Word problems: a review of linguistic and numerical factors contributing to their difficulty
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paper_title: Extending the Mental Number Line A Review of Multi-Digit Number Processing

paper_content:

Multi-digit number processing is ubiquitous in our everyday life - even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single- digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language- related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi- digit number processing research may or should be heading in following years.
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paper_title: What underlies successful word problem solving? A path analysis in sixth grade students

paper_content:

Two component skills are thought to be necessary for successful word problem solving: (1) the production of visual-schematic representations and (2) the derivation of the correct relations between the solution-relevant elements from the text base. The first component skill is grounded in the visual–spatial domain, and presumed to be influenced by spatial ability, whereas the latter is seated in the linguistic–semantic domain, and presumed to be influenced by reading comprehension. These component skills as well as their underlying basic abilities are examined in 128 sixth grade students through path analysis. The results of the path analysis showed that both component skills and their underlying basic abilities explained 49% of students’ word problem solving performance. Furthermore, spatial ability and reading comprehension both had a direct and an indirect relation (via the component skills) with word problem solving performance. These results contribute to the development of instruction methods that help students using these components while solving word problems.
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paper_title: Comprehension of arithmetic word problems: Evidence from students’ eye fixations

paper_content:

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
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paper_title: The Consistency Effect Depends on Markedness in Less Successful but Not Successful Problem Solvers: An Eye Movement Study in Primary School Children.

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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.
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paper_title: Language and the Performance of English-Language Learners in Math Word Problems

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In this article, Maria Martiniello reports the findings of a study of the linguistic complexity of math word problems that were found to exhibit differential item functioning for English-language learners (ELLs) and non-ELLs taking the Massachusetts Comprehensive Assessment System (MCAS) fourth-grade math test. It builds on prior research showing that greater linguistic complexity increases the difficulty of Englishlanguage math items for ELLs compared to non-ELLs of equivalent math proficiency. Through textual analyses, Martiniello describes the linguistic features of some of the 2003 MCAS math word problems that posed disproportionate difficulty for ELLs. Martiniello also uses excerpts from children's think-aloud transcripts to illustrate the reading comprehension challenges these features pose to Spanish-speaking ELLs. Through both DIF statistics and the voices of children, the article scrutinizes the appropriateness of inferences about ELLs' math knowledge based on linguistically complex test items.
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paper_title: ANALYSIS OF PRECURRENT SKILLS IN SOLVING MATHEMATICS STORY PROBLEMS

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We conducted an analysis of precurrent skills (responses that increase the effectiveness of a subsequent or "current" behavior in obtaining a reinforcer) to facilitate the solution of arithmetic word (story) problems. Two students with developmental disabilities were taught four precurrent responses (identifying the initial value, change value, operation, and resulting value) in a sequential manner. Results of a multiple baseline design across behaviors showed that the teaching procedures were effective in increasing correct performance of each of the precurrent behaviors with untaught problems during probes and that once the precurrent behaviors were established, the number of correct problem solutions increased.
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paper_title: FREQUENCY NORMS AND STRUCTURAL ANALYSIS OF ALGEBRA STORY PROBLEMS INTO FAMILIES, CATEGORIES, AND TEMPLATES*

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The goal of this study was to develop a framework for classifying algebra story problems and to determine observed frequencies for each problem type. A total of 1097 algebra story problems were selected from ten standard algebra textbooks. These are divided into eight families based on the nature of the source formula involved; for example, nearly 300 problems were classified in the “time rate family” because they were based on the source formula, “distance = rate x time” or “output = rate x time.” Each family was divided into problem categories based on the general form of the story line; for example, the time rate family consisted of “motion,” “current,” and “work” categories. Each category was divided into templates based on the specific propositional structure of the problem; for example, there were a dozen templates for motion problems such as “overtake,” “closure,” “round trip,” etc. This article describes the procedure for generating families, categories, and templates and provides frequency counts for each observed template. Implications for fostering productive research and instruction are discussed.
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paper_title: Automatic activation of addition facts in arithmetic word problems

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Studies of mental arithmetic have shown that adults solve simple arithmetic problems by retrieving an answer automatically from a network of stored associations. However, most studies have been limited to single-digit addition and multiplication problems. In this article, we examine whether retrieval is also automatic in the context of more complex arithmetic tasks, such as arithmetic word problems. To test this hypothesis, we used a priming procedure with a target-naming task, in which the primes were the numbers included in two sentences containing the numerical information of an arithmetic word problem (e.g., 3 and 2 in “Joe had 3 marbles. Then Tom gave him 2 marbles”), and the targets were either congruent (e.g., 5) or incongruent (e.g., 8) with the prime. A neutral prime was also used replacing the numbers of the problem by capital letters (e.g., X and Y). Manipulating the relationship between the prime and the target and the duration of time that separates these two events, the overall results revea...
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paper_title: Extraneous information and extra steps in arithmetic word problems

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Abstract To determine how middle school students cope with some of the demands imposed on them by arithmetic word problems, 140 eighth graders were asked to solve word problems modeled after those used by the National Assessment of Educational Progress. Processing demands were imposed on the students by adding extraneous information and extra steps to the problems. Results indicated that the presence of extraneous information and extra steps reduced the accuracy of students' solutions. Thinking-out-loud protocols also revealed several misconceptions that students have about solving word problems.
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paper_title: The Language Factor in Mathematics Tests.

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In this study we investigated the importance of language in student test performance on mathematics word problems. Students were given released items from the National Assessment of Educational Progress mathematics assessment, along with parallel items that were modified to reduce their linguistic complexity. In interviews, students typically preferred the revised items over the original counterparts. Paper-and-pencil tests containing original and revised items were administered to 1,174 8th grade students. Students who were English language learners (ELLs) scored lower on the math test than proficient speakers of English. There were also differences in math performance with respect to socioeconomic status (SES) but not gender. Linguistic modification of test items resulted in significant differences in math performance; scores on the linguistically modified version were slightly higher. Some student groups benefited more from the linguistic modification of items-in particular, students in low-level and a...
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paper_title: Remediating Number Combination and Word Problem Deficits Among Students With Mathematics Difficulties: A Randomized Control Trial

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The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students’ math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20–30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students’ mathematics difficulty status. The tutoring protocols proved transportable across sites.
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paper_title: Algebraic word problems: Role of linguistic and structural variables

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No adequate theory of instruction suitable for constructing instructional programs in problem solving has yet been created. One of the reasons for this is the fact that despite intense investigation, much about the sources of problems difficulty still remains to be learned. However, there are a number of studies that examine the effect that problem variables may have on the difficulty of word problems in arithmetic. A basic assumption of this approach is that the structure of the problem itself, to a large measure determines its difficulty level. It was assumed that the relative difficulty of word problems in arithmetic could be determined by analyzing the construction of the problem statement itself. Most of such variables fall into one of the two following groups: (1) those that describe the textual statement of the problems, and (2) those that describe a standard solution algorithm for the problem. A more detailed sets of variables for arithmetic word problems are presented, among others, in Suppes, Loftus and Jerman (1969); Jerman and Rees (1972); Jerman and Mirman (1974). The main purpose of this paper is to report the attempts made to identify and define variables that affect performance of studies on algebraic word problems. If arithmetic word problems are solved using one or more arithmetic operations, then algebraic word problems demand the use of formulae or equations for translating the problem situation from natural language to the corresponding mathematical expression. The variables considered were many, but the variables defined and used for the arithmetic problems in above mentioned studies were examined. It was decided to concentrate on defining variables which describe the textual statements of the problem. In order to describe a solution algorithm for the problem several addtional variables were formulated.
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paper_title: Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers

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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 problemmodel 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. Why are some students successful in solving word problems whereas others are unsuccessful? To help answer this question, we begin with the well-established observation that many students from kindergarten through adulthood have difficulty in solving arithmetic word problems that contain relational statements, that is, sentences that express a numerical relation between two variables (Hegarty, Mayer, & Green, 1992; Lewis & Mayer, 1987; Riley, Greeno, & Heller, 1983; Verschaffel, De Corte, & Pauwels, 1992). For example, Appendix A shows a successful and an unsuccessful solution to a two-step word problem containing a relational statement about the price of butter at two stores. We refer to this as an inconsistent version of the problem because the relational keyword (e.g., "less") primes an inappropriate arithmetic operation (subtraction rather than addition), whereas in a consistent problem, the relational term in the second problem statement primes the required arithmetic operation (e.g., "more" when the required operation is addition). A substantial proportion of college students, who could be called unsuccessful problem solvers, use the wrong arithmetic operation on inconsistent problems but perform correctly on consistent problems (Hegarty et al., 1992; Lewis, 1989; Lewis & Mayer, 1987; Verschaffel et al., 1992). We interpret this finding as evidence that problem comprehension processes play an important role in the solution of arithmetic word problems.
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paper_title: Error Analysis of Mathematical Word Problem Solving Across Students with and without Learning Disabilities

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Solving word problems is a common area of struggle for students with learning disabilities (LD). In order for instruction to be effective, we first need to have a clear understanding of the specific errors exhibited by students with LD during problem solving. Error analysis has proven to be an effective tool in other areas of math but has had little application to errors in word problems. Using an error analysis approach, this study aimed to investigate in depth the various types and frequency of errors made by students with LD and their AA peers during math problem solving. The resulting similarities and differences between the two groups of students are discussed with insight into underlying cognitive processes, and implications for future research.
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paper_title: Pre-service teachers' conceptions and beliefs about the role of real-world knowledge in mathematical modelling of school word problems

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Recent research has documented that many pupils show a strong tendency to exclude real-world knowledge from their solutions of school arithmetic word problems. In the present study, a test consisting of 14 word problems-half of which were problematic From a realistic point of view-was administered to a large group of students from three different leacher training institutes in Flanders. For each word problem, the student-teachers were first asked to solve the problem themselves, and afterwards to evaluate four different answers given by pupils. The results revealed a strong tendency among student-teachers to exclude real-world knowledge from their own spontaneous solutions of school word problems as well as from their appreciations of the pupils' answers. (C) 1997 Elsevier Science Ltd.
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paper_title: Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

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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.
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paper_title: Seeing the Math in the Story On How Abstraction Promotes Performance on Mathematical Word Problems

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The negative social, health, financial, and other life outcomes associated with mathematical proficiency deficits highlight the need to understand the underlying cognitive operations entailed in solving math problems. We focus specifically on mathematical word problems and propose that abstraction can enhance performance by helping people see beyond the incidental details described in word problems and to recognize instead the underlying mathematical relationships. Three studies manipulated abstraction as a procedural mind-set (i.e., inducing abstraction in one task and observing its “carry-over” effect in subsequent unrelated tasks) and observed performance on both numeric and word problems. Participants in the abstract, relative to concrete, mind-set condition were more successful in translating word problems into their analogous numeric forms, resulting in improved performance. We discuss implications of these findings for understanding individual and group differences in mathematics proficiencies, whi...
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paper_title: Why does placing the question before an arithmetic word problem improve performance? A situation model account

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The aim of this paper is to investigate the controversial issue of the nature of the representation constructed by individuals to solve arithmetic word problems. More precisely, we consider the relevance of two different theories: the situation or mental model theory (Johnson-Laird, 1983; Reusser, 1989) and the schema theory (Kintsch & Greeno, 1985; Riley, Greeno, & Heller, 1983). Fourth-graders who differed in their mathematical skills were presented with problems that varied in difficulty and with the question either before or after the text. We obtained the classic effect of the position of the question, with better performance when the question was presented prior to the text. In addition, this effect was more marked in the case of children who had poorer mathematical skills and in the case of more difficult problems. We argue that this pattern of results is compatible only with the situation or mental model theory, and not with the schema theory.
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paper_title: An exploration of aspects of language proficiency and Algebra learning

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We have attempted to investigate whether 3 cognitive components of language proficiencymetalinguistic awareness of symbol, syntax, and ambiguity-are associated with students' success in learning the notation of algebra. Pencil-and-paper tests were given to assess students' metalinguistic awareness and their ability to use algebraic notation. In a total sample of more than 1500 students, aged 11 to 15, who were in their 1st to 4th years of algebra learning, we found that very few students with low metalinguistic awareness scores achieved high algebra scores. We discuss implications of this finding for the school algebra curriculum.
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paper_title: The Impact of Language Characteristics in Mathematics Test Items on the Performance of English Language Learners and Students With Disabilities

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The impact of language characteristics in mathematics test items on student performance was evaluated for students with disabilities (SWD) and English language learners (ELL) as well as a large general student sample. Relationships were examined for test items and students at 4th, 7th, and 10th grades. The individual test item was the unit of analysis. Student performance was represented by item difficulty, or the probability of answering the item correctly. Regression analyses were conducted to examine relationships between item linguistic characteristics as independent variables with item difficulty as the dependent variable. Language characteristics had moderate effects on item difficulty at 4th grade, dropping to small-to-medium effects at 10th grade. ELL and SWD groups were not disproportionately affected by language characteristics in these test items. Difficult mathematics vocabulary had a consistent effect on performance for all students at all grades. Ambiguous or multiple-meaning words increased...
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paper_title: Arithmetic word problem solving: a Situation Strategy First framework

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Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing the numerical solution is the representation of the problem modified so that the relevant arithmetic knowledge might be used. Three experiments were conducted with Year 3 and Year 4 children. Subtraction, multiplication and division problems were created in two versions involving the same wording but different numerical values. The first version could be mentally solved with a Situation strategy (Si version) and the second with a Mental Arithmetic strategy (MA version). Results show that Si-problems are easier than MA-problems even after instruction, and, when children were asked to report their strategy by writing a number sentence, equations that directly model the situation were predominant for Si-problems but not for MA ones. Implications of the Situation Strategy First framework regarding the relation between conceptual and procedural knowledge and the development of arithmetic knowledge are discussed.
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paper_title: Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

paper_content:

