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Preparing Pre-Service Teachers to Use Visual Representations as Strategy to Solve Mathematics Problems: What Did They Learn?
Abstract
Recognizing the multiple interacting layers of pedagogical content knowledge, in this article, we present the findings from a study that examined the extent to which participation in case-based instruction about self-generated visual representations (VRs) supported special education pre-service teachers ( N = 25) in enhancing their (a) knowledge of VRs, specifically what and how to teach, as a strategy for solving mathematics problems, and (b) applied instructional practices, specifically analysis of student knowledge and instructional planning, about VRs for students with disabilities (SWDs) to solve mathematics problems. The data revealed that case-based instruction about VRs in a special education mathematics methods course enriched pre-service teachers personal understanding about VRs and their ability to target SWDs learning needs regarding VRs. Implications for pre-service teacher preparation and research are provided.
... Written symbols refer to mathematical symbols and equations (Mainali, 2021). Visual representation in mathematics represents essential information about mathematical problems, including diagrams, drawings, graphs, etc. (Van Garderen et al., 2021). Therefore, representation should be treated as essential in supporting students' understanding of mathematical concepts and relationships in communicating mathematical approaches and arguments and understanding oneself and others (NCTM, 2000). ...
... This is done to make it easier for both subjects to understand and see the relationship of each information used to answer the problem. In line with Van Garderen et al., (2021), visual representations in the form of tables are tools I'in Triana Agustiningtyas, Purwanto, Sudirman used to assist students in solving verbal problems. Therefore, this needs to be done because verbal representations have a relatively low density of information, so it is more prone to confusion. ...
Translation of representation is translating information presented from the source representation to the target representation based on the translation stages, including unpacking the source, preliminary coordination, constructing the target, and determining equivalence. Translation of representation is one process that plays an essential role in learning mathematics because it can be used as a strategy to solve mathematical problems. Translation of representations of verbal problems needs to be changed to other forms, such as mathematical models or tables, to facilitate understanding so that solutions to given problems can be determined. This study aims to describe the translation of high school student representation in solving HOTS-based SPtLDV problems. Data was collected in class X-1 of SMAN 1 Situbondo for the 2023/2024 school year by providing a representation translation test consisting of 1 HOTS-based SPtLDV problem item, which contained a translation of representations from verbal to graphs. Then, interviews were conducted with two selected subjects. Then, interviews were conducted with two selected subjects. The results showed that in solving the problem, both subjects had gone through four stages of translation representation. At the stage of constructing the target, a subject provides arguments related to the solution obtained by describing the results of solving the problem he encounters through representation in the form of written words. In this case, the subject can produce reasoned criticism without being instructed on the question or the teacher.
... Garderen et al. highlighted the impact of case-based learning on visual representation skills. At the same time, Giancola examined PESTS' mathematical cognitive abilities, reinforcing the importance of critical thinking and problemsolving for effective teaching [25,26]. Despite these insights, existing studies tend to treat critical thinking and mathematical representation as separate constructs and do not account for how cognitive styles, such as fielddependent (FD) and field-independent (FI), influence these competencies. ...
This study examines the critical thinking and mathematical representation abilities of pre-service elementary school teachers’ students (PESTS) through the lens of field-dependent (FD) and field-independent (FI) cognitive styles. Using a qualitative descriptive approach, 39 participants completed the Group Embedded Figures Test (GEFT), problem-solving tasks, and semi-structured interviews to assess cognitive styles and representation skills. Results indicate that FD participants struggled with problem-solving, relying on prior experiences, providing superficial solutions, and using primarily symbolic and numerical representations with limited abstraction. In contrast, FI participants demonstrated more assertive critical thinking, systematically employing diverse representations including symbols, mathematical expressions, and verbal explanations to solve problems independently. These differences underscore the need for differentiated instruction, with the FD learners requiring structured support to enhance critical thinking and broaden their representation skills. The FI learners should be challenged with abstract and complex tasks to enhance their skills further. Personalized, adaptive learning strategies are essential to preparing future educators for the challenges of 21st-century teaching.
... Other researchers have proposed that the ability to innovate in terms of teaching patterns (e.g., online collaborative learning and flipped classrooms) and instructional strategies is necessary for pre-service teachers (Brodahl et al., 2011;H. J. Tang, 2019;van Garderen et al., 2021). ...
Plain Language Summary
Many-Faceted Rasch Model (MFRM) analysis using instructional plans written by pre-service teachers and scores form nine researchers to more objectively and accurately identify the levels and differences in pre-service teachers’ instructional innovation capabilities
Purpose: Using Many-Faceted Rasch Model (MFRM) to create a profile of pre-service teachers’ with regard to their instructional innovation capability. Methods: This study selected nine researchers to evaluate the instructional innovation capabilities of 60 pre-service teachers from a Chinese normal university. The MFRM was used to measure and analyze the rater severity, difficulty of evaluation criteria, and instructional innovation capability of pre-service teachers. Conclusions: Most participants had moderate instructional innovation capability, and only certain individuals demonstrated a high level. In addition, innovation related to teaching content and learning activities was easier than innovation related to instructional and evaluation strategies. Implications: Different instructional plans, the severity of following to the plan and rater severity might have caused differences in the evaluation results.
