Mathematical development: the role of broad cognitive processes

This study investigated the role of broad cognitive processes in the development of mathematics skills among children and adolescents. Four hundred and forty-seven students (age mean [M] = 10.23 years, 73% boys and 27% girls) from an elementary school district in the US southwest participated. Structural equation modelling tests indicated that calculation complexity was predicted by long-term retrieval and working memory; calculation fluency was predicted by perceptual processing speed, phonetic coding, and visual processing; problem solving was predicted by fluid reasoning, crystallised knowledge, working memory, and perceptual processing speed. Younger students’ problem solving skills were more strongly associated with fluid reasoning skills, relative to older students. Conversely, older students’ problem solving skills were more strongly associated with crystallised knowledge skills, relative to younger students. Findings are consistent with the theoretical suggestion that broad cognitive processes play specific roles in the development of mathematical skills among children and adolescents. Implications for educational psychologists are discussed.     

Investigating the Unique Predictors of Word-Problem Solving Using Meta-Analytic Structural Equation Modeling

The purpose of the present study is to clarify the contributions of cognitive skills (nonverbal reasoning, language comprehension, working memory, attention, processing speed) and academic skills (mathematics facts retrieval, mathematics computation, mathematics vocabulary, reading comprehension) in performing mathematics word problems among elementary school students. With the two-stage meta-analytic structural equation modeling approach, I synthesized 112 correlation matrices from 98 empirical studies (N?=?111,346) and fitted the hypothesized partial mediation model. Overall, path analysis indicated that language comprehension, working memory, attention, mathematics vocabulary, and mathematics computation were unique predictors of word-problem solving. Subgroup analysis demonstrated different unique predictors for younger and older students to perform word problems (K-2nd grades versus 3rd–5th grades). Implications, limitations, and future directions are discussed.

     

Effects of Manipulative Instruction on Solving Area and Perimeter Problems by Students with Learning Disabilities

Abstract. A multiple baseline design was employed to test the effect of manipulative instruction on the perimeter and area problem‐solving performance of middle and high school students who had been diagnosed with LD in the area of mathematics. Modeling, prompting/guided practice, and independent practice in conjunction with manipulative training were employed to teach both perimeter and area problem‐solving skills. Analysis of data revealed that the students rapidly acquired the problem‐solving‐skills, maintained these skills over a two‐month period, and transferred these skills to a paper and pencil problem‐solving format. This research extends previous findings by revealing that use of concrete manipulatives promotes the long‐term maintenance of skills.     

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.     

An Examination of the Relationship between Computation,Problem Solving,and Reading

The purpose of this study is to evaluate the relationship of mathematics calculation rate (curriculum-based measurement of mathematics; CBM-M), reading rate (curriculum-based measurement of reading; CBM-R), and mathematics application and problem solving skills (mathematics screener) among students at four levels of proficiency on a statewide test. It was hypothesized that CBM-M provides insufficient information to make good screening decisions and that other measures with content more similar to that of large-scale tests of mathematics would function to improve screening. One hundred and seventy students in third grade from a rural elementary school in the Midwestern United States participated. Structural equation modeling was used to evaluate direct, mediator, and latent growth models. In general, CBM-R mediated the relationship between the mathematics ability screener and passing the state assessment, while CBM-M did not have any significant paths within these models. Results are discussed in terms of the utility of CBM-M and CBM-R procedures in screening for success on state test performance in mathematics.     

论有效渗透小学数学思想方法

数学课堂中,教师不仅要关注学生习得的基本知识和基本技能,更要适时采用不同策略有效渗透一些数学思想方法,培养学生的思维能力。现结合小学数学广角教学实践,从情境创设、知识形成、问题思考、知识应用及课外实践五个维度进行了渗透策略的研究,让学生获得更广泛的数学活动经验,领悟分析问题、解决问题的学习方法,进而实现渗透目的。     

职前中学数学教师解题能力的培养

解题是中学数学教师的基本功之一,但目前职前中学数学教师在解题方面的能力比较弱。因此,需要有效地、全面地提高职前中学数学教师解题的能力。具体来说,需要系统地、有计划地培养职前中学数学教师解题能力。在第一学年研读中学数学课标和教材,强化中学数学的概念性知识;在第二学年熟练掌握中学数学解题中的解题思想和方法;在第三、四学年对解题中的有意识控制的知识进行训练。同时,将解题的书面表达、口头表达的训练融入每一个阶段当中。     

