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Special Education Teachers’ Perceptions of Students’ With Disabilities Ability, Instructional Needs, and Difficulties Using Visual Representations to Solve Mathematics Problems
Abstract
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.
... Like other members of society, teachers consume media information on autism and so are vulnerable to developing certain behaviors and feelings based on the media representation of autism. Teachers' perceptions of disabled students in general are informed by history of disability, cultural orientation, their training (Samsel & Perepa, 2013), schoolwide culture, which is influenced by whether the school stakeholders value or devalue the disability community, which determines many things such as funding of disability programs (Danforth, 2017;Gabel & Connor, 2016;van Garderen, 2019). Because of these factors, educators have ambivalent perceptions of disabled learners, ranging from low expectation and fear to empathy and care, majorly influenced by the catchment culture (Murray, 2008;Sinclair, 2005). ...
Contemporary US media increasingly portray autism “positively.” Based on critical realism and guided by the Disability Studies in Education ( dse ) framework, three television shows—Atypical, Touch, and The Good Doctor—with fictitious Autism Spectrum Disorder ( asd ) character(s) are qualitatively analyzed to understand the impact of the media’s portrayal of autism on the perceptions of neurotypical educators from the perspective of a disabled teacher educator. Autism in the three comedydrama series is portrayed as a savant syndrome of White heterosexual male experience affecting middle-class families. These portrayals of asd are less representative of the autism community and therefore lead to two prominent television strategies of misleading information—false balance and false identity. Since media are not neutral informers, entertainers, educators, and persuaders, it is vital for consumers especially educators to engage in dse informed critical literacy to ensure the consumption of meaningful information about autism.
... Πιο συγκεκριμένα οι εκπαιδευτικοί δήλωσαν πως ο χρόνος που διαθέτουν, το μέγεθος της τάξης και η ευκολία με την οποία προσαρμόζεται η εκάστοτε μέθοδος σε αυτά αλλά και στην διδακτέα ύλη είναι λόγοι που τους ωθούν στο να πάρουν διδακτικές αποφάσεις.Η έλλειψη χρόνου και η ανάγκη κάλυψης της ύλης προέκυψε και από την έρευνα τωνVantassel-baska et al., (2008), στην οποία οι εκπαιδευτικοί αναφέρθηκαν στις αιτίες βάση των οποίων ενεργούν και τόνισαν τον περιορισμένο χρόνο και την μεγάλη έκταση του αναλυτικού προγράμματος. ΟιHandal & Herrington, (2003) αναφέρουν πως οι εκπαιδευτικοί τείνουν να επιλέγουν την ευκολότερη μέθοδο διδασκαλίας και αντιδρούν στην εφαρμογή καινοτόμων και ως εκ τούτου απαιτητικότερων μεθόδων.Η τρίτη κατηγορία παραγόντων σχετίζεται με τις προσωπικές γνώσεις των εκπαιδευτικών, καθώς οι εκπαιδευτικοί φαίνεται να καθορίζουν τις διδακτικές επιλογές τους βάσει των όσων γνωρίζουν και των όσων έχουν διδαχθεί κατά την διάρκεια των σπουδών τους.H προσαρμογή της μεθόδου διδασκαλίας με βάση τις προσωπικές γνώσεις των εκπαιδευτικών επιβεβαιώνεται και από την έρευνα τωνvan Garderen, Scheuermann, & Poch (2018). Αξίζει να αναφερθούμε στην προσπάθεια τουRaymond (1997), που κατάρτησε ένα σχεδιάγραμμα στο οποίο αποτυπώνονται οι σχέσεις μεταξύ των απόψεων των εκπαιδευτικών για τα μαθηματικά και των διδακτικής μεθοδολογίας που ακολουθούν. ...
