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Contreras, José N., and Armando M. Martínez-Cruz. "Solving Problematic Addition and Subtraction Word Problems." Teaching Children Mathematics 13, no. 9 (May 2007): 498–503. http://dx.doi.org/10.5951/tcm.13.9.0498.
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Word problems can play a prominent role in elementary school mathematics because they can provide practice with real-life problems and help students develop their creative, critical, and problem-solving abilities. However, word problems as currently presented in instruction and textbooks fail to accomplish these goals (Gerofsky 1996; Lave 1992). This failure is due, in part, to the unrealistic approach needed to solve them: the straightforward application of one arithmetic operation. Consequently, when faced with word problems in which context is critical to the solution, students fail to connect school mathematics with their real-world knowledge. Problems that cannot be solved by applying a straightforward arithmetic operation are called problematic. Several researchers have examined children's lack of use of their real-world knowledge to solve problematic word problems (Greer 1997; Reusser and Stebler 1997; Verschaffel and De Corte 1997).
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Soylu, Firat, David Raymond, Arianna Gutierrez, and Sharlene D. Newman. "The differential relationship between finger gnosis, and addition and subtraction: An fMRI study." Journal of Numerical Cognition 3, no. 3 (January 30, 2018): 694–715. http://dx.doi.org/10.5964/jnc.v3i3.102.
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The impact of fingers on numerical cognition has received a great deal of attention recently. One sub-set of these studies focus on the relation between finger gnosis (also called finger sense or finger gnosia), the ability to identify and individuate fingers, and mathematical development. Studies in this subdomain have reported mixed findings so far. While some studies reported that finger gnosis correlates with or predicts mathematics abilities in younger children, others failed to replicate these results. The current study explores the relationship between finger gnosis and two arithmetic operations—addition and subtraction. Twenty-four second to third graders participated in this fMRI study. Finger sense scores were negatively correlated with brain activation measured during both addition and subtraction. Three clusters, in the left fusiform, and left and right precuneus were found to negatively correlate with finger gnosis both during addition and subtraction. Activation in a cluster in the left inferior parietal lobule (IPL) was found to negatively correlate with finger gnosis only for addition, even though this cluster was active both during addition and subtraction. These results suggest that the arithmetic fact retrieval may be linked to finger gnosis at the neural level, both for addition and subtraction, even when behavioral correlations are not observed. However, the nature of this link may be different for addition compared to subtraction, given that left IPL activation correlated with finger gnosis only for addition. Together the results reported appear to support the hypothesis that fingers provide a scaffold for arithmetic competency for both arithmetic operations.
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Hikmah, Shofaul. "Teknik Friendly Match Man to Man Untuk Menyelesaikan Operasi Hitung Penjumlahan Dan Pengurangan Bilangan Bulat." Jurnal Edutrained : Jurnal Pendidikan dan Pelatihan 4, no. 1 (July 6, 2020): 27–34. http://dx.doi.org/10.37730/edutrained.v4i1.53.
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The Man to Man Friendly Match Technique is a new breakthrough created to help Madrasah Ibtidaiyah students complete the operation of adding and subtracting integers. This technique emerged as a result of the writer's concern as one of the subjects of Mathematics in Madrasah Ibtidaiyah who saw the low interest and learning outcomes of students in the material operations of calculating the addition and subtraction of integers. Man to Man Friendly Match Technique is a technique that combines the game as an activity that is very closely related to the daily lives of students with the concept of arithmetic operations of addition and subtraction of integers
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Rizkiana, Alif. "Peningkatan Kemampuan Operasi Hitung Penjumlahan dan Pengurangan Dengan Media Konkret Pada Siswa Kelas 1 SD Negeri Bantarkawung 03." Social, Humanities, and Educational Studies (SHEs): Conference Series 3, no. 4 (December 30, 2020): 556. http://dx.doi.org/10.20961/shes.v3i4.54359.
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<em>The arithmetic operations of addition and subtraction are basic mathematics that must be learned since the 1st grade of elementary school to make it easier for students to do mathematics in advanced grades. In this research, the aim is to improve the ability to do arithmetic operations of addition and subtraction with concrete media for grade 1 students at SD Negeri Bantarkawung 03. Through classroom action research, the quality of learning can be improved because the teacher immediately knows what needs to be improved. The number of respondents studied in this study were all first grade elementary school students, totaling 22 students. This research was carried out in 2 cycles. Based on the description of the implementation of the action, the results of the research and discussion, data were obtained that there was an increase in the ability to do arithmetic addition and subtraction operations in each cycle. It can be seen from the application of the pre-cycle, that is, 40% has increased to 20%, so the total is 60% in the first cycle, then it has increased in the second cycle, which is an increase of 27%, the total increase is 87%. The conclusion is that using concrete objects media can improve the ability of addition and subtraction arithmetic operations in grade 1 students of SD Negeri Bantarkawung 03</em>
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Kuhn, Malcus Cassiano, and Arno Bayer. "As Operações de Adição e Subtração nas Aritméticas Editadas para as Escolas Paroquiais Luteranas do Século XX no Rio Grande do Sul." Jornal Internacional de Estudos em Educação Matemática 10, no. 3 (February 6, 2018): 141. http://dx.doi.org/10.17921/2176-5634.2017v10n3p141-153.
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O artigo discute as operações de adição e subtração com números naturais nas aritméticas editadas pela Igreja Evangélica Luterana do Brasil, por meio da Casa Publicadora Concórdia de Porto Alegre, para as escolas paroquiais luteranas do século XX no Rio Grande do Sul. Baseando-se na pesquisa histórica e no conceito de cultura escolar, analisaram-se a Primeira Aritmética da série Ordem e Progresso, a Primeira Aritmética da série Concórdia e duas edições da Segunda Aritmética da série Concórdia. Essas aritméticas apresentam algumas propostas de ensino alicerçadas no método intuitivo, enquanto outras refletem a tradição pedagógica da memorização, com ênfase no desenvolvimento de habilidades para o cálculo mental e escrito, com precisão e foco nos algoritmos e procedimentos de cálculo das operações de adição e subtração. As edições da Segunda Aritmética ainda trazem exercícios e problemas contextualizados com a realidade dos alunos das escolas paroquiais luteranas gaúchas e apresentam as provas reais da adição e subtração, destacando-se a ideia da adição e subtração como operações inversas e a prova dos 9.Palavras-chave: História da Educação Matemática. Adição. Subtração. Livros de Aritmética. Cultura Escolar.AbstractThe article discusses the operations of addition and subtraction with natural numbers in the arithmetic edited by Evangelical Lutheran Church of Brazil, through Concordia Publishing House of Porto Alegre, to the Lutheran parochial schools of the 20th century in Rio Grande do Sul. Basing on historical research and on concept of school culture, analyzing the First Arithmetic of the Order and Progress series, the First Arithmetic of the Concordia series and two editions of the Second Arithmetic of the Concordia series. These arithmetic present some teaching proposals grounded in the intuitive method, while others reflect the pedagogical tradition of memorization, with emphasis on the development of skills for the mental and written calculation, with precision and focus in the algorithms and procedures of calculation of the operations of addition and subtraction . The editions of the Second Arithmetic still bring exercises and problems contextualized with the reality of the students of the gaucho Lutheran parochial schools and present the real proofs of addition and subtraction, emphasizing the idea of addition and subtraction as inverse operations and the proof of the 9.Keywords: History of the Mathematics Education. Addition. Subtraction. Arithmetic Books. School Culture.
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Kurland, Theodore E. "The Number Line And Mental Arithmetic." Arithmetic Teacher 38, no. 4 (December 1990): 44–46. http://dx.doi.org/10.5951/at.38.4.0044.
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Some students have easily recognized difficulties with addition and subtraction. Some have no trouble adding or subtracting single-digit numbers when the sums are less than ten (7 + 2, 5 + 4, etc.) but have to resort to their fingers for sums greater than ten (7 + 5, 8 + 6, etc.). Other students have no difficulty adding numbers whose sums are greater than ten, such as 7 + 5, but have difficulty determining their differences, like 12 − 7. Finally, some students have no difficulty adding 7 + 5 or 8 + 4 but cannot mentall y add 17 + 5 or 18 + 4 or recognize the connection between the sum of singledigit numbers and numbers inc reased by orders of ten. Any one or a combination of these difficulties may appear when students compute; moreover, these problems will continue to plague them, curtailing their confidence and development in mathematics and mental arithmetic.
