Gender Differences in Arithmetic Strategy Use: A Function of Skill and Preference

This study was designed to examine whether first-grade boys' use of retrieval and first-grade girls' use of manipulatives reflected gender differences in their abilities to use these strategies or gender differences in preferences for strategy use. Eighty-four first-grade students, 42 boys and 42 girls, from two suburban elementary schools participated in this study. The children solved basic arithmetic problems under two conditions: a free-choice condition in which they were allowed to solve the problems any way they preferred and a game condition in which the children's strategy use was constrained so that all children used the same strategies on the same arithmetic problems. Strategy use during the free-choice session replicated the findings of earlier research indicating that girls tend to use strategies utilizing manipulatives and boys tend to use retrieval. During the game condition, when we controlled the types of strategies children used on different problems we found that boys were as able as girls to calculate solutions using manipulatives. Girls, however, were not as capable as boys in their retrieval of answers to arithmetic problems from memory. No differences were found in error rates or speed of retrieval. Gender differences were found in the variability of correct retrieval, with boys being significantly more variable than girls.     

Deaf and hard of hearing students' problem-solving strategies with signed arithmetic story problems

The use of problem-solving strategies by 59 deaf and hard of hearing children, grades K-3, was investigated. The children were asked to solve 9 arithmetic story problems presented to them in American Sign Language. The researchers found that while the children used the same general types of strategies that are used by hearing children (i.e., modeling, counting, and fact-based strategies), they showed an overwhelming use of counting strategies for all types of problems and at all ages. This difference may have its roots in language or instruction (or in both), and calls attention to the need for conceptual rather than procedural mathematics instruction for deaf and hard of hearing students.     

Across the great divide: bridging the gap between understanding of toddlers' and older children's thinking

Research on very young children's cognitive development differs greatly from research on cognitive development in older children. The differences include the questions that are asked, the methods that are used to address them, the measures that are employed to provide relevant evidence, and the level of detail at which children's knowledge is represented. The research approaches are so different that they create an impression that infants' and toddlers' thinking differs qualitatively from that of preschoolers and older children. This impression, however, may reflect differences in research approaches rather than differences in children's thinking. In the present study, we attempted to bridge this gap by applying to toddlers a type of process analysis that has proved fruitful in studies of older children. Overlapping waves theory, trial-by-trial strategy assessments, and microgenetic methods were used to analyze 1.5- and 2.5-year-olds' problem solving and learning. The results demonstrated that changes in toddlers' strategies could be assessed reliably on a trial-by-trial basis, that the changes followed the basic form predicted by the overlapping waves model, and that analyses of toddlers' strategies could tell us a great deal about both qualitative and quantitative aspects of their learning. A componential analysis of learning that previously had been applied to older children also proved useful for understanding toddlers' learning. The analysis specified that cognitive change frequently involves five components: acquisition of new strategies; strengthening of the strategies in their original context; improved mapping of strategies onto novel problems; increasingly refined choices among variants of the strategies; and increasingly skillful execution of the strategies. Independent measures of these components indicated that strategic development in toddlers involves improvements in all five components. Analyses of individual differences in learning showed that the effects of distal variables, such as age and sex, could be partially explained in terms of their influence on mastery of the components, but that the distal variables exercised additional direct effects as well. The process of learning in toddlers closely resembled that of older children in other ways as well. Like older children, toddlers use multiple strategies over the course of learning; their choices among strategies are quite adaptive from early on; their choices become progressively more adaptive as they gain experience with the task; they switch strategies not only from trial to trial but within a single trial; their transfer of learning from one problem to the next is primarily influenced by structural relations between problems but also is influenced by superficial features; they show utilization deficiencies early in learning that they gradually overcome; and they show individual differences in learning that fall into a few qualitatively distinct categories. Perhaps most striking, the 1.5- and 2.5-year-olds emerged as active learners, who continued to work out the lessons of previous instruction in the absence of further instruction. That is, they integrated the lessons of their own problem-solving efforts with the previous instruction in ways that magnified the initial effects of the instruction. Overall, the findings indicated that the gap can be bridged; that theories, methods, measures, and representations of knowledge typically used with older children can improve our understanding of toddlers' problem solving and learning as well.     