Η εκπαίδευση μαθητών με μαθησιακές δυσκολίες αποτελεί τις τελευταίες δεκαετίες ένα ερευνητικό πεδίο, στο οποίο καταγράφονται και δοκιμάζονται διαφορετικές θεωρίες και πρακτικές, με στόχο την αναβάθμιση της παρεχόμενης εκπαίδευσης σε αυτούς τους μαθητές. Δεδομένου πως και αυτοί φοιτούν στα γενικά σχολεία και πως η ένταξη τους στην εκπαιδευτική διαδικασία είναι πρωταρχικός στόχος, οι διδακτικές επιλογές των εκπαιδευτικών που εργάζονται στην πρωτοβάθμια εκπαίδευση αποκτούν βαρύνουσα σημασία, καθώς δεν επηρεάζουν μόνο την επίδοση τους, αλλά όλη την παρουσία τους στην σχολική ζωή. Η παρούσα έρευνα αποπειράται να αναζητήσει το κατά πόσον οι διδακτικές επιλογές των εκπαιδευτικών στην διδασκαλία των Μαθηματικών σε μαθητές με μαθησιακές δυσκολίες αντικατοπτρίζουν τεχνικές παραδοσιακής, συμβατικής διδασκαλίας ή ανήκουν στο φάσμα της διαφοροποιημένης διδασκαλίας.
Η έρευνα απευθυνόταν σε εκπαιδευτικούς Γενικής και Ειδικής Αγωγής που υπηρετούν
στην πρωτοβάθμια εκπαίδευση και διενεργήθηκε με ηλεκτρονικό ερωτηματολόγιο. Σκοπός της έρευνας, εκτός από το να καταγράψει ποιο προφίλ διδασκαλίας επιλέγουν οι εκπαιδευτικοί όταν διδάσκουν σε μαθητές με μαθησιακές δυσκολίες, ήταν να αναζητήσει τις αιτίες που ωθούν τους εκπαιδευτικούς να προβαίνουν στις εκάστοτε επιλογές. Από την επεξεργασία των δεδομένων, προέκυψε πως οι εκπαιδευτικοί προτιμούν διαφοροποιημένες τεχνικές διδασκαλίας των μαθηματικών σε μαθητές με μαθησιακές δυσκολίες και οι λόγοι έχουν να κάνουν κυρίως με την αποτελεσματικότητά τους και την καλύτερη επίδοση που επιτυγχάνουν
οι μαθητές. Δεν προέκυψαν σημαντικές στατιστικά διαφορές ανάμεσα στους δύο κλάδους των εκπαιδευτικών και τα λοιπά δημογραφικά στοιχεία, με εξαίρεση τα χρόνια υπηρεσίας στην Ειδική Αγωγή, αλλά τα αποτελέσματα της έρευνας παρουσιάζουν ιδιαίτερο ενδιαφέρον, ειδικά για την περαιτέρω διερεύνηση του θέματος.
Objective: This study aimed to investigate the lived experiences of the teachers who have been dealing with students with Special Learning Disorders (SLD) for years. The research employed a phenomenological-qualitative approach.
Methods: The data was collected using semi-structural interviews with 9 teachers who had the required criterion and worked at learning disorder centers in Bandar Abbas. The study participants were selected through purposeful sampling until the data saturation stage was reached. Then, the data were analyzed through four stages employing the method proposed by Smith.
Results: The findings implied six main categories (Moral and behavioral characteristics of SLD Children, Family Role, School Performance, Teaching Requirements, the Effect of Job on the Teacher’s Life, Remedial and Educational Facilities) and nineteen sub-categories.
Conclusion: Investigating teachers’ experiences gives families, teachers, and special teaching educators valuable information to recognize and intervene on time. It also provides multilateral support and helps improve the status and conditions of children with SLD.
