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Effects of Mathematical Word Problem Solving by Students At Risk or With Mild Disabilities
The Journal of Educational Research
Abstract
The differential effects of 2 instructional strategies, an explicit schema-based strategy and a traditional basal strategy, on the acquisition, maintenance, and generalization of mathematical word problem solving were examined. Thirty-four elementary-aged students with mild disabilities or at risk for mathematics failure were randomly assigned to each of the 2 treatment conditions (schema and traditional). Results indicated that both groups' performance increased from the pretest to the posttest. All students were able to maintain their use of word problem-solving skills and generalized the strategy effects to novel word problems. However, the differences between groups on the posttest, delayed posttest, and generalization test were statistically significant, favoring the schema group. In addition, scores on the immediate posttest (77% correct) and delayed posttest (81% correct) for the schema group approached those of a normative sample of 3rd graders (M = 82% correct).
... In CSI, students learn explicit word problem-solving procedures. Intervention studies showed CSI's positive effects with elementary students with LD (Cassel & Reid, 1996;Jitendra et al., 1998;Moran, Swanson, Gerber, & Fung, Running head: MODEL DRAWING STRATEGY The final, definitive version is available at http://online.sagepub.com/. 5 2014; Swanson, Lussier, & Orosco, 2013), and secondary students with LD (Montague et al., 1993;Montague, Enders, & Dietz, 2011;Montague, Krawec, Enders, & Dietz, 2014). ...
... For example, Jitendra and colleagues conducted a series of studies using a variant of CSI named schema-based instruction (SBI) that taught students with LD to identify problem types (e.g. change problems and group problems) called schema, and to apply a heuristic for solving them (Jitendra et al., 1998). Specifically, students using SBI identified semantic relations in word problems that indicated schema type then selected a schema-specific diagram to organize problem information into a solution. ...
... Specifically, students using SBI identified semantic relations in word problems that indicated schema type then selected a schema-specific diagram to organize problem information into a solution. SBI was effective for improving word problem solving among students with LD (Jitendra et al., 1998;Jitendra, Hoff, & Beck, 1999;Jitendra, DiPipi, & Perron-Jones, 2002). ...
This study used a multiple probe across participants design to examine the effects of a model drawing strategy (MDS) intervention package on fraction comparing and ordering word problem–solving performance of three Grade 4 students. MDS is a form of cognitive strategy instruction for teaching word problem solving that includes explicit instruction in drawing bar diagrams to represent problem components. Results suggest the intervention package was effective for improving the fraction word problem solving of students with learning disabilities and that effects were maintained 2 and 4 weeks after intervention. Implications of these findings and indications for future research are discussed.
... For this category, studies were evaluated for documenting use of multiple measures and for measuring outcomes at appropriate times. The majority of studies (n = 19; 76%) used multiple outcome measures, and more than half the studies (n = 15; 60%) included measures of generalized performance or experimenterdeveloped outcome measures (Butler et al., 2003, 2009Jitendra et al., 1998Jitendra et al., , 2013Van Luit & Schopman, 2000;Woodward, 2006;Woodward & Brown, 2006;Xin et al., , 2011;Xin, Jitendra, & Deatline-Buchman, 2005). With the exception of one study that did not report when measures were administered at pretest and posttest, the remaining studies met the minimum criteria for documenting collection of data at appropriate times. ...
... Based on our analysis, eight studies (61.5%) provided evidence of meeting four (Fuchs et al., 2013(Fuchs et al., , 2014Manalo et al., 2000;Woodward, 2006;Xin et al., 2011) or more (Jitendra et al., 1998(Jitendra et al., , 2013Xin et al., 2005) desirable indicators. As such, the research reported in these studies is considered high quality. ...
... One study (Woodward & Brown, 2006) met all but one of the essential QIs; this study failed to provide sufficient information on teachers delivering the intervention. Only eight of the 13 studies that met the essential QIs provided evidence of meeting at least four desirable indicators to be classified as high quality (Fuchs et al., 2013(Fuchs et al., , 2014Jitendra et al., 1998Jitendra et al., , 2013Manalo et al., 2000;Woodward, 2006;Xin et al., 2005Xin et al., , 2011. ...
The purpose of the present review was to evaluate the quality of the research and evidence base for representation of problems as a strategy to enhance the mathematical performance of students with learning disabilities and those at risk for mathematics difficulties. The authors evaluated 25 experimental and quasiexperimental studies according to the Gersten et al. (2005) guidelines for group research studies. Results suggest that the representation of mathematical problems as a strategy is an evidence-based practice based on the criteria set by Gersten et al. Implications for research are discussed.
... As a result, all 13 of the studies included either a DI or explicit instruction component; specifically, 12 studies used explicit instruction and one used DI. Wilson and Sindelar (1991) x x Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp, et al. (2003) x x x Owen and Fuchs (2002) x x x Fuchs, Seethaler, Powell, Hamlett, and Fletcher (2008) x x x x Fuchs, Fuchs, Craddock, Hollenbeck, Hamlett, and Schatschneider (2008) x x x x Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, and Schroeter (2003) x x x x Cassel and Reid (1996) x x x Jitendra and Hoff (1996) x x x Jitendra et al. (1998) x x x x x x Griffin and Jitendra (2009) x x x Seo and Bryant (2012) x x x Leh and Jitendra (2013) x x x Though dated, Wilson and Sindelar (1991) conducted the only study investigating a DI-based intervention for solving math word problems. Sixty-two students with LD across grades two to five participated in the study. ...
... While Jitendra and colleagues (Griffin & Jitendra, 2009;Jitendra, Griffin, Haria, et al., 2007;Jitendra & Hoff, 1996;Jitendra et al., 1998;Leh & Jitendra, 2013) also maintained a similar team to implement their intervention, they widely varied the location of their studies, lending generalizability to their findings and addressing this concern in previous research. In these studies, explicit instruction was again used to teach third graders strategies for solving word problems. ...
... Jitendra and Hoff (1996) used a single-subject design with intervention delivered in a small-group setting, which proved effective in increasing the problem-solving skills of students with LD. The other studies (Griffin & Jitendra, 2009;Jitendra, Griffin, Haria, et al., 2007;Jitendra et al., 1998;Leh & Jitendra, 2013), which used large group designs and targeted a diverse population of students, ran the intervention in an inclusive class-wide format. The majority of the studies (Jitendra, Griffin, Haria, et al., 2007;Jitendra et al., 1998;Leh & Jitendra, 2013) found that students in the condition using explicit instruction outperformed their control group peers. ...
Though research on effective instruction in math word problem solving is prominent at the middle and secondary levels, much less work has been done in elementary grades. In this article, we review the research on varied problem-solving instructional interventions at the third-grade level for students across ability levels. Third grade was chosen as the focus due to the fact that word problem-solving requirements are first introduced into the curriculum and standardized assessment at this point in time. Drawing on quantitative studies using single subject, quasi-experimental, and randomized controlled trial designs, we examine the instructional components and instructional content identified as effective across the 13 studies that met search criteria. Conclusions focus on current understanding of best practices, limitations of the existing research, and important considerations for future research.
... Problem-solving skills are part of the transferable skills that students need to work with other students to find, analyze, synthesize, and apply knowledge to new situations [1]. Furthermore, Jitendra et al. [2] defined problem-solving as a process of designing a strategy and completing steps to solve a problem. Some literature showed that problem-solving is associated with how students are actively involved in solving mathematical problems that support the development of their problem-solving competencies [3,4]. ...
... The details of the MANOVA test results are presented in Table 14. Based on Table 14, the results showed that there were statistically significant mean differences between the experimental and control groups in terms of combined dependent variables of problem-solving and critical thinking after the intervention [F (2,201) = 16.864,p < 0,05;Wilks' Lambda = .726]. ...
... This approach to strategy instruction primes, or prepares, students to "move from focusing on the superficial features of the problem (e.g., format, vocabulary, irrelevant information) to understanding the problem type or structure and applying known solution methods" (Jitendra & Lein, 2015, p. 1). Three variations of this instructional strategy are found in the research literature: (a) schema-based instruction (SBI; Jitendra et al., 2007;Jitendra et al., 1998;Xin, 2008), (b) schema-broadening instruction (Fuchs et al., 2008), and (c) conceptual modelbased problem solving (COMPS; Xin, Wiles, & Lin, 2008). This instruction is grounded in schema theories of cognitive psychology and supported by studies revealing that understanding the relations among the pieces of information and quantities in word problems is much more important than the arrangement of the arithmetic problem on children's problem-solving strategies (Carpenter, Hiebert, & Moser, 1983;Carpenter & Moser, 1984;García, Jiménez, & Hess, 2006). ...
... Of studies identified, mathematics strategy instruction that primes the common underlying structures of word problems using additive structures involving addition and subtraction were found more frequently (Fuchs, Fuchs, Finelli, Courey, & Hamlet, 2004;Fuchs et al., 2009;Fuchs et al., 2008;Griffin & Jitendra, 2009;Jitendra et al., 2007;Jitendra et al., 1998;Rockwell, Griffin, & Jones, 2011) than studies of multiplicative structures applying multiplication and division (Jitendra, DiPipi, & Perron-Jones, 2002;Xin, 2008;Xin, Jitendra, & Deatline-Buchman, 2005;Xin & Zhang, 2009;Xin et al., 2011). In general, both additive and multiplicative studies noted above suggest that elementary and middle school students with or at risk for mathematics learning disabilities improved problem-solving performance on both researcher-designed assessments and standardized tests of achievement. ...
