The purpose of this chapter is to describe effective mathematics instructional programs delivered in general education settings by elementary classroom and middle school mathematics teachers. All of the programs described can be used within tier I of a response-to-intervention (RTI) service delivery model to improve access to core mathematics instruction and promote mathematics success for all students. Although significant advances have been made in developing effective reading programs aimed at improving students’ reading ability, the research base on core mathematics (i.e., tier I) programs for elementary and middle grades that could be used in an RTI framework is limited (Clarke et al. 2010; Clarke et al. 2008; Crawford and Ketterlin-Geller 2008; Gersten et al. 2009a). This chapter includes information related to the nature and emphasis of the available instructional programs, the instructional settings in which they have been examined, and how all students, including those struggling in mathematics or at risk for mathematics difficulties, responded to core mathematics instruction. The authors note that while effective tier 1 instruction is critical to meet the instructional needs of a range of learners, paying particular attention to struggling students is important because these students may need subsequent tier 2 or tier 3 instruction, and it would be inappropriate to place students in tier 2 if there is no evidence to suggest they had the opportunity to learn from well-designed tier 1 instruction.

Much effort has been invested in the last two decades on improving mathematics instruction so that all students meet the high standards and expectations as measured by state-administered achievement tests (see NMAP 2008). As such, it is crucial that students at risk for mathematics difficulties, who vary considerably in ability, achievement, and motivation, develop the necessary mathematical knowledge to meet grade-level benchmarks. Within an RTI framework, the standards serve as the primary source of learning objectives for tier 1 instruction and play a key role in determining which students need extra support to prevent mathematics difficulties. Further, there is a need for effective professional tools (e.g., instructional programs) to ensure that tier I instruction adequately supports the development of critical mathematical concepts and skills for all students. This chapter begins with a discussion of critical mathematical concepts and skills to be included in instructional programs and the role of instructional practices in enhancing learning for all students and in preventing mathematics difficulties before they become intractable and present persistent problems for struggling learners. Instructional design features that are supported by research in promoting the mathematical development of struggling students are identified and described. Next, the nature and results of specific mathematics instructional programs in elementary and middle schools are described to understand the instructional conditions that need to be in place to promote mathematics success. Finally, the chapter concludes with a summary of limitations and recommendations for future research.

Promoting Mathematics Success Through Effective Core Instruction: Content and Focus of Instructional Programs and Practices

What elements need to be present for a tier 1 core mathematics instructional program to be successfully implemented in elementary and middle grades? Clarke et al. (2011) identify two key components that are critical for an effective tier 1 core program—(a) a focus on relevant content and (b) use of “research-based instructional design principles” (p. 565). To enhance the learning of all students and prevent serious mathematics difficulties, it is essential to not only examine the general education content and instructional environment (e.g., standards and expectations, materials) but also ensure that research-based practices are implemented to maximize the likelihood that all children have an opportunity to learn.

Reform activities in mathematics education in the first decade of 2000 focused primarily on mathematics curricula, attempting to reduce the number of topics and cover those topics in greater depth (Schmidt and Houang 2007). With the release of the curriculum focal points (2006) by the National Council of Teachers of Mathematics, the focus at each grade level was on critical content areas (e.g., number and operation, geometry, measurement) and instructional standards that emphasized problem-solving, reasoning, and critical thinking. The intent of the curriculum focal points was to circumvent the fragmenting of learning expectations and standards, and to focus curricula and instruction on areas that are foundational to mathematical learning. For example, several researchers and policy documents report that core instruction in the early elementary grades should ensure student development of number sense, which is a foundation for more advanced mathematics encountered in later grades (Milgram and Wu 2005; NMAP 2008). Core instructional programs used as part of tier I instruction should incorporate critical content that develop students’ mathematics knowledge. The National Mathematics Advisory Panel (NMAP; 2008) discussed the crucial role of algebra in the overall mathematics curriculum for grades K–8 and provided useful benchmarks for the foundations of algebra (i.e., proficiency with whole numbers, fractions, and particular aspects of geometry and measurement such as similar triangles and properties of two- and three-dimensional shapes) to guide mathematics curricula, instruction, and assessments. With the recent implementation of the common core state standards for mathematics (National Governors Association Center for Best Practices, Council of Chief State School Officers 2010), the ongoing emphasis on providing coverage of a relatively smaller number of topics at each grade level was intentional to ensure robust mathematical understanding. For students struggling with mathematics, the US Department of Education’s What Works Clearinghouse (Gersten et al. 2009a) specifically recommended that instructional materials focus on in-depth treatment of whole numbers in elementary grades and on rational numbers (i.e., a number that can be expressed as an integer or a quotient of integers, excluding zero as a denominator) in upper elementary and middle grades. One of the recommendations regarding content to prevent mathematics difficulties was focused “instruction on solving word problems that is based on common underlying structures” (Gersten et al. 2009a, p. 6).

