Showing papers on "Number sense published in 2010"

Elementary and Middle School Mathematics: Teaching Developmentally

Abstract: Section I Teaching Mathematics: Foundations and Perspectives chapter 1 Teaching Mathematics in the Era of the NCTM Standards chapter 2 Exploring What It Means to Do Mathematics chapter 3 Developing Understanding in Mathematics chapter 4 Teaching Through Problem Solving chapter 5 Planning in the Problem-Based Classroom chapter 6 Building Assessment into Instruction chapter 7 Teaching Mathematics Equitably to All Children chapter 8 Technology and School Mathematics Section II Development of Mathematical Concepts and Procedures chapter 9 Developing Early Number Concepts and Number Sense chapter 10 Developing Meanings for the Operations chapter 11 Helping Children Master the Basic Facts chapter 12 Whole-Number Place-Value Development chapter 13 Strategies for Whole-Number Computation chapter 14 Computational Estimation with Whole Numbers chapter 15 Algebraic Thinking: Generalizations, Patterns, and Functions chapter 16 Developing Fraction Concepts chapter 17 Computation with Fractions chapter 18 Decimal and Percent Concepts and Decimal Computation chapter 19 Proportional Reasoning chapter 20 Developing Measurement Concepts chapter 21 Geometric Thinking and Geometric Concepts chapter 22 Concepts of Data Analysis chapter 23 Exploring Concepts of Probability chapter 24 Developing Concepts of Exponents, Integers, and Real Numbers appendix A Principles and Standards for School Mathematics: Content Standards and Grade Level Expectations appendix B Professional Standards for Teaching Mathematics: Teaching Standards appendix C Guide to Blackline Masters References Index

Validating a Number Sense Screening Tool for Use in Kindergarten and First Grade: Prediction of Mathematics Proficiency in Third Grade.

Abstract: Using a longitudinal design, children were given a brief number sense screener (NSB) screener (N 204) over six time points, from the beginning of kindergarten to the middle of first grade. The NSB is based on research showing the importance of number competence (number, number relations, and number operations) for success in mathematics. Children's mathematics achievement on a validated high-stakes state test was measured 3 years later, at the end of third grade. Test-retest reliability estimates were obtained for the NSB. Two criterion groups were then formed on the basis of the third-grade achievement test (chil- dren who met and who did not meet mathematics standards). Diagnostic validity analyses for the NSB were completed using repeated measures analyses of variance and receiver operator curve analyses. Results from all analyses revealed that scores on the NSB in kindergarten and first grade predicted mathematics proficiency in third grade. Areas under the receiver operator curve indicated that the NSB has high diagnostic accuracy (areas under the receiver operator curve 0.78-0.88). Findings suggest that kindergarten and first-grade performance on the NSB is meaningful for predicting which children experience later mathematics difficulties.

Exploring Kindergarten Teachers’ Pedagogical Content Knowledge of Mathematics

Abstract: The purpose of this study was to assess 81 kindergarten teachers’ pedagogical content knowledge of mathematics on six subcategory areas such as number sense, pattern, ordering, shapes, spatial sense, and comparison. The data showed participants possessed a higher level of pedagogical content knowledge of “number sense” (M = 89.12) compared to other mathematics pedagogical content areas. The second highest scores among six subcategories of pedagogical content knowledge of mathematics was for the pedagogical content area of “pattern” (M = 82.33). The lowest scores among those six subcategories of kindergarten teachers’ pedagogical content knowledge were obtained from the subcategory of “spatial sense” (M = 44.23), which involved the means to introduce children to spatial relationships. The second lowest score was obtained for the subcategory of “comparison” (M = 50.40) which involved the means to introduce the concept of graphing and the use of a balance scale for measurement.