One of the five content strands for students across kindergarten through twelfth grade of the Principles and Standards for School Mathematics is algebra (National Council of Teachers of Mathematics, 2000). Algebraic reasoning is also a Common Core domain area in the elementary grades, an integral part of several domain areas in middle school, and one of six important areas of study for high school students (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). As the standards of these national organizations press for algebra instruction for all students across elementary, middle, and high school, as more states require all students to take and pass algebra courses in high school (Stein, Kaufman, Sherman, & Hillen, 2011), and given that many secondary students, including students with disabilities, are unprepared for algebra courses (Faulkner, Crossland, & Stiff, 2013; National Mathematics Advisory Panel, 2008), the elementary school curriculum needs to emphasize algebraic reasoning more than ever. ::: ::: The widespread adoption of the Common Core indicates that districts are amending curricula to reflect the standards outlined in the Common Core. The Common Core emphasizes algebraic reasoning in the elementary grades with standards listed under the “Operations and Algebraic Thinking” domain. For example, at kindergarten, when given a number 1 through 9, students have to determine the other number (i.e., unknown) that adds to 10. In first grade, students solve addition and subtraction equations with a unknown (e.g., 8 + ? = 11; 5 = + 3). In grades 1, 2, 3, and 4, students use equations with a symbol for the unknown to represent and solve word problems. By grade 3, students solve multiplication and division equations with an unknown (e.g., 3 × = 27; 9 = 81 ÷ ?). In grades 4 and 5, students generate patterns to describe the relationship between sets of numbers (e.g., n + 3). In the middle school grades, the Common Core standards explicitly related to algebraic reasoning fall under the “Expressions and Equations” domain (grades 6, 7, and 8) and the “Functions” domain (grades 8). Many algebraic reasoning skills, especially those related to problem solving, fall under other domains (e.g., Ratios and Proportional Relationships, Measurement) at grades 6, 7, and 8. ::: ::: The Common Core standards influence statewide assessments, which in turn influence mathematics textbooks. Districts want to purchase curriculum materials aligned with the assessments to provide a cohesive mathematics program that enables students to succeed on the assessment. As more and more districts revise their mathematics curriculum to meet or exceed Common Core standards and as all students, including those with learning disabilities, are held accountable on statewide assessments (Yell, Katsiyannis, & Shiner 2006), it is important to understand how students, especially those with mathematics difficulties, interpret and solve algebraic problems. ::: ::: In this introduction, we explain how most researchers approach algebraic reasoning in the elementary grades. We discuss solving algebraic problems using equations and functions. Then, we describe students with mathematics difficulties; the two main areas (i.e., calculations and word problems) in which they struggle during the elementary grades; and how algebraic reasoning may be connected to calculations and to word problems. Finally, we describe the rationale, purpose, and hypotheses for the present study.
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paper_title: What underlies successful word problem solving? A path analysis in sixth grade students

paper_content:

Two component skills are thought to be necessary for successful word problem solving: (1) the production of visual-schematic representations and (2) the derivation of the correct relations between the solution-relevant elements from the text base. The first component skill is grounded in the visual–spatial domain, and presumed to be influenced by spatial ability, whereas the latter is seated in the linguistic–semantic domain, and presumed to be influenced by reading comprehension. These component skills as well as their underlying basic abilities are examined in 128 sixth grade students through path analysis. The results of the path analysis showed that both component skills and their underlying basic abilities explained 49% of students’ word problem solving performance. Furthermore, spatial ability and reading comprehension both had a direct and an indirect relation (via the component skills) with word problem solving performance. These results contribute to the development of instruction methods that help students using these components while solving word problems.
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paper_title: Error analysis and the corresponding cognitive activities committed by year five primary students in solving mathematical word problems

paper_content:

Abstract The purpose of this study was to identify errors and the corresponding cognitive activities committed by the 374 year five primary school students in solving the mathematical word problems for the ‘Fraction’ topic. The design of this study used Newman Error Analysis method and interviews. The findings showed that 52.91% of the errors made were due to lack of understanding, followed by transformation skills (22.37%), process skills (15.55%), encoding (8.84%) and reading (0.34 Based on these findings, the researchers proposed that teachers should ask students explicitly what problems they have in solving the mathematical word problems, observe students reactions and analyze errors committed by them.
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paper_title: The relationship between different kinds of students' errors and the knowledge required to solve mathematics word problems

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The main objective of this research is to examine the relationship between different kinds of errors and the knowledge required to solve word problems in Arithmetic, Algebra and Geometry. Kinfong's and Holtan's framework supports the analysis of the errors, and Mayer's theory was implemented to understand the necessary knowledge for solving math word problems. The research methodology follows a semi-experimental method. Research tools comprise both a descriptive math test and a directed interview. The research findings revealed that students' errors when solving arithmetic word problems result from the lack of linguistic, semantic, structural and communicational knowledge; when solving the geometric word problems, the lack of semantic, intuition and structural knowledge were the cause of the students' errors. Regarding algebra word problems, miscalculation was the reason for the higher error rate. Results show that the highest deficiency is mainly related to the lack of semantic, structural and communicational knowledge.
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paper_title: Solving arithmetic word problems: an analysis of classification as a function of difficulty in children with and without arithmetic LD.

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This study was designed to determine a word problem difficulty classification in children with arithmetic learning disabilities (ALD; n = 104) in comparison with typically achieving students (n = 44). We tested variables such as (a) semantic structure (Change, Combine, Compare, and Equalize), (b) operation (subtraction and addition), and (c) position of the unknown quantity in the problem. Facet theory with multidimensional scaling techniques (MINISSA) was used to analyze the underlying dimensions in the responses of each group of participants. Our results indicate that although the word problem difficulty classifications for the 2 groups of children were different, the position of the unknown quantity had a greater influence on the level of difficulty of story problems than other variables. The noncanonical problems--specifically, those with the unknown term in the first place--although difficult for both groups of children, were the most difficult problems for children with ALD.
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paper_title: Problem solving beyond the logic of things: contextual effects on understanding and solving word problems

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Arguments are put forward in this paper that classroom word problem solving is more-and also less-than the urgent analysis of a factual structure, in the sense that it is essentially a species of a social-cognitive activity. Word-or story-problems, presented in classroom contexts, represent textual and pragmatic patterms of a certain grammaticality. To present a problem verbally to a student means to organize a fact in some way for the attention of a problem solver. There is not only the structure of the problem text itself by which situations are denoted, but there is also the stimulative nature of the social-pragmatic context which shapes the student's textbook-problem solving behavior over a long period of time.
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paper_title: The International Assessment of Mathematical Literacy: PISA 2012 Framework and Items

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The OECD PISA international survey of mathematical literacy for 2012 is based on a new Framework and has several new constructs. New features include an improved definition of mathematical literacy; the separate reporting of mathematical processes involved in using mathematics to solve real world problems; a computer-based component to assess mathematical literacy as it is likely to be encountered in modern workplaces; and new questionnaire items targeting mathematics. Procedures for quality assurance that arise in the preparation of an assessment for use in many countries around the world are illustrated with some items and results from the 2011 international field trial. The paper will provide background for the interpretation of the results of the PISA 2012 survey, which are to be published in December 2013.
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paper_title: The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning

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This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem-solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. We conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than anoth...
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paper_title: Comprehension of arithmetic word problems: Evidence from students’ eye fixations

paper_content:

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
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paper_title: Influence of situational and conceptual rewording on word problem solving

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Se analiza la doble naturaleza semantica y textual de la tarea de resolver un problema matematico en el aula.
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paper_title: Language affects symbolic arithmetic in children: The case of number word inversion

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Abstract Specific language influences have been observed in basic numerical tasks such as magnitude comparison, transcoding, and the number line estimation task. However, so far language influences in more complex calculations have not been reported in children. In this translingual study, 7- to 9-year-old German- and Italian-speaking children were tested on a symbolic addition task. Whereas the order of tens and units in Italian number words follows the order of the Arabic notation, the order is inverted in German number words. For both language groups, addition problems were more difficult when a carry operation was needed, that is, when a manipulation within the place-value structure of the Arabic number system was particularly important. Most important, this carry effect was more pronounced in response latencies for children speaking German, a language with inverted verbal mapping of the place-value structure. In addition, independent of language group, the size of the carry effect was significantly related to verbal working memory. The current study indicates that symbolic arithmetic and the carry effect in particular are modulated by language-specific characteristics. Our results underline the fact that the structure of the language of instruction is an important factor in children’s mathematical education and needs to be taken into account even for seemingly nonverbal symbolic Arabic tasks.
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paper_title: Structural variables affecting CAI performance on arithmetic word problems of disadvantaged and deaf students

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SummaryWe have shown that it is possible to account for a substantial portion of variability in student responses to arithmetic word problems using variables that describe structural features of the problems. However, the results obtained at this stage in our investigations are situation-dependent. The greater variance in observed proportion correct compared with the variance in the predicted proportions, and the differing sets of variables contributing significantly to the regressions come as no surprise. First, the population for this study differed from that used in the pilot study. Second, it is clear that characteristics of the problem set, for example, the frequency of occurrence of exemplars for the range of values for each variable and the way variable values are combined in problem types, affect the weighting for each variable in the regression analysis. Thus, differences were expected because different problem sets were used for the pilot study and the present study.In the light of these differences, the similarity in performance of the two disadvantaged groups gains in significance, and deserves further study. We believe we can increase the generalizability of our results by redesigning the basic problem set to exemplify in a balanced fashion the full range of variables found to account for problem difficulty. Given, however, the difficulty of making accurate predictions about problem-solving results, the correctness of this belief needs to be explicitly tested.
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paper_title: Text Integration and Mathematical Connections: A Computer Model of Arithmetic Word Problem Solving

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Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence-level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence-level and text integration processes. Working memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K-3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process-oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships.
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paper_title: An fMRI study of the Interplay of Symbolic and Visuo-spatial Systems in Mathematical Reasoning

paper_content:

An fMRI study of the Interplay of Symbolic and Visuo-spatial Systems in Mathematical Reasoning Atsushi Terao (atsushi@edu.hokudai.ac.jp) Graduate School of Education, Hokkaido University, Kita 11 Nishi 7 Sapporo, 060-0811 JAPAN Kenneth R. Koedinger (koedinger@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3890 USA Myeong-Ho Sohn (mhsohn@andrew.cmu.edu) Yulin Qin (yulinQ@andrew.cmu.edu) John R. Anderson (ja+@cmu.edu) Department of Psychology, Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3890 USA Cameron S. Carter (CarterCS@msx.upmc.edu) Imaging Research Center, University of California at Davis 1 Shields Ave. Davis, CA 95616 USA Abstract The purpose of this fMRI study was to provide evidence for the mathematician’s belief that mathematical thinking emerges from the interplay between symbolic and visuo- spatial systems. Twelve participants were given algebra word problems and depicted the quantitative relations on a mental number line or made parts of an equation. The regions activated in depicting the picture were also recruited to make an equation. Mathematics is a language. Many scientists say that mathematics is a language to describe the nature of phenomena they are looking at. It is well known that Nicolas Burubaki, a group of mathematicians, stressed the crucial role of formal symbol systems in mathematics. On the other hand, many mathematicians and physicists emphasize the role of visuo-spatial reasoning in mathematics, which recruits qualitative, language- independent representations. For example, Albert Einstein stated “Words and language, whether written or spoken, do not seem to play any part in my thought process.” As Dehaene, Spelke, Pinel, Stanescu, and Tsivkn (1999) suggested, mathematical thinking may emerge from the interplay between symbolic and visuo-spatial systems. In this fMRI study, we approach this problem and provide evidence for this kind of mathematical thinking. Psychological studies have revealed that if a problem apparently looks like a pure symbolic task, it can require students to have some visuo-spatial representations. For example, Griffin, Case and Siegler (1994) showed that the mental “number line”, a qualitative representation of the number system, is crucial readiness for early arithmetic. Lewis used a number-line-like diagram to train undergraduate students having difficulty to solve “compare” word problems (problems containing more-than or less-than relations), and succeeded in improving their performance. Paige and Simon (1966) proposed that solving word problems is not a simple translation of problem sentences into equations, as Bobrow’s (1966) STUDENT did, but needs “physical cues,” a visuo-spatial representation. The 6 th grade students who used our “Picture Algebra” strategy (Koedinger & Terao, 2002) to solve the compare word problem showed relatively high performance. We expect that using this strategy may better prepare students to learn formal algebra. Functional magnetic resonance imaging (fMRI) gives us a new source of information about the mental representations used in mathematics. Dehaene et al. (1999) showed two different mental representations are used for different tasks. Exact calculation (e.g., 4+5=9) elicited left-lateralized activation in the left inferior frontal lobe, together with left angular gyrus and left anterior cingulate. This pattern was interpreted as suggesting that the participants recruited their symbolic systems and did language dependent encoding. Approximation (e.g., 4+5 is closer to 8 than 6), on the contrary, elicited bilateral parietal lobes activation. This pattern was interpreted as suggesting that the participants recruited visuo-spatial systems and did language independent encoding. Dehaene, Piazza, Pinel and Cohen (2003) reviewed neuro-imaging and neuropsychological evidence concerning various numerical tasks and proposed a hypothesis that three parietal circuits are related to number processing. The horizontal segment of the intraparietal sulcus (HIPS) appears to be a core quantity system, analogous to a mental number line. This area seems to be supplemented by two other circuits. One is the bilateral posterior superior parietal lobule (PSPL), which is considered to be involved in attention orientation on the mental number line. The other is the left angular gyrus, which is likely to support manipulations of numbers in a symbolic form (e.g., exact calculation).
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paper_title: FREQUENCY NORMS AND STRUCTURAL ANALYSIS OF ALGEBRA STORY PROBLEMS INTO FAMILIES, CATEGORIES, AND TEMPLATES*

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The goal of this study was to develop a framework for classifying algebra story problems and to determine observed frequencies for each problem type. A total of 1097 algebra story problems were selected from ten standard algebra textbooks. These are divided into eight families based on the nature of the source formula involved; for example, nearly 300 problems were classified in the “time rate family” because they were based on the source formula, “distance = rate x time” or “output = rate x time.” Each family was divided into problem categories based on the general form of the story line; for example, the time rate family consisted of “motion,” “current,” and “work” categories. Each category was divided into templates based on the specific propositional structure of the problem; for example, there were a dozen templates for motion problems such as “overtake,” “closure,” “round trip,” etc. This article describes the procedure for generating families, categories, and templates and provides frequency counts for each observed template. Implications for fostering productive research and instruction are discussed.
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paper_title: Compare word problems: Consistency hypothesis revisited☆

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Abstract Compare problems involve relationships between quantities. In inconsistent language (IL) formats, relational terms conflict with solution operations. Lewis and Mayer’s (1987) consistency hypothesis, which explains likely translations of IL problems to match problem solvers’ schemas, is examined Using think-aloud and recall data. Supporting hypotheses, students committed more errors overall and reversal errors specifically (i.e., used opposite operation) on IL than their consistent language (CL) counterparts and recalled IL problems as CL problems especially following reversal errors. Counter to expectations, language consistency did not result in differences in behaviors, reading or response times, or quality of recall. Additional analyses indicate a strong impact of fraction-of-a-number relational terms and provide evidence of difficulties forming representations and solving problems involving fraction-of-a-number construction.
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paper_title: Strategies of Discourse Comprehension

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rhetorical schemata to be discussed in what follows. Finally, schemata are descriptions, not definitions. The ‘bus’ schema contains information that is nor-
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paper_title: Error Analysis of Mathematical Word Problem Solving Across Students with and without Learning Disabilities

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Solving word problems is a common area of struggle for students with learning disabilities (LD). In order for instruction to be effective, we first need to have a clear understanding of the specific errors exhibited by students with LD during problem solving. Error analysis has proven to be an effective tool in other areas of math but has had little application to errors in word problems. Using an error analysis approach, this study aimed to investigate in depth the various types and frequency of errors made by students with LD and their AA peers during math problem solving. The resulting similarities and differences between the two groups of students are discussed with insight into underlying cognitive processes, and implications for future research.
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paper_title: Effects of Cognitive Strategy Interventions and Cognitive Moderators on Word Problem Solving in Children at Risk for Problem Solving Difficulties

