Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1701 primary and secondary school students. Eight components, which all involve representational transformations, were encompassed under the construct of representational flexibility in fraction and decimal number addition. This structure reveals that, for both concepts, the representational transformation competences of recognition and conversion, and therefore representational flexibility as well, were affected by the complexity of the concepts involved and the direction of the conversion, respectively. Results also showed that two first-order factors were needed to explain the problem-solving ability in fraction and decimal number addition, indicating the differential effect of the modes of representation that is diagrammatic and verbal form on problem-solving ability irrespective of the concepts involved, as in the case of the conversions. Representational flexibility and problem-solving ability were found to be major components of students’ representational thinking of fraction and decimal number addition. The proposed framework was invariant across the primary and secondary school students. Theoretical and practical implications are discussed.

     

Tracing the development of representational flexibility and problem solving in fraction addition: a longitudinal study

The study models the development of students’ multiple-representation flexibility and the use of problem-solving strategies and representations in fraction addition. A test administered three times, with breaks of 3–4?months between successive measurements to 108 students at a transition within primary school (Grade 5–6), 132 students at a transition from primary to secondary education (Grade 6–7) and 148 students at a transition within secondary school (Grade 7–8). Multivariate analysis of variance for repeated measures and dynamic structural equation modelling were carried out in order to analyse the data. Findings suggested that students’ performance improve through measurements. Dynamic modelling provided evidence for the strong interrelation between representational flexibility and problem solving at the three measurements. The results indicated the students’ established pre-existent knowledge and the important role the initial state of the aforementioned cognitive parameters plays on their advancement. Didactical implications are discussed.     

小学数学解决问题方法多样化的策略分析

引导小学生独立解决数学问题是目前我国小学数学教学的目标,而数学因其本身的属性及特点,因此要求学生能够充分发挥逻辑思维能力去寻找更多的解题方法,这就对教师的教学提出了更高的要求。教师在教学过程中要不断创新教学模式,通过现代化教学理念来锻炼学生的自主学习能力,拓宽学生的思维路径,从而使其可以衍生出更加多样化的解题方法。     

Primary school text comprehension predicts mathematical word problem-solving skills in secondary school

This longitudinal study aimed to investigate the extent to which primary school text comprehension predicts mathematical word problem-solving skills in secondary school among Finnish students. The participants were 224 fourth graders (9–10 years old at the baseline). The children’s text-reading fluency, text comprehension and basic calculation ability were tested in grade four. In grade seven and nine, their skills in solving mathematical word problems were assessed. Overall, the results showed that text comprehension in grade four of primary school predicts math word problem-solving skills in secondary school, after controlling for text-reading fluency and basic calculation ability. Among boys, good text comprehension skills in grade four predicted good math word problem-solving skills in grade seven. Among girls, good text comprehension skills in grade four predicted their subsequent mathematical word problem performance in grade nine. The practical implications of the results are discussed as well.     

对初中历史材料分析题解题技巧的思考和认识

随着教育改革、素质教育的不断深入,初中教育教学更加注重对学生综合能力的培养,这使初中历史材料题愈加体现出开放性、实用性、知识性和灵活性的特征.从历史材料题目内容上看,资料内容明显增多,更侧重对学生独立思考能力和知识应用能力的考核.由于历史材料的题型变化灵活,解题方法不断更新,所以许多初中生的历史材料分析题解题能力较为薄...     

The role of rational number density knowledge in mathematical development

Many students still have not developed a robust understanding of rational number concepts at the end of primary school, despite several years of instruction on the topic. The present study aims to examine the patterns, predictors, and outcomes of the development of rational number knowledge in lower secondary school. Latent transition analysis revealed that rational number development from primary to lower secondary school (N = 362) appears to follow similar patterns as in younger students. In particular, a majority of students had poor knowledge of the density of the rational number set. Whole number magnitude knowledge appeared to be an important predictor of the development of rational number size knowledge, but not density knowledge. Finally, fraction density knowledge appeared to be related to concurrent algebra knowledge. Together these results point to an important role for density knowledge in mathematical development.     

Learning Multi‐unit Number Concepts and Understanding Decimal Place Value

Research indicates that many errors in secondary school mathematics can be attributed to lack of understanding of decimal place value. Recently it has been suggested that this arises from an underlying weakness in integer place value concepts and perception of multidigit numbers in unitary terms rather than base ten, multi‐unit structures. Development of place value and multidigit number concepts was examined in individual interviews with 262 primary school pupils, aged between 7 and 12 years, using subjects’ concrete representations of written numerals and associated verbal explanations. Almost all the children had some understanding of integer place value; most older children had multi‐unit concepts of three‐ or four‐digit numbers; but two‐digit numbers, especially values of 20 or less, were frequently perceived in terms of units by all age groups. This was particularly common for the value ten where 55% of subjects displayed unitary concepts. Implications for understanding and teaching decimal place value are discussed.     