This study examined mathematics strategy instruction that primes the common underlying structures of word problems using explicit instruction in a rural elementary classroom with fourth- and fifth-grade students with and without disabilities (n = 27). Although intervention students did not outperform control condition students on a word problem solving (WPS) measure (p = .054), a statistically significant interaction effect was found (p = .003). Follow-up analyses revealed that the WPS pretest–posttest gain was significant for the intervention group only. Students with disabilities in both conditions did not improve performance. Implications for practice and research in rural school classrooms are presented.
... The systematic use of representations has yielded positive results on students' achievement in arithmetic (Fuchs et al., 2005), solving linear algebraic equations (Witzel, 2005;Witzel, Mercer, & Miller, 2003), and word problems (Woodward, 2006). Furthermore, concrete and visual representations have been found to be effective in paving the way for students to understand more abstract symbolic representations and topics (Jitendra et al., 1998;Walker & Poteet, 1989;Woodward, 2006). For example, Woodward (2006) found that fourth-grade students taught to use number lines were better able to understand the meaning of symbolic representations of wholenumber multiplication (e.g., 6 × 7). ...
... First, results indicated that both teaching treatments (MR and TA) had a positive effect on student performance. While this finding is consistent with previous research that found that students' learning benefitted from instruction that used multiple representations and/or traditional algorithms with symbolic representations (Cai & Lester, 2005;Jitendra et al., 1998;Walker & Poteet, 1989;Woodward, 2006), this study went further with the use of a 2 × 2 crossover design. This study found significant treatment effect differences in performance in favor of the TA treatment over the MR treatment after accounting for period effects. ...
This study examined the effects of two teaching treatments [multiple-representation instruction (MR) and traditional algorithmic instruction (TA)] on students’ performance on mathematics problems that integrated fractions, decimals, and percent. Using a 2 × 2 crossover design, the effect of the teaching treatments and their sequences on student performance were compared. Students in the MR/TA treatment sequence group received instruction in which multiple representations were integrated followed by traditional algorithmic instruction, and students in the TA/MR treatment sequence group received treatments in the reverse order. Participants included 89 seventh-grade students enrolled in a pre-algebra course in an urban middle school in the Midwestern United States. Results indicated gains in performance from both teaching treatments (MR and TA), however, statistically significant higher performance was found with the traditional algorithmic treatment. In terms of the order of teaching treatments, no significant performance difference was found.
... CSI typically involves a representational aspect. Research also supports schematic-based instruction (SBI) as an effective intervention in supporting students in word problem solving (Jitendra et al., 1998;Xin, 2008;Xin, Jitendra, & Deatline-Buchman, 2005). SBI typically integrates cognitive strategies within the explicit instruction of the strategy. ...
... Swanson, Lussier, and Orosco (2013) contended that visual-schematic strategies supported the visual-spatial WM of students with MD. SBI has produced positive outcomes by supporting students with MD in math word problem solving across problem types and student age groups (Jitendra et al., 1998;Xin, 2008;Xin et al., 2005). ...
For students with mathematics difficulties (MD), math word problem solving is especially challenging. The purpose of this study was to examine the effects of a problem-solving strategy, bar model drawing, on the mathematical problem-solving skills of students with MD. The study extended previous research that suggested that schematic-based instruction (SBI) and cognitive strategy instruction (CSI) delivered within an explicit instruction framework can be effective in teaching various math skills related to word problem solving. A multiple-baseline design replicated across groups was used to evaluate the effects of the intervention of bar model drawing on math problem-solving performance of students with MD. Student achievement was measured in terms of increased correct use of cognitive strategies and overall accuracy of math word problem solving. Results showed that bar modeling drawing is an effective strategy for increasing elementary students’ accuracy in solving math word problems and their ability to use cognitive strategies to solve the problems.
... Fuchs and colleagues and Jitendra and colleagues) have addressed some of these limitations in the problem-solving research. The work of these teams, and a few others, has supported the use of word problem-solving interventions that incorporate explicit strategy instruction, teaching students to identify problem-type information before making a plan to solve, using multiple exemplars, encouraging self-strategies, and using visual representations (Cassel & Reid, 1996;Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, Hosp, et al., 2003;Fuchs, Fuchs, Prentice, Burch, Hamlett, Owen, & Schroeter, 2003;Fuchs, Seethaler, Powell, Fuchs, & Hamlett, 2008;Griffin & Jitendra, 2009;Jitendra, Griffin, et al., 2007;Jitendra et al., 1998;Jitendra & Hoff, 1996;Leh & Jitendra, 2013;Owen & Fuchs, 2002;Seo & Bryant, 2012;Wilson & Sindelar, 1991). For a full review of this literature, including a breakdown of intervention components in each study, please refer to Kingsdorf and Krawec (2016). ...
... Jitendra and colleagues (e.g. Jitendra, Griffin, et al., 2007;Jitendra et al., 1998;Jitendra & Hoff, 1996) also found that using an explicit instruction format was most beneficial when implementing a visualization intervention with a similar population. Both researchers utilized interventions that included multiple components but identified explicit instruction as a necessary piece. ...
In third grade the focus on math word problems becomes prominent. In the limited third grade research, teacher-mediated explicit instruction with multiple exemplars, teaching students to use visual representations, and the incorporation of self-strategies, have proven effective. For these practices to reach their full potential though, their content must be relevant and provide for growth to mature mathematical concepts. Based on these conclusions, additional research was needed. Therefore, the focus of this study was to evaluate the effectiveness of a multi-component word problem-solving intervention with explicit instruction strategies, multiple exemplars, the teaching of student-generated visual representations, incorporation of a self-monitoring checklist, and Common Core State Standards’ appropriate curriculum. Within a multiple baseline across behaviors design, the study evaluated the paraphrasing, visualizing, and computing word problem-solving responses of 10 third graders identified as learning disabilities, at-risk, and/or ESOL. The study revealed that all students made gains in some behaviors related to problem solving. Results are discussed in relation to a cognitive-behavioral framework and individual student characteristics, including discussions of limitations and educational significance.
... when tutoring was not used). Other studies looked at math interventions and problem solving skills (Jitendra et al., 1998) or children with learning disabilities (Swanson & Hoskyn, 1998;Swanson, Hoskyn, & Lee, 1999). Most of these studies, however, are based on samples of older children. ...
... Other studies looked at math interventions and problem solving skills (Jitendra et al., 1998) or children with learning disabilities (Swanson & Hoskyn, 1998;Swanson et al., 1999). There have also been at least 2 meta-analyses with younger (i.e., preschool) children but in the area of literacy. ...
... Problem-solving occupies a paramount position in education (Rott, 2020), evolving into the primary objective of mathematical learning within schools (NCTM, 2000), assuming a pivotal role within the domain of mathematical education (Li et al., 2020;Peng et al., 2020;Smith & Mancy, 2018;Wilson et al., 2011). It involves the ability to sift through extraneous information, devise strategies, execute essential steps, translate word problems into equations, and apply computational methods to resolve problems (Flores et al., 2016;Jitendra et al., 1998). Problem-solving entails the application of mathematical concepts (Ofsted, 2012;Williams & Williams, 2021) to tackle unfamiliar predicaments (Krulik & Rudnick, 1998), where individuals do not possess immediate access to the correct solution or strategy (Isam, 2020;Kojo et al., 2018). ...
In the rapidly changing global landscape, adaptability to new challenges is essential. Mathematics, recognized as a key contributor to problem-solving skills, holds significant importance in navigating unfamiliar situations. This study delves into the correlation between mathematics interest, reading comprehension, and mathematical problem-solving abilities among elementary school students. Employers and academic institutions increasingly emphasize the application of mathematical knowledge in diverse scenarios, prompting educational reforms prioritizing problem-solving skills.The investigation involved 76 elementary students in Majalengka Regency, Indonesia, exploring mathematics interest, reading comprehension, and problem-solving skills using questionnaires and tests. Findings unveiled a noteworthy positive correlation between reading comprehension and mathematical problem-solving, surpassing the correlation with mathematics interest. Regression analysis reinforced this, highlighting the collective substantial impact of mathematics interest and reading comprehension on students' problem-solving proficiency.This study underscores the crucial role of reading comprehension in interpreting narrative-based mathematical problems. Understanding textual content significantly influences problem-solving strategies. Moreover, affective abilities, particularly mathematics interest, contribute significantly to problem-solving skills. A heightened interest in mathematics enhances motivation, encourages deeper conceptual learning, and fosters creativity in problem-solving.These results underscore the necessity of a holistic approach to mathematics education. Addressing not only cognitive aspects but also affective and reading proficiency facets can substantially enhance students' mathematical problem-solving abilities. Understanding the interrelationship among mathematics interest, reading comprehension, and affective abilities can pave the way for more effective teaching methodologies and improved academic outcomes in mathematics.