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Castro, Sofía, and Pedro Macizo. "All roads lead to Rome: Semantic priming between language and arithmetic." Journal of Numerical Cognition 7, no. 1 (March 31, 2021): 42–65. http://dx.doi.org/10.5964/jnc.6167.
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This study evaluated the existence of universal principles of cognition, common to language and arithmetic. Specifically, we analysed cross-domain semantic priming between affirmative sentences and additions, and between negative sentences and subtractions. To this end, we developed and tested a new priming procedure composed of prime sentences and target arithmetic operations. On each trial, participants had to read an affirmative or negative sentence (e.g., “The circle is red”, “The square is not yellow”) and select, between two images, the one that matched the meaning of the sentence. Afterwards, participants had to solve a one-digit addition or subtraction (e.g., 7 + 4, 6 – 3), either by selecting the correct result between two possible alternatives (Experiment 1), or by verbalizing the result of the operation (Experiment 2). We manipulated the task difficulty of both the sentences and the operations by varying the similarity between the response options for the sentence (Experiment 1 and 2), and the numerical distance between the possible results for the operation (Experiment 1). We found semantic priming for subtractions, so that participants solved subtractions faster after negative versus affirmative sentences, and this effect was modulated by the difficulty of the operation. This is the first study reporting semantic priming effects between language and arithmetic. The outcomes of this work seem to suggest a shared semantic system between both cognitive domains.
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Kolesnikova, Yu V. "Comparative Analysis of the Effectivenessof the Using of Direct and Generalized Conditional Reinforcement in the Development of a Skill of Solving of Simple Arithmetic Problems in a Child with ASD." Autism and Developmental Disorders 71, no. 2 (2021): 52–58. http://dx.doi.org/10.17759/autdd.2021710206.
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Within the framework of the applied behavior analysis, a comparison of the effectiveness of the direct and the generalized reinforcement was made during the teaching the skill of distinguishing arithmetic operations in mathematical problems. The study was conducted in two phases over two weeks with a 9-year-old girl with autism spectrum disorder (ASD). The first phase included training of multiplication and addition tasks, using tangible reinforcement, compared to the training of the arithmetic performance in division and subtraction tasks, using generalized reinforcement. The second phase included the training of discrimination between different arithmetic operations, but tangible and generalized reinforcements were used in variable mode. The results showed no differences in the effectiveness of both generalized and tangible reinforcements in the teaching process. The participant successfully learned to discriminate between different arithmetic operations as addition, multiplication, subtraction and division in single-component tasks.
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Novokshonov, A. K. "Performance analysis of arithmetic algorithms implemented in C++ and Python programming languages." PROBLEMS IN PROGRAMMING, no. 2-3 (June 2016): 026–31. http://dx.doi.org/10.15407/pp2016.02-03.026.
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This paper presents the results of the numerical experiment, which aims to clarify the actual performance of arithmetic algorithms implemented in C ++ and Python programming languages using arbitrary precision arithmetic. "Addition machine" has been chosen as a mathematical model for integer arithmetic algorithms. "Addition machine" is a mathematical abstraction, introduced by R. Floyd and D. Knuth. The essence of "addition machine" is the following: using only operations of addition, subtraction, comparison, assignment and a limited number of registers it is possible to calculate more complex operations such as finding the residue modulo, multiplication, finding the greatest common divisor, exponentiation modulo with reasonable computational efficiency. One of the features of this implementation is the use of arbitrary precision arithmetic, which may be useful in cryptographic algorithms.
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Georges, Carrie, and Christine Schiltz. "Number line tasks and their relation to arithmetics in second to fourth graders." Journal of Numerical Cognition 7, no. 1 (March 31, 2021): 20–41. http://dx.doi.org/10.5964/jnc.6067.
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Considering the importance of mathematical knowledge for STEM careers, we aimed to better understand the cognitive mechanisms underlying the commonly observed relation between number line estimations (NLEs) and arithmetics. We used a within-subject design to model NLEs in an unbounded and bounded task and to assess their relations to arithmetics in second to fourth grades. Our results mostly agree with previous findings, indicating that unbounded and bounded NLEs likely index different cognitive constructs at this age. Bounded NLEs were best described by cyclic power models including the subtraction bias model, likely indicating proportional reasoning. Conversely, mixed log-linear and single scalloped power models provided better fits for unbounded NLEs, suggesting direct estimation. Moreover, only bounded but not unbounded NLEs related to addition and subtraction skills. This thus suggests that proportional reasoning probably accounts for the relation between NLEs and arithmetics, at least in second to fourth graders. This was further confirmed by moderation analysis, showing that relations between bounded NLEs and subtraction skills were only observed in children whose estimates were best described by the cyclic power models. Depending on the aim of future studies, our results suggest measuring estimations on unbounded number lines if one is interested in directly assessing numerical magnitude representations. Conversely, if one aims to predict arithmetic skills, one should assess bounded NLEs, probably indexing proportional reasoning, at least in second to fourth graders. The present outcomes also further highlight the potential usefulness of training the positioning of target numbers on bounded number lines for arithmetic development.
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Ohlsson, Stellan. "Simulating the Understanding of Arithmetic: A Response to Schoenfeld." Journal for Research in Mathematics Education 23, no. 5 (November 1992): 474–82. http://dx.doi.org/10.5951/jresematheduc.23.5.0474.
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In our article “The cognitive complexity of doing and learning arithmetic,” Andreas Ernst, Ernest Rees, and I compare two computer models that learn arithmetic from one-on-one tutoring. One model simulates rote performance, whereas the other model simulates performance based on understanding of arithmetic concepts such as place value. Our results show that the conceptual model has to work more than the rote model to learn subtraction. Also, both models have to work more to learn the regrouping algorithm than the alternative augmenting (equal addition) algorithm.
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Beishuizen, Meindert. "Mental Strategies and Materials or Models for Addition and Subtraction up to 100 in Dutch Second Grades." Journal for Research in Mathematics Education 24, no. 4 (July 1993): 294–323. http://dx.doi.org/10.5951/jresematheduc.24.4.0294.
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Dutch mathematics programs emphasize mental addition and subtraction in the lower grades. For two-digit numbers up to 100, instruction focuses on “counting by tens from any number” (N10), a strategy that is difficult to learn. Therefore, many children prefer as an easier alternative “decomposition” in tens (1010) and units. Instead of the use of arithmetic blocks (BL), the hundredsquare (HU) was introduced in the 1980s because of a (supposed) better modeling function for teaching N10. In a field study with several schools, (a) we compared the strategies N10 and 1010 on procedural effectiveness and error types, and (b) we assessed the influence of the support conditions BL versus HU on the acquisition of mental strategies (we had also a control condition NO with no extra materials or models). Results confirmed the greater effectiveness of N10 but also the preference of many weaker children for 1010. Support for BL or HU had differential effects on mental strategies. Differences are discussed in terms of cognitive psychology: the role of declarative knowledge and the relation between conceptual and procedural knowledge. New Dutch proposals for the 1990s emphasize teaching both strategies N10 and 1010 to enhance the flexibility of students' mental arithmetic.
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Hartmann, Matthias, Jochen Laubrock, and Martin H. Fischer. "The visual number world: A dynamic approach to study the mathematical mind." Quarterly Journal of Experimental Psychology 71, no. 1 (January 2018): 28–36. http://dx.doi.org/10.1080/17470218.2016.1240812.
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In the domain of language research, the simultaneous presentation of a visual scene and its auditory description (i.e., the visual world paradigm) has been used to reveal the timing of mental mechanisms. Here we apply this rationale to the domain of numerical cognition in order to explore the differences between fast and slow arithmetic performance, and to further study the role of spatial-numerical associations during mental arithmetic. We presented 30 healthy adults simultaneously with visual displays containing four numbers and with auditory addition and subtraction problems. Analysis of eye movements revealed that participants look spontaneously at the numbers they currently process (operands, solution). Faster performance was characterized by shorter latencies prior to fixating the relevant numbers and fewer revisits to the first operand while computing the solution. These signatures of superior task performance were more pronounced for addition and visual numbers arranged in ascending order, and for subtraction and numbers arranged in descending order (compared to the opposite pairings). Our results show that the “visual number world”-paradigm provides on-line access to the mind during mental arithmetic, is able to capture variability in arithmetic performance, and is sensitive to visual layout manipulations that are otherwise not reflected in response time measurements.
