Showing papers on "Number sense published in 2012"
TL;DR: It is found that population trends suggest that the precision of one’s number sense improves throughout the school-age years, peaking quite late at ∼30 y, and the large individual differences and prolonged development of number sense hold promise for the impact of educational interventions that target the number sense.
Abstract: It has been difficult to determine how cognitive systems change over the grand time scale of an entire life, as few cognitive systems are well enough understood; observable in infants, adolescents, and adults; and simple enough to measure to empower comparisons across vastly different ages. Here we address this challenge with data from more than 10,000 participants ranging from 11 to 85 years of age and investigate the precision of basic numerical intuitions and their relation to students’ performance in school mathematics across the lifespan. We all share a foundational number sense that has been observed in adults, infants, and nonhuman animals, and that, in humans, is generated by neurons in the intraparietal sulcus. Individual differences in the precision of this evolutionarily ancient number sense may impact school mathematics performance in children; however, we know little of its role beyond childhood. Here we find that population trends suggest that the precision of one’s number sense improves throughout the school-age years, peaking quite late at ∼30 y. Despite this gradual developmental improvement, we find very large individual differences in number sense precision among people of the same age, and these differences relate to school mathematical performance throughout adolescence and the adult years. The large individual differences and prolonged development of number sense, paired with its consistent and specific link to mathematics ability across the age span, hold promise for the impact of educational interventions that target the number sense.
394 citations
TL;DR: Regression analyses showed that non-symbolic skills in kindergarten were predictively related to arithmetical achievement 1 year later and fact retrieval 2 years later and AN comparison was predictivelyrelated to procedural calculation2 years later.
Abstract: Background. The ability to compare numbers, as the most basic form of number sense, has been related to arithmetical achievement.
Aims. The current study addressed the predictive value of non-symbolic and symbolic (number word (NW) and Arabic number (AN)) comparison for arithmetics by means of a longitudinal design.
Sample. Sixteen children with mathematical disabilities (MD), 64 low achievers (LA), and 315 typical achieving (TA) children were followed from kindergarten till grade 2.
Method. The association of comparison skills with arithmetical skills in grades l and 2 was studied. The performances of MD, LA and TA children were compared.
Results. Regression analyses showed that non-symbolic skills in kindergarten were predictively related to arithmetical achievement 1 year later and fact retrieval 2 years later. AN comparison was predictively related to procedural calculation 2 years later. In grade 2, there was an association between both symbolic tasks and arithmetical achievement. Children with MD already had deficits in non-symbolic and symbolic AN comparison in kindergarten, whereas in grade 2 the deficits in processing symbolic information remained.
Conclusions. The combination of non-symbolic and symbolic deficits represents a risk of developing MD.
240 citations
TL;DR: The results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education, but performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-Symbolic quantities was significantly better the higher mathematics achievement.
Abstract: In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.
129 citations
TL;DR: Accounting for initial skill level in mathematical knowledge, children who received the number sense intervention performed better than controls at immediate post test, with meaningful effects on measures of number competencies and general math achievement.
Abstract: Math achievement in elementary school is mediated by performance and growth in number sense during kindergarten. The aim of the present study was to test the effectiveness of a targeted small-group number sense intervention for high-risk kindergartners from low-income communities. Children were randomly assigned to 1 of 3 groups (n = 44 in each group): a number sense intervention group, a language intervention group, or a business-as-usual control group. Accounting for initial skill level in mathematical knowledge, children who received the number sense intervention performed better than controls at immediate posttest, with meaningful effects on measures of number competencies and general math achievement. Many of the effects held 8 weeks after the intervention was completed, suggesting that children internalized what they had learned. There were no differences between the language and control groups on any math-related measures.
123 citations
Book•
15 Aug 2012
TL;DR: This teaching guide discusses the need for Purposeful, Inquiry-Based and Meaningful Instruction Processes of Mathematical Inquiry, Organizing Instruction to Enhance Mathematical Power, Professional Development and Epilogue Appendices.
