Taking the relational structure of fractions seriously: Relational reasoning predicts fraction knowledge in elementary school children

Understanding and using symbolic fractions in mathematics is critical for access to advanced STEM concepts. However, children and adults consistently struggle with fractions. Here, we take a novel perspective on symbolic fractions, considering them within the framework of relational structures in cognitive psychology, such as those studied in analogy research. We tested the hypothesis that relational reasoning ability is important for reasoning about fractions by examining the relation between scores on a domain-general test of relational reasoning (TORR Jr.) and a test of fraction knowledge consisting of various types of fraction problems in 194 s grade and 145 fifth grade students. We found that relational reasoning was a significant predictor of fractions knowledge, even when controlling for non-verbal IQ and fractions magnitude processing for both grades. The effects of relational reasoning also remained significant when controlling for overall mathematics knowledge and skill for second graders but was attenuated for fifth graders. These findings suggest that this important subdomain of mathematical cognition is integrally tied to relational reasoning and opens the possibility that instruction targeting relational reasoning may prove to be a viable avenue for improving children’s fractions skills.     

Examining the unique contributions and developmental stability of individual forms of relational reasoning to mathematical problem solving

Relational reasoning, a higher-order cognitive ability that identifies meaningful patterns among information streams, has been suggested to underlie STEM development. This study attempted to explore the potentially unique contributions of four forms of relational reasoning (i.e., analogy, anomaly, antinomy, and antithesis) to mathematical problem solving. Two separate samples, fifth graders (n = 254) and ninth graders (n = 198), were assessed on their mathematical problem solving ability and the different forms of relational reasoning ability. Linear regression analysis was conducted, with participants’ age, working memory, and spatial skills as covariates. The results showed that analogical and antithetical reasoning abilities uniquely predicted mathematical problem solving. This pattern demonstrated developmental stability across a four-year time frame. The findings clarify the unique significance of individual forms of relational reasoning to mathematical problem solving and call for a shift of research direction to reasoning abilities when exploring dissimilarity-based relations (opposites in particular).     

Patterns in children's ability to recall explicit,implicit and metaphorical information

The purpose of this study was to examine the effects of grade level and recall task type on children's memory for explicit, implicit and metaphorical information following reading. Forty second and 40 fifth grade subjects balanced as to gender read four prose passages and were randomly assigned to cued and free recall conditions. A MANOVA produced significant main effects for grade level and task type on each of three dependent variables; EXPLICIT, IMPLICIT and METAPHOR recall measures. As predicted, fifth graders fared better than second graders and the cued condition exceeded free recall. Of greater significance was the finding that second graders were able to engage in metaphorical reasoning when the task was appropriately structured. The results suggest that providing externally generated, structured probes can greatly enhance children's reading recall at both grade levels and can enable children to demonstrate metaphorical comprehension prior to the age at which it spontaneously appears. In addition, significantly different patterns of recalled information were found within the two task conditions, patterns that remained stable across age groups.     

Children's concepts of gears and their promotion through play

The mechanical transmission of motion, which is observable in gears’ turning direction (TD) and turning speed (TS) can be understood as part of the core concept “force and motion”. Previous studies have suggested that most children before the age of nine have naïve concepts of gears’ TD and TS. However, these studies relied on children's explanation data and might have therefore underestimated young children's potential. Moreover, these studies did not examine whether preschoolers can overcome their naïve concepts through teaching. Recently, educators have proposed guided play as an educational approach that combines children's play with teacher guidance in a purposefully designed environment. We conducted two subsequent studies. In Study 1, we investigated the children's (naïve) concepts of gears’ TD and TS with a cross‐sectional approach using a nonverbal test procedure. This study comprised 248 children aged 5–10 years and 73 adults. The results showed that the proportion of children with adequate concepts increased with age. More specifically, 7‐ to 8‐year‐olds differed significantly from 5‐ to 6‐year‐olds, indicating a developmental shift around this age. However, naïve concepts of TS were more persistent with age than naïve concepts of TD. Altogether, the results indicated the potential to foster 5‐ to 6‐year‐olds’ concepts of both TD and TS in an intervention. Thus, in Study 2, we developed and tested a guided play intervention to foster 5‐ to 6‐year‐old children's concepts of TD and TS. We conducted a prepost‐intervention design with a guided play group (n = 19) and a free play group (n = 21). The results suggested stronger learning gains in the guided play group (d_TD = 0.337, d_TS = 0.758) than in the free play group (d_TD = 0.224, d_TS = 0.158).     

