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.     

The role of the working memory and language skills in the prediction of word problem solving in 4- to 7-year-old children

The aim of this study was to analyse the role of verbal and visuo-spatial working memory (WM) and language skills (vocabulary, listening comprehension) in predicting preschool and kindergarten-aged children’s ability to solve mathematical word problems presented orally. The participants were 116 Finnish-speaking children aged 4–7?years. The results showed that verbal WM (VWM) did not have a direct effect on word problems in young children but was indirectly related to word problems through vocabulary and listening comprehension. These results suggest that in young children, VWM resources support language skills which, furthermore, contribute to variation in solving orally presented word problems. The results also showed that visuo-spatial WM had a direct effect on performance in word problems, suggesting that it plays an important role in word problem solving among this age group.     

Deaf college students' comprehension of relational language in arithmetic compare problems

In this study of deaf college students' performance solving compare word problems, relational statements were either consistent or inconsistent with the arithmetic operation required for the solutions. The results support the consistency hypothesis Lewis and Mayer (1987) proposed based on research with hearing students. That is, deaf students were more likely to miscomprehend a relational statement and commit a reversal error when the required arithmetic operation was inconsistent with the statement's relational term (e.g., having to add when the relational term was less than). Also, the reversal error effect with inconsistent word problems was magnified when the relational statement was a marked term (e.g., a negative adjective such as less than) rather than an unmarked term (e.g., a positive adjective such as more than). Reading ability levels of deaf students influenced their performance in a number of ways. As predicted, there was a decrease in goal-monitoring errors, multiple errors, and the number of problems left blank as the reading levels of students increased. Contrary to expectations, higher reading skills did not affect the frequency of reversal errors.     

Handwriting Survey to Determine Grade Standards

Abstract

One purpose of this study was to determine whether a selected number of spelling series used high- frequency words for their lists when compared to a standard measure of word frequency. Another purpose was to determine whether students misspelled high-frequency words in their writing. Data collected from the spelling series were analyzed using a two-factor analysis of variance and the Tukey method of multiple comparisons. Data collected from children’s writing samples were analyzed using at test. Analysis showed differences in the number of high-frequency words among the series and within the grade levels. There also was a difference between the number of misspelled words in children’s writing at Grade 5 but not at Grade 3.     

Visualisation to support children with attention-deficit/hyperactivity disorder learning to solve mathematical word problems: A randomised controlled trial

This randomised controlled trial experimental study compared the effectiveness of using two different types of visualisation – self-constructed visualisation (SCV) and passively received visualisation (PRV) – to help children with attention-deficit/hyperactivity disorder (ADHD) solve mathematical word problems. While SCV refers to drawings that children create to help solve given word problems, PRV refers to pre-made images that accompany word problems. Twenty children with ADHD in Kuwait, aged nine to 11 years, were randomly assigned to either the SCV or PRV group, where they were taught to use either SCV or PRV to solve word problems across 20 daily one-to-one sessions. The results showed that regardless of the visualisation type, children's word problem-solving ability significantly improved. Children with ADHD should thus be encouraged to use visualisation to help make the word problem-solving process more accessible to them.     

Using Self-Generated Drawings to Solve Arithmetic Word Problems

Abstract

In this article, two intervention studies are described that were set up to investigate whether encouraging elementary students to generate drawings of arithmetic word problems facilitates problem-solving performance. The interventions consisted of 60 to 90 min of practice and showed the usefulness of self-generated drawings for solving word problems. The subjects in the first study were first and second graders, and in the second study, fifth graders. The results indicated that the fifth graders improved problem solutions after the intervention, whereas the first and second graders did not. Unlike the first and second graders, the fifth graders generated lots of drawings of word problems. The findings suggest that the nature of the difficulties children experience when solving arithmetic word problems influences their decision to generate drawings.     

Should the Simple View of Reading Include a Fluency Component?

The Simple View of Reading states that reading comprehension is the product of word recognition and listening comprehension. Whereas much research has focused on word recognition accuracy, recent attention has been directed toward word recognition fluency. The current study investigated whether a separate fluency component should be added to the Simple View of Reading. A battery of reading and language measures was administered to 604 children in second, fourth, and eighth grades. Approximately half these children had language and/or nonverbal cognitive impairments in kindergarten, but weighting procedures were used to reduce the potential bias this sampling characteristic may have entailed. Structural equation modeling was used to determine whether fluency accounted for unique variance in reading comprehension after controlling for word recognition accuracy and listening comprehension. Individual profile analyses were conducted to determine the number of individual participants who␣had poor fluency in the spite of good word recognition accuracy and listening comprehension. Results showed that fluency did not account for unique variance in reading␣comprehension and that few individuals had problems in fluency separate from word recognition accuracy or listening comprehension. Thus, it does not appear that a separate fluency component should be added to the Simple View of Reading.     