In addition to the content or focus of instructional programs, educators should be cognizant of the instructional practices or instructional design principles that are characteristic of successful programs for all students, including students at risk for mathematics difficulties. The following section provides a summary of the key findings from several meta-analyses on effective instructional practices for students with or at risk for mathematics difficulties (see Jayanthi and Gersten 2011). Results of these syntheses have indicated that instructional programs that incorporated key instructional or curricula design elements benefitted students who struggled with mathematics (Baker et al. 2002; Gersten et al. 2009b; Kroesbergen and Van Luit 2003; Swanson and Hoskyn 1998; Xin and Jitendra 1999). Effective interventions included the following characteristics: (a) providing explicit instruction to teach mathematical concepts and procedures (Baker et al. 2002; Gersten et al. 2009b; Kroesbergen and Van Luit 2003; Swanson and Hoskyn 1998; Xin and Jitendra 1999); (b) teaching students to use heuristics (Gersten et al. 2009b); (c) encouraging students to think aloud while solving a problem (Gersten et al. 2009b); (d) using visual representations of mathematical ideas (Gersten et al. 2009b; Xin and Jitendra 1999); and (e) providing a range of examples and sequencing examples (e.g., concrete to abstract; Gersten et al. 2009b). Research support is unequivocal about the value of explicit and systematic instruction and teaching students to use heuristics (Gersten et al. 2009b; NMAP 2008). Explicit instruction includes “providing models of proficient problem- solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review” (Gersten et al. 2009a, p. 6). Another promising instructional approach involves increasing student motivation by providing opportunities for solving real-world problems (Gersten et al. 2009b). Finally, other instructional practices found to be effective for students at risk for mathematics difficulties included providing teachers with ongoing feedback about their students’ performance using formative assessments, providing feedback only to students, and cross-age tutoring (Gersten et al. 2009b; Baker et al. 2002). Given that tier I instruction is designed to meet the needs of students across the levels of achievement from below grade, at grade, and above grade, understanding the extent to which these instructional practices help to develop student competence in mathematics is critical.

When research-based instructional design elements are incorporated into mathematics instructional programs, a broad range of learners (i.e., students with and without mathematics difficulties) can be expected to achieve important mathematics knowledge and skills. Unfortunately, most tier 1 curricula in general education classrooms do not incorporate critical instructional design elements that are necessary to support the learning of students struggling with mathematics (Bryant et al. 2008; Doabler et al. 2012; Sood and Jitendra 2007). In reviewing studies in the next section, it is important to consider that even well-designed classrooms may not be sufficient to address the instructional needs of all students and that there could still be students who may not respond adequately to core instruction and need more intensive interventions and supports. Furthermore, variability in outcomes for students struggling with mathematics can be attributed to not only the different types of instructional programs but also to differences in the complexity of the content and intensity of instructional programs.

Relevant Research and Evidence of Effectiveness of Tier 1 Instructional Programs

In this section, the focus is on studies examining tier 1 core mathematics instruction, and studies are organized in terms of the intensity of the program described. For the purpose of this chapter, intensity is defined in terms of the total amount of time spent on tier 1 mathematics instruction. The authors arbitrarily categorized studies into three groups—high, medium, and low intensity—on the basis of the total amount of instructional time. Due to space limitations and relevance, studies that used a supplementary tutoring program (e.g., Fuchs et al. 2008, 2009; Jitendra et al. 2013a, b), had a limited content focus (e.g., basic arithmetic facts or procedures; e.g., VanDerHeyden et al. 2012), were delivered for a short duration (e.g., 10 min per day), or linked tier 1 instruction with tier 2 intervention (e.g., VanDerHeyden et al. 2007) were excluded. Furthermore, studies were not included if they used a class-wide mathematics instructional program, but did not discuss the findings for students struggling in mathematics (e.g., Codding et al. 2011; Fuchs et al. 2010; Jitendra et al. 2011). One note of caution about the findings is that not all possible tier 1 studies were reviewed in this chapter, and the studies selected included two studies of an early numeracy program, two studies targeting number sense using a class-wide peer tutoring approach, and seven studies focusing on word problem-solving.

The next section describes each study in terms of the participants and overall school context, as well as the nature and effectiveness of the instructional program. RTI is discussed in terms of the strength of the impact of the instructional program on mathematics outcomes using an effect size (ES) index. Although the studies reviewed reported the effects of the instructional programs for all students, this chapter focuses additionally on students at risk for mathematics difficulties (e.g., scoring low on mathematics tests) identified as such by the authors. ESs were computed for each study using Hedges’ g as “the difference between pretest–posttest means for the treatment group and the pretest–posttest means for the control group” (see Flynn et al. 2012, p. 24) divided by the pooled posttest standard deviation. Table 1 provides a summary of implications for practice in terms of the resources necessary to implement the interventions.

Table 1 Summary table of implications for practice
Full size table

High Intensity, Explicit Instruction

Two high-intensity studies provided promising findings for a comprehensive mathematics curriculum specially designed for a wide range of learners (Chard et al. 2008; Clarke et al. 2011). Each study evaluated the efficacy of a core kindergarten program, Early Learning in Mathematics (ELM), on the mathematics achievement of kindergartners. The ELM curriculum provides instruction in critical content (i.e., number and operations, measurement, and geometry) and mathematics vocabulary using research-based instructional practices (e.g., explicit and systematic instruction, frequent and cumulative review). The program systematically introduces mathematics topics and develops student understanding of critical concepts. A defining feature of ELM is instruction that is scaffolded, with initial instruction involving extensive modeling and representations of mathematics concepts and gradually removing the supports as students become more and more independent in their work.

