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Language plays an important role in word problem solving. Accordingly, the language in which a word problem is presented could affect its solution process. In particular, East-Asian, non-alphabetic languages are assumed to provide specific benefits for mathematics compared to Indo-European, alphabetic languages. By analyzing students’ eye movements in a cross-linguistic comparative study, we analyzed word problem solving processes in Chinese and German. 72 German and 67 Taiwanese undergraduate students solved PISA word problems in their own language. Results showed differences in eye movements of students, between the two languages. Moreover, independent cluster analyses revealed three clusters of reading patterns based on eye movements in both languages. Corresponding reading patterns emerged in both languages that were similarly and significantly associated with performance and motivational-affective variables. They explained more variance among students in these variables than the languages alone. Our analyses show that eye movements of students during reading differ between the two languages, but very similar reading patterns exist in both languages. This result supports the assumption that the language alone is not a sufficient explanation for differences in students’ mathematical achievement, but that reading patterns are more strongly related to performance.

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... However, when research investigated these characteristics, isolated influences are often analyzed (Daroczy et al. 2015). In contrast, we assume that the variety of these characteristics causes manifold interactions in CWP solving, for example in solution strategies (Leiss et al. 2019;Strohmaier et al. 2020). Thus, this paper aims to address a broad selection of cognitive abilities-i.e., spatial, verbal, numerical, and general reasoning abilities-as well as gender, and investigates how they can jointly account for individual differences in CWP performance. ...

... They may further gain importance with increasing linguistic complexity of the mathematical content that has to be processed. For (complex) word problem solving in particular, verbal abilities are assumed to contribute to task performance (Abedi 2006;Boonen et al. 2016;Daroczy et al. 2015;Leiss et al. 2019;Strohmaier et al. 2020). ...

... Although the items were designed to assess mathematical literacy of 15-yearold students, choosing items of above-average difficulty (i.e., given their item characteristics in the PISA field trials, these items can be considered rather complex in terms of item difficulty) resulted in an overall solution rate of 56.6% for our sample (see the "Gender differences in cognitive abilities" section), indicating that the items reflected a reasonable challenge for undergraduate students. Similar observations had been previously reported (Ehmke et al. 2005;Strohmaier et al. 2019Strohmaier et al. , 2020. ...

This study analyzed the relative importance of different cognitive abilities for solving complex mathematical word problems (CWPs)—a demanding task of high relevance for diverse fields and contexts. We investigated the effects of spatial, verbal, numerical, and general reasoning abilities as well as gender on CWP performance among N = 1282 first-year university engineering students. Generalized linear mixed models unveiled significant unique effects of spatial ability, β = 0.284, verbal ability, β = 0.342, numerical ability, β = 0.164, general reasoning, β = 0.248, and an overall gender effect in favor of male students, β = 0.285. Analyses revealed negligible to small gender effects in verbal and general reasoning ability. Despite a gender effect in spatial ability, d = 0.48, and numerical ability, d = 0.30—both in favor of male students—further analyses showed that effects of all measured cognitive abilities on CWP solving were comparable for both women and men. Our results underpin that CWP solving requires a broad facet of cognitive abilities besides mere mathematical competencies. Since gender differences in CWP solving were not fully explained by differences in the four measured cognitive abilities, gender-specific attitudes, beliefs, and emotions could be considered possible affective moderators of CWP performance.

... showed that students with a pattern of eye movements that is typical for struggling reading also performed worse in the word problems (Strohmaier, Schiepe-Tiska, Chang, et al., 2020). ...

... Our results support the assumption that these affective-motivational outcomes are relevant when addressing complex word problem solving, as they were uniquely connected to eye movements and thus, their solution process. This indicates that for example, students with higher mathematical anxiety approach complex word problem differently, irrespective of their mathematical abilities (Strohmaier, Schiepe-Tiska, Chang, et al., 2020;Strohmaier et al., 2017). Again, this calls for a comprehensive view on word problem solving that focuses not solely on the cognitive mathematical aspects of word problem solving but takes into account how affective and motivational student characteristics influence the way that complex word problems are approached and solved. ...

... Again, this calls for a comprehensive view on word problem solving that focuses not solely on the cognitive mathematical aspects of word problem solving but takes into account how affective and motivational student characteristics influence the way that complex word problems are approached and solved. For example, students with negative affective-motivational characteristics might tend to apply a direct translation strategy in order not to miss any crucial information, which might in turn hinder them in building a comprehensive problem model (Strohmaier, Schiepe-Tiska, Chang, et al., 2020). ...