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Mandal, Sourav, and Sudip Kumar Naskar. "Solving Arithmetic Word Problems by Object Oriented Modeling and Query-Based Information Processing." International Journal on Artificial Intelligence Tools 28, no. 04 (June 2019): 1940002. http://dx.doi.org/10.1142/s0218213019400025.
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The paper presents an Object Oriented Analysis and Design (OOAD) approach to modeling, reasoning and a database query based approach to processing and solving addition-subtraction (Add-Sub) type arithmetic Mathematical Word Problems (MWP) of elementary school level. The system identifies and extracts the key entities in a word problem like owners, items and their attributes and quantities, verbs, from all the input sentences, using a rule based Information Extraction (IE) approach based on Semantic Role Labeling (SRL) technique. These information are then stored in predefined templates which are further modeled to represent an MWP in the object-oriented paradigm and processed using query based approach to generate the answer. These kind of applications are based on Natural Language Processing (NLP), Natural Language Understanding (NLU) and Artificial Intelligence (AI), and can be used as intelligent dynamic mathematical tutoring tools as part of E-Learning systems, Learning Management Systems, on-line education, etc. The proposed object oriented mathematical word problem solver can solve arithmetic MWPs involving only addition-subtraction operations and it has produced an accuracy of 94.35% on a subset of the AI2 arithmetic questions dataset.
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Klein, Anton S., Meindert Beishuizen, and Adri Treffers. "The Empty Number Line in Dutch Second Grades: Realistic Versus Gradual Program Design." Journal for Research in Mathematics Education 29, no. 4 (July 1998): 443–64. http://dx.doi.org/10.5951/jresematheduc.29.4.0443.
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In this study we compare 2 experimental programs for teaching mental addition and subtraction in the Dutch 2nd grade (N = 275). The goal of both programs is greater flexibility in mental arithmetic through use of the empty number line as a new mental model. The programs differ in instructional design to enable comparison of 2 contrasting instructional concepts. The Realistic Program Design (RPD) stimulates flexible use of solution procedures from the beginning by using realistic context problems. The Gradual Program Design (GPD) has as its purpose a gradual increase of knowledge through initial emphasis on procedural computation followed by flexible problem solving. We found that whereas RPD pupils showed a more varied use of solution procedures than the GPD pupils, this variation did not influence the procedural competence of the pupils. The empty number line appears to be a very powerful model for the learning of addition and subtraction up to 100.
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Carpenter, Thomas P., Elizabeth Fennema, Penelope L. Peterson, and Deborah A. Carey. "Teachers' Pedagogical Content Knowledge of Students' Problem Solving in Elementary Arithmetic." Journal for Research in Mathematics Education 19, no. 5 (November 1988): 385–401. http://dx.doi.org/10.5951/jresematheduc.19.5.0385.
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This study investigated 40 first-grade teachers' pedagogical content knowledge of children's solutions of addition and subtraction word problems. Most teachers could identify many of the critical distinctions between problems and the primary strategies that children used to solve different kinds of problems. But this knowledge generally was not organized into a coherent network that related distinctions between problems, children's solutions, and problem difficulty. The teachers' knowledge of whether their own students could solve different problems was significantly correlated with student achievement.
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Agushaka, Jeffrey O., and Absalom E. Ezugwu. "Advanced arithmetic optimization algorithm for solving mechanical engineering design problems." PLOS ONE 16, no. 8 (August 24, 2021): e0255703. http://dx.doi.org/10.1371/journal.pone.0255703.
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The distributive power of the arithmetic operators: multiplication, division, addition, and subtraction, gives the arithmetic optimization algorithm (AOA) its unique ability to find the global optimum for optimization problems used to test its performance. Several other mathematical operators exist with the same or better distributive properties, which can be exploited to enhance the performance of the newly proposed AOA. In this paper, we propose an improved version of the AOA called nAOA algorithm, which uses the high-density values that the natural logarithm and exponential operators can generate, to enhance the exploratory ability of the AOA. The addition and subtraction operators carry out the exploitation. The candidate solutions are initialized using the beta distribution, and the random variables and adaptations used in the algorithm have beta distribution. We test the performance of the proposed nAOA with 30 benchmark functions (20 classical and 10 composite test functions) and three engineering design benchmarks. The performance of nAOA is compared with the original AOA and nine other state-of-the-art algorithms. The nAOA shows efficient performance for the benchmark functions and was second only to GWO for the welded beam design (WBD), compression spring design (CSD), and pressure vessel design (PVD).
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Kullberg, Angelika, Camilla Björklund, Irma Brkovic, and Ulla Runesson Kempe. "Effects of learning addition and subtraction in preschool by making the first ten numbers and their relations visible with finger patterns." Educational Studies in Mathematics 103, no. 2 (December 10, 2019): 157–72. http://dx.doi.org/10.1007/s10649-019-09927-1.
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AbstractIn this paper, we report how 5-year-olds’ arithmetic skills developed through participation in an 8-month-long intervention. The intervention program aimed to enhance the children’s ways of experiencing numbers’ part-part-whole relations as a basis for arithmetic skills and was built on principles from the variation theory of learning. The report is based on an analysis of assessments with 103 children (intervention group n = 65 and control group n = 38) before and after the intervention and a follow-up assessment 1 year after the intervention. Our findings show that the learning outcomes of the intervention group were significantly higher compared to those of the control group after the intervention and that differences between the groups remained even 1 year after the intervention. In particular, the results show that children participating in the intervention group learned to recognize and use part-part-whole relations in novel arithmetic tasks.
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ÖZKILBAÇ, Bahadır. "Implementation and Design of 32 Bit Floating-Point ALU on a Hybrid FPGA-ARM Platform." Brilliant Engineering 1, no. 1 (December 13, 2019): 26–32. http://dx.doi.org/10.36937/ben.2020.001.005.
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FPGAs have capabilities such as low power consumption, multiple I/O pins, and parallel processing. Because of these capabilities, FPGAs are commonly used in numerous areas that require mathematical computing such as signal processing, artificial neural network design, image processing and filter applications. From the simplest to the most complex, all mathematical applications are based on multiplication, division, subtraction, addition. When calculating, it is often necessary to deal with numbers that are fractional, large or negative. In this study, the Arithmetic Logic Unit (ALU), which uses multiplication, division, addition, subtraction in the form of IEEE754 32-bit floating-point number used to represent fractional and large numbers is designed using FPGA part of the Xilinx Zynq-7000 integrated circuit. The programming language used is VHDL. Then, the ALU designed by the ARM processor part of the same integrated circuit was sent by the commands and controlled.
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Middleton, James A., and Marja van den Heuvel-Panhuizen. "The Ratio Table." Mathematics Teaching in the Middle School 1, no. 4 (January 1995): 282–88. http://dx.doi.org/10.5951/mtms.1.4.0282.
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The middle grades offer unique challenges to the mathematics teacher, especially in this time of transition from traditional to reformed curricula and methods. The range and conceptual quality of mathematical knowledge that students have as they enter grades 5 and 6 vary greatly. Many students have been accelerated through textbooks, resulting in a high degree of proficiency at arithmetic computation but sometimes with little conceptual understanding of the underlying mathematics. Many other students will enter the middle grades with only rudimentary understanding of addition and subtraction. This disparity of skills and understanding creates a difficult dilemma for middle school teachers. Should they review the arithmetic that students have already experienced, or should they forge ahead to a higher level of more difficult mathematics? This decision need not be perceived as a dichotomy. Methods exist for exploring higher-order mathematical topics conceptually that allow understanding by students of varying knowledge levels whatever their base knowledge may be.
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Gomides, Mariuche Rodrigues de Almeida, Isabella Starling-Alves, Giulia Moreira Paiva, Leidiane da Silva Caldeira, Ana Luíza Pedrosa Neves Aichinger, Maria Raquel Santos Carvalho, Julia Bahnmueller, Korbinian Moeller, Júlia Beatriz Lopes-Silva, and Vitor Geraldi Haase. "The quandary of diagnosing mathematical difficulties in a generally low performing population." Dementia & Neuropsychologia 15, no. 2 (April 2021): 267–74. http://dx.doi.org/10.1590/1980-57642021dn15-020015.