Radical sensitivity is the key to understanding Chinese character acquisition in children

This study investigated Chinese children’s development of sensitivity to positional (orthographic), phonological, and semantic cues of radicals in encoding novel Chinese characters. A newly designed picture-novel character mapping task, along with nonverbal reasoning ability, vocabulary, and Chinese character recognition were administered to 198 kindergartners, 172 second graders and 165 fifth graders. Children’s strategies in using positional, phonological, and semantic cues of radicals varied across grades. The higher the children’s grade level, the more commonly children used semantic and positional cues of radicals. Regression analyses showed that the contribution of semantic radical awareness for explaining Chinese character reading increased as children’s grade increased, whereas the contribution of positional regularity awareness decreased. These findings suggest that learning Chinese characters involves a transition from a sound- and position-based approach to a meaning-based approach.     

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.     

Mixed-Domain Events: The Influence of Moral and Conventional Components on the Development of Social Reasoning

The purpose of the current study was to examine children's reasoning about mixed-domain events containing both conventional and moral Components (i.e., violating a conventional rule and negatively affecting others). The participants were preschoolers, first graders, and third graders (N = 100). Children evaluated (a) the legitimacy of an authority to permit mixed-domain acts to occur, and (b) the acceptability of the mixed-domain events when permitted and prohibited by an authority. In addition, children rated the seriousness of mixed-domain rule violations. Results showed that children with increasing age were able to identify the moral components of the mixed-domain events and combined moral and conventional issues in their reasoning about the events. Preschoolers and first graders were more likely than third graders to view the mixed-domain acts as only conventional.     

The Development of Children’s Understanding of Speed Change: A Contributing Factor Towards Commonsense Theories of Motion

Previous research indicates children reason in different ways about horizontal motion and motion in fall. At the same time, their understanding of motion down inclines appears to result from an interaction between horizontal and vertical motion understanding. However, this interaction is still poorly understood. Understanding of speed change may shed further light due to its critical role in natural object motion. This is addressed by the present two studies. Children (n = 144) aged 5–11 years predicted whether a ball, either heavy or light, would accelerate, decelerate or move at unchanging speed along a horizontal, in fall and down an incline, both in a real-object task (Study 1) and a computer-presented task (Study 2). The results suggest understanding of speed change is typically limited to the expectation that change takes place between a point of no motion and any subsequent point in motion, but not between two subsequent points. Despite improvements with age, predictions of continued speed change barely exceeded chance levels in the oldest age group. Modest effects of object mass were noted. Response time data provide further insight regarding children’s predictions, highlighting similarities and differences in reasoning between motion dimensions. The overall findings are used to develop clearer ideas about the development of children’s understanding of speed change as well as to advance commonsense theories of motion.     

Recognition as support for reasoning about horizontal motion: a further resource for school science?

Background: Even infants can recognize whether patterns of motion are or are not natural, yet an acknowledged challenge for science education is to promote adequate reasoning about such patterns. Since research indicates linkage between the conceptual bases of recognition and reasoning, it seems possible that recognition can be engaged to support reasoning.

Purpose: Noting the theoretical and practical significance of showing that recognition can support reasoning, the reported research aimed to examine the possibility in relation to horizontal motion.

Sample: The research was conducted with 167 children (mean age = 9.51 years) from Years 4, 5 and 6 of an English-medium school located in Lisbon, Portugal.