Objective: This study aimed to investigate the lived experiences of the teachers who have been dealing with students with Special Learning Disorders (SLD) for years. The research employed a phenomenological-qualitative approach. Methods: The data was collected using semi-structural interviews with 9 teachers who had the required criterion and worked at learning disorder centers in Bandar Abbas. The study participants were selected through purposeful sampling until the data saturation stage was reached. Then, the data were analyzed through four stages employing the method proposed by Smith. Results: The findings implied six main categories (Moral and behavioral characteristics of SLD Children, Family Role, School Performance, Teaching Requirements, the Effect of Job on the Teacher’s Life, Remedial and Educational Facilities) and nineteen sub-categories. Conclusion: Investigating teachers’ experiences gives families, teachers, and special teaching educators valuable information to recognize and intervene on time. It also provides multilateral support and helps improve the status and conditions of children with SLD.
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.
Although much research on teachers' knowledge and attitudes towards their students with dyslexia has been accumulated, students with dyscalculia have not gained the same attention. Teachers' knowledge about the nature and characteristics of students with dyscalculia seems to be limited and this has a strong impact on their instructional decisions. In secondary education, where teachers' pre-service educational programs are more focused on the scientific subject instead of appropriate instructional methods, teachers' knowledge about dyscalculia has not yet been taken into account. The aim of this study was to examine the extent to which mathematics teachers know what dyscalculia is, and what its features are. Possible differences that may occur between teachers with and without special education training were also examined, as well as differences that may occur as a result of their working experience. One hundred and fourteen secondary mathematics teachers, with an average service time of twelve years, completed an electronic questionnaire in which they had to respond to 19 questions about the definition, the content and the major characteristics of dyscalculia. Although they seemed to understand the innate profile of dyscalculia, 31% of them attributed dyscalculia to learning gaps resulted by student absence from school while 67% of the teachers felt that mistakes of students with dyscalculia in solving algorithms may be reduced if more time is provided to the students. Furthermore, confusion prevailed regarding skills of students with dyscalculia to solve word problems. The findings in this study contribute to the ongoing discussion on the appropriate education and training of secondary mathematics teachers, which should not neglect the special characteristics and difficulties of students with dyscalculia. The teachers' knowledge about dyscalculia is suggested as the base for the design of appropriate teaching practices to address specific learning disabilities in math.
The effects of a mathematical problem-solving intervention on students’ problem-solving performance and math achievement were measured in a randomized control trial with 1,059 7th-grade students. The intervention, Solve It!, is a research-based cognitive strategy instructional intervention that was shown to improve the problem-solving performance of 8th-grade students with and without learning disabilities (LD). The purpose of the present study was to determine whether the effectiveness of the intervention could be replicated with younger students. Forty middle schools in a large urban school district were included in the study, with one 7th-grade math teacher participating at each school (after attrition, n = 34). Solve It! was implemented by the teachers in their inclusive math classrooms. Problem-solving performance was assessed using curriculum-based math problem-solving measures, which were administered as a pretest and then monthly over the course of the 8-month intervention. Students who received the intervention (n = 644) embedded in the district curriculum showed a significantly greater rate of growth on the curriculum-based measures than students in the comparison group (n = 415) who received the district curriculum only. Results of the Bayesian analyses indicated that the intervention effect was somewhat stronger for low-achieving students than for average-achieving students. Overall, findings from the present study as well as the previous study with 8th-grade students indicate that the intervention was effective across ability groups and is an appropriate program to use in inclusive classrooms with students of varying math ability. (PsycINFO Database Record (c) 2014 APA, all rights reserved)
This study of early-career teachers identified a significant relationship between upper-elementary teachers' mathematical content knowledge and their students' mathematics achievement, after controlling for student- and teacher-level characteristics. Further, the mathematical content and pedagogical knowledge of middlegrades teachers were each directly and positively related to their students' mathematics achievement, with and without teacher-level controls. Significant interactions emerged between teachers' perceptions and knowledge influencing student achievement. Teachers' claimed awareness of their students' dispositions toward mathematics interacted with upper-elementary teachers' content knowledge; middle-grades teachers' beliefs regarding modeling mathematical solutions and organizing instruction to support incremental mastery of skills interacted with both content and pedagogical knowledge. Findings provide evidence of the relevance of teacher knowledge and perceptions for teacher preparation and professional development programs.
