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

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

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

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 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.]

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.]

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.

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

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

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

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.

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

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

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

This study 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.

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

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

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

Students who fail algebra are significantly less likely to graduate on time, and algebra failure rates are consistently high in urban districts. Identifying effective credit recovery strategies is critical for getting students back on track. Online courses are now widely used for credit recovery, yet there is no rigorous evidence about the relative efficacy of online versus face-to-face credit recovery courses. To address this gap, this study randomly assigned 1,224 ninth graders who failed algebra in 17 Chicago public high schools to take an online or face-to-face algebra credit recovery course. Compared to students in face-to-face credit recovery, students in online credit recovery reported that the course was more difficult, were less likely to recover credit, and scored lower on an algebra posttest. There were no statistically significant differences by condition on any outcomes measured during the second year of high school (standardized mathematics test and algebra subtest scores, likelihood of passing subsequent math classes, cumulative math credits, or on-track rates). The benefits and challenges of online learning for credit recovery are discussed in light of the findings to date.

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

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

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

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

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

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.]

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.

The purpose of this large-scale, national study is to determine whether some early elementary school math curricula are more effective than others at improving student math achievement, thereby providing educators with information that may be useful for making adequate yearly progress (AYP). This report presents results from the first cohort of first grade in 39 schools participating in the evaluation during the 2006-2007 school year, with the goal of determining the relative effects of different early elementary math curricula on student math achievement in disadvantaged schools. The report also examines whether curriculum effects differ for student subgroups in different instructional settings. A competitive process was used to select four curricula Investigations in Number, Data, and Space; Math Expressions; Saxon Math; and Scott Foresman-Addison Wesley Mathematics) that represent many of the approaches used to teach elementary school math in the United States. An experimental design randomly assigned schools in each participating district to the four curricula, setting up an experiment in each district. The relative effects of the curricula were calculated by comparing math achievement of students in the four curriculum groups. Hierarchical linear modeling (HLM) techniques were used to conduct statistical tests to assess the significance of all the results. Curriculum implementation findings include: (1) All teachers received initial training from the publishers and 96 percent received follow-up training; combined training varied by curriculum and ranged from 1.4 to 3.9 days; (2) Nearly all teachers reported using their assigned curriculum as their core math curriculum and about a third reported supplementing their curriculum with other materials; (3) Eighty-eight percent of teachers reported completing at least 80 percent of their assigned curriculum; and (4) On average, Saxon Math teachers reported spending one more hour on math instruction per week than did teachers of the other curricula. Achievement findings include: (1) Student math achievement was significantly higher in schools assigned to Math Expressions and Saxon Math, than in schools assigned to Investigations in Number, Data, and Space and Scott Foresman-Addison Wesley Mathematics; and (2) Math achievement in schools assigned to the two more effective curricula was not significantly different, nor was math achievement in schools assigned to the two less effective curricula. Another 71 schools joined the study during the 2007-2008 school year and curriculum implementation occurred in both the first and second grades in all participating schools. A follow-up report is planned that will present results based on all 110 schools participating in the evaluation, and for both the first and second grades. The study also is supporting curriculum implementation and data collection during the 2008-2009 school year in a subset of schools, in which implementation will be expanded to the third grade. A third report is planned that will present those results. Four appendixes are included; (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 Analysis Samples and Estimating Curriculum Effects. (Contains 66 footnotes, 10 figures and 43 tables.) A table of acronyms is included. [For Executive Summary of this report, see ED504419.]

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.

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

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

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

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

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.)

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

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