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Teaching Students with Learning Problems in Math to Acquire, Understand, and Apply Basic Math Facts
Abstract
Students with learning problems (i.e., students with learning disabilities or emotional disabilities and those considered at-risk for school failure) are not making acceptable math progress in the nation's schools. Fortunately, instructional practices exist that help these students achieve in math. Ten instructional components that have research support for promoting math achievement are presented. A math curriculum, the Strategic Math Series, which incorporates the research-based teaching practices, is described. Next, the results from field-testing the Strategic Math Series with 22 teachers are presented. The field-test results indicate that 109 students with learning problems were able to (a) acquire computational skills across facts, (b) solve word problems with and without extraneous information, (c) create word problems involving facts, (d) apply a mnemonic strategy to difficult problems, (e) increase their rate of computation, and (f) generalize math skills across examiners, settings, and tasks. Finally, issues in math instruction are discussed, and the need to include best practices within instructional materials is highlighted.
... Kajian lepas menunjukkan terdapat peningkatan yang ketara apabila pendekatan KGA dijalankan dalam pengajaran dan pembelajaran (Flores 2009;Flores 2010;Flores et al. 2014b;Flores et al. 2014a;Kaffer & Miller 2011;Mancl et al. 2012;Mercer & Miller 1992a;Mercer & Miller 1992b;Miller & Mercer 1993;Strozier et al. 2012;Witzel 2005;Witzel et al. 2003;Witzel et al. 2008). Kajian penggunaan pendekatan KGA digunakan dalam pelbagai topik seperti fakta asas dan nilai tempat (Flores et al. 2014a), fakta asas (Flores 2009;Flores 2010;Flores et al. 2014b;Mercer & Miller 1992a;Miller & Mercer 1993;Mancl et al. 2012;Mercer & Miller 1992b;Witzel et al. 2008) dan algebra (Stroizer et al. 2012;Witzel 2005;Witzel et al. 2003). ...
... Kajian lepas menunjukkan terdapat peningkatan yang ketara apabila pendekatan KGA dijalankan dalam pengajaran dan pembelajaran (Flores 2009;Flores 2010;Flores et al. 2014b;Flores et al. 2014a;Kaffer & Miller 2011;Mancl et al. 2012;Mercer & Miller 1992a;Mercer & Miller 1992b;Miller & Mercer 1993;Strozier et al. 2012;Witzel 2005;Witzel et al. 2003;Witzel et al. 2008). Kajian penggunaan pendekatan KGA digunakan dalam pelbagai topik seperti fakta asas dan nilai tempat (Flores et al. 2014a), fakta asas (Flores 2009;Flores 2010;Flores et al. 2014b;Mercer & Miller 1992a;Miller & Mercer 1993;Mancl et al. 2012;Mercer & Miller 1992b;Witzel et al. 2008) dan algebra (Stroizer et al. 2012;Witzel 2005;Witzel et al. 2003). Cadangan pendekatan KGA bagi menyelesaikan masalah murid memahami konsep luas yang selalu dihadapi di dalam kelas adalah kerana kaedah pembelajarannya melibatkan konsep yang mendalam, berbanding hanya menggunakan rumus luas iaitu panjang darab lebar. ...
... Bagi menyokong kajian Stroizer et al. (2012), kaedah menjawab soalan adalah melalui langkah abstrak yang hanya menggunakan simbol dan berayat bagi memastikan profisiensi dalam topik luas dapat diukur. Keputusan yang diperoleh telah menyokong kajian-kajian menggunakan pendekatan KGA terdahulu (Flores 2009;Flores 2010;Flores et al. 2014b;Flores et al. 2014a;Kaffer & Miller 2011;Mancl et al. 2012;Mercer & Miller 1992a;Mercer & Miller 1992b;Miller & Mercer 1993;Strozier et al. 2012;Witzel 2005;Witzel et al. 2003;Witzel et al. 2008 Keputusan menunjukkan bahawa kumpulan yang berada dalam kitaran ketiga CLR adalah mempunyai perbezaan min skor tertinggi berbanding kitaran pertama CLR yang mempunyai perbezaan min skor terendah. Perbandingan antara tiga kitaran CLR bagi luas segi empat sama, luas segi empat tepat dan luas segi tiga bagi ketiga-tiga kumpulan juga menunjukkan terdapat perbezaan yang signifikan antara ujian pasca ketiga-tiga kumpulan tersebut dengan kumpulan yang berada dalam kitaran ketiga CLR mempunyai min skor yang paling tinggi berbanding kumpulan yang berada dalam kitaran pertama CLR. ...
Kajian Pengajaran Kolaboratif (CLR) adalah penambahbaikan kepada Lesson Study bagi meningkatkan kecekapannya. Kajian ini bertujuan untuk menilai sama ada pendekatan konkrit-gambar-abstrak (KGA) yang dibina berdasarkan kitaran CLR dapat memberi impak positif terhadap profisiensi murid Tahun Empat dalam topik luas. Reka bentuk kajian adalah counterbalanced dengan ujian pra dan pasca diberikan pada setiap kitaran CLR. Sampel kajian adalah terdiri daripada 115 orang murid yang dipilih melalui pensampelan rawak berkelompok dan dibahagikan kepada tiga kumpulan bersama tiga orang guru. Ujian-t sampel berpasangan dijalankan bagi menguji sama ada terdapat perbezaan yang signifikan antara ujian pra dan pasca bagi setiap kitaran CLR yang terlibat, manakala ujian ANOVA satu hala dijalankan untuk menganalisis sekiranya terdapat peningkatan min skor profisiensi daripada kitaran CLR pertama, kedua, dan ketiga. Dapatan menunjukkan terdapat peningkatan min skor ujian pasca berbanding min skor ujian pra, serta perbezaan yang siginifikan antara min skor antara kitaran CLR pertama, kedua, dan ketiga. Kesimpulannya, kitaran CLR yang dijalankan membantu guru dalam membentuk rancangan pengajaran yang lebih baik berdasarkan pendekatan KGA, di samping meningkatkan profisiensi murid dan kemahiran guru dalam topik luas. Implikasinya, kajian ini memberikan pendedahan berkaitan amalan CLR dalam kajian dalam meningkatkan profesionalisme guru.
... Una secuencia de enseñanza-aprendizaje progresiva que ha resultado ser efectiva para que estudiantes con dificultades adquieran conocimientos matemáticos elementales es la denominada secuencia Concreta-Representacional-Abstracta (en adelante, CRA). CRA consiste en generar conocimiento avanzando progresivamente en tres fases: (1) el empleo de materiales manipulativos (fase concreta); (2) el uso de representaciones pictóricas (fase representacional); y, (3) la ejecución de operaciones aritméticas (fase abstracta) Mercer y Miller, 1992). ...
... : Jones y Tiller, 2017Mindish, 2021;Peltier et al., 2020). Existen evidencias que apoyan que esta secuencia de instrucción facilita que estudiantes con dificultades aprendan a ejecutar con éxito operaciones aritméticas básicas (Flores y Franklin, 2014;Mercer y Miller, 1992) y los problemas que las involucran (Flores et al., 2016;Miller y Mercer, 1993). Es el caso, por ejemplo, de las investigaciones de Flores y colaboradores en las que, empleando la secuencia CRA, muestran que estudiantes de infantil y primaria que presentan alteraciones emocionales y comportamentales y/o que reciben apoyo educativo en matemáticas aprendieron aspectos relacionados con la aritmética; por ejemplo: el número natural, las fracciones o el redondeo (cf. ...
... Lo anterior muestra que para este estudiante ejecutar restas fue más difícil que ejecutar sumas y permitió corroborar que el estudiante tuvo que realizar más de tres sesiones en la fase de representacional-abstracta y abstracta para adquirir con firmeza la ejecución de sumas y retas (cf. Mercer y Miller, 1992). ...
En este artículo indagamos en las posibilidades de la secuencia Concreta-Representacional-Abstracta-Integrada como metodología de enseñanza-aprendizaje de la suma y la resta en el segundo ciclo de educación infantil. Concretamente, llevamos a cabo una experiencia de naturaleza exploratoria centrada en un estudiante de cuatro años con retraso madurativo del lenguaje y conducta disruptiva. Se organizaron sesiones individuales, de acuerdo con las fases de la secuencia, distribuidas de esta forma: evaluación inicial, intervención, evaluación final y mantenimiento. El estudiante participó en todas ellas, logrando ejecutar progresivamente sumas y restas, sin disponer de conocimientos previos. Además, esta metodología resultó de utilidad para que el estudiante lograse mantener su aprendizaje pasadas cuatro semanas desde la intervención. Por último, consideramos de interés las implicaciones que el presente artículo pueda tener para el profesorado de educación infantil.
... Skills in mathematics are built on top of each other to create new skills. Therefore, it is not possible to switch to a higher skill without learning a basic skill in mathematics (Gurganus, 2017;Hasselbring, Goin and Bransford, 1987;Hinton, Strozier and Flores, 2014;Mercer and Miller, 1992;Woodward, 2006). Basic addition, multiplication and division facts are fundamental skills of mathematics (Baykul, 2006;McCallum and Schmitt, 2011;Stein, Kinder, Silbert and Carnine, 2006). ...
... As a prerequisite for learning high level math skills, the presentation of basic facts skills fluently is just as important as the acquisition (Burns, Codding, Boice and Lukito;2010;Cates and Ryhmer, 2003;Shapiro, 2011). In addition to facilitating (Gagne, 1982;Geary, 2011;Mercer and Miller, 1992;Woodward, 2006) the learning of high level math skills, the presentation of basic facts skills fluently makes individuals to be perceived as normal by others in the community where they live (Johnson and Layng, 1996;Özyürek, 2009;Tekin-Iftar and Kırcali-Iftar 2016;Wolery Ault and Doyle, 1992). Individuals who solve basic facts rapidly increase their chances to get reinforcement and this increases their participation in learning activities (Cates and Ryhmer, 2003;Mercer and Miller, 1992). ...