Word problems are a vital part of mathematics education. In the last decades, the goals of mathematics education have shifted towards real-world applications, modelling, and a functional use of mathematics. Complex word problems are a feasible tool for the learning and assessment of these mathematical competencies. They typically include both functional and redundant contextual information, substantial amounts of text, a complex syntax, and multiple representations. Thus, complex word problems provide specific challenges to both students and researchers. In particular, linguistic factors play a pivotal role in complex word problem solving, and mathematical thinking and reading are tightly interwoven in the solution process.
In the research project presented here, the method of eye tracking was used to address a number of these challenges. Knowledge about the process of word problem solving was integrated with previous research on eye movements, taking into account both mathematical thinking and processes of reading as a foundation for a meaningful interpretation of global measures of eye movements during complex word problem solving. This integration had not been utilized in this context before. Based on these considerations, it was investigated how eye movements were associated with solution strategies and how they were related to per-formance as well as motivational-affective variables. Moreover, the approach provided the possibility to directly compare complex word problem solving in two fundamentally different languages, namely Chinese and German.
This dissertation summarizes two publications that report a total of three studies. The studies all analyzed eye movements during complex word problem solving. The first (N = 17) and second study (N = 42) are reported in Paper A and included the development and evaluation of appropriate global measures of eye movements. Further, they investigated their associa-tion with task difficulty and students’ performance. The third study (N = 139) reported in Paper B compared word problem solving in Chinese and German and associated emerging reading patterns with achievement, mathematical self-concept, mathematics anxiety and flow experience. Furthermore, studies that were associated with the research project but were not part of this dissertation are briefly discussed.
The results indicated that patterns of eye movements were associated with the solution pro-cess of complex word problems and with cognitive and motivational-affective outcomes. Moreover, this association is very similar throughout Chinese and German. Therefore, the approach presented here provides specific possibilities to analyze and interpret complex word problems and offers unique insights into the close interplay between reading and mathematics.

... Problem solvers who utilize this second approach deliberately construct a situation model of the situation described in the text and are therefore able to detect inconsistencies between their situation model and the given information by applying their solution process, a practice that is necessary for correctly validating the solution. In line with these results, Strohmaier et al. (2020) identified eye movements corresponding with these two approaches, plus a third approach that characterized readers who struggled while solving word problems. According to Strohmaier et al., utilizing the direct-translation strategy manifested in "a very linear and intense reading pattern with long mean fixation durations, short saccades and few regressions"(p. ...

... On the other hand, the problem model strategy manifested in a shorter mean fixation duration and a high frequency of regressions. The third pattern Strohmaier et al. (2020) identified, which was linked to struggling readers, consisted of shorter saccades, higher fixation counts, as well as a higher regression count, resulting in overall longer reading times. ...

Text comprehension is a key aspect to consider when designing modelling problems. One important feature of mathematical problems is where the question is placed in the text. We present a theoretical background on text comprehension and modelling problems, and we discuss the pros and cons of placing the question before the text rather than placing it after the text. A review of the research revealed that placing the question before the text is more likely to result in improved comprehension. Further, we propose consequences for task design and future research.

With its context-independent rules valid in any setting, mathematics is considered to be the champion of abstraction, and for a long time human mathematical reasoning was thought to follow nothing but the laws of logic. However, the idea that mathematics is grounded in nature has gained traction over the past decades, and the context-independency of mathematical reasoning has come to be questioned. The thesis we defend concerns the role played by general, non-mathematical knowledge on individuals' understanding of numerical situations. We propose that what we count has a crucial impact on how we count, in the sense that human's representation of numerical information is dependent on the semantic context in which it is embedded. More specifically, we argue that general, non-mathematical knowledge about the entities described in a mathematical word problem can shape its interpretation and foster one of two representations: either a cardinal encoding, or an ordinal encoding. After introducing a new framework of arithmetic word problem solving accounting for the interactions between mathematical knowledge and world knowledge in the encoding, recoding and solving of arithmetic word problems, we present a series of 16 experiments assessing how world knowledge about specific quantities can promote one of two problem representations. Using isomorphic arithmetic word problems involving either cardinal quantities (weights, prices, collections) or ordinal quantities (durations, heights, number of floors), we investigate the pervasiveness of the cardinal-ordinal distinction in a wide range of activities, including problem categorization, problem comparison, algorithm selection, problem solvability assessment, problem recall, sentence recognition, drawing production and transfer of strategies. We gather data using behavioral measures (success rates, algorithm use, response times) as well as eye tracking (fixation times, saccades, pupil dilation), to show that the difference between problems meant to foster either a cardinal or an ordinal encoding has a far-reaching influence on participants from diverse populations (N = 2180), ranging from 2nd graders and 5th graders to lay adults, expert mathematicians and math teachers. We discuss the general educational implications of these effects of semantic (in)congruence, and we propose new directions for future research on this crucial issue. We conclude that these findings illustrate the extent to which human reasoning is constrained by the content on which it operates, even in domains where abstraction is praised and trained.