纠错复原型机器人教学案例研究

中小学机器人课程作为一门典型的工程教育类课程备受关注,如何在机器人教育中培养学生的工程设计思维和问题解决能力值得研究者思考。通过纠错任务解决机器人产品中的问题,能够促进学生理解机器人内部组成结构及功能,培养问题解决能力,获取更深层的学习体验。结合已有研究,文章尝试将纠错复原型教学模式应用于中小学机器人教育,通过感知与操作、拆分与分析、识别与诊断、测试与总结等步骤引导学生发现问题、思考问题并解决问题,旨在为机器人教育提供实践参考。     

Fraction Instruction for Students with Mathematics Disabilities: Comparing Two Teaching Sequences

Abstract. This study investigated the effects of teaching middle school students with mathematics disabilities equivalent fraction concepts and procedures using the concrete‐representational‐abstract (CRA) instructional sequence or the representational‐abstract (RA) instructional sequence. Twenty‐six students formed the CRA group, and 24 students formed the RA group. The two treatment groups received carefully sequenced instruction over 10 lessons. The only difference between the two treatment groups was that the CRA group used concrete manipulative devices for the first three lessons while the RA group used representational drawings. Analyses of the data indicated that students in both treatment groups improved overall in their understanding of fraction equivalency from pretest to posttest. On all achievement measures, students in the CRA group had overall higher mean scores than did students in the RA group. Implications for classroom instruction and suggestions for further research are discussed.     

中小学教育策略探析

中小学年龄段的学生,还没有形成良好的世界观、人生观与价值观,其逻辑思维能力以及综合能力的发展也不够健全。因此,学校应有效加强对中小学教育教学策略的重视,积极更新中小学教育的管理理念,建立科学有效的教育思想,结合先进的教学手段,摒弃传统落后的教育模式,实现对中小学教育策略的创新,从而满足当今中小学生的学习需求,调动中小学生学习的积极性与热情。本文就中小学教育策略进行探讨与分析,以供参考。     

The relationship between students’ approaches to learning and the assessment of learning outcomes

The purpose of the present study is to gain more insight into the relationship between students’ approaches to learning and students’ quantitative learning outcomes, as a function of the different components of problem-solving that are measured within the assessment. Data were obtained from two sources: the revised two factor study process questionnaire (R-SPQ-2F) and students’ scores in their final multiple-choice exam. Using a model of cognitive components of problem-solving translated into specifications for assessment, the multiple-choice questions were divided into three categories. Three aspects of the knowledge structure that can be targeted by assessment of problem-solving were used as the distinguishing categories. These were: understanding of concepts; understanding of the principles that link concepts; and linking of concepts and principles to application conditions and procedures. The 133 second year law school students in our sample had slightly higher scores for the deep approach than for the surface approach to learning. Plotting students’ approaches to learning indicated that many students had low scores for both deep and surface approaches to learning. Correlational analysis showed no relationship between students’ approaches to learning and the components of problem-solving being measured within the multiple choice assessment. Several explanations are discussed.     

小学数学教学中培养学生应用能力探析

小学数学教学在小学课程的编排体系当中占据了文化课的半壁江山,是教学中的重中之重,因此上小学数学教学当中,培养学生的数学应用能力是一大教学难点。众所周知,小学生因为年龄小,数学思维能力有限,这就需要教师全力以赴培养学生的数学应用能力,全面提高学生的数学素养。首先要增强学生丰富的基础知识,有了丰富的数学知识,才能在生活当中加以应用,提高自身的数学思维逻辑能力,掌握最基本的解决问题的方法技巧,为今后的学习打下扎实的基础。     

浅谈小学数学课堂教学中问题意识的培养

数学既是小学阶段就开始接触的学科,也是对小学生数学思维运用和解题能力要求较高的学科,没有问题可“问”的小学数学课堂无疑是枯燥和无趣的,难以真正实现对小学生思维能力的培养,因此,本文结合目前小学数学课堂出现的问题,探究怎样在小学数学教学中将学生问题意识培养起来。     

Improving students’ representational flexibility in linear-function problems: an intervention

This study evaluates the effects of an intervention aimed at improving representational flexibility in linear-function problems. Forty-nine students aged 13–16 participated in the study. A pretest–intervention–posttest design with an experimental and control group was used. At pretest, both groups solved a choice test, where they could freely select a table, a graph, or a formula, and three no-choice tests, where they used predetermined representations. Twenty-five students in the experimental group were exposed to the intervention, where they learnt to fine-tune their choices both to the task at hand and to their own characteristics as representational users. The control group was not exposed to any intervention. The posttest was similar to the choice pretest. A flexibility score was calculated per student based on Siegler and Lemaire’s score. At posttest, the experimental group became more flexible than the control group. The impact of increased flexibility on problem-solving accuracy and speed is discussed.     

