Promoting Strategic Learning by Eighth‐Grade Students Struggling in Mathematics: A Report of Three Case Studies

Abstract. This article presents three in‐depth case studies focused on supporting students with learning challenges to learn math strategically. Participants were three eighth‐grade students enrolled in a learning assistance classroom who were of at least average intelligence but who were performing significantly below grade level in mathematics. These case studies document the processes by which these students were supported to self‐regulate their learning in mathematics more effectively. We begin by outlining important instructional foci in mathematics education for intermediate or secondary students with learning disabilities, along with what research indicates are effective instructional processes. In that context, we introduce the theoretical principles underlying the instructional model used here—Strategic Content Learning (SCL). Based on analyses of case study data, we describe how SCL instruction was structured to promote strategic learning. Throughout the discussion, intervention processes are described in sufficient detail to be of use to practitioners.     

Prospective teachers’ skills of attending,interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos

This study explores prospective teachers’ skills of attending, interpreting and responding to content-specific characteristics of mathematics instruction in classroom videos. Prospective teachers analyzed the mathematics instruction of two teachers through four video clips and proposed alternative instructional ways to support the teaching and learning of mathematics. The results indicated that as prospective teachers examined the teachers’ instructional practices, they increased their level of attending and interpretation to content-specific aspects of instruction rather than focusing on generic dimensions of the instruction. When they watched and compared different characteristics of teachers’ mathematics instruction, they provided more detailed and mathematical instructional suggestions.     

Recognition and development of academic talent in educationally disadvantaged students

Abstract

We studied 36 students identified as “educationally disadvantaged” who scored above average on standardized achievement tests and completed a program to reinforce their academic skills in either language arts or mathematics. We pretested and posttested an additional (control) group of 28 students who received no instructional intervention. We found a significant effect of the instructional intervention for both achievement and aptitude test scores in language arts as well as mathematics. Gains in mathematics were significantly greater than in language arts. After instructional intervention, the majority of students were eligible and academically qualified for challenging gifted‐talented programs. These findings are consistent with two other similar research projects using the same instructional program and model, suggesting stability of results. We discuss implications of this study for identifying and developing academic talent in such a population.     

A Follow‐Up of Mathematics Interventions for Secondary Students with Learning Disabilities

Abstract This article extends a previous review of the literature ( Maccini & Hughes, 1997 ) on mathematics interventions for secondary school students with learning disabilities (LD). A systematic review of the literature from 1995 to 2006 yielded 23 articles that met the criteria for inclusion. It was determined that a number of practices demonstrated significant gains for secondary school students with LD in math, including mnemonic strategy instruction, graduated instructional approach, cognitive strategy instruction involving planning, schema‐based instruction, and contextualized videodisc instruction. We also discuss the nature and focus of math interventions and implications for both research and practice based on the findings.     

Teaching Preschoolers to Count: Effective Strategies for Achieving Early Mathematics Milestones

Attention to early childhood mathematics instructional strategies has sharpened due to the relatively poor mathematics performance of U.S. students in comparison to students from other countries and research evidence that early mathematics skills impact later achievement. Early Childhood counting skills form the foundation for subsequent mathematics learning. In this article, we discuss the milestones of counting development and examine preschool classroom mathematics observations through the lenses of two CLASS Dimensions, Concept Development and Instructional Learning Formats. Recommendations for effective instructional strategies around counting and suggestions for incorporating mathematics instruction into storybook reading are provided.     

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.     

Fraction interventions for struggling elementary math learners: A review of the literature

Understanding fractions has been a pervasively difficult skill for struggling math learners, yet it is essential for success in secondary level mathematics skills. The present systematic review examined the evidence base of fraction interventions for elementary level students identified as struggling math learners. Twelve studies met inclusion criteria and were reviewed for (a) instructional focus (b) instructional components, and (c) effectiveness of the intervention on fraction performance. The majority of studies had a primary instructional focus on conceptual knowledge of fraction learning. Results of the study also indicated that all interventions included multiple evidence‐based instructional components (e.g., concrete and visual representations, range and sequence of examples, etc.). These multicomponent interventions improved performance on a variety of proximal fraction outcome measures. Intervention effects were mixed for generalized outcome measures and minimal for distal outcome measures. Limitations, implications for practice, and future research directions are discussed.     

Inquiry-Oriented Instruction: A Conceptualization of the Instructional Principles

Research has highlighted that inquiry-based learning (IBL) instruction leads to many positive student outcomes in undergraduate mathematics. Although this research points to the value of IBL instruction, the practices of IBL instructors are not well-understood. Here, we offer a characterization of a particular form of IBL instruction: inquiry-oriented instruction. This characterization draws on K-16 research literature in order to explicate the instructional principles central to inquiry-oriented instruction. As a result, this conceptualization of inquiry-oriented instruction makes connections across research communities and provides a characterization that is not limited to undergraduate, secondary, or elementary mathematics education.     