The paper informs about eye-tracking and its potental for mathematics education and discusses its challenges as well as current and future trends of eye-tracking research in mathematics education.

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.

Eye tracking is an increasingly popular method in mathematics education. While the technology has greatly evolved in recent years, there is a debate about the specific benefits that eye tracking offers and about the kinds of insights it may allow. The aim of this review is to contribute to this discussion by providing a comprehensive overview of the use of eye tracking in mathematics education research. We reviewed 161 eye-tracking studies published between 1921 and 2018 to assess what domains and topics were addressed, how the method was used, and how eye movements were related to mathematical thinking and learning. The results show that most studies were in the domain of numbers and arithmetic, but that a large variety of other areas of mathematics education research was investigated as well. We identify a need to report more methodological details in eye-tracking studies and to be more critical about how to gather, analyze, and interpret eye-tracking data. In conclusion, eye tracking seemed particularly beneficial for studying processes rather than outcomes, for revealing mental representations, and for assessing subconscious aspects of mathematical thinking.

Blink rate is a behavioral index highly correlated with frontostriatal dopaminergic activity. The present research was aimed at studying the modulation of spontaneous blink rate in function of the increasing attentional load induced by the Mackworth Clock Test. Since blinking interferes with sensory processing, we expected a decreasing blink rate with increasing attentional demand. Three tasks of 7-min each and different difficulties were administered: the Mackworth had a red dot moving in a circle with intervals varying from 500ms, 350ms to 200ms, corresponding to increasing task difficulty. Participant had to detect the rare jumps of one position by the red dot (targets). The blink rate was recorded from thirty-three female students starting from vertical oculogram recording of the right eye. The time course of blink rate across the 7-min task was also analyzed to test the hypothesis that fatigue arises also during brief tasks depending on the difficulty level. Results showed that the Hard task (200ms dot intervals) was associated with greater percentage of missed targets, faster response times and smaller blink rates with respect to the Medium and Easy ones. Analysis of the time course within the task revealed an increase of blink rate, indexing larger fatigue, starting in the 4th minute, independent from the difficulty level. In addition, trial-by-trial analysis showed that under strong attentional demand dopamine-related blink activity was inhibited throughout the whole task. Results point to the use of blink rate as an ecological index of dopaminergic component of attentional load and fatigue and revealed how human attentiondrops after relatively brief intervals of about 4min.

Keith Rayner’s extraordinary scientific career revolutionized the field of reading research and had a major impact on almost all areas of cognitive psychology. In this article, we review some of his most significant contributions. We begin with Rayner’s research on eye movement control, including the development of paradigms for answering questions about the perceptual span and its relationship to attention, reading experience, and linguistic variables. From there we proceed to lexical processing, where we summarize Rayner’s work on effects of word frequency, length, predictability, and the resolution of lexical ambiguity. Next, we turn to syntactic and discourse processing, covering the well-known garden-path model of parsing and briefly reviewing studies of pronoun resolution and inferencing. The next section shifts from language to visual cognition and reviews research which makes use of eye movement techniques to investigate object and scene processing. Next, we summarize Rayner and colleagues’ approach to computational modeling, with a description of the E-Z Reader model linking attention and lexical processing to eye movement control. The final section discusses the issues Rayner and his colleagues were focused on most recently and considers how Rayner’s legacy will continue into the future.

Within the field of mathematics education, the central role language plays in the learning, teaching, and doing of mathematics is increasingly recognized, but there is not agreement about what this role (or these roles) might be or even about what the term 'language' itself encompasses. In this issue of ZDM we have compiled a collection of scholarship on language in mathematics education research, representing a range of approaches to the topic. In this introduction we outline a categorisation of ways of conceiving of language and its relevance to mathematics education, the theoretical resources drawn upon to systematise these conceptions, and the methodological approaches employed by researchers. We will also identify some outstanding issues and questions and suggest some ways of building upon the diversity in order to strengthen the coherence of the field and the utility of its outcomes.

Word problems (WPs) belong to the most difficult and complex problem types that pupils encounter during their elementary-level mathematical development. In the classroom setting, they are often viewed as merely arithmetic tasks; however, recent research shows that a number of linguistic verbal components not directly related to arithmetic contribute greatly to their difficulty. In this review, we will distinguish three components of WP difficulty: (i) the linguistic complexity of the problem text itself, (ii) the numerical complexity of the arithmetic problem, and (iii) the relation between the linguistic and numerical complexity of a problem. We will discuss the impact of each of these factors on WP difficulty and motivate the need for a high degree of control in stimuli design for experiments that manipulate WP difficulty for a given age group.