Chard et al. (2008) conducted a pilot study to investigate the feasibility and promise of ELM in improving students’ mathematics achievement. All 254 kindergarten students (52 % were eligible for free or reduced-cost lunch, 14 % were minority students, and 6 % were English language learners) and their teachers in six treatment and five comparison schools in an urban school district participated in the study. Of the schools in the treatment condition, approximately half of them had implemented an initial version of ELM in the year prior to this study, while the remaining treatment condition schools served as comparison schools in the previous year so that treatment schools in this study had a differential amount of exposure to the ELM program. Although assignment to condition was not random, pretest performance on the measure of mathematics achievement was comparable between treatment and comparison schools. The 100-lesson ELM curriculum was implemented at least 4 days a week, 30 min per session, until all 100 lessons were completed. Students in treatment schools outperformed students in comparison schools on the SESAT-2 (Stanford Early School Achievement Test—Fourth Edition), a standardized measure of achievement ( g = 0.32). Using a mixed-effect ANCOVA (analysis of covariance) model where schools were treated as the unit of analysis, Chard et al. found the difference between treatment and comparison schools to be statistically significant. Chard et al. did not separately report scores for higher- or lower-scoring students in the treatment and comparison groups. However, their analyses found no statistically significant differences between lower- and higher-scoring students, suggesting that ELM was equally effective for students in both lower- and higher-scoring subgroups. As Chard et al. noted, because the school represented the unit of analysis, the design was underpowered with the small number of schools in the study. Nevertheless, it appears that ELM may be promising in improving student learning.

In the next study, Clarke et al. (2011) extended the work of Chard et al. (2008) by investigating the efficacy of the ELM program when compared to standard district practice on the mathematics achievement of all students, including students at risk for mathematics difficulties, using a rigorous randomized controlled trial. Blocking by school, 64 kindergarten classrooms (matched on full or partial day) were randomly assigned to treatment ( n = 34) or control ( n = 30) conditions. Sixty-five teachers (one classroom included two teachers) and their students participated (56 % were eligible for free or reduced-price lunch, 51 % were minority students, 38 % were English language learners, and 8 % were receiving special education services). A cut score of the 40th percentile on the Test of Mathematics Abilities—Third Edition (TEMA-3) was used to identify students as at risk or not at risk for mathematics difficulties. The ELM curriculum was implemented daily for 60 min.

Regarding students not at risk for mathematics difficulties, results indicated no statistically significant differences between treatment and control students on the TEMA-3 ( g = 0.03) or on the early numeracy curriculum-based measurement (EN-CBM; g = 0.04). In contrast, for students identified as at risk for mathematics difficulties, statistically significant differences favored students in the treatment classrooms compared to students in the control classrooms on the TEMA-3 ( g = 0.27) and on the EN-CBM ( g = 0.24). On both measures, a comparison of the performance of students at risk for mathematics difficulties in relation to the performance of their peers not at risk for mathematics difficulties in ELM and control classrooms showed that the gains made by ELM students at risk for mathematics difficulties were greater than the gains made by control students at risk for mathematics difficulties. Furthermore, it is encouraging that “9.2 % more students in ELM classrooms transitioned from the at-risk to not-at-risk categories than in control classrooms, a difference that was statistically significant” (Clarke et al. 2011, p. 577). In sum, although the effects for both ELM studies (Chard et al. 2008; Clarke et al. 2011) are small and modest, these results provide preliminary evidence regarding the effectiveness of the ELM program to improve student mathematics achievement, particularly for students at risk for mathematics difficulties. At the same time, it might be argued that there is insufficient evidence for ELM benefitting students across all levels of achievement and that ELM might be a better tier 2 intervention based on positive effects for low-performing students only as seen in the Clarke et al. (2011) study. On the one hand, tier 1 instruction that incorporates research-based instructional practices may be effective for students performing at grade and above grade levels only when instruction is appropriately differentiated such that the content is sufficiently challenging for these students. On the other hand, ELM as a tier 1 intervention may be necessary for students at risk for mathematics difficulties to benefit from tier 2 instruction. The authors recently tested the added value of tier 2 tutoring intervention over and above tier I instruction with the ELM curriculum. Positive benefits were seen for tutored students compared to control students receiving only tier I instruction with the ELM curriculum (Clarke et al. 2013).

Medium Intensity, Explicit Instruction

Peer-Assisted Learning Strategies (PALS)

To understand whether peer-mediated instruction can promote mathematics success for students, two of the most recent and related studies from the peer-mediated, class-wide intervention research were examined. The rationale for focusing on only two studies is based on space limitations, and also because peer-mediated instruction is less common than teacher-directed instruction in general education classrooms (Fuchs et al. 2001, 2002). In each study, Fuchs and colleagues assessed the effects of peer-assisted learning strategies (PALS), a version of class-wide peer tutoring, on children’s mathematical development (see Fuchs et al. 1995 for a review of PALS). The studies contrasted PALS with conventional classroom mathematics instruction, in which teachers relied on the instructional activities in their district-adopted mathematics textbook to teach critical mathematics content. During PALS, teachers selected critical content and skills from the district’s mathematics textbook and replaced parts of their mathematics instruction (teacher-directed instruction and student workgroup activities) with peer-mediated class-wide instruction by pairing higher- and lower-achieving students to practice applying the learned content (e.g., number sense) and skills.

