The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities.

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.     

Teaching students with LD to use diagrams to solve mathematical word problems

This study examined the effectiveness of instruction focused on teaching students with learning disabilities (LD) to solve 1- and 2-step word problems of varying types. Three students with LD in Grade 8 participated in the study. During the treatment, students received instruction in diagram generation and a strategy that incorporates diagrams as a part of the procedure to solve word problems. The results indicated that all students improved in the number of diagrams they used and in their ability to generate diagrams. Their word problem solving performance increased. Moreover, the students generated and used diagrams to solve other types of problems. Overall, the students were very satisfied with the instruction and would continue to use the diagrams and the strategy to solve word problems in other classroom settings.     

Video Modeling and Explicit Instruction: A Comparison of Strategies for Teaching Mathematics to Students with Learning Disabilities

As classrooms begin to adopt a greater number of digital technologies such as computers and tablets, it is important for educators to understand how effective such tools can be in aiding in the delivery of instruction to students who struggle in mathematics, such as those identified with a learning disability in mathematics. One digital‐based instructional strategy with a limited research base for students with a learning disability is video modeling. Through a single subject alternating treatments design, this study compared the use of video modeling to face‐to‐face explicit instruction for teaching geometry word problems to three secondary students with a learning disability in mathematics. Across 10 sessions of intervention, all three students demonstrated improved performance on all dependent variables with both interventions, while the explicit instruction condition produced slightly greater accuracy scores for two of the three students. The results and their implications for the field of mathematics are discussed.     

App-Based Manipulatives and Explicit Instruction to Support Division with Remainders

ABSTRACT

In theory, both virtual manipulatives and explicit instruction are viable options to support students with disabilities as they learn mathematics. This study explored the effect of a treatment package—an app-based virtual manipulative (Cuisenaire® Rods) in conjunction with explicit instruction—on students’ acquisition and generalization of solving problems involving division of whole numbers with remainders. Three middle school students with disabilities participated in this multiple baseline, multiple probe across participants single case design study. Each of the students acquired the mathematical behavior of being able to solve division with remainders problems. In other words, a functional relation existed between the intervention package of explicit instruction and the Cuisenaire® Rods app-based manipulative and students’ accuracy in solving division with remainders problems. Yet, two students failed to generalize the skill without the explicit instruction and use of the app-based manipulative.     

策略教学的有效性及教学设计保障

策略教学的有效性决定学习策略辅导的质量,其实质是学生对所学策略的持续保持和迁移运用。策略内容选择和教学方法设计乃策略教学有效性的两大制约因素。基于"为有效而教"的要求,教师在设计策略教学时,应着重考量策略内容的明确性、组织性和针对性,并着力引导学生对所学策略进行积极的自主建构。     

Using Number Lines to Solve Math Word Problems: A Strategy for Students with Learning Disabilities

Students with learning disabilities (LD) consistently struggle with word problem solving in mathematics classes. This difficulty has made curricular, state, and national tests particularly stressful, as word problem solving has become a predominant feature of such student performance assessments. Research suggests that students with LD perform poorly on word problem‐solving items due primarily to deficits in problem representation. Therefore, it is imperative that teachers provide these students with supplemental problem‐solving instruction that specifically targets the development of representational strategies. This article describes how one representational strategy, using number lines, can be used to model word problems as part of a comprehensive problem‐solving intervention to improve the conceptual understanding of math word problems and, subsequently, the problem‐solving performance of students with LD.     

Solving Word Problems Using Schemas: A Review of the Literature

Solving word problems is a difficult task for students at‐risk for or with learning disabilities (LD). One instructional approach that has emerged as a valid method for helping students at‐risk for or with LD to become more proficient at word‐problem solving is using schemas. A schema is a framework for solving a problem. With a schema, students are taught to recognize problems as falling within word‐problem types and to apply a problem solution method that matches that problem type. This review highlights two schema approaches for second‐ and third‐grade students at‐risk for or with LD: schema‐based instruction and schema‐broadening instruction. A total of 12 schema studies were reviewed and synthesized. Both types of schema approaches enhanced the word‐problem skill of students at‐risk for or with LD. Based on the review, suggestions are provided for incorporating word‐problem instruction using schemas.     