The purpose of the study was to improve mathematical problem solving for middle school students with learning disabilities by implementing a research-based instructional program in inclusive general education math classes. A total of 40 middle schools in a large urban district were matched on state assessment performance level (low, medium, and high performing) and socioeconomic status. One school from each pair was randomly assigned to the intervention condition, and one eighth grade math teacher participated at each school (n = 40). Because of attrition at the outset, 24 schools completed the study (8 intervention, 16 comparison). The intervention, Solve It!, a research-based cognitive strategy instructional program, was implemented for 7 months, and periodic progress monitoring was conducted. A cluster-randomized design was used, and the data were consistent with a three-level model in which repeated measures were nested within students and students were nested within schools. The results indicated that students who received the intervention (n = 319) showed significantly greater growth in math problem solving over the school year than students in the comparison group (n = 460) who received typical classroom instruction. Moreover, the intervention effects did not differ for students with learning disabilities, low-achieving students, and average-achieving students. Thus, the findings were positive and support the efficacy of the intervention when implemented by general education math teachers in inclusive classrooms.
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 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.
Research has shown that high expectations of teachers about their students’ academic development have a positive influence on how these students actually develop. Therefore, when aiming to improve students’ learning results it is essential to know how teachers think about their students’ abilities. The present study was meant to investigate what perceptions primary school teachers in special education have of their students’ potential in mathematics and what possibilities they see to reveal this potential. Data were collected through an online questionnaire. Surprisingly, the responses showed that, although the teachers teach students with low achievement scores in mathematics, most of the teachers were positive about the mathematical potential of their students. The teachers often attributed unused potential to causes outside the student and they underpinned this view with observations from school practice. The article concludes with discussing the consequences of these findings for the professional development of teachers.
A random sample of 167 secondary special and general educators who taught math to students with emotional and behavioral disorders (EBD) and learning disabilities (LD) responded to a mail survey. The survey examined teacher perceptions of (a) definition of math; (b) familiarity with course topics; (c) effectiveness of methods courses; (d) preparation to use and frequency of use of effective instructional strategies; and (e) factors contributing to the use of effective instructional strategies. The number of methods courses taken by teachers uniquely and significantly contributed to the number of effective instructional approaches that both general and special educators reported using. Significantly more special educators than general educators reported the use of specific, empirically validated instructional practices. Additional results, implications, and suggestions for future research are provided.
This paper reports on the beliefs of a group of K-8 mathematics teachers about appropriate goals and methods of mathematics teaching for students with mathematics learning difficulties and for students generally. The teachers were involved in a brief professional learning program that aimed to provide them with effective strategies for mathematics teaching for numeracy, and to influence their relevant beliefs towards a more inclusive view of mathematics teaching. The questionnaire used in the study revealed differences between teachers' beliefs in relation to students generally and those with mathematics learning difficulties, and provided evidence that carefully designed professional learning may be able to reduce these differences. Numeracy has its foundations in the discipline of mathematics. Definitions of numeracy typically emphasise the use of mathematics in everyday life and highlight the importance of affect (Australian Association of Mathematics Teachers, 1997). The Department of Education Tasmania (DoET) (2002) refers explicitly to the application of mathematics to everyday life and acknowledges affect in its use of the terms disposition and confidence in its definition of numeracy, which includes the following: Being numerate involves having those concepts and skills of mathematics that are required to meet the demands of everyday life. It includes having the capacity to select and use them appropriately in real settings. Being truly numerate requires the knowledge and disposition to think and act mathematically and the confidence and intuition to apply particular mathematical principles to everyday problems … it also involves the critical and life-related aspects of being able to interpret information thoughtfully and accurately when it is presented in numerical and graphic form. (p. 21) Interpreting the Tasmanian definition necessitates some understanding of what the mathematical demands of everyday life might be and it seems self evident that such demands differ from individual to individual depending upon, among other things, occupation. Being numerate can thus be regarded as requiring different mathematical skills and concepts for different people. Nevertheless, the Australian Government remains committed to ensuring that "all students attain sound foundations in literacy and numeracy" (Department of Education, Training and Youth Affairs (DETYA), 2000, p. 6). DETYA (2000) also made clear that this goal was based upon a firm belief that all students can acquire the mathematical skills necessary for life in modern society and recognised that
This guide for teachers is a companion piece to the meta-analysis from the Center on Instruction, "Mathematics Instruction for Students with Learning Disabilities or Difficulty Learning Mathematics: A Synthesis of the Intervention Research". Based on the findings of this report, seven effective instructional practices were identified for teaching mathematics to K-12 students with learning disabilities. It describes these practices and, incorporating recommendations from "The Final Report of The National Mathematics Advisory Panel" as well, specifies research-based recommendations for students with learning disabilities and for students who are experiencing difficulties in learning mathematics but are not identified as having a math learning disability. [To access "The Final Report of The National Mathematics Advisory Panel" see (ED500486) This publication was created by Instructional Research Group.]