Distributed by Design: On the Promises and Pitfalls of Collaborative Learning with Multiple Representations

This article presents a designed learning environment intended to engage students in learning about the relationships among multiple representations as they work together on a shared task. Over the course of several extended problem-solving sessions, groups developed several successive alignments of participants and representations as they learned to solve increasingly difficult tasks. Our findings highlight the emergent and often unexpected meanings that learners established for representational tools as their groups reorganized into increasingly effective problem-solving ensembles. Our findings echo those of prior research regarding learners' considerable competence and creativity in interpreting and applying distributed representational tools, as well as the careful coordination among learners involved in establishing and acting on those interpretations. Challenges in this design space include instances in our data where students capitalized on connections among representations without really trying to understand those connections, temporarily undermined the distributed character of the representations, and worked more efficiently by reducing the number of participants actively involved in breaking codes. Our findings indicate that managing these challenges requires presenting groups with regular opportunities to reconsider and reorganize their roles, and to experiment with different meanings and uses of flexible tools in the context of tasks with carefully sequenced levels of difficulty.     

STEM教育理念融入小学科学教学实践探究——以“电路出故障了”一课为例

STEM教育理念融入小学科学教学既有可行性也有必要性。教师在将STEM教育理念融入小学科学教学实践过程中,可采取以下四环节教学策略:第一,引入生活,创设情境,提出科学探究问题;第二,设定方案,选择材料,制作解决问题工具;第三,小组合作,科学验证,技术检测解决问题;第四,归纳分析,得出结论,联系实际归纳提升。这个过程中,可培养学生的科学素养、工程素养、技术素养和数学素养,进而在综合性的科学课堂教学中提升学生的动手能力和创新思维。     

中小学教师多媒体艺术修养

中小学教师的艺术修养直接影响多媒体教学的质量。中小学教师多媒体艺术修养内涵由艺术审美能力和艺术呈现能力组成。艺术审美能力的培养以美育教学理念为前提,以学生审美认知需要为中心;艺术呈现能力的提高以教师艺术知识技能为基础,以教学艺术与多媒体教学相结合为手段。中小学教师只有通过"知"、"孕"、"化"的艺术体验过程,其多媒体艺术修养才能在实践中得以提高。     

GENDER DIFFERENCES IN LEARNING OF THE CONCEPT OF FORCE, REPRESENTATIONAL CONSISTENCY, AND SCIENTIFIC REASONING

This quantitative case study used a pre- and posttest design for exploring the gender differences in secondary school students’ (n?=?131, 45 males and 86 females) learning of the force concept when an interactive engagement type of teaching was used. In addition, students’ ability to interpret multiple representations (i.e., representational consistency) was documented by a pre- and posttest and scientific reasoning ability by a pretest only. Males significantly outperformed females in learning of the force concept, pre- and posttest representational consistency, and pretest scientific reasoning. However, the gender difference in learning of the force concept was not significant when ANCOVA was conducted using pretest results of representational consistency and scientific reasoning as covariates. This appeared to indicate that the gender difference in learning gain was related to students’ abilities before the instruction. Thus, the teaching method used was equally effective for both genders. Further, our quantitative finding about the relation between representational consistency and learning of the force concept supports the assumption that multiple representations are important in science learning.     

基于核心素养的少数民族中小学生自我认知发展现状及其影响因素分析

自我认知是向内的自我探索,是人自我意识的认知部分,包括自我观察和自我评价等。本研究通过编制《基于核心素养的少数民族中小学生自我认知发展状况调查》问卷,以青海、内蒙古等少数民族聚居区为主要调查省份,研究少数民族中小学生自我认知发展现状。研究结果表明:少数民族中小学生自我认知能力整体处于中等偏上水平,其中自我评价能力略高于自我观察能力;出生地和就学地在城市的少数民族中小学生自我认知能力发展较好;父母亲职业稳定,家庭条件优越的少数民族中小学生的自我认知能力更高。基于此,为提高少数民族中小学生的自我认知能力,首先,应重视少数民族中小学生自我认知的发展,营造良好氛围;其次,充分利用少数民族中小学生自身优势,加强教育的针对性;最后,指导少数民族中小学生学会自我分析和自我体验,增强自我了解,提高对自身的满意度。     

三角函数最值问题探讨

三角函数作为基本函数之一,在数学中占有很重要的地位,是中专数学的重点研究内容。本文主要介绍解三角函数题的方法,三解函数在中专数学中和其它一些知识的联系及应用,以此来培养学生的解题能力和使知识系统化的能力。