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1.
Floyd R. Vest 《Educational Studies in Mathematics》1971,3(2):220-228
Summary The above catalog contains fifteen headings, each of which indicates a collection of families of models for multiplication and division of whole numbers. The catalog refers to somewhat more than sixteen families of models which are easily distinguished one from the other.Not included in the catalog thus far developed are several interpretations of multiplication and division that are also of interest. Among these are models based on the equivalency of denominations of money and various units of measurement. Other interpretations which are of historical interest are those of McLellan and Dewey [15] and Thorndike [24]. The relation between models of operations on whole numbers and models of operations defined on larger universal sets is also of interest. One aspect of this area of interest is the process of constructing models of multiplication and division of whole numbers from such models by altering the rules of the model or delimiting its universal set. For example, one can begin with one of Diénès' models of multiplication of integers [8, pp. 57–58] and make approapriate adjustments and result in a model of multiplication of whole numbers. Other interpretations developed by Diénès are of interest because they involve concretizations of whole numbers which are operators as opposed to states [8, pp. 12, 30; 9, p, 36].These are a great many strategies available for the use of models in teaching the operations on whole numbers. In one such strategy, an educator can define either multiplication or division on some basis (most likely in terms of a model) and then the other can be defined as its inverse.Another strategy is to define each operation in terms of a different model. For example, one might define multiplication in terms of the repeated addition model and division in terms of the repeated subtraction model.Still another type of procedure involves a multiple embodiment strategy in which several interpretations are taught as representing each operation.The choice of a particular strategy would depend upon a great many factors. Some of the factors would be the type of culture and students for which the program is written, the psychological assumptions adopted by the writer, and the writer's knowledge of the domain of models for the operations as well as their relation to the abstract mathematical domain which they represent. This article has contributed to a basis for intelligent decisions in this area by presenting a characterization of the domain of models for multiplication and division of whole numbers and their relation to the abstract operations. 相似文献
2.
Emrullah Erdem 《International Journal of Science and Mathematics Education》2017,15(6):1157-1174
The current study examines the mental computation performance owned by students at fifth grade. This study was carried out with 118 fifth graders (11–12-year-olds) studying at 3 randomly selected primary schools that served low and middle socio-economic areas in a city of Turkey. “Mental Computation Test (MCT)” has been used to reach how participants mentally compute. In analysing data, participants’ scores on the test were calculated by using a statistical package and which mental computation level they were in was determined. Sample responses of the some students regarding some questions in the test were presented directly and discussed. Evidence was found that the total percentage of the students at quite low and low level is 7.6 % and at high and quite high level is calculated to be 78.8 %. It was also revealed that the mental computation performances of the students in addition and subtraction are better than multiplication and division. On the other hand, it was found out that the performances of the students in all operations decrease as the number of the digits increases. 相似文献
3.
函数图形的运算除迭加外 ,还应有相乘、相除和乘方的运算 ,若一函数是由几个基本初等函数加、减、乘、除和乘方运算而成的 ,那么它的图形就可以采用几何画法而得到 相似文献
4.
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. 相似文献
5.
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. 相似文献
6.
When we consider the gap between mathematics at elementary and secondary levels, and given the logical nature of mathematics
at the latter level, it can be seen as important that the aspects of children's logical development in the upper grades in
elementary school be clarified. In this study we focus on the teaching and learning of “division with decimals” in a 5th grade
classroom, because it is well known to be difficult for children to understand the meaning of division with decimals, caused
by certain conceptions which children have implicitly or explicitly. In this paper we discuss how children develop their logical
reasoning beyond such difficulties/misconceptions in the process of making sense of division with decimals in the classroom
setting. We then suggest that children's explanations based on two kinds of reversibility (inversion and reciprocity) are
effective in overcoming the difficulties/misconceptions related to division with decimals, and that they enable children to
conceive multiplication and division as a system of operations. 相似文献
7.
8.
Xenia Vamvakoussi Wim Van Dooren Lieven Verschaffel 《Educational Studies in Mathematics》2013,82(2):323-330
This study tested the hypothesis that intuitions about the effect of operations, e.g., “addition makes bigger” and “division makes smaller”, are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated adult participants were asked to judge the correctness of statements about the effect of operations. Their accuracy and reaction times were measured. For items where the correct answer was not in line with the assumed intuitions, more mistakes were observed; moreover, we found longer reaction times for correct responses, indicating that these intuitions interfere in participants’ reasoning on these tasks, even when they provide a correct response. 相似文献
9.