Algebraic Problem Solving for Middle School Students with Autism and Intellectual Disability

Problem solving is an important yet neglected mathematical skill for students with autism spectrum disorder and intellectual disability (ASD/ID). In addition, the terminology and vocabulary used in mathematical tasks may be unfamiliar to students with ASD/ID. The current study evaluated the effects of modified schema-based instruction (SBI) on the algebra problem solving skills of three middle school students with ASD/ID. Mathematics vocabulary terms were taught using constant time delay. Participants were then taught how to use an iPad that displayed a task analysis with embedded prompts to complete each step of solving the word problems. This study also examined participant’s ability to generalize skills when supports were faded. Results of the multiple probe across participants design showed a functional relation between modified SBI and mathematical problem solving as well as constant time delay and acquisition of mathematics vocabulary terms. Implications for practice and future research are discussed.     

An exploratory study contrasting high- and low-achieving students' percent word problem solving

This study evaluated whether schema-based instruction (SBI), a promising method for teaching students to represent and solve mathematical word problems, impacted the learning of percent word problems. Of particular interest was the extent that SBI improved high- and low-achieving students' learning and to a lesser degree on the indirect effect of SBI on transfer to novel problems, as compared to a business as usual control condition. Seventy 7th grade students in four classrooms (one high- and one low-achieving class in both the SBI and control conditions) participated in the study. Results indicate a significant treatment by achievement level interaction, such that SBI had a greater impact on high-achieving students' problem solving scores. However, findings did not support transfer effects of SBI for high-achieving students. Implications for improving the problem-solving performance of low achievers are discussed.     

Effectiveness of Schema-Based Instruction for Improving Seventh-Grade Students’ Proportional Reasoning: A Randomized Experiment

Abstract

This study examined the effect of schema-based instruction (SBI) on 7th-grade students’ mathematical problem-solving performance. SBI is an instructional intervention that emphasizes the role of mathematical structure in word problems and also provides students with a heuristic to self-monitor and aid problem solving. Using a pretest-intervention–posttest-retention test design, the study compared the learning outcomes for 1,163 students in 42 classrooms who were randomly assigned to treatment (SBI) or control condition. After 6 weeks of instruction, results of multilevel modeling indicated significant differences favoring the SBI condition in proportion problem solving involving ratios/rates and percents on an immediate posttest (g = 1.24) and on a 6-week retention test (g = 1.27). No significant difference between conditions was found for a test of transfer. These results demonstrate that SBI was more effective than students’ regular mathematics instruction.     

Heart of the Vergilian Simile

Current educational needs call for learners who are self-directed and able to engage in problem solving and higher order thinking. However, most students are not getting explicit and systematic instruction to promote development of learning strategies that can advance these goals. Educators need to cultivate strategy instruction techniques that facilitate students’ acquisition of learning strategies and consider how structured opportunities for teacher scaffolding, peer collaboration, and parental involvement can further this effort.     

Cognitive Strategy Instruction for Mathematical Word Problem-solving of Students with Mathematics Disabilities in China

This study investigated the impact of cognitive strategy instruction (CSI) on mathematical word problem solving of students with mathematics disabilities. A sample of fourth-grade students in a Chinese primary school was divided into a treatment group (75 students) and a comparison group (75 students). The sample consisted of students with mathematics disabilities only, students with both mathematics and reading disabilities, as well as average- and high-achieving students. Results showed that students at all ability levels (except high-achieving students) in the treatment group outperformed significantly their counterparts in the comparison group; the intervention effect was stronger for students with mathematics disabilities only than for those with both mathematics and reading disabilities. The present study indicates that CSI is a contextually and pedagogically appropriate model that has a strong potential to improve mathematical word problem solving.     

Improving seventh grade students’ learning of ratio and proportion: The role of schema-based instruction

The present study evaluated the effectiveness of an instructional intervention (schema-based instruction, SBI) that was designed to meet the diverse needs of middle school students by addressing the research literatures from both special education and mathematics education. Specifically, SBI emphasizes the role of the mathematical structure of problems and also provides students with a heuristic to aid and self-monitor problem solving. Further, SBI addresses well-articulated problem solving strategies and supports flexible use of the strategies based on the problem situation. One hundred forty eight seventh-grade students and their teachers participated in a 10-day intervention on learning to solve ratio and proportion word problems, with classrooms randomly assigned to SBI or a control condition. Results suggested that students in SBI treatment classes outperformed students in control classes on a problem solving measure, both at posttest and on a delayed posttest administered 4 months later. However, the two groups’ performance was comparable on a state standardized mathematics achievement test.     

