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Psychometric Approaches Help Resolve Competing Cognitive Models: When Less Is More Than It Seems
Abstract
Simple arithmetic word problems are often featured in elementary school education. One type of problem, "compare with unknown reference set," ranks among the most difficult to solve. Differences in item difficulty for compare problems with unknown reference set are observed depending on the direction of the relational statement (more than vs. less than). Various cognitive models have been proposed to account for these differences. We employed item response theory (IRT) to compare competing cognitive models of student performance. The responses of 100 second-grade students to a series of compare problems with unknown reference set, along with other measures of individual differences, were fit to IRT models. Results indicated that the construction integration model (Kintsch, 1988, 1998) provided the best fit to the data. We discuss the potential contribution of psychometric approaches to the study of thinking.
... A research strategy of comparing simulated with real data is only possible when an exact theory is at hand. A good example for the value of theories in the context of the dimensionality question of arithmetic skills is presented by Arendasy, Sommer, and Ponocny (2005), who tested the predictions of four theories about solving different arithmetic word problems of the " compare " type (Riley, Greeno, & Heller, 1983). Compare word problems in general are the most difficult of all basic word problems involving addition and subtraction (Riley et al., 1983; Stern, 1994). ...
... of the " compare " type (Riley, Greeno, & Heller, 1983). Compare word problems in general are the most difficult of all basic word problems involving addition and subtraction (Riley et al., 1983; Stern, 1994). However, there are subtle differences in difficulty between various subtypes, which have been explained differently by a number of theories. Arendasy et al. (2005) reviewed these theories and derived specific predictions regarding person homogeneity and item homogeneity of a set of different compare word problems. Person homogeneity means that an IRT model (such as the Rasch model) estimates the same difficulty parameters of the items, regardless of the subsample the estimation is based on. Item h ...
... At the same time, person homogeneity (based on different sample partitioning criteria) is predicted, because the differences between the subtypes of problems are assumed to be the same for all subjects. For testing these predictions, Arendasy et al. (2005) used nonparametric goodness-of-fit tests (T-statistics) that were introduced by Ponocny (2001). In a study with n=100 second graders the application of T-statistics revealed person homogeneity with respect to nine criteria for splitting the sample, but rejected item homogeneity between two specific subtypes of compare word problems. ...
The question of the dimensionality of intelligent performance has kept researchers occupied for decades. We investigate this question in the context of learning elementary arithmetic. Our assumption of a polyhierarchy of skills in arithmetic (HiSkA) predicts a multidimensional structure of test data. This seems to contradict findings that data collected to validate the HiSkA conformed to the Rasch model. To resolve this seeming contradiction, we analysed test data from two samples of third graders with a number of methods ranging from factor analysis and Rasch analysis to multidimensional item response theory (MIRT). Additionally we simulated data sets based on different unidimensional and multidimensional models and compared the results of some of the analyses that were also applied to the empirical data. Results show that a multidimensional generating structure can produce data conforming to the Rasch model under certain conditions, that a general factor explains a substantial amount of variance in the empirical data, but that the HiSkA is capable of explaining much of the residual variance.
... Similarly, a variety of item difficulty modeling methods, including LLTM, have provided evidence for the response process aspect of validity for several tests. These tests include English-language assessments, such as TOEFL (Freedle & Kostin, 1993;Rupp, Garcia, & Jamieson, 2001;Sheehan & Ginther, 2001) and the GRE (Enright, Morley, & Sheehan, 1999;Gorin & Embretson, 2006), as well as tests of nonverbal intelligence (Arendasy, Sommer, & Ponocny, 2005;Bejar, 1993;Embretson, 1996Embretson, , 1998Embretson, , 1999Hornke & Habon, 1986). Although the predictability of item difficulty in these studies is often far from perfect, the results may be useful for test development as noted by Fischer and Pendl (1980). ...
... The factors that underlie the difficulty of mathematics test items have been studied by several researchers, but usually the emphasis is to isolate the effects of a few important variables (e.g., Arendasy et al., 2005;Birenbaum, Tatsuoka, & Gurtvirtz, 1992;Singley & Bennett, 2002). In contrast, Mayer, Larkin, and Kadane's (1984) model is sufficiently broad to be applicable to many types of mathematical problems. ...