... In addition to facilitating (Gagne, 1982;Geary, 2011;Mercer and Miller, 1992;Woodward, 2006) the learning of high level math skills, the presentation of basic facts skills fluently makes individuals to be perceived as normal by others in the community where they live (Johnson and Layng, 1996;Özyürek, 2009;Tekin-Iftar and Kırcali-Iftar 2016;Wolery Ault and Doyle, 1992). Individuals who solve basic facts rapidly increase their chances to get reinforcement and this increases their participation in learning activities (Cates and Ryhmer, 2003;Mercer and Miller, 1992). The individual has the opportunity to practice more. ...
The aim of this study is to determine the effect of Cover-Copy-Compare (CCC) interventions to increase the level of fluency in basic multiplication facts of a student having low performance in math, whether the student can maintain the level of fluency that she attained after a period of time and the social significance of the obtained data. A multiple-probes-across tasks (sets) design was employed in this study. The participant is a 9-year-old female student who is attending the second grade in general education (in the last two months of the second semester). Besides, she receives four hours of individual special education per week in a research center providing special education services for students with developmental disabilities at a state university. The findings of the study indicate that the education done with the CCC technique is effective in increasing the level of fluency of the student in basic multiplication operations in all sets, and she maintains the fluency performance after a certain period of time. The subjective evaluation findings of the study on social validity suggest that the student and the implementer who participated in the study had positive opinions about the effects and the obtained results of the CCC technique on the student. In addition, the social comparison findings of the study regarding social validity show that the student's level of fluency in multiplication facts reached the level of her peers. These findings were discussed together with the findings of other studies.
Bu araştırmanın amacı, Keşfet-Kopyala-Karşılaştır (KKK) ile yapılan öğretim uygulamalarının matematik başarısı düşük bir öğrencinin temel çarpma işlemlerinde akıcılık düzeyini artırmaya olan etkisini, öğrencinin ulaşmış olduğu akıcılık düzeyini aradan belli bir süre geçtikten sonra da sürdürüp sürdürmediğini ve elde edilen sonuçların sosyal açıdan önemini belirlemektir. Araştırmada tek denekli desenlerden beceriler (setler) arası çoklu yoklama modeli kullanılmıştır. Katılımcı, 9 yaşında, genel eğitimde ikinci sınıfa devam eden (ikinci dönemin son iki ayında) bir kız öğrencidir. Aynı zamanda bir devlet üniversitesinde gelişimsel yetersizliği olan öğrenciler için özel eğitim hizmeti veren bir araştırma merkezinde haftada dört saat bireysel özel eğitim hizmeti almaktadır. Araştırma bulguları, KKK tekniği ile yapılan öğretimlerin tüm setlerde öğrencinin temel çarpma işlemlerindeki akıcılık düzeyini artırmada etkili olduğunu, öğrencinin ulaşmış olduğu akıcılık performansını aradan belli bir süre geçtikten sonra da sürdürdüğünü göstermektedir. Araştırmanın sosyal geçerliğe yönelik öznel değerlendirme bulguları, araştırmaya katılan öğrenci ve uygulamacının KKK tekniği ve elde edilen sonuçların öğrenci üzerindeki etkilerine yönelik olumlu
... CRA research also includes mnemonic strategies; however, it is imperative that students achieve firm conceptual understanding prior to emphasis on procedural knowledge and fluency (Miller, Harris, Strawser, Jones, & Mercer, 1998). Mercer and Miller (1992) used the CRA sequence to teach place value, and basic addition, subtraction, multiplication, and division to 109 students who struggled in mathematics. Through individualized instruction, students learned multiplication and division using plates and counters at the concrete level, pictures and drawings at the representational level, and a mnemonic strategy and numbers at the abstract level. ...
... One purpose of the current study was to capture qualitative changes in students' understanding of basic division as they engaged in CRA instruction and, as anticipated based on the literature (C. A. Harris et al., 1995;Miller, 1992 andMiller et al., 1998), improved their accuracy and fluency. The current study developed out of observed deficits in students' attempts in learning division. ...
... After the intervention, students exceeded the criterion of 30 correct digits. The accuracy and fluency results are similar to those found by previous researchers (Harris et al., 1995;Mercer & Miller, 1992;Miller et al., 1998;Miller & Mercer, 1993). However, this study presented instruction in the two operations at the same time and this did not appear to interfere with students' learning. ...
To meet increasingly complex mathematics standards in late elementary school, students must conceptually understand and be fluent in the operations of multiplication and division. This includes understanding the operations’ inverse relation. The purpose of the study was to investigate the effects of alternating concrete–representational–abstract (CRA) multiplication and division instruction on students’ mastery of unknown facts and on their conceptual understanding. Fourth through sixth-grade students with learning disabilities who had failed to master all multiplication facts participated in the study. The researchers used a mixed method design, measuring accuracy and fluency of facts with a multiple probe across students design and qualitative methods to capture changes in students’ explanations of their computation. The researchers demonstrated a functional relation between CRA instruction and accuracy and fluency in multiplication and division. Qualitative results indicated differences in students’ understanding of the operations. Implications of the results will be discussed further.
... Knowledge of what addition means and how to add numbers is necessary before students can progress to problems requiring higher level thinking such as application type word problems. Many students with learning disabilities fail to achieve an understanding of basic math facts such as addition [3]. ...
... Researchers in this study and other studies [19,20] have followed a concrete-representationalabstract (CRA) model used by Mercer and Miller [3] to help young children learn basic math facts such as addition, subtraction, multiplication, and division concepts. See Figure 1 below. ...
... The lessons should include giving an advanced organizer, describing and modeling, conducting guided practice, conducting independent practice, conducting problem-solving practice, administering facts review, and providing feedback [3]. These procedures have been shown to be effective with teaching basic facts to students with learning difficulties. ...
... Alanyazındaki çalışmalar, S-YS-S öğretiminin uygulanmasında öğretmen-öğrenci etkileşiminin önemini vurgulamaktadır. Öğretmen olarak uygulamacının, öğrenciye model olması, rehberlik etmesi, bu rehberliği giderek azaltması ve çok sayıda fırsat sunması gerekmektedir (Hudson vd., 2006;Mercer & Miller, 1992). Öğretmen-öğrenci etkileşimine dayalı bir yöntem olmasından dolayı, doğrudan öğretim yönteminin bu çalışmalarda sıklıkla tercih edildiği düşünülmektedir. ...
... Studies in the literature emphasize the importance of teacher-student interaction in the implementation of CRA instruction. As a teacher, the intervention agent must be a model and guide the student, gradually reduce this guidance, and offer many opportunities (Hudson et al., 2006;Mercer & Miller, 1992). Since it is a method based on teacher-student interaction, it is thought that the direct instruction method is frequently preferred in these studies. ...
Giriş: Bu çalışmada özel gereksinimli bireylere matematik becerilerinin öğretiminde Somut-Yarı Somut-Soyut (SY-YS-S) öğretim uygulamalarının kanıta dayalı olma durumunun değerlendirmesi amaçlanmaktadır.Yöntem: Bu çalışmada, 1980-2020 yılları arasında ulusal ve uluslararası kaynaklarda yayımlanan çalışmaların, betimsel analiz ve kanıta dayalı olma standartlarına göre analiz süreçleri gerçekleştirilmiştir. İlk taramalar sonucunda toplamda 52 çalışmaya ulaşılmıştır. Bu çalışmalardan dâhil etme ölçütlerini karşılayan toplam 21 çalışma betimsel analiz sürecine dâhil edilmiştir. Ardından, tek-denekli araştırmalar için belirlenmiş olan niteliksel ölçütler dikkate alınarak bu çalışmalar yöntemsel açıdan değerlendirilmiştir. Niteliksel ölçütlerin tamamını karşılayan 17 çalışma görsel ve meta-analiz sürecine alınmıştır.Bulgular: Betimsel analiz bulgularına göre, çalışmaların 2011-2019 yılları arasında yoğunlaştığı ve sıklıkla özel öğrenme güçlüğü olan çocuklara yönelik gerçekleştirildiği görülmüştür. Yöntemsel özelliklere bakıldığında; araştırma deseni olarak sıklıkla denekler arası yoklama denemeli çoklu yoklama modelinin kullanıldığı; bağımlı değişken olarak problem çözme alanında en çok onluk bozma gerektiren çıkarma işlemi içeren problemleri çözme becerisi ve dört işlem becerileri alanında ise sıklıkla çarpma işlemi becerisine yer verildiği görülmüştür. S-YS-S stratejisinin sıklıkla doğrudan öğretim yöntemi ile sunulduğu ve RENAME stratejisi ile desteklendiği görülmektedir. İncelenen araştırmaların tümünde grafiksel analiz kullanılmış; ancak istatistiksel analizlere tüm çalışmalarda yer verilmemiştir. Kanıta dayalı değerlendirmeye ilişkin bulgulara bakıldığında, niteliksel ölçütlerin tamamını karşılayan 17 çalışmadan 16’sında olumlu etki görülmüştür. Meta-analiz çalışmalarında, ÖVY ve Tau-U etki büyüklüğü analiz sonuçları ise bulgular başlığında ayrıntılandırılmıştır.Tartışma: Elde edilen bu bulgular, S-YS-S öğretiminin kanıta dayalı bir uygulama olduğunu gösterir niteliktedir. Elde edilen bulgular alanyazın dikkate alınarak tartışılmış, araştırmacılara ve uygulamacılara önerilerde bulunulmuştur.