Reaching the Mountaintop: Addressing the Common Core Standards in Mathematics for Students with Mathematics Difficulties

The Common Core State Standards provide teachers with a framework of necessary mathematics skills across grades K‐12, which vary considerably from previous mathematics standards. In this article, we discuss concerns about the implications of the Common Core for students with mathematics difficulties (MD), given that students with MD, by definition, struggle with mathematical skills. We suggest that instruction centered on the Common Core will be challenging and may lead to problematic outcomes for this population. We propose that working on foundational skills related to the Common Core standards is a necessary component of mathematics instruction for students with MD, and we provide teachers with a framework for working on foundational skills concurrent with the Common Core standards. We caution, however, that implementation of the Common Core is in its infancy, and the implications of the Common Core for students with MD need to be monitored carefully.     

First Principles of Attitudinal Change: a Review of Principles,Methods and Strategies

Knowing how to effectively design attitudinal change instruction for various learning settings is essential for instructional designers. However, the research on instructional design for attitudinal change instruction has been relatively dispersed and lacks cohesion, despite its increasing importance. The purposes of this paper are to 1) reignite the scholarly conversation on designing attitudinal change instruction, and 2) provide directions for future research. Using Merrill’s first principles of instruction as a guide and framework, we review the existing research to identify and discuss the principles for designing attitudinal change instruction.     

Extending Responsiveness to Intervention to Mathematics at First and Third Grades

Responsiveness to intervention (RTI) is an innovative approach to the identification of learning disabilities (LD). The central assumption is that RTI can differentiate between two explanations for low achievement: poor instruction versus disability. If the child responds poorly to validated instruction, then the assessment eliminates instructional quality as a viable explanation for poor academic growth and instead provides evidence of a disability. For children who do respond nicely, RTI serves a critical prevention function. Most of RTI research has been focused on early reading. In this article, we describe two ongoing programs of research on RTI in the area of mathematics: one on a comprehensive mathematics curriculum at first grade and the other focused on word problems at third grade. For each research program, we describe the sample, explain how students are identified as at risk for mathematics disability, provide an overview of the interventions to which responsiveness is gauged, and describe some results to date.     

A Root Awakening: Vocabulary Instruction for Older Students with Reading Difficulties

There is reliable evidence that new vocabulary is primarily acquired through wide independent reading. However, struggling readers tend to avoid reading, resulting in limited word encounters and inadequate vocabulary growth, and they often have difficulties inferring the meanings of new words from context. While there are no clear solutions to the problem of vocabulary acquisition for older students with reading difficulties, there are instructional approaches that have some evidence of effectiveness for this population. We describe the research base and promising practices related to three aspects of vocabulary instruction: (1) creating a verbal learning environment that fosters word consciousness, (2) selecting and teaching specific words, and (3) teaching an independent word learning strategy through a combination of contextual and morphemic analysis. These instructional approaches are grounded in overarching principles recognized by researchers as being characteristic of effective instruction for students with learning difficulties, including explicit instruction, promoting cognitive and collaborative engagement, and providing many opportunities for practice, including distributed practice, with teacher feedback. Finally, we discuss the possibilities inherent in a cross‐content schoolwide approach to vocabulary instruction at the secondary level. We conclude with a call for additional research examining the effectiveness of instructional approaches to vocabulary development for secondary school students with reading difficulties, including a schoolwide collaborative model.     

The research locus and conceptual basis for MULTILIT: Why we do what we do

Abstract

MULTILIT is predicated upon contemporary, evidence‐based best practice and research‐based models for effective instruction of reading and related skills for low‐progress readers. The conceptual foundations and operating principles of MULTILIT are described in Wheldall and Beaman (2000), while three government funded reports from Australia, the United States and the United Kingdom have reviewed the effectiveness of different approaches to the teaching of reading. This paper draws on these resources to explain ‘why we do what we do’ in MULTILIT instructional programs, showing how MULTILIT relates to, and is consistent with, international research and best practice in this area.     

Selecting Students and Services for Compensatory Education: Lessons from Aptitude--Treatment Interaction Research

In this keynote review, we draw from recent theory and research on individual differences, learning, and instruction to answer questions about the selection of students and the provision of services in compensatory education. In addressing the question regarding provision of services to Chapter 1 students, we focus on five issues: (a) fragmentation versus integration of what is to be learned, (b) time as a variable, (c) the provision of direct instruction and possible alternatives to direct instruction, (d) compensatory education for aptitude processes, and (e) instructional grouping decisions. Current Chapter 1 services may promote fragmentation rather than integration of the academic content for the student. Moreover, Chapter 1 services focus more on teaching basic skills than on promoting higher-order thinking in reading and mathematics. We discuss problems associated with current selection procedures such as misclassification and "labeling" of Chapter 1 students. We conclude with recommendations for educational researchers and policymakers.     

