The Effectiveness of Teaching Geometry to Enhance Mathematical Understanding in Children with Down Syndrome

ABSTRACT

It is widely known that people with Down syndrome have difficulties transitioning from a basic understanding of counting and cardinality to more advanced arithmetic skills. This is commonly addressed by resorting to the mechanical use of algorithms, which hinders the acquisition of mathematical concepts. For this reason some authors have recently proposed a shift in the focus of learning from arithmetic to more fertile fields, in terms of understanding. In this paper we claim geometry fits this profile, especially suited for initiating children with Down syndrome into mathematics. To support this we resort to historical, epistemological, and cognitive reasons: the work of Séguin and his intuition on the central role of geometry in the development of abstract thinking in the so-called idiot children, the ideas of René Thom about the role of continuum intuition in the emergence of conscious thinking, and finally the two strengths people with Down syndrome display: visual learning abilities and interest in abstract symbols. To support these ideas we present the main findings of qualitative research on elementary mathematics teaching to a group of seven children (3–8) with Down syndrome in Spain. The didactic method used, naturally enhance their naïve geometrical conceptions.     

Subtypes of mathematical learning disability and their antecedents: A cognitive diagnostic approach

Using cognitive diagnostic modeling (CDM), this study identified subtypes of mathematics learning disability (MLD) based on children's numerical skills and examined the language and spatial precursors of these subtypes. Participants were 99 MLD children and 420 low achievers identified from 1839 Finnish children (966 boys) who were followed from preschool (age 6) to fourth grade (2007–2011). Five subtypes were identified: the arithmetic fluency deficit only subtype, the counting deficit subtype, the pervasive deficit subtype, the symbolic deficit subtype, and the counting and concept deficit subtype. Different subtypes depended on different constellations of language and spatial deficits. Findings highlight the effectiveness of CDM in identifying MLD subtypes and underscore the importance of understanding the specific deficits and antecedents of the subtypes.     

Representation and Process in Learning Environments for Mathematics: A Commentary on Three Systems

Three computer‐based systems for teaching arithmetic and algebra are discussed. The systems embody several pedagogical tactics: To provide students with augmented repre‐sentations that reveal the structure of problem solutions; to make mathematical symbols meaningful by giving students concrete referents for those symbols; to provide students with enriched feedback about the consequences of their mathematical operations; and to adapt instruction to the cognitive processes of the students. Weaknesses in the three systems are discussed and two general problems, called complexity trade‐off and scaffold removal, are put forward as possible explanations for why the systems are not more effective. The alternative view that mathematics is difficult because it requires abstract thinking is outlined and some of its implications discussed briefly.     

Early Numerical Competencies in 5- and 6-Year-Old Children With Autism Spectrum Disorder

Research Findings: To date, studies comparing the mathematical abilities of children with autism spectrum disorder (ASD) and typically developing children are scarce, and results remain inconclusive. In general, studies on this topic focus on mathematical abilities learned from elementary school onward, with little attention for possible precursors at younger ages. The current exploratory study focused on the important developmental period of preschool age, investigating 5 early numerical competencies in 30 high-functioning children with ASD and 30 age-matched control children: verbal subitizing, counting, magnitude comparison, estimation, and arithmetic operations. Children were examined at 5 or 6 years of age, attending the 3rd and final year of preschool. Overall, rather similar early number processing was found in children with and without ASD, although marginally significant results indicated a weaker performance of children with ASD on verbal subitizing and conceptual counting. Practice or Policy: Given the pervasiveness and impact of ASD on other domains of functioning, it is important to know that no general deficits in early numerical competencies were found in this study. However, some downward trends in mathematics performance were identified in children with ASD, which can serve as the basis for additional research in this field.     

From Parental Involvement to Children's Mathematical Performance: The Role of Mathematics Anxiety

This study examined whether children's mathematics anxiety serves as an underlying pathway between parental involvement and children's mathematics achievement. Participants included 78 low-income, ethnic minority parents and their children residing in a large urban center in the northeastern United States. Parents completed a short survey tapping several domains of parental involvement, and children were assessed on mathematics anxiety, whole number arithmetic, word problems, and algebraic reasoning. Research Findings: The results indicated that parents influence children's mathematics achievement by reducing mathematics anxiety, particularly for more difficult kinds of mathematics. Specifically, the mediation analyses demonstrated that parental home support and expectations influenced children's performance on word problems and algebraic reasoning by reducing children's mathematics anxiety. Mathematics anxiety did not mediate the relationship between home support and expectations and whole number arithmetic. Practice or Policy: Policies and programs targeting parental involvement in mathematics should focus on home-based practices that do not require technical mathematical skills. Parents should receive training, resources, and support on culturally appropriate ways to create home learning environments that foster high expectations for children's success in mathematics.     

