• Journal for History of Mathematics
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• v.25 no.4
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• pp.1-10
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• 2012

In Chinese mathematics, there have been two numeral systems, namely one in spoken language for recording and the other by counting rods for computations. They concerned with problems dealing with practical applications, numbers in them are concrete numbers except in the process of basic operations. Thus they could hardly develop a pure theory of numbers. In Song dynasty, 0 and TianYuanShu were introduced, where the coefficients were denoted by counting rods. We show that in this process, counting rods took over the role of the numeral system in spoken language and hence counting rod numeral system plays the role of that for abstract numbers together with the tool for calculations. Decimal fractions were also understood as denominate numbers but using the notions by counting rods, decimals were also admitted as abstract numbers. Noting that abacus replaced counting rods and TianYuanShu were lost in Ming dynasty, abstract numbers disappeared in Chinese mathematics. Investigating JianJie YiMing SuanFa(簡捷易明算法) written by Shen ShiGui(沈士桂) around 1704, we conclude that Shen noticed repeating decimals and their operations, and also used various rounding methods.

• Communications of Mathematical Education
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• v.36 no.4
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• pp.469-486
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• 2022

Counting occupies a fundamental and important position in mathematical learning due to its relation to number concepts and numeral operations. In particular, counting up to large numbers is an essential learning element in that it is structural counting that includes the understanding of place values as well as the one-to-one correspondence and cardinal principles required by counting when introducing number concepts in the early stages of number learning. This study aims to derive didactical implications by investigating the possibility of and the strategies for counting large numbers that is expected to have no students' experience because it is not composed of current textbook activities. To do this, 89 second-grade elementary school students who learned the three-digit numbers and experienced group-counting and skip-counting as textbook activities were provided with questions asking how many penguins were in a picture where 260 penguins were irregularly arranged and how to count. As a result of analyzing students' responses in terms of the correct answer rate, the strategy used, and their cognitive characteristics, the incorrect answer rate was very high, and the use of decimal principles, group-counting, counting by one, and partial sum strategies were confirmed. Based on these analysis results, several didactical implications were derived, including the need to include counting up to large numbers as textbook activities.

• The Mathematical Education
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• v.49 no.4
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• pp.437-462
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• 2010

The purpose of this research is to analyze the textbook contents about the natural number concepts in the Korean National Elementary Mathematics Curriculum. Understanding a concept of natural number is crucial in school mathematics curriculum planning, since elementary students start their basic learning with natural number system. The concepts of natural number have various meaning from the perspectives of pedagogical research, and the philosophy of mathematics. The natural number concepts in the elementary math curriculum consist of four aspects; counting numbers, cardinal numbers, ordinal numbers, and measuring numbers. Two research questions are addressed; (1) How are the natural number concepts focusing on counting, cardinal, ordinal, measuring numbers are covered in the national math curriculum? ; (2) What suggestions can be made to enhance the teaching and learning about the natural number concepts? Findings reveal that (1) the national mathematics curriculum properly reflects four aspects of natural number concepts, as the curriculum covers 50% of the cardinal number system; (2) In the aspect of the counting number, we hope to add the meaning about 'one, two, three, ......, and so on' in the Korean Mathematics curriculum. In the ordinal number, we want to be rich the related meaning in a set. Further suggestions are made for future research to include them ensuing number in the curriculum.

• Education of Primary School Mathematics
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• v.21 no.4
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• pp.415-430
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• 2018

The purpose of this study was to get reflections on teaching numbers and operations for special education from analyzing lessons for counting, addition and subtraction of natural number with counting board for students with autism. In order to attain these purposes, this study analyzed the lessons for counting, addition and subtraction of natural number to students with autism in 4th and 6th graders in special class at regular elementary school using counting board for one hour per week for 30 weeks. According to the analysis, counting board that reveals the structure of numbers becomes an effective mathematical materials, and using the counting strategy and computation strategy can be an effective method of teaching, and it is possible to teach mathematical communication to students with autism. From this result, this study presented suggestions for teaching counting, addition and subtraction for students with disabilities.