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This study investigated the role of strategy instruction and cognitive abilities on word problem solving accuracy in children with math difficulties (MD). Elementary school children (N = 120) with and without MD were randomly assigned to 1 of 4 conditions: general-heuristic (e.g., underline question sentence), visual-schematic presentation (diagrams), general-heuristic + visual-schematic, and an untreated control. When compared to the control condition that included children with MD, an advantage at posttest was found for children with MD for the visual-schematic-alone condition on measures of problem solving and calculation accuracy, whereas all strategy conditions facilitated posttest performance in correctly identifying problem solving components. The results also suggested that strategy conditions drew upon different cognitive resources. The General-heuristic condition drew primarily upon the executive component of working memory (WM), Visual-schematic condition drew upon the visual component of WM and the combined strategies condition drew upon number processing skills.
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paper_title: Seeing the Math in the Story On How Abstraction Promotes Performance on Mathematical Word Problems

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The negative social, health, financial, and other life outcomes associated with mathematical proficiency deficits highlight the need to understand the underlying cognitive operations entailed in solving math problems. We focus specifically on mathematical word problems and propose that abstraction can enhance performance by helping people see beyond the incidental details described in word problems and to recognize instead the underlying mathematical relationships. Three studies manipulated abstraction as a procedural mind-set (i.e., inducing abstraction in one task and observing its “carry-over” effect in subsequent unrelated tasks) and observed performance on both numeric and word problems. Participants in the abstract, relative to concrete, mind-set condition were more successful in translating word problems into their analogous numeric forms, resulting in improved performance. We discuss implications of these findings for understanding individual and group differences in mathematics proficiencies, whi...
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paper_title: The Theory of Conceptual Fields

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The theory of conceptual fields is a developmental theory. It has two aims: (1) to describe and analyse the progressive complexity, on a long- and medium-term basis, of the mathematical competences that students develop inside and outside school, and (2) to establish better connections between the operational form of knowledge, which consists in action in the physical and social world, and the predicative form of knowledge, which consists in the linguistic and symbolic expressions of this knowledge. As it deals with the progressive complexity of knowledge, the conceptual field framework is also useful to help teachers organize didactic situations and interventions, depending on both the epistemology of mathematics and a better understanding of the conceptualizing process of students.
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paper_title: The strategic use of alternative representations in arithmetic word problem solving

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Multiple-step arithmetic problems can be solved by diverse strategies depending on the mental representation constructed by individuals from the situation described in the text of the problem. This representation will indeed determine the organization of subgoals to be reached or in other words the order of completion of calculations. This study aims at determining the conditions under which specific strategies are set up by adults. Using a paradigm that allows us to assess when calculations are performed, we show that adults usually organize their subgoals as they are explicitly mentioned in the problem, even though a strategy that is less demanding on working memory could have been used. However, we show that increasing the difficulty of the problem leads individuals to set up more economic strategies. Moreover, these economic strategies are even more likely to be used when the cognitive cost of the construction of the representation they rely on is low.
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paper_title: Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

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One of the five content strands for students across kindergarten through twelfth grade of the Principles and Standards for School Mathematics is algebra (National Council of Teachers of Mathematics, 2000). Algebraic reasoning is also a Common Core domain area in the elementary grades, an integral part of several domain areas in middle school, and one of six important areas of study for high school students (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). As the standards of these national organizations press for algebra instruction for all students across elementary, middle, and high school, as more states require all students to take and pass algebra courses in high school (Stein, Kaufman, Sherman, & Hillen, 2011), and given that many secondary students, including students with disabilities, are unprepared for algebra courses (Faulkner, Crossland, & Stiff, 2013; National Mathematics Advisory Panel, 2008), the elementary school curriculum needs to emphasize algebraic reasoning more than ever. ::: ::: The widespread adoption of the Common Core indicates that districts are amending curricula to reflect the standards outlined in the Common Core. The Common Core emphasizes algebraic reasoning in the elementary grades with standards listed under the “Operations and Algebraic Thinking” domain. For example, at kindergarten, when given a number 1 through 9, students have to determine the other number (i.e., unknown) that adds to 10. In first grade, students solve addition and subtraction equations with a unknown (e.g., 8 + ? = 11; 5 = + 3). In grades 1, 2, 3, and 4, students use equations with a symbol for the unknown to represent and solve word problems. By grade 3, students solve multiplication and division equations with an unknown (e.g., 3 × = 27; 9 = 81 ÷ ?). In grades 4 and 5, students generate patterns to describe the relationship between sets of numbers (e.g., n + 3). In the middle school grades, the Common Core standards explicitly related to algebraic reasoning fall under the “Expressions and Equations” domain (grades 6, 7, and 8) and the “Functions” domain (grades 8). Many algebraic reasoning skills, especially those related to problem solving, fall under other domains (e.g., Ratios and Proportional Relationships, Measurement) at grades 6, 7, and 8. ::: ::: The Common Core standards influence statewide assessments, which in turn influence mathematics textbooks. Districts want to purchase curriculum materials aligned with the assessments to provide a cohesive mathematics program that enables students to succeed on the assessment. As more and more districts revise their mathematics curriculum to meet or exceed Common Core standards and as all students, including those with learning disabilities, are held accountable on statewide assessments (Yell, Katsiyannis, & Shiner 2006), it is important to understand how students, especially those with mathematics difficulties, interpret and solve algebraic problems. ::: ::: In this introduction, we explain how most researchers approach algebraic reasoning in the elementary grades. We discuss solving algebraic problems using equations and functions. Then, we describe students with mathematics difficulties; the two main areas (i.e., calculations and word problems) in which they struggle during the elementary grades; and how algebraic reasoning may be connected to calculations and to word problems. Finally, we describe the rationale, purpose, and hypotheses for the present study.
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paper_title: Problem solving beyond the logic of things: contextual effects on understanding and solving word problems

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Arguments are put forward in this paper that classroom word problem solving is more-and also less-than the urgent analysis of a factual structure, in the sense that it is essentially a species of a social-cognitive activity. Word-or story-problems, presented in classroom contexts, represent textual and pragmatic patterms of a certain grammaticality. To present a problem verbally to a student means to organize a fact in some way for the attention of a problem solver. There is not only the structure of the problem text itself by which situations are denoted, but there is also the stimulative nature of the social-pragmatic context which shapes the student's textbook-problem solving behavior over a long period of time.
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paper_title: The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning

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This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem-solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. We conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than anoth...
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paper_title: Influence of situational and conceptual rewording on word problem solving

paper_content:

Se analiza la doble naturaleza semantica y textual de la tarea de resolver un problema matematico en el aula.
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paper_title: Fifth-grade students’ approaches to and beliefs of mathematics word problem solving: a large sample Hungarian study

paper_content:

Mathematical word problems used in Verschaffel et al.’s (Learning and Instruction 7:339–359, 1994) study were applied in several follow-up studies. The goal of the present study was to replicate and extend the results of this line of research in a large sample of Hungarian students using an alternative set of data-gathering and data-analysis techniques. 4,037 students forming a nationwide representative sample of the Hungarian fifth-grade student population (aged 10–11) completed the test. The test contained five word problems from the list of 10 P(‘problematic)-items from Verschaffel et al.’s test. In contrast to all previous research in this domain, we used a multiple-choice format, where three options were given for each task: (a) routine-based, non-realistic answer, (b) numerical response that does take into account realistic considerations, (c) a realistic solution stating that the task cannot be solved. The hypotheses of this study were: (1) Students’ responses will confirm previous results, i.e. upper elementary school students prefer to respond to P-items by means of the routine-based answer; (2) Most students will demonstrate a more or less consistent preference for a given answer type (a, b or c) over problems; (3) Students’ school math marks will have low correlation indices with students’ achievement on these word problems. Our results confirm student’s overall tendency to follow non-realistic approaches when doing school word problem solving. The tendency even holds when confronting students with various kinds of realistic answers. Our results show that students demonstrate response patterns over problems, and that the correlation with math school performance is significant but small.
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paper_title: Text Integration and Mathematical Connections: A Computer Model of Arithmetic Word Problem Solving

paper_content:

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence-level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence-level and text integration processes. Working memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K-3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process-oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships.
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paper_title: Linguistic Diversity and the First Settlement of the New World.

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The oft-noted genetic and typological diversity of the languages of the New World is due not to accidents of history but to the operation of regular principles of linguistic geography. Four building blocks of a theory of linguistic diversity are presented here: a classification and measures of diversity, an account of the causes of diversity, measures of rates of differentiation, and means of estimating the age of a linguistic population. On the most conservative archeological view, the New World was first colonized some 12,000 years ago; on the received view, up to 20,000 years ago. But no high-latitude area like Siberia, Beringia, Alaska, or northern Canada could have contributed or transmitted the linguistic diversity to seed the New World in these time frames, and no single dispersing family could have given rise to the attested diversity in these time frames. The unmistakable testimony of the linguistic evidence is that the New World has been inhabited nearly as long as Australia or New Guinea, perhaps some 35,000 years. Genetic unity for 'Amerind' is incompatible with the chronology demanded by the linguistic facts.*
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paper_title: Defining and operationalising L2 complexity

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This chapter takes a critical look at complexity in L2 research. We demonstrate several problems in the L2 literature in terms of how complexity has been defined and operationalised as a construct. In the first part of the chapter we try to unravel its highly complex, multidimensional nature by presenting a taxonomic model that identifies major types, dimensions and components of L2 complexity, each of which can be independently analysed or measured. The second part evaluates how complexity has been measured in empirical SLA research. Using the taxonomy of L2 complexity from part 1 as a framework, we inventory the measures of L2 complexity that have been used in a sample of forty recent L2 studies. Next we discuss the construct validity of several widely used measures of grammatical complexity by identifying their underlying logic as well as the methodological and practical challenges that their computation presents.
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paper_title: Language and the Performance of English-Language Learners in Math Word Problems

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In this article, Maria Martiniello reports the findings of a study of the linguistic complexity of math word problems that were found to exhibit differential item functioning for English-language learners (ELLs) and non-ELLs taking the Massachusetts Comprehensive Assessment System (MCAS) fourth-grade math test. It builds on prior research showing that greater linguistic complexity increases the difficulty of Englishlanguage math items for ELLs compared to non-ELLs of equivalent math proficiency. Through textual analyses, Martiniello describes the linguistic features of some of the 2003 MCAS math word problems that posed disproportionate difficulty for ELLs. Martiniello also uses excerpts from children's think-aloud transcripts to illustrate the reading comprehension challenges these features pose to Spanish-speaking ELLs. Through both DIF statistics and the voices of children, the article scrutinizes the appropriateness of inferences about ELLs' math knowledge based on linguistically complex test items.
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paper_title: The Language Factor in Mathematics Tests.

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In this study we investigated the importance of language in student test performance on mathematics word problems. Students were given released items from the National Assessment of Educational Progress mathematics assessment, along with parallel items that were modified to reduce their linguistic complexity. In interviews, students typically preferred the revised items over the original counterparts. Paper-and-pencil tests containing original and revised items were administered to 1,174 8th grade students. Students who were English language learners (ELLs) scored lower on the math test than proficient speakers of English. There were also differences in math performance with respect to socioeconomic status (SES) but not gender. Linguistic modification of test items resulted in significant differences in math performance; scores on the linguistically modified version were slightly higher. Some student groups benefited more from the linguistic modification of items-in particular, students in low-level and a...
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paper_title: Algebraic word problems: Role of linguistic and structural variables

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No adequate theory of instruction suitable for constructing instructional programs in problem solving has yet been created. One of the reasons for this is the fact that despite intense investigation, much about the sources of problems difficulty still remains to be learned. However, there are a number of studies that examine the effect that problem variables may have on the difficulty of word problems in arithmetic. A basic assumption of this approach is that the structure of the problem itself, to a large measure determines its difficulty level. It was assumed that the relative difficulty of word problems in arithmetic could be determined by analyzing the construction of the problem statement itself. Most of such variables fall into one of the two following groups: (1) those that describe the textual statement of the problems, and (2) those that describe a standard solution algorithm for the problem. A more detailed sets of variables for arithmetic word problems are presented, among others, in Suppes, Loftus and Jerman (1969); Jerman and Rees (1972); Jerman and Mirman (1974). The main purpose of this paper is to report the attempts made to identify and define variables that affect performance of studies on algebraic word problems. If arithmetic word problems are solved using one or more arithmetic operations, then algebraic word problems demand the use of formulae or equations for translating the problem situation from natural language to the corresponding mathematical expression. The variables considered were many, but the variables defined and used for the arithmetic problems in above mentioned studies were examined. It was decided to concentrate on defining variables which describe the textual statements of the problem. In order to describe a solution algorithm for the problem several addtional variables were formulated.
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paper_title: Why does placing the question before an arithmetic word problem improve performance? A situation model account

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The aim of this paper is to investigate the controversial issue of the nature of the representation constructed by individuals to solve arithmetic word problems. More precisely, we consider the relevance of two different theories: the situation or mental model theory (Johnson-Laird, 1983; Reusser, 1989) and the schema theory (Kintsch & Greeno, 1985; Riley, Greeno, & Heller, 1983). Fourth-graders who differed in their mathematical skills were presented with problems that varied in difficulty and with the question either before or after the text. We obtained the classic effect of the position of the question, with better performance when the question was presented prior to the text. In addition, this effect was more marked in the case of children who had poorer mathematical skills and in the case of more difficult problems. We argue that this pattern of results is compatible only with the situation or mental model theory, and not with the schema theory.
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paper_title: The Impact of Language Characteristics in Mathematics Test Items on the Performance of English Language Learners and Students With Disabilities

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The impact of language characteristics in mathematics test items on student performance was evaluated for students with disabilities (SWD) and English language learners (ELL) as well as a large general student sample. Relationships were examined for test items and students at 4th, 7th, and 10th grades. The individual test item was the unit of analysis. Student performance was represented by item difficulty, or the probability of answering the item correctly. Regression analyses were conducted to examine relationships between item linguistic characteristics as independent variables with item difficulty as the dependent variable. Language characteristics had moderate effects on item difficulty at 4th grade, dropping to small-to-medium effects at 10th grade. ELL and SWD groups were not disproportionately affected by language characteristics in these test items. Difficult mathematics vocabulary had a consistent effect on performance for all students at all grades. Ambiguous or multiple-meaning words increased...
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paper_title: Structural variables affecting CAI performance on arithmetic word problems of disadvantaged and deaf students

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SummaryWe have shown that it is possible to account for a substantial portion of variability in student responses to arithmetic word problems using variables that describe structural features of the problems. However, the results obtained at this stage in our investigations are situation-dependent. The greater variance in observed proportion correct compared with the variance in the predicted proportions, and the differing sets of variables contributing significantly to the regressions come as no surprise. First, the population for this study differed from that used in the pilot study. Second, it is clear that characteristics of the problem set, for example, the frequency of occurrence of exemplars for the range of values for each variable and the way variable values are combined in problem types, affect the weighting for each variable in the regression analysis. Thus, differences were expected because different problem sets were used for the pilot study and the present study.In the light of these differences, the similarity in performance of the two disadvantaged groups gains in significance, and deserves further study. We believe we can increase the generalizability of our results by redesigning the basic problem set to exemplify in a balanced fashion the full range of variables found to account for problem difficulty. Given, however, the difficulty of making accurate predictions about problem-solving results, the correctness of this belief needs to be explicitly tested.
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paper_title: The Consistency Effect Depends on Markedness in Less Successful but Not Successful Problem Solvers: An Eye Movement Study in Primary School Children.