... Researcher-developed learning sheets constituted one of the materials used for this study, along with an iPad with app manipulatives, and pencils. In each session, the researcher used a learning sheet containing a modeling portion (two problems), a guided instruction portion (two problems), and an independent practice portion (five problems) consistent with prior studies of the CRA or VRA instructional sequence Mercer & Miller, 1992). Each of the portions was on different sheets of paper stapled together. ...
... If the student solved the problem correctly, then the researcher made a positive statement for the student (e.g., you solved it correctly). If the student did not solve the problem correctly, the researcher noted where the error occurred and showed him or her how to perform it correctly during guided instruction (Mercer & Miller, 1992). During independent practice, the student solved five problems independently, meaning the researcher did not provide any help, prompts, or cues. ...
While mathematics education is key to the post-school outcomes of students with disabilities, it has received less attention in research and practice compared with other aspects of educating this population. Skill maintenance is particularly crucial in mathematics because students build upon prior knowledge across grade levels. They also need to be able to apply mathematical skills in everyday life. Hence, this study utilized a multiple probe across-participants single-case experimental design to evaluate the effectiveness of the virtual–representational–abstract (VRA) with overlearning instructional sequence in teaching multiplication and supporting its maintenance among three students with disabilities. For each student, a functional relation existed between the VRA with overlearning instructional sequence and accuracy of solving multiplication problems. Students also maintained the skill up to 8 weeks after the intervention.
... No study focused on children aged 5 years or below, 15 on children aged 6 to 9 years (first through third grades), 16 on children aged 10 to 12 years (fourth through sixth grades), and 16 on adolescents aged 13 years and older (seventh grade and above). In one study [76], the authors collected data from students at the elementary level, but did not indicate the specific ages or grades of the participants (the total number of studies in this count exceeds 38 because some of them included two or more age groups). ...
... Maintenance of gains was measured using delayed tests in 18 of the 38 studies in the sample, consisting of six group and 12 single-case studies. Among these, maintenance ranged from a few days (e.g., [76]) to 11 weeks [61]. Only one group study (i.e., [53]) was classified as being of high quality and reported an effect size of 0.74. ...
Manipulatives are concrete or virtual objects (e.g., blocks, chips) often used in elementary grades to illustrate abstract mathematical concepts. We conducted a systematic review to examine the effects of interventions delivered with manipulatives on the learning of children with Mathematics Learning Disabilities (MLD). The outcomes observed in the sample (N = 38) were learning, maintenance, and transfer in a variety of mathematical domains. Interventions using manipulatives were reported to be effective for a range of learning objectives (e.g., conceptual understanding, computational fluency), but several methodological weaknesses were observed. Analyses also highlighted considerable heterogeneity in the studies reviewed in terms of participant characteristics, intervention approaches, and methodology. We discuss overall effects of interventions with manipulatives in the MLD population, the methodological quality across the sample, and implications for practice.
... Early CRA research focused on basic number operations such as addition, subtraction, place value, multiplication, and division (Peterson et al. 1988;Mercer and Miller 1992). Peterson et al. used explicitly taught place value skills to students with disabilities in elementary and middle schools who received instruction in self-contained special education settings. ...
... Peterson et al. used explicitly taught place value skills to students with disabilities in elementary and middle schools who received instruction in self-contained special education settings. Within small-group settings, teachers used CRA to teach basic addition and division skills (Mercer and Miller 1992). Harris et al. (1995) taught single-digit multiplication to students with disabilities in the general education classroom. ...
Students who have difficulty with mathematics may have trouble understanding underlying concepts of numbers and operations. The concrete-representational-abstract (CRA) instructional sequence of instruction provides a way for teachers to help students gain meaning from numbers and the mathematical concepts those numbers represent. This study addresses evidence-based practices and applies CRA methods to instruction address concepts such as rounding, regrouping, and equivalent fractions. The purpose of this study was to investigate the effects of CRA instruction on the performance of elementary students across varied areas of need related to poor conceptual understanding and proficiency in completing tasks related to numbers and operations. The researchers implemented a multiple baseline across behaviors design for two students who were at risk for mathematics failure. A functional relation was found for CRA intervention and rounding, regrouping, and fraction concepts for the two students. Results and implications are discussed. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.
... By contrast, this tutoring does not address important psychological factors such as motivation, negative emotions, self-esteem, or problem solving. Therefore, not all students will benefit from private tutoring because some students need more help in improving or developing cognitive-behavioral and/or emotional skills, which requires specialized training that cannot be provided by conventional tutoring (Lambert & Spinath, 2014;Mercer & Miller, 1992). ...
... Consistent with previous studies (Campbell & Ramey, 1994;Durlak et al., 2011;Lipsey & Wilson, 1993), our results indicate that PPI combined with private tutoring promotes a greater improvement in academic performance and has a higher impact than private tutoring alone (Berberoğlu & Tansel, 2014). Accordingly, although private tutoring has been shown to enhance academic performance in urban students (Ireson & Rushforth, 2005;Kenny & Faunce, 2004;Smyth, 2008;Zhang, 2013), our results suggest that a combined approach that includes PPI would have an even greater impact, probably because PPIs focus on developing psychological factors such as positive emotions, personal strengths, self-esteem, and intrinsic motivation, none of which are directly developed by the conventional approach of private tutoring (Lambert & Spinath, 2014;Mercer & Miller, 1992). Therefore, PPI could also be considered a multi-tier approach to learning (Sailor et al., 2009) for those students with emotional and behavioral disorders and atrisk for educational failure. ...
In the last decade, positive psychology interventions (PPI) applied in both clinical and non-clinical samples have demonstrated a proven efficacy to increase positive emotions, well-being, and life satisfaction. However, few studies have used objective indicators of performance to explored the efficacy of PPI to increase students' motivation to study or to improve performance. Therefore, we developed and applied a PPI in a sample of high-school students with poor academic achievement. A pre-post study design including both an interventional and a control group was developed to compare the two groups in terms of average grades and number of failed subjects. Average grades increased significantly in both groups (repeated-measures ANOVA), but this increase was higher in the PPI group. Based on regression analyses, the two factors that explained 40% of the motivation to continue studying were allocation to the PPI group and the overall grade average post-intervention. These findings suggest that PPIs are effective in increasing motivation to study and in enhancing the academic performance of poor performing high school students.
... Mercer and Miller [19] used ten lessons. The concrete manipulatives were used in Lessons 1-3, the representational phase was used in Lessons 4-6. ...
... The student had difficulty finding "318" versus "380" and "515" versus "550"; so more time was dedicated to numbers in the hundreds ending with [11][12][13][14][15][16][17][18][19]. This was the same problem that the student had shown throughout the pretest and concrete phase of the lessons. ...
... Thus, while understanding the mathematical aspects of multiplication has a role in the learning process, memorization is necessary to achieve proficiency and automaticity or fluency, which refers to the ability to recall basic arithmetic facts in a precise way with little effort 36 . Indeed, findings from research of mathematical education show the benefits of achieving fluency in basic mathematical facts [36][37][38][39][40] , and it is perhaps essential in order to develop estimation and mental computation skills [41][42][43][44][45] . However, many students show substantial individual differences and have difficulties in achieving fluency in basic arithmetic facts retrieval, such as multiplication facts 41,42,[46][47][48] . ...
Learning of arithmetic facts such as the multiplication table requires time-consuming, repeated practice. In light of evidence indicating that reactivation of encoded memories can modulate learning and memory processes at the synaptic, system and behavioral levels, we asked whether brief memory reactivations can induce human learning in the numeric domain. Adult participants performed a number-fact retrieval task in which they learned arbitrary numeric facts. Following encoding and a baseline test, 3 passive, brief reactivation sessions of only 40 s each were conducted on separate days. Learning was evaluated in a retest session. Results showed reactivations induced learning, with improved performance at retest relative to baseline test. Furthermore, performance was superior compared to a control group performing test-retest sessions without reactivations, who showed significant memory deterioration. A standard practice group completed active-retrieval sessions on 3 separate days, and showed significant learning gains. Interestingly, while these gains were higher than those of the reactivations group, subjects showing reactivation-induced learning were characterized by superior efficiency relative to standard practice subjects, with higher rate of improvement per practice time. A follow-up long-term retention experiment showed that 30 days following initial practice, weekly brief reactivations reduced forgetting, with participants performing superior to controls undergoing the same initial practice without reactivations. Overall, the results demonstrate that brief passive reactivations induce efficient learning and reduce forgetting within a numerical context. Time-efficient practice in the numeric domain carries implications for enhancement of learning strategies in daily-life settings.
... Next, the symbolic step, where all information relating to the results of the representation is saved in symbol form. Several studies have shown that CRA can be used to facilitate students' conceptual understanding [25][26] [27]. ...
... The factors that distress the students' learning process of mathematics at secondary school levels stated by Askew (2008) included overcrowded classrooms, shortage of educational facilities (labs, instructional material, like mathematics geometry box, etc.), lack of suitable guidance (motivation to learning), lack of communication (freedom of feedback or freedom to ask question) and physical punishment. There are some of the environmental factors as well, which are too affecting the students learning process (Mercer and Miller 1992). ...
Mathematics is critical and significant to recognise the computerised world and match with the newly developing information technology knowledge that is penetrating everywhere in the world. By knowing the importance of Mathematics, this research was set out to describe and explain the difficulties of secondary level students of science group who are facing difficulties in Mathematics. The study was conducted at Federal Government Schools of Islamabad. The study investigated questions concerning experiences of high school (grade 9th and 10th) students’ in Mathematics classroom and the possible reasons for these difficulties. The study was carried out to analyse students’ experiences in the subject of mathematics, and whether the teaching method of mathematics enhances the students’ mathematical thinking and prepares them for pedagogical aspects of Mathematics for future teaching.