The Growth and Nature of Research Interest in Arithmetic and Reading

ABSTRACT

The author examined whether mathematics instruction provided by kindergarten teachers is related to children's mathematics learning during the kindergarten year based on the children's socioeconomic status and race. Hierarchical linear modeling was employed using a large sample of kindergarten students to estimate relationships between the teacher's instructional approach (e.g., concrete-spatial, interpersonal, linguistic) and children's mathematics skills. Results showed that the teacher's instructional approach was selectively related to children's mathematics learning. Students in classrooms where teachers frequently employed a concrete-spatial instructional approach learned more during the kindergarten year. Students from lower socioeconomic backgrounds benefitted more from teachers who often employed an interpersonal approach, and Black/African American students benefitted less from classrooms where teachers relied more on a linguistic approach. The findings suggest that kindergarten teachers’ instruction needs to employ varied methods that take into account students’ mathematics skills and background characteristics.     

Developing a Multi-Dimensional Early Elementary Mathematics Screener and Diagnostic Tool: The Primary Mathematics Assessment

There is a critical need to identify primary level students experiencing difficulties in mathematics to provide immediate and targeted instruction that remediates their deficits. However, most early math screening instruments focus only on the concept of number, resulting in inadequate and incomplete information for teachers to design intervention efforts. We propose a mathematics assessment that screens and provides diagnostic information in six domains that are important to building a strong foundation in mathematics. This article describes the conceptual framework and psychometric qualities of a web-based assessment tool, the Primary Math Assessment (PMA). The PMA includes a screener to identify students at risk for poor math outcomes and a diagnostic tool to provide a more in-depth profile of children’s specific strengths and weaknesses in mathematics. The PMA allows teachers and school personnel to make better instructional decisions by providing more targeted analyses.     

Special Education Teacher Education: A Perspective on Content Knowledge

The passage of No Child Left Behind and the Individuals with Disabilities Education Act has strengthened the conditions for instructional accountability for all learners. The standards‐based approach promulgated by curriculum and state assessments is central to the accountability of conditions in today's schools; however, national and state data continue to document a performance gap between students with learning disabilities (LD) and their nondisabled peers. Building on 5 years of experience in developing and testing the Blending Assessment with Instruction Program in mathematics, we present three basic principles that are essential to ensuring that students with LD achieve curriculum standards: (1) instruction must be aligned with curriculum standards, (2) teachers' content knowledge is essential to translating curriculum standards into aligned instruction, and (3) instructional methodology should receive greater attention during teacher preparation.     

Preservice training in response to intervention: Learning by doing an interdisciplinary field experience

Marking major changes in professional role performance, response to intervention (RTI) is now in the Individuals with Disabilities Education Act (IDEA, 2004) as a possible method to improve the identification of specific learning disabilities. Moreover, RTI and related concepts and initiatives have fundamentally influenced more general methods of prevention, instruction and intervention services, and decision making in schools. Although RTI will continue to evolve, core characteristics of RTI that form the basis of state initiatives are well founded in intervention and instructional research and practice. These core characteristics (i.e., data‐based decision making, evidence‐based intervention, teaming, and problem solving) represent skills immediately needed by professionals‐in‐training. This article reports collaboration between school psychology and special education faculty in designing and implementing an integrative field experience in RTI. Training, student, and program outcomes are used summatively for accountability. © 2008 Wiley Periodicals, Inc.     

Effects of Training in the Use of Social and Cognitive Strategies: An Intervention Study in Secondary Mathematics in Co‐Operative Groups

ABSTRACT

This article reports on an intervention study into the effects of a training in the use of social and cognitive strategies on the learning outcomes of students in secondary mathematics education. Special attention is given to differential effects for high‐ and low‐achieving students. The focus on differential effects is derived from studies into learning in small co‐operative groups, and from the results of meta‐analyses into the effects of training in learning strategies. From these studies it can be concluded that in general such programs contribute to learning. However, it seems that low‐achieving students are unable to benefit from interventions of the kind investigated (i.e., co‐operative learning and training in learning strategies). The main question is whether it is possible to design an instructional program from which all students benefit, and from which the low‐achieving students profit more than their counterparts in the control‐program. In the present study three instructional programs for co‐operative learning were compared: (i) an experimental program with special instruction in the use of social strategies; (ii) an experimental program with special instruction in the use of cognitive strategies; and (iii) a control program without training in either cognitive or social strategies. The programs were identical with respect to mathematical content and general instructional settings (a combination of whole‐class instruction, working in co‐operative groups and individual work). The experiment addressed the following research question: what are the general and differential effects of a training in the use of social and cognitive strategies on the results of learning in secondary mathematics? The research was conducted in two schools for secondary education in a total of 21 classes, involving a total of 511 students. The design was a pretest‐posttest control group design, using two experimental groups and one control group. The data were analysed from a multi‐level perspective. The outcomes of the investigation clearly show the effects of the intervention. Teaching cognitive and social strategies has the expected, positive effects. In addition to this main effect, a compensatory effect for the low‐achieving students was found. The low‐achieving students in the experimental conditions outperformed their counterparts in the control group.