Penelitian ini memiliki tujuan guna mengembangkan multimedia animasi Powtoon berbasis konstruktivis sebagai media belajar pada mata pelajaran matematika di kelas IV materi pecahan sederhana. Metode penelitian yang digunakan adalah jenis penelitian pengembangan atau R&D (Research and Development) dengan model yang dipilih adalah 4-D (Define, Design, Development, dan Disseminate). Pada penelitian ini, menggunakan subjek penelitian siswa kelas IV di SDN Salen dengan jumlah siswa 12 orang. Analisis data yang digunakan berupa analisis deskriptif. Teknik pengumpulan data menggunakan observasi, tes (pretest dan posttest), angket validasi (validasi ahli media, validasi ahli materi, respon guru, dan respon siswa), dan dokumentasi. Hasil penelitian menunjukkan bahwa multimedia animasi powtoon dinyatakan sangat layak atau valid, dengan skor yang didapatkan dari ahli materi sebesar 95% dan ahli media sebesar 85%. Dalam penggunaannya, multimedia animasi powtoon mendapatkan respon yang sangat baik dari guru dan siswa dengan perolehan skor pada uji kepraktisan respon guru sebesar 93,75% yang dinyatakan sangat praktis. Hasil untuk respon siswa didapati skor sebesar 91% yang dapat dikatakan multimedia animasi powtoon menarik bagi siswa. Hal ini tentu berdampak pada pemahaman siswa dalam materi pecahan sederhana dengan nilai rata-rata pretest 83,3 dan posttest 90,5. Sehingga multimedia powtoon dapat membantu siswa dalam belajar matematika di sekolah dasar.
Supporting students’ problem-solving skills, solution planning and sequencing of different stages that are involved in successfully developing a meaningful solution to a problem has been a challenge for teachers. This case study was informed by reflective investigation methodology which explored how procedural flowcharts can support student mathematics problem solving in a senior Mathematical Methods subject in Queensland. The paper used thematic analysis to analyse and report on teachers’ perceptions of the utility of procedural flowcharts during problem solving as well as content analysis on how student-developed flowcharts can support their problem-solving skills. Results show that development of procedural flowcharts can support problem solving as it helps with integration of problem-solving stages.
External visualization (i.e., physically embodied visualization) is central to the teaching and learning of mathematics. As external visualization is an important part of mathematics at all levels of education, it is diverse, and research on external visualization has become a wide and complex field. The aim of this scoping review is to characterize external visualizations in recent mathematics education research in order to develop a common ground and guide future research. A qualitative content analysis of the full texts of 130 studies published between 2018 and 2022 applied a deductive-inductive coding procedure to assess four dimensions: visualization product or process, type of visualization, media, and purpose. The analysis revealed different types of external visualizations including visualizations with physical resemblance ranging from pictorial to abstract visualizations as well as three types of visualizations with structural resemblance: length, area, and relational visualizations. Future research should include measures of visualization products or processes to help explain the demands and affordances that different types of visualizations present to learners and teachers.
This article provides a framework and description of pedagogies that may be used in teacher preparation across a range of settings from college classrooms to P-12 settings to support teacher candidates as they learn to use high-leverage practices (HLPs). These “pedagogies of enactment” must include a continuum of opportunities to use teaching practices in increasingly authentic settings, ranging from video analysis, case studies, rehearsal, and virtual simulations to use of practices with coaching support in a classroom (e.g., lesson study, structured tutoring, and aligned field experiences). In this article, we use research on the development of professional expertise and from cognitive science to identify pedagogies from the teacher education research base that have promise for promoting candidates’ learning and describe how these pedagogies might be scaffolded over the duration of a teacher education program to promote learning.
Background: A growing body of research identifies the harmful effects that adverse childhood experiences (ACEs; occurring during childhood or adolescence; eg, child maltreatment or exposure to domestic violence) have on health throughout life. Studies have quantified such effects for individual ACEs. However, ACEs frequently co-occur and no synthesis of findings from studies measuring the effect of multiple ACE types has been done.
Methods: In this systematic review and meta-analysis, we searched five electronic databases for cross-sectional, case-control, or cohort studies published up to May 6, 2016, reporting risks of health outcomes, consisting of substance use, sexual health, mental health, weight and physical exercise, violence, and physical health status and conditions, associated with multiple ACEs. We selected articles that presented risk estimates for individuals with at least four ACEs compared with those with none for outcomes with sufficient data for meta-analysis (at least four populations). Included studies also focused on adults aged at least 18 years with a sample size of at least 100. We excluded studies based on high-risk or clinical populations. We extracted data from published reports. We calculated pooled odds ratios (ORs) using a random-effects model.