Abstract: Prologue - Understanding This Teaching Guide and the Role of Affect in the Teaching-Learning Process Fostering Mathematical Power - the Need for Purposeful, Inquiry-Based and Meaningful Instruction Processes of Mathematical Inquiry - Problem Solving, Reasoning and Communicating Fostering and Evaluating Meaningful Learning - Making Connections and Assessing Understanding Basic Mathematical Tools - Numbers and Numerals Introducing Arithmetic - Understanding the Whole-Number Operations and Mastering the Basic Number Combinations Understanding Base-Ten, Place-Value Skills - Reading, Writing and Arithmetic with Multidigit Numbers Thinking with Whole Numbers - Number Sense, Estimation and Mental Computation Exploring Numbers Further - Number Theory and Integers and Operations on Integers Working with "Parts of a Whole" and Other Meanings of Rational Numbers and Common Fractions Understanding Operations on Common Fractions Place-Value Representations of Fractional Parts - Decimal Fractions, Decimals and Operations on Decimals Comparing Quantities Fairly - Ratios, Proportions and Percent Making Sense of Information and Using it to Make Everyday Decisions - Statistics and Probability The Mathematics of Our Environment - Geometry and Spatial Sense Sizing Up Things - Measurement and Measurement Formulas The Transition from Arithmetic to Algebra - Prealgebra and Functions Reflections on Teaching - Organizing Instruction to Enhance Mathematical Power, Professional Development and Epilogue Appendices.
122 citations
TL;DR: This article examined the association between the performance of children on several basic number processing tasks and their individual math achievement scores on a curriculum-based test measured 1 year later, and found that most of the variance in children's math achievement was predicted by nonsymbolic number line estimation performance and speed of comparing symbolic numbers.
Abstract: Numerical processing has been extensively studied by examining the performance on basic number processing tasks, such as number priming, number comparison, and number line estimation. These tasks assess the innate “number sense,” which is assumed to be the breeding ground for later mathematics development. Indeed, several studies have associated children's performance in these tasks with individual differences in mathematical achievement. To date, however, most of these studies have cross-sectional designs. Moreover, the few longitudinal studies either use complex tasks (e.g., story problems) or investigate only one of these basic number processing tasks at a time. In this study, we examine the association between the performance of children on several basic number processing tasks and their individual math achievement scores on a curriculum-based test measured 1 year later. Regression analyses showed that most of the variance in children's math achievement was predicted by nonsymbolic number line estimation performance (i.e., estimating large quantities of dots) and, to a lesser extent, the speed of comparing symbolic numbers. This knowledge about the predictive value of the performance of 5- to 7-year-olds on these markers of number processing can help with the early identification of at-risk children. In addition, this information can guide appropriate educational interventions.
88 citations
01 Jan 2012
TL;DR: A review of the development of time and number sense across infancy and childhood, and neuropsychological findings as they relate to time and counting discrimination in infants and adults is presented in this article, couched within a mode-control model of timing and counting which assumes time and numbers share a common magnitude representation system.
Abstract: Estimations of time and number share many similarities in both non-humans and man. The primary focus of this review is on the development of time and number sense across infancy and childhood, and neuropsychological findings as they relate to time and number discrimination in infants and adults. Discussion of these findings is couched within a mode-control model of timing and counting which assumes time and number share a common magnitude representation system. A basic sense of time and number likely serves as the foundation for advanced numerical and temporal competence, and aspects of higher cognition—this will be discussed as it relates to typical childhood, and certain developmental disorders, including autism spectrum disorder. Directions for future research in the developmental neuroscience of time and number (NEUTIN) will also be highlighted.
67 citations
TL;DR: A basic sense of time and number likely serves as the foundation for advanced numerical and temporal competence, and aspects of higher cognition—this will be discussed as it relates to typical childhood, and certain developmental disorders, including autism spectrum disorder.
Abstract: Estimations of time and number share many similarities in both non-humans and man. The primary focus of this review is on the development of time and number sense across infancy and childhood, and neuropsychological findings as they relate to time and number discrimination in infants and adults. Discussion of these findings is couched within a mode-control model of timing and counting which assumes time and number share a common magnitude representation system. A basic sense of time and number likely serves as the foundation for advanced numerical and temporal competence, and aspects of higher cognition—this will be discussed as it relates to typical childhood, and certain developmental disorders, including autism spectrum disorder. Directions for future research in the developmental neuroscience of time and number (NEUTIN) will also be highlighted.
