Multiplication—repeated addition?

1967 ◽  
Vol 14 (5) ◽  
pp. 373-376
Author(s):  
Robert D. Bechtel ◽  
Lyle J. Dixon
Keyword(s):  
Rational Number ◽  
School Teacher ◽  
Number System ◽  
Mathematical Principle ◽  
Specific Topic ◽  
Number Systems ◽  
Whole Number ◽  
Graded Material ◽  
Real Number System

The elementary school teacher of today needs to have a comprehensive view of eleme ntary mathematics. The present emphasis on the “new” mathematics requires an understanding of number systems such as the natural number system, the whole number system, the rational number system, and the real number system, together with an understanding of some elementary concepts from geometry. This understanding of mathematics can no longer be limited to specific areas covered at one level in graded material, but should encompass the structures of different number systems. One must recognize the role of a specific topic in mathematics in relation to the overall structure of the systems under consideration. Failure to do this often leads to confusion for a student who must relearn or radically alter a previously learned mathematical principle. Such a situation should be avoided.

Mathematics Developed

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Real Number ◽  
Number System ◽  
Scientific Practices ◽  
Natural Numbers ◽  
Real Numbers ◽  
Sir Isaac Newton ◽  
The Real ◽  
Isaac Newton ◽  
Number Systems ◽  
Real Number System

This chapter considers two crucial shifts in mathematical knowledge: the natural numbers ℕ and the real number system ℝ. ℝ has proved to serve together with the natural numbers ℕ as one of the two core structures of mathematics; together they are what Solomon Feferman described as “the sine qua non of our subject, both pure and applied.” Indeed, nobody can claim to have a basic grasp of mathematics without mastery of the central elements in the theory of both number systems. The chapter examines related theories and conceptions about real numbers, with particular emphasis on the work of J. H. Lambert and Sir Isaac Newton. It also discusses various conceptions of the number continuum, assumptions about simple infinity and arbitrary infinity, and the development of mathematics in relation to the real numbers. Finally, it reflects on the link between mathematical hypotheses and scientific practices.

Zero, the troublemaker

1969 ◽  
Vol 16 (5) ◽  
pp. 365-367
Author(s):  
Boyd Henry
Keyword(s):  
Elementary School ◽  
School Teacher ◽  
Number System ◽  
Elementary School Teacher ◽  
Fifth Grade ◽  
Teaching Career ◽  
Know How ◽  
Grade Teacher

Every elementary school teacher will agree that the concept of zero is difficult for mimy children to grasp. In fact, many teachers themselves are uncomfortable when they must work with numbers involving zero. One veteran fifth-grade teacher was observed drilling her students to repeat that 6 × 0 = 0, but that 0 × 6 = 6. Apparently she didn't really believe the commutative law which she had previously “taught” her students. Moreover, the role of zero in the structure of the number system was at best vague and certainly confusing to her. We will never know how many hundreds of students she had confused about zero over the long years of her teaching career. The fault does not lie entirely with the teacher alone, of course. At least one textbook she had used in years past made quite a point of stating that zero is not a number. Such mjsinformation as this could only confuse both teacher and students. After all, she might reason, if zero is not a number, it is not obliged to follow the laws of numbers such as the commutative Jaw for multiplication.

Nonstandard natural number systems and nonstandard models

1981 ◽  
Vol 46 (2) ◽  
pp. 365-376 ◽  
Author(s):  
Shizuo Kamo
Keyword(s):  
Natural Number ◽  
Ordered Set ◽  
Uniform Space ◽  
Additive Group ◽  
Number System ◽  
Order Type ◽  
The Other ◽  
Number Systems ◽  
Nonstandard Models ◽  
Real Number System