... In response to these chronic mathematical difficulties, special education researchers have identified critical instructional and curricular variables that promote effective programming for students with mathematics learning disabilities (Cawley & Parmar, 1992;Mercer & Miller, 1992;Rivera & Smith, 1987). Drawing from cognitive and developmental psychology and behavioral theory, the knowledge base of what constitutes sensible instructional practices for youngsters with math LD has increased significantly in recent years. ...
... Cette façon de concevoir l'articulation entre les différents modes de représentation remet en question le recours systématique à la séquence linéaire présentée dans plusieurs ouvrages qui proposent de commencer par le concret (matériel de manipulation), pour ensuite utiliser le semi-concret (dessins et schémas) puis l'abstrait (symboles et expressions numériques) (Forbringer et Fuchs, 2014;Mercer et Miller, 1992;Baker, Gersten et Lee, 2002). Cette séquence peut convenir aux premiers apprentissages mathématiques (dénombrement, opérations d'addition et de soustraction, etc.), mais elle est difficilement justifiable pour les concepts plus abstraits tels les nombres négatifs et l'algèbre (MEES, 2019). ...
De nombreux systèmes éducatifs ont amorcé, au cours des dernières années, un virage important vers une gestion axée sur les résultats (Bourgault, 2004). Or, ce type de gestion nous amène à poser un regard sur la réussite des élèves ayant des difficultés comportementales, car ce sont eux les plus susceptibles d’abandonner l’école (MELS, 2006). Par conséquent, il devient essentiel d’implanter des systèmes d’intervention, fondés sur des données probantes, favorisant la prévention de ces difficultés. Le système Positive Behavioural Interventions and Supports (PBIS) est fondé sur des données probantes (Lapointe et Freiberg, 2006). Le PBIS, appelé en français le soutien au comportement positif (SCP), représente un modèle de réponse à l’intervention comportementale. Dans cet article, nous décrivons le SCP. Par la suite, nous montrons ses effets sur la diminution des écarts de conduite majeurs des élèves à l’intérieur d’écoles québécoises contribuant ainsi à la prévention des difficultés comportementales.
... Cette façon de concevoir l'articulation entre les différents modes de représentation remet en question le recours systématique à la séquence linéaire présentée dans plusieurs ouvrages qui proposent de commencer par le concret (matériel de manipulation), pour ensuite utiliser le semi-concret (dessins et schémas) puis l'abstrait (symboles et expressions numériques) (Forbringer et Fuchs, 2014;Mercer et Miller, 1992;Baker, Gersten et Lee, 2002). Cette séquence peut convenir aux premiers apprentissages mathématiques (dénombrement, opérations d'addition et de soustraction, etc.), mais elle est difficilement justifiable pour les concepts plus abstraits tels les nombres négatifs et l'algèbre (MEES, 2019). ...
Dans le présent article, nous présentons un modèle d’accompagnement du développement pédagogique et organisationnel pour soutenir la mise en oeuvre d’un modèle de la réponse à l’intervention (RàI). Ce cadre a été utilisé dans le déploiement des ressources pour appuyer l’apprentissage du langage oral et écrit au préscolaire/premier cycle du primaire dans deux commissions scolaires au Québec. Ses origines, sa nature, ses finalités et des actions fondamentales pour le mettre en oeuvre sont détaillées. Enfin, des pistes de réflexion sont proposées à l’intention des gestionnaires en éducation pour guider leurs décisions et leurs actions.
... Although, it is not right to force students to memorize the multiplication facts, making it easier for them to remember by means of some strategies, is beneficial for them from several respects. Memorization of the multiplication facts and responding it rapidly through this way makes it easier to acquire high level mathematical skills including advance level multiplication, division, solving word problems, fractions and decimal numbers (Bender, 2008;Gagne, 1982;Geary, 2011;Mercer and Miller, 1992;Woodward, 2006). ...
In this study, it was aimed to determine the effects of the Simultaneous Prompting (SP) package jointly with the systematic review and corrective feedback in teaching a student with poor performance in mathematics to recall the multiplication facts. One of the single subject designs, the multiple probes design across behaviors (sets) is used in this study. The participant is an 11 years old female student at the 6th grade who has not been diagnosed with any disability but has been getting support because of her poor performance in mathematics than her peers. The dependent variable of this study is the skill of recall multiplication facts with the sets of 3, 5 and 8. The independent variable of the study is the SP package jointly with the systematic review and corrective feedback. Results show that the SP was effective in teaching the student with low mathematical performance to the skill of recall the multiplication facts. The student generalized the learned multiplication facts to another teacher and different setting and maintained her performance on the 15th and 45th days following the systematic reviews. Considering the findings of social validity, it has been determined that the teacher presented positive opinions, as she became happy with that achievement and the method is a way, which can be adopted and implemented by all teachers. These results indicate the acceptability of the SP package and results of the study are highly meaningful. Implications and future research needs are discussed.
... Early intervention needs to address the broad area of concern (e.g., math) as well as the specific component skill deficit (e.g., conceptual understanding, procedural knowledge, fact fluency; Burns 2011a; Codding et al. 2011). Given the hierarchal nature of math understanding (Woodward 2006), of those component skills, students must master basic math facts and move to direct retrieval of facts (i.e., automaticity) in order to dedicate cognitive resources to more complex tasks (e.g., algebraic operations; Delazer et al. 2005;Mercer and Miller 1992;Pellegrino and Goldman 1987;Van Luit and Naglieri 1999) and improve their overall mathematical proficiency (Kilpatrick 2001, National Mathematics Advisory Panel 2008. Students reach automaticity when they move from slow, step-by-step computation to quick, effortless, and low error responding (Delazer et al. 2005). ...
Examining the use of conceptual frameworks such as the instructional hierarchy (IH) to drive academic interventions represents an important area of inquiry in order to understand why an intervention was effective. However, to date, the IH has only been examined retroactively to explain the effectiveness of math interventions and has not been used to drive intervention implementation based on students’ needs. Therefore, the current study used a multiple baseline single-case design to examine the utility of the IH to determine an appropriately targeted acquisition (incremental rehearsal) or proficiency (timed drill) intervention for difficulties with multiplication facts. Students first received the contraindicated intervention (i.e., non-targeted) and then their correctly targeted intervention. Results showed that all students demonstrated greater fluency growth during their targeted intervention and that students in need of an acquisition intervention also demonstrated greater multiplication fact retention and growth rate during their targeted intervention. The utility of the IH as a heuristic to correctly target interventions based on students’ baseline skill levels is discussed.
... According to research, effective instruction includes guided practice, correction and feedback, independent practice, and weekly or monthly review. Several research studies have extended and supported the instructional planning components first identified and compiled by Rosenshine and Stevens in 1986 (Maccini, McNaughton, & Ruhl, 1999;Maccini & Ruhl, 2000;Mercer & Miller, 1992). Rosenshine and Stevens' original work was based on a review of 100 correlation and experimental studies that examined instructional components leading to effective instruction. ...
... In many cases, the empirical support for existing math interventions is connected to a specific instructional target (e.g., math fact fluency); however, the underlying components of those interventions might be adapted to other subskills (Codding & Martin, 2016). For example, although the empirical support for interventions such as cover, copy, and compare (CCC; Skinner, Turco, Beatty, & Rasavage, 1989), concrete, representational, abstract instruction (CRA; Mercer & Miller, 1992), and cognitive strategy instruction (CSI; Montague, Krawec, Enders, & Dietz, 2014) is often tied to specific instructional targets, the underlying characteristics of those interventions (e.g., modeling, immediate feedback, high opportunities to respond) are generally considered to be evidence-based principles that could be applied to a wider range of content (Codding & Martin, 2016). Relatedly, there is some evidence for supplemental interventions that target multiple, related skills that are involved when working with whole and rational numbers. ...
In the present study, we evaluated the number of attempts required to master specific subskills for working with whole and rational numbers among students at risk for math difficulties. Participants included a subset of students in grades four through eight receiving supplemental math support. Mastery—defined as 85% correct on short tests—was assessed following instruction for each subskill. Using survival analysis, we evaluated the number of attempts required to reach a .50 and a .90 probability of mastery on each subskill. The number of required attempts varied across subskills, with many subskills requiring more than one attempt to demonstrate mastery. Further, some of the most difficult content was aligned with curricular standards below students’ grade level. Thus, among students identified for supplemental support, it may be worthwhile to remediate select subskills that fall outside of the grade‐level curriculum before providing additional instruction on grade‐level content. Implications for math subskill assessment and remediation are discussed along with limitations and directions for future research.
Effective mathematics interventions should be explicit and include students’ active involvement with multiple representations of the mathematical concept. The concrete‐representational‐abstract‐integrated (CRA‐I) sequence includes these characteristics and has been shown as an effective practice for students who struggle in mathematics. The purpose of the current study was to use CRA‐I, to teach the partial products algorithm. Three fifth‐grade students receiving Tier 3 instruction within a multi‐tiered system of support participated in the study, using base 10 blocks, number lines, and arrays. The researchers used a multiple probe across students design and collected data regarding students’ progress, mastery, and conceptual understanding. There were functional relations between CRA‐I and each of the skills related to progress, mastery, and conceptual understanding. Implications and conclusions will be discussed.