This meta-analysis summarizes empirical results on the correspondence between teachers' judgments of students' academic achievement and students' actual academic achievement. The article further investigates theoretically and methodologically relevant moderators of the correlation between the two measures. Overall, 75 studies reporting correlational data on the relationship between teachers' judgments of students' academic achievement and students' performance on a standardized achievement test were analyzed, including studies focusing on different school types, grade levels, and subject areas. The overall mean effect size was found to be .63. The effect sizes were moderated by use of informed versus uninformed teacher judgments, with use of informed judgments leading to a higher correspondence between teachers' judgments and students' academic achievement. A comprehensive model of teacher-based judgments of students' academic achievement is provided in the Discussion. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
This article is a theoretical and empil#cal analysis of the construct of col-lective teacher e ff}'cacjz l*Trsl, a model o/" collective e/'fi'cacy was elaborated for use in schools. 7ben, an operatio~Tal measure qf colleclive teacher e./.'/'i-cacy was developed, tested, and./bund to have strong reliability and rea-sonable validity. Finally, using the instrument to examirte urban elementaly schools in one laqge midwesten7 district, collective teacber eJJ~'cacy was posi-lively associated wilh d!ff'erences between schools in student-level achieve-ment in, both rectding and mathematics. l~,oc;l'l~ D. GODt)Ama is an Assistant Professor in the School ot: Education, Univer-sity of Michig:m, 610 E. University, 4204 SEB, Ann Arl)or, MI 48109-1259; rgoddard@umich.edu. He specializes in o,'ganizational theory and educational ad-ministration. WArm.: K. Hov is the Fawcett Chair in Educational Administ,'ation, School. Fie specializes in organiz'ltional theo,'y and educational admin-istration.
This paper reports on the impact of a brief professional learning program for K-8 teachers of mathematics, on teachers' beliefs about effective numeracy teaching strategies and appropriate goals of numeracy teaching, for students with mathematics learning difficulties and for students generally. Evaluation data indicated that the teachers finished the program less inclined to espouse differing beliefs in relation to the two types of students, and that their final beliefs were more in line with the aims of the program. Numeracy is accepted as having its foundations in mathematics (Australian Education Council, 1990) and is typically defined as also involving affective dimensions and the ability to use mathematics in everyday life (Australian Association of Mathematics Teachers, 1997). More recently, the Tasmanian Department of Education (DoET) (2002) stated that numeracy; requires the knowledge and disposition to think and act mathematically and the confidence and intuition to apply mathematical principles to everyday problems. … it also involves the critical and life-related aspects of being able to interpret information thoughtfully and accurately when it is presented in numerical and graphic form (p. 21).
The goals of this study are to understandelementary school teachers' beliefs andpractices and to unveil factors that influencethe way teachers adapt mathematics reformrhetoric when trying to adopt it. In theresearch, I searched for beliefs beyondmathematics that influence teachers' decisionsand choices for teaching mathematics. Workingwith children from different socioeconomicbackgrounds, teachers interpret reform indifferent ways. Based on their concept of students' needs, teachers select which partsof the reform documents are appropriate fortheir students. While children from uppersocioeconomic backgrounds experience problemsolving, those from lower socioeconomicbackgrounds undergo rote learning. Because notall children have the opportunity to learn thesame quality mathematics, the emerging concernof this study is the issue of equity inmathematics teaching.