The study explored the contribution of working memory to mathematical word problem solving in
children. A total of 69 children in grades 2, 3 and 4 were given measures of mathematical problem
solving, reading, arithmetical calculation, fluid IQ and working memory. Multiple regression
analyses showed that three measures associated with the central executive and one measure
associated with the phonological loop contributed unique variance to mathematical problem solving
when the influence of reading, age and IQ were controlled for in the analysis. In addition, the animal
dual-task, verbal fluency and digit span task continued to contribute unique variance when the effects
of arithmetical calculation in addition to reading, fluid IQ, and age were controlled for. These findings
demonstrate that the phonological loop and a number of central executive functions (shifting,
co-ordination of concurrent processing and storage of information, accessing information from
long-term memory) contribute to mathematical problem solving in children.

Holmqvist, K., Nyström, N., Andersson, R., Dewhurst, R., Jarodzka, H., & Van de Weijer, J. (Eds.) (2011). Eye tracking: a comprehensive guide to methods and measures, Oxford, UK: Oxford University Press.

When comparing Chinese and English language, large differences in orthography, syntax, semantics, and phonetics are found.
These differences may have consequences in the processing of mathematical text, yet little consideration is given to them
when the mathematical abilities of students from these different cultures are compared. This paper reviews the differences
between English and Mandarin Chinese language, evaluates current research, discusses the possible consequences for processing
mathematical text in both languages, and outlines future research possibilities.

The concept of flow is briefly reviewed and several theoretical and methodological problems related to flow research are discussed.
In three studies, we attempted to avoid these problems by measuring the experience of flow in its components, rather than
operationally defining flow in terms of challenge and skill. With this measure, we tested the assumption that experience of
flow substantially depends on the balance of challenge and skill. This assumption could only be partially supported, and,
as expected, this relationship was moderated by the (perceived) importance of the activity and by the achievement motive.
Furthermore, flow predicted performance in two of the three studies.

Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate final questions to incomplete word problems, and (3) modeling performance patterns using a computer simulation. Solution performance was found to be systematically related to recall and question generation performance. Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution “errors” were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error patterns were successfully simulated by manipulating a computer model's language comprehension strategies, as opposed to its knowledge of logical set relations.

The process of fixation identification—separating and labeling fixations and saccades in eye-tracking protocols—is an essential part of eye-movement data analysis and can have a dramatic impact on higher-level analyses. However, algorithms for performing fixation identification are often described informally and rarely compared in a meaningful way. In this paper we propose a taxonomy of fixation identification algorithms that classifies algorithms in terms of how they utilize spatial and temporal information in eye-tracking protocols. Using this taxonomy, we describe five algorithms that are representative of different classes in the taxonomy and are based on commonly employed techniques. We then evaluate and compare these algorithms with respect to a number of qualitative characteristics. The results of these comparisons offer interesting implications for the use of the various algorithms in future work.

It is hypothesized that the number, position, size, and duration of fixations are functions of the metric used for dispersion in a dispersion-based fixation detection algorithm, as well as of the threshold value. The sensitivity of the I-DT algorithm for the various independent variables was determined through the analysis of gaze data from chess players during a memory recall experiment. A procedure was followed in which scan paths were generated at distinct intervals in a range of threshold values for each of five different metrics of dispersion. The percentage of points of regard (PORs) used, the number of fixations returned, the spatial dispersion of PORs within fixations, and the difference between the scan paths were used as indicators to determine an optimum threshold value. It was found that a fixation radius of 1 degrees provides a threshold that will ensure replicable results in terms of the number and position of fixations while utilizing about 90% of the gaze data captured.

Due to the continuous increase in the number of countries participating in international comparative assessments such as TIMSS and PISA, ensuring linguistic and cultural equivalence across the various national versions of the assessment instruments has become an increasingly crucial challenge. For example, 58 countries participated in the PISA 2006 Main Study. Within each country, the assessment instruments had to be adapted into each language of instruction used in the sampled schools. All national versions in languages used for 5 per cent or more of the target population (that is, a total of 77 versions in 42 different languages) were verified for equivalence against the English and French source versions developed by the PISA consortium. Information gathered both through the verification process and through empirical analyses of the data are used in order to adjudicate whether the level of linguistic equivalence reached an acceptable standard in each participating country. The paper briefly describes the procedures typically used in PISA to ensure high levels of translation/adaptation accuracy, and then focuses on the development of the set of indicators that are used as criteria in the equivalence adjudication exercise. Empirical data from the PISA 2005 Field Trial are used to illustrate both the analyses and the major conclusions reached.