This study examined the effects of the concrete-representational-abstract integrated sequence (CRA-I) on teaching place value concepts and their application. The research questions addressed the extent to which CRA-I changed student performance in (a) completing equations that required subtraction with regrouping in the tens place, (b) completing equations that required subtraction with regrouping in the tens and hundreds place, (c) rounding three-digit numbers to the nearest ten or hundred, and (d) using multiple equations to decompose three-digit numbers. Four students in the fourth grade participated. Three students were eligible for services under learning disabilities (LD) or other health impairments (OHI); two students were English language learners. Employing a single-case multiple-probe-across-behaviors design, a functional relation was found between CRA-I and behaviors associated with number concepts. Implications will be discussed.
The study aimed to investigate the validity of confirmatory factor analysis (CFA) of the scale of academic learning difficulties among Basic Education pupils (1-6) in Yemen, whose ages ranged from 5 to 12. To test the validity of the proposed model, using AMOS program, the scale was administered to 450 male and female pupils of grades (1-6) selected from 8 schools in Amanat Al Asema. This was done with the help of teachers and in collaboration with a trained team to observe the pupils’ difficulties. The CFA results showed quality of conformity of data of the theoretical proposed model of the scale with its five principal components, based on different indicators of quality conformity. These include: (CFI)= 0.94; (GFI)= 0.84; (RMSEA) = 0.064; (RMR)= 0.041; (IFI)= 0.94; (TLI)= 0.93; (PCFI)= 0.85; (PGFI)= 0.72. The study also presented evidence of the validity and reliability of the scale which was high, including the complex reliability (CR); validity of the CFA, covering its five dimensions; and the relevance of CRF for assessing academic learning difficulties among Basic Education pupils and for studies concerned with such issues. Keywords: academic learning difficulties, structural equation modeling, confirmatory factor analysis, basic education pupils.
The study aimed to investigate the validity of confirmatory factor analysis (CFA) of the scale of academic learning difficulties among Basic Education pupils (1-6) in Yemen, whose ages ranged from 5 to 12. To test the validity of the proposed model, using AMOS program, the scale was administered to 450 male and female pupils of grades (1-6) selected from 8 schools in Amanat Al Asema. This was done with the help of teachers and in collaboration with a trained team to observe the pupils’ difficulties. The CFA results showed quality of conformity of data of the theoretical proposed model of the scale with its five principal components, based on different indicators of quality conformity. These include: (CFI)= 0.94; (GFI)= 0.84; (RMSEA) = 0.064; (RMR)= 0.041; (IFI)= 0.94; (TLI)= 0.93; (PCFI)= 0.85; (PGFI)= 0.72. The study also presented evidence of the validity and reliability of the scale which was high, including the complex reliability (CR); validity of the CFA, covering its five dimensions; and the relevance of CRF for assessing academic learning difficulties among Basic Education pupils and for studies concerned with such issues. Keywords: academic learning difficulties, structural equation modeling, confirmatory factor analysis, basic education pupils.
This study investigated the effects of a place value intervention with third‐grade students with learning disabilities. The intervention added content to a research‐based intervention using the concrete‐representational‐abstract (CRA) sequence. The added content reflected current mathematics standards for third grade. Students’ place value understanding was measured using probes in which students had to identify the value of each digit within a three‐digit number, round three‐digit numbers to the nearest 10 or 100, and write three equations showing an expanded form of a three‐digit number. A single‐case, multiple‐probe‐across‐students design showed a functional relation between CRA and completion of items requiring place value understanding. Students completed a generalization task by estimating the sum of a given equation.
In recent years, virtual manipulatives have been explored and used as an alternative to concrete manipulatives in mathematics for students on their own and as part of manipulative-based instructional sequences. Researchers examining virtual manipulative-based instructional sequences tend to focus on students documented with disabilities, as opposed to students at-risk or struggling with mathematics, as well as students’ acquisition of the target skill, despite students experiencing learning in four stages: acquisition, fluency, maintenance, and generalization. This study explored the virtual-representational-abstract (VRA) instructional sequence across four stages of learning for three elementary students struggling in mathematics. In the single-case design study, researchers found a functional relationship between the VRA instructional sequence delivered online via explicit instruction and students’ computational accuracy in their targeted area of mathematics need. Researchers also found limited influence on fluency rate or generalization to word problem accuracy but that students did maintain.
The present study aimed to determine the effectiveness of the concrete-representationalabstract instruction strategies employed in the direct instruction of fractions to students
with learning disabilities. Furthermore, the generalization of the instruction to different
settings and tools, the follow-up data for one and three weeks after the instruction, and
the social validity data based on the views of the mothers on concrete-representationalabstract instruction strategies were analyzed. In the study, the inter-behavioral multiple
probe model with a probe stage, a single-subject research model, was employed. The
dependent variable was the level of identification of proper, half and quarter fractions by
the participating students in the study, while the independent variable was the concreterepresentational-abstract introduction strategies implemented with the direct instruction
method. The study was conducted with three male students with learning disabilities,
who resided in Izmir and attended an inclusive primary school. The study findings
demonstrated that the concrete-representational-abstract instruction strategies were
effective in the instruction of proper, half and quarter fractions to the students with
learning disabilities, these skills acquired by the students could be permanent for one to
three weeks after the instruction, and all students could generalize these skills to various settings and instruments, and the views of the mothers on concrete-representational-abstract instruction strategies were positive.
Keywords: instruction of fractions, mathematics instruction, concrete-representational-abstract instruction strategies, students with learning disabilities
Student fluency in mathematics is important, as fluency supports mathematics proficiency and achievement. While fluency can be supported with flashcards and worksheets, it can be supported by games. In this exploratory study, researchers examined the relationship between students’ fluency on addition with regrouping problems and playing a virtual Make 10 s game, which supports the making tens strategy for addition. After seven sessions of playing the virtual Make 10 s games, researchers found Tau–U effects from the single case design study were high for at least three of the four students. Yet, researchers were unable to conclusively determine if a functional relation existed between student accuracy for digits and answers in solving the single–digit addition with regrouping problems within one minute and students playing the virtual Make 10 games.
Virtual manipulatives are a form of technology that support the mathematics teaching and learning of students with high-incidence and low-incidence (or extensive support needs) disabilities. The purpose of this chapter is to present virtual manipulatives as an assistive technology. Access to and use of virtual manipulatives have increased over the last decade, resulting in virtual manipulatives serving as a modern assistive technology to support students with a wide range of disabilities in mathematics. This chapter will present an overview of virtual manipulatives as assistive technology for students with disabilities and the research base supporting the effectiveness of this technology.
Mathematical fluency supports maintenance of mathematical skills (Shurr et al., 2019). Multiple interventions support fluency, including the use of games. However, limited research exists exploring games as a means of increasing student fluency in mathematics. In this single case design study, researchers examined the relation between four elementary students’ fluency with single-digit multiplication problems and playing the Product Game, which is a game focused on whole number multiplication facts. Researchers found a functional relation between the intervention of playing the game for 10 min and student accuracy with digits and answers on a 1 min multiplication fluency probe. Overall, students also maintained higher digit and answer rates.
The concrete–representational–abstract (CRA) sequence of instruction is an explicit methodology for teaching mathematics using multiple representations of concepts. Learning concepts through multiple representations fosters conceptual understanding and mathematical thinking. This article describes how a special education teacher used explicit CRA instruction with two elementary students with emotional and behavioral disorders. Its aims are to describe and provide a rationale for explicit CRA instruction. We describe lesson activities, methods, materials, and procedures. Finally, we offer suggestions for effective implementation.
Zur Aktivierung des Vorwissens kann ein Advance Organizer eingesetzt werden, mit zu Beginn des Unterrichts in einer lehrerzentrierten Präsentation eine Organisationshilfe für neue Lerninhalte zur Verfügung gestellt wird. Befunde aus Metaanalysen bestätigen die generelle Wirksamkeit dieser Methode. Forschungsziel des vorliegenden Projekts ist die Evaluation eines Advance Organizer für heterogene Lerngruppen im Mathematikunterricht. In einer empirisch-quantitativen Untersuchung mit einem randomisierten Kontrollgruppendesign mit Prätest, Intervention, Posttest und zwei Follow-Up-Messungen wird diese Methode hinsichtlich ihrer Wirksamkeit evaluiert. Positive Wirkungen des Advance Organizer zeigen sich besonders im Behalten spezifischer Wissensinhalte für die Gesamtgruppe. Für Schülerinnen und Schüler unter Risikobedingungen werden höhere Effekte im Behalten spezifischer sowie allgemeiner Mathematikleistungen erkennbar. Unabhängig vom Einsatz des Advance Organizer beeinflussen besonders das Vorwissen aber auch das Lernverhalten das Lernen und Behalten der vermittelten Wissensinhalte.
The aims of this study explain the need of chemistry book on acid and base developed by 4S TMD models with STES approach. Problems of education in the last 100 years are often related to students understanding of the concepts taught, one of which is acid-base [1]. In acid-base, there are many applications in daily life. So to overcome these problems, we need a teaching material with the STES approach [2]. The STES approach is used to improve positive attitude towards science and conceptual knowledge as well as improving teacher quality and student learning outcomes [3,4]. This study uses qualitative descriptive research. The instruments of data collection used in this study were structured questions and interview. The study involves 3 senior high schools and 3 vocational high schools with 12 chemistry teachers. The teachers structure question responses was analyzed descriptively for each item. The teacher needs teaching materials that explain the material simply and involve examples in daily life. The teaching material available lack a concept that contains knowledge is associated with technological developments, especially in acid-base material. The research findings indicate the need to develop chemistry book on acid-base to meet the shortcomings of interesting teaching materials. The results of this study are the basis for us in developing a chemistry book on acid and base by 4S TMD models with STES approach that can be used chemistry learning and teaching in upper high schools.