Findings: Of 11 621 references identified by the search, 37 included studies provided risk estimates for 23 outcomes, with a total of 253 719 participants. Individuals with at least four ACEs were at increased risk of all health outcomes compared with individuals with no ACEs. Associations were weak or modest for physical inactivity, overweight or obesity, and diabetes (ORs of less than two); moderate for smoking, heavy alcohol use, poor self-rated health, cancer, heart disease, and respiratory disease (ORs of two to three), strong for sexual risk taking, mental ill health, and problematic alcohol use (ORs of more than three to six), and strongest for problematic drug use and interpersonal and self-directed violence (ORs of more than seven). We identified considerable heterogeneity (I2 of >75%) between estimates for almost half of the outcomes.
Interpretation: To have multiple ACEs is a major risk factor for many health conditions. The outcomes most strongly associated with multiple ACEs represent ACE risks for the next generation (eg, violence, mental illness, and substance use). To sustain improvements in public health requires a shift in focus to include prevention of ACEs, resilience building, and ACE-informed service provision. The Sustainable Development Goals provide a global platform to reduce ACEs and their life-course effect on health.
Funding: Public Health Wales.
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.
An overview of the many types of studies that fall into the qualitative design genre is provided. Strategies that qualitative researchers use to establish the authors' studies as credible and trustworthy are listed and defined. So that readers will recognize the important contribution qualitative studies have made in the field of special education, a range of well-known and lesser known examples of qualitative research are reviewed. The quality indicators that are important in conducting and evaluating qualitative research are identified. Finally, as an example of the evidence that can be produced using qualitative methods, the authors provide a summary of how 3 studies have provided important information that can be used to inform policy and practice.
Students with mathematics difficulties and learning disabilities (LD) typically struggle with solving word problems. These students often lack knowledge about efficient, cognitive strategies to utilize when solving word problems. Cognitive strategy instruction has been shown to be effective in teaching struggling students how to solve word problems that employ specific word problem types. The cognitive strategy, Math Scene Investigator (MSI), is an example of a cognitive strategy for word problem solving. The MSI strategy described in this paper is suitable for primary-level students with mathematics difficulties and LD. Instructional steps are provided along with an example of an interactive lesson.
The purpose of this meta-analysis was to synthesize findings from 42 interventions (randomized control trials and quasi-experimental studies) on instructional approaches that enhance the mathematics proficiency of students with learning disabilities. We examined the impact of four categories of instructional components: (a) approaches to instruction and/or curriculum design, (b) formative assessment data and feedback to teachers on students' mathematics performance, (c) formative data and feedback to students with LD on their performance, and (d) peer-assisted mathematics instruction. All instructional components except for student feedback with goal-setting and peer-assisted learning within a class resulted in significant mean effects ranging from 0.21 to 1.56. We also examined the effectiveness of these components conditionally, using hierarchical multiple regressions. Two instructional components provided practically and statistically important increases in effect size–teaching students to use heuristics and explicit instruction. Limitations of the study, suggestions for future research, and applications for improvement of current practice are discussed.
Teachers notice through the lens of their professional knowledge and views. This study hence focuses not solely on teachers’ noticing, but also on their knowledge and views, which allows insight into how noticing is informed and shaped by professional knowledge. As a discipline-specific perspective for noticing we chose dealing with multiple representations, since they play a double role for learning mathematics: On the one hand they are essential for mathematical understanding, but on the other hand they can also be an obstruction for learning. This comparative study takes into account pre-service as well as in-service teachers in order to explore the role of teaching experience for such professional knowledge, views and noticing. The participants answered a questionnaire addressing different components of specific knowledge and views. For eliciting the teachers’ theme-specific noticing, vignette-based questions were used. The data analysis was done mainly by quantitative methods, but was complemented by a qualitative in-depth analysis focusing on how the teachers’ theme-specific noticing was informed by different components of their professional knowledge. The results suggest that pre-service as well as in-service teachers do not fully understand the key role of multiple representations for learning mathematics in the sense of their discipline-specific significance. The participating in-service teachers distinguished themselves however from the pre-service teachers especially regarding their theme-specific noticing. Moreover, the evidence indicates that teachers’ noticing of critical instances of dealing with multiple representations draws on situated as well as on global knowledge and views.
This study examined various influences on special education preservice teachers' appropriation of pedagogical tools for teaching reading to students with high-incidence disabilities using an activity theory framework. Interview, observation, and artifact data were collected on 6 preservice teachers, their reading methods course instructors, field supervisors, and practicum cooperating teachers. Using grounded theory methods, 4 concepts emerged as chief influences on participants' appropriation of conceptual and practical reading tools: (a) opportunities to appropriate knowledge in practice, (b) personal qualities, (c) motivation for knowledge assimilation, and (d) access to knowledge. Specific information related to these 4 concepts and their relationships are reported with implications for future research and practice in special education teacher education.