64 citations
12 Jun 2012
TL;DR: It is described how Fingu theoretically can support children's development of fundamental arithmetic skills, focusing on conceptual subitizing, the embodiment of numerosity, and finger gnosis, and the results of an exploratory micro-longitudinal study of the game are presented.
Abstract: We describe the design background of the mathematics game Fingu for iPad aimed at 4 to 8 year old children. We first describe how Fingu theoretically can support children's development of fundamental arithmetic skills, focusing on conceptual subitizing, the embodiment of numerosity, and finger gnosis. Then we present the results of an exploratory micro-longitudinal study of the game with 11 5- and 6-year old children playing the game for several weeks and being filmed at three occasions. We discuss how their behavior with the game develops over time and can be related to the development of arithmetic skills. Finally we discuss how we will proceed testing the effectiveness of Fingu in a larger controlled study.
36 citations
TL;DR: In this paper, the developmental and instructional implications of using five-frames and related instructional manipulatives in early childhood mathematics classrooms are discussed. And a set of teaching tips designed to help teachers maximize the potential advantages of integrating five-frame in the pre-K classroom are presented.
Abstract: Teachers in early childhood and elementary classrooms (grades K-5) have been using ten-frames as an instructional tool to support students’ mathematics skill development for many years. Use of the similar five-frame has been limited, however, despite its apparent potential as an instructional scaffold in the early elementary grades. Due to scant evidence of teacher use and a lack of systematic research we know little to nothing about both the developmental and pedagogical implications of using five frames and related instructional manipulatives in early childhood mathematics classrooms. In this paper, we provide an overview of five-frames and specifically demonstrate ways that five-frames, if used in conjunction with concrete manipulatives, can support pre-kindergarten (pre-K) children’s development of Gelman and Gallistel’s (1978) three basic counting principles: the stable-order principle, one-to-one correspondence, and cardinality. We conclude by discussing the developmental and instructional implications of using five-frames, as well as offer a set of teaching tips designed to help teachers maximize the potential advantages of integrating five-frames in the pre-K classroom.
32 citations
TL;DR: In this article, the authors demonstrated that it is possible to teach children aged 5-6 to use decomposition strategy and thus reduced their reliance on counting to solve addition problems, and they further showed that children adopt efficient strategies was related to their systematic knowledge of the part-part-whole relationship of the numbers 1-10.
Journal Article•
Journal Article•
TL;DR: This article examined a framework for developing number sense and explained how it may be expanded to include measurement benchmarks, and showed how a class of Year 5/6 students grappled with some activities designed to promote number sense, and concluded by providing an estimation game that may be applied to any classroom setting.
Abstract: Tracey Muir examines a framework for developing number sense and explains how it may be expanded to include measurement benchmarks. She explains how a class of Year 5/6 students grappled with some activities designed to promote number sense and finishes by providing an estimation game that may be applied to any classroom setting.
TL;DR: In this article, the authors explored strategies used by high-achieving 6th grade students in the United Arab Emirates (UAE) to solve basic arithmetic problems involving number sense, and found that a low percentage of solutions involved aspects of number sense such as appropriate use of benchmarks, using numbers flexibly when mentally computing, estimating, and judging reasonableness of results; understanding relative effect of operations; and decomposing or recomposing numbers to solve problems.
Abstract: The purpose of this study was to explore strategies used by high-achieving 6th grade students in the United Arab Emirates (UAE) to solve basic arithmetic problems involving number sense. The sample for the study consisted of 15 high-achieving boys and 15 high-achieving girls in grade 6 from 2 schools in the Emirate of Abu Dhabi, UAE. Data for the study were collected through individual interviews in which students were presented with 10 basic problems. The results showed that a low percentage of solutions involved aspects of number sense such as appropriate use of benchmarks; using numbers flexibly when mentally computing, estimating, and judging reasonableness of results; understanding relative effect of operations; and decomposing or recomposing numbers to solve problems. It was also found that students were highly dependent on school-taught rules. In many cases, these rules were confused and misused.
TL;DR: In this paper, the authors investigated the number sense of 5th grade Turkish students on the topic of decimal numbers and found that there was no significant difference in number sense between the genders.