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ABSTRACT. Brazilian students’ mathematical achievement was repeatedly observed to fall below average levels of mathematical attainment in international studies such as PISA. Objective: In this article, we argue that this general low level of mathematical attainment may interfere with the diagnosis of developmental dyscalculia when a psychometric criterion is used establishing an arbitrary cut-off (e.g., performance<percentile 10) may result in misleading diagnoses. Methods: Therefore, the present study evaluated the performance of 706 Brazilian school children from 3rd to 5th grades on basic arithmetic operations addition, subtraction, and multiplication. Results: In line with PISA results, children presented difficulties in all arithmetic operations investigated. Even after five years of formal schooling, less than half of 5th graders performed perfectly on simple addition, subtraction, or multiplication problems. Conclusions: As such, these data substantiate the argument that the sole use of a psychometric criterion might not be sensible to diagnose dyscalculia in the context of a generally low performing population, such as Brazilian children of our sample. When the majority of children perform poorly on the task at hand, it is hard to distinguish atypical from typical numerical development. As such, other diagnostic approaches, such as Response to Intervention, might be more suitable in such a context.
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Shcherban, V. "Arithmetic Table as an Integral Part of all Computational Mathematics." Bulletin of Science and Practice 6, no. 6 (June 15, 2020): 31–41. http://dx.doi.org/10.33619/2414-2948/55/04.
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The paper is devoted to studying the following issue as a statement. What do we know and what we don’t know about arithmetic tables. Perhaps there is no mathematical problem as naive or simple as finding a method for creating arithmetic tables. We confirm that the general method has not been found yet. This study provides nonterminal solution to this problem. Why? The presentation of arithmetic material in essence, plus some accompanying ideas, makes it possible to develop them further in the system. Materials and methods. The system looks like this: a numerical table as a Pascal's triangle and a symmetric polynomial in two or three variables. Some arithmetic properties of such tables will be found, studied and proved. All this was made possible only after successful decryption of the entire class of numeric tables of truncated triangles in the cryptographic system. Results. For example, the arithmetic properties of truncated Pascal’s triangle for finding all prime numbers have been found and presented, and then their formulas have been placed. In addition to elementary addition and subtraction tables, unlimited “comparison” tables of numbers are given and presented for the first time. Conclusions. For computer implementation of the objectives set, the rules of real actions that should exist for tables have been laid down. Only recurrent numeric series should be used for this purpose.
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Fritz-Stratmann, Annemarie, Antje Ehlert, and Gabriele Klüsener. "Learning support pedagogy for children who struggle to develop the concepts underlying the operations of addition and subtraction of numbers: the ‘Calculia’ programme." South African Journal of Childhood Education 4, no. 3 (December 30, 2014): 23. http://dx.doi.org/10.4102/sajce.v4i3.232.
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This paper argues for teaching pre-service teachers about remediation strategies for learners who encounter problems in mathematics in the early grades. The premise is that all teachers should be equipped with theory-based practical knowledge to support learning. A few teaching sessions to develop the concepts that underlie the mathematical operations of addition and subtraction are introduced in this paper. An empirically validated, comprehensive model of cumulative arithmetic competence development from the ages of four to eight years forms the basis for the construction of the suggested teaching unit. The model distinguishes five competence levels of arithmetical conceptual development, and proposes that concepts build on one another hierarchically. A ‘part plus part is equivalent to whole’ model was constructed based on this hierarchical structure and the understanding that the concept of addition is a dynamic process. The teaching examples include exercises for all children, not only ones who struggle. Possibilities for adapting the exercises to the individual development level of slower or faster learners are also included. All exercises are accompanied by a reflection on the procedure and strategies applied in order to support meaningful and sustainable learning and to give student teachers the opportunity to use knowledge of mathematical cognition theory during their pre-service years.
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Lenček, Mirjana, and Iva Sladović. "Uspješnost osnovnoškolaca u osnovnim aritmetičkim operacijama." Logopedija 7, no. 1 (June 30, 2017): 13–23. http://dx.doi.org/10.31299/log.7.1.3.
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145 elementary school students from third to eighth grade with no difficulties in mathematics have been tested for their abilities in basic arithmetic operations. The aim of the examination was to gain insight into the level of mastery in mathematics in each of the grades.The observed variable in this experiment is the success in solving arithmetic problems in addition, subtraction, multiplying and dividing, as well as in applying the rules of mathematical operations depending on ascending educational age. The observed results showed that the group of third graders statistically significantly differs from all other educational levels, so it can be concluded that the strategies of problem solving in basic mathematical operations shift exactly in the third grade, which is consistent with other similar studies. It seems that counting based strategies give way to data retrieval strategies. As no statistically significant differences in problem solving of basic arithmetical operations from fourth to eighth grade were observed, it can be concluded that the levels of basic mathematical competences are levelled out in terms of accuracy. As the research has been performed on a relatively small sample in each grade, it is not possible to generalise the results. However, they can provide guidelines for shaping the examination material for recognising difficulties in mathematics, as well as for determining the direction of intervention in those difficulties or in dyscalculia.
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Danuri, Danuri, and Rizki Muhammad Ridho. "Analisis Kesulitan Belajar Operasi Hitung Penjumlahan, Pengurangan, Perkalian Dan Pembagian Pada Siswa Kelas IV SD N Bugel Panjatan Kulon Progo." Edukasi: Jurnal Penelitian dan Artikel Pendidikan 12, no. 2 (December 30, 2020): 67–76. http://dx.doi.org/10.31603/edukasi.v12i2.4067.
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Analysis of learning difficulties in mathematics is one of the important activities in order to determine the obstacles experienced during class learning. This research aims to analyze the difficulty of learning in the material operations of the addition, subtraction, multiplication and division in grade IV students at Bugel state elementary school, Panjatan, Kulon Progo, Yogyakarta. The research method used is a qualitative approach. This research was carried out in May 2020. From the social situation, there were grade IV students at state elementary school in Kepanewonan Panjatan, Kulon Progo Yogyakarta. The subjects in this study were grade IV students and teachers. Data collection used were observation, interview, and test techniques. The collected data were analyzed using the Miles and Huberman model data analysis which included data reduction, presentation, and drawing conclusions. Then the data validity test was done by triangulation of sources, and techniques. The results of the study on the arithmetic operations and operations reduction students did not have difficulty in understanding the material and calculating the matter of addition and subtraction. While in multiplication and division operations, there were still students who had difficulties.
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Alves, Antonio Mauricio Medeiros. "Livros Didáticos Integrados para o Ensino Primário Gaúcho: uma Análise da Abordagem das Operações Aritméticas da Soma e Subtração (1960-1978)." Jornal Internacional de Estudos em Educação Matemática 11, no. 1 (June 27, 2018): 55. http://dx.doi.org/10.17921/2176-5634.2018v11n1p55-63.
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Em meados do século XX um importante movimento de renovação do ensino da Matemática se desenvolveu mundialmente, influenciando as práticas docentes e também a produção didática para o ensino dessa disciplina. Esse texto apresenta um estudo sobre as transformações decorrentes desse movimento na abordagem das operações aritméticas da soma e subtração em três coleções de livros didáticos produzidos no Rio Grande do Sul, para o ensino primário, no período de 1960-1978: Estrada Iluminada e Nossa Terra Nossa Gente (em duas versões). O estudo, de cunho histórico, privilegiou a análise documental de 17 volumes de livros das coleções citadas e adota como referencial teóricometodológico a História Cultural, a partir de autores como Roger Chartier. Verificou-se que as operações aritméticas da soma e da subtração tiveram sua abordagem modificada em função de um novo conteúdo, a Teoria dos Conjuntos.Palavras-chave: Livro Didático. Matemática Moderna. Ensino Primário. Operações Aritméticas.Abstract An important movement of teaching Mathematics renew was developed, in the world, in the middle of the XX century, influencing the teaching practices and also didactic production for the teaching of this discipline. This text presents a study about the results of transformations from this movement in the approach to the arithmetic operations of addition and subtraction in three collections of didactic books produced in Rio Grande do Sul for primary education in the period 1960-1978: Estrada Iluminada and Nossa Terra Nossa Gente (in two versions). The historical study was developed in a methodological approach that privileged the documentary analysis of 17 volumes of books of the aforementioned collections and adopts as a theoretical and methodological studies on the Cultural History from authors such as Roger Chartier. It was verified that the arithmetic operations of addition and subtraction had their approach modified in function of a new content,the Set Theory.Keywords: Didactic Book. Modern Mathematics. Primary Education. Arithmetic Operations.
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Flexer, Roberta J., and Naomi Rosenberger. "Beware of Tapping Pencils." Arithmetic Teacher 34, no. 5 (January 1987): 6–10. http://dx.doi.org/10.5951/at.34.5.0006.
