruction, and control (no tutoring). Tutoring occurred for 45 sessions over 15 weeks for 15-25 minutes per session. Results provided evidence of an interaction between tutoring condition and MD subtype status for assessment of fact retrieval. For MD-only students, students in both fact retrieval conditions achieved comparably and outperformed MD-only students in the control group as well as those in the procedural computation/estimation instruction group. By contrast, for MDRD students, there were no significant differences among intervention conditions.

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.)

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.

This report describes results from field-testing of POWERSOURCE[C] formative assessment alongside professional development and instructional resources. The researchers at the National Center for Research, on Evaluation, Standards, & Student Testing (CRESST) employed a randomized, controlled design to address the following question: Does the use of POWERSOURCE[C] strategies improve 6th-grade student performance on assessments of the key mathematical ideas relative to the performance of a comparison group? Sixth-grade teachers were recruited from 7 districts and 25 middle schools. A total of 49 POWERSOURCE[C] and 36 comparison group teachers and their students (2,338 POWERSOURCE[C], 1,753 comparison group students) were included in the study analyses. All students took a pretest of prerequisite knowledge and a transfer measure of tasks drawn from international tests at the end of the study year. Students in the POWERSOURCE[C] group used sets of formative assessment tasks. POWERSOURCE[C] teachers had exposure to professional development and instructional resources. Results indicated that students with higher pretest scores tended to benefit more from the treatment as compared to students with lower pretest scores. In addition, students in the POWERSOURCE[C] group significantly outperformed control group students on distributive property items and the effect was larger as pretest scores increased. Results, limitations and future directions are discussed. Four appendices are included: (1) Item Analysis Results of POWERSOURCE[C] pretest; (2) POWERSOURCE[C] Pretest questionnaire; (3) POWERSOURCE[C] Pretest questionnaire; and (4) Descriptive Statistics of Posttest scores by District and Treatment, School, Teacher in Between-School Design and Teacher in within-School (W-S) Design. (Contains 12 figures and 29 tables.)

Numerous high school students, including many who are enrolled in career and technical education (CTE) courses, do not have the math skills necessary for today's high-skill workplace or college entrance requirements. This study tests a model for enhancing mathematics instruction in five high school CTE programs (agriculture, auto technology, business and marketing, health, and information technology). The model includes a pedagogy and intense teacher professional development. Volunteer CTE teachers were randomly assigned to an experimental (n = 59) or control (n = 78) group. The experimental teachers worked with math teachers to develop CTE instructional activities that integrated more mathematics into the occupational curriculum. After 1 year of the math-enhanced CTE lessons, students in the experimental classrooms performed equally on technical skills and significantly better than control students on two standardized tests of math ability (TerraNova and ACCUPLACER[R]). (Contains 5 tables, 1 figure, and 9 notes.)

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.)

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.)

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.)

We examined the extent to which use of a technology-enhanced continuous progress monitoring system would enhance the results of math instruction, examined variability in teacher implementation of the program, and compared math results in classrooms in which teachers did and did not use the system. Classrooms were randomly assigned to within-school experimental and control groups. Participating students were pre- and post-tested using two standardized, nationally normed tests of math achievement. When teachers implemented the continuous progress monitoring system as intended, and when they used the data from the system to manage and differentiate instruction, students gained significantly more than those for whom implementation was limited or nil. Failure to take into account intervention integrity would have made it look like continuous progress monitoring did not enhance math results. (Contains 5 tables.)

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 purpose of this study was to assess the effects of schema-broadening instruction (SBI), with and without explicit instruction in strategies for tackling the complexities involved in real-life (RL) math problems, on the math problem solving of third-grade students. Teachers (n = 30) were assigned randomly to 3 16-week conditions: control, SBI, and SBI-RL. Students in their classes (n = 455) were pre- and posttested on problem-solving measures at increasing transfer distances from the problems used for teaching problem solutions, with far transfer mirroring real-life problems. For the most part, SBI students, with and without RL, improved statistically significantly better than control students, with large effect sizes, but comparably to each other. By contrast, on the least structured real-life question, paralleling the ambiguities of real-life problem solving most closely, reliable and large effects favored the SBI-RL students over SBI and control students, suggesting RL's potential.

This study investigated the effectiveness of combining enhanced classroom instruction and intense supplemental intervention for struggling readers in first grade. Further, it compared two supplemental interventions derived from distinct theoretical orientations, examining them in terms of effects on academic outcomes and whether children's characteristics were differentially related to an instructional intervention. One intervention (Proactive Reading) was aligned with behavioral theory and was derived from the model of Direct Instruction. The other intervention (Responsive Reading) was aligned with a cognitive theory and was derived from a cognitive-apprenticeship model. These interventions were provided to small groups of first-grade students at risk for reading difficulties. Students were assessed on various reading and reading-related measures associated with success in beginning reading. Results indicated that (a) first-grade students who were at risk for reading failure and who received supplemental instruction in the Responsive or Proactive interventions scored higher on measures of reading and reading-related skills than students who received only enhanced classroom instruction, (b) enhanced classroom instruction appeared to promote high levels of reading growth for many children at risk for reading failure, (c) the two interventions were essentially equally effective even though they reflected different theoretical perspectives, and (d) children's characteristics did not differentially predict the effectiveness of an intervention.

This study investigated the possibility of enhancing the level of preschoolers number sense by introducing two intervention programmes, "Lets think!" and "Young children with special educational needs count, too!" Forty-five preschoolers, mean age 66.4 months, were randomly assigned to the experimental and control groups. The experimental group received instruction twice a week, for half an hour, and for a period of nine months. Pretest?postest comparison revealed that the experimental group showed enhanced number-sense performance immediately after the instruction ended, but the between-group difference faded after six months. There were no statistically significant differences between the groups in general mathematical thinking abilities (transfer tasks) after the intervention.

