The construction of word meaning in a multicultural classroom. Mediational tools in peer collaboration during mathematics lessons

In this article, we examine the construction of word meaning by students during collaborative activities in a multicultural classroom at a Dutch primary school. The analysis is based on recordings of student talk in small groups of four or five students during mathematics lessons. Difficulties with specific terms and expressions frequently arose during group work. Students could easily ask each other about the meaning of difficult words as a part of the collaborative activities they were accustomed to. In the groups with both Dutch and minority children, the minority students addressed their Dutch classmates as language experts. Conversations about language difficulties also occurred in the groups with only minority students. The conversations about word meaning revealed four patterns: (1) ignoring a question about the meaning of a word, (2) showing the meaning using gestures, (3) explaining, or (4) discussing word meaning. In none of the cases were the language problems solved by referring to the everyday meaning of the word. Instead, the conversations focussed directly on the specialised meaning which the word had in the context of the mathematics lesson. This means that the children used the mathematical discourse as a mediational tool for constructing a mathematical meaning of the words.     

Investigating the Unique Predictors of Word-Problem Solving Using Meta-Analytic Structural Equation Modeling

The purpose of the present study is to clarify the contributions of cognitive skills (nonverbal reasoning, language comprehension, working memory, attention, processing speed) and academic skills (mathematics facts retrieval, mathematics computation, mathematics vocabulary, reading comprehension) in performing mathematics word problems among elementary school students. With the two-stage meta-analytic structural equation modeling approach, I synthesized 112 correlation matrices from 98 empirical studies (N?=?111,346) and fitted the hypothesized partial mediation model. Overall, path analysis indicated that language comprehension, working memory, attention, mathematics vocabulary, and mathematics computation were unique predictors of word-problem solving. Subgroup analysis demonstrated different unique predictors for younger and older students to perform word problems (K-2nd grades versus 3rd–5th grades). Implications, limitations, and future directions are discussed.

     

Does immigration background matter? How teachers’ predictions of students’ performance relate to student background

Accurate teacher evaluations of student performance are crucial for effective teaching. This study examined whether students’ immigration and language background affect teachers’ evaluations. Multilevel analyses tested whether teachers overestimate the performance of immigrant relative to that of non-immigrant students. As part of the German PISA 2003 assessment, 305 teachers predicted the performance of seven of their students on two mathematics problems of different linguistic complexity. Results revealed an interaction effect of students’ language background and linguistic complexity of the problem on teachers’ predictions. Teachers overestimated the performance of bilingual students more than the performance of monolingual immigrant or non-immigrant students on a linguistically complex problem. Teachers need to consider both students’ language background and linguistic demands of the material used to appropriately support bilingual students.     

Australian Vietnamese Students Learning Mathematics: High Ability Bilinguals and Their Use of Their Languages

Bilingual students have, at times, been thought to be at a disadvantage in learning mathematics because of an assumed interference between their two languages. Earlier research, confirmed again in this study, shows that this is a naive view to take. Although some bilingual students do have a harder time, others seem to be at an advantage. This study explores the use that bilingual students who are succeeding in mathematics make of their two languages. These students seem to have better metalinguistics skills that allow them to self-correct when solving problems, and are perhaps more confident in their approach to solving difficult problems. It also appears that students in this study switched between languages in early years of schooling, but only used English by the time they were completing elementary school. This project was supported by an Australian Research Council Large Grant, an Australian Research Council Small Grant, and by internal research grants from Australian Catholic University.     