Fuchs et al. (2001) randomly assigned 20 kindergarten teachers in three title 1 and two nontitle 1 urban schools to either PALS or a no-PALS condition. In total, 153 students participated (44 % were eligible for free or reduced-price lunch, 63 % were minority students, 2 % were English learners, and 10 % were receiving special education services). The PALS curriculum was adapted from the RightStart program (Griffin et al. 1994) that focuses on number sense. Treatment teachers implemented PALS twice a week for 20 min across 15 weeks. To understand whether or not PALS was effective for all students, Fuchs et al. classified students into three achievement groups (high, average, low) using scores from the SESAT. On the SESAT, both average- and low-achieving students in the PALS condition outperformed their no-PALS peers, with ESs of 0.44 and 0.38, respectively. In contrast, the effectiveness of PALS for high-achieving students compared to high-achieving students in the no-PALS condition was questionable ( g = − 0.16). The authors argued that, perhaps, a number sense curriculum was not sufficiently challenging to address these students’ needs. Relatedly, Fuchs et al. also collected data on the primary 1 level of the Stanford Achievement Test (SAT) to address the issue that high-achieving students would experience a ceiling effect on the SESAT; additionally, the primary 1 is not as closely aligned with the instructional content taught in PALS as is the SESAT. As such, the primary 1 provides a more distal measure of student achievement than the SESAT. On the primary 1 measure, both high- and low-achieving students in the PALS condition outperformed their no-PALS peers, with ESs of 0.66 and 0.30, respectively. Given the earlier explanation that high-achieving students in the PALS condition might have already mastered the majority of the content in the PALS curriculum, why did these children perform better than their high-achieving no-PALS peers on this measure? The authors attributed the positive, moderate-to-large effect to the strategy of explaining to others and reported “pairing children did ensure routine opportunities for HA students to work together on more complex skills using high number sets” (p. 507). Surprisingly, the ES for average-achieving students on the primary 1 measure was small ( g = − 0.21), favoring the no-PALS condition.

In the next study, Fuchs et al. (2002) examined whether PALS would be efficacious for first-grade students in improving their mathematics knowledge. Blocking by school, 20 classrooms were randomly assigned to PALS or no-PALS conditions. All classrooms were part of a single urban district, where eight classrooms in each condition were located in title I schools and the other two classrooms (in each condition) were in nontitle I schools. The final sample included 327 students (62 % were eligible for free or reduced-price lunch, 75 % were minority students, 31 % were English language learners, and 6 % were receiving special education services). PALS teachers led three 30-min sessions a week for 16 weeks. The intensity of PALS in this study was more than twice the total time that PALS was implemented in Fuchs et al. (2001). At the beginning of the study, students were classified as high, average, or low performing based on teacher judgments of students’ classroom performance. To assess student learning, the primary 1 and primary 2 levels of the SAT were administered before and after the intervention was implemented. In addition, each item on the primary 1 and primary 2 was categorized as being aligned with PALS or not aligned with PALS. Results were examined separately for each of the three achievement groups and for the aligned and unaligned items. For the items aligned with PALS, effects were negligible even though they favored the PALS condition for high-, average-, and low-performing students, with ESs of 0.15, 0.16, and 0.19, respectively. In contrast and consistent with the authors’ hypotheses, no statistically significant between-condition (PALS and no PALS) differences were found for items that were not aligned with PALS. ESs for high-, average-, and low-performing students were − 0.05, − 0.07, and 0.01, respectively. In sum, findings about the effectiveness of PALS as a tier 1 intervention are somewhat mixed based on the two studies that provided focused instruction on number sense using a PALS curriculum adapted from the RightStart program (Griffin et al. 1994). It may be that additional studies about the effectiveness of PALS as a tier I intervention are needed in which instruction is appropriately differentiated for students across all levels of achievement.

Schema-Based/Schema-Broadening Instruction

To understand the effectiveness of instructional strategies that build student capacity to solve word problems by focusing on common underlying structures, findings from two teams of researchers led by Fuchs and Jitendra are examined. Fuchs and colleagues’ instructional approach to solving word problems emphasizes schema-broadening instruction and is defined by an explicit focus on transfer in the instructional design. A schema is a specific knowledge structure that individuals use to categorize problem types into groups requiring similar solutions (Gick and Holyoak 1983). All tier 1 studies of schema-broadening instruction by Fuchs and colleagues included the following problem types: shopping list problems, half problems, buying bag problems, and pictograph problems. In an ongoing series of investigations, Fuchs and colleagues have examined the effects of schema-broadening instruction to promote “transfer to problems with unexpected features within the taught problem types (e.g., irrelevant information, relevant information presented outside the narrative in figures or tables, presentation of problems in real-life context)” (Schumacher and Fuchs 2012, p. 611). Transfer occurs when the learner recognizes that novel problems, even though different in certain features, are related to previously solved problems. The authors introduced the concept of transfer (i.e., to move) using several examples from daily life and mathematics (e.g., move from adding two-digit horizontal problems to solving two-digit vertical problems). In all of these studies, schema-broadening instruction was contrasted with conventional classroom problem-solving instruction (i.e., teacher-delivered instruction on problem–solution rules, which was based on the district-adopted mathematics textbook and included the four target problem types that were the focus of instruction in each of the four studies described herein).