Cognitive and Attitudinal Predictors Related to Line Graphing Achievement among Elementary Pre-service Teachers

The purpose of this study was to determine the extent to which six cognitive and attitudinal variables predicted pre-service elementary teachers’ (N = 87) performance on line graphing. Predictors included reading comprehension and mathematics scores, logical thinking performance scores, as well as measures of attitudes toward science, mathematics and graphing. Results indicated that mathematical and logical thinking ability were the most significant predictors of line graph performance among the other variables, accounting for 41% of the total variability. Findings from this study suggest that elementary science education programs augment instruction to include aspects of mathematics and logical thinking.     

Development of Arithmetical Thinking: Evaluation of Subject Matter Knowledge of Pre-Service Teachers in Order to Design the Appropriate Course

One of the key courses in the mathematics teacher education program in Israel is arithmetic, which engages in contents which these pre-service mathematics teachers (PMTs) will later teach at school. Teaching arithmetic involves knowledge about the essence of the concept of “number” and the development thereof, calculation methods and strategies. properties of operations on different sets of numbers, as well as the properties of the numbers themselves. Hence, the question arises: how to educate PMTs in order to supplement their mathematical knowledge with the required components? The present study explored the development of arithmetic thinking among pre-service teachers intending to teach mathematics at elementary school. This was done by matching the van Hiele theory of the development of geometric thinking to arithmetic. Analysis of findings obtained both in the present study and in many studies of geometry teaching indicates that this approach to considering the learners’ level of thinking development might lead to meaningful learning in arithmetic course for PMTs.     

Strategies to Encourage Mathematics Learning in Early Childhood: Discussions and Brainstorming Promote Stronger Performance

This longitudinal study examined the influence of prekindergarten teacher characteristics and classroom instructional processes during mathematical activities on the growth of mathematics learning scores in prekindergarten, kindergarten, and first grade. Participants attended state-funded and Head Start prekindergarten programs. Mathematical performance was measured in fall and spring in prekindergarten and spring in kindergarten and first grade using the Test of Early Mathematics Ability–3 (TEMA-3; Ginsburg & Baroody, 2003). Two dimensions of the Classroom Assessment Scoring System (CLASS; i.e., instructional learning formats and concept development; Pianta, La Paro, & Hamre, 2008) were scored based on observed classroom mathematics activities. Teachers provided information about their education and years of prekindergarten teaching experience. Research Findings: Instructional processes that included elements of the CLASS concept development dimension, such as discussions and brainstorming to encourage children’s understanding, were related to growth of mathematics scores. Neither teacher characteristics nor instructional processes of the CLASS instructional learning formats dimension, such as using different modalities and materials, and learning objectives, were related to growth of mathematics scores. Practice or Policy: The findings extend our understandings of how instructional processes impact children’s early mathematical performance. These findings may be helpful in increasing our understanding of the types of instructional processes that might be emphasized in teacher professional development specifically related mathematical activities. Professional development that focuses on the CLASS concept development dimension may be easier for teachers to remember and implement in their classrooms and, consequently, have a greater impact on mathematics learning.     

Promoting equity in mathematics teacher preparation: a framework for advancing teacher learning of children’s multiple mathematics knowledge bases

Research repeatedly documents that teachers are underprepared to teach mathematics effectively in diverse classrooms. A critical aspect of learning to be an effective mathematics teacher for diverse learners is developing knowledge, dispositions, and practices that support building on children’s mathematical thinking, as well as their cultural, linguistic, and community-based knowledge. This article presents a conjectured learning trajectory for prospective teachers’ (PSTs’) development related to integrating children’s multiple mathematical knowledge bases (i.e., the understandings and experiences that have the potential to shape and support children’s mathematics learning—including children’s mathematical thinking, and children’s cultural, home, and community-based knowledge), in mathematics instruction. Data were collected from 200 PSTs enrolled in mathematics methods courses at six United States universities. Data sources included beginning and end-of-semester surveys, interviews, and PSTs’ written work. Our conjectured learning trajectory can serve as a tool for mathematics teacher educators and researchers as they focus on PSTs’ development of equitable mathematics instruction.     