In mathematical word problem solving, reading and mathematics interact. Previous research used the method of eye tracking to analyze reading processes but focused on specific elements in prototype word problems. This makes it difficult to compare the role of reading in longer, more complex word problems and between individuals. We used global measures of eye movements that refer to the word problem as a whole, similar to methods used in research on eye movements during reading. Global measures allow comparisons of reading processes of word problems of different structure. To test if these global measures are related to cognitive processes during word problem solving, we analyzed the relation between eye movements and the perceived difficulty of a task and its solution rate. We conducted two experiments with adults and undergraduate students (N = 17 and N = 42), solving challenging mathematical word problems from PISA. Experiment 1 showed that more difficult items were read with longer fixations, more saccades, more regressions, and slower, with correlations ranging from r = 0.70 to r = 0.86. Multilevel modelling in experiment 2 revealed that for the number of saccades and the proportion of regressions, the relationship was stronger for low-performing students, with performance explaining up to 37% of the variance between students. These two measures are primarily associated with building a problem model. We discuss how this approach enables the use of eye tracking in complex mathematical word problem solving and contributes to our understanding of the role of reading in mathematics.

Solving reality-based tasks is an important goal in mathematics instruction and is anchored in education standards determined by mathematical modeling skills. These tasks demand a serious examination of the real-world as well as text comprehension to successfully solve them. Therefore, this study empirically reconstructed the comprehension process during the solution of reality-based tasks and examined how it correlates with process-, person-, and task-related attributes. Fifty-five seventh graders using the Think Aloud Method solved reality-based tasks that were varied in their linguistic and situational complexity level. Their mathematical performance as well as their reading ability were measured. Based on detailed analyses of solution processes combined with the performance data, we point out the relevance of comprehension activities and empirically identify factors that influence the comprehension process.

Csikszentmihalyi (1975) was fascinated by artists who spent most of their time working on paintings or sculptures while being completely immersed in the activity. The artists had the feeling that painting and sculpturing were the most important things in the world. However, as soon as they had finished their projects they lost all interest in their work, put it in a corner, and started a new painting or sculpture. How was this possible? Why did they spend most of their time working on a project and then lost all interest after they were done? When Csikszentmihalyi asked the artists what kind of reward their behavior drove, whether they wanted to become rich or famous, they denied it. It seemed that the reward of painting or sculpturing came from the activities themselves.

In mathematical word problem solving, a relatively well-established finding is that more errors are made on word problems in which the relational keyword is inconsistent instead of consistent with the required arithmetic operation. This study aimed at reducing this consistency effect. Children solved a set of compare word problems before and after receiving a verbal instruction focusing on the consistency effect (or a control verbal instruction). Additionally, we explored potential transfer of the verbal instruction to word problems containing other relational keywords (e.g., larger/smaller than) than those in the verbal instruction (e.g., more/less than). Results showed a significant pretest-to posttest reduction of the consistency effect (but also an unexpected decrement on marked consistent problems) after the experimental verbal instruction but not after the control verbal instruction. No significant effects were found regarding transfer. It is concluded that our verbal instruction was useful for reducing the consistency effect, but future research should address how this benefit can be maintained without hampering performance on marked consistent problems.

Besides fostering science achievement, developing positive science-related attitudes is also an important educational goal. Students need to learn to value science, develop an interest in science, and establish positive science-related self-views. Achieving these multidimensional goals enables students to participate in a society based on scientific reasoning, and influences their educational and professional career choices. This is of high significance because the shortage of skilled workers in specific technical and science professions such as engineering and physical science—especially among females—has become a concern in recent years, and is expected to worsen in the future. This chapter provides an overview of important science-related outcomes (e.g., interest in science, enjoyment of science, instrumental motivation, self-concept, self-efficacy, perceived value of science, self-regulation strategies, epistemological beliefs, technology- and environment-related attitudes, career aspirations) and their research backgrounds. However, for international large-scale assessment (ILSA) studies such as the Programme for International Student Assessment (PISA), there are limitations; and selection criteria arise from study characteristic features. These criteria and limitations are discussed, and this chapter describes how ILSAs have covered the topic of science-related attitudes. On the basis of the above considerations, the selected constructs for the PISA 2015 field trial are presented.

Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME.

Research on eye movements during Chinese reading is reviewed. We begin by briefly describing the basic characteristics of the Chinese writing system, and then address five main topics: 1) basic characteristics of eye movements during Chinese reading; 2) the influence of the intrinsic characteristics of Chinese orthography on eye movements (e.g. stroke complexity, frequency, predictability, and neighbourhood size); 3) selection of saccade targets during Chinese reading; 4) parafoveal effects (perceptual span, parafoveal preview effects, and parafoveal-on-foveal effects); and 5) developmental changes in eye movements during Chinese reading. Finally, some important theoretical issues associated with Chinese reading are discussed.