Design and methods: Individual pre-tests were administered to all participants to assess initial reasoning about the direction and speed with which rolling balls travel after collision. Reasoning was assessed in the sense of both predicting and explaining. Thereafter, about two-thirds of the sample worked with software that, via simulations of the incorrect patterns that were typically predicted (and comparison with simulations of correct patterns), engaged recognition as feedback on reasoning. The remaining children became an untutored control group. Replicating characteristic computer use in classrooms, half of the software sample worked as singletons with adult guidance available on request and half worked in pairs without the option of guidance. A few weeks later, all participants were post-tested following pre-test procedures.

Results: Pre- to post-test change amongst the children who worked with the software exceeded pre- to post-test change within the control group, and this was observed with both predictions and explanations. The differences were strongest amongst the children who worked as singletons, even though they seldom requested adult support.

Conclusions: Although issues remain to be addressed before the approach can be optimized, its viability as support for reasoning has been demonstrated, and this may have relevance beyond horizontal motion.     

Mixed-Domain Events: The Influence of Moral and Conventional Components on the Development of Social Reasoning

The purpose of the current study was to examine children's reasoning about mixed-domain events containing both conventional and moral Components (i.e., violating a conventional rule and negatively affecting others). The participants were preschoolers, first graders, and third graders (N = 100). Children evaluated (a) the legitimacy of an authority to permit mixed-domain acts to occur, and (b) the acceptability of the mixed-domain events when permitted and prohibited by an authority. In addition, children rated the seriousness of mixed-domain rule violations. Results showed that children with increasing age were able to identify the moral components of the mixed-domain events and combined moral and conventional issues in their reasoning about the events. Preschoolers and first graders were more likely than third graders to view the mixed-domain acts as only conventional.     

Adolescents' and parents' reasoning about actual family conflict

This study employed a distinct domain perspective on social-cognitive development to assess reasoning about issues of family conflict. Subjects were 102 fifth through twelfth graders from 2-parent families and their parents. Individually interviewed family members described actual family conflicts and, for each, justified their position on the dispute and reasoned about them from the other's perspective. Parents generated fewer conflicts than did children. Preadolescent to late adolescent families generally agreed that conflicts occurred over the mundane, everyday details of family life, but they did not agree on their interpretation. Adolescents understood but rejected their parents' conventional interpretations of conflicts, reasoning instead in terms of personal choice. Boys' understanding of their parents' conventional perspectives increased significantly with age, whereas girls' understanding was significantly lower in early adolescence, as compared to pre- or late adolescence. The results are discussed in terms of adolescent individuation and parent-child conflict during adolescence.     

Conservation Comes of Age

Abstract

This study investigated the effects of instruction in environmental education on the attitudes of elementary school children toward the environment. One class of third graders was provided with two hours of environmental education instruction. A comparable class of third graders served as a control group. Both classes were pre- and posttested using an author-designed environmental attitude questionnaire. Analyses of these data showed that 1) the experimental group had significantly improved attitudes on a pre-post basis and 2) this same group had significantly more positive attitudes toward the environment when compared with the control group. When tested again as fifth graders, the experimental group remained significantly more positive toward the environment than the fifth grade control group.     

The Use of Trait Labels in Making Psychological Inferences

Three studies investigated children's capacity to use trait labels as tools for making inferences about mental states. For example, knowledge that a story character is "nice" as opposed to "mean" could lead to predictions that the character would respond with greater negative affect upon discovering that his or her action had made someone upset. Study 1 (N = 48) examined whether participants (kindergartners, second graders, fifth graders, and adults) would make different psychological inferences based on whether a character was labeled as "nice" versus "mean." Study 2 (N = 30) examined the same issue with 4-year-olds using a simpler methodology. Study 3 (N = 30) extended the results of Study 2, by examining whether describing characters as "shy" versus "not shy" would lead 4-year-olds to make different mental state inferences. Taken together, these findings suggest that even for young children, trait labels can serve as a basis for making nonobvious inferences. Developmental differences are discussed.     

