Showing papers on "Number sense published in 2008"

Individual differences in non-verbal number acuity correlate with maths achievement

Abstract: Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

Using Kindergarten Number Sense to Predict Calculation Fluency in Second Grade

Abstract: Children's number sense in kindergarten was used to predict their calculation fluency in second grade (N = 198). Using block entry regression, usual predictors of age, reading, memory, and verbal and spatial cognition were entered in the first block and number sense measures were added in the second block. Number sense measures contributed a significant amount of variance over and above the more general predictors (26%—42%). Uniquely predictive subareas were active memory for numbers, number knowledge, and number combinations, with number combinations standing out as the strongest single predictor. Number sense screening in kindergarten, using “at-risk” versus “not-at-risk” criteria, successfully ruled out 84% of the children who did not go on to have calculation fluency difficulties and positively identified 52% of the children who later showed fluency difficulties. The relation of early number skills to later calculation fluency has important implications for math screening and intervention.

A Number Sense Assessment Tool for Identifying Children at Risk for Mathematical Difficulties

Abstract: Publisher Summary Although number sense has been defined differently and sometimes is used loosely in connection with math, researchers generally agree that number sense in the 3- to 6-year-old period involves interrelated abilities involving numbers and operations, such as subitizing quantities of 3 or less quickly, without counting; counting items in a set to at least five with knowledge that the final count word indicates how many are in the set; discriminating between small quantities; comparing numerical magnitudes and transforming sets with totals of 5 or less by adding or taking away items. Arguably, number and associated operational knowledge is the most important area of mathematical learning in early childhood. Number sense has three components that include counting, number knowledge and number operations. Counting is a crucial tool for learning about numbers and arithmetic operations, and counting weaknesses have consistently been linked to mathematics difficulties. Early difficulties in counting are precursors for later problems with math operations. Weak number sense can result in poorly developed counting procedures, slow fact retrieval, and inaccurate computation, all characteristics of math learning disabilities. It is difficult to memorize arithmetic ‘facts' by rote, without understanding how combinations relate to one another on a mental number line. Accurate and efficient counting procedures can lead to strong connections between a problem and its solution. It has been suggested that basic number sense is a circumscribed cognitive function and relatively independent from general memory, language and spatial knowledge.

A Study of the Performance of 5th Graders in Number Sense and its Relationship to Achievement in Mathematics

Abstract: In order to investigate the performance of number sense and its relationships with mathematics achievement of Taiwanese students who had just completed the 5th-grade mathematics curriculum, a computerized number sense scale has been developed. This number sense scale includes four factors which are recognizing relative number size, using multiple representations of numbers and operations, judging the reasonableness of estimates of computed results and recognizing the relative effect of operations on numbers. A total of 1,212 students in Taiwan participated in this study. The main findings of this study are summarized as follows. First, the students perform best on “recognizing relative number size” and perform worst on “judging the reasonableness of estimates of computed results”. This finding is consistent with previous studies. It shows that students in Taiwan seem quite poor on judging the reasonableness of estimates of computed results. Second, female students, on average, have higher scores on recognizing the relative number size than male students, even though only a small effect size is found. And, third, the achievements of the students in mathematics are significantly correlated with their number sense, as measured by the average grade for the academic year of 5th-grade students.