Reading is a highly complex skill that is prerequisite to success in many societies in which a great deal of information is communicated in written form. Since the 1970s, much has been learned about the reading process from research by cognitive psychologists. This book summarizes that important work and puts it into a coherent framework.

We examined students' difficulties in comprehending relational statements in arithmetic word problems, such as "Gas at Chevron is 5 cents more per gallon than gas at ARCO." In Experiment 1, students were asked to write their complete solution processes for arithmetic word problems containing relational statements. Students were more likely to miscomprehend a relational statement when the required arithmetic operation was inconsistent with the statement's relational term, such as having to subtract when the relational term was more than. This effect was magnified when the relational term was marked (e.g., less than) rather than unmarked (e.g., more than). In Experiment 2, students were given information about two variables and asked to generate a statement expressing the relation between them. Students tended to produce relational statements by using unmarked rather than marked comparative terms. Finally, we present a model of word problem comprehension processes that uses schemata as guides to comprehension. In this conceptualization, students are more likely to make comprehension errors when the order of terms in the relational statement conflicts with the order of terms in their schemata.

This chapter focuses on the specific autotelic quality and the affective, cognitive, and performance-related consequences of the flow experience. Research findings documenting a positive relationship between skills–demands compatibility (the central precondition of flow experiences) and components of an autotelic experience (intrinsic motivation, enjoyment, and involvement) are discussed. Besides, possible consequences of flow experiences are addressed. A review of currently available findings indicates that flow may foster positive affect and even lead to enhanced performance. Unfortunately, the findings, which are mostly correlational in nature, do not provide conclusive evidence regarding the consequences related to flow experiences—reflecting the fact that the empirical analysis of flow experiences is quite complex. Important intricacies of flow research and theorizing and their implications are discussed—specifically, the lack of methods to test for causal effects of flow experiences and the tendency to equate flow experience with skills–demands compatibility.

Many children read mathematics word problems and directly translate them to arithmetic operations. More sophisticated problem solvers transform word problems into object-based or mental models. Subsequent solutions are often qualitatively different because these models differentially support cognitive processing. Based on a conception of problem solving that integrates mathematical problem-solving and reading comprehension theories and using constant comparative methodology (Strauss & Corbin, 1994), 98 sixth- and seventh-grade students' problem-solving behaviors were described and classified into five categories. Nearly 90% of problem solvers used one behavior on a majority of problems. Use of context such as units and relationships, recording information given in the problem, and provision of explanations and justifications were associated with higher reading and mathematics achievement tests, greater success rates, fewer errors, and the ability to preserve the structure of problems during recall. These results were supported by item-level analyses.

This review evaluates the role of language—specifically, the Chinese-based system of number words and the simplicity of Chinese mathematical terms—in explaining the relatively superior performance of Chinese and other East Asian students in cross-national studies of mathematics achievement. Relevant research is critically reviewed focusing on linguistic and cultural influences. The review (a) provides equivocal findings about the extent to which number words in the Chinese language afford benefits for mathematics learning; (b) indicates that cultural and contextual factors are gaining prominence in accounting for the superior performance of East Asian students in cross-national studies; and (c) yields emerging evidence from neuroscience that highlights interrelationships among language, cultural beliefs, and mathematics learning. Although it is not possible to disentangle the influences of linguistic, cultural, and contextual factors on mathematics performance, language is still seen as contributing to early cross-national differences in mathematics attainment.

A tutoring approach is derived from a model of problem comprehension, based on the van Dijk and Kintsch (1983; Kintsch, 1988) theory of discourse processing. A problem statement is regarded as a text from which the student must glean propositional and situational information and make critical inferences. The competent student must coordinate this information with known problem models so that formal (i.e., algebraic) operations can be applied and exact solutions can be obtained. We argue that this task is a highly reading-oriented one in which poor text comprehension and an inability to access relevant long-term knowledge lead to serious errors. In particular, poor students often omit from their solutions or misspecify necessary mathematical constraints that are based on reading inferences needed to describe fully the problem situation. Furthermore, formal algebraic expressions are so abstract that their meaning is often elusive; this contributes to mistranslations and misinterpretations. The competent approach is teachable, however. We describe experimental results with ANIMATE, a learning environment that knows nothing of the problem at hand or of the student's actions. Subjects encouraged to reason explicitly about the situations described in typical word problems consistently performed as well as or better than those who were not, in both training and transfer task. We conclude that, by using an environment that gives equations situation-based meaning through computer animation, students learn to relate formal expressions to the referent situations. This enhances problem comprehension and gives a stronger representational base to the problem-solving process. A call for evaluation methods beyond just algebra problem-solving performance is made. The implications of this work for the design of future computer-based tutors and other learning environments are also discussed.