Fuchs et al. (2003a) randomly assigned 24 third-grade teachers in six urban schools to one of four conditions (three treatment and a control condition); each condition contained six teachers and conditions were stratified across schools. In total, 375 students provided data at pretest and posttest (45 % were eligible for free or reduced-price lunch, 56 % were minority students, 4 % were English language learners, and 6 % were receiving special education services). The 16-week intervention took one of three forms (i.e., the three treatment conditions): teachers and research assistants provided (a) solution instruction (26 lessons) that taught rules for problem-solving and sorting problems into specific problem types that require the same solution methods, (b) partial solution instruction plus transfer instruction (26 lessons), and (c) full solution instruction plus transfer instruction (36 lessons). Transfer instruction focused on explicitly teaching the concept of transfer, highlighting superficial problem features (i.e., different format, different key word, additional or different question, and problem scope—problem is placed within a larger problem-solving context) that can modify a problem but not the structure or solution, and understanding that novel problems could incorporate these superficial features and yet represent a familiar problem structure or solution. Word problem-solving instruction occurred two times a week for 25-40 min across 16 weeks.

Student learning was assessed using three tests of transfer—immediate (four problem types with novel cover stories), near (four problem types with novel cover stories and with one superficial feature varied per problem), and far (all four problem types “embedded in a real-life problem-solving context, with all four superficial features varied and additional elements of novelty” Fuchs et al. 2003a, p. 299). Students were classified as high, average, or low achieving, and results were analyzed separately for each transfer measure and students’ achievement status. Students in each treatment condition significantly outperformed control students across the three measures and in each achievement group. Across the three treatment conditions, ESs for students identified as high, average, and low achieving on the measure of immediate transfer were large and ranged from 1.29 to 3.26, 1.92 to 5.45, and 1.36 to 3.45, respectively. On the measure of near transfer, although the ESs were large, they were generally smaller than the ESs for the measure of immediate transfer. ESs for high-, average-, and low-achieving students ranged from 1.05 to 2.56, 0.87 to 1.93, and 1.30 to 3.10, respectively. Finally, although ESs on the measure of far transfer were considerably smaller than on the measures of near and immediate transfer, the effect sizes can be considered moderate to large. ESs for high-, average-, and low-achieving students ranged from 0.40 to 1.41, 0.22 to 0.67, and 0.40 to 0.97, respectively.

In a follow-up study, Fuchs et al. (2003b) evaluated the contribution of self-regulated learning strategies (SRL) incorporated into problem-solving transfer instruction. Self-regulation learning strategies consisted of goal setting and self-monitoring. Students set goals for their performance, scored performance on problem-solving tasks, and graphed their scores. Twenty-four third-grade teachers from six urban schools were randomly assigned to one of three conditions (two treatment or control conditions); 395 students participated. Treatment teachers and research assistants either provided (a) transfer instruction that was similar to the Fuchs et al. (2003a) solution plus transfer condition or (b) transfer plus SRL that incorporated goal setting and self-evaluation in addition to transfer instruction. Instruction occurred two times a week, across 16 weeks. As in Fuchs et al. (2003a), students were assessed on three measures of transfer, and students were classified into one of three achievement groups (high, average, low). Students in both treatment conditions outperformed control students on all three measures of transfer. Across measures, ESs for the high-, average-, and low-achieving students ranged from 0.43 to 4.41, 0.57 to 5.65, and 0.95 to 3.17, respectively. As might be expected, in all cases, ESs were largest for the measure of immediate transfer and smallest for the measure of far transfer.

Regarding the effect of SRL, moderate-to-large effects favored the transfer plus SRL condition over the transfer-only condition on the measures of immediate and near transfer; ESs for high-, average-, and low-achieving students were 1.00 and 0.92, 0.81 and 0.51, and 0.31 and 0.32, respectively. On the measure of far transfer, small effects favored the transfer plus SRL condition over the transfer-only condition for high-, average-, and low-achieving students with ESs of 0.23, 0.13, and 0.27, respectively. Together, these results suggest that incorporating goal setting and self-monitoring into transfer instruction produces positive outcomes for all students.

The third study (Fuchs et al. 2004b) examined the added value of practice in sorting problems into specific problem types when embedded in the Fuchs et al. (2003a) transfer instruction. Twenty-four third-grade teachers from six urban schools were randomly assigned to one of three conditions (two treatment or control); 366 students provided pretest and posttest data (37 % were eligible for free or reduced-price lunch, 38 % were minority students, and 8 % were English language learners). Treatment teachers and research assistants either provided (a) schema-based transfer instruction (see Fuchs et al. 2003a) or (b) schema-based transfer instruction plus instruction on sorting problems into specific problem types. Word problem-solving instruction occurred two times a week for 30-40 min across 16 weeks. Student learning was assessed using three measures of transfer (immediate, near, and far). Using pretest scores on the test of immediate transfer, students were categorized into one of three achievement groups (high, average, low), and results were examined separately for each group.