中学数学竞赛中的初等数论问题——以希望杯初中数学竞赛试题为例

以希望杯试题为依托,就初中数学竞赛中所涉及的初等数论问题,根据运用初等数论知识进行解题的方法与初中数学竞赛试题中的解题方法,总结出作为竞赛辅导老师应具备对于相关知识的本质内容的理解并能对不同方法进行甄别、优化,以此作为对初中生进行竞赛辅导的依据。     

初中数学发散性思维能力培养策略

《初中数学课程标准(2011版)》指出,数学课程能使学生掌握必备的基础知识和基本技能,培养学生的抽象思维和推理能力,培养学生的创新意识和实践能力数学的发散性思维能力是"问题解决"的基础,是培养数学推理能力和创新意识前提要求。数学发散性思维作为用学科自身的品质陶冶人、启迪人、充实人。"问题解决"是人的高级数学思维。高级思维的学习,可以使学生充分享受思维的快乐,可以让思维自由飞翔。本文就初中数学发散思维的培养谈几点体会,通过创设问题情景、设置开放性试题、发挥学科优势等教学策略,着力培养初中学生的数学发散性思维能力,实现有效教学。     

基于应用题解题障碍的认知工具模式设计

通过整理近年来关于小学数学应用题的相关研究,总结学生在应用题解题过程中普遍存在的认知障碍,结合现有解题系统分析研究各类型的认知工具对小学应用题解题障碍的支持,为帮助学生选择合适的认知工具提供借鉴,并在此基础上提出基于问题解决障碍的小学数学应用题解题系统的模型构建,为新的认知工具的设计开发提供新的角度。     

Does Early Algebraic Reasoning Differ as a Function of Students’ Difficulty with Calculations versus Word Problems?

According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second‐grade students, we administered: (1) measures of calculations and word problems in the fall and (2) an assessment of prealgebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word‐problem measures, we placed 148 students into one of four difficulty status categories: typically performing, calculation difficulty, word‐problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word‐problem difficulty is more strongly associated with difficulty with prealgebraic reasoning. As an indicator of later algebra difficulty, word‐problem difficulty may be a more useful predictor than calculation difficulty, and students with word‐problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.     

Investigating Elementary and Middle School Students’ Subjective Well-Being and Mathematical Performance in Shanghai

Students in Shanghai ranked at the top in mathematics on the past two assessments of the Program for International Student Assessment (PISA). However, even though Asian students in general, and Chinese students in particular, ranked at the top in mathematics, we know little about Shanghai students’ subjective well-being (SWB). This paper reports two studies that investigated the SWB of elementary and middle school students in Shanghai, as well as their mathematics performance. It is found that the mean scores of elementary school students on all scales of SWB are significantly higher than those of middle school students, representing a large drop for students when moving from elementary to middle school. Furthermore, students’ mathematics performance is moderately correlated with their SWB. And for Grade 6, students’ SWB affects their mathematics performance, especially from academic self-concept, attentiveness in the classroom, and relationship with teachers.     

初中物理比例计算技巧初探

初中物理习题的计算往往步骤比较多,但对于填空题和选择题,学生在解答正确的前提下更需要提高解题效率。文章结合教学经验,总结出初中物理比例计算法,并通过具体的教学案例说明比例计算法的应用技巧,同时也指出其局限性,旨在提高学生解题效率和正确率。     

Visual spatial representation in mathematical problem solving by deaf and hearing students

This research examined the use of visual-spatial representation by deaf and hearing students while solving mathematical problems. The connection between spatial skills and success in mathematics performance has long been established in the literature. This study examined the distinction between visual-spatial "schematic" representations that encode the spatial relations described in a problem versus visual-spatial "pictorial" representations that encode only the visual appearance of the objects described in a problem. A total of 305 hearing (n = 156) and deaf (n = 149) participants from middle school, high school, and college participated in this study. At all educational levels, the hearing students performed significantly better in solving the mathematical problems compared to their deaf peers. Although the deaf baccalaureate students exhibited the highest performance of all the deaf participants, they only performed as well as the hearing middle school students who were the lowest scoring hearing group. Deaf students remained flat in their performance on the mathematical problem-solving task from middle school through the college associate degree level. The analysis of the students' problem representations showed that the hearing participants utilized visual-spatial schematic representation to a greater extent than did the deaf participants. However, the use of visual-spatial schematic representations was a stronger positive predictor of mathematical problem-solving performance for the deaf students. When deaf students' problem representation focused simply on the visual-spatial pictorial or iconic aspects of the mathematical problems, there was a negative predictive relationship with their problem-solving performance. On two measures of visual-spatial abilities, the hearing students in high school and college performed significantly better than their deaf peers.     