01 Jan 2012
TL;DR: The results of these studies show that working memory can be trained and moreover, that counting skills can be improved by training working memory, but for a direct effect, training number sense seems to be the most effective.
Abstract: Previous research has shown that number sense skills can be improved by playing numerical games and that working memory is related to number sense as well. In this chapter, it is investigated if working memory could be trained and what the additional effect could be of a working memory training on children’s number sense skills. Two studies are reported, in which low-performing kindergarten children received training on either number sense or on both number sense and working memory. The results of these studies show that working memory can be trained and moreover, that counting skills can be improved by training working memory. However, for a direct effect, training number sense seems to be the most effective. More research is needed to further examine these results. This chapter provides directions for early interventions for children at risk for mathematical learning problems.
TL;DR: This article argued that the terrain of early numerical understanding is little more than simple counting devoid of cognitive complexity, and that the knowledge upon which early QL develops comprises interdependent dimensions: Number Knowledge, Counting Skills and Principles, Nonverbal Calculation, Number Combinations and Story Problems, summarised as Number Sense.
Abstract: The fundamental meaning of Quantitative Literacy (QL) as the application of quantitative knowledge or reasoning in new/unfamiliar contexts is problematic because how we acquire knowledge, and transfer it to new situations, is not straightforward. This article argues that in the early development of QL, there is a specific corpus of numerical knowledge which learners need to integrate into their thinking, and to which teachers should attend. The paper is a rebuttal to historically prevalent (and simplistic) views that the terrain of early numerical understanding is little more than simple counting devoid of cognitive complexity. Rather, the knowledge upon which early QL develops comprises interdependent dimensions: Number Knowledge, Counting Skills and Principles, Nonverbal Calculation, Number Combinations and Story Problems - summarised as Number Sense. In order to derive the findings for this manuscript, a realist synthesis of recent Education and Psychology literature was conducted. The findings are of use not only when teaching very young children, but also when teaching learners who are experiencing learning difficulties through the absence of prerequisite numerical knowledge. As well distilling fundamental quantitative knowledge for teachers to integrate into practice, the review emphasises that improved pedagogy is less a function of literal applications of reported interventions, on the grounds of perceived efficacy elsewhere, but based in refinements of teachers' understandings. Because teachers need to adapt instructional sequences to the actual thinking and learning of learners in their charge, they need knowledge that allows them to develop their own theoretical understanding rather than didactic exhortations.
01 Jan 2012
TL;DR: The author reveals new approaches to teaching early number skills and to Remediate Number Fact Dyscalculia, as well as investigating the relationships between Motor Learning, the Visual System and Dyslexia, which have implications for interventions for developmental disabilities.
Abstract: Reading . Chapter 1. Introduction to the Reading and Writing Sections Victoria J. Molfese and Zvia Breznitz . Chapter 2. Evidence of Dynamic Changes in Brain Processing from Imaging Techniques: Implications for Interventions for Developmental Disabilities Dennis L. Molfese, Victoria J. Molfese, Krista Garrod and David L. Molfese . Chapter 3. Magnetic Source Imaging: A Suitable Tool of Exploring the Neurophysiology of Typical and Impaired Reading Ability Roozeh Rezaie, Panagiotis G. Simos, Jack M. Fletcher and Andrew Papanicolaou . Chapter 4. ERP Studies of Reading Disabilities Peter J. Molfese . Chapter 5. A Model of Brain Activity of Young as Compared to Adult Dyslexic Readers, and their Outcomes from Intervention Shelley Shaul . Chapter 6. Optimizing Reading Enhancement: Evidence from Brain Research Olga Chuntonov and Zvia Breznitz . Chapter 7. The Error Detection Mechanism among Dyslexic and Skilled Readers: Characterization and Plasticity Tzipi Horowitz-Kraus . Chapter 8. Reading in more than one Language: Behavior and Brain Perspectives Anat Prior .- Writing and Motor Skills . Chapter 9. Spelling Disability - Neurophysiologic Correlates and Intervention Gerd Schulte-Korne . Chapter 10. The Relationships between Motor Learning, the Visual System and Dyslexia Itamar Sela .- Mathematics . Chapter 11. Numerical Cognition: From Development to Intervention (Introduction) Orly Rubinsten . Chapter 12. The Beginning of the Road: Learning Mathematics for the First Time Tamar Ben-Shalom, Andrea Berger and Avishai Henik . Chapter 13. Ordinal Processing of Numerical and Non-numerical Information Dana Sury and Orly Rubinsten . Chapter 14. Diagnostics and Intervention in Developmental Dyscalculia: Current Issues and Novel Perspectives Korbinian Moeller, Martin Fischer, Ursula Cress and Hans-Christoph Nuerk . Chapter 15. New Approaches to Teaching Early Number Skills and to Remediate Number Fact Dyscalculia Liane Kaufmann and Silvia Pixner . Chapter 16. Number Sense in Low-performing Kindergarten Children: effects of a Working Memory and an Early Math Training Evelyn H. Kroesbergen, Jaccoline E. Van 't Noordende and Meijke E. Kolkman
01 Jan 2012
TL;DR: Geronime et al. as mentioned in this paper found that mathematics self-concept mediates the relationship between number sense and mathematics achievement at the middle school level, when controlling for gender, race, socioeconomic status, and special education services.