Solving word problems is a common area of struggle for students with learning disabilities (LD). In order for instruction to be effective, we first need to have a clear understanding of the specific errors exhibited by students with LD during problem solving. Error analysis has proven to be an effective tool in other areas of math but has had little application to errors in word problems. Using an error analysis approach, this study aimed to investigate in depth the various types and frequency of errors made by students with LD and their AA peers during math problem solving. The resulting similarities and differences between the two groups of students are discussed with insight into underlying cognitive processes, and implications for future research.
S pecial education teacher prepa-ration has evolved over the past 150 years, since special educa-tion teachers were first prepared in residential settings. Shifting perspectives on disabilities, effective practice, and providing services to students with disabilities has led to changes in how special education is con-ceptualized and organized, and, consequently, how special education preparation programs are structured. Today, special education teacher preparation has lost focus, and there is enormous heterogeneity among programs (Goe, 2006). Re-defining special education teacher preparation is difficult, especially when the need to do so occurs as serious questions are being raised about the ef-fectiveness of teacher education generally, and when, for students with disabilities, successful ABSTRACT: The authors trace changes in conceptions of special education teacher quality and preparation in response to developments in special education research, policy, and practice. This de-velopmental arc is a backdrop for understanding contemporary special education practice and charting future directions for preparing special education teachers. Federal policy, and recent re-search on teaching and learning, and the response-to-intervention (RTI) movement require a shift in thinking about how to prepare quality special education teachers and the expertise they need to be effective. To function effectively in RTI and fulfill federal highly qualified teacher requirements, special education teachers must master an increasingly complex knowledge base and sophisticated repertoire of instructional practices. The authors contend that preservice preparation is inadequate for this purpose and that preparation for special education teaching should build upon an existing knowledge base and demonstrated competence in classroom practice.
The purpose of this meta-analysis was to synthesize findings from 42 interventions (randomized control trials and quasi-experimental studies) on instructional approaches that enhance the mathematics proficiency of students with learning disabilities. We examined the impact of four categories of instructional components: (a) approaches to instruction and/or curriculum design, (b) formative assessment data and feedback to teachers on students' mathematics performance, (c) formative data and feedback to students with LD on their performance, and (d) peer-assisted mathematics instruction. All instructional components except for student feedback with goal-setting and peer-assisted learning within a class resulted in significant mean effects ranging from 0.21 to 1.56. We also examined the effectiveness of these components conditionally, using hierarchical multiple regressions. Two instructional components provided practically and statistically important increases in effect size–teaching students to use heuristics and explicit instruction. Limitations of the study, suggestions for future research, and applications for improvement of current practice are discussed.
This article describes the essential instructional elements necessary for delivering explicit mathematics instruction to students with mathematics difficulties. Mathematics intervention research indicates that explicit instruction is one of the most effective instructional approaches for teaching students with or at risk for math difficulties. Explicit instruction is a systematic approach that facilitates important instructional interactions between teachers and students around critical math content. This article describes a framework for delivering explicit math instruction in the early grades. Although the explicit framework is relevant across the range of early math content (i.e., measurement, geometry), the focus is on explicit instruction in the context of teaching place value concepts in kindergarten and first grade classrooms. Place value is a critical component of whole-number understanding and a necessary concept for students to develop mathematical proficiency. Key questions associated with explicit math instruction are also addressed.
This article addresses a much understudied topic and concern regarding how students of varying ability levels employ visualization as a strategy in mathematics learning. The importance of this topic can be found in its connection to students’ ability to solve mathematical word problems. Many students, particularly students with learning disabilities, often struggle to use visualization as a strategy and this impacts their mathematics performance. The purpose of this article is to present findings from a study that examined the challenges that students—those identified as learning disabled and high-achieving—displayed when using one visualization form, a diagram, to solve mathematics problems. Overall, nine challenges related to the use of diagram proficiency to solve problems were identified. Further, students with learning disabilities were found to be more likely than their high achieving peers to experience these challenges. Implications for practice are provided.
This study examined students’ understanding of diagrams and their use of diagrams as tools to solve mathematical word problems. Students with learning disabilities (LD), typically achieving students, and gifted students in Grades 4 through 7 (N = 95) participated. Students were presented with novel mathematical word problem-solving tasks and interviewed for their understanding of diagrams for solving problems. Scoring rubrics were designed to assess for levels of problem solving performance, evidence of diagram use, type of diagram generated, how the diagram was used to solve problems, their definition of a diagram, and their reason for why a diagram can be used to solve problems. Results indicated that students with LD typically generated diagrams of a poorer nature and used them in a less strategic manner than their peers. Concerns regarding the impact of content knowledge on the quality of diagram generation as well as implications regarding instruction to use diagrams to understand and solve problems are discussed.
While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
This article explores epistemological awareness and instantiation of methods, as well as uninformed ambiguity, in qualitative methodological decision making and research reporting. The authors argue that efforts should be made to make the research process, epistemologies, values, methodological decision points, and argumentative logic open, accessible, and visible for audiences. To these ends, they discuss two ways of conceptualizing the role of epistemological awareness and instantiation of methods, including (a) a series of decision junctures and (b) a spatial conceptualization of epistemological decision making. Through an analysis of researchers' decision junctures drawn from studies published in high-impact education journals in 2006, the authors illustrate current methodological awareness and instantiation of methods in the field of education research.
