The consistency effect in word problem solving is effectively reduced through verbal instruction

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

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

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

Gender differences in cognitive abilities of learning-disabled females and males

Gender differences in level and pattern of cognitive abilities were examined in 28 LD college-able females (CA 18–25) as compared to 21 LD college-able males (CA 18–25). Both groups were in the average IQ range as measured by the Wechsler Adult Intelligence Scale, with LD males significantly higher on the Full Scale IQ and three out of the four subtests, Picture Completion, Block Design, and Information. The LD females performed significantly better on the Digit Symbol subtest. The hierarchies of subtest performance and Bannatyne and ACID category scores were compared. LD females have strengths in visual-motor abilities and verbal conceptualization, while the LD males’ highest abilities were nonverbal visual-spatial confirming earlier studies on younger LD individuals and non-LD males and females. Performance on the Digit Symbol subtest was the next to the lowest for the males, the highest for females. However, for both groups, short-term and long-term memory for digits and factual knowledge and mental arithmetic problem solving were relative weaknesses. Results indicate different patterns of cognitive abilities in LD females and males which have implications for identification, service, and prognosis for the learning disabled, especially females.     

Inhibiting interference from prior knowledge: Arithmetic intrusions in algebra word problem solving

In Singapore, 6–12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13–17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is correlated with the ability to inhibit intrusions from the earlier arithmetic strategies. Similar to typical school practice in Singapore, we asked 14 year-old students (N = 157) to use only letter-symbolic algebra to solve 9 algebra word problems. After having controlled for algebraic knowledge, working memory, and intelligence, better inhibitory ability still predicted fewer arithmetic intrusions and higher problem solving accuracy. Path analysis revealed 2 types of inhibition. Inhibition-of-reified-processes predicted accuracy through arithmetic intrusions. Inhibition-of-recently-learned-associations predicted accuracy through intelligence. Findings suggest establishing pedagogical links between arithmetic and algebraic methods may facilitate students' transition to letter-symbolic algebra.     

On the Stability of Students'Rules of Operation for Solving Arithmetic Problems

The purpose o f this study was to examine the consistency with which students applied procedural rules for solving signed-number operations across identical items presented in different orders. A test with 64 open-ended items was administered to 161 eighth graders. The test consisted o f two 32-item subtests containing identical items. The items in each subtest were in random order. Students'responses to each subtest were compared with respect to the identified underlying rules o f operation used to solve each problem set. The results indicated that inconsistent rule application was common among students who had not mastered signed-number arithmetic operations. In contrast, mastery level students, those who use the right rules, show a stable pattern o f rule application in signed-number arithmetic. These results are discussed in light of the hypothesis testing approach to the learning process.     

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

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

Verbal counting skill predicts later math performance and difficulties in middle school

This study examined the role of verbal counting skill as an early predictor of math performance and difficulties (at or below −1.5 standard deviation in basic math skills) in middle school. The role of fourth-grade level arithmetical skills (i.e., calculation fluency, multi-digit arithmetic i.e. procedural calculation, and word problem solving) as mediators was also investigated. The participants included 207 children in central Finland who were studied from kindergarten to the seventh grade. Path modeling showed that verbal counting in kindergarten is a strong predictor for basic math performance in seventh grade, explaining even 52% of the variance in these skills after controlling for the mothers’ education levels. This association between early verbal counting skill and basic math performance was partly mediated through fourth-grade procedural calculation and word problem solving skills. Furthermore, verbal counting had an unique predictive relation to middle school math performance above and beyond the basic arithmetical and problem solving skills in fourth grade. Poor kindergarten verbal counting skill was a significant indicator for later difficulties in mathematics.     

Performance on Middle School Geometry Problems With Geometry Clues Matched to Three Different Cognitive Styles

ABSTRACT— This study investigated the relationship between 3 ability‐based cognitive styles (verbal deductive, spatial imagery, and object imagery) and performance on geometry problems that provided different types of clues. The purpose was to determine whether students with a specific cognitive style outperformed other students, when the geometry problems provided clues compatible with their cognitive style. Students were identified as having a particular cognitive style when they scored equal to or above the median on the measure assessing this ability. A geometry test was developed in which each problem could be solved on the basis of verbal reasoning clues (matching verbal deductive cognitive style), mental rotation clues (matching spatial imagery cognitive style), or shape memory clues (matching object imagery cognitive style). Straightforward cognitive style–clue‐compatibility relationships were not supported. Instead, for the geometry problems with either mental rotation or shape memory clues, students with a combination of both verbal and spatial cognitive styles tended to do the best. For the problems with verbal reasoning clues, students with either a verbal or a spatial cognitive style did well, with each cognitive style contributing separately to success. Thus, both spatial imagery and verbal deductive cognitive styles were important for solving geometry problems, whereas object imagery was not. For girls, a spatial imagery cognitive style was advantageous for geometry problem solving, regardless of type of clues provided.     