Abstract: NUMBER SENSE MEDIATED BY MATHEMATICS SELF-CONCEPT IN IMPACTING MIDDLE SCHOOL MATHEMATICS ACHIEVEMENT Lara K. Geronime, B.A., M.A. Marquette University, 2012 The purpose of the current study was to extend the research on number sense to the middle school level and to simultaneously consider socioemotional elements related to the construct at this developmental stage. Its genesis was initially rooted in an ongoing and dramatic emphasis by U.S. policymakers, researchers, and educators on improving mathematics achievement in order to compete globally in technology and innovation. Despite debates about optimal curriculum and instruction, tremendous support exists for the construct of number sense. However, middle school research examining the phenomena has been limited to intervention protocols targeting specific skillsets and better measurement of its domains. Concomitantly, educational research has produced ample evidence of the decline in student mathematics motivation over time, and the corresponding literature establishes a link between mathematics self-concept and mathematics achievement, particularly during adolescence. The Early Childhood Longitudinal Study, Kindergarten Class of 1998-99 provides a sample of 4,425 U.S. eighth graders for the present study, assessed directly and indirectly in cognitive, demographic, and affective domains. Multiple regression analyses confirmed the hypotheses that number sense predicts both mathematics selfconcept and mathematics achievement at the middle school level, when controlling for gender, race, socioeconomic status, and special education services. Additionally, a path analysis with Statistical Analysis Software (SAS) and the Sobel test revealed that mathematics self-concept mediates the relationship between number sense and mathematics achievement. This indirect effect, when combined with the direct effect of number sense, results in a significant, medium total effect value of .35 for the model. By incorporating this knowledge regarding the interconnection of these three constructs into mathematics curriculum and instruction, as well as teacher education, the United States can move closer to bringing about equity of opportunity and motivating students to pursue more complex mathematics coursework and subsequently professions.
01 Jan 2012
TL;DR: In this paper, prospective elementary teachers participate in a mathematics content course designed to support their development of number sense, and they move from being reliant on the mental analogues of the standard algorithms to reasoning more flexibly in mental computation.