Validation and Decision Accuracy of Early Numeracy Skill Indicators

Abstract: . The purpose of this study was to develop short-duration assessment measures hypothesized to be valid samples of early mathematical behavior.The Early Numeracy Skill Indicators were designed using a curriculum-based assessment approach. Participants included 64 kindergarten children from a school district in the rural Northeast. Design components featured longitudinal correlation analyses conducted over a 26-week period. Decision analyses were completed using receiver operating characteristic techniques. Results indicated that selected Early Numeracy Skill Indicators tasks produced reliable, valid, and diagnostically accurate scores in to established criterion measures. Implications focus on the use and relation development of the measures as a means to prevent failure and enhance mathematics competency. Further, these tools complement the growing availability of early mathematics curriculum-based measures. ********** Preventing early learning problems through effective formative assessment has been shown to promote student success and enhance competence (Black & Wiliam, 1998; Daly, Hintze, & Hamler, 2000; Deno, 1989; Fuchs & Fuchs, 2001; Shinn, 1995). Screening young children for readiness skills upon school entry is an established educational tradition continually in need of empirical support (Fuchs & Fuchs, 2001; Gredler, 1992). Many schools feature kindergarten screening processes as means to establish relationships, examine entry-level skills, and prepare instructional support mechanisms. Typically, screening entails the brief examination of children for skill proficiency in foundational areas of academic development such as social behavior, speech, language, early literacy, and early numeracy (Howell & Nolet, 1999). The aim of these examinations is to help educators make accurate placement and instructional support decisions before the onset of schooling (Gredler, 1992). Adopting a data-driven, curriculum-based measurement (CBM) assessment approach at kindergarten entry may reduce the need for higher stakes assessment later in a child's educational career and may serve to equalize opportunity (Black & Wiliam, 1998; Fuchs & Fuchs, 1986, 2001). Stanovich (1986) applied the idea of a "Matthew effect" to children learning to read; he noted that children with strong early literacy skills increasingly outperform less competent peers throughout their educational careers (Good, Simmons, & Smith, 1998). Early assessment and intervention are crucial for preventing this divisive trend. Large-scale data from the National Assessment of Educational Progress and the Early Childhood Longitudinal Study suggest similar Matthew effects. The National Assessment of Educational Progress is a nationally representative assessment examining subject matter proficiency. Fourth-grade results in mathematics from 1996 through 2004 indicate that the percentage of children at or below basic levels of proficiency (approximately 40%) has remained stable, whereas the number of children at or above proficient levels has doubled (National Center for Education Statistics, 2004). These data are consistent with the spirit of the Matthew effect in reading, suggesting that students with stronger skills are more likely to profit from education than lower skilled peers (Kavale, Forness, & Siperstein, 1999; Stanovich, 1986). Factors placing young children at risk for mathematics achievement problems are evident in the results of the Early Childhood Longitudinal Study, which followed a cohort of 10,500 kindergarten students from 1998 to their third-grade year in 2002 (National Center for Education Statistics, 2002). On a standardized measure of number sense, operations, and geometry designed to align with popular curricular "strands" (National Council of Teachers of Mathematics, 2000, 2006), students with no reported risk factors significantly outperformed students with more than one risk factor. …

An Investigation of 3rd-Grade Taiwanese Students' Performance in Number Sense

Abstract: The main purpose of this study was to investigate the number sense performance of 3rd‐graders in Taiwan, and to diagnose areas of weakness or deficiency in number sense development. A total of 808 3rd‐graders participated in this study. The results indicated that these students did not perform well on each of the five number sense components (correct rates approx. 34%), and they appeared worst on the performance of “Judging the reasonableness of computational results”. Boys and girls did not show any appreciable difference in their ability to solve number sense problems. The importance of number sense should be highlighted both by teachers and in textbooks and more time and opportunity provided for students to work on this type of exercise at lower grade levels. This would require that “drill and practice” exercises in mathematics should not indeed be over‐ taught, and the teaching of number sense to children should begin as early as possible.

Early spatial thinking and the development of number sense

Abstract: The author investigates the role of very young children's spatial structuring abilities on the development of their sense of number. It describes several classroom activities that facilitate the simultaneous development of spatial ability and number sense in kindergarten students.

Ideas in Practice: Instructional Design and Delivery for Adult Learners

Abstract: More adult learners are participating in formal education than ever before (Miglietti & Strange, 1998; Wedege, 1999), and these adults often are finding their entrance into higher education through developmental courses (King, 1999; Miglietti & Strange). Therefore, attempts to help students gain fluency in basic skills must incorporate learning experiences that are relevant to adults. What type of learning experiences will be relevant to adult students? Such a question is difficult to answer because the notion of "adult learner" refers to a widely heterogeneous group of people, encompassing several generations (Oblinger, 2003; Wagschal, 1997) and multiple characteristics (Kasworm, 2003). In spite of this diversity, the following discussion intends to address the needs of students who typically are considered nontraditional adult students.Meeting the Needs of Adults in Developmental CoursesAdult students find computers to be an increasingly important component of their educational endeavors. For example, some adults view simple computer functions, such as word processing, as essential to their day-to-day job efficiency and daily tasks (Oblinger, 2003); other adults view education as a viable option only because of the flexibility that online classrooms can provide (Knowlton & Thomeczek, 2007). Because of efficiency, flexibility, and other benefits, integrating computers into learning experiences provides strong opportunities for meeting the needs of adult students (Boud & Presser, 2002).Few practical examples of appropriate learning experiences have been disseminated in the academic literature. Therefore, developmental educators struggle to create efficient and relevant learning experiences for nontraditional students, particularly in mathematics (Wedege, 1999). This article aims to begin filling this gap in the literature.Developmental MathematicsDevelopmental mathematics instructors in community colleges and universities regularly report that many students exhibit an aversion to the study of mathematics (Wedege, 1999), yet many empirical studies suggest that success in college mathematics courses results in a greater likelihood for student retention and graduation (Parker, 2005). When asked to identify a "least favorite" topic, basic mathematics and beginning algebra students often name fractions. In fact, these students regularly experience significant fear and a sense of failure when performing basic operations with fractions (Ruedy & Nirenberg, 1990). Not surprisingly, these students have difficulty understanding fractions and applying them to larger mathematical tasks.In spite of this difficulty, fear, and sense of failure, it is imperative that students successfully come to understand and apply fractions. After all, skill in operations with fractions is necessary for increased number sense leading to quantitative literacy (Keijzer & Terwel, 2001; Mathematical Association of America, 1998); dearly, quantitative literacy is a key aspect of education that will help students throughout their college careers (Harrell & Forney, 2003). Students will only gain these skills if they receive efficient and relevant learning experiences; however, the impact of the method of instructional delivery on student success is seldom investigated.Adult students prefer math study that is active and investigational (Miglietti & Strange, 1998; Wedege, 1999). Many adults express a preference for learning experiences with dearly defined objectives, goals, and expectations (Boud & Presser). Adults also value learning experiences that engage them (Boud & Presser), provide plenty of practice with new skills (Boud & Presser; Oblinger, 2003), and offer continuous feedback (Miglietti & Strange).The challenge is dear: Adult students who are enrolled in developmental math courses must experience efficient and relevant learning experiences about fractions. …