The purpose of this study was to investigate students' use of visual imagery and its relationship to spatial visualization ability while solving mathematical word 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, visual imagery representation, and spatial visualization ability. The results indicated that gifted students performed better on both spatial visualization measures than students with LD and average-achieving students. Use of visual images was positively correlated with higher mathematical word-problem-solving performance. Furthermore, the use of schematic imagery was significantly and positively correlated with higher performance on each spatial visualization measure; conversely, it was negatively correlated with the use of pictorial images.
This article explores elementary school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teacher characteristics. The Learning Mathematics for Teaching project administered a multiple-choice assessment covering topics in number and operation to a nationally representative sample of teachers ( n = 625) and at the same time collected information on teacher and student characteristics. Performance did not vary according to mathematical topic (e.g., whole numbers or rational numbers), and items categorized as requiring specialized knowledge of mathematics proved more difficult for this sample of teachers. There were few substantively significant relationships between mathematical knowledge for teaching and teacher characteristics, including leadership activities and self-reported college-level mathematics preparation. Implications for current policies aimed at improving teacher quality are addressed.
This study investigates relationships between a beginning elementary school teacher's beliefs and mathematics teaching practices. A proposed model of relationships between beliefs and practice provided a conceptual framework for the examination of factors that influence beliefs, practice, and the level of inconsistency between them. Data were gathered over 10 months through audiotaped interviews, observations, document analysis, and a beliefs survey. Analyses included the categorization and comparison of beliefs and practice and the development of a revised model of relationships between beliefs and practice. Findings indicate that this teacher's beliefs and practice were not wholly consistent. Rather, her practice was more closely related to her beliefs about mathematics content than to her beliefs about mathematics pedagogy. Her beliefs about mathematics content were highly influenced by her own experiences as a student and her beliefs about mathematics pedagogy were primarily influenced by her own teaching practice. However, the extent to which her teacher preparation program influenced either her beliefs or practice was limited.
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.
This article is a theoretical and empirical analysis of the construct of collective teacher efficacy. First, a model of collective efficacy was elaborated for use in schools. Then, an operational measure of collective teacher efficacy was developed, tested, and found to have strong reliability and reasonable validity. Finally, using the instrument to examine urban elementary schools in one large midwestern district, collective teacher efficacy was positively associated with differences between schools in student-level achievement in both reading and mathematics.
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 article we articulate a methodology for studying mathematics teacher development in the context of reform. The generation of accounts of teachers' practice, an adaptation of the case study, provides an approach to understanding teachers' current practice and to viewing their current practice in the context of development toward envisioned reforms. The methodology is an alternative both to studies that focus on teachers' deficits and to teachers' own accounts of their practice. Conceptual frameworks developed within the mathematics education research community are applied to the task of investigating the nature of practice developed by teachers in transition. We characterize this methodology as explicating the teacher's perspective from the researchers' perspectives.
This study investigates relationships between a beginning elementary school teacher's beliefs and mathematics teaching practices. A proposed model of relationships between beliefs and practice provided a conceptual framework for the examination of factors that influence beliefs, practice, and the level of inconsistency between them. Data were gathered over 10 months through audio-taped interviews, observations, document analysis, and a beliefs survey. Analyses included the categorization and comparison of beliefs and practice and the development of a revised model of relationships between beliefs and practice. Findings indicate that this teacher's beliefs and practice were not wholly consistent. Rather, her practice was more closely related to her beliefs about mathematics content than to her beliefs about mathematics pedagogy. Her beliefs about mathematics content were highly influenced by her own experiences as a student and her beliefs about mathematics pedagogy were primarily influenced by her own teaching practice. However, the extent to which her teacher preparation program influenced either her beliefs or practice was limited.