This study investigated the differential effects of two problem-solving instructional approaches-schema-based instruction (SBI) and general strategy instruction (GSI)-on the mathematical word problem-solving performance of 22 middle school students who had learning disabilities or were at risk for mathematics failure. Results indicated that the SBI group significantly outperformed the GSI group on immediate and delayed posttests as well as the transfer test. Implications of the study are discussed within the context of the new IDEA amendment and access to the general education curriculum.
The authors propose a framework for teacher preparation programs that aims to help prospective teachers learn how to teach from studying teaching. The framework is motivated by their interest in defining a set of competencies that provide a deliberate, systematic path to becoming an effective teacher over time. The framework is composed of four skills, rooted in the daily activity of teach- ing, that when deployed deliberately and systematically, constitute a process of creating and test- ing hypotheses about cause-effect relationships between teaching and learning during classroom lessons. In spite of the challenges of acquiring these skills, the authors argue that the framework outlines a more realistic and more promising set of beginning teacher competencies than those of traditional programs designed to produce graduates with expert teaching strategies.
The effectiveness of representations used by elementary school teachers, in the first three years of their teaching, is the focus of this paper. I report on findings, from a study which makes use of the Knowledge Quartet framework. I draw on data from the first two years of the study which indicates that the use of representations is a key feature of lessons. Beginning teachers recognised that their use of representations were not always appropriate. There is some evidence that focused reflections have facilitated the development of the teachers' pedagogical knowledge in relation to the use of representations.
In the 2010 issue of Teacher Education and Special Education ( TESE), Sindelar and colleagues examined the current status of research on special education teachers and outlined future work necessary to improve the special education teacher workforce. In this article, the authors focus explicitly on Sindelar and colleagues’ charge to increase the quality of research on teacher education. They begin the article by conducting a literature review of all articles published in TESE from 2010 to 2019, examining papers focused on advancing theory, measurement, and practice in teacher education. They compare patterns in the research to the broader teacher education literature captured in seven special and teacher education journals. They conclude by discussing needs of the field going forward and provide some thoughts about how we might address those needs through a comprehensive research agenda—one that articulates a vision for how we might develop teaching quality at the intersection of general and special education.
This chapter chronicles the developmental journey of a model for teacher pedagogical content knowledge (PCK) in science education, now identified as the Refined Consensus Model (RCM) of PCK, that represents the contributions and collective thinking of two dozen international researchers in science teacher education. This journey starts by recounting the process that led to an update and significant revisions to the model of teacher professional knowledge and skills including PCK (informally known as the 2012 Consensus Model (CM)). Then, we unpack and describe the different components of the model in both diagrammatic form and in explanatory text. The RCM describes the complex layers of knowledge and experiences that shape and inform teachers’ practice and mediate student outcomes. A key feature of this model is the identification of three distinct realms of PCK—collective PCK, personal PCK, and enacted PCK. These realms are used to situate the specialised professional knowledge held by different science educators in different settings ranging from the collected knowledge understood by many to the unique subset of knowledge an individual teacher draws upon. The model also recognises that the broader professional knowledge bases are foundational to teacher PCK while the learning context a teacher is working in can greatly influence the teaching and learning that takes place.
In this article, we present findings that examined special education teachers’ perception of students’ with disabilities ability, instructional needs, and difficulties for using visual representations (VRs) as a strategy to solve mathematics problems. In addition, whether these perceptions differed by instructional grade or setting currently teaching was examined. Survey data from 97 in-service teachers revealed, regardless of instructional setting or grade level taught, that they believe students with disabilities have the ability to learn about and use VRs and need to be taught to use VRs. Furthermore, the special education teachers perceived students with disabilities to have difficulty with all aspects related to using VRs in mathematical problem-solving. Implications for teacher training and development are provided.
In this study, we challenge the deficit perspective on mathematical knowing and learning for children labeled as LD, focusing on their struggles not as a within student attribute, but rather as within teacher-learner interactions. We present two cases of fifth-grade students labeled LD as they interacted with a researcher-teacher during two constructivist-oriented teaching experiments designed to foster a concept of unit fraction. Data analysis revealed three main types of interactions, and how they changed over time, which seemed to support the students’ learning: Assess, Cause and Effect Reflection, and Comparison/Prediction Reflection. We thus argue for an intervention in interaction that occurs in the instructional process for students with LD, which should replace attempts to “fix” ‘deficiencies’ that we claim to contribute to disabling such students.
