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Descriptors:
Arithmetic; Numeracy; Mathematics Skills; Training Methods; Preschool Children; Factor Structure; Early Childhood Education; Numbers; Number Concepts; Computation; Mathematics; Factor Analysis; Scores; Comparative Analysis; Correlation; Intervention

Abstract:
Validating the structure of informal numeracy skills is critical to understanding the developmental trajectories of mathematics skills at early ages; however, little research has been devoted to construct evaluation of the Numbering, Relations, and Arithmetic Operations domains. This study was designed to address this knowledge gap by examining the structure of these three numeracy skill domains and examining the relations among these domains. Three hundred ninety-three children participated in the study (51.7% girls, 55.7% White, 33.8% African American, and 10.5% other). Results indicated that the relations among the informal numeracy skills were best explained by a three-factor model that included Numbering, Relations, and Arithmetic Operations factors, and this factor structure was the same in both younger and older preschool children. (Contains 9 tables, 1 figure, and 5 notes.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Cognitive Processes; Attention; Neonates; Numbers; Visual Discrimination; Role; Visual Stimuli

Abstract:
This study investigated processing of number and extent in newborns. Using visual preference, we showed that newborns discriminated between small sets of dot collections relying solely on implicit numerical information when non-numerical continuous variables were strictly controlled (Experiment 1), and solely on continuous information when numerical variables were controlled (Experiment 2). When number and extent were pitted against each other (Experiment 3), newborns showed no visual preference, suggesting that the two variables play comparable roles in attracting newborns' visual attention. In contrast to reports of dominance of continuous variables, these findings suggest that multiple dimensions attract newborns' attention and guide their visual exploration. (Contains 3 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Numbers; Algebra; Calculus; College Mathematics; Misconceptions; Error Patterns; College Students

Abstract:
The purpose of this study was to determine whether or not certain errors made when simplifying exponential expressions persist as students progress through their mathematical studies. College students enrolled in college algebra, pre-calculus, and first- and second-semester calculus mathematics courses were asked to simplify exponential expressions on an assessment. Persistent errors are identified and characterized. Using quantitative and qualitative methods, we found that the concept of negativity played a prominent role in most of the students' errors. We theorize that an underdeveloped conception of additive and multiplicative inverses is the root of these errors. (Contains 5 tables and 1 figure.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Mathematics; Numbers; Comprehension; College Students; College Mathematics; Case Studies; Student Experience

Abstract:
The aim of this paper is to examine students' understanding of the limiting behavior of a function from [set of real numbers][superscript 2] to [set of real numbers] at a point "P." This understanding depends on which definition is used for a limit. Several definitions are considered; two of these concern the notion of a neighborhood of "P", while another two are directed at the consistency of limits obtained by restricting the function to lines or half-lines passing through "P". A case study is presented involving four university students studying Mathematics. Comments are made about their abilities in working with each definition, associated images that were evoked, and how they related one definition to another (including the issue of logical equivalence). The influence of the students' previous experience in handling limits for real functions of one variable is also discussed. (Contains 5 figures.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Foreign Countries; Program Descriptions; Number Concepts; Numbers; Number Systems; Instructional Materials; Material Development; Indigenous Populations; Elementary School Teachers; Instructional Design; Indigenous Personnel; American Indians; American Indian Languages; American Indian Education; Native Language; Ethnology; Mathematics

Abstract:
Results from a project conducted in Mexico are discussed, in which a group of 17 indigenous teachers analyzed the numeration systems of their first language. The main goal of the project is to develop resources that help teachers in supporting students' understanding of the systems. In the first phase of the project, the central organizing ideas of 14 numeration systems were specified. Each system belonged to a different Mesoamerican language. Three aspects of the systems were identified that would have to be accounted for in instructional design. They include using 20 as a multiplicative base. Examples are presented of the instructional resources that indigenous teachers could use to help their students understand the quantitative rationales of the systems. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Grade 1; Elementary School Students; Screening Tests; Alphabets; Computation; Classification; Phonemic Awareness; Word Recognition; Decoding (Reading); Numbers; Addition; Subtraction; Phonics; Comprehension

Abstract:
The First Grade Pre-Screening is designed to be used at the start of the first grade school year so that teachers can obtain information about their incoming students. This information is intended to give teachers insight about what math and reading skills a student may or may not have at the beginning of the year. The information can aid teachers in planning instruction that will meet the needs of each student. The First Grade Pre-Screening is designed to be given in a short amount of time and to provide a simple snapshot of a student's skills. It is a first step in a relationship between the student and the first grade teacher. As the school year begins, daily interactions will allow the teacher to learn even more about the student. This will allow him or her to tailor instruction most appropriately. The First Grade Pre-Screening addresses skills based on "Indiana's Academic Standards--Kindergarten." The inventory covers skills in: (1) Counting; (2) Sorting and Classifying; (3) Patterns; (4) Phonemic Awareness; (5) Decoding and Word Recognition; (6) Identifying Numerals; (7) Adding and Subtracting; (8) Identifying Letters; (9) Phonics; and (10) Comprehension. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Numbers; Mathematics Instruction; Mathematical Concepts; Education Courses; Preservice Teachers; Elementary School Teachers; Elementary School Mathematics; Language Usage