Cognitive complexity level is important for measuring both aptitude and achievement in large-scale testing. Tests for standards-based assessment of mathematics, for example, often include cognitive complexity level in the test blueprint. However, little research exists on how mathematics items can be designed to vary in cognitive complexity level. In fact, determining the cognitive complexity level of items is usually based on correspondence to definitions rather than on empirically and theoretically justifiable variables that can predict item difficulty. In the current study, mathematical problem-solving items were designed for varying cognitive complexity levels based on a cognitive model of item processing. Structural variants of item models were designed to vary on two aspects of the cognitive model, the equation source and the number of subgoals. Participants were randomly assigned to test forms that contained different structural variants of the item models. Results from the linear logistic test model, the two-parameter-logistic—constrained model, and a corresponding linear mixed modeling procedure indicated that the item design variables affected both item difficulty and response time. Implications of the results for using structural variants in item generation and for the plausibility of the hypothesized cognitive model are discussed.
... In the baseline phase, the types of word problems that received lower percent accuracy were changeunknown change quantity(+), change-unknown start quantity(+), change-unknown start quantity(-), compare-unknown referent quantity(+), compare-unknown difference quantity(+), and compare-unknown compared quantity(-), which were between 0% to 44%. These showed the same tendency that the compare type of word problems was more complicated than other types (Arendasy et al., 2005;Fuson et al., 1996). After the implementation, the types of word problems with over 80% average percent accuracy were change-unknown result quantity(+), change-unknown result quantity(-), and change-unknown start quantity(-), which were 94%, 81%, and 81%, respectively. ...
Improving the ability to solve mathematical word problems is one of the most critical issues facing students with developmental disabilities, because it is directly related to their independent living skills. The purpose of this study was to propose a teaching model that implements augmented reality (AR) and video modeling (VM), and to validate its effectiveness, including its immediate, maintenance, and generalization effects on improving the percent accuracy of solving mathematical word problems. The research design of this study adopted single-case multiple probe across students experimental design. The independent variable of this study was the VM with AR teaching model, and the dependent variable was the percent accuracy of the test subjects in solving mathematical word problems. All three test subjects showed an immediate effect, with two showing maintenance and generalization effects. This VM with AR teaching model provides an alternate way for classroom teachers when teach students with developmental disabilities to solve mathematical word problems. This teaching model allows students with developmental disabilities to improve their mathematical word-problem solving skills.
... Following this paradigm, Okamoto (1996) conducted a study to determine whether conceptual structures adequately described children's performance on quantitative word problems. Previous research has established that quantitative word problems vary in difficulty even when the basic mathematical operations were the same (Arendasy & Sommer, 2005;Carpenter & Moser, 1984;Hudson, 1983;Riley & Greeno, 1988). Because of the differences in difficulty, the linguistic structure of a word problem has been hypothesized to afford a specific mental representation. ...
... er the generative framework has to be extended by an (automated) quality control framework in order to be able to generate items at a high psychometric level (Arendasy, 2004Arendasy, , 2005, in press). The generative framework basically includes a set of radicals derived from cognitive psychology based models of the latent trait to be measured (cf. Arendasy, Sommer, & Ponocny, 2005) and serves as a specification of the construct-related solution strategies. In line with Greeno, Smith, and Moore (1993), and Siegler (1996) , Arendasy (2005, in press) assumes that a test person's solution strategy choice is not merely a characteristic of the test person himself, but arises from an interaction between characteristics ...
This paper deals with three studies on the computer-based, automatic generation of algebra word problems. The cognitive psychology based generative/quality control frameworks of the item generator are presented. In Study I the quality control framework is empirically tested using a first set of automatically generated items. Study II replicates the findings of Study I using a larger set of automatically generated algebra word problems. Study III deals with the generative framework of the item generator by testing construct validity aspects of the item generator produced items. Using nine Rasch-homogeneous subscales of the new intelligence structure battery (INSBAT, Hornke et al., 2004), a hierarchical confirmatory factor analysis is reported, which provides first evidence of convergent as well as divergent validity of the automatically generated items. The end of the paper discusses possible advantages of automatic item generation in general ranging from test security issues and the possibility of a more precise psychological assessment to mass testing and economical questions of test construction.