Using Problem Solving to Assess Young Children’s Mathematics Knowledge

Mathematics problem solving provides a means for obtaining a view of young children’s understanding of mathematics as they move through the early childhood concept development sequence. Assessment information can be obtained through observations and interviews as children develop problem solutions. Examples of preschool, kindergarten, and primary grade children’s approaches to problem solving are provided in the article. Prekindergarten and kindergarten age children discover problems during play. For example, they figure out how to use informal measurement to use construction materials such as unit blocks and Lego to build a desired building or make a desired object. Moldable materials such as clay and play dough provided shape experiences. The daily sequence of activities builds on their concept of time. Primary grade children solve adult- and child-generated problems. They may use manipulatives and/or drawings to generate problem solutions prior to using symbols and notation. Teacher and/or student devised rubrics can be used to guide evaluation.     

Professional uncertainty and professional growth: A case study of experienced teachers

This qualitative study proposes a model of the processes of attaining professional knowledge through which teachers develop strategies for getting comfortable with their image of themselves as an effective teacher. This is in itself a changing concept of effectiveness which has to be able to encompass dealing with professional uncertainty as a legitimate function of their role and not as a reflection of their inadequacies. The 12 experienced teachers in this case study preserved and built a sense of their own competence by developing a concept of professional practice which was both knowledgeable and malleable. The strategies they used to resolve professional uncertainty are described. In this case study, the NUD.IST computer software program was used to support the data analysis and interpretation.     

Family strategies to support and develop resilience in early childhood

Early childhood is an important time for the development of resilience. A recently completed study has followed three cohorts of resilient children and young people living in disadvantaged areas in Victoria, Australia, through different transitions in their educational careers. This paper focuses on the early childhood cohort, where we have followed children from kindergarten/preschool into primary school. Using data gathered primarily through interviews with parents (mothers in each case), this paper presents a qualitative naturalistic sub-study that used deductive thematic analysis to explore the different strategies used by families to support their child’s resilience. Our findings highlight that resilience was a salient concept for all of the mothers in the study and each mother articulated a range of strategies they used within the family to try and support their child’s development and resilience. These strategies were constrained by the settings in which the families lived.     

Analysis of fraction errors by a model of cognitive science

In clinical interviews, grade 7 students of average abilities worked out some problems on the topic of fractions; the problems were given to the children in different forms of representation. From the examples it is illustrated that, even if children are dealing with similiar exercises and even if they had been taught in the same way, their mathematical thinking might be quite different. The differences in the children’s behavior seem to come from differences in the internal representation of their knowledge of the concepts and procedures. To explain why the children handled as they did, a theoretical framework is needed. In this paper, Davis and McKnight’s «hypothetical mechanisms» are used to explain children’s understanding of fractions. A revised version of Minsky’s and Davis’ frame-model seems to be especially useful to explain the relationship between the teacher’s notions, the children’s «concept images» and the real-world aspects of mathematics.     

Fourth graders’ adaptive strategy use in solving multidigit subtraction problems

Using the choice/no-choice methodology we investigated Dutch fourth graders’ (N = 124) adaptive use of the indirect addition strategy to solve subtraction problems. Children solved multidigit subtraction problems in one choice condition, in which they were free to choose between direct subtraction and indirect addition, and in two no-choice conditions, in which they had to use either direct subtraction or indirect addition. Furthermore, children were randomly assigned to mental computation, written computation, or free choice between mental and written computation. One third of the children adaptively switched their strategy according to the number characteristics of the problems, whereas the remaining children consistently used the same strategy. The likelihood to adaptively switch strategies decreased when written computation was allowed or required, compared to mandatory mental computation. On average, children were adaptive to their own speed differences but not to the accuracy differences between the strategies.     

Strategic competence: Applying Siegler’s theoretical and methodological framework to the domain of simple addition

In this study we investigated the variability, frequency, efficiency, and adaptiveness of young children’s strategy use in the domain of simple addition by means of the choice/no-choice method. Seventy-seven beginning second-graders, divided in 3 groups according to general mathematical ability, solved a series of 25 simple additions in 3 different conditions. In the first condition, children could choose whatever strategy they wanted to solve each problem. In the second and third condition, the same children had to solve all problems with one particular strategy, respectively adding up to 10 and retrieval. The results demonstrate that second-graders as a whole choose adaptively between retrieval, decomposition, and counting strategies when solving simple additions, and that they use these strategies neither equally frequently nor equally efficiently. Furthermore, our results indicate that children with different mathematical ability use generally the same strategies to solve these problems, but differ in the frequency, accuracy and adaptiveness with which they apply these strategies. Finally, this study documents the value of the choice/no-choice method to assess the adaptiveness of young children’s strategy use in the domain of early arithmetic.     

