The influence of English-Korean bilingualism in solving mathematics word problems

The purpose of this study was to investigate language related difficulties and the language of cognitive processes of English-Korean bilingual students in solving mathematics word problems. Qualitative case study research methodology was used to collect, analyze, and present data. The principle of purposeful sampling was used to select six English-Korean bilingual students. Different types of bilinguals revealed distinct patterns of difficulties and languages in solving mathematics word problems written in English and Korean. Children in the transition stage that is unstable and changing revealed more difficulties in solving the mathematics word problems overall.This article is a summary of doctoral dissertation under the direction of James W. Wilson at the University of Georgia.     

Effects of consistency and adequacy of language information on understanding elementary mathematics word problems

Two types of elementary mathematics word problems involving different linguistic structures were devised to examine the understanding and solution of these problems by 91 Grade 3, 4, and 5 children divided into “more able” and “less able” subgroups. One task consisted of 12 consistent and 12 inconsistent language problems on the basic processes of addition, subtraction, multiplication and division. Another task consisted of a total of 36 word problems with 12 items each containing adequate, inadequate, and redundant information, respectively, for problem solution. Subsidiary tasks of general ability, vocabulary, reading comprehension, mathematics concepts, reflection on mathematics learning, and working memory were also administered to provide estimates of the contribution of these “nonmathematics” tasks to the solution of elementary mathematics problems. Analyses of variance and covariance of group data showed significant main effects of grade, consistency, and adequacy of linguistic information in problem solution. Word problems containing inconsistent information were more difficult than those with consistent information. Further, word problems containing inadequate and redundant information were more difficult to classify, and for the children to explain, than those items with just enough information. Interviews with 12 individual children provided further insight into their strategies for problem solutions. Both cognitive and developmental perspectives are important for mathematics learning and teaching for children with or without learning disabilities.     

Problem solving in schools and beyond: Transitions from the naive to the neophyte to the master

Techniques emerging from the considerable research on cognitive aspects of survey methodology include various forms of probing and cognitive interviewing. These techniques are used to examine whether respondents' interpretations of self-report items are consistent with researchers' assumptions and intended meanings given the constructs the items are designed to measure. However, although informal procedures are common, such developments have not been systematically applied in educational research. We describe how information derived from the systematic application of cognitive pretesting can contribute to determining the validity—designated cognitive validity—of self-report items. Examples are presented from prominent motivation-related instruments that assess real-world instructional practices, mastery classroom goal structure, and student self-efficacy. The implications and pragmatics of adopting this approach are discussed.     

Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

2493 children aged from 5 to 15 years in 114 classes in 25 primary schools were asked to do a similar set of 22 arithmetic word problems. 1195 of the children were in Grades K through 6 in Victorian schools (Australia) and 1298 were in Grades 4 through 6 in Papua New Guinea (PNG) schools. For both samples the questions were posed in English. This was the first language for most children in the Victorian sample but, for the PNG sample, English was usually the second, third, or even fourth language (even though it is the language of instruction in PNG schools). While the test instrument was based on widely accepted information processing models of how children solve arithmetic word problems, the data obtained were not so much in accord with these models as with psycholinguistic theories on children's acquisition of polarised comparative pairs (like more and less). The data also indicate that children from the two samples used similar strategies and made similar errors, with the order of relative difficulty being the same for both samples, the main factor determining difficulty being the semantic structure of the questions. Differences in performance between corresponding grades from the two samples can be attributed to differences in the degree of English language competence rather than to numerical facility.     

Developing understanding through confronting varying views: The case of solving qualitative physics problems

This study explored high school students' collaborative efforts in solving qualitative physics problems. It aims to investigate whether and how confronting students with varying views help to improve their problem solving skills and develop better understanding of the underlying physics concepts. The varying views were provided to 18 Year 12 students by requiring them (a) to work in dyads on three problems during which they had to consider and confront each other's ideas; and (b) to consider, in a feedback session, multiple solutions to each problem, comparing the solutions with their own and reflecting on their mistakes. The study adopts Marton's emerging theory of awareness as its theoretical underpinning- that the varying views would bring to students' focal awareness and enable them to discern the different critical aspects of the problem situations and this would help them to develop conceptual understanding. The results show that confronting students with varying views have positive effects on students' learning, thus lending support to the theory of awareness.     