... In addition, certain aspects of the word problem itself have been shown to influence problem-solving success (De Corte, Verschaffel, & Pauwels, 1990;Kintsch & Greeno, 1985;Lewis & Mayer, 1987;Pape, 2003;Stern, 1993;Verschaffel, De Corte, & Pauwels, 1992). Among these, consistency (relational term consistent vs. inconsistent with the required arithmetic operation) and markedness (relational term unmarked ['more than'/'times more than'] vs. marked ['less than'/'times less than']) are the most well-established (Arendasy, Sommer, & Ponocny, 2005;Clark, 1969;Hegarty, Mayer, & Green, 1992;Lewis & Mayer, 1987;Verschaffel, 1994;Verschaffel et al., 1992). However, the effects of consistency and markedness have not yet been examined in children differing in problem-solving skill. ...
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.
... The factors that underlie the difficulty of mathematics test items have been studied by several researchers, but usually the emphasis is to isolate the effects of a few important variables (e.g., Singley & Bennett, 2002;Arendasy & Sommer, 2005;Birenbaum, Tatsuoka, & Gurtvirtz, 1992). The goal in the current study was to examine the plausibility of a model that could explain performance in a broad bank of complex mathematical problem solving items. ...
The linear logistic test model (LLTM; Fischer, 1973) has been applied to a wide variety of new tests. When the LLTM application involves item complexity variables that are both theoretically interesting and empirically supported, several advantages can result. These advantages include elaborating construct validity at the item level, defining variables for test design, predicting parameters of new items, item banking by sources of complexity and providing a basis for item design and item generation. However, despite the many advantages of applying LLTM to test items, it has been applied less often to understand the sources of complexity for large-scale operational test items. Instead, previously calibrated item parameters are modeled using regression techniques because raw item response data often cannot be made available. In the current study, both LLTM and regression modeling are applied to mathematical problem solving items from a widely used test. The findings from the two methods are compared and contrasted for their implications for continued development of ability and achievement tests based on mathematical problem solving items.
Young novice drivers are more prone than older drivers to get involved in a risky driving situation. Some young drivers underestimate risk while overestimating their driving abilities, increasing the likelihood of engaging in risky behaviour. Age and inexperience both influence risk estimation, though it is not clear which of these variables is more important. Can drivers’ willingness to engage in risky behaviour be assessed in a similar way to hazard perception skill using video-based risky situations? The aim of the current study was to assess whether a video-based tool could measure the willingness to participate in risky driving situations and whether it can distinguish between different types of risky driving scenarios across gender and driver age groups. We also explored the moderating effect of age and gender on drivers’ experience in relation to the risky manoeuvres and participants’ willingness to engage in risky situations. Participants were presented with naturalistic videos from the perspective of the driver that contained active risky situations (result of driver’s own actions) and were asked to make a decision regarding a potential action (to overtake a bus/bicycle or pass through an amber light) and whether they would accelerate at this point. Participants reported that they were more willing to accelerate and overtake cyclists and buses and less willing to pass a light in amber. Young drivers were more willing to both engage in the risky behaviours and accelerate than older drivers, with young males reporting higher scores than the other groups. Gender differences were observed, with males being more prone to overtake and pass through a light in amber than females; however, this difference was not observed for the intention to accelerate. All the above effects remained when we tested the impact of experience on decision making while controlling for age and gender, although driving experience was no longer significant. These results demonstrate that drivers’ intention to assume risk can indeed be measured in a similar video-based methodology to that used by hazard perception tests. The findings raise the possibility of assessing and training drivers on a wider range of safety-related behaviours.
One of the major weaknesses of the item exposure rates of figural items in Intelligence Quotient (IQ) tests lies in its inaccuracies. In this study, a new approach is proposed and a useful test tool known as the Virtual Item Bank (VIB) is introduced. The VIB combine Automatic Item Generation theory and image processing theory with the concepts of figural tests and Computerized Adaptive Testing (CAT). It is believed that this tool will assist in improving traditional figural tests - in terms of solving previous issues relating to item exposure and allowing a figural test to be more easily developed.