Children's use of response confidence in the processing of instructional feedback

The main purpose of the present study was to extend a model of instructional feedback (Kulhavy & Stock, 1989) to elementary school children. Forty-five third and fifth graders (third graders, N = 21; fifth graders, N = 24) read text passages, answered questions, and either rated response confidences for each question or used an imagery strategy. Although differences were not found between treatment groups, analyses of response patterns revealed differential age effects. Response patterns for fifth graders were similar to those of college populations; however, these patterns did not emerge for third graders.     

Pictures,imagery, and children’s recall of central versus peripheral sentence information

It is fairly clear that story-relevant pictures help children recall story information. But do pictures showing details distract students’ attention from central information, and thus hinder recall? In this study of second and fifth graders, such pictures (as well as subject-generated images) facilitated recall of both central and incidental information contained in single-sentence mini-stories.     

Developing Model‐Based Reasoning

Key elements of the structure and function of models in mathematics and science are identified. These elements are used as a basis for discussing the development of model‐based reasoning. A microgenetic study examines the beginnings of model‐based reasoning in a pair of fourth‐ and fifth‐grade children who solved several problems about chance and probability. Results are reported in the form of a cognitive model of children's problem‐solving performance. The cognitive model explains a transition in children's reasoning from tacit reliance on empirical regularity to a form of model‐based reasoning. Several factors fostering change in children's thinking are identified, including the role of notations, peer interaction, and teacher assistance. We suggest that model‐based reasoning is a slowly‐developing capability that emerges only with proper contextual and social support and that future study should be carried out in classrooms, where these forms of assistance can also be part of the object of study.

Model‐based reasoning is a significant intellectual milestone because it bridges the worlds of personal, intuitive knowledge, on the one hand, and mathematical‐scientific theory, on the other. However, across disciplines, consensus is still forming about what model‐based reasoning comprises, and there is little knowledge about its ontogenetic origins or how it develops. We consider analogy as the core of modeling, because in model‐based reasoning a system in one domain is used to understand a system in another. To understand how models come to play a role in reasoning, it is important to initiate study of their origins. Accordingly, we report a microgenetic study examining the beginnings of model‐based reasoning in a pair of young children solving problems about chance and probability. In this study we are engaged in the enterprise of modeling the development of modeling. That is, we report our results in the form of a cognitive model of children's problem‐solving performance that explains a transition in reasoning from a tacit reliance on empirical regularity to a form of model‐based reasoning. It is important to note the two distinct meanings for the term model used in this article. The first describes how children come to understand and appropriate a system of reasoning exemplified in practices of modeling. The second describes a research tool, a model of human reasoning—specifically, how children in this study began to use models of probability to reason about uncertain events. In this report, we use the terms model or model‐based reasoning to refer to the former interpretation, whereas references to a cognitive model denote the simulation of children's thinking—in this case, implemented as a computer program.

Before describing the empirical work, we first identify some key elements of the structure and function of models. Next, these elements of modeling are used as the basis for generating some conjectures about the development of model‐based reasoning. We describe a task that we used as a window to understanding progression in student reasoning toward reliance on models as tools for thought. We present our rationale for developing cognitive models of student performance and explain some choices concerning the implementation of the cognitive model reported here. Finally, we turn to the children's performance on chance and probability tasks and explain how that performance illuminates both what children do not understand about models and the kinds of relevant knowledge that they are acquiring.     

基于ADAMS的AL4自动变速器运动仿真实现

利用ADAMS软件,参照AL4自动变速器和直齿圆柱齿轮的运动规律,利用简化方法,建立AL4自动变速器传动模型。根据其真实的工作状态,通过添加约束、驱动、设定参数等,实现AL4自动变速器在不同档位下的运动状态仿真,可以真实地判定各档位的转动方向和速度大小,更加直观地了解AL4自动变速器的工作机理。