... In order to investigate the effects of SBI with a larger sample size, Jitendra et al. (1998) conducted a study on 34 second-to-fifth-grade students who were identified as lowachieving in mathematics. Students were randomly assigned to receive SBI in 17-20 small group tutoring sessions versus traditional tutoring sessions. ...
In this study we tested the effects of schema-based instruction (SBI) strategies on 3 rd grade students' mathematics skills. We compared SBI to general strategy instruction (GSI) when teaching multi-step word problems to students in mixed-abilities general education classrooms. SBI strategies were used to instruct students in an attempt to increase understanding of mathematical word problems, assist with planning a strategy, solving, and checking problems. We assessed the overall effectiveness of SBI on students' procedural accuracy and computational accuracy with mixed computation word problems. Students' attitudes toward problem solving were compared before and after the use of SBI. ANCOVA was used to analyse pre/post-test data on overall problem-solving, procedural and computational fluency, and attitudes toward mathematics. The results indicated that the treatment group consistently outperformed the comparison group on all three achievement assessments by a statistically significant margin, on measures of overall problem-solving ability, procedural fluency, and computational fluency, but not on attitudes towards mathematics. The intervention was shown to be beneficial for enhancing student learning of math across a number of academic constructs.
... They suggest that SBI support self-regulation when finding solutions and reduce working memory load by allowing students to break the problem-solving process into a sequence of manageable-sized tasks. The improvement in SBI student's problem-solving performance is similar to the medium to large effect sizes found in previous studies on grade 3 students (Fuchs et al., 2008;Jitendra et al., 1998). Cognitively Focused Interventions employs a structured sequence of steps with a teacher focused on ensuring that students (including ELLs) understand and can use the mathematical ideas that are the focus of teaching. ...
Due to rapid immigration, many children worldwide are learning mathematics in a second or additional language. This language diversity can be challenging for both teachers and students and carries profound implications for mathematics educators. Research shows that teachers use various ways to support English Language Learners. Research on multilingualism in mathematics classrooms has often focused on qualitative research. This meta-analysis aims to explore the statistically effective successful teaching practices from the studies using quantitative or mixed-method research approaches and aims to inform the research field in a cumulative manner. The specific research question that guided this meta-analysis is: What is the evidence regarding successful teaching of mathematics for Year 1–10 English Language Learners from 2009–2019 in countries where curricula are delivered predominantly in English? Four successful intervention categories were identified: Dual Language Programmes, Curriculum integration, Teacher Professional Development, and Cognitively Focused Interventions. The paper concludes with recommendations for practice and further research in this area.
... In early year settings, real objects, physical manipulatives, and representations could be considered as factors which encourage mathematical learning (Cass et al. 2003), Tsuei 2014. Emphasis is placed at the effectiveness of the use of picturebased icons or symbolic representatives (Harrison 2002), of visual and concrete representations in order to develop problem-solving strategies (Jitendra et al. 1998). ...
Early numeracy is related to preschoolers’ acquisition of skills such as understanding and operating with quantities. Consequently, early numeracy has substantial impact on first Graders’ attainments in typical mathematics. Meta-analysis was conducted to address the extent in which early numeracy interventions are effective. Twenty studies were analyzed, including 3.080 young low-performing children (Ncontrol = 1.815, Ntreatment = 1.265). The overall best estimate for programs based on Early Numeracy Interventions odds ratio was moderately effective g = 0.61. Heterogeneity was large. Results of the final meta-regression model predicted larger treatment effects for short-term interventions including 1 to 9 sessions. On average, the interventions included instructional strategies such as explicit instruction, corrective feedback, CRA, concrete manipulatives, and visual representation, and one-to-one instructional arrangement are moderately effective for children aged 5–8. Results of the study are discussed with respect to implications for designing early numeracy interventions.
... Furthermore, it was found that students with problem-solving difficulties may or may not have difficulties in computation (Fuchs et al., 2008). Other findings suggest that some students can master context-free arithmetic problems while still receiving low scores on word problems (Jitendra et al., 1998). These students may have specific difficulties in comprehending problems and integrating information, rather than in performing the computations (Hegarty, Mayer, & Monk, 1995). ...
The current study aimed at identifying the difficulties experienced by children with mathematics learning disability (MLD) in the problem representation phase of arithmetic word problem solving using a novel problem types identification task. An MLD group (n = 66) and a typically achieving control group (n = 139) were recruited for an assessment on problem type identification as well as some domain-general and mathematics-related cognitive abilities. Results from ANCOVA showed that the MLD group scored significantly lower than the typically achieving control group on this assessment, after controlling for the effect of cognitive correlates, reading achievement and arithmetic performance. Furthermore, this assessment significantly predicted MLD membership even after taking children's arithmetic competency into account. The current study confirmed the difficulties in problem representation of arithmetic word problems experienced by students with MLD and provided evidence for the need to introduce schema instructions in mathematics classes.
... In early year settings, real objects, physical manipulatives, and representations could be considered as factors which encourage mathematical learning (Cass et al. 2003), Tsuei 2014. Emphasis is placed at the effectiveness of the use of picturebased icons or symbolic representatives (Harrison 2002), of visual and concrete representations in order to develop problem-solving strategies (Jitendra et al. 1998). ...
... Past research demonstrates that instruction that focuses on problem structure is critical to successful arithmetic problem solving (e.g., Fuchs et al., 2008Fuchs et al., , 2010Fuson & Willis, 1989;Jitendra et al., 1998). The SBI program on proportional reasoning was designed with a purpose of teaching both underlying concepts (e.g., ratios, rates, percent) and procedures for solving proportional problems. ...
This study used integrative data analysis (Curran & Hussong, 2009 Curran, P. J., & Hussong, A. M. (2009). Integrative data analysis: The simultaneous analysis of multiple data sets. Psychological Methods, 14(2), 81–100. doi:10.1037/a0015914[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), which allows combining data from different studies, to examine the generalizability of a research-based mathematics program, schema-based instruction (SBI), with a focus on proportional reasoning. Data were pooled from existing SBI studies spanning three U.S. states in which teachers and their classrooms were randomly assigned to either a SBI or control condition. Results of a series of multilevel (i.e., two-level, students within teachers) models indicated the effectiveness of SBI in improving students’ mathematical problem-solving performance. The pooled data findings clarify that SBI improves students’ mathematical problem solving more so for proximal than distal measures. These findings also show Black and Hispanic students continue to lag behind White students, that teachers with more experience tend to be associated with higher performing classrooms, and that increasing concentrations of students eligible for free or reduced price lunch and students receiving special education services are generally associated with weaker performances. Implications of these findings are discussed.
... In early year settings, real objects, physical manipulatives, and representations could be considered as factors which encourage mathematical learning (Cass et al. 2003), Tsuei 2014. Emphasis is placed at the effectiveness of the use of picturebased icons or symbolic representatives (Harrison 2002), of visual and concrete representations in order to develop problem-solving strategies (Jitendra et al. 1998). ...
Early numeracy is related to preschoolers’ acquisition of skills such as understanding and operating with quantities. Consequently, early numeracy has substantial impact on first Graders’ attainments in typical mathematics. Meta-analysis was conducted to address the extent in which early numeracy interventions are effective. Twenty studies were analyzed, including 3.080 young low-performing children (Ncontrol=1.815, Ntreatment=1.265). The overall best estimate for programs based on Early Numeracy Interventions odds ratio was moderately effective g=0.61 (95% CI=[0.44, 0.78]). Heterogeneity was large. Results of the final meta-regression model predicted larger treatment effects for short-term interventions including 1 to 9 sessions. On average, the interventions included instructional strategies such as Explicit Instruction (g=0.57), Corrective Feedback (g = 0.55), CRA (g =0.64), Concrete Manipulatives (g=0.60) Visual Representations (g=0.57) and one-to-one instructional arrangement g=0.79 are moderately effective for children aged 5-8. Results of the study are discussed with respect to implications for designing early numeracy interventions.
... Le rappresentazioni grafiche aiutano gli studenti a risolvere i problemi, poiché connettono le relazioni tra le quantità del problema e le operazioni necessarie per risolverlo. Secondo diverse ricerche (Jitendra et al., 1998;Terwel, Van Oers, Efficacia di un intervento didattico innovativo per il potenziamento del problem solving 207 za dei significati delle operazioni aritmetiche, assenza di qualsiasi riflessione critica sulla soluzione, scoperta di affinità solo apparenti e ragionamenti non corrispondenti ai calcoli. ...
The aim of this research is to evaluate the effectiveness of an innovative teaching action based on evidence and on improving the problem solving skills of students in last two years of primary school with particular reference to the type of origin country. The methodological approach of the action and the results will be discussed following to a brief overview of the theoretical framework and the adopted learning model. Regardless of the type of origin, the analysis of the results showed significant differences in the outcomes of the students involved in the research. Therefore, it contributes to define the teaching action as particularly effective and inclusive.
... In early year settings, real objects, physical manipulatives, and representations could be considered as factors which encourage mathematical learning (Cass et al. 2003), Tsuei 2014. Emphasis is placed at the effectiveness of the use of picturebased icons or symbolic representatives (Harrison 2002), of visual and concrete representations in order to develop problem-solving strategies (Jitendra et al. 1998). ...