paper_content:

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.
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paper_title: Children's Interpretations of Arithmetic Word Problems

paper_content:

Two experiments were conducted to investigate children's interpretations of standard arithmetic word problems and the factors that influence their interpretations. In Experiment 1, children were required to solve a series of problems and then to draw and select pictures that represented the problems' structures. Solution performance was found to vary systematically with the nature of the representations drawn and chosen. The crucial determinant of solution success was the interpretation a child assigned to certain phrases used in the problems. In Experiment 2, solution and drawing accuracy were found to be significantly improved by rewording problems to avoid ambiguous linguistic forms. Together, these results imply that (a) word-problem solution errors are caused by misinterpretations of certain verbal expressions commonly used in problem texts, and (b) these misinterpretations are the result of missing or inadequate mappings of these verbal expressions to part-whole knowledge.
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paper_title: Extraneous information and extra steps in arithmetic word problems

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Abstract To determine how middle school students cope with some of the demands imposed on them by arithmetic word problems, 140 eighth graders were asked to solve word problems modeled after those used by the National Assessment of Educational Progress. Processing demands were imposed on the students by adding extraneous information and extra steps to the problems. Results indicated that the presence of extraneous information and extra steps reduced the accuracy of students' solutions. Thinking-out-loud protocols also revealed several misconceptions that students have about solving word problems.
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paper_title: Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

paper_content:

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.
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paper_title: Semantic characteristics that make arithmetic word problems difficult

paper_content:

This paper was published in the 2001 Proceedings of the 24th Annual Conference of the Mathematics Education Research Group of Australasia Incorporated held at University of Sydney, Australia from 30 June-4 July, 2001
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paper_title: Comprehension of arithmetic word problems: Evidence from students’ eye fixations

paper_content:

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
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paper_title: Influence of situational and conceptual rewording on word problem solving

paper_content:

Se analiza la doble naturaleza semantica y textual de la tarea de resolver un problema matematico en el aula.
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paper_title: Text Integration and Mathematical Connections: A Computer Model of Arithmetic Word Problem Solving

paper_content:

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence-level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence-level and text integration processes. Working memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K-3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process-oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships.
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paper_title: Adding Apples and Oranges : Alignment of Semantic and Formal Knowledge

paper_content:

We show that the same mechanism that mediates analogical reasoning (i.e., structural alignment) leads to interpretive "content effects" in reasoning about arithmetic word problems. Specifically, we show that both college students and textbook writers tend to construct arithmetic word problems that maintain systematic correspondence between the semantic relations that people infer from pairs of real-world objects (e.g., apples and baskets support the semantic relation CONTAIN [content, container]) and mathematical relations between arguments of arithmetic operations (e.g., DIVIDE [dividend, divisor]). Such relational alignments, to which we refer here as semantic alignments, lead to selective and sensible application of abstract formal knowledge. For example, people usually divide apples among baskets rather than baskets among apples, and readily add apples and oranges but refrain from adding apples and baskets.
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paper_title: ANALYSIS OF PRECURRENT SKILLS IN SOLVING MATHEMATICS STORY PROBLEMS

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We conducted an analysis of precurrent skills (responses that increase the effectiveness of a subsequent or "current" behavior in obtaining a reinforcer) to facilitate the solution of arithmetic word (story) problems. Two students with developmental disabilities were taught four precurrent responses (identifying the initial value, change value, operation, and resulting value) in a sequential manner. Results of a multiple baseline design across behaviors showed that the teaching procedures were effective in increasing correct performance of each of the precurrent behaviors with untaught problems during probes and that once the precurrent behaviors were established, the number of correct problem solutions increased.
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paper_title: Extending the Mental Number Line A Review of Multi-Digit Number Processing

paper_content:

Multi-digit number processing is ubiquitous in our everyday life - even in school, multi-digit numbers are computed from the first year onward. Yet, many problems children and adults have are about the relation of different digits (for instance with fractions, decimals, or carry effects in multi-digit addition). Cognitive research has mainly focused on single-digit processing, and there is no comprehensive review of the different multi-digit number processing types and effects. The current review aims to fill this gap. First, we argue that effects observed in single- digit tasks cannot simply be transferred to multi-digit processing. Next, we list 16 effect types and processes which are specific for multi-digit number processing. We then discuss the development of multi-digit number processing, its neurocognitive correlates, its cultural or language- related modulation, and finally some models for multi-digit number processing. We finish with conclusions and perspectives about where multi- digit number processing research may or should be heading in following years.
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paper_title: Arithmetic word problem solving: a Situation Strategy First framework

paper_content:

Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing the numerical solution is the representation of the problem modified so that the relevant arithmetic knowledge might be used. Three experiments were conducted with Year 3 and Year 4 children. Subtraction, multiplication and division problems were created in two versions involving the same wording but different numerical values. The first version could be mentally solved with a Situation strategy (Si version) and the second with a Mental Arithmetic strategy (MA version). Results show that Si-problems are easier than MA-problems even after instruction, and, when children were asked to report their strategy by writing a number sentence, equations that directly model the situation were predominant for Si-problems but not for MA ones. Implications of the Situation Strategy First framework regarding the relation between conceptual and procedural knowledge and the development of arithmetic knowledge are discussed.
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paper_title: The strategic use of alternative representations in arithmetic word problem solving

paper_content:

Multiple-step arithmetic problems can be solved by diverse strategies depending on the mental representation constructed by individuals from the situation described in the text of the problem. This representation will indeed determine the organization of subgoals to be reached or in other words the order of completion of calculations. This study aims at determining the conditions under which specific strategies are set up by adults. Using a paradigm that allows us to assess when calculations are performed, we show that adults usually organize their subgoals as they are explicitly mentioned in the problem, even though a strategy that is less demanding on working memory could have been used. However, we show that increasing the difficulty of the problem leads individuals to set up more economic strategies. Moreover, these economic strategies are even more likely to be used when the cognitive cost of the construction of the representation they rely on is low.
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paper_title: Automatic activation of addition facts in arithmetic word problems

paper_content:

Studies of mental arithmetic have shown that adults solve simple arithmetic problems by retrieving an answer automatically from a network of stored associations. However, most studies have been limited to single-digit addition and multiplication problems. In this article, we examine whether retrieval is also automatic in the context of more complex arithmetic tasks, such as arithmetic word problems. To test this hypothesis, we used a priming procedure with a target-naming task, in which the primes were the numbers included in two sentences containing the numerical information of an arithmetic word problem (e.g., 3 and 2 in “Joe had 3 marbles. Then Tom gave him 2 marbles”), and the targets were either congruent (e.g., 5) or incongruent (e.g., 8) with the prime. A neutral prime was also used replacing the numbers of the problem by capital letters (e.g., X and Y). Manipulating the relationship between the prime and the target and the duration of time that separates these two events, the overall results revea...
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paper_title: Remediating Number Combination and Word Problem Deficits Among Students With Mathematics Difficulties: A Randomized Control Trial

paper_content:

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students’ math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20–30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students’ mathematics difficulty status. The tutoring protocols proved transportable across sites.
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paper_title: Neighborhood consistency in mental arithmetic: Behavioral and ERP evidence

paper_content:

BackgroundRecent cognitive and computational models (e.g. the Interacting Neighbors Model) state that in simple multiplication decade and unit digits of the candidate answers (including the correct result) are represented separately. Thus, these models challenge holistic views of number representation as well as traditional accounts of the classical problem size effect in simple arithmetic (i.e. the finding that large problems are answered slower and less accurate than small problems). Empirical data supporting this view are still scarce.MethodsData of 24 participants who performed a multiplication verification task with Arabic digits (e.g. 8 × 4 = 36 - true or false?) are reported. Behavioral (i.e. RT and errors) and EEG (i.e. ERP) measures were recorded in parallel.ResultsWe provide evidence for neighborhood-consistency effects in the verification of simple multiplication problems (e.g. 8 × 4). Behaviorally, we find that decade-consistent lures, which share their decade digit with the correct result (e.g. 3 6), are harder to reject than matched inconsistent lures, which differ in both digits from the correct result (e.g. 28). This neighborhood consistency effect in product verification is similar to recent observations in the production of multiplication results. With respect to event-related potentials we find significant differences for consistent compared to inconsistent lures in the N400 (increased negativity) and Late Positive Component (reduced positivity). In this respect consistency effects in our paradigm resemble lexico-semantic effects earlier found in simple arithmetic and in orthographic input processing.ConclusionOur data suggest that neighborhood consistency effects in simple multiplication stem at least partly from central (lexico-semantic') stages of processing. These results are compatible with current models on the representation of simple multiplication facts – in particular with the Interacting Neighbors Model – and with the notion of decomposed representations of two-digit numbers in general.
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paper_title: All for one but not one for all: How multiple number representations are recruited in one numerical task

paper_content:

Number processing recruits a complex network of multiple numerical representations. Usually the components of this network are examined in a between-task approach with the disadvantage of relying upon different instructions, tasks, and inhomogeneous stimulus sets across different studies. A within-task approach may avoid these disadvantages and access involved numerical representations more specifically. In the present study we employed a within-task approach to investigate numerical representations activated in the number bisection task (NBT) using parametric rapid event-related fMRI. Participants were to judge whether the central number of a triplet was also its arithmetic mean (e.g. 23_26_29) or not (e.g. 23_25_29). Activation in the left inferior parietal cortex was associated with the deployment of arithmetic fact knowledge, while activation of the intraparietal cortex indicated more intense magnitude processing, instrumental aspects of calculation and integration of the base-10 structure of two-digit numbers. These results replicate evidence from the literature. Furthermore, activation in the dorsolateral and ventrolateral prefrontal cortex revealed mechanisms of feature monitoring and inhibition as well as allocation of cognitive resources recruited to solve a specific triplet. We conclude that the network of numerical representations should rather be studied in a within-task approach than in varying between-task approaches.
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paper_title: Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

paper_content:

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.
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paper_title: Attention, automaticity, and developmental dyscalculia.

paper_content:

People suffering from developmental dyscalculia (DD) show an abnormal pattern of the size congruity effect. They do not display a facilitation component in a numerical Stroop task. In this task, participants are presented with 2 digits that differ both in physical size and numerical value, and they have to compare the digits while ignoring one of the dimensions. The present study examined performance of those with DD and control participants in the numerical Stroop task under cognitive load. The no-load condition replicated previous findings (i.e., lack of facilitation in the physical task for the DD group). Load had opposite effects on interference and facilitation. Load eliminated facilitation and increased interference in the control group. Load increased interference only in the physical task in the DD group. The opposite effect of load on facilitation and interference suggests that these components are related to different cognitive mechanisms. The fact that load produced a DD-like pattern in the control group could suggest that individuals with DD suffer from difficulty in recruiting attention in addition to the deficits in numerical processing.
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paper_title: Interacting neighbors: A connectionist model of retrieval in single-digit multiplication

paper_content:

For most adults, retrieval is the most common way to solve a single-digit multiplication problem (Campbell & Xue, 2001). Many theories have been proposed to describe the underlying mechanism of arithmetical fact retrieval. Testing their validity hinges on evaluating how well they account for the basic findings in mental arithmetic. The most important findings are the problem size effect (small multiplication problems are easier than larger ones; cf. 3 × 2 and 7 × 8), the five effect (problems with 5 are easier than can be accounted for by their size), and the tie effect (problems with identical operands are easier than other problems; cf. 8 × 8 and 8 × 7). We show that all existing theories have difficulties in accounting for one or more of these phenomena. A new theory is presented that avoids these difficulties. The basic assumption is that candidate answers to a particular problem are in cooperative/competitive interactions and these interactions favor small, five, and tie problems. The theory is implemented as a connectionist model, and simulation data are described that are in good accord with empirical data.
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paper_title: Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

paper_content:

One of the five content strands for students across kindergarten through twelfth grade of the Principles and Standards for School Mathematics is algebra (National Council of Teachers of Mathematics, 2000). Algebraic reasoning is also a Common Core domain area in the elementary grades, an integral part of several domain areas in middle school, and one of six important areas of study for high school students (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). As the standards of these national organizations press for algebra instruction for all students across elementary, middle, and high school, as more states require all students to take and pass algebra courses in high school (Stein, Kaufman, Sherman, & Hillen, 2011), and given that many secondary students, including students with disabilities, are unprepared for algebra courses (Faulkner, Crossland, & Stiff, 2013; National Mathematics Advisory Panel, 2008), the elementary school curriculum needs to emphasize algebraic reasoning more than ever. ::: ::: The widespread adoption of the Common Core indicates that districts are amending curricula to reflect the standards outlined in the Common Core. The Common Core emphasizes algebraic reasoning in the elementary grades with standards listed under the “Operations and Algebraic Thinking” domain. For example, at kindergarten, when given a number 1 through 9, students have to determine the other number (i.e., unknown) that adds to 10. In first grade, students solve addition and subtraction equations with a unknown (e.g., 8 + ? = 11; 5 = + 3). In grades 1, 2, 3, and 4, students use equations with a symbol for the unknown to represent and solve word problems. By grade 3, students solve multiplication and division equations with an unknown (e.g., 3 × = 27; 9 = 81 ÷ ?). In grades 4 and 5, students generate patterns to describe the relationship between sets of numbers (e.g., n + 3). In the middle school grades, the Common Core standards explicitly related to algebraic reasoning fall under the “Expressions and Equations” domain (grades 6, 7, and 8) and the “Functions” domain (grades 8). Many algebraic reasoning skills, especially those related to problem solving, fall under other domains (e.g., Ratios and Proportional Relationships, Measurement) at grades 6, 7, and 8. ::: ::: The Common Core standards influence statewide assessments, which in turn influence mathematics textbooks. Districts want to purchase curriculum materials aligned with the assessments to provide a cohesive mathematics program that enables students to succeed on the assessment. As more and more districts revise their mathematics curriculum to meet or exceed Common Core standards and as all students, including those with learning disabilities, are held accountable on statewide assessments (Yell, Katsiyannis, & Shiner 2006), it is important to understand how students, especially those with mathematics difficulties, interpret and solve algebraic problems. ::: ::: In this introduction, we explain how most researchers approach algebraic reasoning in the elementary grades. We discuss solving algebraic problems using equations and functions. Then, we describe students with mathematics difficulties; the two main areas (i.e., calculations and word problems) in which they struggle during the elementary grades; and how algebraic reasoning may be connected to calculations and to word problems. Finally, we describe the rationale, purpose, and hypotheses for the present study.
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paper_title: Error analysis and the corresponding cognitive activities committed by year five primary students in solving mathematical word problems

paper_content:

Abstract The purpose of this study was to identify errors and the corresponding cognitive activities committed by the 374 year five primary school students in solving the mathematical word problems for the ‘Fraction’ topic. The design of this study used Newman Error Analysis method and interviews. The findings showed that 52.91% of the errors made were due to lack of understanding, followed by transformation skills (22.37%), process skills (15.55%), encoding (8.84%) and reading (0.34 Based on these findings, the researchers proposed that teachers should ask students explicitly what problems they have in solving the mathematical word problems, observe students reactions and analyze errors committed by them.
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paper_title: The relationship between different kinds of students' errors and the knowledge required to solve mathematics word problems

paper_content:

The main objective of this research is to examine the relationship between different kinds of errors and the knowledge required to solve word problems in Arithmetic, Algebra and Geometry. Kinfong's and Holtan's framework supports the analysis of the errors, and Mayer's theory was implemented to understand the necessary knowledge for solving math word problems. The research methodology follows a semi-experimental method. Research tools comprise both a descriptive math test and a directed interview. The research findings revealed that students' errors when solving arithmetic word problems result from the lack of linguistic, semantic, structural and communicational knowledge; when solving the geometric word problems, the lack of semantic, intuition and structural knowledge were the cause of the students' errors. Regarding algebra word problems, miscalculation was the reason for the higher error rate. Results show that the highest deficiency is mainly related to the lack of semantic, structural and communicational knowledge.
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paper_title: What makes multiplication facts difficult. Problem size or neighborhood consistency?

paper_content:

Two current models of arithmetic fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are "consistent," compared with "inconsistent" errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN-model differ in their explanations of the problem-size effect, a hallmark finding robustly observed in arithmetic fact retrieval. Only the IN-model predicts that a correct answer's neighborhood consistency can fully account for the problem-size effect, which was confirmed in our analysis.
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paper_title: The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning

paper_content:

This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem-solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. We conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than anoth...
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paper_title: Decade breaks in the mental number line? Putting the tens and units back in different bins

paper_content:

Abstract Most models of number recognition agree that among other number representations there is a central semantic magnitude representation which may be conceptualized as a logarithmically compressed mental number line. Whether or not this number line is decomposed into different representations for tens and units is, however, controversial. We investigated this issue in German participants in a magnitude comparison (selection) task in which the larger of two visually presented Arabic two-digit numbers had to be selected. Most importantly, we varied unit–decade-compatibility: a number pair was defined as compatible if the decade magnitude comparison and the unit magnitude comparison of the two numbers would lead to the same response (e.g. 52 and 67) and as incompatible if this was not the case (e.g. 47 and 62). While controlling for overall numerical distance, size and other variables, we consistently found compatibility effects. A control experiment showed that this compatibility effect was not due to perceptual presentation characteristics. We conclude that the idea of one single number line representation that does not additionally assume different magnitude representations for tens and units is not sufficient to account for the data. Finally, we discuss why decade effects were not found in other experimental settings.
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paper_title: Parity Influences the Difficulty of Simple Addition and Subtraction but Not Multiplication Problems in Children

paper_content:

Summary.—In the present paper, the results of early 20th century studies of children's difficulties with basic addition, subtraction, and multiplication problems are analyzed for parity effects. Parity influenced the difficulty of addition and subtraction problems, tasks in which any parity influence must be implicit, but did not influence difficulty of multiplication problems. Solution of multiplication problems relies to a greater extent on retrieval of rote-memorized answers. Addition and subtraction problems rely more on non-retrieval strategies. It is these latter strategies that depend on internal representations sensitive to the parity status of the numbers being processed. That parity affected responding in tasks where parity was irrelevant and no overt motor responses were made poses problems for the Markedness of Response Codes (MARC) and polarity explanations of parity effects in reaction time. Both these explanations require that an explicit parity judgment indicated by a binary motor response...
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paper_title: Working memory and phonological processing as predictors of children’s mathematical problem solving at different ages

paper_content:

The study explored the contribution of working memory (WM) to mathematical problem solving in younger (8-year-old) and older (11-year-old) children. The results showed that (1) significant agerelated differences in WM performance were maintained when measures of phonological processing (i.e., digit naming speed, short-term memory, phonological deletion) were partialed from the analysis; (2) WM predicted solution accuracy of word problems independently of measures of problem representation, knowledge of operations and algorithms, phonological processing, fluid intelligence, reading, and math skill; and (3) a second-order WM factor was correlated with problem solving, suggesting that a general or executive system underlies age-related performance. The results were interpreted as support for the notion that the executive system was an important predictor of age-related changes in problem solving beyond the contribution of math and reading skills, and this system operates independently of the phonological system and domain-specific knowledge in predicting solution accuracy.
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paper_title: Neighborhood consistency in mental arithmetic: Behavioral and ERP evidence

paper_content:

BackgroundRecent cognitive and computational models (e.g. the Interacting Neighbors Model) state that in simple multiplication decade and unit digits of the candidate answers (including the correct result) are represented separately. Thus, these models challenge holistic views of number representation as well as traditional accounts of the classical problem size effect in simple arithmetic (i.e. the finding that large problems are answered slower and less accurate than small problems). Empirical data supporting this view are still scarce.MethodsData of 24 participants who performed a multiplication verification task with Arabic digits (e.g. 8 × 4 = 36 - true or false?) are reported. Behavioral (i.e. RT and errors) and EEG (i.e. ERP) measures were recorded in parallel.ResultsWe provide evidence for neighborhood-consistency effects in the verification of simple multiplication problems (e.g. 8 × 4). Behaviorally, we find that decade-consistent lures, which share their decade digit with the correct result (e.g. 3 6), are harder to reject than matched inconsistent lures, which differ in both digits from the correct result (e.g. 28). This neighborhood consistency effect in product verification is similar to recent observations in the production of multiplication results. With respect to event-related potentials we find significant differences for consistent compared to inconsistent lures in the N400 (increased negativity) and Late Positive Component (reduced positivity). In this respect consistency effects in our paradigm resemble lexico-semantic effects earlier found in simple arithmetic and in orthographic input processing.ConclusionOur data suggest that neighborhood consistency effects in simple multiplication stem at least partly from central (lexico-semantic') stages of processing. These results are compatible with current models on the representation of simple multiplication facts – in particular with the Interacting Neighbors Model – and with the notion of decomposed representations of two-digit numbers in general.
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paper_title: Does Understanding Relational Terminology Mediate Effects of Intervention on Compare Word Problems

paper_content:

Abstract The purpose of this study was to assess whether understanding relational terminology (i.e., more, less, and fewer) mediates the effects of intervention on compare word problems. Second-grade classrooms (N = 31) were randomly assigned to one of three conditions: researcher-designed word-problem intervention, researcher-designed calculation intervention, or business-as-usual (teacher-designed) control. Students in word-problem intervention classrooms received instruction on the compare problem type, which included a focus on understanding relational terminology within compare word problems. Analyses, which accounted for variance associated with classroom clustering, indicated that (a) compared with the calculation intervention and business-as-usual conditions, word-problem intervention significantly increased performance on all three subtypes of compare problems and on understanding relational terminology, and (b) the intervention effect was fully mediated by students’ understanding of relational terminology for one subtype of compare problems and partially mediated by students’ understanding of relational terminology for the other two subtypes.
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paper_title: Dissociating the solution processes of small, large, and zero multiplications by means of fMRI

paper_content:

The present study investigated the neuroanatomical basis of different solution processes in single-digit multiplication by means of fMRI. Sixteen participants silently produced the solution of three distinct types of multiplication, i.e., problems involving zero (e.g., 3*0), small (e.g., 2*4), or large operands (e.g., 8*7). Zero and small problems are assumed to be solved by rule application and fact retrieval, respectively, and problems with large operands sometimes involve backup strategies when direct retrieval is not sufficient. Small problems, when compared with a high-level baseline not requiring any kind of calculation, activated a network of parietal, subcortical, and frontal areas. This activation pattern supports the hypothesis that arithmetic fact retrieval is mediated by a verbal processing loop including the angular gyrus and the basal ganglia. Problems with larger operands showed increased activation in the anterior cingulate cortex, SMA, and the left inferior frontal gyrus, which could reflect increased conflict during the fact-retrieval process, but also higher demands for controlling and coordinating multiple processing steps when a problem cannot be solved by direct retrieval. Zero problems, in comparison to multiplications with small operands, activated the caudate nucleus and the right inferior frontal cortex, showing that rule application is separable from fact retrieval on a neuroanatomical level, too.
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paper_title: Error Analysis of Mathematical Word Problem Solving Across Students with and without Learning Disabilities

paper_content:

Solving word problems is a common area of struggle for students with learning disabilities (LD). In order for instruction to be effective, we first need to have a clear understanding of the specific errors exhibited by students with LD during problem solving. Error analysis has proven to be an effective tool in other areas of math but has had little application to errors in word problems. Using an error analysis approach, this study aimed to investigate in depth the various types and frequency of errors made by students with LD and their AA peers during math problem solving. The resulting similarities and differences between the two groups of students are discussed with insight into underlying cognitive processes, and implications for future research.
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paper_title: The Acquisition of Addition and Subtraction Concepts in Grades One through Three.

paper_content:

Children's solutions to simple addition and subtraction word problems were studied in a 3-year longitudinal study that followed 88 children from Grades 1 through 3. The children were able to solve the problems using a variety of modeling and counting strategies even before they received formal instruction in arithmetic. The invented strategies continued to be used after several years of formal instruction. Four levels of problemsolving ability were found. At the first level, children could solve problems only by externally modeling them with physical objects. Modeling strategies were gradually replaced with more sophisticated counting strategies. The results of the study are at variance with important aspects of models of children's performance proposed by Briars and Larkin and by Riley, Greeno, and Heller.
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paper_title: Error analysis and the corresponding cognitive activities committed by year five primary students in solving mathematical word problems

paper_content:

Abstract The purpose of this study was to identify errors and the corresponding cognitive activities committed by the 374 year five primary school students in solving the mathematical word problems for the ‘Fraction’ topic. The design of this study used Newman Error Analysis method and interviews. The findings showed that 52.91% of the errors made were due to lack of understanding, followed by transformation skills (22.37%), process skills (15.55%), encoding (8.84%) and reading (0.34 Based on these findings, the researchers proposed that teachers should ask students explicitly what problems they have in solving the mathematical word problems, observe students reactions and analyze errors committed by them.
---
paper_title: What makes multiplication facts difficult. Problem size or neighborhood consistency?

paper_content:

Two current models of arithmetic fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are "consistent," compared with "inconsistent" errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN-model differ in their explanations of the problem-size effect, a hallmark finding robustly observed in arithmetic fact retrieval. Only the IN-model predicts that a correct answer's neighborhood consistency can fully account for the problem-size effect, which was confirmed in our analysis.
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paper_title: Influence of situational and conceptual rewording on word problem solving

paper_content:

Se analiza la doble naturaleza semantica y textual de la tarea de resolver un problema matematico en el aula.
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paper_title: Separate roles for executive and phonological components of working memory in mental arithmetic

paper_content:

A dual-task methodology was used to investigate the roles played by executive and phonological aspects of working memory in mental arithmetic. Experiment 1 showed that suppression of articulation impaired the ability to add a pair of briefly presented three-digit numbers. Suppression had no effect when the need to store temporarily was minimized by making the numbers visible throughout calculation. Experiment 2 showed that disrupting executive processes by requiring concurrent performance of a Trails task impaired the ability to add numbers that remained permanently visible. Performance on the Trails task deteriorated as the number of carry operations in the addition increased. Experiment 3 showed that this decline in Trails performance was not simply due to the extra time taken by carrying. These and other features of the results suggest that the carrying component of mental arithmetic places substantial demands on executive processes, whereas the need to retain problem information is met by the phonological loop. The results are consistent with an interpretation of executive processes according to which there is a limit on the capacity to inhibit strongly primed routine operations.
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paper_title: To retrieve or to calculate? Left angular gyrus mediates the retrieval of arithmetic facts during problem solving

paper_content:

While there is consistent evidence from neuropsychological and brain imaging studies for an association between the left angular gyrus and mental arithmetic, its specific role in calculation has remained poorly understood. It has been speculated that the angular gyrus mediates the retrieval of arithmetic facts during problem solving, but this hypothesis has not been directly tested. In the present functional Magnetic Resonance Imaging study comprising 28 adults, we used trial-by-trial strategy self-reports to identify brain regions underpinning different strategies in arithmetic problem solving. Analyses revealed stronger activation of the left angular gyrus while solving arithmetic problems for which participants reported fact retrieval whereas the application of procedural strategies was accompanied by widespread activation in a fronto-parietal network. These data directly link the left angular gyrus with arithmetic fact retrieval and show that strategy self-reports can be used to predict differential patterns of brain activation.
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paper_title: Working memory and phonological processing as predictors of children’s mathematical problem solving at different ages

paper_content:

The study explored the contribution of working memory (WM) to mathematical problem solving in younger (8-year-old) and older (11-year-old) children. The results showed that (1) significant agerelated differences in WM performance were maintained when measures of phonological processing (i.e., digit naming speed, short-term memory, phonological deletion) were partialed from the analysis; (2) WM predicted solution accuracy of word problems independently of measures of problem representation, knowledge of operations and algorithms, phonological processing, fluid intelligence, reading, and math skill; and (3) a second-order WM factor was correlated with problem solving, suggesting that a general or executive system underlies age-related performance. The results were interpreted as support for the notion that the executive system was an important predictor of age-related changes in problem solving beyond the contribution of math and reading skills, and this system operates independently of the phonological system and domain-specific knowledge in predicting solution accuracy.
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paper_title: Automatic activation of addition facts in arithmetic word problems

paper_content:

Studies of mental arithmetic have shown that adults solve simple arithmetic problems by retrieving an answer automatically from a network of stored associations. However, most studies have been limited to single-digit addition and multiplication problems. In this article, we examine whether retrieval is also automatic in the context of more complex arithmetic tasks, such as arithmetic word problems. To test this hypothesis, we used a priming procedure with a target-naming task, in which the primes were the numbers included in two sentences containing the numerical information of an arithmetic word problem (e.g., 3 and 2 in “Joe had 3 marbles. Then Tom gave him 2 marbles”), and the targets were either congruent (e.g., 5) or incongruent (e.g., 8) with the prime. A neutral prime was also used replacing the numbers of the problem by capital letters (e.g., X and Y). Manipulating the relationship between the prime and the target and the duration of time that separates these two events, the overall results revea...
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paper_title: Kindergartners' Understanding of Additive Commutativity within the Context of Word Problems

paper_content:

Abstract Baroody and Gannon (1984) proposed that children's understanding of additive commutativity progresses through several levels of understanding based on a unary view of addition (change meaning) before developing a “true” level of understanding based on a binary conception (part-whole meaning). Resnick (1992) implied that children have both a unary and a binary conception of additive commutativity from the earliest stages of development. Fifty-three 5- and 6-year-old (M = 6-0) kindergartners' unary and binary understanding of additive commutativity was investigated using performance on tasks involving change-add-to and part-part-whole word problems, respectively. The data were inconsistent with the predictions of both models and suggest three alternate theoretical explanations. Moreover, the data indicate that success on a task involving change-add-to problems may be a more rigorous test of understanding of additive commutativity than that involving part-part-whole problems.
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paper_title: Remediating Number Combination and Word Problem Deficits Among Students With Mathematics Difficulties: A Randomized Control Trial

paper_content:

The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students’ math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20–30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students’ mathematics difficulty status. The tutoring protocols proved transportable across sites.
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paper_title: Does Understanding Relational Terminology Mediate Effects of Intervention on Compare Word Problems

paper_content:

Abstract The purpose of this study was to assess whether understanding relational terminology (i.e., more, less, and fewer) mediates the effects of intervention on compare word problems. Second-grade classrooms (N = 31) were randomly assigned to one of three conditions: researcher-designed word-problem intervention, researcher-designed calculation intervention, or business-as-usual (teacher-designed) control. Students in word-problem intervention classrooms received instruction on the compare problem type, which included a focus on understanding relational terminology within compare word problems. Analyses, which accounted for variance associated with classroom clustering, indicated that (a) compared with the calculation intervention and business-as-usual conditions, word-problem intervention significantly increased performance on all three subtypes of compare problems and on understanding relational terminology, and (b) the intervention effect was fully mediated by students’ understanding of relational terminology for one subtype of compare problems and partially mediated by students’ understanding of relational terminology for the other two subtypes.
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paper_title: Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

paper_content:

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.
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paper_title: The strategic use of alternative representations in arithmetic word problem solving

paper_content:

Multiple-step arithmetic problems can be solved by diverse strategies depending on the mental representation constructed by individuals from the situation described in the text of the problem. This representation will indeed determine the organization of subgoals to be reached or in other words the order of completion of calculations. This study aims at determining the conditions under which specific strategies are set up by adults. Using a paradigm that allows us to assess when calculations are performed, we show that adults usually organize their subgoals as they are explicitly mentioned in the problem, even though a strategy that is less demanding on working memory could have been used. However, we show that increasing the difficulty of the problem leads individuals to set up more economic strategies. Moreover, these economic strategies are even more likely to be used when the cognitive cost of the construction of the representation they rely on is low.
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paper_title: Solving arithmetic word problems: an analysis of classification as a function of difficulty in children with and without arithmetic LD.

paper_content:

This study was designed to determine a word problem difficulty classification in children with arithmetic learning disabilities (ALD; n = 104) in comparison with typically achieving students (n = 44). We tested variables such as (a) semantic structure (Change, Combine, Compare, and Equalize), (b) operation (subtraction and addition), and (c) position of the unknown quantity in the problem. Facet theory with multidimensional scaling techniques (MINISSA) was used to analyze the underlying dimensions in the responses of each group of participants. Our results indicate that although the word problem difficulty classifications for the 2 groups of children were different, the position of the unknown quantity had a greater influence on the level of difficulty of story problems than other variables. The noncanonical problems--specifically, those with the unknown term in the first place--although difficult for both groups of children, were the most difficult problems for children with ALD.
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paper_title: Extraneous information and extra steps in arithmetic word problems

paper_content:

Abstract To determine how middle school students cope with some of the demands imposed on them by arithmetic word problems, 140 eighth graders were asked to solve word problems modeled after those used by the National Assessment of Educational Progress. Processing demands were imposed on the students by adding extraneous information and extra steps to the problems. Results indicated that the presence of extraneous information and extra steps reduced the accuracy of students' solutions. Thinking-out-loud protocols also revealed several misconceptions that students have about solving word problems.
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paper_title: Error Analysis of Mathematical Word Problem Solving Across Students with and without Learning Disabilities

paper_content:

Solving word problems is a common area of struggle for students with learning disabilities (LD). In order for instruction to be effective, we first need to have a clear understanding of the specific errors exhibited by students with LD during problem solving. Error analysis has proven to be an effective tool in other areas of math but has had little application to errors in word problems. Using an error analysis approach, this study aimed to investigate in depth the various types and frequency of errors made by students with LD and their AA peers during math problem solving. The resulting similarities and differences between the two groups of students are discussed with insight into underlying cognitive processes, and implications for future research.
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paper_title: Seeing the Math in the Story On How Abstraction Promotes Performance on Mathematical Word Problems

paper_content:

The negative social, health, financial, and other life outcomes associated with mathematical proficiency deficits highlight the need to understand the underlying cognitive operations entailed in solving math problems. We focus specifically on mathematical word problems and propose that abstraction can enhance performance by helping people see beyond the incidental details described in word problems and to recognize instead the underlying mathematical relationships. Three studies manipulated abstraction as a procedural mind-set (i.e., inducing abstraction in one task and observing its “carry-over” effect in subsequent unrelated tasks) and observed performance on both numeric and word problems. Participants in the abstract, relative to concrete, mind-set condition were more successful in translating word problems into their analogous numeric forms, resulting in improved performance. We discuss implications of these findings for understanding individual and group differences in mathematics proficiencies, whi...
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paper_title: An fMRI study of the Interplay of Symbolic and Visuo-spatial Systems in Mathematical Reasoning

paper_content:

An fMRI study of the Interplay of Symbolic and Visuo-spatial Systems in Mathematical Reasoning Atsushi Terao (atsushi@edu.hokudai.ac.jp) Graduate School of Education, Hokkaido University, Kita 11 Nishi 7 Sapporo, 060-0811 JAPAN Kenneth R. Koedinger (koedinger@cmu.edu) Human-Computer Interaction Institute, Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3890 USA Myeong-Ho Sohn (mhsohn@andrew.cmu.edu) Yulin Qin (yulinQ@andrew.cmu.edu) John R. Anderson (ja+@cmu.edu) Department of Psychology, Carnegie Mellon University 5000 Forbes Ave. Pittsburgh, PA 15213-3890 USA Cameron S. Carter (CarterCS@msx.upmc.edu) Imaging Research Center, University of California at Davis 1 Shields Ave. Davis, CA 95616 USA Abstract The purpose of this fMRI study was to provide evidence for the mathematician’s belief that mathematical thinking emerges from the interplay between symbolic and visuo- spatial systems. Twelve participants were given algebra word problems and depicted the quantitative relations on a mental number line or made parts of an equation. The regions activated in depicting the picture were also recruited to make an equation. Mathematics is a language. Many scientists say that mathematics is a language to describe the nature of phenomena they are looking at. It is well known that Nicolas Burubaki, a group of mathematicians, stressed the crucial role of formal symbol systems in mathematics. On the other hand, many mathematicians and physicists emphasize the role of visuo-spatial reasoning in mathematics, which recruits qualitative, language- independent representations. For example, Albert Einstein stated “Words and language, whether written or spoken, do not seem to play any part in my thought process.” As Dehaene, Spelke, Pinel, Stanescu, and Tsivkn (1999) suggested, mathematical thinking may emerge from the interplay between symbolic and visuo-spatial systems. In this fMRI study, we approach this problem and provide evidence for this kind of mathematical thinking. Psychological studies have revealed that if a problem apparently looks like a pure symbolic task, it can require students to have some visuo-spatial representations. For example, Griffin, Case and Siegler (1994) showed that the mental “number line”, a qualitative representation of the number system, is crucial readiness for early arithmetic. Lewis used a number-line-like diagram to train undergraduate students having difficulty to solve “compare” word problems (problems containing more-than or less-than relations), and succeeded in improving their performance. Paige and Simon (1966) proposed that solving word problems is not a simple translation of problem sentences into equations, as Bobrow’s (1966) STUDENT did, but needs “physical cues,” a visuo-spatial representation. The 6 th grade students who used our “Picture Algebra” strategy (Koedinger & Terao, 2002) to solve the compare word problem showed relatively high performance. We expect that using this strategy may better prepare students to learn formal algebra. Functional magnetic resonance imaging (fMRI) gives us a new source of information about the mental representations used in mathematics. Dehaene et al. (1999) showed two different mental representations are used for different tasks. Exact calculation (e.g., 4+5=9) elicited left-lateralized activation in the left inferior frontal lobe, together with left angular gyrus and left anterior cingulate. This pattern was interpreted as suggesting that the participants recruited their symbolic systems and did language dependent encoding. Approximation (e.g., 4+5 is closer to 8 than 6), on the contrary, elicited bilateral parietal lobes activation. This pattern was interpreted as suggesting that the participants recruited visuo-spatial systems and did language independent encoding. Dehaene, Piazza, Pinel and Cohen (2003) reviewed neuro-imaging and neuropsychological evidence concerning various numerical tasks and proposed a hypothesis that three parietal circuits are related to number processing. The horizontal segment of the intraparietal sulcus (HIPS) appears to be a core quantity system, analogous to a mental number line. This area seems to be supplemented by two other circuits. One is the bilateral posterior superior parietal lobule (PSPL), which is considered to be involved in attention orientation on the mental number line. The other is the left angular gyrus, which is likely to support manipulations of numbers in a symbolic form (e.g., exact calculation).
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paper_title: An exploration of aspects of language proficiency and Algebra learning

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We have attempted to investigate whether 3 cognitive components of language proficiencymetalinguistic awareness of symbol, syntax, and ambiguity-are associated with students' success in learning the notation of algebra. Pencil-and-paper tests were given to assess students' metalinguistic awareness and their ability to use algebraic notation. In a total sample of more than 1500 students, aged 11 to 15, who were in their 1st to 4th years of algebra learning, we found that very few students with low metalinguistic awareness scores achieved high algebra scores. We discuss implications of this finding for the school algebra curriculum.
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paper_title: Language affects symbolic arithmetic in children: The case of number word inversion

paper_content:

Abstract Specific language influences have been observed in basic numerical tasks such as magnitude comparison, transcoding, and the number line estimation task. However, so far language influences in more complex calculations have not been reported in children. In this translingual study, 7- to 9-year-old German- and Italian-speaking children were tested on a symbolic addition task. Whereas the order of tens and units in Italian number words follows the order of the Arabic notation, the order is inverted in German number words. For both language groups, addition problems were more difficult when a carry operation was needed, that is, when a manipulation within the place-value structure of the Arabic number system was particularly important. Most important, this carry effect was more pronounced in response latencies for children speaking German, a language with inverted verbal mapping of the place-value structure. In addition, independent of language group, the size of the carry effect was significantly related to verbal working memory. The current study indicates that symbolic arithmetic and the carry effect in particular are modulated by language-specific characteristics. Our results underline the fact that the structure of the language of instruction is an important factor in children’s mathematical education and needs to be taken into account even for seemingly nonverbal symbolic Arabic tasks.
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paper_title: Separate roles for executive and phonological components of working memory in mental arithmetic

paper_content:

A dual-task methodology was used to investigate the roles played by executive and phonological aspects of working memory in mental arithmetic. Experiment 1 showed that suppression of articulation impaired the ability to add a pair of briefly presented three-digit numbers. Suppression had no effect when the need to store temporarily was minimized by making the numbers visible throughout calculation. Experiment 2 showed that disrupting executive processes by requiring concurrent performance of a Trails task impaired the ability to add numbers that remained permanently visible. Performance on the Trails task deteriorated as the number of carry operations in the addition increased. Experiment 3 showed that this decline in Trails performance was not simply due to the extra time taken by carrying. These and other features of the results suggest that the carrying component of mental arithmetic places substantial demands on executive processes, whereas the need to retain problem information is met by the phonological loop. The results are consistent with an interpretation of executive processes according to which there is a limit on the capacity to inhibit strongly primed routine operations.
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paper_title: Sixty-four or four-and-sixty? The influence of language and working memory on children's number transcoding

paper_content:

Number transcoding (e.g., writing 64 when hearing “sixty-four”) is a basic numerical skill; rather faultlessly performed in adults, but difficult for children. In the present study, children speaking Dutch (an inversed number language) and French (a non-inversed number language) wrote Arabic digits to dictation. We also tested their IQ and their phonological, visuospatial, and executive working memory. Although the number of transcoding errors (e.g., hearing 46 but writing 56) was equal in both groups, the number of inversion errors (e.g., hearing 46 but writing 64) was significantly higher in Dutch-speaking than in French-speaking children. Regression analyses confirmed that language was the only significant predictor of inversion errors. Working-memory components, in contrast, were the only significant predictors of transcoding errors. Executive resources were important in all children. Less-skilled transcoders also differed from more-skilled transcoders in that they used semantic rather than asemantic transcoding routes. Given the observed relation between number transcoding and mathematics grades, current findings may provide useful information for educational and clinical settings.
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paper_title: Object-based neglect in number processing

paper_content:

Recent evidence suggests that neglect patients seem to have particular problems representing relatively smaller numbers corresponding to the left part of the mental number line. However, while this indicates space-based neglect for representational number space little is known about whether and - if so - how object-based neglect influences number processing.To evaluate influences of object-based neglect in numerical cognition, a group of neglect patients and two control groups had to compare two-digit numbers to an internally represented standard. Conceptualizing two-digit numbers as objects of which the left part (i.e., the tens digit should be specifically neglected) we were able to evaluate object-based neglect for number magnitude processing.Object-based neglect was indicated by a larger unit-decade compatibility effect actually reflecting impaired processing of the leftward tens digits. Additionally, faster processing of within- as compared to between-decade items provided further evidence suggesting particular difficulties in integrating tens and units into the place-value structure of the Arabic number system.In summary, the present study indicates that, in addition to the spatial representation of number magnitude, also the processing of place-value information of multi-digit numbers seems specifically impaired in neglect patients.
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paper_title: Individual differences in solving arithmetic word problems

paper_content:

BackgroundWith the present functional magnetic resonance imaging (fMRI) study at 3 T, we investigated the neural correlates of visualization and verbalization during arithmetic word problem solving. In the domain of arithmetic, visualization might mean to visualize numbers and (intermediate) results while calculating, and verbalization might mean that numbers and (intermediate) results are verbally repeated during calculation. If the brain areas involved in number processing are domain-specific as assumed, that is, that the left angular gyrus (AG) shows an affinity to the verbal domain, and that the left and right intraparietal sulcus (IPS) shows an affinity to the visual domain, the activation of these areas should show a dependency on an individual’s cognitive style.Methods36 healthy young adults participated in the fMRI study. The participants habitual use of visualization and verbalization during solving arithmetic word problems was assessed with a short self-report assessment. During the fMRI measurement, arithmetic word problems that had to be solved by the participants were presented in an event-related design.ResultsWe found that visualizers showed greater brain activation in brain areas involved in visual processing, and that verbalizers showed greater brain activation within the left angular gyrus.ConclusionsOur results indicate that cognitive styles or preferences play an important role in understanding brain activation. Our results confirm, that strong visualizers use mental imagery more strongly than weak visualizers during calculation. Moreover, our results suggest that the left AG shows a specific affinity to the verbal domain and subserves number processing in a modality-specific way.
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paper_title: Relating numeric cognition and language processing: Do numbers and words share a common representational platform?

paper_content:

Abstract Numerical processing and language processing are both grounded in space. In the present study we investigated whether these are fully independent phenomena, or whether they share a common basis. If number processing activates spatial dimensions that are also relevant for understanding words, then we can expect that processing numbers may influence subsequent lexical access to words. Specifically, if high numbers relate to upper space, then they can be expected to facilitate understanding of words such as bird that are having referents typically found in the upper vertical space. The opposite should hold for low numbers. These should facilitate the understanding of words such as ground referring to entities with referents in the lower vertical space. Indeed, in two experiments we found evidence for such an interaction between number and word processing. By eliminating a contribution of linguistic factors gained from additional investigations on large text corpora, this strongly suggests that understanding numbers and language is based on similar modal representations in the brain. The implications of these findings for a broader perspective on grounded cognition will be discussed.
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paper_title: Preschool Origins of Cross-National Differences in Mathematical Competence: The Role of Number-Naming Systems

paper_content:

Differences in mathematical competence between U S and Chinese children first emerge during the preschool years, favor Chinese children, and are limited to specific aspects of mathematical competence The base-10 structure of number names is less obvious in English than in Chinese, differences between these languages are reflected in children's difficulties learning to count Language differences do not affect other aspects of early mathematics, including counting small sets and solving simple numerical problems Because later mathematics increasingly involves manipulation of symbols, this early deficit in apprehending the base-10 structure of number names may provide a basis for previously reported differences in mathematical competence favoring Chinese schoolchildren
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paper_title: Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers

paper_content:

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 problemmodel 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. Why are some students successful in solving word problems whereas others are unsuccessful? To help answer this question, we begin with the well-established observation that many students from kindergarten through adulthood have difficulty in solving arithmetic word problems that contain relational statements, that is, sentences that express a numerical relation between two variables (Hegarty, Mayer, & Green, 1992; Lewis & Mayer, 1987; Riley, Greeno, & Heller, 1983; Verschaffel, De Corte, & Pauwels, 1992). For example, Appendix A shows a successful and an unsuccessful solution to a two-step word problem containing a relational statement about the price of butter at two stores. We refer to this as an inconsistent version of the problem because the relational keyword (e.g., "less") primes an inappropriate arithmetic operation (subtraction rather than addition), whereas in a consistent problem, the relational term in the second problem statement primes the required arithmetic operation (e.g., "more" when the required operation is addition). A substantial proportion of college students, who could be called unsuccessful problem solvers, use the wrong arithmetic operation on inconsistent problems but perform correctly on consistent problems (Hegarty et al., 1992; Lewis, 1989; Lewis & Mayer, 1987; Verschaffel et al., 1992). We interpret this finding as evidence that problem comprehension processes play an important role in the solution of arithmetic word problems.
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paper_title: Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

paper_content:

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.
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paper_title: The Cultural Number Line: A Review of Cultural and Linguistic Influences on the Development of Number Processing

paper_content:

Approximate processing of numerosities is a universal and preverbal skill, while exact number processing above 4 involves the use of culturally acquired number words and symbols. The authors first review core concepts of numerical cognition, including number representation in the brain and the influential view that numbers are associated with space along a “mental number line.” Then, they discuss how cultural influences, such as reading direction, finger counting, and the transparency of the number word system, can influence the representation and processing of numbers. Spatial mapping of numbers emerges as a universal cognitive strategy. The authors trace the impact of cultural factors on the development of number skills and conclude that a cross-cultural perspective can reveal important constraints on numerical cognition.
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paper_title: On the language specificity of basic number processing: Transcoding in a language with inversion and its relation to working memory capacity

paper_content:

Transcoding Arabic numbers from and into verbal number words is one of the most basic number processing tasks commonly used to index the verbal representation of numbers. The inversion property, which is an important feature of some number word systems (e.g., German einundzwanzig [one and twenty]), might represent a major difficulty in transcoding and a challenge to current transcoding models. The mastery of inversion, and of transcoding in general, might be related to nonnumerical factors such as working memory resources given that different elements and their sequence need to be memorized and manipulated. In this study, transcoding skills and different working memory components in Austrian (German-speaking) 7-year-olds were assessed. We observed that inversion poses a major problem in transcoding for German-speaking children. In addition, different components of working memory skills were differentially correlated with particular transcoding error types. We discuss how current transcoding models could account for these results and how they might need to be adapted to accommodate inversion properties and their relation to different working memory components.
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paper_title: One language, two number-word systems and many problems: numerical cognition in the Czech language.

paper_content:

Comparing numerical performance between different languages does not only mean comparing different number-word systems, but also implies a comparison of differences regarding culture or educational systems. The Czech language provides the remarkable opportunity to disentangle this confound as there exist two different number-word systems within the same language: for instance, "25" can be either coded in non-inverted order "dvadsetpät" [twenty-five] or in inverted order "pätadvadset" [five-and-twenty]. To investigate the influence of the number-word system on basic numerical processing within one culture, 7-year-old Czech-speaking children had to perform a transcoding task (i.e., writing Arabic numbers to dictation) in both number-word systems. The observed error pattern clearly indicated that the structure of the number-word system determined transcoding performance reliably: In the inverted number-word system about half of all errors were inversion-related. In contrast, hardly any inversion-related errors occurred in the non-inverted number-word system. We conclude that the development of numerical cognition does not only depend on cultural or educational differences, but is indeed related to the structure and transparency of a given number-word system.
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paper_title: Solving arithmetic word problems: an analysis of classification as a function of difficulty in children with and without arithmetic LD.

paper_content:

This study was designed to determine a word problem difficulty classification in children with arithmetic learning disabilities (ALD; n = 104) in comparison with typically achieving students (n = 44). We tested variables such as (a) semantic structure (Change, Combine, Compare, and Equalize), (b) operation (subtraction and addition), and (c) position of the unknown quantity in the problem. Facet theory with multidimensional scaling techniques (MINISSA) was used to analyze the underlying dimensions in the responses of each group of participants. Our results indicate that although the word problem difficulty classifications for the 2 groups of children were different, the position of the unknown quantity had a greater influence on the level of difficulty of story problems than other variables. The noncanonical problems--specifically, those with the unknown term in the first place--although difficult for both groups of children, were the most difficult problems for children with ALD.
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paper_title: Numerical development—from cognitive functions to neural underpinnings

paper_content:

Living at the beginning of the 21st century requires being numerate because numerical abilities are not only essential for life prospects of individuals but also for economic interests of post-industrial knowledge societies (e.g., Butterworth et al., 2011). In recent years, numerical development has gained increasing research interest. Following this trend, we invited empirical and theoretical contributions for a Research Topic on Numerical development—from cognitive functions to neural underpinnings. We are grateful to all authors for their high-quality contributions, the reviewers for their constructive comments and suggestions in the interactive peer-review, and the publisher's editorial team for their excellent support. ::: ::: The different contributions nicely illustrate that the construct numerical development does not denote a unitary, clearly circumscribed, and comprehensive entity. Instead, the empirical, review, opinion, and commentary articles clearly suggest that it is important to consider different empirical and theoretical perspectives evaluating cross-domain (e.g., language or spatial abilities) but also domain-specific [e.g., basic numerical competencies, approximate number system (ANS), spatial-numerical associations (SNA)] determinants of and influences on typical but also atypical numerical development. ::: ::: A first set of studies investigated cross-domain as well as domain-specific influences on typical numerical development. With respect to cross-domain influences LeFevre et al. (2013) showed a reliable impact of children's spatial abilities on numerical skills. Additionally, Durkin et al. (2013) observed that language ability is a unique predictor of actual and future numerical achievement. Getting closer to the domain of numerical cognition, Lafay et al. (2013) found that finger counting may be useful but not necessary to develop accurate symbolic numerical competencies. As regards basic numerical precursor competencies, two studies investigated the influence of the ANS on numerical development. Mejias and Schiltz (2013) suggest that the ANS may be targeted by educational strategies as it seems to be associated with socio-economic status. Lonnemann et al. (2013) observed that children's addition performance is associated with different markers of the ANS during development. For secondary school children, Huber et al. (2013) found the relation of multiplication and division to be stronger for easier problems and more skilled (i.e., higher grade) students. Moreover, two fMRI studies investigated neural correlates of numerical development. Mussolin et al. (2013) observed differential developments of the contributions of the right and left intraparietal sulcus (IPS) to magnitude comparison over age, while Gullick and Wolford (2013) found a fronto-parietal shift of activation with age in general, but also specific effects on the lateralization of IPS involvement for processing negative numbers. Finally, Lambrechts et al. (2013) observed that processing the continuous quantities time and space seems resilient to healthy aging similar to numerosity. ::: ::: With a particular focus on the determinants of atypical numerical processing in mathematics learning disability (MLD) or developmental dyscalculia (DD), Mazzocco et al. (2013) showed that specific basic whole number misconceptions reliably predict atypical performance on Grade 8 arithmetic tests. Furthermore, Chu et al. (2013) found that even though inacuity of the ANS is a reliable predictor of risk for MLD it may not be its primary source. Nevertheless, Landerl (2013) evaluated the influences of other basic numerical competencies on numerical development and observed that children with DD exhibited specific impairments. Importantly, however, Haase et al. (2014) concluded that subtypes of MLD may not only be associated with content-related deficits but also more general impairments of information processing should be considered. In line with this view, Van Viersen et al. (2013) suggested that evaluating eye-fixation behavior may provide interesting new insights into the mechanisms underlying DD. In their Opinion, Kaufmann et al. (2013) argue that MLD/DD is a heterogeneous disorder resulting from individual differences in development or function at neuroanatomical, neuropsychological, behavioral, and interactional levels. Finally, Kaser et al. (2013) described the development and first successful evaluation of a multimodal and adaptive computer-based training program for children with DD. ::: ::: Another set of studies investigated interactions in the processing of numbers and space. With respect to SNA, Knops et al. (2013) observed a reversed operational momentum (OM) effect in children with their degree of attentional control predicting the propensity to exhibit the OM effect. Kucian et al. (2013) commented that the left–right associations underlying the OM effect may dependent on development, reflecting an interaction of visuo-spatial and attentional processes with number related skills which might account for the non-observation of an OM effect in children (with DD). Moreover, Link et al. (2014) observed that unbounded number line estimation may be a valuable tool for assessing primary school children's spatial representation of number magnitude in an unbiased manner. Goldman et al. (2013) argued that the development of an analog comparison process and the specific processing of end stimuli contribute to the emergence of the mental number line. In their review, Patro et al. (2014) suggested a taxonomy for the classification of SNA from infancy to late preschool years. On the other hand, another two studies addressed direct interactions between the processing of numerical and physical magnitude as reflected by the size congruity effect (Henik and Tzelgov, 1982). Leibovich et al. (2013) found that numerical and physical magnitudes are represented by different, yet interactive systems. Ben-Shalom et al. (2013) observed that even preschoolers were able to process number magnitude information automatically. Finally, unrelated to space, Gabriel et al. (2013) suggested that conceptual knowledge about fractions and procedural knowledge about how to manipulate them should be distinguished. Faulkenberry (2013) commented that the evaluation of solution strategies might be beneficial to differentiate procedural and conceptual knowledge. ::: ::: Extending the focus on language influences Imbo et al. (2014) and Klein et al. (2013) specifically investigated influences of number word inversion on numerical development. They found that language, but not working memory capacity, predicted the number of inversion errors and conclude that inversion-related difficulties do not fade over time, respectively. Lopes-Silva et al. (2014) observed the more basic perceptual phonemic awareness to predict number transcoding reliably, whereas magnitude processing and working memory did not. ::: ::: As documented by this broad range of studies dealing with different aspects of numerical development—from behavioral performance to underlying neural substrates, from cross-sectional to longitudinal evaluations, from healthy to clinical populations—the current Research Topic brought together the expertise of researchers from different backgrounds and clearly advanced our understanding of numerical development—a topic with both scientific and every-day relevance.
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paper_title: Preschool Children’s Narratives and Performance on the Peabody Individualized Achievement Test – Revised: Evidence of a Relation between Early Narrative and Later Mathematical Ability

paper_content:

In this study, different measures derived from 41 3- to 4-year-old children’s selfgenerated picture-book narratives and their performance on a general measure of language development (TELD-2, Hresko, Reid & Hammill, 1991) were evaluated with respect to their possible predictive relation two years later with 5 areas of academic achievement (General information, Reading recognition, Reading comprehension, Math, Spelling) assessed using the Peabody Individualized Achievement Test – Revised (PIAT-R, Markwardt, 1998). Children’s TELD-2 scores were significantly predictive of their General information scores. The narrative measures of conjunction use, event content, perspective shift, and mental state reference were significantly predictive of later Math scores. Post-hocanalyses revealed that, for the same children, the observed relations with Math achievement did not arise with nonspontaneous adult-prompted narrations.
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paper_title: Spatial associations in numerical cognition—From single digits to arithmetic

paper_content:

The literature on spatial associations during number processing is dominated by the SNARC (spatial–numerical association of response codes) effect. We describe spatial biases found for single digits and pairs of numbers, first in the “original” speeded parity task and then extending the scope to encompass different tasks, a range of measures, and various populations. Then we review theoretical accounts before surveying the emerging evidence for similar spatial associations during mental arithmetic. We conclude that the mental number line hypothesis and an embodied approach are useful frameworks for further studies.
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paper_title: The Whorfian hypothesis and numerical cognition: is `twenty-four' processed in the same way as `four-and-twenty'?

paper_content:

Recent theoretical developments have redefined a Whorfian effect as a processing difference due to the language of the individual, and no longer as a marker for or against linguistic determinism. Within this framework, Whorfian effects can be used to investigate whether a particular part of the cognitive system is penetrable by language processes or forms an encapsulated module, provided the experimenter ensures that the target language difference is not caused by peripheral input or output processes. In this article, we examine the possibility of a Whorfian effect in numerical cognition by making use of the: fact that in the Dutch number naming system the order of tens and units is reversed (i.e. 24 is read 'four-and-twenty'). In a first experiment, we asked native French- and Dutch-speaking students to name the solution of addition problems with a two-digit and a single-digit operand (e.g. 20 + 4 = ?, 24 + 1 = ?). The order of the operands was manipulated (20 + 4 vs. 4 + 20) as well as the presentation modality (Arabic vs. verbal). Three language differences emerged from this study. Experiment 2, however, showed that these differences were all due to input or output processes rather than differences in the addition operation (i.e. the differences between Dutch and French disappeared when subjects were asked to type the answer rather than pronounce it). On the basis of these findings, we question the idea that mathematical operations are based on verbal processes. (C) 1998 Elsevier Science B.V.
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paper_title: The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning

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This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem-solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. We conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than anoth...
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paper_title: On the relation between grammatical number and cardinal numbers in development

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This mini-review focuses on the question of how the grammatical number system of a child's language may help the child learn the meanings of cardinal number words (e.g., "one" and "two"). Evidence from young children learning English, Russian, Japanese, Mandarin, Slovenian, or Saudi Arabic suggests that trajectories of number-word learning differ for children learning different languages. Children learning English, which distinguishes between singular and plural, seem to learn the meaning of the cardinal number "one" earlier than children learning Japanese or Mandarin, which have very little singular/plural marking. Similarly, children whose languages have a singular/dual/plural system (Slovenian and Saudi Arabic) learn the meaning of "two" earlier than English-speaking children. This relation between grammatical and cardinal number may shed light on how humans acquire cardinal-number concepts. There is an ongoing debate about whether mental symbols for small cardinalities (concepts for "oneness," "twoness," etc.) are innate or learned. Although an effect of grammatical number on number-word learning does not rule out nativist accounts, it seems more consistent with constructivist accounts, which portray the number-learning process as one that requires significant conceptual change.
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paper_title: READING HABITS FOR BOTH WORDS AND NUMBERS CONTRIBUTE TO THE SNARC EFFECT

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This study compared the spatial representation of numbers in three groups of adults: Canadians who read both English words and Arabic numbers from left to right; Palestinians who read Arabic words and Arabic-Indic numbers from right to left; and Israelis who read Hebrew words from right to left but Arabic numbers from left to right. Canadians associated small numbers with left and large numbers with right space (the SNARC effect), Palestinians showed the reverse association, and Israelis had no reliable spatial association for numbers. These results suggest that reading habits for both words and numbers contribute to the spatial representation of numbers.
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paper_title: Linguistic Influences on Mathematical Development: How Important Is the Transparency of the Counting System?