Pathways to word problem solving: The mediating roles of schema construction and mathematical vocabulary

When solving word problems, many children encounter difficulties in making sense of the information and integrate it into a meaningful schema. This is the fundamental phase on which subsequent problem solution depends. To better understand the processing underlying this fundamental phase, this study examined the roles of schema construction and knowledge of mathematical vocabularies in word problem solving. The participants were 139 Chinese third graders studying in Hong Kong. Path analysis showed that there were two kinds of pathways to word problem solving: language-related and number-related. In particular, reading fluency was related to word problem solving in two mediated language-related pathways: one via schema construction, the other via knowledge of mathematical vocabularies. In the number-related pathway, arithmetic concept was related to word problem solving via knowledge of mathematical vocabularies. These findings highlight the specific roles of schema construction and mathematical vocabulary in word problem solving, thereby providing useful implications of how best to support children in understanding and integrating the information from the problem.     

Predicting Spelling Behaviour in 7 to 11 Year-olds

There is an increasing number of studies which demonstrate that readers in more transparent orthographies than English, such as Italian, Spanish, Turkish, Greek and German have little difficulty in decoding written words, while English children have many more problems. Increasingly, lack of orthographic transparency in English is seen as having a powerful negative effect on the development of reading skills in English-speaking children. There is evidence that English-speaking children who fail to acquire reading skills may fall into two distinct categories: those who would succeed in languages, other than English, that have greater orthographic consistency; and, those who would still have problems even with perfect orthographic transparency. The first, larger, group are let down by the interaction of poor teaching methods and an incomprehensible system of orthography. At Hull University’s Institute for Learning, word factors associated with poor spelling and reading have been identified. We have found three factors which account for the relative ease with which pupils can spell words: frequency of the word in the English language, length of the word, and the presence of “tricky” letters or letter combinations. Data is presented illustrating our predictive model of spelling which enables word difficulty to be calculated from the characteristics of English words. The implications the model has for teaching and learning English is elaborated, with particular reference to the most common 150 words in the English language.     

部件数和字频在汉字识别中的作用

采用词汇判断作业,设置了频率(高频和低频)和部件数(2部件和3部件)两个变量,考察了部件数和字频对汉字加工的影响。结果发现高、低频真字均有部件数效应,字频效应也显著,但字频和部件数不存在交互作用。这表明汉字是以末级部件为心理表征的,部件数对高频和低频汉字的识别都有影响。     

Teaching students with LD to use diagrams to solve mathematical word problems

This study examined the effectiveness of instruction focused on teaching students with learning disabilities (LD) to solve 1- and 2-step word problems of varying types. Three students with LD in Grade 8 participated in the study. During the treatment, students received instruction in diagram generation and a strategy that incorporates diagrams as a part of the procedure to solve word problems. The results indicated that all students improved in the number of diagrams they used and in their ability to generate diagrams. Their word problem solving performance increased. Moreover, the students generated and used diagrams to solve other types of problems. Overall, the students were very satisfied with the instruction and would continue to use the diagrams and the strategy to solve word problems in other classroom settings.     

唐山滦南方言词汇的特点及成因探究

以唐山滦南方言词汇为研究对象,从音节数量、构词语素和构词方式三方面分析了词汇特点。并从语音变化、历史遗留、其他语言或方言影响、认知造词四方面探析了其成因。     

The association between mathematical word problems and reading comprehension

This study aimed to investigate the interplay between mathematical word problem skills and reading comprehension. The participants were 225 children aged 9–10 (Grade 4). The children’s text comprehension and mathematical word problem‐solving performance was tested. Technical reading skills were investigated in order to categorise participants as good or poor readers. The results showed that performance on maths word problems was strongly related to performance in reading comprehension. Fluent technical reading abilities increased the aforementioned skills. However, even after controlling for the level of technical reading involved, performance in maths word problems was still related to reading comprehension, suggesting that both of these skills require overall reasoning abilities. There were no gender differences in maths word problem‐solving performance, but the girls were better in technical reading and in reading comprehension. Parental levels of education positively predicted children’s maths word problem‐solving performance and reading comprehension skills.     

What is the unknown? The ability to identify the semantic role of the unknown from word problems longitudinally predicts mathematical problem solving performance

The current study aimed to investigate children’s difficulties in word problem solving through assessing their ability to mathematize, or to identify the semantic role of the unknown from word problems. Fifth graders (n = 213) were given an advanced word problem reasoning task in which they had to match word problems with schematic diagrams that depict different processes (multiplication versus division) and the unknown being in different semantic roles (e.g., unit size, number of units, or total in an equal group problem). They were also tested on their mathematical problem solving as well as some potential confounding variables (i.e., intelligence and working memory) and mediators. The ability to identify the semantic role of the unknown was shown to be longitudinally predictive of children’s mathematical problem solving performance even after controlling for the effects of covariates and autoregressor. Such a relation was partially mediated by children’s ability to convert word problems into the correct number sentences/equations. The findings not only highlight the importance of unknown identification in mathematical problem solving process, but also provide a practical tool to assess such an ability.     