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Advocates of a new, “magical” method for teaching addition, subtraction, multiplication, and division are attracting the attention of many elementary chool teachers. The method involves teaching children to tap reference points on numeral to count out sums, differences, and products. Teacher like it because it's easy. Student catch on quickly and get the right answers. Even students who might have had trouble with arithmetic before can tap out correct answers. What's the problem, then? To a mathematics educator, the problem with the new “magic” is that a mechanical technique for getting answers is displacing learning with understanding.
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Gutiérrez-Naranjo, Miguel A., and Alberto Leporati. "First Steps Towards a CPU Made of Spiking Neural P Systems." International Journal of Computers Communications & Control 4, no. 3 (September 1, 2009): 244. http://dx.doi.org/10.15837/ijccc.2009.3.2432.
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We consider spiking neural P systems as devices which can be used to perform some basic arithmetic operations, namely addition, subtraction, comparison and multiplica- tion by a fixed factor. The input to these systems are natural numbers expressed in binary form, encoded as appropriate sequences of spikes. A single system accepts as inputs num- bers of any size. The present work may be considered as a first step towards the design of a CPU based on the working of spiking neural P systems.
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Lin, Chun-Ling, Melody Jung, Ying Choon Wu, Hsiao-Ching She, and Tzyy-Ping Jung. "Neural Correlates of Mathematical Problem Solving." International Journal of Neural Systems 25, no. 02 (February 12, 2015): 1550004. http://dx.doi.org/10.1142/s0129065715500045.
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This study explores electroencephalography (EEG) brain dynamics associated with mathematical problem solving. EEG and solution latencies (SLs) were recorded as 11 neurologically healthy volunteers worked on intellectually challenging math puzzles that involved combining four single-digit numbers through basic arithmetic operators (addition, subtraction, division, multiplication) to create an arithmetic expression equaling 24. Estimates of EEG spectral power were computed in three frequency bands — θ (4–7 Hz), α (8–13 Hz) and β (14–30 Hz) — over a widely distributed montage of scalp electrode sites. The magnitude of power estimates was found to change in a linear fashion with SLs — that is, relative to a base of power spectrum, theta power increased with longer SLs, while alpha and beta power tended to decrease. Further, the topographic distribution of spectral fluctuations was characterized by more pronounced asymmetries along the left–right and anterior–posterior axes for solutions that involved a longer search phase. These findings reveal for the first time the topography and dynamics of EEG spectral activities important for sustained solution search during arithmetical problem solving.
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Björklund, Camilla, Ference Marton, and Angelika Kullberg. "What is to be learnt? Critical aspects of elementary arithmetic skills." Educational Studies in Mathematics 107, no. 2 (March 20, 2021): 261–84. http://dx.doi.org/10.1007/s10649-021-10045-0.
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AbstractIn this paper, we present a way of describing variation in young children’s learning of elementary arithmetic within the number range 1–10. Our aim is to reveal what is to be learnt and how it might be learnt by means of discerning particular aspects of numbers. The Variation theory of learning informs the analysis of 2184 observations of 4- to 7-year-olds solving arithmetic tasks, placing the focus on what constitutes the ways of experiencing numbers that were observed among these children. The aspects found to be necessary to discern in order to develop powerful arithmetic skills were as follows: modes of number representations, ordinality, cardinality, and part-whole relation (the latter has four subcategories: differentiating parts and whole, decomposing numbers, commutativity, and inverse relationship between addition and subtraction). In the paper, we discuss particularly how the discernment of the aspects opens up for more powerful ways of perceiving numbers. Our way of describing arithmetic skills, in terms of discerned aspects of numbers, makes it possible to explain why children cannot use certain strategies and how they learn to solve tasks they could not previously solve, which has significant implications for the teaching of elementary arithmetic.
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Wright, Robert J. "Assessing Young Children's Arithmetical Strategies and Knowledge: Providing Learning Opportunities for Teachers." Australasian Journal of Early Childhood 27, no. 3 (September 2002): 31–36. http://dx.doi.org/10.1177/183693910202700307.
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Two interrelated initiatives in early numeracy are briefly described—the Count Me In Too Project in New South Wales, and Mathematics Recovery. The article then focuses on an approach to student assessment which is used in both initiatives. This approach enables teachers to better understand young children's early arithmetical strategies and knowledge. The approach is described in three parts: Part A focuses on initial strategies for addition or subtraction; for example, counting-from-one, counting-on, using finger patterns, and using strategies other than counting by ones. Part B focuses on strategies to solve two-digit subtraction and strategies for incrementing by tens and ones. And Part C focuses on strategies for early multiplication and division. Following this, the strategies which have been described in each part are discussed in terms of their relative sophistication. The discussion includes links to relevant literature and issues relevant to the teaching of early numbers. The conclusion lists six important points about early number teaching and learning.
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Vassilev, Vassil, Aleksandr Efremov, and Oksana Shadura. "Automatic Differentiation in ROOT." EPJ Web of Conferences 245 (2020): 02015. http://dx.doi.org/10.1051/epjconf/202024502015.
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In mathematics and computer algebra, automatic differentiation (AD) is a set of techniques to evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.), elementary functions (exp, log, sin, cos, etc.) and control flow statements. AD takes source code of a function as input and produces source code of the derived function. By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program. This paper presents AD techniques available in ROOT, supported by Cling, to produce derivatives of arbitrary C/C++ functions through implementing source code transformation and employing the chain rule of differential calculus in both forward mode and reverse mode. We explain its current integration for gradient computation in TFormula. We demonstrate the correctness and performance improvements in ROOT’s fitting algorithms.
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Khotimah, Husnul, Besse Intan Permatasari, and Nur Ismiyati. "Pengajaran Perkalian dan Pembagian dengan Metode Jarimatika." Abdimas Universal 2, no. 2 (September 29, 2020): 86–89. http://dx.doi.org/10.36277/abdimasuniversal.v2i2.77.
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One of the subjects that requires planting concepts from an early age is mathematics. At school age, especially grade 3 students, the student's age has reached 9 to 10 years. Students at that age, among others, are happy with activity exercises and are happy with activities in the form of competitions. Based on this, learning should be packaged in the form of activities. At the elementary level, one of the materials that is the problem for students is multiplication and division. This material is basic in mathematics but it is often found that there are junior high school students who do not know the results of multiplication under 10. Jarimatika is a method for teaching addition, subtraction, multiplication and division to students, especially at the elementary level. Jarimatika is an abbreviation of finger and arithmetic. As the name implies, the media in this method are the fingers. Based on this community service activity, it can be concluded that the students are enthusiastic and happy with the teaching of multiplication and division with Jarimatika. In addition, students are motivated to answer questions from the teacher.
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Östergren, Rickard, and Joakim Samuelsson. "Is Repeated Testing of Declarative Knowledge in Mathematics Moderated by Feedback?" Journal of Education and Culture Studies 2, no. 3 (August 23, 2018): 209. http://dx.doi.org/10.22158/jecs.v2n3p209.
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<p>This study set out to examine the effects of repeated testing of students’ declarative knowledge in mathematics in grade 7 (13-14 years old) and to what extent feedback moderates the effect of continually testing students’ declarative knowledge. Students who have automated the 400 basic arithmetical combinations (200 addition combinations and 200 subtraction combinations) have gained declarative knowledge. Mastering these combinations gives students an advantage where doing various calculations and performing different mathematical procedures are concerned (Dowker, 2012). If a student has automated the basic combinations, their attention will not be diverted from the procedure when solving calculation tasks, and there is thereby less risk of incorrect answers (Dowker, 2012). Previous studies have also shown that declarative knowledge in mathematics predict future results in more advanced mathematics (Hassel Bring, Goin, & Bransford, 1988; Gersten, Jordan, & Flojo, 2005; Rathmell & Gabriele, 2011).</p>
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Gajiyeva, F. "Correct Assessment of Students’ Achievements in Elementary Classes and Content Lines." Bulletin of Science and Practice 7, no. 7 (July 15, 2021): 322–30. http://dx.doi.org/10.33619/2414-2948/68/43.