浅谈数学概念教学

数学是由概念与命题等内容组成的知识体系,是一门以抽象思维为主的学科,因此概念是数学基础知识和基本技能教学的核心。学生数学素养差,关键表现在对数学概念的理解、应用和转化等方面的不足。文章指出了现行中学数学概念教学中可能出现的弊端,并根据学生的认知特点及数学本身的特征介绍了如何进行数学概念教学的几种方法。     

Mathematics Teacher Educators' Perceptions and Use of Cognitive Research

Instructors (N = 204) of elementary mathematics methods courses completed a survey assessing the extent to which they value cognitive research and incorporate it into their courses. Instructors' responses indicated that they view cognitive research to be fairly important for mathematics education, particularly studies of domain‐specific topics, and that they emphasize topics prominent in psychology studies of mathematical thinking in their courses. However, instructors reported seldom accessing this research through primary or secondary sources. A mediation analysis indicated that mathematics methods instructors' perception of the importance of the research predicts their incorporation of it in their courses, and that this relation is partially mediated by their accessing of it. Implications for psychologists who have an interest in education and recommendations for facilitating the use of cognitive research in teacher preparation are discussed.     

Cognitive play and mathematical learning in computer microworlds

Based on the constructivist principle of active learning, we focus on children's transformation of their cognitive play activity into what we regard as independent mathematical activity. We analyze how, in the process of this transformation, children modify their cognitive play activities. For such a modification to occur, we argue that the cognitive play activity has to involve operations of intelligence which yield situations of mathematical schemes.We present two distinctly different cases. If the first case, the medium of the cognitive play activity was a discrete computer microworld. we illustrate how two children transformed the playful activity of making pluralities into situations of their counting schemes. In the second case, the medium was a continuous microworld. We illustrate two children's transformation of the playful activity of making line segments (sticks) into situations of their counting schemes. We explain one child's transformation as a generalizing assimilation because it was immediate and powerful. The transformation made by the other child was more protracted, and social interaction played a prominent role. We specify several types of accommodations induced by this social interaction, accommodations we see as critical for understanding active mathematics learning. Finally, we illustrate how a playful orientation of independent mathematical activity can be inherited from cognitive play.     

Becoming numerate with information and communications technologies in the twenty‐first century

This article draws on data from a three‐year Australian Research Council‐funded study that examined the ways in which young children become numerate in the twenty‐first century. We were interested in the authentic problem‐solving contexts that we believe are required to create meaningful learning. This being so, our basic tenet was that such experiences should involve the use of information and communications technologies (ICT) where relevant, but not in tokenistic ways. This article highlights learning conditions in which young children can become numerate in contemporary times. We consider ‘academic’ or ‘school‐based’ mathematical tasks in the context of a Mathematical Tasks Continuum. This continuum was conceptualised to enable focused and detailed thinking about the scope and range of mathematical tasks that young children are able to engage within contemporary school contexts. The data from this study show that most of the tasks the children experienced in early years mathematics classes were unidimensional in their make up. That is, they focus on the acquisition of specific skills and then they are practiced in disembedded contexts. We suggest that the framework created in the form of the Mathematical Tasks Continuum can facilitate teachers’ thinking about the possible ways in which they could extend children’s academic work in primary school mathematics, so that the process of becoming numerate becomes more easily related to authentic activities that they are likely to experience in everyday life.     

后现代教育理念视野下的数学教师教学行为

后现代教育理念对数学教育目的观、课程观、教学观的反思和批判产生一定的影响。关于数学教学观的反思主要体现于数学教学行为的改变：构建数学认知结构,强调学生个体的＂意识的建构＂;呈现静态知识方式,创设学生个体的＂认知的情境＂;重构思维发展模式,塑造学生个体的＂茎块式思维＂;培养问题质疑能力,强化学生个体对＂问题的反思＂;突破师生交往方式,注重师生之间的＂对话的互动＂。     

Ability stereotyping in mathematics

Ability is a concept central to the current practice of mathematics teaching. However, the widespread view of mathematics learning as an ordered progression through a hierarchy of knowledge and skill, mediated by the stable cognitive capability of the individual pupil, can be sustained only as a gross global model, and is of limited value in describing and understanding the particular cognitive capabilities of individual pupils in order to plan, promote and evaluate their learning. In effect, individual pupils, and groups of pupils, are subject to ability stereotyping; characterisation in terms of a summary global judgement of cognitive capability, associated with overgeneralised and stereotyped expectations of mathematical behaviour, and stereotyped perceptions of an appropriate mathematics curriculum.     