History of Science in the Physics Curriculum: A Directed Content Analysis of Historical Sources

Although history of science is a potential resource for instructional materials, teachers do not have a tendency to use historical materials in their lessons. Studies showed that instructional materials should be adaptable and consistent with curriculum. This study purports to examine the alignment between history of science and the curriculum in the light of the facilitator model on the use of history of science in science teaching, and to expose possible difficulties in preparing historical materials. For this purpose, qualitative content analysis method was employed. Codes and themes were defined beforehand, with respect to levels and their sublevels of the model. The analysis revealed several problems with the alignment of historical sources for the physics curriculum: limited information about scientists’ personal lives, the difficulty of linking with content knowledge, the lack of emphasis on scientific process in the physics curriculum, differences between chronology and sequence of topics, the lack of information about scientists’ reasoning. Based on the findings of the analysis, it would be difficult to use original historical sources; educators were needed to simplify historical knowledge within a pedagogical perspective. There is a need for historical sources, like Harvard Case Histories in Experimental Science, since appropriate historical information to the curriculum objectives can only be obtained by simplifying complex information at the origin. The curriculum should leave opportunities for educators interested in history of science, even historical sources provides legitimate amount of information for every concepts in the curriculum.     

英语词义发展的转喻解释

对词义发展的原因,传统的词汇语义学偏向于将词义的变化归于历史的、社会的因素。认知语言学认为语言是受认知环境和社会环境制约的,词义发展和变化的内因来源于语言使用者的认知思维,通过人类认知手段（如转喻、隐喻）由一词的基本意义向其他意义延伸。本文拟从认知语言学的角度对词义发展现象进行转喻解释,并讨论其对英语词汇教学的启示。     

Teaching children how to include the inversion principle in their reasoning about quantitative relations

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research on improving children’s understanding and use of the principle of inversion through interventions has only been applied to the solving of a + b − b = ? sums. The main aim of the intervention described in this article was to study the effects of teaching children about the explicit use of inversion as part of the relational calculus needed to solve inverse addition and subtraction problems using a calculator. The study showed that children taught about relational calculus differed significantly from those who were taught numerical procedures, and also that effects of the intervention were stronger when children were taught about relational calculus with mixtures of indirect and direct word problems than when these two types of problem were given to them in separate blocks.     

Learning from physics instruction

Forty high school students participated in a study to investigate learning and remembering from physics instruction. At pretest, students received aptitude tests, an alternate form of achievement test (counterbalanced) and a word association (WA) test. Then the 40 students were divided into instruction (N = 28) and control (N = 12) groups. The control group received all subsequent tests but no instruction. The instruction group received physics instruction over five 2-hour sessions. At the end of each session, the WA test was administered; the alternate form of the achievement test was administered at the end of the last session. The instruction and control groups did not differ significantly in achievement at pretest. The instruction group showed a significant gain from pre- to posttest; no such gain was found for the control group. The instruction and control groups did not differ on the number of responses generated on the WA test at pretesting. The number of responses given by students in the instruction group increased significantly from test 1 (pretest) to test 6 (posttest). For the control group, the number of responses increased initially and then leveled out well below the instruction group data. When WA test responses were tallied for “constrained” responses (i.e., tallying only those concepts found in equations defining the stimulus word), the number of constrained responses increased consistently for the instruction group. For the control group, constrained responses showed an initial increase and then leveled out well below instruction group data. Finally, the aptitude, achievement, and WA data were intercorrelated. This analysis suggests that for students who perform well in solving problems at the end of instruction, the physics concepts became meaningful soon after they were introduced. Subsequently, these students were able to consolidate, in memory, the meaning of the concepts and the interrelations among these concepts in the form of equations. Students not able to solve problems at the end of instruction seemed to be completing the process of acquiring the meaning of the concepts.     

居延汉简否定副词的平面描写

居延汉简否定副词有"不、罔、未、非、莫、毋、勿、無、未尝、未曾"10个,全部承袭先秦汉语而来,虽然有新用法产生,但总体来说稳定性很强.用法方面突出的特点是否定副词的集中化趋势以及否定副词的交叉融合现象.     

Beyond additive and multiplicative reasoning abilities: how preference enters the picture

While previous studies mainly focused on children’s additive and multiplicative reasoning abilities, we studied third to sixth graders’ preference for additive or multiplicative relations. This was investigated by means of schematic problems that were open to both types of relations, namely arrow schemes containing three given numbers and a fourth missing one. In study 1, children had to fill out the missing number, while in study 2, children had to indicate all possibly correct answers among a set of given alternatives. Both studies explicitly showed the existence of a preference for additive relations in some children, while others preferred multiplicative relations. Mainly younger children preferred additive relations, whereas mainly children in upper primary education preferred multiplicative relations. Number ratios also impacted children’s preference, especially in fifth grade. Moreover, the results of study 2 provided evidence for the strength of children’s preference and showed that calculation skills do not coincide with preference, and hence, that preference and calculation skills are two distinct child characteristics. The results of both studies using these open problems resembled previous research results using classical multiplicative or additive word problems. This supports the hypothesis that children’s preferred type of relations may be at play in solving classical word problems as well—besides their abilities—and may hence be an additional factor explaining the mistakes that children make in those word problems. This research line thus seems promising for further research as well as educational practice.