Students have difficulty solving arithmetic word problems containing a relational term that is inconsistent with the required arithmetic operation (e.g., containing the term
less, yet requiring addition) rather than consistent. To investigate this consistency effect, students' eye fixations were recorded as they read arithmetic word problems on a computer monitor and stated a solution plan for each problem. As predicted, low-accuracy students made more reversal errors on inconsistent than consistent problems, students took more time for inconsistent than consistent problems, this additional time was localized in the integration/planning stages of problem solving rather than in the initial reading of the problem, these response-time patterns were obtained for high-accuracy but not for low-accuracy students, and high-accuracy students required more rereadings of previously fixated words for inconsistent than for consistent problems. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

According to A. Lewis and R. E. Mayer's (1987) simulation model for understanding compare problems, students make more comprehension errors when the order of the terms in the relational statement is not consistent with the preferred order. Three eye movement experiments, designed to test a number of hypotheses directly derived from this model, are discussed. The 1st experiment with university students solving 1-step compare problems revealed no evidence in favor of the model; the data of the 2nd experiment with 3rd graders, on the other hand, provided good support. To explain the results of the 1st experiment, a 3rd experiment was carried out in which university students were given a set of 2-step compare problems. The results from that study also fit with the model well, suggesting that the model holds true only when the task puts some cognitive demands on the S. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

This study used the technique of eye-movement registration to examine the influence of the semantic structure of one-step addition and subtraction word problems (simple vs. complex) on the eye-fixation patterns of 10 high-ability and 10 low-ability second graders. We investigated the effects of the factors semantic complexity and ability level on response time and on the amount and proportion of fixation time spent on words and on numbers in the problem. For a subgroup of 10 children, we analyzed the data for two substages separately: an initial systematic reading phase and the rest of the solution process. The data provide additional evidence that the semantic factor plays a crucial role in word problem solving. The findings concerning the effect of the factor ability level are less straightforward. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

It is proposed that when solving an arithmetic word problem, unsuccessful problem solvers base their solution plan on numbers and keywords that they select from the problem (the direct translation strategy), whereas successful problem solvers construct a model of the situation described in the problem and base their solution plan on this model (the problem-model strategy). Evidence for this hypothesis was obtained in 2 experiments. In Experiment 1, the eye fixations of successful and unsuccessful problem solvers on words and numbers in the problem statement were compared. In Experiment 2, the degree to which successful and unsuccessful problem solvers remember the meaning and exact wording of word problems was examined. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

In Experiment 1, students were asked to write their complete solution processes for arithmetic word problems containing relational statements. Students were more likely to miscomprehend a relational statement when the required arithmetic operation was inconsistent with the statement's relational term, such as having to subtract when the relational term was
more than. This effect was magnified when the relational term was marked (e.g.,
less than) rather than unmarked (e.g.,
more than). In Experiment 2, students were given information about two variables and asked to generate a statement expressing the relation between them. Students tended to produce relational statements by using unmarked rather than marked comparative terms. Finally, we present a model of word problem comprehension processes that uses schemata as guides to comprehension. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

In a study within the DISUM research project, we investigated the role that the construction of situation models plays as an essential prerequisite for understanding a given mathematical modelling task, using a sample of 21 9th grade classes (N=416). Specific task characteristics, general mathematical competence, reading competence, and teacher interventions aiming at understanding the situation model were analyzed as crucial factors influencing students’ ability to solve modelling tasks. The results show that: (1) strategies for constructing an adequate situation model have a significant influence on modelling competence, (2) mathematical reading competence and intra-mathematical competence can explain almost one third of the variance of the performance on the modelling test, (3) teacher interventions may encourage students to adopt strategies facilitating the construction of situation models, but an increase of modelling competence requires separate strategy training.

This study examined the effects of consistency (relational term consistent vs. inconsistent with required arithmetic operation) and markedness (relational term unmarked [‘more than’] vs. marked [‘less than’]) on word problem solving in 10–12 years old children differing in problem-solving skill. The results showed that for unmarked word problems, less successful problem solvers showed an effect of consistency on regressive eye movements (longer and more regressions to solution-relevant problem information for inconsistent than consistent word problems) but not on error rate. For marked word problems, they showed the opposite pattern (effects of consistency on error rate, not on regressive eye movements). The conclusion was drawn that, like more successful problem solvers, less successful problem solvers can appeal to a problem-model strategy, but that they do so only when the relational term is unmarked. The results were discussed mainly with respect to the linguistic–semantic aspects of word problem solving.