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The article discusses the acquisition of knowledge, skills and habits by students in general education schools, the adoption of high levels of standards, the correct assessment of student achievement, the results of the correct assessment carried out by the teacher, interaction in the process of assessment and learning, the correct construction of teacher–student–parent relationships to improve the quality of teaching in the educational process. The article also talks about the importance of content lines in primary grades, about the five content lines of the mathematics course in grades I–IV, about inter-subject and intra-subject connections in long-term school practice, about modern information and communication technologies, elements of statistics and probability. The article discusses in detail the actions of addition and subtraction, written multiplication and division, the concept of numbers, fractions, parts, 4 arithmetic operations within a million, used in classes’ I–IV. It also mentions 5 content lines of the secondary school mathematics course: cognitive (cognitive) activity, emotional (affective) activity, motor (psycho-motor) activity, as well as the need for these actions.
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Siuciak, Mirosława. "Jako Jan jechał samotrzeć przez las półczwarta dnia - czyli o dawnych sposobach wyrażania relacji ilościowych." Białostockie Archiwum Językowe, no. 9 (2009): 281–92. http://dx.doi.org/10.15290/baj.2009.09.19.
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A comparison of old and present indexes for quantitative assessment shows very significant lexical and formal differences. A quantification manner, inherited from the Proto-Slavic language, was unstable and multidimensional, which was apparent in Old Polish texts, whose old complex structures already had a synthesized form, but nevertheless, their inflections still revealed a primary structure. Lack of formal stabilization, which was still noticeable in the Middle Polish period, made numerals and numerical structures directly reflect mathematical operations they had been founded upon. Complex numerical structures often expressed arithmetical operations explicite: 1) addition, e.g. thirty and five, two and twenty, 2) subtraction, e.g. a hundred without one, 3) multiplication, e.g. three-fold one hundred thousand. An essential feature of such structures was stimulation of utterance recipient’s mental activity. One-word numerals in a form of composites, which were constructed on a basis of a mechanism of subtraction, i.e. collective numerals like samotrzeć ‘one in a group of three’, and partitive like półczwarta ‘four minus half, enforced similar behavior. Disappearance of the above mentioned structures and lexemes occurred in New Polish period in result of numerous linguistic tendencies, among which the most vital trend in the history of a language was communication improvement and reduction of mental activity to a necessary minimum, as well as in effect of a tendency to simplify a system. Only the items that directly determine a sequence of numbers in an arithmetic system have remained in the Polish language from a formally and formatively varied class of lexemes naming numbers, i.e. cardinal numbers have specialized as numerical quantifiers supported only residually by collective numbers with greatly restricted context.
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Eaves, Joanne, Camilla Gilmore, and Nina Attridge. "Investigating the role of attention in the identification of associativity shortcuts using a microgenetic measure of implicit shortcut use." Quarterly Journal of Experimental Psychology 73, no. 7 (April 20, 2020): 1017–35. http://dx.doi.org/10.1177/1747021820905739.
Full textAbstract:
Many mathematics problems can be solved in different ways or by using different strategies. Good knowledge of arithmetic principles is important for identifying and using strategies that are more sophisticated. For example, the problem “6 + 38 − 35” can be solved through a shortcut strategy where the subtraction “38 − 35 = 3” is performed before the addition “3 + 6 = 9,” a strategy that is derived from the arithmetic principle of associativity. However, both children and adults make infrequent use of this shortcut and the reasons for this are currently unknown. To uncover these reasons, new sensitive measures of strategy identification and use must first be developed, which was one goal of our research. We built a novel method to detect the time-point when individuals first identify an arithmetic strategy, based on trial-by-trial response time data. Our second goal was to use this measure to investigate the contribution of one particular factor, attention, in the identification of the associativity shortcut. In two studies, we found that manipulating visual attention made no difference to the number of people who identified the shortcut, the trial number on which they first identified it, or their accuracy and response time for solving shortcut problems. We discuss the theoretical and methodological contribution of our findings and argue that the origin of people’s difficulty with associativity shortcuts may lie beyond attention.
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Eriksson, Kenneth, Don Estep, Peter Hansbo, and Claes Johnson. "Introduction to Adaptive Methods for Differential Equations." Acta Numerica 4 (January 1995): 105–58. http://dx.doi.org/10.1017/s0962492900002531.
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Knowing thus the Algorithm of this calculus, which I call Differential Calculus, all differential equations can be solved by a common method (Gottfried Wilhelm von Leibniz, 1646–1719).When, several years ago, I saw for the first time an instrument which, when carried, automatically records the number of steps taken by a pedestrian, it occurred to me at once that the entire arithmetic could be subjected to a similar kind of machinery so that not only addition and subtraction, but also multiplication and division, could be accomplished by a suitably arranged machine easily, promptly and with sure results…. For it is unworthy of excellent men to lose hours like slaves in the labour of calculations, which could safely be left to anyone else if the machine was used…. And now that we may give final praise to the machine, we may say that it will be desirable to all who are engaged in computations which, as is well known, are the managers of financial affairs, the administrators of others estates, merchants, surveyors, navigators, astronomers, and those connected with any of the crafts that use mathematics (Leibniz).
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Rufiana, Intan Sari, Wahyudi Wahyudi, and Dwi Avita Nurhidayah. "Optimalisasi mutu lulusan dengan pembekalan keterampilan berhitung model MARS (matematika dan aritmatika sederhana)." Transformasi: Jurnal Pengabdian Masyarakat 15, no. 1 (June 30, 2019): 44–52. http://dx.doi.org/10.20414/transformasi.v15i1.471.
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[Bahasa]: Tujuan kegiatan pengabdian kepada masyarakat ini (PKM) adalah memberikan keterampilan terkait dengan kemampuan aritmetika sederhana pada mahasiswa Program Studi Pendidikan Matematika Universitas Muhammadiyah Ponorogo khususnya pada Operasi Aljabar (Penjumlahan dan Pengurangan). Keterampilan ini termasuk di luar kompetensi utama yang mendukung keilmuan. Dengan adanya kegiatan ini diharapkan dalam implementasinya, mahasiswa mampu bersaing dengan lulusan dari Perguruan Tinggi lain. Kegiatan ini dilaksanakan untuk (1) mempersiapkan dan membekali para calon pendidik khususnya mahasiswa program studi matematika Fakultas Keguruan dan Ilmu Pendidikan Universitas Muhammadiyah Ponorogo Semester VI dan Semester VIII dalam hal berhitung cepat, (2) memberikan motivasi pada mahasiswa untuk memiliki keterampilan lain yang mendukung keilmuan. Kegiatan ini dilaksanakan melalui kegiatan pelatihan keterampilan berhitung dengan menggunakan jarimatika. Dapat disimpulkan bahwa kegiatan pelatihan ini dapat membangkitkan motivasi dan membekali mahasiswa dengan keterampilan lain yang mendukung keilmuannya. Dari 30 mahasiswa yang mengikuti kegiatan PKM, sebanyak 50% berpendapat bahwa model berhitung ini sangat mudah diterapkan, 30% menyatakan mudah diterapkan, dan sisanya menyatakan cukup mudah diterapkan. Selain itu, 53% dari 30 peserta sangat paham terhadap konsep berhitung menggunakan jarimatika, 27% dari 30 peserta menyatakan paham, 17% menyatakan cukup paham dan 3% menyatakan sulit.
Kata kunci: optimalisasi keterampilan, berhitung matematika, jarimatika
[English]: The purpose of this community service program is to provide students with simple arithmetic skills in the Mathematics Education Study Program at Universitas Muhammadiyah Ponorogo, especially in Algebra Operations (addition and subtraction) which include as the supporting competencies. This program is expected to prepare students for competing with graduates from other tertiary institutions. It was carried out to (1) prepare and equip prospective mathematics teachers in terms of fast counting, (2) provide motivation for students to have other skills that support their main competences. The program was in the form of numeracy skills training using Jarimatika. Of the 30 students who took part; 50% of the students agreed that the numeracy model is very easy to implement, 30% said it was easy to apply, and the rest stated it was quite easy to apply. In addition, 53% of the 30 participants understood the concept of arithmetic using Jarimatika, 27% expressed understanding, 17% stated that they understood well, and 3% stated that it was difficult. The program can arouse motivation and equip students with additional skills that can support their knowledge.
Keywords: optimization of skills; math counting; jarimatika
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Alpert, S. I. "The basic arithmetic operations on fuzzy numbers and new approaches to the theory of fuzzy numbers under the classification of space images." Mathematical machines and systems 3 (2020): 49–59. http://dx.doi.org/10.34121/1028-9763-2020-3-49-59.