Early numeracy is related to preschoolers’ acquisition of skills such as understanding and operating with quantities. Consequently, early numeracy has substantial impact on first Graders’ attainments in typical mathematics. Meta-analysis was conducted to address the extent in which early numeracy interventions are effective. Twenty studies were analyzed, including 3.080 young low-performing children (Ncontrol=1.815, Ntreatment=1.265). The overall best estimate for programs based on Early Numeracy Interventions odds ratio was moderately effective g=0.61 (95% CI=[0.44, 0.78]). Heterogeneity was large. Results of the final meta-regression model predicted larger treatment effects for short-term interventions including 1 to 9 sessions. On average, the interventions included instructional strategies such as Explicit Instruction (g=0.57), Corrective Feedback (g = 0.55), CRA (g =0.64), Concrete Manipulatives (g=0.60) Visual Representations (g=0.57) and one-to-one instructional arrangement g=0.79 are moderately effective for children aged 5-8. Results of the study are discussed with respect to implications for designing early numeracy interventions.
... Swanson, Lussier, and Orosco (2013) asserted that visual-schematic strategies which form the basis of SBI support the visual-spatial WM of students with MD. SBI has produced favorable results for increasing outcomes for students with MD in word problem solving across a variety of grade levels and problem types Jitendra et al., 1998;Xin, 2008;Xin, Jitendra, & Deatline-Buchman, 2005). ...
The purpose of this review is to determine the evidence base classification of schema-based instruction (SBI) as an intervention to improve word problem-solving outcomes in mathematics for students with learning or mathematics disabilities in Grades K–12. Using the Council for Exceptional Children’s quality indicators (QIs) and standards, we reviewed both single-case and group design studies to classify the evidence of SBI. Results of this review indicate that SBI is a potentially evidence-based practice (EBP) for students with learning disabilities. Implications and directions for research and practice are presented.
... Soyut olan matematik problemi çözme becerisini, şemalar kullanarak görselleştiren bu stratejinin, yetersizliğe sahip ya da matematikte risk altında olan öğrencilerde etkili olduğunu gösteren pek çok araştırma mevcuttur. Alanyazında şemaya dayalı öğretim stratejisi ile yapılan öğretimlerin; öğrenme güçlüğü (Garderen, 2007;Jitendra ve Hoff, 1996;Jitendra, Hoff ve Beck, 1999;Jitendra, DiPipi ve Perron-Jones, 2002;Na, 2009;Xin, Jitendra ve Deatline-Buchman, 2005;Xin, 2008), zihin yetersizliği (Baki, 2014;Jitendra, Griffin, McGoey, Gardill, Bhat ve Riley, 1998;Kha, 2012;Karabulut ve ark., 2015;Tufan, 2016), otizm spektrum bozukluğu (James, 2015; Kasap, 2015;Rockwell, Griffin ve Jones, 2011;Rockwell, 2012;Root, Browder, Saunders ve Lo, 2017), duygu ve davranış bozukluğu (Hill, 2012;Jitendra ve ark., 2010), görme yetersizliği (Tuncer, 2009) olan çocuklarda problem çözme becerilerini artırdığına ilişkin bulgulara rastlanmışken ayrıca matematikte risk altında olan (Leh, 2011; Karabulut ve ark., 2015). Ayrıca alan öğretmenlerinin doğrudan öğretim yöntemine hâkim olmasından ve doğrudan öğretim yönteminin akademik becerilerin öğretiminde etkililiğini gösteren pek çok çalışma (Bayram, 2006;Çelik, 2007;Çalık, 2008;Dağseven, 2008;İlik, 2009;Kahyaoğlu, 2010;Karakoç, 2002) yapılmasından dolayı, doğrudan öğretim yöntemi ile sunulan şemaya dayalı öğretim stratejisinin temel toplama ve temel çıkarma işlemlerini içeren gruplama problemleri çözümü öğretiminde etkili olup olmayacağının araştırılmasının önemli olacağı düşünülmektedir. ...
... External representations (see Figure 1) encompass physically embodied, observable configurations -such as pictures, concrete materials, tables, equations, diagrams, and drawings of one-, two-, or three-dimensional figures or various forms of schemata (Jitendra et al., 1998). All these embodiments can be provided in the form of drawings or can be digitalized by computer programs. ...
The current study provides a meta-analysis of global research on using representations to support the learning of mathematics in Pre-K through Grade 5. A total of 13 primary studies encompassing 1,941 subjects was analyzed. The weighted mean effect size for the 13 studies was reported to be ES = 0.53 (SE = 0.05). A 95% confidence interval around the overall mean – Clower = 0.42 and Cupper = 0.63 – proved its statistical significance and its relative precision. The calculated effect size signifies strong, robust support for the use of representations in Pre-K through Grade 5 mathematics classes and highlights the importance of providing students with opportunities to construct and explore transitioning between various forms of representations. Moderator analysis revealed differences among the effects due to a different type of representation, grade levels, and concepts taught. A synthesis of moderator effects allowed for a formulation of a general way of applying representations that produces maximum learning effects and that the teachers can adopt in their school practice. While the effect sizes provided a means of determining the most effective ways of applying representations, questions about how to develop students’ transitioning from one representation to another remain unsolved. A further discussion of the impact of the study findings beyond the boundaries of elementary mathematics classrooms follows.
... Studies in the existing literature report that the schema approach is effective in teaching verbal mathematics problems to typically developing individuals (Griffin & Jitendra, 2009;Jitendra et al., 2009) as well as those with learning disabilities (Griffin & Jitendra, 2009;Jitendra, DiPipi, & Perron-Jones, 2002;Jitendra & Hoff, 1996;Jitendra, Hoff, & Beck,1999;Jitendra et al., 2007;Na, 2009;Xin, 2008;Xin, Jitendra, & Deatline-Buchman, 2005), emotional and behavioral disorders (Jitendra, George, Sood, & Price, 2010), visual impairment (Tuncer, 2009), and intellectual disorders (Baki, 2014;Jitendra et al., 1998;Karabulut, Yıkmış, Özak, & Karabulut, 2013;Kot, 2014). However, only a limited number of studies have been conducted with individuals with ASD (Rockwell, 2012;Rockwell, Griffin, & Jones, 2011). ...
Teaching verbal mathematics problem-solving skills to individuals with developmental disorders is important for these individuals to understand cause and effect relations. The present study aimed to determine the effectiveness of a schema approach for individuals with Autism Spectrum Disorders (ASD) acquiring and sustaining verbal mathematics problem-solving skills. The study also aims to investigate the participants’ verbal mathematics problem-solving skills, their retention levels after the termination of the application, and generalization of this skill to different types of mathematics problems. Three individuals with ASD aged 9, 11, and 14 participated in the study. A single-subject, multiple-probe design with probe conditions across participant research method was used. The findings demonstrated that instruction with the schema approach increased the participants’ verbal mathematics problem-solving performance. This increase was retained 1, 3, and 5 weeks after the instruction was completed. Furthermore, two participants were able to generalize the verbal mathematics problem-solving skills for comparison-type problems with unknown results in comparison-type problems with unknown difference amounts. Social validity data collected from the participants’ mothers and classroom teachers showed that both groups had positive views on the use of the schema approach in the instruction of verbal mathematics problem-solving skills.
... One study did not report the number of sessions preventing a calculation of the total duration (Jitendra & Hoff, 1996). However, the same author team conducted a similar study (Jitendra et al., 1998) with almost identical procedures; the decision was made to use the same total duration for the missing value. ...
A variety of instructional practices have been recommended to increase the problem-solving (PS) performance of elementary school children. The purpose of this meta-analysis was to systematically review research on the use of schema instruction to increase the PS performance of elementary school–age students. A total of 21 studies, with 3,408 elementary school student participants, and 324 students identified with disabilities, met criteria. Moderator analysis includes assignment, implementer, instruction, problem type, and treatment duration. Results indicate an overall effect size (Hedge’s g) for schema instruction of 1.57 (.043), CI95 [1.52, 1.61] for immediate PS and 1.09 (.046), CI95 [1.04, 1.14] for transfer PS. Moderator analysis, future research, and implications for practice are discussed.
... Notably, for elementary mathematical problems within the domain of operations and algebraic thinking, the CCSS-M requires students to both represent and solve problems (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). The requirement for students to represent problems is aligned to the use of visual representations (Jitendra et al., 1998;Witzel, Mercer, & Miller, 2003;Woodward, 2006) and the use of underlying structure to solve word problems (Darch, Carnine, & Gersten, 1984;Fuchs et al., 2003;Xin, Jitendra, & Deatline-Buchman, 2005) which are both prevalent in the literature base of effective mathematic instruction. Given that the use of representation and structure is so prevalent in the research base, standards, and curriculumit is conspicuously absent in current approaches to CBM in mathematics. ...
Standardized formative mathematics assessments typically fail to capture the depth of current standards and curricula. Consequently, these assessments demonstrate limited utility for informing the instructional implementation choices of teachers. This problem is particularly salient as it relates to the mathematical problem solving process. The purpose of this study was to develop and evaluate the psychometric characteristics of Structural-Symbolic Translation Fluency, a curriculum-based measure (CBM) of mathematical problem solving. The development of the assessment was based on previous research describing the cognitive process of translation (Mayer, 2002) as well as mathematical concept development at the quantitative, structural, and symbolic levels (Dehaene, 2011; Faulkner, 2009; Griffin, 2004).