As in the Fuchs studies described above, large and statistically significant effects were seen on all measures for students in all achievement groups, favoring the treatment conditions over the control condition. Across measures and both treatment–control comparisons, the ESs for high-, average-, and low-achieving students ranged from 0.76 to 4.66, 0.91 to 4.21, and 1.84 to 8.02, respectively. However, results comparing the treatment conditions were more mixed. On the measure of immediate transfer, although there was no effect for adding sorting instruction to schema-based transfer instruction for high- ( g = − 0.08) and low-achieving ( g = 0.03) students, a moderate, positive effect was found for average-achieving students ( g = 0.54). In contrast, on the measure of near transfer, results revealed moderate, negative effects for high- ( g = − 0.64) and low-achieving ( g = − 0.59) students, but no effect for average-achieving students ( g = 0.06). Finally, on the measure of far transfer, results revealed small negative effects for high- ( g = − 0.28) and low-achieving ( g = − 0.20) students, and a negligible effect for average-achieving students ( g = 0.10). Together, these results suggest that both high- and low-achieving students did not benefit from sorting instruction. In fact, the inclusion of sorting instruction for these students appears to have inhibited learning, particularly on the measure of near transfer. For average-achieving students, the results were more positive; however, significant effects were found only on the measure of immediate transfer. In sum, significant positive effects favoring the inclusion of sorting instruction were found only for average-achieving students on one measure, suggesting little to no benefit of including sorting instruction in schema-based transfer instruction for low- and high-achieving students.

In a fourth study, Fuchs et al. (2004a) estimated the value of an expanded schema-based transfer instruction that included three additional real-life transfer features—“irrelevant information, combining of problem types, and mixing of superficial features” (p. 423) that are considered more challenging than the transfer features (different format, different vocabulary, different question) in previous schema-based transfer instruction studies. Blocking by school, 24 third-grade teachers from seven urban schools were randomly assigned to one of three conditions (two treatment or control); 351 students participated (48 % were eligible for free or reduced-price lunch, 55 % were minority students, 4 % were English language learners, and 8 % were receiving special education services). The treatment conditions included either (a) schema-based transfer instruction (see Fuchs et al. 2003a) or (b) expanded schema-based transfer instruction that included three additional transfer features. Word problem-solving instruction was conducted for 16 weeks, and included 34 lessons with each lesson occurring for 25–40 min. To assess student learning, four measures of transfer problem-solving, all of which included novel problems, were administered at pretest and posttest. The measures were labeled transfer 1 through transfer 4, with greater numbers representing greater transfer distance. As in previously described studies, students were identified as high, average, or low achieving, and results were analyzed separately for each measure and for each student achievement group.

When comparing the treatment conditions to the control condition, large positive effects favoring the treatment conditions were found on all measures for students in all achievement groups. Across measures and treatment–control comparisons, the ESs for high-, average-, and low-achieving students ranged from 0.72 to 5.91, 0.78 to 7.93, and 1.40 to 4.67, respectively. In all cases, ESs were largest on the measure of nearest transfer (i.e., transfer 1) and smallest on the measure of farthest transfer (i.e., transfer 4). In contrast, while the results comparing expanded schema-based transfer instruction to schema-based transfer instruction were mixed across student achievement groups on transfer 1 and transfer 2, moderate-to-large, positive effects ( g = 0.55–1.28) favored expanded schema-based transfer instruction on the transfer 3 and transfer 4 measures for all student achievement groups.

To understand the short- and long-term effects of schema-based instruction (SBI) in enhancing students’ word problem-solving performance and mathematics achievement, findings from Jitendra et al. (2007) are examined. SBI is a multicomponent intervention based on schema theories of cognitive psychology, research on expert problem solvers, and research regarding effective instructional practices (e.g., explicit instruction) for students at risk for mathematics difficulties. Specifically, SBI for elementary grades includes four critical elements: (a) priming the underlying problem structure, (b) using visual representations (schematic diagrams), (c) explicitly teaching of problem-solving heuristics, and (d) providing instruction in metacognitive strategy knowledge. Jitendra et al. (2007) randomly assigned 88 third-grade students (49 % were eligible for free or reduced-price lunch, 50 % were minority students, 5 % were English language learners, and 13 % were receiving title 1 services) and their six teachers in one of the lowest-achieving schools in an urban school district to one of two conditions: SBI or comparison (general strategy instruction, GSI). Students in the GSI condition were taught a heuristic to (a) understand the problem, (b) plan to solve the problem, (c) solve the problem, and (d) look back or check. Problem-solving strategies such as using objects, acting out the problem or drawing a diagram, choosing an operation—writing a number sentence, and using data from a graph or table were incorporated in the planning step of the problem-solving heuristic. Students in both conditions received word problem-solving instruction on one-step and two-step addition problem structures (i.e., change, group, and compare) 5 days a week for 25 min in addition to 25 min of daily core mathematics instruction across 12 weeks of the study. To assess student learning, a measure of word problem-solving was administered at pretest, posttest, and delayed posttest (6 weeks following the end of the intervention). In addition, data were collected on a state-administered test of mathematics achievement. Across measures, findings are presented separately for all students and a subgroup of at-risk students (i.e., learning disabilities, math title 1, English language learners) identified as such by the authors.

On the measure of word problem-solving, results revealed small-to-moderate, positive effects favoring SBI at posttest and moderate-to-large, positive effects favoring SBI on the delayed posttest. On the immediate posttest, ESs for not-at-risk and at-risk students were 0.39 and 0.41, respectively; on the delayed posttest, ESs for not-at-risk and at-risk students were 0.51 and 0.85, respectively. Similarly, results revealed moderate-to-large, positive ESs favoring SBI for both not-at-risk ( g = 0.46) at-risk ( g = 0.80) students on the state-administered test of mathematics achievement. Together, these results provide evidence supporting the use of SBI (i.e., priming the underlying problem structure using schematic diagrams, explicitly teaching a problem-solving heuristic, emphasizing metacognitive strategy use) for all students, including those at risk for mathematics difficulties.