In the past decade and a half, measurable outcomes of teacher performance and student achievement have moved to the forefront of education policy. Researchers have identified a gap in special education teacher preparation research and have called for the re-envisioning of current educational models, emphasizing a focus on effective methods in special education teacher preparation. Prior studies and literature reviews have demonstrated a need for experimental research on effective methods that move preservice teachers (PSTs) from knowledge to application. In this article, the authors present a model of learning and transfer based on the How People Learn theoretical framework. Guided by this framework, a review of literature resulted in 12 experimental, quantitative studies of instructional methods delivered primarily within university classroom-based settings, measuring PSTs’ outcomes at increasingly deeper levels of learning and transfer. Findings indicate various instructional methods within university coursework lead to strong, positive learning outcomes for PSTs, with most studies measuring knowledge acquisition and conceptual application of knowledge. Yet, more studies are needed to evaluate the effectiveness of coursework on teacher candidates’ application for and within classroom settings, as well as students’ outcomes. Implications and future research are discussed.
The use of visual representations (VRs) in mathematics is a strongly recommended practice in special education. Although recommended, little is known about special educators’ knowledge of and instructional emphasis about VRs. Therefore, in this study, the authors examined special educators’ own knowledge of and their instructional emphasis with VRs in mathematics for students with disabilities (SWDs) in Grades K-12. A total of 146 teachers (pre- and in-service) responded to an online survey. A mixed methods triangulation research design was utilized. Findings include the following: (a) teachers hold conceptions about VRs and the roles they serve in problem solving; their ideas, however, lack depth and are narrow in perspective; (b) what teachers emphasize about VRs to SWDs tends to match their own conceptions, but the explanations they provide for SWDs need expanding and refinement; and (c) given their narrow conceptions, VRs may be limited to a peripheral role in special educators’ instruction.
The authors consider the future of special education personnel preparation by responding to an overarching question: What frameworks might teacher educators use as a basis to promote special education teacher effective performance now and in the future? In answering this question, they summarize current trends in the context of schooling and special education (i.e., Common Core State Standards [CCSS], multi-tiered systems of support [MTSS]) and what these contexts demand of special education teachers. The authors propose a practice-based model for fostering effective special education teacher performance. Grounded in the science of learning, the model includes approaches in teacher education that align with this literature. Implications for implementing the model are provided, which recognize current constraints on schools and colleges of education, to better promote this model for fostering effective performance.
There is widespread agreement that effective teachers have unique knowledge of students' mathematical ideas and thinking. However, few scholars have focused on conceptualizing this domain, and even fewer have focused on measuring this knowledge. In this article, we describe an effort to conceptualize and develop measures of teachers' combined knowledge of content and students by writing, piloting, and analyzing results from multiple-choice items. Our results suggest partial success in measuring this domain among practicing teachers but also identify key areas around which the field must achieve conceptual and empirical clarity. Although this is ongoing work, we believe that the lessons learned from our efforts shed light on teachers' knowledge in this domain and can inform future attempts to develop measures.
In this study, researchers operated from cognitive and situated perspectives to understand how individual qualities and contextual factors influenced elementary special education teachers' learning in a multifaceted professional development (PD) project, Literacy Learning Cohort, focused on word study and fluency instruction. Grounded theory methodology was used to analyze qualitative interviews, cohort meetings, and classroom observations. Participants included five special educators who taught reading to students with disabilities in Grades 3 to 5. Results highlighted the central role of teachers' ability to analyze their current instructional practice in developing integrated knowledge of word study and fluency instruction and crafting more integrated instruction. Teachers' individual qualities, contextual factors, and PD components also worked in concert with teachers' propensity to analyze instruction and ultimately influenced teacher learning (i.e., degree of integrated knowledge and practice demonstrated). Implications of these findings for designing effective PD efforts are discussed.
This study examined the role of visual representation type, spatial ability, and reading comprehension in word problem solving in 128 sixth-grade students by using primarily an item-level approach rather than a test-level approach. We revealed that compared to students who did not make a visual representation, those who produced an accurate visual-schematic representation increased the chance of solving a word problem correctly almost six times. Inaccurate visual-schematic and pictorial representations, on the other hand, decreased students’ chance of problem solving success. Noteworthy, reading comprehension was related to word problem solving at the test-level but not at the item-level. In interpreting the results, we advocate the use of item-level analyses since they are able to disclose such level-of-analysis discrepancies.
In this article, the authors propose an agenda for special education teacher education researchers, with particular attention to policy work and studies of innovations in pre-service preparation, induction and mentoring, and professional development. Because previous research is limited and unfocused, the foundation for future research is weak, but opportunities to study questions of importance and interest are seemingly limitless. The authors discuss strategies to bolster the research foundation, namely, by oversampling special education teachers in the Schools and Staffing Survey and the Teacher Follow-Up Survey and by fostering the development of models of teacher development and related measures of teacher quality.
This study determined the effects of the Explicit Inquiry Routine (EIR), a teaching routine, on the math performance of 14 middle-school students with math learning disabilities. The routine integrates validated mathematical teaching practices from general education (inquiry, dialogue) and special education (intensive, explicit instruction) to engage students in an interactive inquiry process across multiple modes (concrete, representation, and abstract) of illustration and manipulation to develop an understanding of the one-variable equation. A multiple-baseline-across-students design was employed. Both statistical analysis and visual inspection indicated that students' scores increased and were maintained for up to 11 weeks after instruction was terminated. In addition, students transferred their skills to textbook word problems and standardized math achievement measures.