Data on the Structural-Symbolic Translation Fluency assessment were collected from 11 mathematics and psychometrics experts and 42 second grade students during the spring of 2016. Data were analyzed through descriptive statistics, frequencies, Spearman-Brown correlation, joint probability of agreement, Pearson correlation, and hierarchical multiple regression. Psychometric features of interest included internal consistency, inter-rater reliability, test-retest reliability, content validity, and criterion-related validity. Testing of the 9 research questions revealed 9 significant findings. Despite significant statistical findings, several coefficients did not meet pre-established criteria required for validation. Hypothesized modifications to improve the psychometric characteristics are suggested as the focus of future research. In addition, recommendations are made concerning the role of assessing the translation process of mathematical problem solving.
... General Structure of SBSI SBSI is composed of four procedural steps (FOPS): (1) Find the problem type (problem schema identification, schema knowledge), (2) Organize the information in the problem using the schema-diagram (representation, elaboration knowledge), (3) Plan to solve the problem (strategic knowledge), and (4) Solution (execution knowledge) (Jitendra et al., 1998;Schwab, Tucci, & Jolivette, 2013). The FOPS heuristic is distinguishable from general heuristics in that it is linked to particular problem types (e.g., additive, multiplicative, compare, proportion). ...
While deficits for students with learning disabilities (LD) are prevalent in almost all aspects of mathematics, difficulty in the application and understanding of problem-solving tasks are much more challenging to remediate than computational and procedural skills. Given the complexities involved in authentic problem-solving activities emphasized in current mathematics standards and the inherent challenges presented to students with LD, the importance of using strategies and techniques guided by evidence-based practices is paramount. Yet, ineffective instructional strategies for problem solving are still widespread in both mathematics curricula and available teacher resources. In this chapter, we provide a description of a commonly used ineffective problem-solving strategy (i.e., the keyword strategy), an overview of the keyword research, and an explanation for its ineffectiveness. We conclude with a description of three evidenced-based problem-solving approaches and practices that significantly improve the mathematical performance of students with LD.
... Multiple studies have demonstrated the efficacy of schema instruction (SI) for students with MD (Fuchs et al., 2008;Fuchs et al., 2009;Jitendra et al., 1998;Powell & Fuchs, 2010). In SI, a schema is used as a framework in which students are taught to identify additive problem types (e.g., group, change, compare; Griffin & Jitendra, 2009). ...
Word problems are prevalent on high-stakes assessments, and success on word problems has implications for grade promotion and graduation. Unfortunately, English Language Learners (ELLs) continue to perform significantly below their native English-speaking peers on mathematics assessments featuring word problems. Little is known about the instructional needs and performance of ELLs at risk of mathematics difficulty (MD). In the present study, an exploratory quasi-experimental design was used to investigate word-problem instruction for ELLs in a culturally and linguistically diverse public elementary school. Specifically, we studied the efficacy of a word-problem intervention for ELLs with MD (N = 9) that combined culturally and linguistically responsive practices with schema instruction (CLR-SI). The study is unique in that it combines research on effective instruction for ELLs and students with MD; CLR-SI has not been investigated for either ELLs or students with MD. Results have implications for teachers, administrators, and researchers of ELLs with MD.
This meta-analysis synthesized the findings of experimental and quasi-experimental studies to determine the efficacy of math interventions for enhancing the word problem-solving accuracy of students with mathematics difficulties in Grades 4 to 12. Further, we identified intervention and design characteristics that moderate the effectiveness of interventions. We appraised study quality using Council for Exceptional Children (CEC's) evidence standards. We also used random-effects meta-regression models with robust variance estimation (RVE) to compute the overall weighted treatment effect and investigate the moderating impact of 15 variables. Results from 36 intervention studies published between 1989 and 2020 reveal a relatively large and positive mean effect size (g = 0.71; SE = 0.11, 95% CI [0.49, 0.92]) across 88 outcomes, suggesting the examined interventions were effective. The moderator analyses' findings showed the treatment efficacy varied as a function of five intervention characteristics, including intervention model, interventionist, grade level, math topic, and intervention duration. None of the design features we examined led to significant moderator effects. We discuss findings and limitations and provide directions for future research and practice.
Pengajaran matematik yang efektif perlu mengandungi unsur-unsur penyampaian isi pelajaran yang berkesan kepada pelajar agar pelajar dapat memahami dan seterusnya maklumat tersebut boleh diaplikasikan untuk menyelesaikan pelbagai masalah matematik dalam pelbagai situasi. Keberkesanan penyampaian pengajaran matematik boleh ditingkatkan melalui pengaplikasian kaedah-kaedah pembelajaran seperti worked example dan self-regulation techniques yang telah dikenalpasti semasa kajian literatur. Oleh itu, kajian tindakan ini telah dijalankan untuk memenuhi tujuan ini. Instrumen yang digunakan dalam kajian tindakan ini termasuklah soal selidik, Kuiz Pra, Kuiz Pos dan Ujian. Kajian ini membolehkan pensyarah mengadaptasi kaedah worked example dan self-regulation techniques bagi meningkatkan kefahaman pelajar dalam algebra. Dengan ini, kelemahan pelajar dalam algebra dapat diatasi. Kedua-dua kaedah ini telah membantu para pelajar memahami konsep dan meningkatkan kemahiran penyelesaian algebra dengan lebih baik dan berkesan. Abstract Effective mathematics teaching should contain elements of effective content delivery to students so that students can understand and then the information can be applied to solve various mathematical problems in various situations. The effectiveness of mathematics teaching delivery can be enhanced through the application of learning methods such as worked examples and self-regulation techniques that have been identified during the literature review. Therefore, this action research has been conducted to fulfill this purpose. Instruments used in this action research include questionnaires, Pre Quiz, Post Quiz, and Test. This study allows lecturers to adapt worked example methods and self-regulation techniques to improve students' understanding of algebra. With this, students' weaknesses in algebra can be overcome. Both of these methods have helped students understand the concept and improve their algebra solution skills better and more effectively.
This meta-analysis assessed the effect of word-problem-solving interventions on the word-problem-solving accuracy of
students identified as having a learning disability (LD) or at risk for an LD in kindergarten to the sixth grade. Eighteen
randomized control group designed studies met the inclusion criteria. Overall, word-problem-solving interventions yielded
a significant positive effect on the word-problem-solving accuracy of students in elementary grades with LD (effect size
[ES] = 1.08). Instructional components that underlie effective studies were also identified. Results suggest that peer
interaction and transfer instructions yielded large effects on treatment outcomes. Results also suggested that intensive
interventions (50-min sessions, 34 total sessions) in Grade 3 regardless of instructional setting yielded the largest ESs.
These findings support the need to develop and implement quality evidence-based instruction in classroom settings (Tier
1 instruction) prior to utilizing additional resources for more intensive and individualized intervention.
The purpose of this research study was to examine the effectiveness of culturally appropriate problem-solving instruction (CAPSI) incorporating elements from schema instruction, video modeling, and vocabulary instruction. Culturally relevant information and the students’ native language were explicitly incorporated into math problem-solving instruction throughout the duration of intervention implementation. Four second-grade Latinx English language learner (ELL) students attending a P–6 elementary school took part in this study. The primary outcome data for this study were collected using word problem probes consisting of 30 word problems selected from grade-level textbooks. In addition to the word problem probes, the effects of the intervention on general problem solving and vocabulary acquisition were analyzed via pre- and posttest measures. The overall effectiveness of CAPSI was analyzed using a multiple baseline design across participants. Results from this study demonstrated improvements in problem solving and math vocabulary acquisition after the implementation of CAPSI.
The analysis of word problem types and schema-based diagrams are assumed useful for helping students and thus, important for preservice teachers to learn in their preparation programs. Drawing on pre-and post-assessments and videotaped demonstrations, this study analyzed elementary preservice teachers’ understanding of addition and subtraction word problem types and how well they could demonstrate their understanding after their exposure in a mathematics methods course. Results suggest that while participants’ knowledge of word problem types and their ability to solve word problems were substantially improved, their identification, diagram representations, and solutions for different types of word problems were not substantially developed. Policy and research implications for these findings are then discussed and proposed.
This meta-analysis synthesized the available empirical evidence on the impact of interventions on the development and enhancement of mathematical word problem solving skills for K–12 students with learning disabilities (LD) and those with mathematics difficulties (MD). A total of 33 studies met inclusion criteria for this meta-analysis. After removing 2 influential data points, 31 true-experimental or quasi-experimental studies (34 independent effect sizes) contributed effect sizes for this meta-analysis. Results of a random effects model showed that word problem solving interventions yielded a moderate positive mean effect size (g = 0.56, 95% confidence interval [0.40, 0.72]). The relation between word problem solving interventions and student learning was moderated by grade level, year of publication, type of measure, intervention model type, and implementer. On average, intervention effects were larger for (a) elementary grades than secondary grades, (b) studies published between 2000 and 2009, (c) researcher-developed than norm-referenced measures, (d) researcher implementers than teacher implementers. Results also showed intervention effects for schema-based transfer instruction were larger than those for schema-based instruction, strategy instruction, and other. These results support mathematics word problem solving interventions for students with LD and/or MD. We conclude by discussing directions for future research and implications for practice.