Overcoming Obstacles to Understanding and Solving Word Problems in Mathematics

ABSTRACT The trouble students have with solving word problems often comes from the difficulty in understanding the problem structure embedded in the problem text. This research was conducted to study the interactive effects of learner and instructional variables on understanding and solving word problems among Filipino‐English bilingual grade school students in the Philippines. The results showed better understanding and solution performance (1) when problems were written in the students’ first language, (2) when the problems were re‐worded to state more explicitly the relationship among the known and unknown quantities, (3) for students in higher levels of schooling and (4) for students with higher levels of academic achievement. The effects of re‐wording were mediated by the effects of the language; and the effects of language also varied between the highand low‐achievement students. In most cases where the students’ performance improved, the improvement seems to be due to better comprehension of the text (or fewer basic comprehension errors); except with the improvement due to re‐wording, which seems to be due to fewer misinterpretation errors. The results of the study provide further support for earlier findings, but also extend and modify the theoretical ramifications of the earlier findings and point to some important implications for educational practice.     

Reading,syntactic, and working memory skills of bilingual Portuguese-English Canadian children

The purpose of the study was to examine the nature of language, memory, and reading skills of bilingual students and to determine the relationship between reading problems in English and reading problems in Portuguese. The study assessed the reading, language, and memory skills of 37 bilingual Portuguese-Canadian children, aged 9–12 years. English was their main instructional language and Portuguese was the language spoken at home. All children attended a Heritage Language Program at school where they were taught to read and write Portuguese. The children were administered word and pseudoword reading, language, and working memory tasks in English and Portuguese. The majority of the children (67%) showed at least average proficiency in both languages. The children who had low reading scores in English also had significantly lower scores on the Portuguese tasks. There was a significant relationship between the acquisition of word and pseudoword reading, working memory, and syntactic awareness skills in the two languages. The Portuguese-Canadian children who were normally achieving readers did not differ from a comparison group of monolingual English speaking normally achieving readers except that the bilingual children had significantly lower scores on the English syntactic awareness task. The bilingual reading disabled children had similar scores to the monolingual reading disabled children on word reading and working memory but lower scores on the syntactic awareness task. However, the bilingual reading disabled children had significantlyhigher scores than the monolingual English speaking reading disabled children on the English pseudoword reading test and the English spelling task, perhaps reflecting a positive transfer from the more regular grapheme phoneme conversion rules of Portuguese. In this case, bilingualism does not appear to have negative consequences for the development of reading skills. In both English and Portuguese, reading difficulties appear to be strongly related to deficits in phonological processing.     

Language and Mathematical Performance: a Comparison of Lower Secondary School Students with Different Level of Mathematical Skills

This study investigated the connection between language (i.e., word comprehension, reading comprehension and spelling skills) and mathematical performance. The sample consisted of grade nine students (N?=?810) in 14 lower secondary schools in the Swedish speaking areas of Finland. Standardized tests for reading and writing skills, and mathematical performance were used. Based on the mathematics test the students were categorized into eight performance groups. Many students had problems in both mathematics and language performance. On the whole data level reading skills were a powerful predictor for math performance, the reading factor explained 52% of the variance in the model. Hence, the reading skills focusing on understanding of the text are important in solving mathematical tasks at the end of compulsory school.     

Effectiveness of bilingual education in Cambodia: a longitudinal comparative case study of ethnic minority children in bilingual and monolingual schools

There is little research in the developing countries of South East Asia on the effectiveness of bilingual education programmes that use first language instruction for ethnic minority children. This study investigated the effectiveness of a bilingual education programme involving ethnic minority children in Cambodia by comparing their performance in mathematics, Khmer literacy and oral Khmer to their ethnic minority peers whose education is in the national language only. The findings show that students in the bilingual schools performed better in mathematics than their peers in the monolingual schools, but the differences in Khmer literacy and oral Khmer test scores were statistically insignificant. The study suggests that bilingual education using first language instruction could benefit academic development among ethnic minority students in Cambodia, at least in terms of mathematics achievement.     