The Study of Number Sense: Realistic Activities Integrated into Third-Grade Math Classes in Taiwan

Abstract: The authors compared the performance of students who received integration of number sense activities in instruction with students who received instruction using regular mathematics textbooks. Two classes of third-grade students (N = 60) were randomly assigned to experimental and control groups. Students in each group were given a pretest, post I test, and post II test, and were interviewed individually across 20 class periods of intervention. Results showed that the performance of the students in the experimental group was significantly higher than those in the control group. The interview data also showed that the students in the experimental group performed better than students in the control group in flexible and effective ways to use number sense.

Promoting Sixth Graders' Number Sense and Learning Attitudes via Technology-based Environment

Abstract: A quasi-experimental design was adopted to investigate the effect of integrating technology into mathematics teaching on students’ number sense and their learning attitudes. Two sixth-grade classes were selected from an elementary school in Taiwan for participation in this study. The control group with 32 students followed their usual mathematics instruction on number sense without using technology. The experimental group with 32 students learned number sense in a technology-based environment. The ANCOVA results showed a statistically significant difference between the control group and the experimental group on number sense performance. The t- test results showed no significant difference before and after the instruction for the control group. However, there was a statistically significant difference before and after the instruction for the experimental group. Furthermore, data also showed a significant difference in the students’ learning attitudes between the control group and the experimental group after the teaching. These results indicated that integrating technology into number sense teaching and learning not only promote students’ number sense, but also has a positive effect on attitudes towards learning number sense.

Number Sense in Kindergarten: A Factor-Analytic Study of the Construct.

Abstract: Mathematics proficiency is becoming increasingly important for all individuals in today's society. Specifically, successful math achievement is necessary for the technological jobs of the 21st century and for many other daily activities (Mazzocco & Thompson, 2005). As a result of the growing emphasis on technology in today's workplace, employees are expected to display a higher skill level in mathematics than in the past (Clarke & Shinn, 2004). Furthermore, according to the National Science Board (2003), the careers with the highest rate of growth will require individuals who are proficient in math and science. Therefore, students with mathematics deficiencies may have limited career opportunities. In addition to the limited job opportunities associated with mathematics deficiency, there are also economic consequences of poor mathematics achievement. Because math-oriented jobs tend to yield higher salaries, differences in mathematics achievement levels may contribute to the growing economic disparity among various groups in the United States. As such, there has been increased concern for individuals of lower socioeconomic status, minorities, and females because these groups generally tend to exhibit lower levels of mathematics achievement (Arnold, Fisher, Doctoroff, & Dobbs, 2002; Business-Higher Education Forum, 2005; National Science Foundation, 2003). As a result, these patterns of mathematics proficiency and career opportunities may contribute to the gender gap in salaries and to intergenerational cycles of poverty (Arnold et al., 2002). Given the importance of mathematics proficiency in today's society, it is necessary to understand current mathematics achievement among students in the United States. Children's Achievement in Mathematics Recent research shows that many students in the United States are failing to acquire the mathematics skills necessary to succeed both within and outside the classroom (Clarke & Shinn, 2004; National Research Council [NRC], 2001; Reese, Miller, Mazzeo, & Dossey, 1997). Furthermore, national studies examining students' math achievement indicate that students may not possess the skills necessary to meet the changing technological demands of the workplace (Clarke & Shinn, 2004). One such assessment is the National Assessment of Educational Progress (NAEP), which is an ongoing assessment of the proficiency of America's students in various skill areas (e.g., reading, mathematics, writing). NAEP provides insights regarding the academic achievement, instructional experiences, and school environment for students in Grades 4, 8, and 12. The 2007 NAEP results in mathematics showed that approximately 18% of students in Grade 4 and 29% of students in Grade 8 were below the basic level of achievement. Approximately 37% of fourth-graders and 32% of eighth-graders were at the basic level. The basic level is defined by NAEP as "partial mastery of knowledge and skills that are fundamental for proficient work" (Reese et al., 1997, p. 53). In the same assessment, 39% of fourth-graders and 32% of eighth-graders were at or above the proficient level. Proficiency is defined by NAEP as students having "demonstrated competency over challenging subject matter" and being "well-prepared for the next level of schooling" (Reese et al., 1997, p. 53). Overall, results of the NAEP show that the majority of students are failing to meet the national standards set for proficiency in mathematics. The demand for mathematical skills in the workplace, coupled with current low levels of mathematics proficiency, suggest a need to examine how math skills develop and which early math skills are most important for later mathematics achievement. In addition, results of such research would inform early identification and intervention efforts to promote early mathematics development and prevent later failure. Early Development of Mathematics Skills The importance of early identification of children who are likely to experience later academic difficulties in school has been highlighted by many scholars and school-based practitioners. …