The purpose of this within‐subject comparisons exploratory study was to examine the influence of a graphic representational strategy on the problem solving performance of fourth graders, including special education students with learning problems. We employed a preliminary design experiment, prior to conducting a formal experimental or quasi‐experimental study, to gain insights into factors that may inhibit or enhance implementation of the intervention, especially in the context of real world of classroom (Gersten, Baker, & Lloyd, 2000). Students received teacher‐led strategy instruction in problem solving using a whole group (8 to 9 students) format followed by guided practice in applying the strategy during cooperative groups. Results indicate that students’ word problem solving performance increased from the pretest to posttest on multiplication and division problems. In addition, some students were able to generalise the skill to untaught problems. Implications of the representational strategy for solving word problems by elementary students and special education students with learning problems are discussed.
The purpose of this review was to conduct a meta-analysis of 25 years of mathematics interventions for students with mathematics difficulty or disability in Grades 4 through 12. A search of the literature yielded 25 studies that met the inclusion criteria. Studies were coded to extract important study information (e.g., participant information, research design, description of treatment, and comparison groups) and data needed to calculate Hedge’s g. We used robust variance estimation (RVE) to address dependence resulting from multiple outcomes per study. The RVE random-effects model estimated a treatment effect of 0.85. After adjusting for small-study effects, the final model estimated an underlying, moderate effect of 0.49 with a large amount of unexplained heterogeneity between studies. Studies with more than 15 hr of treatment and those focused on fraction content significantly moderated mathematics outcomes. Findings are limited by extreme variability across study estimates, the lack of standardized mathematics measures, and a limited number of studies across 25 years of research.
Promising findings from controlled research studies often fail to be transferred to and implemented in schools successfully. This problem is particularly apparent when considering implementation of evidence-based practices related to complex systems such as response to intervention (RTI) and other multitiered intervention frameworks in mathematics. This article addresses the challenges schools face when implementing intensive intervention in mathematics with a data-based individualization (DBI) framework. Preliminary findings from the first year of a formative evaluation study that addressed school-based implementation of DBI include (a) factors that impacted readiness for DBI, (b) the application of DBI at school sites, and (c) the importance of ongoing coaching and consultation. We then discuss practical implications for schools and districts when planning for intensive intervention in mathematics.
The purpose of the present study was to test the efficacy of a modified cognitive strategy instructional intervention originally developed to improve the mathematical problem solving of middle and high school students with learning disabilities (LD). Fifth and sixth grade general education mathematics teachers and their students of varying ability (i.e., average-achieving [AA] students, low-achieving [LA] students, and students with LD) participated in the research study. Several features of the intervention were modified, including (a) explicitness of instruction, (b) emphasis on meta-cognition, (c) focus on problem-solving prerequisites, (d) extended duration of initial intervention, and (e) addition of visual supports. General education math teachers taught all instructional sessions to their inclusive classrooms. Curriculum-based measures (CBMs) of math problem solving were administered five times over the course of the year. A multilevel model (repeated measures nested within students and students nested within schools) was used to analyze student progress on CBMs. Though CBM scores in the intervention group were initially lower than that of the comparison group, intervention students improved significantly more in the first phase, with no differences in the second phase. Implications for instruction are discussed as well as directions for future research.
The purpose of this chapter is to describe several advances and trends in teaching mathematics to students with learning disabilities. Mathematics disabilities affect between 5% and 8% of students in the general population. These students generally experience serious difficulties in developing requisite concepts, skills, and strategies for success in mathematics and, consequently, are at risk for school failure and poor post-secondary outcomes. Research in special education has identified a number of evidence-based practices that have proven effective for teaching students with learning, attention, and/or behavioral disorders that interfere with success in mathematics. Following a discussion of student characteristics and evidence-based practices, two examples of research programs focusing on mathematical problem-solving interventions for students with learning disabilities are highlighted.
Mathematical word problems are the most common form of mathematics problem solving implemented in K–12 schools. Identifying key words is a frequent strategy taught in classrooms in which students struggle with problem solving and show low success rates in mathematics. Researchers show that using the concrete-representational-abstract (CRA) sequence with explicit instruction improves students' computational skills. Researchers also show that schema-based instruction increases students' problem-solving performance. This study combined CRA and schema-based instruction for three students receiving tertiary interventions for mathematics instruction. A functional relation was found for the three students' problem-solving performance. The researchers also interviewed each student to gain a more complete picture of students' thinking. Results and implications are discussed.
Technology involves application of research to solve practical problems, and it may include bodies of knowledge and processes as well as tools. Whether technology impacts learning, and if it does, how it may be best used are questions that have been debated for several decades. This chapter discusses the effect technology has had on schooling and argues that any attempt to improve student learning must stand on relevant, well-designed curricula and evidence-based instructional methods. With this in place, technologies can help support RTI efforts around learning, assessment, teaching, and productivity. The use of technology in instruction and intervention across tiers is discussed, including its affordances, limitations, and barriers, with research and recommendations for the use of technology in reading and mathematics specifically addressed.
This article reports results from a study investigating the efficacy of a proportional problem-solving intervention, schema-based instruction (SBI), in seventh grade. Participants included 806 students with mathematical difficulties in problem solving (MD-PS) from an initial pool of 1,999 seventh grade students in a larger study. Teachers and their students in the larger study were randomly assigned to an SBI or control condition and teachers in both conditions then provided instruction on the topics of ratio, proportion, and percent. We found that students with MD-PS in SBI classrooms scored on average higher than their counterparts in control classrooms on a posttest and delayed posttest administered 9 weeks later. Given students’ difficulties with proportional problem-solving and the consequences of these difficulties, an important contribution of this research is the finding that when provided with appropriate instruction, students with MD-PS are capable of enhanced proportional problem-solving performance.
In this chapter, intervention research on mathematics learning of students with learning disabilities is reviewed. Much of the early research has focused on learning of computational skills, basic facts and procedures, employing behavioral approaches including reinforcement and drill-and-practice. Other investigations examined the effectiveness of metacognitive strategy training in improving attention to task and basic problem solving. More recent intervention research focuses on higher-level mathematics functioning using metacognitive strategies involving problem solving, motivational, self-regulation, and affective factors in improving mathematics achievement. Some but not all of these studies meet the evidence-based criteria. Finally, some recent efforts have begun to examine the role of early mathematics instruction within a Response to Intervention framework. Implications of this research with respect to National Council for Teachers of Mathematics standards are provided.
Students with Emotional and Behavioral Disorders (E/BD) have been consistently experiencing dismal outcomes. The purpose of this chapter is to provide a brief overview of outcomes for this population, examine school-based instructional and behavioral strategies, and discuss transition related practices intended to improve present and future outcomes. It is recommended that while transition-specific practices are essential in maximizing the potential for success in post-school environments, it is also necessary to ensure that students with E/BD are engaged in school through evidence-based practices in early intervention/prevention, instructional, and behavioral interventions.
This chapter is concerned with the development of an important aspect of children's problem-solving skiU in arithmetic-the ability to solve arithmetic word problems. There are several factors that might enable older children to perform better in problem-solving tasks than younger children, including the complexity of conceptual knowledge about the problem domain and the sophis tication of problem-solving procedures . The studies reviewed here suggest that , with age, children's improved ability to solve word problems primarily involves an increase in the complexity of conceptual knowledge required to understand the situations described in those problems . We will describe these findings in this chapter and consider some general issues about the development of problem solving skill .
The problems diverse learners have in mathematics are analyzed in terms of teacher access to high quality educational tools — textbooks, videos, software, activity guides, and so on. The first section discusses emerging tools implied by recent reform efforts in mathematics education. The second section critiques current textbooks in terms of the rate at which content is introduced, the efficacy of the explanations and activities, and the adequacy of practice and review. Four attributes of mathematics educational tools that may help teachers reduce the failure of diverse learners associated with increasing expectations, current practices, and the expectations of reform are: (a) teaching the big ideas of mathematics; (b) teaching conspicuous strategies for solving problems; (c) scaffolding the acquisition of meaningful learning; and (d) providing review that is sufficient, distributed over time, varied across contexts, and cumulative.
Performance on mathematics word problems having varying structures was compared for Grade 3-8 students with and without mild disabilities. Students with disabilities performed at significantly lower levels than did those without disabilities in four types of word problems, even when the problems involved only single-digit computation. Significant effects were also evident for grade and operation (addition, subtraction, multiplication, and division). The findings highlight the need for math instruction to move from a focus on computation to problem-solving activities, including word problems of varying structures. Further, test developers must make more effort to present a variety of reasoning and problem-solving activities in standardized tests.