基于应用题解题障碍的认知工具模式设计

通过整理近年来关于小学数学应用题的相关研究,总结学生在应用题解题过程中普遍存在的认知障碍,结合现有解题系统分析研究各类型的认知工具对小学应用题解题障碍的支持,为帮助学生选择合适的认知工具提供借鉴,并在此基础上提出基于问题解决障碍的小学数学应用题解题系统的模型构建,为新的认知工具的设计开发提供新的角度。     

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

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

The effects of using drawings in developing young children’s mathematical word problem solving: A design experiment with third-grade Hungarian students

The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students?? knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment involved five experimental and six control classes (N?=?106 and 138, respectively) of third-grade students. The experiment comprised 20 lessons with 73 word problems, providing a systematic overview of the basic word problem types. Teachers of the experimental classes received a booklet containing lesson plans and overhead transparencies with different types of visual representations attached to the word problems. Students themselves were invited to make drawings for each task, and group work and teacher-led discussion shaped their beliefs about the role of visual representations in word problem solving. The effect sizes of the experiment were calculated from the results of two tests: an arithmetic skill and a word problem test, and the unbiased estimates for Cohen??s d proved to be 0.20 and 0.62. There were significant changes also in experimental group students?? beliefs about mathematics. The experiment pointed to the possibility, feasibility, and importance of learning about visual representations in mathematical word problem solving as early as in grade?3 (around age 9?C10).     

Mathematics word problem solving for deaf students: a survey of practices in grades 6-12

One hundred and thirty-three mathematics teachers of deaf students from grades 6-12 responded to a survey on mathematics word problem-solving practices. Half the respondents were teachers from center schools and the other half from mainstream programs. The latter group represented both integrated and self-contained classes. The findings clearly show that regardless of instructional setting, deaf students are not being sufficiently engaged in cognitively challenging word problem situations. Overall, teachers were found to focus more on practice exercises than on true problem-solving situations. They also emphasize problem features, possibly related to concerns about language and reading skills of their students, rather than analytical and thinking strategies. Consistent with these emphases, teachers gave more instructional attention to concrete visualizing strategies than to analytical strategies. Based on the results of this study, it appears that in two of the three types of educational settings, the majority of instructors teaching mathematics and word problem solving to deaf students lack adequate preparation and certification in mathematics to teach these skills. The responses of the certified mathematics teachers support the notion that preparation and certification in mathematics makes a difference in the kinds of word problem-solving challenges provided to deaf students.     

跨文化数学教育研究综述

从我国民族地区数学课程教材改革、双语教学、学生数学学习心理研究、跨文化数学教育理论研究与方法等方面综述了我国近20年来跨文化数学教育研究取得的成就。目前,对我国少数民族学生数学学习心理研究已达到较高水准,数学课堂的跨文化教育实验研究——中小学＂数学情境与提出问题＂教学实验颇具中国特色并在实践层面产生重要影响。进一步还需研究构建符合少数民族学生思维特点的数学课程体系,编写民族文字数学教材,加强少数民族双语数学教学研究与高素质双语教师队伍建设。     

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

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

The Effects of Paraphrasing Interventions on Problem‐Solving Accuracy for Children at Risk for Math Disabilities

The purpose of this study was to examine effectiveness of paraphrasing interventions on mathematics word problem‐solving accuracy in third grade children (N = 72) at risk for mathematics disabilities (MD). Three instructional conditions directed students’ attention through paraphrasing, via writing, to different propositions within word problems. Students were randomly assigned to one of four intervention conditions: paraphrase question propositions (restate), paraphrase relevant propositions (relevant) and paraphrase all propositions (complete) or an untreated control. A mixed ANCOVA indicated that paraphrasing relevant and complete propositions significantly increased posttest accuracy when compared to the control and restate condition. Results from the study provide support for the effectiveness of paraphrasing interventions that directs students to restate/paraphrase propositions of mathematics word problems relative to the control condition.     