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Wales uses languages with both regular (Welsh) and irregular (English) counting systems. Three groups of 6- and 8-year-old Welsh children with varying degrees of exposure to the Welsh language—those who spoke Welsh at both home and school; those who spoke Welsh only at home; and those who spoke only English—were given standardized tests of arithmetic and a test of understanding representations of two-digit numbers. Groups did not differ on the arithmetic tests, but both groups of Welsh speakers read and compared 2-digit numbers more accurately than monolingual English children. A similar study was carried out with Tamil/English bilingual children in England. The Tamil counting system is more transparent than English but less so than Welsh or Chinese. Tamil-speaking children performed better than monolingual English-speaking children on one of the standardized arithmetic tests but did not differ in their comparison of two-digit numbers. Reasons for the findings are discussed.
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paper_title: Language affects symbolic arithmetic in children: The case of number word inversion

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Abstract Specific language influences have been observed in basic numerical tasks such as magnitude comparison, transcoding, and the number line estimation task. However, so far language influences in more complex calculations have not been reported in children. In this translingual study, 7- to 9-year-old German- and Italian-speaking children were tested on a symbolic addition task. Whereas the order of tens and units in Italian number words follows the order of the Arabic notation, the order is inverted in German number words. For both language groups, addition problems were more difficult when a carry operation was needed, that is, when a manipulation within the place-value structure of the Arabic number system was particularly important. Most important, this carry effect was more pronounced in response latencies for children speaking German, a language with inverted verbal mapping of the place-value structure. In addition, independent of language group, the size of the carry effect was significantly related to verbal working memory. The current study indicates that symbolic arithmetic and the carry effect in particular are modulated by language-specific characteristics. Our results underline the fact that the structure of the language of instruction is an important factor in children’s mathematical education and needs to be taken into account even for seemingly nonverbal symbolic Arabic tasks.
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paper_title: Grammatical number elicits SNARC and MARC effects as a function of task demands

paper_content:

Despite the robustness of the spatial–numerical association of response codes (SNARC) and linguistic markedness of response codes (MARC) effect, the mechanisms that underlie these effects are still under debate. In this paper, we investigate the extraction of quantity information from German number words and nouns inflected for singular and plural using two alternative forced choice paradigms. These paradigms are applied to different tasks to investigate how access to quantity representation is modulated by task demands. In Experiment 1, we replicated previous SNARC findings for number words—that is, a relative left-hand advantage for words denoting small numbers and a right-hand advantage for words denoting large numbers in semantic tasks (parity decision and quantity comparison). No SNARC effect was obtained for surface or lexical processing tasks (font categorization and lexical decision). In Experiment 2, we found that German words inflected for singular had a relative left-hand advantage, and German ...
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paper_title: Language effects in addition: How you say it counts

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This study aimed to investigate the role of language in calculation. Two populations were compared, one with a base-10 language, and another (Basque) in which number words are constructed by combining multiples of 20 and units or teens (e.g., “35” is said “twenty and fifteen”). Experiment 1 asked participants to verbally solve additions presented as Arabic digits. Basque participants solved the additions that consisted of a multiple of 20 and a teen (e.g., 20 + 15) faster than controls with identical answers (e.g., 25 + 10). No differences were found in the base-10 language group. Experiment 2 replicated this result even if participants had to type the answer on a numerical keypad, instead of saying it. Hence, the structure of number words in each of the languages influenced the way additions were solved, even if language was not necessary for conducting the task. Finally, in Experiment 3, both language groups performed a numerical comparison task in which no effects of the structure of number words were ...
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paper_title: Adding Apples and Oranges : Alignment of Semantic and Formal Knowledge

paper_content:

We show that the same mechanism that mediates analogical reasoning (i.e., structural alignment) leads to interpretive "content effects" in reasoning about arithmetic word problems. Specifically, we show that both college students and textbook writers tend to construct arithmetic word problems that maintain systematic correspondence between the semantic relations that people infer from pairs of real-world objects (e.g., apples and baskets support the semantic relation CONTAIN [content, container]) and mathematical relations between arguments of arithmetic operations (e.g., DIVIDE [dividend, divisor]). Such relational alignments, to which we refer here as semantic alignments, lead to selective and sensible application of abstract formal knowledge. For example, people usually divide apples among baskets rather than baskets among apples, and readily add apples and oranges but refrain from adding apples and baskets.
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paper_title: Children's Mathematical Development: Research and Practical Applications

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This study examines number-skills developments from infancy through adolescence. It identifies universals in early mathematical development and describes how they are expressed in different cultures. Cross-national differences in maths achievement are considered.
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paper_title: The Consistency Effect Depends on Markedness in Less Successful but Not Successful Problem Solvers: An Eye Movement Study in Primary School Children.

paper_content:

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.
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paper_title: Compare word problems: Consistency hypothesis revisited☆

paper_content:

Abstract Compare problems involve relationships between quantities. In inconsistent language (IL) formats, relational terms conflict with solution operations. Lewis and Mayer’s (1987) consistency hypothesis, which explains likely translations of IL problems to match problem solvers’ schemas, is examined Using think-aloud and recall data. Supporting hypotheses, students committed more errors overall and reversal errors specifically (i.e., used opposite operation) on IL than their consistent language (CL) counterparts and recalled IL problems as CL problems especially following reversal errors. Counter to expectations, language consistency did not result in differences in behaviors, reading or response times, or quality of recall. Additional analyses indicate a strong impact of fraction-of-a-number relational terms and provide evidence of difficulties forming representations and solving problems involving fraction-of-a-number construction.
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paper_title: Strategies of Discourse Comprehension

paper_content:

rhetorical schemata to be discussed in what follows. Finally, schemata are descriptions, not definitions. The ‘bus’ schema contains information that is nor-
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paper_title: Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers

paper_content:

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 problemmodel 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. Why are some students successful in solving word problems whereas others are unsuccessful? To help answer this question, we begin with the well-established observation that many students from kindergarten through adulthood have difficulty in solving arithmetic word problems that contain relational statements, that is, sentences that express a numerical relation between two variables (Hegarty, Mayer, & Green, 1992; Lewis & Mayer, 1987; Riley, Greeno, & Heller, 1983; Verschaffel, De Corte, & Pauwels, 1992). For example, Appendix A shows a successful and an unsuccessful solution to a two-step word problem containing a relational statement about the price of butter at two stores. We refer to this as an inconsistent version of the problem because the relational keyword (e.g., "less") primes an inappropriate arithmetic operation (subtraction rather than addition), whereas in a consistent problem, the relational term in the second problem statement primes the required arithmetic operation (e.g., "more" when the required operation is addition). A substantial proportion of college students, who could be called unsuccessful problem solvers, use the wrong arithmetic operation on inconsistent problems but perform correctly on consistent problems (Hegarty et al., 1992; Lewis, 1989; Lewis & Mayer, 1987; Verschaffel et al., 1992). We interpret this finding as evidence that problem comprehension processes play an important role in the solution of arithmetic word problems.
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paper_title: Linguistic and pedagogical factors affecting children's understanding of arithmetic word problems: A comparative study

paper_content:

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.
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paper_title: Using Retelling Data to Study Elementary School Children's Representations and Solutions of Compare Problems.

paper_content:

In an attempt to explain subjects' difficulties with particular types of compare problems, Lewis and Mayer (1987) have presented a model that simulates the comprehension processes when solving these problems. The crucial component of their model is the "consistency hypothesis," according to which students are more likely to experience representational difficulties-and consequently to make errors-when the presentation of the terms in the relational statement of the compare problem is not consistent with their preferred format. To test this model, an experiment was carried out in which 40 fifth graders were asked to solve and retell a set of one-step compare problems. The accuracy data, solution times, and, especially, the retelling data yielded strong evidence in favor of the consistency hypothesis.
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paper_title: Short-Term Memory, Working Memory, and Inhibitory Control in Children with Difficulties in Arithmetic Problem Solving

paper_content:

The relations between short-term memory, working memory, inhibitory control, and arithmetic word problem solution were studied in children who were poor in arithmetic problem solving (n = 23). The children were compared with a group of good problem solvers (n = 26), matched for vocabulary, age, and gender. The results corroborate the hypothesis of poor problem solvers' general deficit in inhibitory processes. They had lower scores and made more intrusion errors in a series of working memory tasks requiring inhibition of irrelevant information. The results showed that problem solving performance is related to the ability of reducing the accessibility of nontarget and irrelevant information in memory. Span tasks that imply passive storage of information showed that poor problem solvers were impaired when they have to retain numerical information, but they did not differ from children who did not have difficulty with mathematics when the material included words.
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paper_title: What underlies successful word problem solving? A path analysis in sixth grade students

paper_content:

Two component skills are thought to be necessary for successful word problem solving: (1) the production of visual-schematic representations and (2) the derivation of the correct relations between the solution-relevant elements from the text base. The first component skill is grounded in the visual–spatial domain, and presumed to be influenced by spatial ability, whereas the latter is seated in the linguistic–semantic domain, and presumed to be influenced by reading comprehension. These component skills as well as their underlying basic abilities are examined in 128 sixth grade students through path analysis. The results of the path analysis showed that both component skills and their underlying basic abilities explained 49% of students’ word problem solving performance. Furthermore, spatial ability and reading comprehension both had a direct and an indirect relation (via the component skills) with word problem solving performance. These results contribute to the development of instruction methods that help students using these components while solving word problems.
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paper_title: The Influence of Working Memory and Classification Ability on Children′s Word Problem Solution

paper_content:

Abstract The purpose of this study was to investigate the relationship between problem classification (problem schemata) and working memory span on children′s solution accuracy of arithmetic word problems. Measures of working memory, classification ability, knowledge of processing operations, verbatim recall of word problem propositions, reading comprehension, and arithmetic performance were correlated with solution accuracy. A regression analysis indicated that reading comprehension and knowledge of processing operations were the best predictors of solution accuracy. Working memory and classification ability were significant predictors only when forced into the equation first. A confirmatory factor analysis, with the effect of reading comprehension partialed out, indicated that working memory and classification ability loaded on separate factors with solution accuracy. Overall, the results provide support for the hypothesis that the amount of working memory resources activated is independent of schemata activation. Consistent with an independence hypothesis (Brainerd & Reyna, 1992), the results also show a weak relationship between memory and problem solving accuracy.
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paper_title: Comprehension of arithmetic word problems: Evidence from students’ eye fixations

paper_content:

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
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paper_title: Working memory and literacy as predictors of performance on algebraic word problems

paper_content:

Previous studies on individual differences in mathematical abilities have shown that working memory contributes to early arithmetic performance. In this study, we extended the investigation to algebraic word problem solving. A total of 151 10-year-olds were administered algebraic word problems and measures of working memory, intelligence quotient (IQ), and reading ability. Regression results were consistent with findings from the arithmetic literature showing that a literacy composite measure provided greater contribution than did executive function capacity. However, a series of path analyses showed that the overall contribution of executive function was comparable to that of literacy; the effect of executive function was mediated by that of literacy. Both the phonological loop and the visual spatial sketchpad failed to contribute directly; they contributed only indirectly by way of literacy and performance IQ, respectively.
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Title: Word Problems: A Review of Linguistic and Numerical Factors Contributing to Their Difficulty

Section 1: Introduction
Description 1: This section introduces the topic of word problems (WPs), their importance in the school curriculum, and sets the stage by highlighting the linguistic and numerical complexities that contribute to their difficulty.

Section 2: Individual Differences and Social Factors
Description 2: This section examines how individual differences and social factors influence performance in solving WPs, including cognitive abilities, social background, educational environment, and motivational aspects.

Section 3: Subcategories of Word Problems and Solution Strategies
Description 3: This section categorizes different types of WPs and discusses various solution strategies used by students, detailing how these categories and strategies impact the difficulty level and error types.

Section 4: Linguistic Complexity and Linguistic Studies
Description 4: This section explores the various dimensions of linguistic complexity in WPs, including vocabulary, syntax, and discourse structures, and their effects on comprehension and problem-solving.

Section 5: Structural Factors
Description 5: This section delves into the micro-level structural elements of WPs such as word and sentence forms, and macro-level elements like discourse structure, discussing their impact on solving WPs.

Section 6: Semantic Factors
Description 6: This section investigates the role of semantic elements like verbal cues, object relations, and the presence of distractors, in influencing the difficulty and solving strategies of WPs.

Section 7: Numerical Complexity and Numerical Studies
Description 7: This section reviews the numerical aspects of WPs, including number properties, required operations, mathematical solution strategies, and their combined influence on cognitive load and performance.

Section 8: Number Properties
Description 8: This section focuses specifically on the properties of numbers used in WPs such as parity, digit size, and problem size, discussing their roles in influencing task performance and strategy selection.

Section 9: Required Operation
Description 9: This section discusses the arithmetic operations required in WPs, how different types of operations (addition, subtraction, multiplication, etc.) affect problem difficulty, and cognitive processes related to these operations.

Section 10: Mathematical Solution Strategies
Description 10: This section examines various mathematical solution strategies employed in WPs and how numerical factors such as number magnitude and position influence these strategies.

Section 11: Information Relevance and Step-Wise Problem Processing
Description 11: This section explores the importance of information relevance in WPs, the challenges posed by extra or irrelevant information, and the impact of multiple steps on problem-solving accuracy.

Section 12: Other Numerical Processes and Representation
Description 12: This section discusses other numerical processes and representations in WPs including the presence of carry operations, number sense, and symbolic understanding.

Section 13: Connecting Linguistic and Mathematical Factors
Description 13: This section highlights the interrelation between linguistic and numerical factors, emphasizing the need for integrated studies to understand their combined effects on WP difficulty and solution processes.

Section 14: Lexical Consistency Effect
Description 14: This section reviews the effects of lexical consistency in WPs, how inconsistent linguistic markers can lead to errors, and the strategies employed by successful and unsuccessful problem solvers.

Section 15: Future Direction, Open Questions
Description 15: This section outlines future research directions, highlighting the need for further collaboration between linguists and numerical cognition researchers to systematically investigate the interactions between linguistic and numerical factors in WPs.