Leren vermenigvuldigen met meercijferige getallen

Abstract: The reason for this study lies in the fact that efforts to create a learning trajectory for the domain of multidigit multiplication based on the realistic approach of mathematics education and building on children’s own informal strategies with the aim of transforming these strategies into efficient notation-supported calculation strategies, have not been quite successful. That this is the case, can not only be concluded from an analysis of common learning trajectories in recent Dutch mathematics textbooks, but also from a number of personal experiences of the researcher with the development of one of these textbooks. Moreover recent research in the field of multidigit multiplication in The Netherlands, shows that the success rates being obtained by 11-12 year old Dutch students with solving problems in this field such as the one below, have dropped over the last ten years substantially from about .75 to .60. Research also shows that the number of students that use mental strategies without supporting those strategies with appropriate auxiliary notations, is increasing considerably. With these facts in mind a research was conducted that addresses the following questions: (1) How can a learning trajectory for multidigit multiplication be shaped that builds on students’ own, informal strategies and that helps them to develop theses strategies into efficient, notation-supported calculation strategies in the domain of stylized mental computation? (2) How do the solutions and the notations of the students develop under influence of such a learning trajectory? And to what extent does this learning trajectory provide also weaker students with sufficient support to develop the intended strategies? (3) What are the key elements of a local instruction theory that constitutes the rationale for such a learning trajectory? The research fits in with the tradition of developmental research that has been established in The Netherlands as a type of research in which, on the basis of an overarching domain specific theory (known as Realistic Mathematics Education, RME), an experimental learning trajectory is developed together with a corresponding local instruction theory that is being put to the test in a classroom teaching experiment. The whole process of designing, trying and revising is recorded and analyzed as carefully as possible in order to be able to develop an empirically grounded instruction theory. An important background of the research was constituted by the fact that in recent documents on curriculum development in The Netherlands, there is clear tendency to pay less attention to the traditional standard written algorithms in education, and to focus more on topics like number sense, mental calculation and measurement. In the light of this tendency it is important to dispose of a well outlined learning trajectory that is primarily aiming at stylized mental computation and that can offer also weaker students that do not arrive at the ultimate goals of most abbreviated procedures, sufficient opportunities to arrive at satisfactory achievements. The results of the research primarily consist of a description of the outline of a learning trajectory with a corresponding local instruction theory for multidigit multiplication. Within this theory special attention is paid to the level structure of the learning process. Next to this, a number of more general findings and recommendations is presented that refer to actual aspects of the RME-approach and the corresponding process of progressive mathematization.

Development of a Computerized Number Sense Scale for 3-rd Graders: Reliability and Validity Analysis

Abstract: This study was to develop a computerized number sense scale (CNST) to assess the performance of students who had already completed the 3rd-grade mathematics curriculum. In total, 808 students from representative elementary schools, including cities, country and rural areas of Taiwan, participated in this study. The results of statistical analyses and content analysis indicated that this computerized number sense scale demonstrates good reliability and validity. Cronbach’s α coefficient of the scale was .8526 and its construct reliability was .805. In addition, the 5-factor number sense model was empirically and theoretically supported via confirmatory factor analysis and literature review.