美国中小学数学教育的现状及思考

美国中小学数学课程计划充分体现多样化、自主化的特色,尤其是高中会提供丰富的课程计划供不同学生选择.高中数学必修课程一般至少修3个学分的数学,包括代数1、几何、代数2等内容.美国高中数学选修课程很多.美国中小学数学教材呈现教材多样、目标明确、利于教学、资源丰富等特色.美国学校课堂教学重视多种学习方式、重视基础知识与基本技能教学、重视基本方法教学、重视问题解决教学、注重分层教育.。     

Algebraic Problem Solving for Middle School Students with Autism and Intellectual Disability

Problem solving is an important yet neglected mathematical skill for students with autism spectrum disorder and intellectual disability (ASD/ID). In addition, the terminology and vocabulary used in mathematical tasks may be unfamiliar to students with ASD/ID. The current study evaluated the effects of modified schema-based instruction (SBI) on the algebra problem solving skills of three middle school students with ASD/ID. Mathematics vocabulary terms were taught using constant time delay. Participants were then taught how to use an iPad that displayed a task analysis with embedded prompts to complete each step of solving the word problems. This study also examined participant’s ability to generalize skills when supports were faded. Results of the multiple probe across participants design showed a functional relation between modified SBI and mathematical problem solving as well as constant time delay and acquisition of mathematics vocabulary terms. Implications for practice and future research are discussed.     

新课标下中职学生数学问题意识的培养

数学是一门以问题为核心的学科,对于中职学生来讲,他们的问题意识非常薄弱,其数学教学氛围十分低迷,学生们的数学基础差厌学情绪较高,整个数学课堂教学陷入了举步维艰的困境之中,针对这些情况的发生,教师以积极探索的创新态度以剖析如何让学生在数学学习中培养寻求、探索和解决问题的意识,掌握提出问题的技能,培养学生们自己发现问题解决问题的意识.     

An empirical investigation of differences between mathematics specialists and non-specialists at the high school level in Cyprus: A Logistic regression Approach

AN EMPIRICAL INVESTIGATION OF DIFFERENCES BETWEEN MATHEMATICS SPECIALISTS AND NON-SPECIALISTS AT THE HIGH-SCHOOL LEVEL in Cyprus – The data obtained from high-school seniors for the Third International Mathematics and Science Study (TIMSS) for the country of Cyprus appear to be contradictory. Although Cypriot students did not perform well in mathematics in elementary school, middle school, and in the non-advanced sectors of high school, students in advanced mathematics courses in high school managed to perform exceptionally well. In seeking to account for this apparent disparity, the present study examines the differences between mathematics specialists and non-specialists at the high-school level and discusses the implications that these have for teaching practice. It shows how students educated in an environment that might not be optimal for producing high-achieving students in mathematics and science in elementary and middle school (according to the TIMSS) might nonetheless manage to excel in these fields at the end of their schooling. In conclusion, the authors address the implications of their study for similar educational systems in other developing countries.     

Using Performance Task Data to Improve Instruction

Two well-accepted ideas among educators are (a) performance assessment is an effective means of assessing higher-order thinking skills and (b) data-driven instruction planning is a valuable tool for optimizing student learning. This article describes a locally developed performance task (LDPT) designed to measure critical thinking, problem solving, and written communication skills of elementary and middle school students in a large school district. An investigation was conducted to determine how middle school teachers use LDPT responses to inform instruction and to ascertain how students are reflecting on their performance. Results from an analysis of documents indicated teachers were engaged in formal data-informed practices within professional learning communities. Moreover, educators examined student work samples and planned instruction based on student-specific needs. Strategies to reinforce proficiency in the skills assessed by the district's LDPT are reported.     

An empirical investigation of differences between mathematics specialists and non-specialists at the high school level in Cyprus: A Logistic regression Approach

AN EMPIRICAL INVESTIGATION OF DIFFERENCES BETWEEN MATHEMATICS SPECIALISTS AND NON-SPECIALISTS AT THE HIGH-SCHOOL LEVEL in Cyprus – The data obtained from high-school seniors for the Third International Mathematics and Science Study (TIMSS) for the country of Cyprus appear to be contradictory. Although Cypriot students did not perform well in mathematics in elementary school, middle school, and in the non-advanced sectors of high school, students in advanced mathematics courses in high school managed to perform exceptionally well. In seeking to account for this apparent disparity, the present study examines the differences between mathematics specialists and non-specialists at the high-school level and discusses the implications that these have for teaching practice. It shows how students educated in an environment that might not be optimal for producing high-achieving students in mathematics and science in elementary and middle school (according to the TIMSS) might nonetheless manage to excel in these fields at the end of their schooling. In conclusion, the authors address the implications of their study for similar educational systems in other developing countries.