A Local Instruction Theory for the Development of Number Sense

Abstract: Gravemeijer's (1999, 2004) construct of a local instruction theory suggests a means of offering teachers a framework of reference for designing and engaging students in a set of sequenced, exemplary instructional activities that support students' mathematical development for a focused concept. In this paper we offer a local instruction theory to guide the design of a set of instructional activities in support of the development of number sense. We make explicit the goals, assumptions, underlying rationale, and related instructional activities and provide examples from a mathematics content course for preservice elementary teachers. In this way, we contribute to an elaboration of the construct of local instruction theory.

Assessing preschool number sense: skills demonstrated by children prior to school entry

Abstract: Components of early number sense, as identified in two Delphi studies and in the number sense literature related to mathematics difficulties, were assessed for 176 children in preschools and childcare centres across one local government area in Sydney, Australia, using tasks or modifications of tasks reported in the number sense literature. In addition, the children’s receptive vocabulary was measured using The Peabody Picture Vocabulary Test (third edition) and math reasoning was measured using Woodcock‐Johnson III Tests of Achievement. Although the children demonstrated a broad range of skills, there were no significant differences between children attending childcare and preschools for any of the measures. However, boys performed significantly better than girls in quantitative concepts and girls performed better than boys in subitising. In discussing the data, a comparison is made of the skills demonstrated by children and skills that were highlighted in the two Delphi studies and in the early number s...

Feeling number: grounding number sense in a sense of quantity

Abstract: Drawing on results from psychology and from cultural and linguistic studies, we argue for an increased focus on developing quantity sense in school mathematics. We explore the notion of “feeling number”, a phrase that we offer in a twofold sense—resisting tendencies to feel numb-er (more numb) by developing a feeling for numbers and the quantities they represent. First, we distinguish between quantity sense and the relatively vague notion of number sense. Second, we consider the human capacity for quantity sense and place that in the context of related cultural issues, including verbal and symbolic representations of number. Third and more pragmatically, we offer teaching strategies that seem helpful in the development of quantity sense coupled with number sense. Finally, we argue that there is a moral imperative to connect number sense with such a quantity sense that allows students to feel the weight of numbers. It is important that learners develop a feeling for number, which includes a sense of what numbers are and what they can do.

Examining eighth grade kuwaiti students’ recognition and interpretation of reasonable answers

Abstract: This research documents Kuwaiti eighth grade students’ performance in recognizing reasonable answers and the strategies they used to determine reasonableness. The results from over 200 eighth grade students show they were generally unable to recognize reasonable answers. Students’ performance was consistently low across all three number domains (whole numbers, fractions, and decimals). There was no significant difference in students’ performance on items that focused on the practicality of the answers or on items that focused on the relationships of numbers and the effect of operations, or on both. Interview data revealed that 35% of the students’ strategies were derived from two criteria for judging answers for reasonableness: the relationships of numbers and the effect of operations, and the practicality of the answers. They used strategies such as estimation, numerical benchmarks, real-world benchmarks, and applied their understanding of the meaning of operations. However, over 60% of the students’ strategies were procedurally driven. That is, they relied on algorithmic techniques such as carrying out paper-and-pencil procedures. Additionally, some of the students’ strategies reflected misunderstandings of how and when to apply certain procedures. Given these findings, mathematics education in Kuwait should shift the emphasis from paper-and-pencil procedures and provide systematic attention to the development of number sense and computational estimation so Kuwaiti students will be more adept at recognizing reasonable answers.