This study investigated the effectiveness of Math Explorer at enhancing the word problem-solving skills of students with mathematics difficulties (MD). The study, which had a multiple-probe-across-subjects design, was conducted over 18 weeks. Four students with MD in Grades 2 and 3 participated. All students were able to use the four-step cognitive and three-step metacognitive strategies to solve addition and subtraction word problems and improved their word problem-solving performance on the computer- and paper/pencil-based tests. After the intervention phase, three out of the four students successfully maintained their improved word problem-solving performance levels during the follow-up phase. Implications for teachers and future software programmers are discussed, and directions for future computer-assisted instruction (CAI) research are suggested.
This chapter is concerned with the development of an important aspect of children's problem-solving skiU in arithmetic-the ability to solve arithmetic word problems. There are several factors that might enable older children to perform better in problem-solving tasks than younger children, including the complexity of conceptual knowledge about the problem domain and the sophis tication of problem-solving procedures . The studies reviewed here suggest that , with age, children's improved ability to solve word problems primarily involves an increase in the complexity of conceptual knowledge required to understand the situations described in those problems . We will describe these findings in this chapter and consider some general issues about the development of problem solving skill .
This article describes developmental models of word problem solving that are grounded in the notion of general developmental constraints of the mind. These models were constructed based on the assumption that differences in children's word problem-solving performance are due, at least in part, to developmental differences in their conceptual structures in the quantitative domain. Three levels of knowledge were identified and modeled. The simplest model represents quantitative relations as an ordered array of mental objects. The next level of the model represents numbers as objects of manipulations open two mental number lines that are coordinated in a tentative fashion. The most complex model represents numerical operations as objects of manipulations on two mental number lines that are well-coordinated with explicit, functional rules. These models were implemented as production systems. The accuracy of the predictions resulting from the simulations of the models was tested in an empirical study. Global tests of the models found a good fit of the data to the models. The results wee consistent with the theoretical analysis that the three levels of knowledge were internally coherent and qualitatively different from each other, and that the models could predict children's performance differences to a satisfactory degree.
The hypothesis was tested that older chidren, fifth/sixth graders versus third graders, who are more accurate in word problem solution, would show evidence of problem differentiation consistent with schema theories of problem representation and solution. A mathematics word problem sorting task and an accuracy test were used to assess problem representation and accuracy of solution across 16 different problem types. Cluster analyses of sorting data revealed that older children demonstrated sorting patterns consistent with a refined schema theory. Younger children were both less accurate and less systematic in their sorting. Idealized sorting patterns representing schema theories as well as surface structure problem similarity were tested against actual sorting patterns. Regression analyses confirmed that the fifth/sixth graders' and third graders' sorting data were best fit by a schematic sort and by a surface structure sort, respectively. Additionally, an English versus Spanish language contrast showed no effect of language on either accuracy of solution or the sorting patterns of the math word problems.
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.
Develops a theory to specify in part how a person stores and searches through information retained from sentences. The theory states that (1) functional relations, like the abstract subject-predicate relation which underlies sentences, are more available from memory than other, less basic kinds of information; (2) certain "positive" adjectives, like long, are stored in memory in a less complex and more accessible form than their opposites, like short; and (3) listeners can only retrieve, from memory, information which is congruent at a deep level to the information they are searching for. The present theory, unlike previous ones, correctly predicts the principal differences in the solution times of 8 types of 2-term series problems and 32 types of 3-term series problems (e.g., If John isn't as bad as Pete, and Dick isn't as good as Pete, then who is worst?). It also accounts for previous observations on children solving these problems and explains other phenomena in deductive reasoning. (46 ref.) (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Recent research has led to a significant reconceptualization of the nature of children's number knowledge development. This article outlines infants' and preschoolers' implicit protoquantitative reasoning schemas and shows how these combine with early counting knowledge to produce mathematical concepts of number. Research on elementary school children's informal and invented arithmetic is reviewed, and implications for mathematics education are evaluated. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate final questions to incomplete word problems, and (3) modeling performance patterns using a computer simulation. Solution performance was found to be systematically related to recall and question generation performance. Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution “errors” were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error patterns were successfully simulated by manipulating a computer model's language comprehension strategies, as opposed to its knowledge of logical set relations.