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.

Mathematical problem solving is a transfer challenge requiring children to develop schemas for recognizing novel problems as belonging to familiar problem types for which they know solutions. Schema-based transfer instruction (SBTI) explicitly teaches transfer features that change problems in superficial ways to make them appear novel even though they still require known solution strategies. This study assessed the effects of an expanded version of SBTI incorporating more challenging transfer features for broadening schemas and helping children recognize real-life math problems as solvable. Teachers were assigned randomly to 16-week control, SBTI, or expanded SBTI conditions. Students completed pretests and posttests focusing on increasing transfer distances. On a measure approximating real-life problem solving, the expanded SBTI group outperformed the SBTI group, which in turn outperformed the control group. (Contains 1 figure, 3 tables, and 3 footnotes.)

The purposes of this study were to assess the effects of schema-based instruction (SBI) in promoting mathematical problem solving and to investigate schema induction as a mechanism in the development of mathematical problem solving. Twenty-four 3rd-grade teachers, with 366 students, were assigned randomly to conditions that provided instruction on 4 types of word problems. The 3 16-week conditions were contrast, SBI, and SBI with practice in sorting word problems into schemas. Students were pre- and posttested on mathematical problem-solving tests and were posttested on schema development. Students receiving SBI, with and without sorting practice, improved more than the contrast group on problem-solving measures. Concurrently, the SBI groups' schema development exceeded that of the contrast group, and schema development explained a substantial portion of unique variance in students' posttreatment problem-solving performance. Results also suggested the need for additional research testing the contribution of practice in sorting word problems.

Studied the effects of four instructional methods on the mathematical reasoning and metacognitive knowledge of 384 eighth graders. Results show that cooperative learning plus metacognitive training outperformed other combinations of individual and cooperative learning and metacognitive instruction or learning without metacognitive instruction. (SLD)

Assesses the effects of explicitly teaching for transfer by broadening the categories by which students group problems requiring the same solution methods and prompting students to search novel problems for these broad categories. This transfer treatment was combined with instruction on solution methods. Improvement on immediate- and near-transfer measures supported the utility of solution instruction. (Contains 44 references, 2 tables, and 2 appendixes.) (GCP)

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)

Assesses the contribution of self-regulated learning strategies (SRL), when combined with problem-solving transfer instruction, on 3rd-graders' mathematical problem solving. SRL incorporated goal setting and self-evaluation. Contrasts the effectiveness of transfer plus SRL to the transfer treatment alone and to teacher-designed instruction. SRL positively affected performance. (Contains 39 references and 5 tables.) (GCP)

Developed a heuristic strategy (SOLVED) to explain the processes involved in understanding and solving mathematical word problems (problem translation, problem interpretation, solution planning, solution execution, and solution monitoring) among upper elementary students. Results indicated that SOLVED was more effective in aiding both short-term and delayed problem solving than was traditional problem solving instruction. Accuracy in problem solving significantly correlated with metacognitive processing. (SM)

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)

Examines the effects of a dyadic peer-mediated treatment, Peer-Assisted Learning Strategies (PALS), on first-grade children's mathematics development. Results indicated that treatment implementation was strong; teachers judged PALS to be effective and generally feasible; and students with and without disabilities, at all points along the achievement continuum, benefited from PALS. Implications for research and practice are discussed. (Contains 35 references and 3 tables.) (GCP)

Teachers implemented a 6-week classroom intervention designed to promote emergent math skills and math interest in preschool-aged children. After the program, experimental children scored significantly higher than control children on a standardized test of math ability and enjoyed math activities more than the control children, as measured by both teacher and self-report. (Contains 94 references and 2 tables.) (GCP)

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)

Examined the effects of a dyadic peer-mediated treatment on kindergarten children's mathematics development. Found that treatment implementation was strong for most teachers; teachers judged the treatment to be effective and feasible for implementation on their own; and students with and without disabilities, at all points along the achievement continuum, benefited from the treatment. (Author)

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)

Develops and tests a learning environment for teaching and learning how to model and solve mathematical application problems using four classes of fifth graders. Indicates that the intervention had a positive effect on different aspects of pupils' mathematical modeling and problem solving abilities. (Contains 62 references.) (Author/ASK)

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)

Four fifth and sixth grade students with learning disabilities were taught a strategy for comprehending word problems and devising appropriate solutions. Following instruction performance on mixed sets of addition and subtraction word problems improved. Although generalization to a different setting occurred, maintenance was mixed. (Author/DB)

Six students (ages 12-14) with learning disabilities received either cognitive or metacognitive strategy instruction for mathematical problem solving, followed by instruction in the complementary component. Results indicated that cognitive and metacognitive strategies were more effective than either cognitive or metacognitive strategy instruction alone. Students did not, however, maintain the strategy over time. (JDD)

Four fifth and sixth grade students with learning disabilities were taught a strategy for comprehending word problems and devising appropriate solutions. Following instruction performance on mixed sets of addition and subtraction word problems improved. Although generalization to a different setting occurred, maintenance was mixed. (Author/DB)

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)

This study compared the effectiveness of 3 procedures for teaching 62 elementary students with learning disabilities to identify the correct algorithm in solving addition and subtraction word problems. The group receiving strategy teaching and sequencing practice problems and the group receiving strategy teaching only scored higher than sequencing-only students. (Author/JDD)

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)

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)

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)