We developed a cognitive-emotional strategy training (CEST) intervention to teach fifth-grade students (N = 57) self-regulated learning strategies that can be used when confusion is experienced during mathematics problem solving in addition to strategies they can implement during learning to help solve them. Fifth-grade students were randomly assigned to the intervention condition or the control condition. A think-emote-aloud protocol was administered to capture self-regulatory processes and emotions as students solved a complex mathematics problem. Using an explanatory mixed methods design, results revealed that, compared to students in the control condition, students who received the intervention scored significantly higher on the mathematics problem, implemented more cognitive and metacognitive learning strategies across the four phases of self-regulated learning, expressed more positive emotions and fewer negative emotions, and were better able to regulate and resolve their confusion when it occurred. These results extend previous findings from the strategy instruction literature by incorporating consideration of the role of emotions during learning.
Bu araştırmanın amacı; hafif düzey zihinsel yetersizliği olan öğrencilere matematiksel kavram ve becerilerinin öğretimine yönelik öğretmenlerin sınıflarında yaptıkları uygulamaları incelemektir. Araştırmanın örneklem grubunu Sakarya’da özel eğitim okulunda çalışan 10 özel eğitim öğretmeni oluşturmaktadır. Verilerin toplanmasında, görüşme türlerinden yarı yapılandırılmış görüşme tekniğinden yararlanılmıştır. Araştırmacılar tarafından hazırlanan yarı yapılandırılmış görüşme soruları kullanılarak veriler toplanmıştır. Verilerin çözümlenmesinde betimsel analiz kullanılmıştır. Araştırmanın sonunda özel eğitim okulunda çalışan öğretmenler, matematiksel kavram ve becerilerin öğretimi sırasında bireyselleştirilmiş eğitim ve öğretim planı yaptıklarını ifade etmişlerdir. Öğretmenler, matematiksel kavram ve becerilerin öğretimi sırasında öğrencilerin dikkatini çekmeye ve sürdürmeye yönelik öğrencinin yakın çevresinden örnekler verdiğini ifade etmişlerdir.
Elementary standards include multiplication of single‐digit numbers and students advance to solve complex problems and demonstrate procedural fluency in algorithms. The ability to illustrate procedural fluency in algorithms is dependent on the development of understanding and reasoning in multiplication. Development of multiplicative reasoning provides the foundation for advanced mathematics and algebraic reasoning. For students who struggle in mathematics, instruction in multiplication algorithms should ensure conceptual understanding so that students have a foundation for success in advanced mathematics. The concrete representational abstract (CRA) sequence addresses conceptual understanding and the strategic instruction model (SIM) supports procedural knowledge. The current pilot study combined these methods to teach elementary students the partial products algorithm. Twelve students in grades four and five participated in the study, receiving instruction from teachers in their school during an intervention period. Within a pre‐experimental design, using pre‐ and postintervention data, students showed a significant change in performance. The article will describe and show how teachers implemented the CRA‐SIM interventions and discuss implications for practice.
The purpose of this study was to investigate effectiveness of teaching multiplication facts by using the Concrete-Representational-Abstract (CRA) Teaching Strategy presented by direct teaching method to children with intellectual disabilities. This study also examined generalization effects across different people and settings, maintenance effects after 10 and 20 days and social validity about teachers’ views with CRA Teaching Strategy. A single-subject design with multiple-probe design with probe trials across subjects was employed in this study. The dependent variable of this study was the level of achievement of the multiplication facts of students with intellectual disabilities, whereas the independent variable of this research was CRA Teaching Strategy which was used to teach multiplication facts to the subjects. This research was conducted with two girls and a boy with intellectual disabilities. Subjects, whose ages ranged from 9 to 10 years, already had prerequisite skills such as reacting to written and verbal instructions, object manipulating, drawing simple shapes, identifying numbers, object (rational) counting and skip counting, basic addition skills and the concept of equality. Findings of the study indicated that CRA Teaching Strategy was effective in teaching multiplication facts to students with intellectual disabilities. Subjects were able to retain their skills 10 and 20 days after the completion training sessions, and they were also able to generalize the newly acquired target skills across different settings and people. In addition, interviews with two special education teachers and three mainstream teachers revealed that their ideas were generally positive on the use of Concrete-Representational-Abstract Teaching Strategy.
Keywords: Concrete-Representational-Abstract Teaching Strategy; Intellectual disabilities; Teaching mathematics.
Mathematical thinking as an important instrument in science is a stumbling block for many students in the first years. A lot of investigations occur to help the students understanding the principles of mathematics. The proposed tutorial system for the basics focuses on the analysis and visualizations of the solution algorithms and solution processes with Boolean Networks and Self-Enforcing Networks. The students can check not only the correctness of their results, but also if the solution steps are complete. In addition, in case of wrong results the students can check in which step of the solution they made a mistake and what kind of mistake. The goal is to promote the explorative learning and to help understanding the problems through self-recognition.
Bir derleme çalışması olan bu makalenin amacı, matematikte işlem
akıcılığının geliştirilmesinde kullanılan tekniklerden biri olan dinleyerek
işlem yapmanın (DİY) temel uygulama basamaklarını açıklamak ve öğrencilere ve
uygulamacılara sağladığı katkıları alanyazında yer alan bilimsel araştırmalar
ışığında tartışmaktır. Çalışmada öncelikle alanyazın araştırmaları ışığında
DİY’in uygulama basamakları, uygulama öncesi hazırlıklar, uygulama ve
değerlendirme başlıkları altında ele alınarak açıklanmıştır. Uygulama öncesi
hazırlıklarda öğrencinin kullanacağı çalışma yaprakları ve ses kayıtlarının
nasıl hazırlanması gerektiği açıklanmıştır. Sonra birebir ya da gruba yönelik
uygulamaların nasıl yapılması gerektiği açıklanarak değerlendirmede yapılması
gereken önemli noktalara değinilmiştir. Daha sonra alanyazında, DİY uygulamalarının
etkilerini belirlemeye yönelik araştırmalar, katılımcı sayısı, cinsiyeti, yaşı
ve sınıf düzeyi, araştırma deseni, hedeflenen beceri (bağımlı değişken),
uygulanan prosedür (bağımsız değişken), uygulama biçimi ve elde edilen sonuçlar
ele alınarak açıklanmıştır. Bu değişkenler ışığında sonuçların öğrenciler ve
uygulamacılar açısından yararları tartışılmış ve ileri araştırmalara yönelik
önerilerde bulunulmuştur.
Manipulatives are considered a common tool for mathematics teaching and learning, for both students with and without disabilities. Yet, a systematic review of the current state of research regarding manipulatives for students with disabilities did not exist prior to this article. This manuscript presents A systematic review of the literature regarding manipulatives for mathematics teaching and learning for students with disabilities. A total of 36 articles involving manipulatives and mathematics for students with disabilities were found: 7 that examined the impact of student learning with manipulatives and 29 that explored the impact of manipulatives use through the delivery of the concrete-representational-abstract (CRA) approach. This article presents the characteristics of the current state of research on manipulatives for students with disabilities as well as the implications of the research base for teachers and students.
Mathematical problem solving is a central theme of K-12 mathematics and an essential skill for college and career readiness. However, problem solving is a challenging task for many young students, especially for students with cognitive difficulties because it requires not only mathematics skills but also reading comprehension, reasoning, and ability to transform words and numbers into the appropriate operations. This chapter is designed to provide helpful suggestions for teaching students with autism spectrum disorder (ASD) who have math problem solving difficulties. The emphasis is made on the visual supports that aid students with ASD to develop abstract and conceptual understanding and apply the learned concepts in real-life problem solving. The following are topics included in this chapter: (1) the general guidelines for math instruction targeting students with ASD; (2) teaching problem solving based on the concrete-representational-abstract (CRA) approach; (3) teaching students in concrete level using manipulative; and (4) teaching students at semiabstract level: schema-based instruction (SBI) for students with ASD. Several lesson ideas are also included in the chapter to illustrate the visual strategies and SBI for problem solving skills.
Current research on direct instruction is reviewed with a particular emphasis on two strands—studies of instructional design and technology and research on effective staff development. The latter research suggests that three components are essential to the position of instructional supervisor or resource consultant, including a working knowledge of the research on effective teaching and the ability to effectively coach teachers in the classroom setting. Extensive training is required for individuals to become competent in these components. The technology research indicates that the same instructional variables that enhance learner performance with traditional written materials appear to enhance learning with computer-assisted instruction and interactive videodisc instruction. Several examples of technology applications are presented. The article concludes with a discussion of future directions in direct instruction research.