Complex mental arithmetic: the contribution of the number sense.

Abstract: Young adults were asked to solve two-digit addition problems and to say aloud the result of each calculation step to allow the identification of computation strategies. We manipulated the position of the largest addend (e.g., 25 + 48 vs. 48 + 25) to assess whether strategies are modulated by magnitude characteristics. With some strategies, participants demonstrated a clear preference to take the largest addend as the starting point for the calculation. Hence, rather than applying strategies in an inflexible manner, participants evaluated and compared the operands before proceeding to calculation. Further, mathematically skilled participants tended to use those magnitude-based strategies more often than less skilled ones. The findings demonstrate that magnitude information plays a role in complex arithmetic by guiding the process of strategy selection, and possibly more so for mathematically skilled participants.

Development and Validation of a Computer‐Administered Number Sense Scale for Fifth‐Grade Children in Taiwan

Abstract: To investigate the structure of number sense and then to assess its uses in fifth-grade children's number sense development, a computerized number sense scale was developed and evaluated. The findings of the study indicate that the newly developed scale, with four dominant factors identified and reconfirmed, is internally consistent and substantially valid. It can be reliably used as a screening measure for a quick check of students' number sense development via online self-assessment. Compared with our previous study, both qualitative and quantitative changes were detected in students' number sense development at different grades. The qualitative change in number sense development is manifested in different numbers of factor components produced at different grades. The quantitative change is manifested in different amounts of factor variance explained at different grades. Furthermore, among the four aspects of number sense, Taiwanese students perform best on recognizing the relative number size yet relatively worse on judging the reasonableness of computational results.

The impact of the Infinite Mathematics Project on teachers' knowledge and teaching practice: A case study of a Title IIB MSP professional development initiative

Abstract: Ongoing, effective professional development is viewed as an essential mechanism for eliciting change in teachers’ knowledge and practice in support of enacting the vision of NCTM’s Principles and Standards of School Mathematics. This case study of the Infinite Mathematics Project, a Title IIB MSP professional development initiative, seeks to provide a qualitative examination of the characteristics and strategies used in the project and their impact on teacher learning and practice. The project embodied many features and strategies of effective professional development such as: mathematics content focus; sustained over time; reform activities (e.g., lesson study, teacher collaboration); active learning opportunities (e.g., implementing an action plan; developing differentiated instruction activities for a mathematics classroom); coherence with NCTM and state standards; and collective participation by IHE facilitators and participant K-12 teachers from partner districts. The findings reveal teachers gained both content knowledge (knowledge about mathematics, substantive knowledge of mathematics, pedagogical content knowledge, and curricular knowledge) and pedagogical knowledge (knowledge about strategies for differentiating instruction in a mathematics classroom, for supporting students’ reading in the content area, for fostering the development of number sense, for implementing standards-based teaching, and for critically analyzing teaching). The study also provides some evidence that the project had an impact on teaching practice. In addition, an implication of the study suggests the positive impact of Title IIB MSP partnership requirements. THE IMPACT OF THE INFINITE MATHEMATICS PROJECT ON TEACHERS' KNOWLEDGE AND TEACHING PRACTICE: A CASE STUDY OF A TITLE IIB MSP PROFESSIONAL DEVELOPMENT INITIATIVE

Improving Basic Math Skills Using Technology.