Low Intensity, Explicit Instruction

Two final studies by Jitendra and colleagues provide findings about the effectiveness of SBI for seventh-grade students. These two pilot studies were conducted in an urban, middle school. The first study (e.g., Jitendra et al. 2009) focused on ratio and proportion problem-solving and occurred in early January; the second study (Jitendra and Star 2012) focused on percent problem-solving and was conducted in May of the same year. In both studies, Jitendra and colleagues reviewed the district-adopted mathematics textbook, identified specific concepts and problem-solving skills, and mapped the relevant topics to the SBI units on ratio, proportion, and percent. Jitendra et al. (2009) examined the potential impact of a 10-day intervention (SBI) for 148 students (42 % were eligible for free or reduced-price lunch, 54 % were minority students, 3 % were English language learners, and 10 % were receiving special education services) from eight seventh-grade mathematics classrooms. Blocking by classroom ability level (high, average, and low) based on grades in mathematics from the previous school year and scores on mathematics subtests of SAT-10, classrooms were randomly assigned to either SBI or a “business-as-usual” control condition that received the same amount of instruction on the same topics (i.e., ratio and proportion). Mathematics teachers provided all instruction that occurred during the regularly scheduled mathematics instructional period, 5 days a week for 40 min across 10 school days. In addition to priming the underlying problem structure, using visual representations, explicitly teaching problem-solving heuristics, and providing instruction in metacognitive strategy use, SBI emphasized multiple solution strategies (cross multiplication, unit rate, and equivalent fractions). To assess student learning, data were collected on a measure of problem-solving immediately following completion of the intervention (posttest), and again 4 months later to assess maintenance effects (delayed posttest). In addition, data were collected on a state-administered test of mathematics achievement. Students were categorized as low, average, or high achieving, and the results are presented separately for each student achievement group.

On the test of problem-solving, moderate-to-large ESs were found for both average- and high-achieving students, favoring SBI on the immediate and delayed posttests. For average-achieving students, the ESs on the immediate posttest and delayed posttest were 0.86 and 0.47, respectively; for high-achieving students, the ESs for the immediate and delayed posttests were 0.75 and 1.31, respectively. On the test of mathematics achievement, results revealed no effect for average-achieving students ( g = − 0.03) and a moderate effect favoring SBI for high-achieving students ( g = 0.52). In contrast, the findings for low-achieving students were mixed. No effect was found for SBI at posttest ( g = 0.01) for low-achieving students, whereas a small effect favoring SBI was found at delayed posttest ( g = 0.26) for low-achieving students. On the test of mathematics achievement, a small effect ( g = − 0.22) favored the control condition for low-achieving students. As Jitendra et al. (2009) suggested “the value of integrating metacognitive strategy knowledge as an instructional feature in schema-based instruction, particularly using schematic diagrams to represent information, may not have been realized in the short-term (10-day) intervention for low-ability students” (p. 260). This feature of the intervention and the fact that these students’ reported difficulty in mastering the two conceptually based strategies (unit rate, equivalent fractions) may explain their persistent problem-solving difficulties, suggesting the need for more practice, time, and scaffolding of instruction to positively impact learning.

In the second study (Jitendra and Star 2012), data were collected from four (two high-ability and two low-ability classrooms) of the eight seventh-grade mathematics classrooms in the Jitendra et al. (2009) study. A total of 70 students (36 % were eligible for free or reduced-price lunch, 59 % were minority students, 4 % were English language learners, and 7 % were receiving special education services) participated in the study that examined the effects of SBI on student learning of percent word problems. Teachers led daily 40-min sessions for 9 days in SBI and control classrooms. The content focused on percent problem-solving, which is a difficult topic for many middle school students (see Parker and Leinhardt 1995). Results revealed that while high-achieving students in the SBI condition statistically outperformed high-achieving students in the control condition on percent problem-solving posttest ( g = 0.97), no statistically significant differences were found for low-achieving students ( g = − 0.39); and a small-to-medium ES favored students in the control condition. On the transfer test, no statistically significant differences were found for either high- ( g = − 0.01) or low- ( g = − 0.10) achieving students. Jitendra and Star (2012) reported that low-achieving students did not respond successfully to the SBI intervention in this study. Unlike previous studies (e.g., Jitendra et al. 2007) that implemented SBI for 12 weeks on average, the 9-day intervention was likely not sufficient for low achievers, who may have needed more time and support to recognize the underlying problem structure and “show gains in flexible knowledge of procedures for solving a wide range of problems” (p. 157).