Abstract:
This article examines the ways in which prospective elementary teachers' develop an understanding of language use for defining the whole throughout a 9-day rational number unit. Student work samples and classroom conversations are used to illustrate their difficulties and growth with defining the whole and corresponding language use for describing fractional amounts. The results indicate that three mathematical ideas became "taken-as-shared" by the class. The first was that fractions depend on a group or whole. The second included defining an "of what." The third was developing language in terms of what the denominator represents. Difficulties prospective teachers had conceptualizing language included distinguishing among the phrases "of a," "of one," "of the," and "of each." Implications for mathematics education courses and future research studies are also discussed. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Academic Achievement; Numbers; Reflection; Preservice Teachers; Interviews; Models; Mathematics; Mathematics Education; Tests; Mathematics Instruction

Abstract:
The aim of this paper is to propose a theoretical model to analyze prospective teachers' reasoning and knowledge of real numbers, and to provide an empirical verification of it. The model is based on Sierpinska's theory of theoretical thinking. Data were collected from 59 prospective teachers through a written test and interviews. The data indicated that mathematical tasks on real numbers, based on Sierpinska's theory, could be categorized according to whether they require reflective, systemic or analytic thinking. Analysis of the data identified three different groups of prospective teachers reflecting different types of theoretical thinking about real numbers. The interviews confirmed the empirical data from the written test, and provided a better insight into the thinking and characteristic features of the prospective teachers in each group. The analysis also indicated that the participants were more successful in tasks requiring systemic and analytic thinking, and only when this was achieved were they able to solve problems which required reflective thinking. Implications for teaching related to the findings of the study are discussed. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Numbers; Number Concepts; Comparative Analysis; Arithmetic; Symbols (Mathematics); Mathematical Concepts

Abstract:
Much of the research on mathematical cognition has focused on the numbers 1, 2, 3, 4, 5, 6, 7, 8, and 9, with considerably less attention paid to more abstract number classes. The current research investigated how people understand decimal proportions--rational numbers between 0 and 1 expressed in the place-value symbol system. The results demonstrate that proportions are represented as discrete structures and processed in parallel. There was a "semantic interference effect": When understanding a proportion expression (e.g., "0.29"), both the correct proportion referent (e.g., 0.29) and the incorrect natural number referent (e.g., 29) corresponding to the visually similar natural number expression (e.g., "29") are accessed in parallel, and when these referents lead to conflicting judgments, performance slows. There was also a "syntactic interference effect," generalizing the unit-decade compatibility effect for natural numbers: When comparing two proportions, their tenths and hundredths components are processed in parallel, and when the different components lead to conflicting judgments, performance slows. The results also reveal that zero decimals--proportions ending in zero--serve multiple cognitive functions, including eliminating semantic interference and speeding processing. The current research also extends the distance, semantic congruence, and SNARC effects from natural numbers to decimal proportions. These findings inform how people understand the place-value symbol system, and the mental implementation of mathematical symbol systems more generally. (Contains 3 figures and 1 table.) Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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Descriptors:
Number Systems; Mathematics Instruction; Preservice Teachers; Elementary School Teachers; Interviews; Mathematics; Mathematics Education; Undergraduate Study; Numbers; Mathematics Curriculum

Abstract:
Research shows that students, and sometimes teachers, have trouble with fractions, especially conceiving of fractions as numbers that extend the whole number system. This paper explores how fractions are addressed in undergraduate mathematics courses for prospective elementary teachers (PSTs). In particular, we explore how, and whether, the instructors of these courses address fractions as an extension of the whole number system and fractions as numbers in their classrooms. Using a framework consisting of four approaches to the development of fractions found in history, we analyze fraction lessons videotaped in six mathematics classes for PSTs. Historically, the first two approaches--part-whole and measurement--focus on fractions as parts of wholes rather than numbers, and the last two approaches--division and set theory--formalize fractions as numbers. Our results show that the instructors only implicitly addressed fraction-as-number and the extension of fractions from whole numbers, although most of them mentioned or emphasized these aspects of fractions during interviews. Note:The following two links are not-applicable for text-based browsers or screen-reading software. Show Hide Full Abstract

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