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

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

A comparison of updating processes in children good or poor in arithmetic word problem-solving

This study examines the updating ability of poor or good problem solvers. Seventy-eight fourth-graders, 43 good and 35 poor arithmetic word problem-solvers, performed the Updating Test used in Palladino et al. [Palladino, P., Cornoldi, C., De Beni, R., and Pazzaglia F. (2002). Working memory and updating processes in reading comprehension. Memory and Cognition, 29, 344–354.]. The participants listened to wordlists, each comprising 12 words referring to objects or animals of different sizes. At the end of each list participants were asked to recall the 3 or 5 words denoting the smallest objects/animals in the list. Results show that poor problem-solvers recalled fewer correct words and made more intrusion errors (recall of non-target words) than good problem-solvers. Results support the hypothesis that the ability to select and update relevant, and suppress irrelevant information, is related to problem-solving, even when the influence of reading comprehension is controlled for. With reference to Baddeley's, and other recent WM models [Miyake, A., and Shah, P. (Eds.), (1999). Models of working memory: Mechanisms of active maintenance and executive control. New York: Cambridge University Press.], our results point to the idea that problem-solving relies on the central executive for processing and updating information contained in the problems.     

Counting Processes in Simple Addition1

Abstract: Svenson, O. & Hedenborg, M. L. 1980. Counting processes in simple addition. Scandinavian Journal of Education Research 24,93‐104. Verbal protocols and response latencies were used in this study to investigate the thought processes used by children solving simple arithmetic problems (I + J = , where 0≤I≤13 and 0< J ≤13). To specify, the verbal reports were used to classify the solutions into one of two main groups: (a) answers retrieved from long term memory and (b) answers involving active manipulation of the problem in working memory (reconstructive solutions). The response latencies in each of these groups were analyzed separately for each child. As expected, latencies for retrieved answers were shorter than for reconstructed solutions. The retrieval from long term memory of the answer to a problem with the smaller number first (e.g., 3+4) required about 0.1 sec longer time than when the numbers were in the reverse order (4 + 3). Grouping latencies according to verbal protocols made it possible to refine the analysis of the response latencies. To exemplify, counting strategies with greater units than one (e.g., 6 + 4: 6, 8, 10) were identified and parameters describing the cognitive processes listed.     

The effects of using drawings in developing young children’s mathematical word problem solving: A design experiment with third-grade Hungarian students

The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students?? knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment involved five experimental and six control classes (N?=?106 and 138, respectively) of third-grade students. The experiment comprised 20 lessons with 73 word problems, providing a systematic overview of the basic word problem types. Teachers of the experimental classes received a booklet containing lesson plans and overhead transparencies with different types of visual representations attached to the word problems. Students themselves were invited to make drawings for each task, and group work and teacher-led discussion shaped their beliefs about the role of visual representations in word problem solving. The effect sizes of the experiment were calculated from the results of two tests: an arithmetic skill and a word problem test, and the unbiased estimates for Cohen??s d proved to be 0.20 and 0.62. There were significant changes also in experimental group students?? beliefs about mathematics. The experiment pointed to the possibility, feasibility, and importance of learning about visual representations in mathematical word problem solving as early as in grade?3 (around age 9?C10).     

Using Self-Generated Drawings to Solve Arithmetic Word Problems

Abstract

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

Generalizability and Validity of a Mathematics Performance Assessment

The QUASAR Cognitive Assessment Instrument (QCAI) is designed to measure program outcomes and growth in mathematics. It consists of a relatively large set of open-ended tasks that assess mathematical problem solving, reasoning, and communication at the middle-school grade levels. This study provides some evidence for the generalizability and validity of the assessment. The results from the generalizability studies indicate that the error due to raters is minimal, whereas there is considerable differential student performance across tasks. The dependability of grade level scores for absolute decision making is encouraging; when the number of students is equal to 350, the coefficients are between .80 and .97 depending on the form and grade level. As expected, there tended to be a higher relationship between the QCAI scores and both the problem solving and conceptual subtest scores from a mathematics achievement multiple-choice test than between the QCAI scores and the mathematics computation subtest scores.     