The purpose of this study was to investigate students' with learning disabilities (LD), average-achieving students' and gifted students' use of visual imagery while solving mathematical word problems. Additionally, the relationships between visual imagery types, spatial visualization ability and mathematical problem solving were examined.The sample included sixth-grade students (N = 66) from four urban South Florida elementary and middle schools. Students were assessed on measures of mathematics achievement, mathematical word problem-solving, visual-spatial representation, and spatial visualization. Visualization, or visual imaging, was defined as the construction and formation of images either internally (e.g., mental imagery) and/or externally (e.g., with the aid of pencil and paper).The results indicated that gifted students use significantly more visual images than average-achieving students and students with LD. Five types of visual-spatial representations were used by students while solving the mathematical word problems. They included: (a) concrete imagery, (b) action imagery, (c) kinesthetic imagery, (d) number forms, and (e) pattern imagery. Visual-spatial representations were coded as either schematic, representations that encode the spatial relations described within the problem, or pictorial, representations that encode objects or persons described in the problem. Success in mathematical word problem solving was correlated with the use of schematic representations, whereas use of pictorial representations was negatively correlated with success. Schematic representations were positively correlated with spatial visualization ability, whereas pictorial representations were negatively correlated with spatial visualization ability. Students with LD used significantly more pictorial representations than the other students. Students with LD and average-achieving students performed significantly more poorly on measures of spatial visualization ability than gifted students.This research further clarified the relationships among visual imagery, spatial ability, and mathematical problem solving. Additionally, the findings contribute to the literature related to visual imagery use and mathematical word problem solving by students with LD. The results of this study have important instructional implications for all students, but especially for students who have difficulty solving mathematical word problems.
One can use classroom cases in at least three different ways: as instructional materials; as raw data in research on teacher cognition; and as catalysts that can promote change, particularly among experienced teachers. Each of these uses is based on a different epistemological tradition. Case-based instruction is an old concept in teacher education, the first casebook having been published in 1927. Classroom cases continued to be used to educate teachers over the next 50 years: as critical incidents, vignettes, protocols, and simulations. In contrast, the other two uses of classroom cases appear to be novel concepts in the field of teacher education, suggesting significant changes in the way we perceive classroom teaching and educational research. They appear to mark a growing acceptance of the ''qualitative" in teaching and in research. I use this term as Eisner (1991) does, to include that which is naturalistic, interpretive, expressive, and attentive to the particular.
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.
Lee S. Shulman builds his foundation for teaching reform on an idea of teaching that emphasizes comprehension and reasoning, transformation and reflection. "This emphasis is justified," he writes, "by the resoluteness with which research and policy have so blatantly ignored those aspects of teaching in the past." To articulate and justify this conception, Shulman responds to four questions: What are the sources of the knowledge base for teaching? In what terms can these sources be conceptualized? What are the processes of pedagogical reasoning and action? and What are the implications for teaching policy and educational reform? The answers — informed by philosophy, psychology, and a growing body of casework based on young and experienced practitioners — go far beyond current reform assumptions and initiatives. The outcome for educational practitioners, scholars, and policymakers is a major redirection in how teaching is to be understood and teachers are to be trained and evaluated.
This article was selected for the November 1986 special issue on "Teachers, Teaching, and Teacher Education," but appears here because of the exigencies of publishing.
This article provides a review of research in cognitive strategy instruction for improving mathematical problem solving for students with learning disabilities (LD). The particular focus is on one of the salient components of this instructional approach - self-regulation. Seven studies utilizing this approach for teaching problem solving to students with LD were previously evaluated to determine its status as evidence-based practice. The results of this evaluation are described, and the self-regulation component embedded in the cognitive routine for each of the studies is pre- sented. The article concludes with a discussion of several princi- ples associated with research and practice in strategy instruction and some practical considerations for implementation in schools.
The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an essential starting point for studying and improving one's teaching. Thirty K-8 preservice teachers completed 4 written tasks. Each task specified a learning goal and then asked the preservice teachers to complete a teaching activity with this goal in mind. For example, preservice teachers were asked to evaluate whether a student's responses to a series of mathematics problems showed understanding of decimal number addition. The results indicate that preservice teachers can identify mathematical subconcepts of learning goals in supportive contexts but do not spon- taneously apply a strategy of unpacking learning goals to plan for, or evaluate, teaching and learning. Implications for preservice education are discussed.