The use of the empty number line in England and the Netherlands

Abstract: This paper contrasts the development of the Empty Number Line (ENL) in the Netherlands with its use in the teaching of mental calculations in England. In doing so the paper investigates the Dutch adoption of Soviet socio-cultural philosophies, in particular Gal’perin’s theory, and relates the notion of internalisation to the analysis of the ENL and its use within the English Primary National Strategy (PNS). By contrasting the English and Dutch approaches a better understanding of the specificity of the development of number sense in relation to mental calculation strategies is attained.

Addressing Issues Related to Technology and Engineering: An Interview with Michael Hacker and David Burghardt, Codirectors of Hoftra University's Center for Technological Literacy

Abstract: [ILLUSTRATION OMITTED] Over the past decade, you have done considerable research pertaining to the overlapping principles, concepts, and activities related to technology and engineering. What are a few of the key characteristics that you have found? We've done a research-based comparison of the professional competencies required by ABET for engineers and by NCATE for technology teachers. The comparison in Table 1 shows a focus on rigorous technical content preparation for both groups--an emphasis on mathematics and science for engineers, and on pedagogy for teachers. There is a high degree of alignment with respect to other competencies, and both professional groups are well prepared in areas of professional practice, design and problem solving, team functioning, ethical and professional responsibility, communication skills, social and cultural impacts, and professional growth. A major area of congruence is the focus on design as the core process that underlies engineering and technological development. ABET defines engineering design as "the process of devising a system, component, or process to meet desired needs. It is a decision-making process (often iterative), in which the basic sciences, mathematics, and the engineering sciences are applied to convert resources optimally to meet these stated needs (ABET, 2008)." A clear difference is how engineers are rigorously prepared in mathematics and science. In the area of knowledge application, engineers are well prepared to solve real-world design problems requiring mathematics, science, and engineering topical knowledge, whereas teachers are well prepared to design instructional environments. When it comes down to it, isn't the level of rigor (such as with mathematics; i.e., calculus) one of the key components that separates the teaching of technology from the teaching of engineering due to the differences in the level of reasoning that takes place? You're exactly right about the mathematics. Technology teachers don't take very much mathematics or science as undergraduates. But there is a real opportunity for our teachers to make a real contribution to core disciplinary knowledge, particularly in mathematics. Because mathematics is often taught in an algorithmic way, students question its value; and it's true that some of the mathematics that is required of students, particularly at the middle level, is not easily related to grade-appropriate contexts in other subjects. Some math, however, that is difficult for students and occurs frequently on standardized assessments can indeed be contextualized within a technology education program. And it doesn't rise to the level of calculus. It's algebra and geometry and number sense; ratio and proportion and scale. It's a matter of our teachers first knowing what math kids are responsible for, and second, knowing how to teach it. I'll give you an example. A math assessment item that kids have real difficulty with is this one: "Solve multistep equations by combining like terms, using the distributive property, or moving variables to one side of the equation." Note: The distributive property is an algebraic property that is used when you multiply terms within parentheses by a term outside the parentheses. As an example, 4(5 + 6) = 20 + 24 = 44 (the 4 is distributed across the terms in the parentheses). This math concept appears frequently on standardized tests. OK, say the kids are designing an emergency shelter for victims of an air crash on a snowy mountain top where a cargo plane was carrying materials to be delivered to a home center distribution facility, and these materials are now strewn around the mountain. Makes for a pretty good design problem if the kids are the four- or five-person crew that survives and they have to build a shelter that will sustain them until a rescue team (that they radioed for help) is able to reach them. …

Research and development in the teaching and learning of number systems and arithmetic

Abstract: The purpose of TSG 10 was to gather participants interested in research and development in the teaching and learning of number systems and arithmetic, including operations in the number systems, ratio and proportion, and rational numbers. The focus was broad and included issues such as the development of number concepts and ‘number sense’ in students, the meaning of the arithmetic operations, the role of contexts and models in the development of numerical and arithmetical knowledge, and the development of teaching/learning units that connect basic arithmetic skills with higher order thinking skills. From an international perspective, we discussed advances in research and practice, new trends, and the state-of-the-art. We put together a Proceedings (ISBN/EAN 978-90-807827-4-7) that could be downloaded from the TSG 10 ICME-11 website.

The impact on computational fluency through instruction in number sense

Abstract: Thesis [M.Ed.] - Wichita State University, College of Education, Dept. of Curriculum and Instruction