Construction play and cognitive skills associated with the development of mathematical abilities in 7-year-old children

Construction play is thought to develop logico-mathematical skills, however the underlying mechanisms have not been defined. In order to fill this gap, this study looks at the relationship between Lego construction ability, cognitive abilities and mathematical performance in 7-year-old, Year 2 primary school children (N = 66). While studies have focused on the relationship between mathematics performance and verbal memory, there are limited studies focussing on visuospatial memory. We tested both visuospatial and verbal working memory and short term memory, as well as non-verbal intelligence. Mathematical performance was measured through the WIAT-II numerical operations, and the word reading subtest was used as a control variable. We used a Lego construction task paradigm based on four task variables found to systematically increase construction task difficulty. The results suggest that Lego construction ability is positively related to mathematics performance, and visuospatial memory fully mediates this relationship. Future work of an intervention study using Lego construction training to develop visuospatial memory, which in turn may improve mathematics performance, is suggested.     

Connaissances utilisées par des elèves de 8 à 12 ans dans la formulation de problèmes arithmétiques concrets

This study analyses children development of semantic, linguistic, procedural and schematic knowledge in the context of writing arithmetic word problems. 139 children aged between 8 and 12 years old were presented with a task which consisted in writing arithmetic word problems, according to some contraints: words, questions or measures to include in their problems; type of problems to write. Results show the relevance of actual theoritical models of problem solving (Mayer, 1983; Kintsch & Greeno, 1985). Schematic knowledge seem indeed more important than other knowledge in the process of writing arithmetic word problems; semantic knowledge are also used to choose relevant numbers or measures; the roles of linguistic and procedural knowledge seem less evident. Finally, some hypotheses related with the development of mental models of arithmetic word problems are formulated.     

Analyzing difficulties with mole-concept tasks by using familiar analog tasks

This study was conducted to determine which skills and concepts students have that are prerequisites for solving moles problems through the use of analog tasks. Two analogous tests with four forms of each were prepared that corresponded to a conventional moles test. The analogs used were oranges and granules of sugar. Slight variations between test items on various forms permitted comparisons that would indicate specific conceptual and mathematical difficulties that students might have in solving moles problems. Different forms of the two tests were randomly assigned to 332 high school chemistry students of five teachers in four schools in central Indiana. Comparisons of total test score, subtest scores, and the number of students answering an item correctly using appropriate t-test and chi square tests resulted in the following conclusions: (1) the size of the object makes no difference in the problem difficulty; (2) students understand the concepts of mass, volume, and particles equally well; (3) problems requiring two steps are harder than those requiring one step; (4) problems involving scientific notation are more difficult than those that do not; (5) problems involving the multiplication concept are easier than those involving the division concept; (6) problems involving the collective word “bag” are easier to solve than those using the word “billion”; (7) the use of the word “a(n)” makes the problem more difficult than using the number “1”.     

The consistency effect depends on markedness in less successful but not successful problem solvers: An eye movement study in primary school children

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.     

Expertise,inhibitory control and arithmetic word problems: A negative priming study in mathematics experts

Solving arithmetic word problems such as “Mary has 25 marbles. She has 5 more marbles than John. How many marbles does John have?”, in which the relational term (more than) interferes with the arithmetic operation (subtraction), relies in part on the ability to inhibit an overlearned ‘add if more or subtract if less’ heuristic in children, adolescents and adults. Here, we used a negative priming (NP) paradigm to investigate whether experts in mathematics need to inhibit this heuristic when solving this type of arithmetic word problem. We found NP effects in experts in mathematics, but with a smaller amplitude than those in non-experts (N = 40). We replicate these results in a second experiment (N = 62) in which we matched experts and non-experts on general intelligence and inhibitory control ability. This suggests that experts also need to inhibit the ‘add if more or subtract if less’ heuristic to solve such problems but were more efficient at inhibiting the heuristic than non-experts.     

大班幼儿加减运算能力的发展特点——以武汉市8所幼儿园为例

选取武汉市8所幼儿园112名大班幼儿作为被试,对幼儿加减运算能力进行测试。结果发现:(1)幼儿笔算、口算、解答应用题和自编应用题四种能力之间均显著相关,解答应用题和自编应用题的能力低于笔算和口算能力。(2)幼儿解答应用题能力的发展有显著的性别差异,男孩的能力优于女孩。(3)省示范园和市示范园幼儿的加减运算能力发展优于私立园幼儿。