USING COMBINATORIAL APPROACH TO IMPROVE STUDENTS’ LEARNING OF THE DISTRIBUTIVE LAW AND MULTIPLICATIVE IDENTITIES

This article reports an alternative approach, called the combinatorial model, to learning multiplicative identities, and investigates the effects of implementing results for this alternative approach. Based on realistic mathematics education theory, the new instructional materials or modules of the new approach were developed by the authors. From the combinatorial activities based on the things around daily life, the teaching modules assisted students to establish their concept of the distributive law, and to generalize it via the process of progressive mathematizing. The subjects were two classes of 8th graders. The experimental group (n = 32) received a combinatorial approach to teaching by the first author using a problem-centered with double-cycles instructional model, while the control group (n = 30) received a geometric approach to teaching, from the textbook by another teacher who uses lecturing. Data analyses were both qualitative and quantitative. The findings indicated that the experimental group had a better performance than the control group in cognition, such as for the inner-school achievement test, mid-term examination, symbol manipulation, and unfamiliar problem-solving: also in affection, such as the tendency to engage in the mathematics activities and enjoy mathematical thinking.     

The Remarkable Number “1”

In human history, the origin of the numbers came from definite practical needs. Indeed, there is strong evidence that numbers were created before writing. The number “1”, dating back at least 20,000 years, was found as a counting symbol on a bone. The famous statement by the German mathematician Leopold Kronecker (1823–1891), “God made the integers; all else is the work of man,” has spawned a lively modern philosophical discussion, and this discussion begins by trying to get a philosophical handle on “1.” This approach remains under heavy discussion, and is more-or-less unresolved (Frege in Die Grundlagen der Arithmetik (English: The foundations of arithmetic). Polhman, 1884). In this note, we consider the many facets of “one” in it many guises and applications. Nonetheless, “one” has multiple meanings, from the very practical to the abstract, from mathematics to science to basically everything. We examine here a mere slice of mathematical history with a focus on the most basic and applicable concept therein. It troubles many, particularly students, even today.     

Epistemic quality for equitable access to quality education in school mathematics

This paper reports on a study that aims to address the challenges of UN Sustainable Development Goal 4 to ensure inclusive and equitable quality education for all. The study focuses on school mathematics in particular. With regard to ensuring equitable access to quality education, it is argued that there is a need to consider the epistemic quality of what students come to know, make sense of and be able to do in school mathematics. Accordingly, the aim is to maximize the chances that all pupils will have epistemic access to school mathematics of high epistemic quality. The study is based on the theoretical framework of Joint Action Theory in Didactics (JATD). Associated research questions focus on the quality of teacher-student(s) joint action and on the epistemic quality of the content. The paper draws on empirical research findings of the Developing Mathematical Thinking in the Primary Classroom (DMTPC) project (2010–12) and also on the findings of a parallel study of mathematics teachers’ assessment practices in Ghana. One teacher’s action research project is used as an exemplar to illustrate how mathematics can become more accessible and inclusive thus leading to an evolution in mathematical thinking and high-quality epistemic access for all.     

体验数学的重要性:幼儿数学素养发展的起点

数学素养是现代社会公民应具备的基本素养,对于个体的终身发展具有重要的意义,应从幼儿园教育阶段就开始培养幼儿的数学素养。让幼儿体验到数学的重要性是促进幼儿数学素养发展的起点。由于幼儿尚不具备抽象思维能力,所以他们通常并不能自发体验到数学的重要性,而需要幼儿园教师的适当引导。教师应遵循兴趣激发原则、基本概念原则、自然引导原则,在幼儿园一日生活、区域游戏、集体教学中让幼儿充分感知和体验数学的重要性。只有引导幼儿体验到数学很重要,才能激发幼儿学习数学的信心与热情,提高幼儿对数学活动的兴趣与参与度,帮助幼儿开启真正愿意学习数学的过程,为幼儿数学素养的终生发展打下良好的基础。     

赵爽的数学哲学思想

赵爽是我国三国时期的数学家.他在数学上的贡献主要有三个方面:一是给出了勾股定理简洁、漂亮的证明;二是奠定了重差术的理论基础;三是给出了一元二次方程的一个求根公式,以及根与系数的关系.赵爽的数学哲学思想主要有五个方面:数学起源于人类的实践活动;数形统一的思想;归纳和演泽并重的思想;变中有不变的思想;实用思想.