Word problems and mathematical reasoning-A study of children''s mastery of reference and meaning in textual realities

The mastery of word problems is seen as an important test of mathematical ability. When solving such problems, students supposedly go beyond rote learning and mechanical exercises to apply their knowledge to realistic problem situations in which mathematical reasoning becomes an important instrument for making concrete judgements. Research shows that performance on word problems is often surprisingly poor. Non-realistic, and even logically inconsistent, answers to word problems are often accepted by students, and there are many signs that students seldom make so-called realistic considerations when applying their mathematical knowledge to real world events. The study reported is a follow-up of the work by Verschaffel, De Corte, and Lasure (1994) in which the difficulties students have in making realistic considerations were clearly illustrated. In the present study, students (10–12 years of age) worked in groups, and the tasks given (estimating distances) were introduced as part of a general discussion of how to calculate distances to school. Results show that the participants were clearly able to entertain different assumptions regarding how to measure distances, and they make distinctions between alternative options when discussing, for instance, the distance between two villages as indicated on a road sign on the one hand, and when talking about the shortest possible distance on the other. It is argued that the problem of what constitutes a realistic consideration when solving word problems is far from simple but has to be understood in context.     

Equity and Access: All Students are Mathematical Problem Solvers

Often mathematical instruction for students with disabilities, especially those with learning disabilities, includes an overabundance of instruction on mathematical computation and does not include high-quality instruction on mathematical reasoning and problem solving. In fact, it is a common misconception that students with learning disabilities are not strong problem solvers in general. This article highlights the inherent problem solving strengths that students with learning disabilities possess; how they use those skills to address everyday barriers and challenges, and how teachers can relate these skills to academic mathematical instruction. Additionally, practical classroom examples, suggested teaching strategies, and questions for further examinations are discussed.     

Empowering educationally disadvantaged mathematics students through a strategies-based problem solving approach

A major impediment to problem solving in mathematics in the great majority of South African schools is that disadvantaged students from seriously impoverished learning environments are lacking in the necessary informal mathematical knowledge to develop their own strategies for solving non-routine problems. A randomized pretest–posttest control group design was used to empirically investigate the effectiveness of a strategies-based problem solving approach on the problem solving performance of 9th grade disadvantaged students. In this approach students receive explicit instruction on a wide repertoire of problem solving strategies. The results reported in this study show a significant improvement in problem solving performance when a strategies-based approach to problem solving was being implemented. Quantitative and qualitative analyses of the responses to the items showed how the treatment group students had internalized as part of their problem solving repertoire the strategies on which they had been explicitly instructed on. The findings of this study make a case for the adoption of this approach so that the gap between the student’s existing problem solving competence, and the cognitive demands of a problem solving task can be bridged.     

‘I like it instead of maths’: how pupils with moderate learning difficulties in Scottish primary special schools intuitively solved mathematical word problems

This study by Lio Moscardini of the University of Strathclyde shows how a group of 24 children in three Scottish primary schools for pupils with moderate learning difficulties responded to word problems following their teachers' introduction to the principles of Cognitively Guided Instruction (CGI). CGI is a professional development programme in mathematics instruction based on constructivist principles developed at the University of Wisconsin‐Madison. The study found that the sample group of pupils were able to develop their understanding of mathematical concepts through actively engaging in word problems without prior explicit instruction and with minimal teacher adjustments. The pupils' conceptual understandings demonstrated by their solution strategies within CGI activities were generally not consistent with classroom records of assessment. The results were encouraging in illustrating the capacity of the sample group of pupils with moderate learning difficulties to reveal their mathematical thinking and considers the importance of this insight for instructional decision making.     

Slower implicit categorical learning in adult poor readers

We investigated the relationship between reading and explicit and implicit categorical learning by comparing university students with poor reading to students with normal reading abilities on two categorical learning tasks. One categorical learning task involved sorting simple geometric shapes into two groups according to a unidimensional rule. The sorting rule was easily stated by the participants, consistent with explicit learning, and all participants attained criterion levels of performance. The second task involved the integration of features on different dimensions with a more complex rule that could not be described by participants, even though most could attain criterion levels of performance consistent with implicit learning. Poor readers performed as well as those without reading problems in explicit learning but not in implicit learning. Implicit learning was correlated with word reading, phonological decoding, and orthographic skill, independent of verbal ability. We consider the role of implicit learning in reading, and how a deficit could impair phonological and orthographic representation and processing.