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.
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.
The purpose of this article is to offer teacher training and professional development recommendations in mathematics based on the findings of a federally funded 3-year intervention study that improved the problem solving of middle school students with a focus on students with learning disabilities. Over the 3-year project, 29 seventh and eighth grade teachers implemented a problem-solving intervention based on cognitive strategy instruction. Though the intervention was successful in improving students’ problem-solving performance, several issues related to teaching effectiveness and teacher training came to light. The article will: (1) describe the intervention and its implementation, (2) present the findings of the study, and (3) discuss the issues of effectiveness and possible solutions via teacher education and professional development.
This article explores elementary school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teacher characteristics. The Learning Mathematics for Teaching project administered a multiple-choice assessment covering topics in number and operation to a nationally representative sample of teachers (n = 625) and at the same time collected information on teacher and student characteristics. Performance did not vary according to mathematical topic (e.g., whole numbers or rational numbers), and items categorized as requiring specialized knowledge of mathematics proved more difficult for this sample of teachers. There were few substantively significant relationships between mathematical knowledge for teaching and teacher characteristics, including leadership activities and self-reported college-level mathematics preparation. Implications for current policies aimed at improving teacher quality are addressed.
This study investigated the effects of a two-phase cognitive strategy on algebra problem solving of adolescents with learning disabilities. The strategy was designed to enable students to represent and solve three types of word problems. The study used a modified multiple baseline with 11 replications as well as a two-group design. Conditions of the multiple-baseline design included baseline, instruction to mastery, transfer, and maintenance. Visual analysis of the single-subject data showed the strategy to be an effective intervention for this sample of students with deficits in algebra problem solving, but with criterial knowledge of basic operations and one-step problems. Statistical analyses of the two-group data showed that the instructed students had significantly higher posttest scores than the comparison group. Overall, the instructed students demonstrated improved performance on algebra word problems. Maintenance and transfer of the strategy were evident. This study has implications for teaching complex problem solving to adolescents with learning disabilities in secondary schools.
Good discusses the types of teacher expectation effects evi denced in the classroom. Particular attention is focused on the research that addresses teachers' expectations for and interac tions with individuals believed to be of high or low potential. Good presents a model for use in understanding the dynamics of expectation communication in the classroom and highlights numerous studies relating teacher expectations with student behavior. The differential treatment of students by teachers is described by the author, with special attention given to how teachers express low expectations. The article concludes with a description of future research directions.
Attention to the beliefs of teachers and teacher candidates should be a focus of educational research and can inform educational practice in ways that prevailing research agendas have not and cannot. The difficulty in studying teachers’ beliefs has been caused by definitional problems, poor conceptualizations, and differing understandings of beliefs and belief structures. This article examines the meaning prominent researchers give to beliefs and how this meaning differs from that of knowledge, provides a definition of belief consistent with the best work in this area, explores the nature of belief structures as outlined by key researchers, and offers a synthesis of findings about the nature of beliefs. The article argues that teachers’ beliefs can and should become an important focus of educational inquiry but that this will require clear conceptualizations, careful examination of key assumptions, consistent understandings and adherence to precise meanings, and proper assessment and investigation of specific belief constructs. Implications of findings and directions for future research are offered.
During the past 15 years or so, teacher education research has made significant strides in studying the complex relationships between teacher beliefs and practices. This new line of research has generated important findings that are of practical implications for teacher education. This article reviews this small body of research and, in so doing, elucidates the two competing theses (i.e. ‘consistency’ vs ‘inconsistency') that are recurring in the literature on the relationships between teacher beliefs and practices. It begins with an overview of traditional research on teaching in general, followed by a discussion of teacher cognition under which teacher beliefs/theories are subsumed. After introducing the notion of ‘the Missing Paradigm’ in the mainstream teacher education research, the article examines the theoretical frameworks underlying teacher beliefs and practices research. Next it provides a synthesis of recent research on teacher beliefs and practices, addressing critical issues germane to the research findings. After a brief discussion of several critical methodological issues, the article outlines six possible directions for future research.