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Classification in remote sensing is a very difficult procedure, because it involves a lot of steps and data preprocessing. Fuzzy Set Theory plays a very important role in classification problems, because the fuzzy approach can capture the structure of the image. Most concepts are fuzzy in nature. Fuzzy sets allow to deal with uncertain and imprecise data. Many classification problems are formalized by using fuzzy concepts, because crisp classes represent an oversimplification of reality, leading to wrong results of classification. Fuzzy Set Theory is an important mathematical tool to process complex and fuzzy da-ta. This theory is suitable for high resolution remote sensing image classification. Fuzzy sets and fuzzy numbers are used to determine basic probability assignment. Fuzzy numbers are used for detection of the optimal number of clusters in Fuzzy Clustering Methods. Image is modeled as a fuzzy graph, when we represent the dissimilitude between pixels in some classification tasks. Fuzzy sets are also applied in different tasks of processing digital optical images. It was noted, that fuzzy sets play an important role in analysis of results of classification, when different agreement measures between the reference data and final classification are considered. In this work arithmetic operations of fuzzy numbers using alpha-cut method were considered. Addition, subtraction, multiplication, division of fuzzy numbers and square root of fuzzy number were described in this paper. Moreover, it was illustrated examples with different arithmetic operations of fuzzy numbers. Fuzzy Set Theory and fuzzy numbers can be applied for analysis and classification of hyperspectral satellite images, solving ecological tasks, vegetation clas-sification, in remote searching for minerals.
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FAUSER, BERTFRIED, and P. D. JARVIS. "THE DIRICHLET HOPF ALGEBRA OF ARITHMETICS." Journal of Knot Theory and Its Ramifications 16, no. 04 (April 2007): 379–438. http://dx.doi.org/10.1142/s0218216507005269.
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Many constructs in mathematical physics entail notational complexities, deriving from the manipulation of various types of index sets which often can be reduced to labelling by various multisets of integers. In this work, we develop systematically the "Dirichlet Hopf algebra of arithmetics" by dualizing the addition and multiplication maps. Then we study the additive and multiplicative antipodal convolutions which fail to give rise to Hopf algebra structures, but form only a weaker Hopf gebra obeying a weakened homomorphism axiom. A careful identification of the algebraic structures involved is done featuring subtraction, division and derivations derived from coproducts and chochains using branching operators. The consequences of the weakened structure of a Hopf gebra on cohomology are explored, showing this has major impact on number theory. This features multiplicativity versus complete multiplicativity of number theoretic arithmetic functions. The deficiency of not being a Hopf algebra is then cured by introducing an "unrenormalized" coproduct and an "unrenormalized" pairing. It is then argued that exactly the failure of the homomorphism property (complete multiplicativity) for non-coprime integers is a blueprint for the problems in quantum field theory (QFT) leading to the need for renormalization. Renormalization turns out to be the morphism from the algebraically sound Hopf algebra to the physical and number theoretically meaningful Hopf gebra (literally: antipodal convolution). This can be modelled alternatively by employing Rota–Baxter operators. We stress the need for a characteristic-free development where possible, to have a sound starting point for generalizations of the algebraic structures. The last section provides three key applications: symmetric function theory, quantum (matrix) mechanics, and the combinatorics of renormalization in QFT which can be discerned as functorially inherited from the development at the number-theoretic level as outlined here. Hence the occurrence of number theoretic functions in QFT becomes natural.
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Lubin, Amélie, Sandrine Rossi, Nicolas Poirel, Céline Lanoë, Arlette Pineau, and Olivier Houdé. "The Role of Self-Action in 2-Year-Old Children: An Illustration of the Arithmetical Inversion Principle before Formal Schooling." Child Development Research 2015 (February 24, 2015): 1–7. http://dx.doi.org/10.1155/2015/879258.
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The importance of self-action and its considerable links with cognitive activity in childhood are known. For instance, in arithmetical cognition, 2-year-olds detected an impossible arithmetical outcome more accurately when they performed the operation themselves (actor mode) than when the experimenter presented it (onlooker mode). A key component in this domain concerns the understanding of the inversion principle between addition and subtraction. Complex operations can be solved without calculation by using an inversion-based shortcut (3-term problems of the form a+b-b must equal a). Some studies have shown that, around the age of 4, children implicitly use the inversion principle. However, little is known before the age of 4. Here, we examined the role of self-action in the development of this principle by preschool children. In the first experiment, 2-year-olds were confronted with inversion (1+1-1=1 or 2) and standard (3-1-1=1 or 2) arithmetical problems either in actor or onlooker mode. The results revealed that actor mode improved accuracy for the inversion problem, suggesting that self-action helps children use the inversion-based shortcut. These results were strengthened with another inversion problem (1-1+1=1 or 2) in a second experiment. Our data provide new support for the importance of considering self-action in early mathematics education.
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Sidney, Pooja Gupta, and Martha Wagner Alibali. "Creating a context for learning: Activating children’s whole number knowledge prepares them to understand fraction division." Journal of Numerical Cognition 3, no. 1 (July 21, 2017): 31–57. http://dx.doi.org/10.5964/jnc.v3i1.71.
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When children learn about fractions, their prior knowledge of whole numbers often interferes, resulting in a whole number bias. However, many fraction concepts are generalizations of analogous whole number concepts; for example, fraction division and whole number division share a similar conceptual structure. Drawing on past studies of analogical transfer, we hypothesize that children’s whole number division knowledge will support their understanding of fraction division when their relevant prior knowledge is activated immediately before engaging with fraction division. Children in 5th and 6th grade modeled fraction division with physical objects after modeling a series of addition, subtraction, multiplication, and division problems with whole number operands and fraction operands. In one condition, problems were blocked by operation, such that children modeled fraction problems immediately after analogous whole number problems (e.g., fraction division problems followed whole number division problems). In another condition, problems were blocked by number type, such that children modeled all four arithmetic operations with whole numbers in the first block, and then operations with fractions in the second block. Children who solved whole number division problems immediately before fraction division problems were significantly better at modeling the conceptual structure of fraction division than those who solved all of the fraction problems together. Thus, implicit analogies across shared concepts can affect children’s mathematical thinking. Moreover, specific analogies between whole number and fraction concepts can yield a positive, rather than a negative, whole number bias.
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HNIZDILOVA, Olena, Olga GRISHKO, and Lesya KLEVAKA. "DEVELOPMENT LOGICO-MATHEMATICAL REPRESENTATIONS AND SKILLS IN PRESCHOOL CHILDREN IN THE PROCESS OF USING DENESH LOGICAL BLOCKS AND THE KEYZENER STICK." Cherkasy University Bulletin: Pedagogical Sciences, no. 4 (2020): 199–206. http://dx.doi.org/10.31651/2524-2660-2020-4-199-206.
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Modern preschool education is based on a personality-oriented model. Mathematics occupies a leading place in solving the problem of development of logical and mathematical ideas and skills in preschool children. It sharpens the child's mind, develops flexibility of thinking, teaches logic. In the pedagogical practice of a modern preschool institution, Dienes' logical blocks and Kuizener's wands are actively used by modern educators. The teacher's work with this material significantly contributes to the development of children's logical and mathematical ideas and skills.The set of Dienes blocks consists of 48 geometric figures of different colors (red, blue, yellow), shape (round, square, triangular, rectangular), size (large, small), thickness (thick, thin). There are no two identical figures in the set, each characterized by the four properties mentionedabove. Together with the logical blocks in the work of the educator of the preschool institution, cards are used on which the properties of the blocks (color, shape, size, thickness) are conditionally determined. The use of cards allows children to develop the ability to substitute and model properties, the ability to encode and decode information about them. These properties and skills develop in the process of performing a variety of subject-game actions. In the process of working with Dienes blocks in the game, children not only consolidate ideas about geometric shapes, signs of objects, form mental actions, but also develop mental processes: thinking, memory, attention, imagination, speech.Kuizener's sticks are didactic material for the development of children's mathematical abilities. The set contains quadrangular sticks of 10 different colors and a length of 1 to 10 cm (this can usually be stripes). George Kuizener designed sticks so that sticks of the same length are made in the same color and denote a certain number. The greater the length of the stick, the greater the numerical value it expresses. Sticks allow you to translate practical external actions into the internal plan; to master spatial relations. Didactic material gives the chance to train kids in addition of number from units and two smaller numbers; learn to measure objects; learn arithmetic operations (addition, subtraction, division, multiplication); learn to divide the whole into parts; to bring them to the realization of the relations "less – more", "less by ... – more by ...".In working with Kuizener sticks, the spatial and quantitative characteristics are not as obvious to children as color, shape, size when working with Dienes blocks. But to open these characteristics will help the joint activities of an adult with a child.