Limitations and Implications for Future Research

The studies reviewed suggest that well-designed, core mathematics instructional programs applied within an RTI framework could address the diverse needs of a range of learners when instruction is differentiated in ways that not only is the content appropriately matched to the instructional needs of students performing below grade level but also the content is sufficiently challenging for students above grade level. Common features of instructional programs in the studies reviewed were explicit and systematic instruction, scaffolded instruction with teachers or peers first modeling followed by teacher-guided practice with corrective feedback until students are able to work independently, and development of student initial understanding of critical concepts and procedures using a range of examples. Several programs also included other features such as teaching to mastery (e.g., Clarke et al. 2011), providing adequate opportunities in terms of frequent cumulative review to learn the concepts and skills (e.g., Clarke et al. 2011; Fuchs et al. 2003a, b; Jitendra et al. 2007), having students verbalize their thinking (e.g., Clarke et al. 2011; Fuchs et al. 2003b; Jitendra et al. 2007, 2009; Jitendra and Star 2012) using mathematical representations (e.g., Clarke et al. 2011; Jitendra et al. 2007, 2009; Jitendra and Star 2012), teaching the use of heuristics to solve word problems (Jitendra et al. 2007, 2009; Jitendra and Star 2012). It is important that adequate progress monitoring and universal screening measures are used during and at the end of tier 1 intervention to detect children who are still struggling and provide them with tier 2 or tier 3 intervention that incorporates these key instructional features.

An important finding across the studies reviewed is that differences in responsiveness to instruction for students at risk for mathematics difficulties appeared to be associated with variations in the intensity of the instructional programs and content. In several of the medium-intensity studies, moderate-to-large ESs for students at risk for mathematics difficulties (MD) favored schema-based/schema-broadening instruction that focused on word problem-solving; the effects for peer-mediated instruction that focused on number sense were mixed and ranged from no effect to small-to-moderate effects. In contrast, the effects for students at risk for mathematics difficulties in the high-intensity studies that focused on the entire kindergarten content (e.g., number sense, measurement, geometry) were small, and the effects were mixed for students at risk for mathematics difficulties in the low-intensity studies that addressed complex content (e.g., ratio and proportions).

While the research base described above provides important evidence about the effectiveness of various core mathematics instructional programs in improving the mathematics success of a range of learners, there are a number of limitations of the current research that suggest important avenues for future research. First, little is known about how best to support the development of advanced mathematics (e.g., proportional reasoning, algebra, geometry). The majority of studies examined elementary school student populations and content, with only two studies conducted in middle schools and none at the high school level. Of note, none of the high- or medium-intensity interventions that were reviewed examined the effectiveness of instructional programs for students at the secondary grades. A lack of research in secondary settings is of concern for two reasons. First, it is critical that instructional programs implemented within an RTI framework are evidence based, and currently, there is a deficit in the evidence about effective core mathematics programs for the secondary grades. Second, the complexity of the instructional content covered in secondary school (e.g., ratios and proportions, probability) is greater than that covered in elementary school (e.g., whole numbers), which may limit the generalizability of findings about programs tested in elementary grades to the secondary grades. As such, more research is needed about the effectiveness of instructional programs designed to address more advanced content (e.g., rational numbers, algebra, geometry).

A second limitation of the current research involves fidelity of implementation. The National Center on Response to Intervention (NCRI) defines fidelity of implementation as “the delivery of content and instructional strategies in the way in which they were designed and intended to be delivered: accurately and consistently. Although interventions are aimed at learners, fidelity measures focus on the individuals who provide the instruction” (NCRI n.d.). It is critical that researchers conducting studies on the effectiveness of instructional programs develop valid and reliable fidelity measures that can be used to not only assess the extent to which programs are implemented as intended but also the extent to which fidelity affects program effectiveness. In all the studies reviewed, fidelity measures were used to assess implementation; however, none of the studies directly related the fidelity data to outcomes. As such, these studies only provide descriptions of the extent to which programs were implemented with fidelity, and fail to address the extent to which fidelity affected program effectiveness in promoting mathematics success for all students.

A related limitation involves the professional training of teachers, specifically the quality and intensity of the professional development provided. In the studies described above, teachers received anywhere from 2 to 12 h of professional development. For example, in three studies, teachers participated in a single 2-h workshop, while in another study teachers attended three half-day professional development sessions throughout the duration of the study. What is less known from the studies reviewed is how much training is necessary for teachers to implement these programs with fidelity to improve student learning.

Finally, a methodological limitation that is inherent to all classroom-based research is the nesting of students within classrooms and classrooms within schools. Failure to account for this nesting and the resulting dependency between units of analysis produces biased estimates of relevant effects. In the studies considered, the respective researchers chose a variety of approaches designed to address the issues (e.g., power, independence) associated with nested designs. Unfortunately, in most of the studies, the authors did not utilize the most advanced statistical methods available (e.g., multilevel modeling); the result in most cases was a loss of statistical power. A related issue involves disagreement between the unit of assignment and the unit of inference. For example, in most of the studies described above, the instructional program of interest was assigned to whole classrooms and in some cases whole schools; however, the unit of inference in all cases was students. This disconnect between the unit of assignment and the unit of inference leads to what is known as the “ecological fallacy.” The ecological fallacy is a logical fallacy specific to the interpretation of statistical data and indicates that correlations between variables at the aggregate level (e.g., classrooms) are not equal to correlations (between the same variables) at the individual level (e.g., students). This means that fitting a regression model to group-level data will not necessarily produce the same effects as fitting that same regression model to individual-level data.

In conclusion, based on the studies summarized in this chapter, several lessons were learned about the instructional conditions that need to be in place to insure that all students, including student at risk for mathematics difficulties, are successful in meeting the requirements for mathematics content involving whole numbers. However, less is known about what is needed to prepare students in meeting the standards and expectations for advanced mathematics in late elementary, middle, and high school.