A Classroom Investigation of the Growth of Metacognitive Awareness in Kindergarten Children Through the Writing Process

This study investigated the presence and growth of kindergarten children's metacognition as they engaged in the writing process. The study was conducted in an environment that surrounded children with books, language, and print. Twice a month the teacher/researcher interviewed the children as they finished writing, asking questions designed to help them reflect on their thinking and strategies they used in their writing. Anecdotal records, observations, and individual writing folders were used to complete a checklist of writing strategies for each child. Interviews with the children confirmed that they were exhibiting and showing growth in their metacognition. They were able to provide appropriate answers to questions that required them to talk about their thinking and identify strategies that helped them in their writing. The study provides a model that could be used in classrooms to help children in the development of their growing metacognition and writing in an authentic learning environment.     

The Relation between Reading Comprehension and Task‐specific Strategies used in Arithmetical Word Problems

The study was concerned with the relation between reading comprehension and task‐specific strategies used in arithmetical word problems. Two hypotheses were formulated in the study. H1 stated that subjects good at arithmetic as well as reading (Group 1) would emphasize deductive strategies to a greater extent than subjects in the remaining groups. H2 considered IQ to be a stronger predictor for subjects’ strategies in arithmetical word tasks than reading comprehension. Four groups of 9‐year‐old children with different competencies in arithmetic and reading were sampled for the investigation. An intelligence test (WISC) and three other tests, Mathematical tests 1‐3, were constructed and administered to the 20 subjects. The tests were used to examine subjects’ strategies in completed task solutions. The findings indicated that great similarities existed between Groups 1 (good at both skills) and 2 (good at arithmetic and poor at reading) as to strategies used in addition and subtraction problems. Whereas subjects in groups 1 and 2 mainly used deductive strategies, those in Groups 3 (good at reading and poor at arithmetic) and 4 (poor at both skills) rather used procedural strategies in solving the problems. Concerning strategies applied in multiplication and division tasks a more diverse pattern of strategies was revealed in all of the four groups, but the majority of subjects emphasized deductive strategies. Multiple regression analyses of the data indicated strong linear relationships between the predictors and the criteria used in the analyses. However, IQ was the only significant single predictor found in the regression analyses. H1 had to be suspended, but H2 could not be rejected.     

Proportions in elementary school

Proportion problems were given to pupils in grades 3, 4, and 5. The number structure of the problems was kept at an elementary level and the contexts were varied. The success rates were high, which shows that young children have some understanding of the concept of proportion. Elementary strategies based on addition were often used. Familiarity with the use of ratios in the context, the presence of a mixture, and the use of manipulatives were context variables that influenced performance. The abilities required to solve proportion problems in various contexts were essentially independent from each other. The cognitive characteristics that predict success were found to be different from those that predict use of multiplicative strategies.     

The Role of Drawing in Young Children’s Construction of Science Concepts

It has been observed that many young children like making marks on paper and that they enjoy the activity. It is also known that children’s drawings are vehicles for expression and communication. Therefore, it would be logical and reasonable for teachers to incorporate children’s drawings into building science concepts. To demonstrate how drawings are utilized to help a child to acquire a science concept, the article first presents a vignette of an interaction between an adult and a 5-year-old boy, focusing on the science concept of the physical characteristics of a spider. It is then followed by several analytical explanations of how drawings build children’s understandings. Not only are the introduced strategies useful for one-on-one interactive communication, but also applicable to a small group of young children. The article ends with the specifics of how these strategies were applied to a group of four children in their acquisition of the science concept of the water cycle.     

Tutoring conjunctions and construction of geometry concepts in the early childhood education: The case of the angle

Recent research concerning the understanding of geometrical concepts, such as the concept of angles, shows that children of pre-school age, as far as it concerns problems of angle comparison, systematically focus on visual comparative procedures. We suggest, however, that social interaction of a child having the role of tutor in diverse guided environments, could allow the building of this concept in the nursery school on strategies that have formed in natural settings. Sixty-four (64) subjects were accessed according to their performance in a test of angle comparison. They were introduced for an equal period to two experimental guidance conditions. In the first condition, the code of guidance is exclusively verbal and, in the second, the symbolic code of the Logo environment suceeds the verbal communication. Results show, on one hand, much more systematic improvement in a subjects performance where different contexts of guidance were used. On the other hand, in the verbal guidance experimental condition, children’s correct answers showed a dinstinct progress to the stage of finally advancing their performance.     