The mathematical achievement of youth with learning disabilities, attention deficit disorders, and attention-deficit hyperactive disorders is lower than that of their peers. Cognitive ability (including memory) and reading contribute to the comprehension skills needed to eliminate extraneous information, handle multiple operations, and transform verbal information within problems. Further, slow computation affects problem solving by increasing attentional load. Through our work and a review of other studies, we have documented that when IQ and reading are controlled, “true” math deficits are specific to mathematical concepts and problem types. Implications for instruction are drawn from learner characteristics as these interact with the current mathematics curriculum.
This article reviews published research on mathematical word-problem-solving instruction involving students with mild disabilities and students at risk for math failure. The purpose of the review was to summarize 14 word-problem-solving intervention studies with regards to participants, procedures, and findings and provide implications for future research and classroom practice in teaching word-problem solving. Interventions gleaned from the review entailed representational techniques (e.g., diagramming), strategy-training procedures (e.g., cognitive and metacognitive), task variations (sequencing and word-problem context), and computer-assisted instruction. With the exception of one study, all studies reported positive effects of interventions on students' mathematical word-problem solving.
Learning disabled, attention-disordered, and normal adolescents were assessed with 2 performance measures and 3 behavioral measures, recorded while the Ss worked on math problems. Differences were found between the combined experimental and comparison groups in retrieval speed for each operation, but not in accuracy. Off-task attention contributed to the slower speeds of Ss only during multiplication. Rapid fact retrieval was the only significant predictor of the number of correct answers to word problems. Because of the statistical and procedural controls used (IQ, reading skill, and initial math ability), experimental vs control group differences in problem solving were demonstrated only for specific types of word problems. Overall, these findings support theoretical predictions that attention-disordered and learning disabled children have difficulty with repetitive stimuli, which contributes to basic- and advanced-level math deficits. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Explored why US schoolchildren score below their Japanese cohorts on math tests. Fifth-graders in Japan and the US took a test of math achievement that evaluated computational skill and a test of math problem solving that evaluated problem-representation and solution-planning skills. Consistent with previous international comparisons, children in Japan scored highest on both tests. However, students in the US performed relatively better on problem solving than on computation, whereas students in Japan showed the reverse trend. Most importantly, when students of equivalent achievement level were compared, students in the US scored higher than their Japanese cohorts on problem-solving skills. The results are consistent with an exposure hypothesis, that is, the idea that international differences in mathematical performance are caused by differences in the amount and kind of exposure to math. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Direct instruction has come to have many different meanings, all of which are associated with some form of structured teaching. In this article, Direct Instruction refers primarily to the work done by Engelmann and his colleagues. Further, this article focuses on curriculum analysis, an aspect of Direct Instruction that is frequently neglected or obscured by an emphasis in the literature on classroom management and teacher performance. We discuss explicit strategies as they are used to teach higher cognitive skills, the importance of teacher training to an appropriate use of Direct Instruction curricula, and the supporting empirical research—from the mildly to the severely handicapped—for this highly successful approach to instruction.
A decade of heightened emphasis in the United States on mathematics and science education has had little influence on academic achievement or parental attitudes. American elementary school children in 1990 lagged behind their Chinese and Japanese peers to as great a degree as they did in 1980. Comparison of the performance of elementary and secondary school students between 1980 and 1990 reveals a decline from first to eleventh grade in the relative position of American students in mathematics. Parental satisfaction with American students' achievement and education remains high and standards remain low. Innate ability continues to be emphasized by Americans as a basis for achievement. American eleventh graders report more indications of stress than do their Chinese and Japanese counterparts.
Schemas in Problem Solving explores a theory of schema development and studies the applicability of the theory as a unified basis for understanding learning, instruction and assessment. The theory's prescriptions for teaching are direct, and its application to assessment suggests new directions for tests. After examining the roots of the theory in earlier work by philosophers and psychologists, Marshall illustrates the main features of her theory with experimental evidence from students who are learning to recognize and solve arithmetic story problems. She describes individual performance with traditional empirical studies as well as computer simulation. The computer simulation reflects an approach in modelling cognition. Marshall's model links neural networks with symbolic systems to form a hybrid model that uses pattern matching of sets of features as well as logical step-by-step rules.
This study examined effects of two types of instruction on the word-problem solving performance of postsecondary students with learning disabilities. We used an analysis of error patterns to determine the effects of explicit instructions when word-problem language did not directly correspond (i.e., was inconsistent) with required arithmetic operations. Thirty-eight students randomly participated in either a translation training group, a diagram training group, or an attention-control group. Analyses of variance revealed that the diagram group outperformed both the attention-control and the translation group. We interpret these findings as showing the importance of procedural as well as declarative forms of math word-problem solving knowledge.
This study investigated the effects of a two-phase cognitive strategy on algebra problem solving of adolescents with learning disabilities. The strategy was designed to enable students to represent and solve three types of word problems. The study used a modified multiple baseline with 11 replications as well as a two-group design. Conditions of the multiple-baseline design included baseline, instruction to mastery, transfer, and maintenance. Visual analysis of the single-subject data showed the strategy to be an effective intervention for this sample of students with deficits in algebra problem solving, but with criterial knowledge of basic operations and one-step problems. Statistical analyses of the two-group data showed that the instructed students had significantly higher posttest scores than the comparison group. Overall, the instructed students demonstrated improved performance on algebra word problems. Maintenance and transfer of the strategy were evident. This study has implications for teaching complex problem solving to adolescents with learning disabilities in secondary schools.
Using a cognitive-metacognitive theoretical framework, this study examined the verbalizations of middle school students as they thought aloud while solving three mathematical word problems. Middle school subjects who had been identified as learning disabled, average achieving, and gifted were given demonstrations and practice in thinking aloud and were then instructed to solve one-step, two-step, and three-step word problems while thinking aloud. A priori categories were developed based on information-processing theory and a cognitive-metacognitive perspective of mathematical problem solving. Verbalizations were coded, and between-group and grade comparisons of the number of verbalizations within categories were conducted. The relationship between type of verbalization and overall performance was also analyzed. The findings support Ericcson and Simon's (1980) view of concurrent verbalization as a technique for assessing cognitive processes and Swanson's (1988) notion that students with learning disabilities may use a qualitatively different approach to problem solving than their nondisabled peers.
Mathematics performance is conceptualized as a problem-solving situation consisting of cognitive and metacognitive processes associated with (a) representing the problem, (b) planning a solution, (c) carrying out the operations entailed by the plan, and (d) monitoring the course of solution. Strategies for accomplishing these activities have been instructed within a variety of instructional models, three of which are discussed: direct instruction, self-instruction, and mediated performance (guided learning). Strategy instruction studies representative of each method have been applied to mathematics computation tasks and to word problem solving. These studies are reviewed with a focus on what makes for effective strategy instruction for learning disabled children. Conclusions center on the need to deal explicitly with problem representation and conceptual understanding of the task, task-specific planning and computation skills, and general organizational frameworks for systematically monitoring solution progress. Finally, it is recommended that strategy instruction studies make provision for component practice sufficient to develop efficient access to and use of the available knowledge base as well as of the instructed strategies.
The purpose of this study was to investigate the problem-solving performance of learning disabled students and their peers. Irrelevant linguistic and numerical information embedded within addition word problems was presented to matched groups of subjects. Whereas irrelevant numerical information negatively affected pupil performance, irrelevant linguistic information did not affect performance. Learning disabled students exhibited lower accuracy levels and problem-solving rates than grade-age peers. Implications of these findings are discussed in terms of students' problem-solving processes as well as educational programming for learning disabled pupils.
Problem solving is receiving increased attention from professionals working with learning disabled adolescents because of its obvious relation to success within the secondary school curriculum and life adjustment in the post-school environment. This study examines the problem-solving characteristics of learning disabled adolescents through a paradigm that uses verbalized self-instructions. Havertape and Kass delineate specific characteristics of learning disabled adolescents which differentiate their problem-solving skills from those of their normal peers. Implications for intervention are discussed.
In the past few years, it has been argued that the international assessments are seriously flawed by sampling and test bias and that conclusions about U.S. educational inferiority are unwarranted. Alternatively, it has been argued that the differences in national performances are related to cultural differences, institutional arrangements such as curriculum and length of the school year, and ineffective pedagogy. To clarify these issues, I review the history of the assessments and evaluate the major explanations of the achievement differences. I examine, in depth, the theoretical and empirical underpinnings of curriculum-centered explanations because of their relevance to school reform (Westbury, 1992, 1993).
Four students with learning disabilities, whose primary difficulty in solving simple word problems involved performing the wrong operation, were taught a strategy for comprehending the problem and devising an appropriate solution. Students learned to apply the strategy first to addition word problems, then to subtraction word problems. Upon completion of instruction, students' overall performance on mixed sets of addition and subtraction word problems improved, and they were much less likely to perform the wrong operation. Although the effects of instruction generalized to a different setting, maintenance of strategy effects was mixed, perhaps due to administration of the maintenance probes during summer vacation.
The purpose of this study was to investigate mathematical problem-solving characteristics of middle school students with learning disabilities. Cognitive, metacognitive, and affective characteristics of middle school students with learning disabilities and average-achieving and gifted students (n = 90) were studied to determine similarities and differences among good, average, and poor problem solvers. We included measures of mathematical achievement, reasoning, mathematical word problems, and strategy knowledge, use, and control. Results indicated that despite a positive attitude toward mathematics, students with learning disabilities differed significantly from average-achieving and gifted students in general mathematical achievement and problem solving. Results also suggested that students' poor mathematical problem solving may be related less to solution errors than to an ability to represent problems and predict appropriate solution equations and operations. Implications for research and practice are discussed.