Abstract: The students of the targeted fourth, fifth, sixth, and ninth grade classes exhibited difficulties with number sense that interfered with understanding and recall of basic math facts. Evidence for the existences of the problem included teacher observation, test scores, and student and teacher surveys. The research participants included 42 children under 12 years old, 50 children over 12 years old and 20 teachers. The dates of the study were January 11, 2010 – May 07, 2010. During the course of this intervention, teacher researcher(s) attempted to enhance students’ basic math skills by re-teaching basic math skills with an emphasis on number sense using computers, calculators, and other technological devices. The students accessed mathematical websites and software via computers weekly. The rationale for the action research project was to enhance basic math skills of the targeted fourth, fifth, sixth, and ninth graders. Deficiencies in basic math skills in the classroom typically lead to inaccurate computation that created obstacles when problem solving. After a thorough review of the literature, three common probable causes were found to be lack of prior knowledge, negative attitude towards math, varied teaching methods. After a review of the literature, the teacher researchers selected four possible solutions for the lack of retention of basic math skills from the variety of solutions proposed by researchers. These possible solution strategies included: early screening, implementing the use of manipulatives, cooperative learning, and integrating technology. The teacher researchers recommend that all mathematics teachers, regardless of grade level, take some time to reinforce basic math skills. To reinforce and explore these topics, the teacher researchers recommend using technology such as software programs, PowerPoint, Elmos, Smart Boards, projectors, calculators, internet websites, You Tube videos and DVDs, and music CDs. Overall, targeted students in the fourth, fifth, sixth, and ninth grades improved their understanding of basic math skills by using technology. Their post intervention test scores indicated a noticeable increase in student mastery of basic mathematics. More students earned scores of 70% or higher when compared to the pre assessment scores.

Using Number Sense to Compare Fractions.

Abstract: Third-grade teachers found that giving particular attention to the use of real-world contexts, mental imagery, and manipulatives brought success to problem solving as students moved from using models to reasoning.

Preschoolers' Number Sense.

Abstract: Consider using a game-based assessment of number sense in young children, including those at risk because of socioeconomic level, disability, or the necessity of learning a second language.

Connecting the Numbers in the Primary Grades Using an Interactive Tool

Abstract: The different ways in which the Square Tool, a free web-based tool can be used to enhance students' number sense across the primary schools grades is discussed. The tool offers users the opportunities to explore more abstract ideas, including identifying primes, composites, and multiple numbers.

The Relationship between the Number Sense and Problem Solving Abilities of Year 7 Students.

Abstract: This paper reports on a component of a large yearlong study in three Year 7 classes in three different schools. The aim of this research component was to determine the relationship between students’ number sense and their problem-solving ability by means of paper-andpencil tests, classroom observations, and interviews of students and teachers. The results revealed a strong correlation between these two aspects of school mathematics, with important implications for classroom teachers.

An Investigation on the Error Patterns in Computation of Whole Numbers Committed by Singaporean Children with Dyscalculia

Abstract: In Singapore, very little research studies have been done on children with dyscalculia and mathematics-related anomalies (also known as mathematics learning disabilities) despites its problems are widespread. The writers have chosen to work with these children identified to have difficulties with mathematics at the Learning Disabilities Center. They narrowed their focus on the error patterns in computation (i.e., addition, subtraction, multiplication, and division) of whole numbers made by these children. The rationale is that children struggling with basic arithmetical computations also have difficulty completing arithmetic problems that involve multi-steps. The underlying key factor in poor computation is the inadequate or poor concept of number sense.

Playing numerical board games improves number sense in children from low-income backgrounds

Abstract: Arguments. We hypothesized that playing a linear, numerical, board game similar to the first row of Snakes and Ladders would produce substantial gains in numerical understanding. In such games, the greater the number in a square, the greater the distance that the child has moved the token, the number of discrete moves the child has made, the number of number names the child has spoken and heard, and the amount of time since the game began. Thus, playing the game provides visual–spatial, kinesthetic, auditory, and temporal cues to numerical magnitude. Playing the game for four 15–20min sessions produced large gains in low-income children’s understanding of numerical magnitudes, counting, number recognition, and ability to learn answers to novel arithmetic problems. The gains in knowledge of numerical magnitudes endured for 9 weeks and exceeded gains produced by engaging in other numerical activities or by playing a circular version of the board game.