The literature on effective instruction for difficult-to-teach students is reviewed. Preinstructional variables include classroom management planning and planning for instructional tasks (student assessments, communication of learning goals, pacing decisions, and time allocations). Instructional delivery variables include engagement time, success rate, academic learning time, and monitoring. Postinstructional variables include testing and feedback. (JDD)
Current knowledge about cognitive development in mathematics is reviewed, specific effective strategies for presenting materials in class are described, and effective strategies for mathematics program content and organization are outlined. Substantial information on recent research and on instructional materials is included. (MSE)
Results from a review of laboratory and field studies on the effects of goal setting on performance show that in 90% of the studies, specific and challenging goals led to higher performance than easy goals, "do your best" goals, or no goals. Goals affect performance by directing attention, mobilizing effort, increasing persistence, and motivating strategy development. Goal setting is most likely to improve task performance when the goals are specific and sufficiently challenging, Ss have sufficient ability (and ability differences are controlled), feedback is provided to show progress in relation to the goal, rewards such as money are given for goal attainment, the experimenter or manager is supportive, and assigned goals are accepted by the individual. No reliable individual differences have emerged in goal-setting studies, probably because the goals were typically assigned rather than self-set. Need for achievement and self-esteem may be the most promising individual difference variables. (3½ p ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Investigated the effectiveness of an experimental mathematics teaching program. The treatment program was primarily based on a naturalistic study of 40 relatively effective 4th-grade mathematics teachers. Students were tested before and after with a standardized test and a content test (posttest only), which had been designed to approximate the actual instructional content that students had received during the treatment. Observational measures revealed that teachers generally implemented the treatment, and analyses of product data showed that students of treatment teachers generally outperformed those of control teachers on both the standardized and content tests. Since strong efforts were made to control for Hawthorne effects, it seems reasonable to conclude that teachers and/or teaching methods can exert a significant difference on student progress in mathematics. (4 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Using error analysis and individual interviews, the problem-solving actions of 176 1st and 2nd graders were analyzed in Exp I. Shortcomings of Ss' knowledge and solution strategies were discovered. It seemed that these shortcomings could be overcome by instruction; therefore, a teaching experiment (Exp II; 52 2nd graders) was undertaken wherein instruction was given for 2 wks to an experimental class, while in a control group, the usual arithmetic program was taught. Experimental instruction related mainly to 3 topics: the equality sign, the part–whole relation, and verification of the outcome of an arithmetic operation. Results show that the experimental teaching program led to a decrease in Ss' thinking errors on elementary addition and subtraction problems. (35 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Math textbooks, which usually represent the mathematics curriculum, seem to be linked to the poor math performance of U.S. students. The major shortcomings of math textbooks are described in this article; then an alternative perspective is offered (the sameness analysis), along with research conducted with students with learning disabilities and at-risk students. The article then presents a detailed illustration of the sameness analysis--how to teach the addition-subtraction and multiplication-division relationships and their interrelationships in the context of solving word problems in mathematics.
Editor's Comment: The articles in this special series on curriculum make significant contributions, first, by proposing that the curriculum employed for all students, including those with learning disabilities, should emphasize thinking, problem solving, and reasoning, a point few would disagree with, given the complexities of life in this postindustrial-information age. The authors also provide some evidence that students with learning disabilities can learn and apply sophisticated concepts, rules, and strategies. Furthermore, the authors describe one instructional process that emphasizes higher order thinking, that is, "sameness analysis." This process facilitates the integration of concepts, rules, strategies, schema, systems, heuristics, and algorithms. The authors believe that sameness analysis fosters a holistic understanding of a content area. Over the last several years this journal has contained numerous articles that cogently argued for a more holistic approach to education. Sameness analysis is one that lends itself to the holistic approach. Finally, the series of articles quite appropriately addresses the importance of efficient teaching. Some recent research indicates that students with learning disabilities may be receiving less instruction than their nonhandicapped peers-this despite the fact that they have problems in learning. Efficient teaching takes on increased importance with the recent call (by some) for increased or total integration of students with learning disabilities into general classrooms.---JLW
This study evaluated the relative effectiveness of a curriculum that incorporated three empirically derived principles of curriculum design with a basal approach in teaching basic fractions concepts to students with learning disabilities and other low performing students in high school remedial math classes. The components of effective mathematics instruction articulated by Good and Grouws (1979) were implemented in both conditions. Thus, the curriculum design variables were isolated by keeping all other aspects of instruction constant. Results indicated that, although both programs were reasonably successful in teaching the material, the curriculum program utilizing sophisticated principles of curriculum design was significantly more effective. Mean scores on a curriculum-referenced test were 96.5% for that group and 82.3% for the basal group. Secondary analyses of item clusters revealed that areas of weakness in the performance of the basal group could be directly linked to hypothesized flaws in its curriculum design.
The study compared basic and elaborated corrections within the context of otherwise identical computer-assisted instruction (CAI) programs that taught reasoning skills. Twelve learning disabled and 16 remedial high school students were randomly assigned to either the basic-corrections or elaborated-corrections treatment. Criterion-referenced test scores were significantly higher for the elaborated-corrections treatment on both the post and maintenance tests and on the transfer test. Time to complete the program did not differ significantly for the two groups.
This article is the second of a twopart series designed to review the critical features of facilitating generalization and adaptation of learning strategies. In Part 1, a model of generalization was presented along with research supportive of the model and identification of research needs. Essentially, the model views instruction for generalization not as something that comes at the end of an instructional sequence, but rather as consisting of four levels of generalization that transverse instruction—antecedent, concurrent, subsequent, and independent. The purpose of this article is to consolidate a number of studies that describe a unified set of specific instructional techniques that can be used while addressing generalization and to present them as part of an overall instructional approach for learning strategies. Within each level, specific procedures mediated by the special (or remedial) education teacher, regular content teacher, peer, and/or student are illustrated. Each category is followed by a synthesis of related teaching practices. The procedures identified here should not be considered definitive because demonstration of efficacy awaits additional validation; however, they do serve as a basis for planning instruction consistent with what has been learned about generalization to date.
The purpose of this paper is to review research on the use of formative evaluation with mildly handicapped pupils. First, the importance of formative evaluation to special education is described. Then, four critical issues in formative evaluation methodology are discussed: focus of measurement, frequency of measurement, data display, and data-utilization methods. Finally, a proposal for additional related research is advanced.
A survey of resource room teachers (N = 101) was used to determine: (a) the amount of time resource room teachers spend in mathematics instruction, (b) the amount of those teachers' formal training in mathematics and mathematics instruction, (c) their self-reported degrees of competence in assessing and teaching mathematics skills and concepts, and (d) the importance these teachers assign to general knowledge of mathematics, mathematics assessment, and mathematics instruction. Mathematics instruction was found to occupy about one-third of the average resource room teacher's teaching time. Although subjects reported considerable college coursework in mathematics and mathematics instructional methods, they felt inadequate at several competencies considered necessary for teachers of learning disabled students. Differences between elementary and secondary teachers are reported.
The relationship between automatization ability, as measured by the Rapid Automatic Naming Test (RAN), and proficiency in arithmetic basic fact computation was investigated. Subjects included 120 learning disabled and 120 nondisabled children between 8 and 13 years of age; 60 subjects in each group were designated as either younger or older. Significant correlations were obtained between RAN performance and basic fact proficiency for both the learning disabled and nondisabled groups. In addition, learning disabled subjects were found to be less proficient in basic fact computation and slower on RAN than their nondisabled peers at both younger and older age levels. Correlations were substantial enough to further inquire whether LD youngsters' lack of proficient basic fact skills may be due, in part at least, to weak automatization. The construct of automatization, or automaticity, has applicability to academic skills beyond those previously investigated.
A total of 114 junior-high, middle-grade, and high-school LD teachers responded to a survey concerning various aspects of mathematics disabilities in students presently on their caseload. The most common deficit areas reported included division of whole numbers, basic operations involving fractions, decimals, percent, fraction terminology, multiplication of whole numbers, place value, measurement skills, and language of mathematics. The remediation approach most often utilized consisted of teacher adaptation of mainstream texts, followed by commercial and teacher-made materials. Respondents indicated the need for a commercial mathematics program based on the mainstream mathematics curriculum with an emphasis on systematic, extended practice for students with learning disabilities in mathematics. The results hold implications for teachers working with secondary students with learning disabilities in mathematics.
This investigation focused on validating two feedback routines for use by special education teachers to enhance the performance of students with learning disabilities. One routine (the Feedback Routine) involved teacher-delivered elaborated feedback, the other (the Feedback-Plus-Assistance Routine) consisted of elaborated feedback plus a student-acceptance routine, which included setting goals for the next practice trial. Two experimental designs were employed: one to determine whether teachers could learn the routines, the other to determine the effects on student learning. Dependent measures were (a) teacher and student performance of the routines, (b) student trials to mastery, and (c) student errors across trials. Measures of teacher and student satisfaction and teacher maintenance were also gathered. Results indicated that the special education teachers effectively integrated the routines into their teaching repertoires. Further, the routines significantly reduced the number of student trials to mastery and the number of student errors in practice attempts following feedback sessions. The two routines appeared equally powerful in terms of teacher and student learning; however, the teachers continued to maintain the routine requiring student involvement in goal setting for a longer period.
Modeling through demonstration is a viable teaching strategy for helping students acquire a variety of skills. In the present study, modeling is examined as it relates to learning disabled students' acquisition and generalization of computational skills. Data from a number of learning disabled students in three research locations are presented and summarized to validate the effectiveness of this technique.
The literature offers many suggestions for remediating errors in mathematics, however, research on what teaching methods are most effective for remediation of specific errors is almost nonexistent. This study was designed to investigate the acquisition, generalization, and maintenance of arithmetic borrowing skills among learning disabled students who made systematic inversion errors in subtraction. Blankenship's teaching technique consisting of demonstration plus feedback is simple and applicable to both special and regular class settings. An important contribution of this study is that it considers the long-term effects of the teaching procedure by assessing the generalization and maintenance effects of the students' performance.
The fruits of a decade of research on teaching are discussed in this article. The topics focussed on are time utilization, classroom management, teacher expectations and teacher effectiveness research. The difficulty of translating these findings into practice is discussed. It is argued that the complexities and uniqueness of each classroom make it impossible to follow a simple research‐into‐practice model. Research needs to become more integrative — studying teachers, students, and curriculum simultaneously — and the technology to change practice needs to be better developed.
Three cognitive factors have been suggested as responsible for children's learning problems in simple arithmetic: (a) the encoding of numbers, (b) the efficiency of operation execution, and (c) strategies for carrying out the operations. The study reported here compares these cognitive components across three groups of Grade 5 children: a group having problems in arithmetic, a group having problems in reading, and a control group. The method of subtraction is applied to response times in various arithmetic tasks to yield measures of the cognitive components of single digit addition and subtraction. Results indicate that children with arithmetic learning problems are characterized by very slow operation execution; there is less support for inefficient number encoding, and none for inappropriate strategies. The implications for the design of remedial instruction and for further studies of learning problems are discussed.