This paper examines what it would take to expand the number of teacher leaders in mathematics, noting that one of the major obstacles facing reform in mathematics education is the lack of adequate preparation among teachers currently teaching mathematics. The paper highlights experiences that have promoted the development of three middle school mathematics teacher leaders. First, it describes the professional journey of an exceptional teacher leader with 40 years of experience in diverse school environments. Next, it examines the training of two others who are still emerging as teacher leaders. The stories underscore the reality that efforts to promote improvements in mathematics education programs will not succeed unless they are coupled with opportunities that target teachers' need to grow as professionals. All three teachers share a strong commitment to raising the level of student mathematics achievement, making personal changes and improvements to positively affect their teaching quality. All three work to evaluate their teaching and strengthen their weaknesses. Two of the teacher leaders realized that they did not know enough math to truly engage their students in meaningful mathematics discourse. The results emphasize the need for support from district leadership and continuous professional development. (Contains 11 references.) (SM)
In this study, the authors examine the influence of teacher and student communication patterns, instructional practices, and teacher pedagogical content knowledge on students' mathematics learning in both general and special education mathematics classrooms. Five pre-service special education teachers in field placements and 43 students with varying disabilities participated in this study. Observations of teachers during mathematics lessons and follow-up interviews were conducted. Students in these settings completed a pretest and posttest of mathematics content taught over 6 weeks of instruction. Results reveal two sets of instructional practices, communication patterns, and teacher understandings of mathematics for teaching that differentially affected student performance. Implications are discussed for teacher education and further research. (Contains 5 tables and 1 figure.)
This study involved an investigation of the relationship between the kinds of solution representations Chinese and U.S. students use and the sorts of pedagogical representations Chinese and U.S. teachers use during instruction. The findings suggest that the representations teachers use influence the representations their students use and, hence, have an impact upon the students’ problem solving. One of the practical implications of the findings is that if students are given the opportunity to construct their own representations of mathematical concepts, rules, and relationships, they also should be encouraged to develop the ability to use symbolic representations, rather than to rely on concrete ones. In addition, the finding that the Chinese teachers in this study overwhelmingly used symbolic representations for the solutions of instructional tasks, whereas the U.S. teachers relied almost exclusively on verbal explanations and pictorial representations, indicates that pedagogical practice is constrained by social and cultural factors.
The purpose of this study was to investigate students' use of visual imagery while solving mathematical problems. Students with learning disabilities (LD), average achievers, and gifted students in sixth grade (N= 66) participated in this study. Students were assessed on measures of mathematical problem solving and visual-spatial representation. Visual-spatial representations were coded as either primarily schematic representations that encode the spatial relations described in the problem or primarily pictorial representations that encode persons, places, or things described in the problem. Results indicated that gifted students used significantly more visual-spatial representations than the other two groups. Students with LD used significantly more pictorial representations than their peers. Successful mathematical problem solving was positively correlated with use of schematic representations; conversely, it was negatively correlated with use of pictorial representations.
Current reform efforts call for an emphasis on the use of representation in the mathematics classroom across levels and topics.
The aim of the study was to examine teachers’ conceptions of representation as a process in doing mathematics, and their perspectives
on the role of representations in the teaching and learning of mathematics at the middle-school level. Interviews with middle
school mathematics teachers suggest that teachers use representations in varied ways in their own mathematical work and have
developed working definitions of the term primarily as a product in problem solving. However, teachers’ conception of representation
as a process and a mathematical practice appears to be less developed, and, as a result, representations may have a peripheral
role in their instruction as well. Further, the data suggested that representation is viewed as a topic of study rather than
as a general process, and as a goal for the learning of only a minority of the students—the high-performing ones. Implications
for mathematics teacher education, prospective and practicing, are discussed.
KeywordsRepresentation-Middle school mathematics teachers
This paper describes a model for generating andaccumulating knowledge for both teaching andteacher education. The model is applied firstto prepare prospective teachers to learn toteach mathematics when they enter theclassroom. The concept of treating lessons asexperiments is used to explicate theintentional, rigorous, and systematic processof learning to teach through studying one's ownpractice. The concept of planning teachingexperiences so that others can learn fromone's experience is used to put into practicethe notion of contributing to a sharedprofessional knowledge base for teachingmathematics. The same model is then applied tothe work of improving teacher preparationprograms in mathematics. Parallels are drawnbetween the concepts emphasized for prospectiveteachers and those that are employed byinstructors who study and improve teacherpreparation experiences. In this way, parallelsalso are seen in the processes used to generatean accumulating knowledge base for teaching andfor teacher education.
The purpose of this study was to examine differences in math problem solving among students with learning disabilities (LD, n = 25), low-achieving students (LA, n = 30), and average-achieving students (AA, n = 29). The primary interest was to analyze the processes students use to translate and integrate problem information while solving problems. Paraphrasing, visual representation, and problem-solving accuracy were measured in eighth grade students using a researcher-modified version of the Mathematical Processing Instrument. Results indicated that both students with LD and LA students struggled with processing but that students with LD were significantly weaker than their LA peers in paraphrasing relevant information. Paraphrasing and visual representation accuracy each accounted for a statistically significant amount of variance in problem-solving accuracy. Finally, the effect of visual representation of relevant information on problem-solving accuracy was dependent on ability; specifically, for students with LD, generating accurate visual representations was more strongly related to problem-solving accuracy than for AA students. Implications for instruction for students with and without LD are discussed.