AbstractIt is known (see [1, 3.1.5]) that the order type of the nonstandard natural number system *N has the form ω + (ω* + ω) θ, where θ is a dense order type without first or last element and ω is the order type of N. Concerning this, Zakon [2] examined *N more closely and investigated the nonstandard real number system *R, as an ordered set, as an additive group and as a uniform space. He raised five questions which remained unsolved. These questions are concerned with the cofinality and coinitiality of θ (which depend on the underlying nonstandard universe *U). In this paper, we shall treat nonstandard models where the cofinality and coinitiality of θ coincide with some appropriated cardinals. Using these nonstandard models, we shall give answers to three of these questions and partial answers to the other to questions in [2].

scholarly journals Probing the neural basis rational numbers: the role of inhibitory control and magnitude representations

2021 ◽  
Author(s):  
Miriam Rosenberg-Lee
Keyword(s):  
Inhibitory Control ◽  
Rational Number ◽  
Mathematics Curriculum ◽  
Parietal Lobe ◽  
Rational Numbers ◽  
Number Processing ◽  
Neural Basis ◽  
Good Correspondence ◽  
Whole Number

Rational numbers, such as fractions, decimals and percentages, are a persistent challenge in the mathematics curriculum. An underappreciated source of rational number difficulties are whole number properties that apply to some, but not all, rational numbers. I contend that mastery of rational numbers involves refining and expanding whole number representations. Behavioral evidence for the role inhibitory control and magnitude-based processing of rational numbers support this hypothesis, although more attention is needed to task and stimuli selection, especially among fractions. In the brain, there is scant evidence on the role of inhibitory control in rational number processing, but surprisingly good correspondence, in the parietal lobe, between the handful of neuroimaging studies of rational numbers and the accumulated whole number literature. Decimals and discrete nonsymbolic representations are fruitful domains for probing the neural basis role of whole number interference in rational number processing.

Study on Block Segmenting Method of House Number

2013 ◽  
Vol 718-720 ◽  
pp. 2522-2527
Author(s):  
Huan Ju Yu ◽  
Yun Ling Li ◽  
Qing Wen Qi
Keyword(s):  
Number System ◽  
City Management ◽  
Digital City ◽  
Urbanization Process ◽  
Advantages And Disadvantages ◽  
Number Systems ◽  
Information Construction ◽  
City Construction ◽  
House Number

House number is one important city geocoding method, which has the most fundamental positioning role and also is the main information carrier in digital city construction. Because traditional house number system has faultiness and city spatial layout has changed a lot with urbanization process in china, house numbers are chaotic in many cities. It is very inconvenient for city management and information construction. It is urgent to reconstruct scientific house number system. First, the authors introduce the advantages and disadvantages of main domestic and foreign house number systems, analyzed the spatial characters of house number, and compared the merits and demerits of block geocoding and street geocoding method. Then, they point out that street is the appropriate spatial framework for house number system of our country. Next they analyzed the spatial localization characters of the house number geocoding within street framework, and pointed out the shortages of traditional serial number and distance number. Then, they put forward and elaborate the block segmenting method of house number. At last, on the assumption that each block has 100 numbers, the figures of block segmenting are displayed. The block segmenting method can ensure the robustness of house number system. It further strengthens the order and localization role of house number and realizes binding house numbers to street block segment. House numbers will not become chaos with the increases and decreases or changing hands of the buildings in the system.

In My Opinion: Fractions in the Early-Years Curriculum: More Needed, Not Less

2003 ◽  
Vol 10 (1) ◽  
pp. 6-7
Author(s):  
Carol A. Powell ◽  
Robert P. Hunting
Keyword(s):  
Young Children ◽  
Curriculum Development ◽  
Rational Number ◽  
Mathematics Curriculum ◽  
Number System ◽  
Early Years ◽  
Instructional Materials ◽  
Primary Mathematics ◽  
Whole Number ◽  
Multiplicative Structures

Watanabe (2001) has argued that the teaching of fractions should be eliminated from the primary mathematics curriculum, based on issues related to curriculum, development, and instructional materials. We disagree, for the following main reasons: First, this approach overlooks young children's developing multiplicative structures, which have their roots in part-whole relationships. Second, although we agree that the teaching of formal symbolism and notation for fractions can be delayed, conversations between teachers and children can establish important ideas from which formal symbols later will flow naturally. Third, sharing situations can help young children develop whole-number knowledge and can lay foundations for the rational-number system.