Through a glass darkly: A naturalistic study of students' understanding of mathematical word problems

A father is now 20 years older than his son. In 8 years, the father's age will be 5 years more than twice the son's age. Find their present age.Word problems, such as this one, are a perennial source of difficulty for students of school mathematics. Unfortunately, very little is known about why they are problematic, mainly because so little is known about how students understand such problems or the strategies they use in their efforts to solve them. Traditional research into word problems has shed precious little light on this question owing, in no small part, to its almost singular preoccupation with results of pupils' activities—as expressed in some sort of test score, and to its tendency to all but ignore what students actually do when confronted with problems of this kind.This study was carried out as one facet of a larger research project designed to gain more insight into some of the ways in which students understand school mathematics. It focuses on the efforts of one pupil, a twelve-year-old girl in grade seven, to come to terms with solving word problems using an algebraic approach. Strategies associated with both the structured and the unstructured clinical interview were used in order to reveal what was involved in her attempts to make sense of the word problems in her grade seven mathematics textbook.Based on the information gained in the interview, a rational reconstruction of the student's problem-solving strategy is proposed, and compared with the strategies normally prescribed in contemporary school mathematics textbooks. What emerges from this comparison is the finding that, while there appear to be systematic and fundamental differences between the procedures prescribed by the text and those actually used by the pupil in working through certain problems, these differences are undetectable in the finished product; either in the answer itself or in the rough or finished work. What this suggests, among other things, is that if, as educators and/or researchers, we limit our attempts to understand how students go about learning to solve word problems (or how they approach any other part of the school mathematics curriculum, for that matter) to examining what they commit to paper, we are apt to be seriously misled concerning what they genuinely understand and what they fail to understand. In short, if we are to learn more about why pupils experience difficulties with word problems we must begin to pay serious attention to what they say and do as they work their way through them.This study was supported by a Research and Development Grant from the Faculty of Education, Queen's University.     

Realistic considerations in solving problematic word problems: Do Japanese and Belgian children have the same difficulties?

The aim of this study was to compare Japanese and Belgian elementary school pupils' (lack of) activation of real-world knowledge during understanding and solving arithmetic word problems in a school context. The word problem test used in a study by Verschaffel, De Corte, and Lasure (1994) was collectively administered to 91 Japanese fifth graders. Besides standard problems which can be modeled in a straightforward way by one or two basic arithmetic operations with the given numbers, this test contained a series of problematic items which cannot be modeled and solved in such a way, at least if one seriously takes into account the realities of the context evoked by the problem statement. The results of the study revealed that Japanese pupils, similarly to Belgian children, have a strong tendency to neglect commonsense knowledge and realistic considerations during their solution of word problems. Moreover, a comparison of Japanese pupils with and without extra hints aimed at improving the disposition towards more realistic mathematical problem solving revealed that these extra hints had only a small effect.     

Taken out of context: differential processing in contextual and isolated word reading

Three experiments are reported that investigate the cognitive processes underlying contextual and isolated word reading. In Phase 1, undergraduate participants were exposed to 75 target words under three conditions. The participants generated 25 words from definitions, read 25 words in context and read 25 in isolation. In Phase 2, volunteers completed either an explicit recall task (Experiment 1), an implicit word stem completion task (Experiment 2) or both tasks (Experiment 3). Our findings provide converging evidence that contextual and isolated word reading elicit different patterns of cognitive processing. Specifically, Experiments 1–3 demonstrated that words read in context were remembered similarly to words generated from definitions: words from both conditions were recalled more frequently in the surprise memory task and selected less often to complete the word stems in the implicit memory task. The opposite pattern was noted for words read in isolation. Reading in context is discussed in terms of its greater reliance on semantic processing, whereas isolated word reading is discussed in relation to perceptually driven processes.     

谈对语用意义的多层面语境理解

语境是语言形式与语用环境中非语言特征的联系，可分为语言语境和情景语境，语言和语境相互作用产生一定的语用意义，语言认知意义之外的语用意义主要是通过语境实现的，语境中蕴含了语用的多种因素，一方面制约着语言的语用条件，另一方面又提供了语言语用意义生成条件。对各种环境中信息的处理和理解，很大程度上是对语境的分析和认知。     

Impact of authenticity on sense making in word problem solving

The study presented in this paper seeks to investigate the impact of authenticity on the students’ disposition to make necessary real world considerations in their word problem solving. The aim is also to gather information about the extent to which different reasons for the students’ behaviors are responsible for not providing solutions that are consistent with the ‘real’ situations described in the word problems. The study includes both written solutions to word problems and interview data from 161 5th graders. The results show an impact of authenticity on both the presence of ‘real life’ considerations in the solution process and on the proportion of written solutions that were really affected by these considerations. The students’ frequent use of superficial solution strategies and their beliefs about mathematical word problem solving were found to be the main reasons for providing solutions that are inconsistent with the situations described in the word problems.     

