Cognitive models: The missing link to learning fraction multiplication and division

This quasi-experimental study aims to streamline cognitive models on fraction multiplication and division that contain the most worthwhile features of other existing models. Its exploratory nature and its approach to proof elicitation can be used to help establish its effectiveness in building students’ understanding of fractions as compared to the traditional algorithmic way of teaching, vis-à-vis the students’ negative notions about learning fractions. Interestingly, the study showed the benefits and drawbacks of using these cognitive models in the teaching and learning of mathematics.     

Effectiveness of the MASTER program for teaching special children multiplication and division

In this study, the utility of a MAthematics Strategy Training for Educational Remediation (MASTER) program, (Van Luit, Kaskens, & Van der Krol, 1993) was examined. The effectiveness of the program, designed to encourage strategy utilization with multiplication and division problems, was investigated for 84 students with poor mathematics skills, some of whom had learning disabilities (n = 42) and others with mild levels of mental retardation (n = 42). The results showed that the use of the self-instruction program resulted in significant improvement over the general instruction program. Furthermore, far transfer was found for the children with learning disabilities in the experimental group when they used effective problem-solving strategies on nontrained tasks. The results are consistent with previous findings suggesting the importance of self-instruction in mathematics training programs.     

Preservice elementary teachers' explicit beliefs about multiplication and division

This study, conducted in the United States, was designed to assess the extent to which the beliefs, multiplication always makes bigger and division always makes smaller, are explicitly held by preservice elementary teachers. Paper and pencil instruments were administered to 136 preservice elementary teachers in a large university in the southeastern United States. Responses on the instruments and in interviews provide insight into the sources of preservice teachers' beliefs and into the relationship among preservice teachers' beliefs, computational skills, and performance on word problems.     

Semantic structures of one-step word problems involving multiplication or division

From classifications of word problems in international discussion of elementary mathematics instruction as well as from conceptual elaborations of didactical analyses in Germany, a classification of semantic structures of one-step word problems involving multiplication or division is proposed, comprehending four main classes: Forming the n-th multiple of measurers, combinatorial multiplication, composition of operators, and multiplication by formula. This classification is more comprehensive and differentiated than the classifications of Vergnaud (1983), Nesher (1988), and Bellet al. (1989) — aiming at a better assignment between diverse contextual circumstances and conceptual demands of mathematics and at compatibility with the well-known semantic structures of addition and subtraction word problems.     

Insights fourth and fifth graders bring to multiplication and division with decimals

This study was designed to gain information about the understandings children in Israel and the United States have about multiplication and division of whole numbers that may be useful in building accurate understandings of these operations with decimals and the extent to which they hold conceptions about these operations that may interfere with their work with decimals. Data from interviews of the fourth and fifth graders indicate that students of this age already hold misconceptions such as multiplication always makes bigger. However they also hold conceptions that are prerequisite to understanding the area model of multiplication and the measurement model of division. These early conceptions might be used to build understanding of multiplication and division by decimals. Implications for the content and sequencing of instructional activities are presented.     

How reform curricula in the USA and Korea present multiplication and division of fractions

In order to give insights into cross-national differences in schooling, this study analyzed the development of multiplication and division of fractions in two curricula: Everyday Mathematics (EM) from the USA and the 7th Korean mathematics curriculum (KM). Analyses of both the content and problems in the textbooks indicate that multiplication of fractions is developed in KM one semester earlier than in EM. However, the number of lessons devoted to the topic is similar in the two curricula. In contrast, division of fractions is developed at about the same time in both curricula, but due to different beliefs about the importance of the topic, KM contains five times as many lessons and about eight times as many problems about division of fractions as EM. Both curricula provide opportunities to develop conceptual understanding and procedural fluency. However, in EM, conceptual understanding is developed first followed by procedural fluency, whereas in KM, they are developed simultaneously. The majority of fraction multiplication and division problems in both curricula requires only procedural knowledge. However, multistep computational problems are more common in KM than in EM, and the response types are also more varied in KM.     

Examining the relation between whole numbers and fractions: A meta-analytic structural equation modeling approach

With a Meta-Analytic Structural Equation Modeling approach, we investigated the role of whole-number knowledge on fraction performance, accounting for domain-general skills, age, and mathematics-difficulty status. We conducted analyses based on 6,096 students from 39 independent samples within 29 studies. Findings suggested conceptual whole-number knowledge emerged as a significant and stable predictor of conceptual and procedural fraction knowledge. More importantly, students experiencing mathematics difficulty demonstrated a distinct pattern of performance compared to typically-achieving students, including: conceptual whole-number knowledge had lesser impacts on conceptual fraction knowledge; procedural whole-number knowledge had greater impacts on procedural fraction knowledge; and working memory and fluid intelligence made fewer contributions to fraction knowledge.     