The purpose of this qualitative study was to investigate typical middle school general education mathematics teachers' beliefs and knowledge of students with learning disabilities and inclusive instruction and to gain an understanding of the process of inclusion as it is implemented in middle school classrooms. In-depth interviews, surveys, and classroom observations were conducted with seven teachers. The constant comparative method was used to analyze all interview and observation data. The findings reveal that even teachers who believe that inclusion is being successfully implemented are unclear about their responsibilities towards included students and the learning characteristics and specific mathematics teaching approaches that would be effective. The general educators feel that they were grossly under-prepared during preservice and inservice for the realities of inclusion teaching. The study provides insights that can be used to enhance preservice and inservice programs for teachers and underscores the necessity for building teamwork and collaboration among general and special education middle school teachers.
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 reports the findings of a study that sought to identify particular centrally held beliefs of secondary mathematics
teachers that underpinned the establishment of classroom environments that were consistent with the principles of constructivism.
The nine crucial beliefs identified were held by one or other of two teachers and emerged from teacher and student surveys,
interviews with the teachers and classroom observations. As is the case with all beliefs, these beliefs were contextually
bound but since the contexts in which they applied were broader than particular classrooms it is argued that they may be generalisable
to other contexts and even predictive of teachers likely to create similar classroom environments.
The goal of this study is to characterize the patterns of teachers’ beliefs regarding low-achieving students and instruction of higher order thinking. Subjects are 40 Israeli teachers. Results show that 45% of the teachers believe that higher order thinking is inappropriate for low-achieving students. Findings suggest that teachers’ beliefs in this context are related to their general theory of instruction: viewing learning as hierarchical in terms of students’ academic level was found to be related to a traditional view of learning, i.e., seeing learning as progressing from simple, lower order cognitive skills to more complex ones. Implications for teacher education are discussed.
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.
Research findings regarding general self-concept, academic self-concept, and self-awareness in students with learning disabilities have varied, and results are still inconclusive regarding the consistency between students' and teachers' judgments of academic performance. The current study focuses on students' and teachers perceptions of the students' strategy use and performance in nine different academic and organizational domains. Six hundred sixty-three students and their 57 teachers were involved in the study. Findings indicated that the students with learning disabilities considered themselves appropriately strategic and competent in the five domains of reading, writing, spelling, math, and organization. These students also rated their academic performance and organization as average to above-average in seven of nine domains, with the exception of checking and planning their work. Nevertheless, the self-ratings of the students with learning disabilities were still significantly lower than the self-ratings of average achievers in virtually all domains. The second major set of findings revealed a sharp discrepancy between the self-assessments of the students with learning disabilities and their teachers' judgments. Teachers rated the students with learning disabilities as weak in their strategy use and below-average in their performance in all nine academic and organizational domains. Finally, gender differences were not evident in eight of the nine domains. These results have added to the increasing body of literature indicating that students with learning disabilities frequently perceive themselves as capable and effective and often rate themselves as academically stronger than their teachers judge them to be.
Mathematics learning disabilities (LD) have gained increased attention over the last decade from both researchers and practitioners. A large percentage of students receiving learning disability services experience difficulties with mathematics, but little research has examined the specific mathematics behaviors of students with LD who have teacher-identified math weaknesses. This study examines the literature on mathematics LD and identifies specific behaviors from that body of research for the purpose of determining the extent to which those behaviors are observed in students with LD. Data are presented from observations of 391 special education professionals on 1724 students with LD, 870 of whom had identified math weaknesses and 854 of whom did not. Our results validate the existing literature and provide implications for teachers, researchers, and others interested in studying mathematics LD.
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.
Improving mathematical problem solving in grades 4 through 8: A practice guide (NCEE 2012-4055)
- J Woodward
- S Beckmann
- M Driscoll
- M Franke
- P Herzig
- A Jitendra
- P Ogbuehi