Abstract: In order to support children's learning of elementary mathematics meaningfully, elementary teachers need to understand that mathematics deeply and flexibly (Ball, 1990; Ma, 1999). In other words, they need good number sense (Reys & Yang, 1998). However, researchers have found that prospective elementary teachers tend to reason inflexibly, relying heavily on standard algorithms (e.g., Ma, 1999; Newton, 2008; Yang, 2007). Previous research has provided single snapshots or comparisons of pre/post snapshots of number sense. In this study, I analyzed prospective elementary teachers' number sense development. In earlier work, Nickerson and I created a local instruction theory (Gravemeijer, 1999) for the development of number sense (Nickerson & Whitacre, 2010). In a previous classroom teaching experiment, we found that prospective elementary teachers enrolled in a mathematics content course informed by the local instruction theory developed improved number sense (Whitacre & Nickerson, 2006). They moved from being reliant on the mental analogues of the standard algorithms to reasoning more flexibly in mental computation. In the present study, I duplicated analyses from the previous study and found similar results. I also moved beyond the previous study by investigating number sense development as a microgenetic, sociogenetic, and ontogenetic process (Saxe & Esmonde, 2005). I asked the following research questions: As prospective elementary teachers participate in a mathematics content course designed to support their development of number sense, 1. How does the number sense of individuals evolve? 2. What ideas come to function as if shared? What classroom mathematical practices emerge and become established? I approached this study from a situated perspective (Cobb & Bowers, 1999). The emergent perspective informed my approach to the research in terms of taking both social and individual lenses to the analysis of number sense development (Cobb & Yackel, 1996). I made innovations in the analysis of number sense. I documented collective activity in the class in terms of progressions through classroom mathematical practices. I also analyzed two case studies of individuals' number sense development. These analyses provide insights into the phenomenon of prospective elementary teachers' number sense development, which will inform revisions and elaboration to the local instruction theory
Dissertation•
22 Mar 2012
TL;DR: This article explored the number sense of 47 primary school pre-service teachers in Namibia and was motivated by the poor performance of Namibian primary school learners in both national and international standardised assessment tests.
Abstract: This study explored the number sense of 47 final year primary school pre-service teachers in Namibia and was motivated by the poor performance of Namibian primary school learners in both national and international standardised assessment tests. The literature review revealed that learner performance is linked to teacher subject knowledge (Ball, 1990, Ma, 1999) and that teachers’ confidence in doing and teaching mathematics influences the way they teach and their willingness to learn mathematics (Ball, 1990; Graven 2004). Number sense studies of pre-service teachers (Kaminski, 1997; Tsao, 2004; Veloo, 2010; Yang, Reys & Reys, 2009) have indicated that the development of number sense should be a focus of primary pre-service teacher education.
The data in this mixed method research design were obtained from a Number Sense Questionnaire, a Written Computations Questionnaire and a Mental Calculations Questionnaire. These questionnaires were adapted from instruments developed by Professor Der-Ching Yang for 6th and 8th grade learners in Taiwan. Teacher confidence was measured by the McAnallen Confidence in Mathematics and Mathematics Teaching Survey. Six randomly selected pre-service teachers were interviewed to determine their use of number-sensible strategies.
The correlation analysis shows a strong relationship between number sense and mental calculations; between number sense and confidence in both the ability to do and the ability to teach mathematics and between mental and written calculations.
The overall results of this study reveal that the final year primary pre-service teachers demonstrate limited number sense and possess very few of the indicators of number sense that were described by Kalchman, Moss and Case (2001). The findings expose a lack of conceptual understanding of the domain numbers and operations, particularly in the domain of rational numbers and the operations of multiplication and division. The pre-service teachers have little or no access to a variety of flexible number-sensible strategies to solve problems and calculate mentally. They lack the fluency in basic facts and procedures to perform written calculations efficiently and correctly. Unexpectedly, the analysis of the confidence survey shows that they are confident in both their ability to do mathematics and their ability to teach mathematics.
It is recommended that mental calculations and computational estimation should become a focus of primary school mathematics education. Institutions responsible for teacher training should develop the number sense of pre-service teachers and research effective and long-term professional development programmes. The confidence and willingness of the teachers to learn can be used as an important resource.
Journal Article•
01 Jan 2012
TL;DR: In this article, the authors investigated the understanding of in-service elementary school teachers in Taiwan about number sense, teaching strategies of number sense and the development of the number sense of students.
Abstract: The goal of this study was to investigate understanding of in-service elementary school teachers in Taiwan about number sense, teaching strategies of number sense and the development of number sense of students. Data were gathered through interviews of nine elementary mathematics teachers, regarding their understanding about number sense. The data indicated the followings: (1) The intuitive of number sense of teachers described as good intuition or sensitive about number sense. They stress on understanding number meanings and ignore these relationships about quantity and operation; (2) The teachers have similar views about teaching strategies of number sense; (3) The teachers believe that children with good number sense have well-understood number meanings and relationships, awareness that multiple strategies exist, and recognize reasonableness of data and calculation; (4) The teachers lacked for knowledge of number sense, so that they cannot address the complete teaching strategies of number sense; and (5) The background variables of teachers are related to the understanding of number sense.