Learning to solve addition and subtraction word problems in English as an imported language

This paper reports an intervention phase of a design study aimed to assist second-grade Filipino children in solving addition word problems in English, a language they primarily encounter only in school. With Filipino as the medium of instruction, an out-of-school pedagogical intervention providing linguistic and representational scaffolds was implemented with 17 children. Pre-intervention, children experienced linguistic difficulties and were limited to conceptualising and solving simple additive structures. Post-intervention interviews revealed improved performance and understanding of more complex structures, but only when linguistic difficulties were minimised. The study identified socially and culturally driven barriers to learning: superficial strategies, children’s engagement, and learning in an urban poor context.     

Do children with mathematics learning disability in Hong Kong perceive word problems differently?

The current study aimed at identifying the difficulties experienced by children with mathematics learning disability (MLD) in the problem representation phase of arithmetic word problem solving using a novel problem types identification task. An MLD group (n = 66) and a typically achieving control group (n = 139) were recruited for an assessment on problem type identification as well as some domain-general and mathematics-related cognitive abilities. Results from ANCOVA showed that the MLD group scored significantly lower than the typically achieving control group on this assessment, after controlling for the effect of cognitive correlates, reading achievement and arithmetic performance. Furthermore, this assessment significantly predicted MLD membership even after taking children's arithmetic competency into account. The current study confirmed the difficulties in problem representation of arithmetic word problems experienced by students with MLD and provided evidence for the need to introduce schema instructions in mathematics classes.     

关于双语教学热点问题的冷静思考

双语教学指的是用2种语言作为教学媒介语,通过学习科目知识（例如,数学课、物理课以及其他专业基础课或专业课等）来达到掌握该语言的目的.随着中国对外开放步伐的加快,双语教学的学科范围日渐扩大.但很多学校在双语教学的试行过程中,存在一些错误做法.为此,我们要特别提醒：双语教学切忌走入误区.双语教学的理论定位和实践定位都是相当高的,双语教学对学校、师资、课程、教材和学生都有着极高的要求.双语教学在全国各院校如火如荼的开展起来,这对于推动社会进步,促进经济发展,提高学生综合素质和办学效果无疑是一件很有意义的事情,但需要广大教育工作者对双语教学热点问题进行冷静思考.双语教学需要我们大胆实践,勇于探索,认真总结,困难面前勇者胜.     

One speaker,two languages: Learning opportunities in the mathematics classroom

The data reported in this article come from a large project whose goal was to explore how Latin American students in Catalonia, Spain use their two languages—Spanish and Catalan—to support their learning of mathematics in small groups with other students who are Spanish- or Catalan-dominant. For 5 years, lessons from bilingual mathematics classrooms in three public secondary schools were video-recorded and transcribed. In the presentation of findings, I discuss three language practices that emerged from the analyses of several classroom instances as follows: (1) caution with mathematical vocabulary, (2) invention of terms, and (3) word-for-word translation. One example is chosen to represent each practice and some of its situated effects. The first two examples support the view that the experience of language difficulties, either real or presumed, contributes to generating opportunities that may be beneficial to mathematics learning. The third example, where the focus on the mathematics is hindered, points to a contrasting finding. Unlike other studies in the field, which have reported the difficulties and obstacles that arise in learning and teaching due to bilingualism in the classroom, I propose a change of focus through the conceptualization of language as a potential for thinking and doing, and particularly for learning and teaching mathematics.     

Language Proficiency and the Relation to Word-Problem Performance in Emergent Bilingual Students with Mathematics Difficulties

As students enter the upper elementary grades, word problems become a main component of mathematics instruction, increasing in complexity as students advance through the curriculum. For students identified as emergent bilinguals with mathematics difficulty (MD), the linguistic complexity inherent in word problems may serve as a barrier to word-problem proficiency. The current study investigated the potential relation between academic English proficiency and word-problem outcomes for emergent bilinguals with MD. After analyzing data from 241 third-grade students, results indicated students who participated in an evidence-based word-problem intervention outperformed students who did not receive the intervention. Moreover, students’ academic English-language proficiency scores in the domains of reading and writing positively correlated with higher scores on a measure of word-problem solving.