Organisation du produit de deux ensembles: Influence des interactions sociales entre pairs sur les procédures de résolution et les performances individuelles

This study investigates the efficacy of socio-cognitive confrontations between children upon their individual abilities to solve problems involving the organization of products of two sets. The focus is more precisely on the analysis of both the effects of interaction setting conditions, and the mechanisms that may yield progress. 67 5 to 6-year-old children were assigned to a control group in which they had to work alone and two experimental groups in which they had to work in pairs on two problems. The difference between the two experimental groups was where each child sat in relation to the other and to the matrix they had to fill. Results show higher proportions of progress and of changes of problem-solving procedures among children belonging to the experimental groups. The role played by the interaction setting conditions is also evidenced. Moreover, the «destabilization» of solving procedures appears to be a determining element for the efficacy of peer interactions. Finally in the discussion, we attempt at showing how a solving processes analysis enables us to understand better how activities carried out in socio-cognitive confrontation and individual cognitive activities can be articulated.     

主动性学习理念下教师支持幼儿发展适宜性游戏基本策略

区域活动是幼儿在幼儿园内参与游戏的主要形式之一。主动性学习理念下的区域活动对教师支持幼儿游戏发展提出了新的要求。以大班的区域活动为例,通过创设支持性的游戏环境、观察与评估幼儿的游戏、在鼓励与欣赏中引导幼儿的游戏往更为复杂高阶的方向发展等策略,可以促进幼儿适宜性游戏的发展。     

Algebraic Generalization Strategies Used by Kuwaiti Pre-service Teachers

This study reports on the algebraic generalization strategies used by elementary and middle/high school pre-service mathematics teachers in Kuwait. They were presented with 9 tasks that involved linear, exponential, and quadratic situations. The results showed that these pre-service teachers had difficulty in generalizing algebraic rules in all 3 types of tasks. The most difficult problems were those involving exponents; they seemed to confuse exponential growth with the multiplication of n. Although they used several strategies such as explicit, recursive, and chunking, they tended to rely on explicit reasoning. They saw the explicit rule as a quick and easy rule to be used. In order to improve students’ understanding of algebraic reasoning, we must improve the understating of generalization. Teachers’ programs should put more emphasis on the type of content that teachers need to learn rather than on how much mathematics credits teachers need.     

Do They Know What They Are Doing? Children's Use of Economical Addition Strategies and Knowledge of Commutativity

Children's methods of solving addition problems progress towards strategies that reverse the order of addends. The role knowledge of commutativity plays in this development is not clear. Previous studies suggest some use the strategies without believing addend order is irrelevant to addition. The numbers of such children are very small and the tests of commutativity may be inaccurate. Two studies using new tasks compare strategy use and knowledge. In the first, 47/48 children between 6 and 10 years old predicted order of addends would not alter quantity in some contexts. They also predominantly used strategies that reversed addend order. In the second study, 5‐year‐olds (18/24) were more likely to know commutativity than to use a strategy that reversed addend order or to answer sums correctly. Knowing commutativity precedes using strategies that presuppose it and does not derive from doing sums. Representing the addends with objects induced more arithmetic errors. Children used less accurate strategies when they could see the objects.     

Medicus interruptus in the behaviour of children in disadvantaged contexts in Scotland

The medicalisation of the behaviour of children is a phenomenon that is attracting growing attention, with particular concern about the increased likelihood of children living in disadvantaged contexts receiving a medical diagnosis, such as attention-deficit hyperactivity disorder, and treatment. This paper reports on a study of professionals involved with children experiencing behavioural problems. The professionals interviewed in this study articulated their own reservations about the medicalisation of children’s behaviour and revealed a number of strategies for interrupting the process towards diagnosis. These interruptions, analysed using Deleuze and Guattari’s concept of deterriorialisation, took place along linguistic, visual and affective planes and were successful in encouraging teachers and head teachers to see alternatives to the medical route. The findings have implications for existing practice in the response to, and support for, behavioural problems and for teacher education.