Item response theory has become an essential component in the toolkit of every researcher in the behavioral sciences. It provides a powerful means to study individual responses to a variety of stimuli, and the methodology has been extended and developed to cover many different models of interaction. This volume presents a wide-ranging handbook to item response theory - and its applications to educational and psychological testing. It will serve as both an introduction to the subject and also as a comprehensive reference volume for practitioners and researchers. It is organized into six major sections: the nominal categories model, models for response time or multiple attempts on items, models for multiple abilities or cognitive components, nonparametric models, models for nonmonotone items, and models with special assumptions. Each chapter in the book has been written by an expert of that particular topic, and the chapters have been carefully edited to ensure that a uniform style of notation and presentation is used throughout. As a result, all researchers whose work uses item response theory will find this an indispensable companion to their work and it will be the subject's reference volume for many years to come.
According to Lewis and Mayer's (1987) simulation model for understanding compare problems, students make more comprehension errors when the order of the terms in the relational statement is not consistent with the preferred order. Three eye movement experiments, designed to test a number of hypotheses directly derived from this model, are discussed. The 1st experiment with university students solving 1-step compare problems revealed no evidence in favor of the model; the data of the 2nd experiment with 3rd graders, on the other hand, provided good support. To explain the results of the 1st experiment, a 3rd experiment was carried out in which university students were given a set of 2-step compare problems. The results from that study also fit with the model well, suggesting that the model holds true only when the task puts some cognitive demands on the subject.
This study investigated children's ability to write number sentences for simple addition and subtraction word problems. First graders taught to represent problems with open number sentences (e.g., 5+□ =8) represented a variety of problems with number sentences that directly modeled the action in the problems. First graders taught to represent all problems with number sentences in standard form (a+b=□ , a-b=□ ) were limited in the problems they could represent. Second graders could represent problems directly with open number sentences or transform them to number sentences in standard form. The results, consistent with research on solutions using modeling and counting strategies, suggest that open number sentences may provide mathematical symbolism that allows young children to build upon informal strategies for representing and solving simple word problems.
We present information-processing models of different levels of knowledge for understanding the language used in texts of arithmetic word Problems, for forming semantic models of the situations that the texts describe, and for making the inferences needed to answer the questions in the problems. In the simplest cognitive models, inferences are limited to properties of sets that exist in a semantic model. In more complex cognitive models, relations between sets are represented internally and support more complex reasoning. Performance on three sets of problems by kindergarten through third-grade students was used to test the models. Global tests provided support for the models. These included measures of scalability and frequencies of individual children's patterns of solutions that agreed with predictions of the models. Performance on problems involving combinations and changes of sets was explained better by the cognitive models than performance on problems involving comparisons. Comparisons may require more advanced understanding of numbers as values of operators rather than only as cardinalities of sets.
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 problem-solving 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.
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.
The following values have no corresponding Zotero field:
Research Notes: 20090703 Wenn ich das richtig verstehe, zeigt Fischer, dass das Rasch-Modell notwendig aus einigen Axiomen folgt, zu denen insb. die spezifische Objektivität gehört. Das heißt, die spezifische Objektivität ist ein Postulat, nicht eine Folge des Raschmodells.
ID - 500
This article presents an analysis of conceptual and linguistic complexities of matching situations expressed as word problems and describes possible ways of conceptualizing and solving such problems. Data from first and second graders suggest a progression of four levels in conceptualizing and solving these problems. In the first-Relational-level, children can answer "Who has more/less?" but not "How much more/less?" In the second-Language Cue-level, children are more likely to solve problems with action, Equalizing language ("If he gets 2 more cats, he will have as many cats as dogs") than with static, Compare language ("He has 2 more dogs than cats"). They are especially likely to solve problems in which finding the unknown compared quantity is directed by keywords in the relational sentence. At the third-Understand Matching Situations-level, children find Inconsistent problems (those in which the relational sentence is opposite to the needed solution action) considerably more difficult than other types. Children overwhelmingly solve problems in which one compared quantity is unknown by using an Equalizing approach in which the Extra quantity is added to or taken from the other known quantity. They predominantly solve problems in which the difference between two known compared quantities is unknown by using a Matching conception in which the Small quantity is taken from the Big quantity. At the fourth-Solve Inconsistent-level, children come to be able to solve Inconsistent problems, primarily by using Equalizing conceptions in which the relation given in the relational sentence is reversed.