Identifying and defining practicable school learning environments that effectively promote educational excellence for all students–including those with special needs and academically-at-risk students who have poor prognoses for schooling success–has been the ongoing goal of researchers, practitioners, and policy makers. Recent research and school improvement efforts have led to advances in defining the specific features affecting individual learning, including those variables determining successful programs, practices, and learning environments. Buttressed by recent studies in effective schooling from virtually all areas of research in human development and learning, this paper emphasizes the importance of systematically broadening the data base in the areas of program description and student outcomes as a way of significantly improving programmatic, implementation, and strategic planning efforts in effective schooling.
Educators have long been concerned with generalization of cognitive interventions. Typically, educators view generalization as a stage of instruction that follows acquisition of a new skill. In an effort to shed light on the problem of generalization with regard to learning strategies, this paper presents generalization as a concept that should be addressed prior to, during, and subsequent to instruction in use of a strategy. A model for generalization is described that emphasizes elements of remedial teacher-, regular teacher-, peer-, and student-mediated techniques for facilitating generalization during all phases of instruction. Studies that illustrate components of the model are reviewed, and future research needs in this area are identified. This article is the first of a two-part series.
Instructional effectiveness is a popular topic, and a crucial one for educating students with mild handicaps. In this paper, the literature on effective instruction is integrated, and implications for instructing students with handicaps are provided. Based on the instructional effectiveness data base and research on quantity and quality of instruction for different categories of students, the authors contend that systematic use of an organized instructional cycle characterized by match, structure, and monitoring is needed to improve academic outcomes for students with mild handicaps.
Traditionally, discrimination has been understood as an active process, and a technology of its procedures has been developed and practiced extensively. Generalization, by contrast, has been considered the natural result of failing to practice a discrimination technology adequately, and thus has remained a passive concept almost devoid of a technology. But, generalization is equally deserving of an active conceptualization and technology. This review summarizes the structure of the generalization literature and its implicit embryonic technology, categorizing studies designed to assess or program generalization according to nine general headings: Train and Hope; Sequential Modification; Introduce to Natural Maintaining Contingencies; Train Sufficient Exemplars; Train Loosely; Use Indiscriminable Contingencies; Program Common Stimuli; Mediate Generalization; and Train “To Generalize”.
What can classroom teachers do to help children who have difficulty with computational skills? Some people may argue that teachers need not worry about computational skills because of the increasing availability of miniaturized calculators. Such an argument assumes that an individual will (1) know what operation should be used; (2) have the financial resources to buy hand calculators; (3) have a hand calculator at his fingertips every time he needs to add, subtract, multiply, or divide; and (4) be able to determine the reasonableness of an answer once it has been derived. These assumptions are difficult to satisfy. The development of basic computational skills clearly should not be neglected; it remains one of the objectives of elementary school mathematics programs.
The aim of the study is to investigate the informal and formal mathematical knowledge of children suffering from "mathematics difficulty" (MD). The research involves comparisons among three groups: fourth-grade children performing poorly in mathematics but normal in intelligence; fourth-grade peers matched for intelligence but experiencing no apparent difficulties in mathematics; and a randomly selected group of third graders. These children were individually presented with a large number of tasks designed to measure key mathematical concepts and skills. The findings suggest that: (1) MD children are not seriously deficient in key informal mathematical concepts and skills; (2) MD children seem to have elementary concepts of base ten notation but experience difficulty in related enumeration skills, particularly when large numbers are involved; (3) MD children's calculational errors often result from common error strategies; (4) MD children display severe difficulty in recalling common addition facts; and (5) in the area of problem solving, MD children are capable of "insightful" solutions and can solve simple forms of word problems, but experience difficulty with complex word problems. MD children are in many respects similar to normal, younger peers; an hypothesis of "essential cognitive normality" is advanced. The only and dramatic exception occurs in the area of number facts. While clinical experience corroborates this finding, its explanation is not evident.
Investigated types of arithmetic errors made by 213 learning disabled 9–18 yr olds. Written responses to multiplication and division problems were analyzed. More than 25% of the errors were due to addition difficulties and 60% of the multiplication errors were due to factors other than proficiency with multiplication facts. Error classifications are discussed, and emphasis is placed on the importance of thorough investigation of errors by teachers. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
The primary purpose of this book is to prepare special education professors and teachers, resource room teachers, remedial education teachers, and regular classroom teachers for the challenges of individualized programming for students with learning or behavioral problems.
Individualized programming requires an understanding of subject matter, assessment, teaching approaches for each content area, instructional activities, seatwork activities, and commercial programs. . . . This text provides a comprehensive, practical text for special education and remedial education methods courses; a resource for special education and remedial education in-service programs; and a handbook for individual teachers.
As a result of feedback from reviewers and users of the second edition, this third edition features some noteworthy changes, including additional coverage of research-based teaching principles with expanded coverage of learning strategies, generalization training, self-monitoring techniques, and techniques for increasing the time students spend on academic tasks. More in-depth discussions of peer tutoring, motivational techniques, teacher coaching, and computer-assisted instruction are presented. Descriptions of tests, software, and materials have been updated throughout this edition. Other significant changes include an expanded coverage of study skills and a discussion of the stages of learning. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Via a dmonstration-plus-modl technique, the equal additions method of subtraction was used to teach subtraction requiring regmuping. A multiple baseline design acmss elementary aged studnts labeled learning disabled was med to evaluate the equal adfitions method. Within subjects, a multiple probe design across problem types was utilized to determine generalization to new problem types and maintenance of trained problem types. Results suggest that instruction in the equal adfitions method of subtraction was effective in pducing an increase in student's computation of subtraction with regmuping. In adfition, all studnts showed some generalization to untrained pblem gpes.
This study examines the mathematical performance of 220 children from 8 years through 17 years of age diagnosed as having learning disabilities. Student records were searched for data indicating performance on standardized test instruments relating to mathematics. Data for the Woodcock-Johnson Psycho-Educational Achievement Battery math subtests and for the IQ scores from the Wechsler Intelligence Scale for Children-Revised were obtained. Comparisons were made among children at different ages and among specific age clusters. Primary attention was directed toward calculations and applications of math concepts and principles. Developmental patterns across the ages studied were discovered. Implications for long-term comprehensive programming are presented.
The influence of a modeling technique on the acquisition of long division by eight students with learning disabilities was studied. The instructional intervention, which included demonstration, imitation, and key guide words, was found to be effective. Initial assessment scores of division problems were 0%. With the application of the intervention, students mastered (2 out of 3 days at 100) each long division skill in minimal time (ranging from 2 to 9 days).
This article details a process to develop and test an instructional program designed to teach students with learning disabilities to solve four kinds of arithmetic story problems. The program was developed in response to the recognition that some students who have adequate reading and computation skills lack the procedural, process, and task-specific knowledge necessary to solve these problems. In the course of developing the unit, the literature on information processing, mathematics education, and instructional theory provided important guidelines for content and teaching approaches. Both single subject and group research designs were employed to test the effectiveness of the problem-solving unit.
This study investigated the effect of an eight-step cognitive strategy on verbal math problem solving performance of six learning disabled adolescents. The cognitive strategy was designed to enable students to read, understand, carry out, and check verbal math problems that are encountered in the general math curriculum at the secondary level. Conditions of the multiple baseline desing included baseline, treatment, generalization, maintenance, and retraining. During treatment, students received strategy acquisition training, strategy application practice, and testing. Visual analysis of the data indicated this eight-step cognitive strategy to be an effective intervention for this sample of students who had deficits in verbal math problem solving. Overall, the students demonstrated impoved performance on twostep verbal math problems. Maintenance and generalization of the strategy were evident. This study has implications for an alternative teaching methodology that focuses on cognitive strategy training to improve the verbal math problem solving of learning disabled students.
The purpose of this study was to explore how student achievement relates to ambitiousness of goal setting and to goal mastery. The subjects were 58 special education students for whom teachers assessed baseline performance and set reading goals employing a standard format. On the basis of the relation between baseline and the anticipated goal performance, students were assigned to goal ambitiousness groups. For 18 weeks, teachers implemented students' goals, end-of-treatment goal mastery was determined, and pretest and posttest achievement scores were entered into multivariate and univariate analyses of covariance, with pretest reading scores as the covariate. Analyses revealed that ambitiousness of goals was associated positively with achievement; goal mastery was not. Implications for goal setting and IEP evaluation procedures are discussed.
In a field that has long been dominated by studies of elementary school--aged children, it is gratifying to see that we now have enough information about interventions with learning-disabled adolescents to fill two Topical Reviews. This month's article is the first of two parts on recent innovations and research on educational programs for LD children in middle- and high--schools. Although the authors note that the study of educational programming for LD adolescents is only in its infancy, it is encouraging to see the wealth of empirical information that is now available in this area. The information provided in this and next month's Topical Review provides a solid starting point, indeed, for the development of effective educational programs that meet a broad variety of goals for adolescents with learning disabilities.-J.K.T.
Mathe-matics instruction for learning disabled students: A review of research
- M A Mastropieri
- T E Scruggs
- S Shiah
Mastropieri, M.A., Scruggs, T.E., & Shiah, S. (1991). Mathe-matics instruction for learning disabled students: A review of research. Learning Disabilities Research & Practice, 6, 89-98.
Students with learning disabilities
- Cd Mercer
Mercer, CD. (1992). Students with learning disabilities (4th ed.). New York: Macmillan.
Strategic math series: Multiplication facts 0-81
- Cd Mercer
- S P Miller
Mercer, CD., & Miller, S.P. (1991). Strategic math series: Multiplication facts 0-81. Lawrence, KS: Edge Enter-prises.