The division operation is not frequent relatively in traditional applications, but it is increasingly indispensable and important
in many modern applications. In this paper, the implementation of modified signed-digit (MSD) floating-point division using
Newton-Raphson method on the system of ternary optical computer (TOC) is studied. Since the addition of MSD floating-point
is carry-free and the digit width of the system of TOC is large, it is easy to deal with the enough wide data and transform
the division operation into multiplication and addition operations. And using data scan and truncation the problem of digits
expansion is effectively solved in the range of error limit. The division gets the good results and the efficiency is high.
The instance of MSD floating-point division shows that the method is feasible. 相似文献
10.
Matthew Clarke 《教育政策杂志》2020,35(2):151-167
ABSTRACTThis paper examines the tensions between education policy’s attachment to notions such as excellence and inclusion and its investments in managerial tropes of competition, continuous quality improvement, standards and accountability that are at odds with and which undermine its attachments. In order to explore these tensions, I draw on the psychoanalytic notion of fantasy, explained through Stanley Kubrick’s final film, Eyes wide shut. My argument is that while the individual and society are both constituted through unavoidable division, antagonism and opacity, these notions are obscured through the operations of fantasy which holds out the promise of wholeness, harmony and redemption. In particular, education serves as a key site in which these fantasmatic ideals are promoted and pursued, a claim I substantiate via an analysis of the UK government’s 2016 White Paper, Educational Excellence Everywhere. Specifically, I read the White Paper in terms of five fantasies of: control; knowledge and reason; inclusion; productivity; and victimhood. My argument is that while fantasy is an inescapable element that inevitably structures what we take to be ‘reality’, education policy might strive to inhabit fantasy differently, thereby finding ways of escaping its current mode of seeing education with eyes wide shut. 相似文献
11.
We report a multidimensional test that examines middle grades teachers’ understanding of fraction arithmetic, especially multiplication and division. The test is based on four attributes identified through an analysis of the extensive mathematics education research literature on teachers’ and students’ reasoning in this content area. We administered the test to a national sample of 990 in‐service middle grades teachers and analyzed the item responses using the log‐linear cognitive diagnosis model. We report the diagnostic quality of the test at the item level, mastery classifications for teachers, and attribute relationships. Our results demonstrate that, when a test is grounded in research on cognition and is designed to be multidimensional from the onset, it is possible to use diagnostic classification models to detect distinct patterns of attribute mastery. 相似文献
12.
This study examined 361 Chinese and 345 Singaporean sixth-grade students’ performance and problem-solving strategies for solving 14 problems about speed. By focusing on students from two distinct high-performing countries in East Asia, we provide a useful perspective on the differences that exist in the preparation and problem-solving strategies of these groups of students. The strategy analysis indicates that the Chinese sample used algebraic strategies more frequently and more successfully than the Singaporean sample, although the Chinese sample used a limited variety of strategies. The Singaporean sample’s use of model-drawing produced a performance advantage on one problem by converting multiplication/division of fractions into multiplication/division of whole numbers. Several suggestions regarding teaching and learning of mathematical problem solving, algebra, and problems about speed and its related concepts of ratio and proportion are made. 相似文献
13.
We propose a novel high-performance hardware architecture of processor for elliptic curve scalar multiplication based on the Lopez-Dahab algorithm over GF(2163) in polynomial basis representation. The processor can do all the operations using an efficient modular arithmetic logic unit, which includes an addition unit, a square and a carefully designed multiplication unit. In the proposed architecture, multiplication, addition, and square can be performed in parallel by the decomposition of computation. The point addition and point doubling iteration operations can be performed in six multiplications by optimization and solution of data dependency. The implementation results based on Xilinx Virtexll XC2V6000 FPGA show that the proposed design can do random elliptic curve scalar multiplication GF(2163) in 34.11 μs, occupying 2821 registers and 13 376 LUTs. 相似文献
14.