Working with Text-Reasoning in Third Grade

2021 ◽  
pp. 26-30
Author(s):  
E. Korochkina
Keyword(s):  
Elementary School ◽  
Teaching Methods ◽  
School Teacher ◽  
Practical Experience ◽  
Third Graders ◽  
Elementary School Teacher ◽  
Russian Language ◽  
Different Types ◽  
Independent Work

The article reveals the practical experience of an elementary school teacher in shaping knowledge of different types of speech (types of text) among third-graders: text-description; narration text; text-reasoning. An example of organizing a Russian language lesson to familiarize with the text-reasoning is given. The role of such teaching methods as observing the characteristics of texts of different types, conducting an educational dialogue, and independent work on creating texts of different types is emphasized.

Multiplication by 10 base-5: Making Sense of Place Value Structure Through an Alternate Base

2015 ◽  
Vol 3 (2) ◽  
pp. 83-98
Author(s):  
Jodi Fasteen ◽  
Kathleen Melhuish ◽  
Eva Thanheiser
Keyword(s):  
Preservice Teachers ◽  
Conceptual Understanding ◽  
Sense Of Place ◽  
Number System ◽  
Place Value ◽  
Mathematical Activity ◽  
Making Sense ◽  
Procedural Fluency ◽  
Whole Number

Prior research has shown that preservice teachers (PSTs) are able to demonstrate procedural fluency with whole number rules and operations, but struggle to explain why these procedures work. Alternate bases provide a context for building conceptual understanding for overly routine rules. In this study, we analyze how PSTs are able to make sense of multiplication by 10five in base five. PSTs' mathematical activity shifted from a procedurally based concatenated digits approach to an explanation based on the structure of the place value number system.

scholarly journals O Papel do Apoiador Escolar na Educação Inclusiva / The Importance of the School Supporter in Inclusive Education

2021 ◽  
Vol 15 (58) ◽  
pp. 592-600
Author(s):  
Sara Mariza dos Santos ◽  
Kennya De Lima Almeida
Keyword(s):  
Inclusive Education ◽  
School Environment ◽  
School Teacher ◽  
Research Work ◽  
Public And Private ◽  
Educational Networks ◽  
Educational Inclusion ◽  
Random Groups ◽  
The City

Resumo:  A educação inclusiva é conhecida como uma forma de trabalhar com crianças com necessidades especiais no ambiente escolar. Vista de forma mais ampla, ela tem o papel de acolher a diversidade e dar assistência a todos os estudantes, pois o objetivo da inclusão educacional é acabar com a exclusão social. O trabalho de pesquisa tem como objetivo avaliar as dificuldades encontradas pelos professores apoiadores das salas de aula, saber qual o suporte e formação que recebe para atuar. Além disso, a pesquisa possibilita compreender a realidade da inclusão a partir de redes de ensino diferentes, a pública e a privada. A metodologia aplicada incluiu dados da observação da sala de aula no intuito de narrar e analisar o cotidiano do “professor apoiador escolar”. O trabalho foi realizado na Cidade de Salgueiro/PE, e em Umãs/PE, com apoiadores escolares de três escolas, os participantes foram apoiadores escolhidos em turmas aleatórias, em um total de 10 apoiadores de sala de aula.---Inclusive education is known as a way of working with children with special needs in the school environment. Viewed more broadly, it has the role of welcoming diversity and providing assistance to all students, as the objective of educational inclusion is to end social exclusion. The research work aims to assess the difficulties encountered by supportive teachers in the classroom, to know what support and training they receive to act. In addition, the research makes it possible to understand the reality of inclusion from different educational networks, public and private. The methodology applied included data from classroom observation in order to narrate and analyze the daily life of the “supporting school teacher”. The work was carried out in the City of Salgueiro/PE, and in Umãs/PE, with school supporters from three schools, the participants were supporters chosen in random groups, in a total of 10 classroom supporters.

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