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Li, Hai-Sheng, Yusi Xu, Yunbai Qin, Deli Fu, and Hai-Ying Xia. "The addition and subtraction of quantum matrix based on GNEQR." International Journal of Quantum Information 17, no. 07 (October 2019): 1950056. http://dx.doi.org/10.1142/s0219749919500564.
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The efficient quantum circuits of arithmetic operations are important to perform quantum algorithms. To implement efficient matrix operations, we first modify the generalized model of the novel enhanced quantum representation of digital images (GNEQR) to store unsigned and signed integer matrices. Next, we design the circuits of the circuits of quantum addition, quantum modulo addition, quantum subtraction, and quantum modulo subtraction, these operations all keeping two operands unchanged. Then, we propose the circuits of quantum matrix addition, quantum matrix modulo addition, quantum matrix subtraction, and quantum matrix modulo subtraction for the first time. Furthermore, we present a simulation method to verify the correctness of the proposed arithmetic operations of matrix. The results of simulation experiment show that the propose arithmetic operations of matrix are efficient and correct.
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Zhu, Rongjuan, Xuqun You, Shuoqiu Gan, and Jinwei Wang. "Spatial Attention Shifts in Addition and Subtraction Arithmetic: Evidence of Eye Movement." Perception 48, no. 9 (July 19, 2019): 835–49. http://dx.doi.org/10.1177/0301006619865156.
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Recently, it has been proposed that solving addition and subtraction problems can evoke horizontal shifts of spatial attention. However, prior to this study, it remained unclear whether orienting shifts of spatial attention relied on actual arithmetic processes (i.e., the activated magnitude) or the semantic spatial association of the operator. In this study, spatial–arithmetic associations were explored through three experiments using an eye tracker, which attempted to investigate the mechanism of those associations. Experiment 1 replicated spatial–arithmetic associations in addition and subtraction problems. Experiments 2 and 3 selected zero as the operand to investigate whether these arithmetic problems could induce shifts of spatial attention. Experiment 2 indicated that addition and subtraction problems (zero as the second operand, i.e., 2 + 0) do not induce shifts of spatial attention. Experiment 3 showed that addition and subtraction arithmetic (zero as the first operand, i.e., 0 + 2) do facilitate rightward and leftward eye movement, respectively. This indicates that the operator alone does not induce horizontal eye movement. However, our findings support the idea that solving addition and subtraction problems is associated with horizontal shifts of spatial attention.
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Panggabean, R. F. Setia Budi, and Kimura Patar Tamba. "KESULITAN BELAJAR MATEMATIKA: ANALISIS PENGETAHUAN AWAL [DIFFICULTY IN LEARNING MATHEMATICS: PRIOR KNOWLEDGE ANALYSIS]." JOHME: Journal of Holistic Mathematics Education 4, no. 1 (November 13, 2020): 17. http://dx.doi.org/10.19166/johme.v4i1.2091.
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<p>In line with constructivism theory, prior knowledge is important in learning. Prior knowledge is the basis for a person being able to accept any new information that has been given. The background of this paper is the recognition that a number of students have difficulty learning mathematics. The results of observations when teaching grade 12 students about the matrix show that students still have difficulty in performing addition and subtraction arithmetic operations. This fact shows that there is a problem in students' prior knowledge. This paper aims to see the importance of the position of initial knowledge in student learning difficulties. This paper is a literature review. The results obtained show that initial knowledge is a source of student learning difficulties. This can be seen from the nature of mathematics and the thinking process of students in learning mathematics. From the nature of mathematics, according to constructivism, knowledge is acquired in a progressive constructive manner. Difficulties will arise when the prior knowledge is epistemologically different from the new knowledge. From the thinking process, students use their prior knowledge to construct new knowledge or respond to new information. The implication is that learning difficulties will arise when there is a conflict between prior knowledge and new knowledge.</p><p><strong>BAHASA INDONESIA ABSTRACT: </strong>Sejalan dengan teori konstruktivisme, pengetahuan awal (<em>prior knowledge</em>) merupakan hal penting dalam pembelajaran. Pengetahuan awal menjadi landasan bagi seseorang untuk mampu menerima informasi baru yang telah diberikan. Hal yang melatar belakangi tulisan ini adalah banyaknya anak didik yang kesulitan belajar matematika. Hasil observasi ketika mengajar SMA kelas XII tentang matriks menunjukkan siswa masih kesulitan dalam melakukan operasi hitung penjumlahan dan pengurangan. Hal tersebut menunjukkan ada masalah dalam pengetahuan awal siswa. Tujuan penulisan adalah untuk melihat pentingnya posisi pengetahuan awal dalam kesulitan belajar siswa. Tulisan ini merupakan kajian literatur. Hasil yang diperoleh menunjukkan pengetahuan awal merupakan sumber kesulitan belajar siswa. Hal ini dilihat dari natur matematika dan proses berpikir siswa dalam belajar matematika. Dari natur matematika, menurut teori konstruktivisme, pengetahuan diperoleh secara konstruktif progresif. Implikasinya, kesulitan akan muncul ketika secara epistemology pengetahuan awal berbeda dengan pengetahuan yang akan dipelajari. Dari proses berpikir, dalam membentuk pengetahuan baru atau merespon informasi yang dihadapkan padanya, siswa menggunakan pengetahuan awalnya. Implikasinya, kesulitan belajar akan muncul ketika terjadi konflik antara pengetahuan awal dengan pengetahuan yang akan dipelajari.</p><div><hr align="left" size="1" width="33%" /></div>
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Thompson, Charles S., and A. Dean Hendrickson. "Verbal Addition and Subtraction Problems: Some Difficulties and Some Solutions." Arithmetic Teacher 33, no. 7 (March 1986): 21–25. http://dx.doi.org/10.5951/at.33.7.0021.
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Many of the difficulties that children have in solving verbal (story) problems involving addition and subtraction arise because of their limited understanding of the arithmetic operations that are involved. They don't know when to use addition or subtraction because they lack specific knowledge regarding the various situations that give rise to these operations. Often, children are taught addition only as “putting together” and subtraction only as “taking away,” but many other settings involve addition and subtraction operations. Children need to receive specific instruction in different contexts if they are to become good solvers of verbal addition and subtraction problems. This article describes the contexts and then explains a successful sequence of activities that teach verbal problems.
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A, RAJKUMAR, JOSE PARVIN PRAVEENA N, and DHANUSH C. "Intuitionistic decagonal fuzzy number and its arithmetic operations." Journal of Management and Science 7, no. 1 (June 30, 2017): 51–58. http://dx.doi.org/10.26524/jms.2017.7.
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This paper introduced a new conception Intuitionistic Decagonal fuzzy Number and defines fundamental arithmetic operations like addition, subtraction. Numerical examples for addition and subtraction between two Intuitionistic Decagonal fuzzy Numbers are given.Score function and accuracy function are also defined.
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Michaux, Nicolas, Nicolas Masson, Mauro Pesenti, and Michael Andres. "Selective Interference of Finger Movements on Basic Addition and Subtraction Problem Solving." Experimental Psychology 60, no. 3 (February 1, 2013): 197–205. http://dx.doi.org/10.1027/1618-3169/a000188.
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Fingers offer a practical tool to represent and manipulate numbers during the acquisition of arithmetic knowledge, usually with a greater involvement in addition and subtraction than in multiplication. In adults, brain-imaging studies show that mental arithmetic increases activity in areas known for their contribution to finger movements. It is unclear, however, if this truly reflects functional interactions between the processes and/or representations controlling finger movements and those involved in mental arithmetic, or a mere anatomical proximity. In this study we assessed whether finger movements interfere with basic arithmetic problem solving, and whether this interference is specific for the operations that benefit the most from finger-based calculation strategies in childhood. In Experiment 1, we asked participants to solve addition, subtraction, and multiplication problems either with their hands at rest or while moving their right-hand fingers sequentially. The results showed that finger movements induced a selective time cost in solving addition and subtraction but not multiplication problems. In Experiment 2, we asked participants to solve the same problems while performing a sequence of foot movements. The results showed that foot movements produced a nonspecific interference with all three operations. Taken together, these findings demonstrate the specific role of finger-related processes in solving addition and subtraction problems, suggesting that finger movements and mental arithmetic are functionally related.