Minimum competency testing is currently a reality in American education; the practice presents a number of problems for regular class and handicapped students. In this research, the mathematics performance of learning disabled (LD) students and their nondisabled peers on the Florida State Student Assessment Test-II (SSAT-II) was evaluated, and employers' opinions about the importance of mathematical skills assessed on the SSAT-II were analyzed. It was reasoned that information such as this would provide valuable benchmarks for use in planning, organizing, and implementing educational programs for high school students. Skill performance and mastery scores were tabulated and compared for adolescents classified as learning disabled and their regular class peers; the relative importance of various skills to employers was also tabulated and compared. Strengths and weaknesses in basic math competencies were identified and relations among these skill performances and employer opinions were identified. In general, LD students performed better on skills requiring literal use of numbers and worse on skills requiring application of mathematical knowledge. Employers' ratings of the extent to which various skills were required in the world of work supported the importance of competence in these basic skills.
Learning to solve arithmetic story problems is a difficult task, particularly for students with learning disabilities. The comprehensive curriculum presented here recognizes the complexity of the problem‐solving process, is responsive to the instructional needs of learning disabled students, and has proven effective in improving their performance in solving arithmetic story problems. It provides direct instruction in obtaining relevant information from the problem, as well as how to proceed through the problem‐solving process. Students are taught to use a plan, to test relevant hypotheses, and to evaluate their work.
In recent years, a new perspective has challenged the conventional wisdom that U.S. students do poorly in international assessments. It holds that sampling biases and a different curricular arrangement have made U.S. achievement appear low and that a focus on rankings has masked what are really small achievement differences among countries. To assess this perspective, I review evidence from the major international assessments of the past two decades, including the recent Third International Math and Science Study (TIMSS). I argue for a middle ground. U.S. performance has not been consistently poor, but varies by subject and grade. Ranks sometimes have exaggerated differences, but even developed countries often differ greatly in achievement. Contrary to a common misconception, the assessments have not unfairly compared the mass of U.S. students to small academic elites in other countries. Our poor math performance is not a simple matter of selection bias or course sequencing, but also reflects real deficiencies in curricular focus and teaching. These have been linked to our assembly-line cultural conception of knowledge and schooling. To ground the international findings, I summarize evidence on domestic indicators. Although trends have been generally stable, U.S. student achievement has been weak for several decades. Fundamental school reform is warranted.
The aim of the study is to investigate the informal and formal mathematical knowledge of children suffering from "mathematics difficulty" (MD). The research involves comparisons among three groups: fourth-grade children performing poorly in mathematics but normal in intelligence; fourth-grade peers matched for intelligence but experiencing no apparent difficulties in mathematics; and a randomly selected group of third graders. These children were individually presented with a large number of tasks designed to measure key mathematical concepts and skills. The findings suggest that: (1) MD children are not seriously deficient in key informal mathematical concepts and skills; (2) MD children seem to have elementary concepts of base ten notation but experience difficulty in related enumeration skills, particularly when large numbers are involved; (3) MD children's calculational errors often result from common error strategies; (4) MD children display severe difficulty in recalling common addition facts; and (5) in the area of problem solving, MD children are capable of "insightful" solutions and can solve simple forms of word problems, but experience difficulty with complex word problems. MD children are in many respects similar to normal, younger peers; an hypothesis of "essential cognitive normality" is advanced. The only and dramatic exception occurs in the area of number facts. While clinical experience corroborates this finding, its explanation is not evident.
Presents an approach to teaching math word problems to both regular and learning disabled elementary students. The approach begins with 5 readiness tasks, in which students manipulated objects, pictures, numbers, number sentences, and story combinations. The next set of activities begins with the story problem and involves the interpretation of its parts, building on the mental images that the students learned to develop in the previous activities. These readiness activities promote the specialized reading skills and thought processes necessary for the solution of story problems. (1 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Reviews 25 mathematics intervention studies (published during 1973–1988) that used a total of 519 mentally retarded Ss (aged 6.33–53 yrs). The studies focus on 3 areas: basic skills and concepts, rule learning and problem-solving, and applications such as use of time, money, and measurement skills. All studies report positive findings after implementation of their respective interventions. Other topics that are relevant to these studies are discussed: sample issues, design issues, and task/intervention issues. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Incl. bibliographical references
This study investigated the effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of six middle school students with learning disabilities. Conditions of the multiple baseline, across-subjects design included baseline, two levels of treatment, setting and temporal generalization, and retraining. For Treatment 1, subjects received either cognitive or metacognitive strategy instruction. Treatment 2 consisted of instruction in the complementary component of the instructional program so that all subjects eventually received both cognitive and metacognitive strategy instruction. This design allowed a componential analysis of the content as well as sequence of instruction. Generally, subjects improved their mathematical problem solving as measured by performance on one-, two-, and three-step word problems. Discussion focused on effectiveness of treatment, acquisition and application of strategic knowledge, error pattern analysis, and the need to tailor instruction to the learner's individual characteristics.
The major purpose of this study was to attempt to understand some of the reasons for the high academic achievement of Chinese and Japanese children compared to American children. The study was conducted with first and fifth graders attending elementary schools in the Minneapolis metropolitan area, Taipei (Taiwan), and Sendai (Japan). 1,440 children (240 first graders and 240 fifth graders in each city) were selected as target subjects in the study. The children were selected from 20 classrooms at each grade in each city and constituted a representative sample of children from these classrooms. In a follow-up study, first graders were studied again when they were in the fifth grade. The children were tested with achievement tests in reading and mathematics constructed specifically for this study, the children and their mothers were interviewed, the children's teachers filled out a questionnaire, and interviews were held with the principals of the schools attended by the children. In the follow-up study, achievement tests were administered, and the children and their mothers were interviewed. Background information about the children's everyday lives revealed much greater attention to academic activities among Chinese and Japanese than among American children. Members of the three cultures differed significantly in terms of parents' interest in their child's academic achievement, involvement of the family in the child's education, standards and expectations of parents concerning their child's academic achievement, and parents' and children's beliefs about the relative influence of effort and ability on academic achievement. Whereas children's academic achievement did not appear to be a central concern of American mothers, Chinese and Japanese mothers viewed this as their child's most important pursuit. Once the child entered elementary school, Chinese and Japanese families mobilized themselves to assist the child and to provide an environment conducive to achievement. American mothers appeared to be less interested in their child's academic achievement than in the child's general cognitive development; they attempted to provide experiences that fostered cognitive growth rather than academic excellence. Chinese and Japanese mothers held higher standards for their children's achievement than American mothers and gave more realistic evaluations of their child's academic, cognitive, and personality characteristics. American mothers overestimated their child's abilities and expressed greater satisfaction with their child's accomplishments than the Chinese and Japanese mothers. In describing bases of children's academic achievement, Chinese and Japanese mothers stressed the importance of hard work to a greater degree than American mothers, and American mothers gave greater emphasis to innate ability than did Chinese and Japanese mothers.
This study examines the mathematical performance of 220 children from 8 years through 17 years of age diagnosed as having learning disabilities. Student records were searched for data indicating performance on standardized test instruments relating to mathematics. Data for the Woodcock-Johnson Psycho-Educational Achievement Battery math subtests and for the IQ scores from the Wechsler Intelligence Scale for Children-Revised were obtained. Comparisons were made among children at different ages and among specific age clusters. Primary attention was directed toward calculations and applications of math concepts and principles. Developmental patterns across the ages studied were discovered. Implications for long-term comprehensive programming are presented.
This study investigated the effect of an eight-step cognitive strategy on verbal math problem solving performance of six learning disabled adolescents. The cognitive strategy was designed to enable students to read, understand, carry out, and check verbal math problems that are encountered in the general math curriculum at the secondary level. Conditions of the multiple baseline desing included baseline, treatment, generalization, maintenance, and retraining. During treatment, students received strategy acquisition training, strategy application practice, and testing. Visual analysis of the data indicated this eight-step cognitive strategy to be an effective intervention for this sample of students who had deficits in verbal math problem solving. Overall, the students demonstrated impoved performance on twostep verbal math problems. Maintenance and generalization of the strategy were evident. This study has implications for an alternative teaching methodology that focuses on cognitive strategy training to improve the verbal math problem solving of learning disabled students.
This study examined the effects of a schema-based direct instruction strategy on the word-problem-solving performance of three third- and fourth-grade students (2 girls, 1 boy) with learning disabilities. An adapted multiple-probe-across-students design was used. Results indicated that the intervention was successful in increasing the percentage of correct solutions to word problems for all 3 students. In addition, maintenance of word-problem solving was seen 2 to 3 weeks after the study. Student interviews indicated that the strategy was beneficial. Further research with different students and problem types (e.g., multistep) and an investigation of the long-term effects of the strategy and its use in novel settings appear warranted.
The assessment of schema knowledge for arithmetic story problems: A cognitive science perspective
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