In contrast to expectation-based, predictive views of discourse comprehension, a model is developed in which the initial processing is strictly bottom-up. Word meanings are activated, propositions are formed, and inferences and elaborations are produced without regard to the discourse context. However, a network of interrelated items is created in this manner, which can be integrated into a coherent structure through a spreading activation process. Data concerning the time course of word identification in a discourse context are examined. A simulation of arithmetic word-problem under- standing provides a plausible account for some well-known phenomena in this area. Discourse comprehension, from the viewpoint of a computa- tional theory, involves constructing a representation of a dis- course upon which various computations can be performed, the outcomes of which are commonly taken as evidence for com- prehension. Thus, after comprehending a text, one might rea- sonably expect to be able to answer questions about it, recall or summarize it, verify statements about it, paraphrase it, and
It is shown that the problem of evaluating model fit can be solved within the framework of the general multinomial model, and it is shown how tests for this framework can be adapted to the Rasch model. Four types of tests are considered: generalized Pearson tests, likelihood ratio tests, Wald tests, and Lagrange multiplier tests. The statistics presented not only support the purpose of a global overall model test, but also provide information with respect to specific model violations, such as violation of sufficiency of the sum score, strictly monotone increasing and parallel item response functions, unidimensionality, and differential item functioning.
Solving arithmetic word problems links the ability to execute arithmetic operations and the ability to apply these operations in real-word situations. An appropriate recent interest in word problems has produced much data on the relative difficulty of different kinds of word problems, and some data on the strategies children use to solve these problems together with the kinds of errors they make. We also have accounts of problem difficulty based on descriptions of the problem, that is, on their structural variables or on their semantic structure. We present here and account of problem difficulty based on a description of the psychological processes of the child. The purpose of this article is to outline a model of these processes, made explicit in the form of a computer-implemented model. The model solves many common word problems by acting them out with representations of physical counters. More difficult problems require augmenting this procedure first with knowledge that one object is a member of both a set and its superset, and second with knowledge that processes can be "undone" and that subsets can be exchanged. We characterize problems by the kind of knowledge the model uses to solve them. A comparison of these knowledge types with data on problem difficulty allows us to estimate how much the need to use each kind of knowledge contributes to a problem's difficulty. We also observe and relate to children's performance the sequence of steps the program produces a solution, and the errors the program makes when it lacks certain knowledge. Finally, we contrast the knowledge in the program with the knowledge we believe is acquired by children in school.
outline the real world domain of whole number addition and subtraction situations / describe the developmental progression of conceptual structures that children between the ages of 2 and 8 construct to interpret and solve these situations / the unitary conceptual structures built for numbers up to one hundred will be considered first, and then the multiunit conceptual structures built for multi-digit whole numbers will be discussed
a vision of what might be possible in preschool and primary school classrooms will then be summarized / most of our present knowledge about U.S. children's understanding of addition and subtraction is based on research done with children who have received traditional mathematics school instruction; many of the limitations of this instruction have been pointed out in this chapter (PsycINFO Database Record (c) 2012 APA, all rights reserved)
Research conducted in several countries has shown consistent patterns of performance on change, combine and compare word problems involving addition and subtraction. This paper interprets these findings within a theoretical framework that emphasizes the development of empirical, logical and mathematical knowledge.
The Rasch model is an item analysis model with logistic item characteristic curves of equal slope,i.e. with constant item discriminating powers. The proposed goodness of fit test is based on a comparison between difficulties
estimated from different scoregroups and over-all estimates.
Based on the within scoregroup estimates and the over-all estimates of item difficulties a conditional likelihood ratio is
formed. It is shown that—2 times the logarithm of this ratio isx
2-distributed when the Rasch model is true.
The power of the proposed goodness of fit test is discussed for alternative models with logistic item characteristic curves,
but unequal discriminating items from a scholastic aptitude test.
Anteckningar fran seminarier lasaret 1969-1970 utarbetade av Rolf Sundberg
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A developmental theory of number understanding
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Hamburg-Welcher-Intelligenztest fur Kinder
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Erwerb mathematischer Kompetenzen: Literaturuberblick [The development of mathematical competencies: A literature overview
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Die Entwicklung des mathematischen Verstandnisses im Kindesalter [The development of mathematical competences in childhood
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Signifikanz und Relevanz von Modellabweichungen beim Rasch-Modell [Significance and relevance of deviations of the Rasch Model
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