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Multiplication And Repeated Addition
In mathematics education, there was a debate on the issue of whether the operation of multiplication should be taught as being a form of repeated addition. Participants in the debate brought up multiple perspectives, including axioms of arithmetic, pedagogy, learning and instructional design, history of mathematics, philosophy of mathematics, and computer-based mathematics. Background of the debate In the early 1990s Leslie Steffe proposed the counting scheme children use to assimilate multiplication into their mathematical knowledge. Jere Confrey contrasted the counting scheme with the splitting conjecture. Confrey suggested that counting and splitting are two separate, independent cognitive primitives. This sparked academic discussions in the form of conference presentations, articles and book chapters. The debate originated with the wider spread of curricula that emphasized scaling, zooming, folding and measuring mathematical tasks in the early years. Such tasks both require and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Education
In contemporary education, mathematics education—known in Europe as the didactics or pedagogy of mathematics—is the practice of teaching, learning, and carrying out Scholarly method, scholarly research into the transfer of mathematical knowledge. Although research into mathematics education is primarily concerned with the tools, methods, and approaches that facilitate practice or the study of practice, it also covers an extensive field of study encompassing a variety of different concepts, theories and methods. List of mathematical societies, National and international organisations regularly hold conferences and List of mathematics education journals, publish literature in order to improve mathematics education. History Ancient Elementary mathematics were a core part of education in many ancient civilisations, including ancient Egypt, Babylonia, ancient Babylonia, ancient Greece, ancient Rome, and Vedic civilization, Vedic Ancient India, India. In most cases, formal edu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soviet Union
The Union of Soviet Socialist Republics. (USSR), commonly known as the Soviet Union, was a List of former transcontinental countries#Since 1700, transcontinental country that spanned much of Eurasia from 1922 until Dissolution of the Soviet Union, it dissolved in 1991. During its existence, it was the list of countries and dependencies by area, largest country by area, extending across Time in Russia, eleven time zones and sharing Geography of the Soviet Union#Borders and neighbors, borders with twelve countries, and the List of countries and dependencies by population, third-most populous country. An overall successor to the Russian Empire, it was nominally organized as a federal union of Republics of the Soviet Union, national republics, the largest and most populous of which was the Russian SFSR. In practice, Government of the Soviet Union, its government and Economy of the Soviet Union, economy were Soviet-type economic planning, highly centralized. As a one-party state go ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance imaging and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix (mathematics)
In mathematics, a matrix (: matrices) is a rectangle, rectangular array or table of numbers, symbol (formal), symbols, or expression (mathematics), expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object. For example, \begin1 & 9 & -13 \\20 & 5 & -6 \end is a matrix with two rows and three columns. This is often referred to as a "two-by-three matrix", a " matrix", or a matrix of dimension . Matrices are commonly used in linear algebra, where they represent linear maps. In geometry, matrices are widely used for specifying and representing geometric transformations (for example rotation (mathematics), rotations) and coordinate changes. In numerical analysis, many computational problems are solved by reducing them to a matrix computation, and this often involves computing with matrices of huge dimensions. Matrices are used in most areas of mathematics and scientific fields, either directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations. The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. Definition Let ''V'' be a vector space of dimension ''n'' over a field ''F'' and let : B = \ be an ordered basis for ''V''. Then for every v \in V there is a unique linear combination of the basis vectors that equals '' v '': : v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Numbers
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathematics), fields, module (mathematics), modules, vector spaces, lattice (order), lattices, and algebra over a field, algebras over a field. The term ''abstract algebra'' was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variable (mathematics), variables to represent numbers in computation and reasoning. The abstract perspective on algebra has become so fundamental to advanced mathematics that it is simply called "algebra", while the term "abstract algebra" is seldom used except in mathematical education, pedagogy. Algebraic structures, with their associated homomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Numbers
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Embodied Cognition
Embodied cognition represents a diverse group of theories which investigate how cognition is shaped by the bodily state and capacities of the organism. These embodied factors include the motor system, the perceptual system, bodily interactions with the environment (situatedness), and the assumptions about the world that shape the functional structure of the brain and body of the organism. Embodied cognition suggests that these elements are essential to a wide spectrum of cognitive functions, such as perception biases, memory recall, comprehension and high-level mental constructs (such as meaning attribution and Categorization, categories) and performance on various cognitive tasks (reasoning or judgment). The embodied mind thesis challenges other theories, such as Cognitivism (psychology), cognitivism, computationalism, and Dualism (philosophy of mind), Cartesian dualism. It is closely related to the extended mind thesis, situated cognition, and enactivism. The modern version dep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vygotsky Circle
The Vygotsky Circle (also known as Vygotsky–Luria CircleYasnitsky, A. & van der Veer, R. (Eds.) (2015)Revisionist Revolution in Vygotsky Studies London and New York: RoutledgeYasnitsky, A., van der Veer, R., Aguilar, E. & García, L.N. (Eds.) (2016) Buenos Aires: Miño y Dávila Editores) was an influential informal network of psychologists, educationalists, medical specialists, physiologists, and neuroscientists, associated with Lev Vygotsky (1896–1934) and Alexander Luria (1902–1977), active in 1920–1940s in the Soviet Union (Moscow, Leningrad and Kharkiv). The work of the Circle contributed to the foundation of the integrative science of mind, brain, and behavior in their cultural and bio-social development also known under somewhat vague and imprecise name of cultural-historical psychology. The Vygotsky Circle, also referred to as "Vygotsky boom" incorporated the ideas of social and interpersonal relations, the practices of empirical scientific research, and "Stalinist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathematics), product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from term (arithmetic), ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication As Scaling Integers
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a '' product''. Multiplication is often denoted by the cross symbol, , by the mid-line dot operator, , by juxtaposition, or, in programming languages, by an asterisk, . The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''; both numbers can be referred to as ''factors''. This is to be distinguished from ''terms'', which are added. :a\times b = \underbrace_ . Whether the first factor is the multiplier or the multiplicand may be ambiguous or depend upon context. For example, the expression 3 \times 4 , can be phrased as "3 times 4" and evaluated as 4+4+4, where 3 is the multiplier, but a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