• Honam Mathematical Journal
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• v.34 no.3
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• pp.423-438
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• 2012

In this paper, we discuss numbers of downwards and upwards in generalized paper folding sequences. We compute the exact number of downwards and upwards in $R^n_p$ and $(R_pR_q)^n$ by using the properties of recursive sequences where n, p and q are natural numbers with $p{\geq}2$ and $q{\geq}2$.

• East Asian mathematical journal
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• v.25 no.3
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• pp.247-262
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• 2009

Permutation and combination are the part of mathematics which can be introduced the pliability and diversity of thought. In prior studies of permutation and combination, there treated difficulties of learning, strategy of problem solving, and errors that students might come up with. This paper provides the method so that meaningful teaching and learning might occur through the systematic approach of permutation and combination. But there were little prior studies treated counting numbers that basic of mathematics' action. Therefore this paper tries to help the understanding of permutation and combination with the process of changing from informal knowledge to formal knowledge.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.6
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• pp.95-102
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• 2015

Generally, there is no sorting algorithm much faster than O(nlogn). The quicksort has a best performance O(nlogn) in best and average-case, and $O(n^2)$ in worst-case. This paper suggests virtual radix counting bucket sort such that counting the frequency of numbers in each radix digit, and moves the arbitrary number to proper virtual bucket in the array, and divides the array into radix digit numbers virtually. Also, this algorithm moves the data to proper location within an array for using the minimum auxiliary memory. This algorithm performs k-times such that the number of k digits in given data and the time complexity is O(n). Therefore, this algorithm has a O(kn) time complexity.

• Communications of Mathematical Education
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• v.28 no.3
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• pp.375-394
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• 2014

The purpose of this study is to examine the transition of mathematical representation in Joseon mathematics, which is focused on numbers and operations, letters and expressions. In Joseon mathematics, there had been two numeral systems, one by chinese character and the other by counting rods. These systems were changed into the decimal notation which used Indian-Arabic numerals in the late 19th century passing the stage of positional notation by Chinese character. The transition of the representation of operation and expressions was analogous to that of representation of numbers. In particular, Joseon mathematics represented the polynomials and equations by denoting the coefficients with counting rods. But the representation of European algebra was introduced in late Joseon Dynasty passing the transitional representation which used Chinese character. In conclusion, Joseon mathematics had the indigenous representation of numbers and mathematical symbols on our own. The transitional representation was found before the acceptance of European mathematical representations.

• The Journal of the Institute of Internet, Broadcasting and Communication
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• v.15 no.5
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• pp.61-68
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• 2015

Among comparison sorts, no algorithm excels a current set lower bound of O(nlogn) in operation. Quicksort, the fastest of its kind, has a complexity of O(nlogn) at its best and on average and $O(n^2)$ at worst. This paper thus presents two methods: first is an O(n+k) simple counting sort which operates much more speedily than an O(n+k), (k=maximum value) counting sort, and second is an O(ln) radix counting sort which counts the frequency of numbers in the digit l of a data and saves it in a corresponding virtual bucket in an array, only to virtually divide the array into radix digit numbers. For the 6 experimental data, the proposed algorithm makes O(nlogn) or $O(n^2)$ of Quicksort simple into O(n+k) or O(ln). After all, the proposed sorting algorithm has proved to be much faster than the counting sort and Quicksort.

• Journal of Aquaculture
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• v.19 no.3
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• pp.178-182
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• 2006

We studied new observation method about take process of rotifer by marine fish larvae. Till now, we can not accurate observation and count of first rotifer feeding day and/or feeding numbers of rotifer by marine fish larvae. Because take rotifer is ingested and disappeared in the digestive system of fish larvae. However we suggest possible observation method for these problems. The trophi is mastication organ of rotifer, and has only one in each rotifer individual. The trophi is left in the mastication organ because sole indigestible organ of rotifer. Therefore we can accurate observation and count of first rotifer feeding day and/or feeding rotifer numbers of marine fish larvae by trophi observation method (RTCM; Rotifer Trophi Counting Method).