Fun with numbers: Alternative models for predicting salary levels

The increasing awareness and concern with equity issues in higher education, along with the escalating litigation, has prompted institutions to undertake salary prediction studies. Four prediction models (built on a males only and total sample) were compared: (1) entering all variables, (2) excluding rank and tenure, (3) using predicted rank and tenure, and (4) using only objective variables. Models were tested using all permanent full-time faculty at a large midwestern university. Using predicted rank and tenure was the most suitable for equity studies. Including all variables yielded the best results for explaining/predicting reward systems. The other two models did not appear appropriate for either purpose. The males only sample consistently produced the largest bias effects. Institutions considering a salary prediction study should find these outcomes helpful in determining appropriate analytical strategies.     

The inverse relation between multiplication and division: Concepts, procedures, and a cognitive framework

Researchers have speculated that children find it more difficult to acquire conceptual understanding of the inverse relation between multiplication and division than that between addition and subtraction. We reviewed research on children and adults’ use of shortcut procedures that make use of the inverse relation on two kinds of problems: inversion problems (e.g., 9 ×24 ?24 {9} \times {24} \div {24} ) and associativity problems (e.g., 9 ×24 ?8 {9} \times {24} \div {8} ). Both can be solved more easily if the division of the second and third numbers is performed before the multiplication of the first and second numbers. The findings we reviewed suggest that understanding and use of the inverse relation between multiplication and division develops relatively slowly and is difficult for both children and adults to implement in shortcut procedures if they are not flexible problem solvers. We use the findings to expand an existing model, highlight some similarities and differences in solvers’ use of conceptual knowledge across operations, and discuss educational implications of the findings.     

从Fibonacci数的恒等变换到Lucas数的同余式

研究了Fibonacci数的和式∑a b=nUa^mUb^m/a!b!,得出了一些关于Fibonacci数与Lucas数的恒等变换和一些有趣的Lucas数的同余式。     

关于广义费马数的一个结论

设b为偶数.本文给出了广义费马数F(6,n)=b2n+1是合数的一个充要条件.     

The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.     

A comparison of strategic development for multiplication problem solving in low-, average-, and high-achieving students

The present study investigated the differences of strategy use between low-, average-, and high-achieving students when solving different multiplication problems. Nineteen high-, 48 average-, and 17 low-achieving students participated in this study. All participants were asked to complete three different multiplication tests and to explain how they solved these problems. Results suggested that low achievers used incorrect operation strategies more frequently, indicating a lack of conceptual understanding of multiplication. High-achieving students demonstrated greater flexibility in problem-solving and were more accurate in performing direct retrieval or math algorithm strategies. Results were discussed about improving low achievers’ use of advanced strategies, enhancing their flexibility in choosing strategies and improving students’ accuracy in using direct retrieval or math algorithms.     

A TOM1 homologue is required for multiplication of Tobacco mosaic virus in Nicotiana benthamiana

The AtTOM1 gene of Arabidopsis thaliana had been shown to be essential for the efficient multiplication of Tobacco mosaic virus(TMV) in A.thaliana.In this study,we cloned an AtTOM1-like gene from Nicotiana benthamiana named as NbTOM1.Sequence alignment showed that NbTOM1 is closely related to AtTOM1 homologues of N.tabacum and Lycopersicon esculentum with 97.2% and 92.6% nucleotide sequence identities,respectively.Silencing of NbTOM1 by a modified viral satellite DNA-based vector resulted in complete inhibition of the multiplication of TMV in N.benthamiana.The result suggests that inhibition of NbTOM1 via RNA silencing is a potentially useful method for generating TMV-resistant plants.     

对作出分子点群乘法表的几点讨论

本文对量子化学中群表示理论的给定分子点群的乘法表进行了讨论.考虑了分子主轴的旋转方向、分子中原子编号的次序以及分子点群中对称操作的先后次序对乘法表结构的影响,并将所举例中可能得到的8张乘法表分为两类.找出了两类表之间相互关系,并给出了合适的方法和建议.