Eleanore Hargreaves 《Teaching Education》2013,24(3):327-344
In this paper, I explore the experiences of secondary teachers in four London schools [UK] who participated in Teacher Learning Communities, defined as meetings in which professional learning was supported as they learned about Assessment for Learning (AfL). The claim for these communities is that they lead to sustained improvements in teaching and learning, where the following design principles are adhered to: where leaders respect and value a need that has been identified by participants as of importance to themselves; they are school-based and integral to school operations; there is teacher collaboration; and there is input from within and beyond the school to support teachers’ theoretical as well as practical learning. The findings from this research project suggest that Teacher Learning Communities’ benefits were compromised specifically: where they were imposed on teachers; where they were not accommodated sufficiently within other school commitments; where leaders were too directive; where meeting formats were adhered to inflexibly; and where practice was emphasized at the expense of theories. My conclusion is that both AfL and Teacher Learning Communities rely for their success on sustained critical reflection among their participants, which can be inhibited where the above limitations apply. 相似文献
15.
In this paper, we focus on Finnish pre-service elementary teachers’ (N?=?269) and upper secondary students’ (N?=?1,434) understanding of division. In the questionnaire, we used the following non-standard division problem: “We know that 498:6?=?83. How could you conclude from this relationship (without using long-division algorithm) what 491:6?=?? is?” This problem especially measures conceptual understanding, adaptive reasoning, and procedural fluency. Based on the results, we can conclude that division seems not to be fully understood: 45% of the pre-service teachers and 37% of upper secondary students were able to produce complete or mainly correct solutions. The reasoning strategies used by these two groups did not differ very much. We identified four main reasons for problems in understanding this task: (1) staying on the integer level, (2) an inability to handle the remainder, (3) difficulties in understanding the relationships between different operations, and (4) insufficient reasoning strategies. It seems that learners’ reasoning strategies in particular play a central role when teachers try to improve learners’ proficiency. 相似文献
16.
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. 相似文献
17.
Parmjit Singh 《Educational Studies in Mathematics》2000,43(3):271-292
The purpose of this study was to construct an understanding of two grade six students' proportional reasoning schemes. The
data from the clinical interviews gives insight as to the importance of multiplicative thinking in proportional reasoning.
Two mental operations, unitizing and iterating play an important role in student's use of multiplicative thinking in proportion
tasks. Unitizing a composite unit and iterating it to its referent point enables one to preserve the invariance of a ratio.
Proportions involved the coordination of two number sequences, keeping the ratio unit invariant under the iteration. In the
iteration process, one needed to explicitly conceptualize the iteration action of the composite ratio unit to make sense of
ratio problems and to have sufficient understanding of the meaning of multiplication and division and its relevance in the
iteration process. One needed to have constructed multiplicative structures and iteration schemes in order to reason proportionally.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
Michael C. Wittmann Virginia J. Flood Katrina E. Black 《Educational Studies in Mathematics》2013,82(2):169-181
We show that students rearranging the terms of a mathematical equation in order to separate variables prior to integration use gestures and speech to manipulate the mathematical terms on the page. They treat the terms of the equation as physical objects in a landscape, capable of being moved around. We analyze our results within the tradition of embodied cognition and use conceptual metaphors such as the path-source-goal schema and the idea of fictive motion. We find that students solving the problem correctly and efficiently do not use overt mathematical language like multiplication or division. Instead, their gestures and ambiguous speech of moving are the only algebra used at that moment. 相似文献
19.
Isabel Maria Bjork Claudine Bowyer-Crane 《European Journal of Psychology of Education - EJPE》2013,28(4):1345-1360
This study investigates the relationship between skills that underpin mathematical word problems and those that underpin numerical operations, such as addition, subtraction, division and multiplication. Sixty children aged 6–7 years were tested on measures of mathematical ability, reading accuracy, reading comprehension, verbal intelligence and phonological awareness, using a mix of standardised and experimenter-designed tests. The experimental hypothesis was that mathematical word problems will call upon cognitive skills that are different and additional to those required by numerical operations, in particular verbal ability and reading comprehension, while phonological awareness and reading accuracy will be associated with both types of mathematical problems. The hypothesis is partly affirmed and partly rejected. Reading comprehension was found to predict performance on mathematical word problems and not numerical operations, and phonological awareness was found to predict performance on both types of mathematics. However, the predictive value of verbal ability and reading accuracy was found to be non-significant. 相似文献
20.
Dake Zhang Yi Ding Dave E. Barrett Yan Ping Xin Ru-de Liu 《European Journal of Psychology of Education - EJPE》2014,29(2):195-214
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. 相似文献