Journal Article•
TL;DR: This paper used a collective case study design, in which three Kindergarten teachers from Greig Heights Primary School participated in a professional learning and development program designed to enhance aspects of their teacher knowledge.
Abstract: Pedagogical content knowledge has been widely acknowledged by researchers and practitioners as a significant factor for improving student knowledge, understanding and achievement. Recently, the knowledge teachers need for teaching has expanded to include teacher horizon content knowledge, “an awareness of how mathematical topics are related over the span of mathematics included in the curriculum” (Ball, Thames, & Phelps, 2008, p. 403). This study uses a collective case study design, in which three Kindergarten teachers from Greig Heights Primary School participated in a professional learning and development program designed to enhance aspects of their teacher knowledge. This paper will provide an emerging description of the nature of teacher knowledge, and discuss the potential implications this has for catering for the needs of students at-risk of experiencing difficulties in acquiring early numeracy skills (i.e., number sense knowledge).
22 Mar 2012
TL;DR: Data from national and international assessments clearly support the existence of the difficulties American students have with fractions and the need for research into student fractional understanding comes from the difficulty teachers themselves have with understanding fractional concepts.
Abstract: Introduction In the spring of 2007, a group of six mathematics educators came together as part of Baylor University's graduate program to design a course related to mathematics education that would be of value to all six of them. The backgrounds of these six were very different. One was a tenured professor who had conducted research on many different areas of mathematics teaching and learning. Two were middle school teachers; one was still teaching and attending school part time, while the other had left teaching to attend graduate school full time to complete a doctorate in education. One was an elementary school teacher with little formal background in mathematics outside the methods courses required for certification. One was a high school teacher with fifteen years of experience, and one was high school certified but had taught adult remedial education for most of her career. In addition, four of the participants had majored in mathematics during their undergraduate careers, while the other two had majored in elementary education with no specific emphasis on mathematics. At first glance, it would be easy to assume that such a diverse group would struggle to reach consensus on what would be a worthwhile investigation. However, it took only a short time to decide to research rational numbers, specifically fractions. The reason for this was that, at all the different levels with which the six of us worked, we had all experienced issues with our students' understanding of fractions, or lack thereof. Data from national and international assessments clearly support the existence of the difficulties American students have with fractions we had all observed in our own experiences. For example, the National Assessment of Educational Progress (NAEP), often referred to in the US as the Nation's Report Card, has historically shown that students struggle with all but the least complex questions involving fractions. Wearne and Kouba (2000) found in their analysis of the 1996 NAEP assessment that students struggled with problems that were multi-step or nonroutine. Kastberg and Norton (2007) furthered this analysis by comparing results from the 1996, 2000 and 2003 NAEPs. Again, students did well on simple questions such as identifying the picture that represents a specified fraction, with 83 percent of tested 4th grade students answering this question correctly in 2003 (89). However, more complex problems such as naming and shading an equivalent fraction remained a struggle for 4th graders, as only 19 percent of students correctly responded correctly (89). An examination of the latest NAEP data shows there are still struggles with these fraction concepts (NCES 2009). In 2009, students improved somewhat in dealing with equivalent fractions, as 55% were able to correctly identify the picture showing that 3/4 and 6/8 are equivalent. However, only 25% could accurately determine which of four fractions was closest to 1/2. Both of these questions were classified as being of low complexity. An examination of international testing data from the Trends in International Math and Science Study (TIMSS) further supports the findings from the NAEP and would seem to indicate that fractions are more of a problem for American children than for children from many other countries. Gonzales et al. (2004) reported results of the 2003 TIMSS study revealing that 4th grade students in the United States had scores that were significantly lower than their counterparts in eleven of the twenty-four participating countries. Only 59% of the questions in the strand related to fractions and number sense were answered correctly on the 1999 TIMSS (Mullis et al. 2000). All this demonstrates further the ongoing trend of the challenges faced by children in the United States regarding fractions. The final proof of the need for research into student fractional understanding comes from the difficulty teachers themselves have with understanding fractional concepts. …
01 Nov 2012
TL;DR: In this paper, the authors investigated the effect of co-teaching versus solo teaching on student's math achievement in elementary school and concluded that both solo teaching and coteaching were beneficial to the two different groups of students within their various learning environments.