The effects of successful versus failure-based cases on argumentation while solving decision-making problems

Solving complex, ill-structured problems may be effectively supported by case-based reasoning through case libraries that provide just-in-time domain-specific principles in the form of stories. The cases not only articulate previous experiences of practitioners, but also serve as problem-solving narratives from which learners can acquire meaning. The current study investigated the effects of different case-types (success, failures) on analogical transfer to similar problems. In the first week, undergraduate sales management students (N = 36) were assigned to different case library treatments (success, failure) and asked to construct a multifaceted argument (initial argument, counterargument, rebuttal) to resolve an ill-structured, decision-making hiring problem. In the following week, students constructed an argument to solve a novel case without the support of the case library. Data analysis revealed the failure-based case library condition produced significantly higher scores on measurements of counterarguments and holistic argumentation scores on both tasks. A discussion of the implications for pedagogy and instructional design are also presented.     

Word learning: An ERP investigation of word experience effects on recognition and word processing

Adults of varying reading comprehension skill learned a set of previously unknown rare English words (e.g., gloaming) in three different learning conditions in which the type of word knowledge was manipulated. The words were presented in one of three conditions: (1) orthography-to-meaning (no phonology); (2) orthography-to-phonology (no meaning); and (3) phonology-to-meaning (no orthography). Following learning, participants made meaning judgments on the learned words, familiar known words, and unpresented (unlearned) rare words while their ERPs were recorded. The behavioral results showed no significant effects of comprehension skill on meaning judgment performance. Contrastingly, the ERP results indicated comprehension skill differences in P600 amplitude; high-skilled readers showed stronger familiarity effects for learned words, whereas less-skilled readers did not distinguish between learned words, familiar words, and unlearned words. Evidence from the P600 and N400 illustrated superior learning of meaning when meaning information was coupled with orthography rather than phonology. These results suggest that the availability of word knowledge (orthography, phonology, and meaning) at learning affects subsequent word identification processes when the words are encountered in a new context.     

Arithmetical word problems: Thinking in the head versus thinking on the table

The paper presents an alternative explanation for the pattern of relative difficulty of arithmetical word problems to that suggested by Riley et al. (1983). Their view is based on the assumption that all symbolic manipulations take place internally. It is argued that for many problems much of the manipulation is accomplished externally.     

Teachers' approaches towards word problem solving: Elaborating or restricting the problem context

This contribution reports about a seven-month long video-based study in two regular Flemish sixth-grade mathematics classrooms. The focus is on teachers' approaches towards problem solving. In our analysis we distinguished between a paradigmatic-oriented (focus on the mathematical structure) and a narrative-oriented (focus on the contextual aspects of a problem) perspective on the problem-solving process. The findings have highlighted that the word problem-solving lessons were more dominated by a paradigmatic than a narrative approach and that interventions in which the relation between the mathematics structure and the realistic constraints of the problem context is addressed, were rare.     

浅析上下文语境因素对俄文广告效果的影响

上下文语境因素对交际的成功与否发挥着重大的作用。本文结合语境理论,简要分析了上下文语境因素对俄文广告效果的影响,并提出了几点更好的实现俄文广告效果的措施。     

Solving word problems: A case of modelling?

According to the researchers who report in this set of papers there are two causes for the phenomenon that primary- and secondary-school students ignore relevant and plausibly familiar aspects of reality in answering word problems. The first cause is the stereotyped character of common word problems. The second cause lies in the classroom climate. In this article, it is argued that the students act sensibly in their situation. Furthermore, it is noted that the use of stereotyped problems and the accompanying classroom climate relate to teacher beliefs about the goals of mathematics education. Therefore, improving the results on problematic word problems will ask for a change in teacher beliefs. Furthermore, a directed effort to change the classroom socio-math norms will be needed. In relation to this, Greer's suggestion for a change towards a modelling perspective is supported. However, what modelling is, is worked out differently. In line with the RME instructional theory, a plea is made for modelling as an activity of organizing, not of translation.     

Problem solving in the rat: Piecemeal acquisition of cognitive maps

The purpose of this experiment was to compare the problem-solving performance of rats allowed to explore either one or two tables of Maier’s three-table-problem apparatus on successive days. The feeding experience and test trial were administered on the day after all tables and runways had been explored in this piecemeal fashion. No rat that explored only one table and runway per day was able to solve the problem, whereas 60% of the rats that explored two tables and their interconnecting runways did solve the problem. All rats that explored the entire apparatus on each exploratory day were able to solve the problem. These data support the notion that animals can conceptually link objects experienced successively into cognitive representations which specify the constant relationships existing between those objects. The existence of such an absolute spatial mechanism makes it unnecessary for an organism to depend upon relative spatial mechanisms such as routes or cues.