Reviews studies of eye movements in reading and other information-processing tasks such as picture viewing, visual search, and problem solving. The major emphasis of the review is on reading as a specific example of the more general phenomenon of cognitive processing. Basic topics discussed are the perceptual span, eye guidance, integration across saccades, control of fixation durations, individual differences, and eye movements as they relate to dyslexia and speed reading. In addition, eye movements and the use of peripheral vision and scan paths in picture perception, visual search, and pattern recognition are discussed, as is the role of eye movements in visual illusion. The basic theme of the review is that eye movement data reflect the cognitive processes occurring in a particular task. Theoretical and practical considerations concerning the use of eye movement data are also presented. (7½ p ref)

A processing model is presented that deals explicitly with both the text-comprehension and problem-solving aspects of word arithmetic problems. General principles from a theory of text processing (van Dijk & Kintsch, 1983) are combined with hypotheses about semantic knowledge for understanding problem texts (Riley, Greeno, & Heller, 1983) in an integrated model of problem comprehension. The model simulates construction of cognitive representations that include information that is appropriate for problem-solving procedures that children use. Several information-processing steps are distinguished, and various levels of representation are described. The model provides an analysis of processing requirements, including requirements for short-term memory that differ among types of problems. Predictions about difficulty of problems based on these processing differences are generally consistent with data that have been reported.

Presents a model of reading comprehension that accounts for the allocation of eye fixations of 14 college students reading scientific passages. The model deals with processing at the level of words, clauses, and text units. Readers made longer pauses at points where processing loads were greater. Greater loads occurred while readers were accessing infrequent words, integrating information from important clauses, and making inferences at the ends of sentences. The model accounts for the gaze duration on each word of text as a function of the involvement of the various levels of processing. The model is embedded in a theoretical framework capable of accommodating the flexibility of reading. (70 ref)

Recent studies of eye movements in reading and other information processing tasks, such as music reading, typing, visual search, and scene perception, are reviewed. The major emphasis of the review is on reading as a specific example of cognitive processing. Basic topics discussed with respect to reading are (a) the characteristics of eye movements, (b) the perceptual span, (c) integration of information across saccades, (d) eye movement control, and (e) individual differences (including dyslexia). Similar topics are discussed with respect to the other tasks examined. The basic theme of the review is that eye movement data reflect moment-to-moment cognitive processes in the various tasks examined. Theoretical and practical considerations concerning the use of eye movement data are also discussed.

Understanding word problems leads to the construction of different levels of representation. Some levels specify the elements which are indispensable for solving the problem (problem model, PM) and others specify the agents, actions and events in everyday concepts (situation model, SM).
By studying how the information is selected, we try to specify the nature of the representations constructed during the reading of a word problem: understanding a problem leads to the construction of two complementary levels of representation (PM and SM) or to the construction of only one representation (PM)?
Ninety-one fifth-grade pupils (mean age 10 years 9 months) took part in this study and were divided into two groups according to their mathematical ability.
As well as the information considered as indispensable for solving the problems (solving information), different types of information (situational information) were introduced into standard word problems. In a first task, participants were asked to select the information in order to 'make the word problem as short as possible' (locate the elements used for developing PM). In a second task, they were asked to select the information in order to 'make the word problem easier to understand' (determine whether the participants developed a SM).
The participants successfully differentiated between the solving information and the situational information. An interaction was also observed between the type of information and the task. The mathematical ability of the participants was seen to have an influence on the selection of situational information.
Understanding leads to the construction of two complementary representation levels: the problem model and the situation model.

Studies on rewording word problems can be grouped into two main groups: situational rewording, in which the situation denoted by the text is described more richly, and conceptual rewording, in which the underlying semantic relations are highlighted.
Our aims are to define and distinguish these two kinds of rewording and to test empirically their relative effectiveness in two different studies.
In the first study, 79 third graders, 64 fourth graders and 65 fifth graders took part; the sample for Study 2 was similar.
In Study 1, children were asked to solve both easy and difficult two-step change problems in three different versions: standard, situational and conceptual rewording. In Study 2, three different versions of the situational version were compared: one with only temporal elaborations, one with only causal elaborations and a 'complete' version combining both elaborations.
In Study 1, conceptually reworded problems elicited the best results, especially among younger children and for difficult two-step problems. Neither in Study 1 nor in Study 2 did the situationally reworded problems yield better performance than standard items.
Only conceptual rewording has proved to be useful for improving children's performance, especially among younger children and for difficult problems. The lack of impact of situational rewording cannot be explained in terms of the length of the resulting text.