Abstract: As education continues to progress schools are constantly seeking innovative ways to cultivate and enhance achievement for all students. As a result many public schools are pushing toward the inclusion model. This model includes co-taught instruction to meet the many needs of special education students. This research study was implemented to investigate the comparative effects of co-teaching versus solo-teaching on student’s math achievement in elementary school. Study participants included two fourth grade classes in an elementary school, one with a regular education (solo-teaching) and the other with the same regular education teacher and a special education teacher for the co-taught class. The independent variable is the teaching arrangement (co taught class vs. a solo-taught class) as considered by the school system and the dependent variable is the math achievement as measured by Number Sense, Multiplication, and Division pre and post test units. Comparison of student math achievement between co-teaching and soloteaching showed that solo teaching was more effective than co-teaching on student’s achievement in Multiplication, co-teaching was more effective on student achievement in the Number sense unit than solo-teaching, and that no statistical difference was shown between coteaching and solo teaching in their effect on student learning in the Division unit. It is concluded that both solo teaching and co-teaching were beneficial to the two different groups of students within their various learning environments. Further experimental research is needed.
02 Feb 2012
TL;DR: The aim of this paper is to demonstrate Web-based courseware for lower elementary level, number sense concepts that incorporates motivational design strategies derived from Keller’s ARCS (Attention, Relevance, Confidence, and Satisfaction) model.
Abstract: This work concerns the development of Web-based courseware for the motivation of elementary school underachievers in mathematics learning. Underachievers having more experience of failure in learning tend to lose their self-confidence and motivation to study. If they achieve more when studying, they will improve selfconfidence and learning motivation, and this will lead to more success in learning accomplishments. Existing Web-based courseware rarely meets the needs of underachievers, mainly by not considering the role that motivation plays for this group. The aim of this paper is to demonstrate Web-based courseware for lower elementary level, number sense concepts. It incorporates motivational design strategies derived from Keller’s ARCS (Attention, Relevance, Confidence, and Satisfaction) model. This program is expected to replace the existing courseware for underachievers learning number sense because the motivational focus for the learners is reflected in the functions and features of the program.
Dissertation•
01 Jan 2012
TL;DR: In this paper, a large-scale genetically sensitive investigation into the origins of number sense and into the nature of its relationship with mathematics was presented, which suggested that individual differences in number sense abilities were mostly driven by non-shared environmental factors, with modest contribution of genetic factors.
Abstract: Number sense is defined as the process of extracting numerical information by estimating numerosity and magnitudes of numerical symbols. Humans show great variability in estimation skills from an early age. Although little is known about the origin of individual differences in number sense, these individual variations positively correlate with mathematics. This thesis presents the first large-scale genetically sensitive investigation into the origins of number sense and into the nature of its relationship with mathematics. The research plan can be devised in two parts. In the first phase, a battery of web-tests age appropriate for 16-year olds, designed to assess number sense (as measured by numerosity and magnitude estimation), mathematics and cognitive abilities was created and validated. The battery was then administered to the large UK representative of twins of the Twins Early Development Study (TEDS).
In the second phase, using data from 7,598 sixteen year-old twins from the TEDS sample, this thesis used univariate and multivariate genetic analyses to investigate the contribution of genes and environment to individual differences in number sense and to its association with mathematics. The results suggested that individual differences in number sense abilities were mostly driven by non-shared environmental factors, with modest contribution of genetic factors. No average or aetiological sex differences were found in number sense. The relationship between mathematics and number sense was largely genetically mediated. However, contrary to the predictions, the genetic relationship between number sense and mathematics was found to be mediated by g. The existing longitudinal data in the TEDS sample was used to investigate the retrospective relationship between number sense, measured at 16, and mathematics and a range of cognitive abilities, measured at 16 and at previous ages as far back as age 7. The results suggest that the relationship between mathematics and number sense may be uneven across development. In particular, numerosity estimation may be important only at the very early stages of mathematical learning. Overall, this investigation did not find evidence that number sense is what is "special" about mathematics. The results support the Generalist Genes Hypothesis that same genes contribute to individual differences in various aspects of cognition and learning.