Posted on June 13, by Patrick A. Trottier The concepts and ideas in the following write-up have been developed from a number of prominent and respected people in the field of OD, especially: The Four Basic Consultative Models: In the following write-up, we first offer a simple description of the first two consultative models the expert and the medical approachand go into more detail of the next two models the process and the emergent.
Phenomenology … in a nutshell The development of mathematics is intimately interwoven with the progress of civilization, influencing the course of history through its application to science and technology.
But mathematics has changed.
Even the mathematics of the s can seem quite strange now, so greatly has mathematics evolved in the past years and so thoroughly has it been reworked in the post-modern approach. Despite its arcane appearance from the outside looking in, the present, abstract and highly specialized state of mathematics is the natural evolution of the subject, and there is much ahead that is exciting.
Here, then, is the story of mathematics, in a nutshell… The Development of Mathematics, in a Nutshell Though mathematical knowledge is ancient, stretching back to the Stone Agethe evolution of mathematics to its current modern state has seen fundamental changes in concepts, organization, scope, outlook, and practice.
Without understanding the evolution of mathematical thought, it is difficult to appreciate modern mathematics in its contemporary, highly specialized state. Seven Periods of Mathematical Practice Roughly speaking, I would identify seven periods in the evolution of mathematics, each with distinct characteristics.
Proto-Mathematics from the mists of ancient time, through the archeological evidence of c. Proto-Mathematics The essence of mathematics, call it proto-mathematics, exists in empirical observations and interactions with the environment.
Even the earliest man had need of basic mathematical understanding: Archaeological evidence for basic mathematical understanding e.
This meant greater food with less work per capita, the impetus for greater specialization craftsthe growth of communities, the development of classes and heirarchies warrior, farmerthe growth of administration, and greater leisure.
Writing allowed man to transmit his knowledge, to teach, and learn, and preserve what he had learned from generation to generation. Ancient Mathematics From empirical mathematics arose, through abstraction, the sciences of arithmetic number and geometry figure.
These were developed into an extremely sophisticated science by the Babylonians and the Egyptians, and reached spectacular heights during their respective civilizations, applied to astronomy, the regulation of time, administration, planning and logistics, land surveying, calculation of areas and volumes, construction, and the engineering of incredible monuments.
Each may have been viewed abstractly, and reasoning given, but the formal structure as a whole was absent. The knowledge and facility of Babylonian and Egyptian mathematics was quite sophisticated: Early Classical Mathematics The Greeks introduced to mathematics a fundamental abstraction: The view of mathematics was of a formal structure as a whole, held together by the laws of thought, with results organized into a linear body of work, each proved in terms of statements already accepted or proved, with the full understanding of the need for first principles, or axioms.
The science of Geometry flourished under the Greeks, including applications to mechanics, machines, astronomy, and engineering, both Greek and Roman. Many challenging problems in curvilinear and solid geometry were obtained through methods of the Calculus: The Encountering of Paradox In the development of arithmetic and the number concept, the Greeks discovered early on the inadequacy of the common notion of number rational number to describe lengths.
Indeed, a simple length, the diagonal of a square, eluded their common notion of number. This was the beginning of the discovery of paradoxes in the theory of mathematics. The fact that the diagonal and side of a square are logically incommensurable is not a problem of reality; it is a problem with the logical theory that had been developed: And here is this theory: And this theory blends arithmetic with geometry, number with measure.
But the theory now, irrefutably, has a problem. These lengths are incommensurable. There is no rational number that can measure that length, no matter how small the scale of measurement is! This blew a fuse in the ancient Greek world and led to all kinds of intellectual searching to try to find the flaw, the problem.
The key point to keep in mind, is that the problem is with construction of the mathematical theory. It is NOT an issue with the world, or with progress, or with science, or with engineering. In the real world, diagonals can be measured, no problem.
In fact, all lengths can be measured up to the precision of the measuring instrument being used. Which means that all measurements are rational, and there is no practical difficulty.
Given the complexities of the concept of number, trouble in attempting to expand it to cover all measurement existence of irrationals, etc. Numbers were regarded as useful, but with suspicion and not always reliable. This way of thinking led to geometry being supreme to the Greeks.
But now a separation had clearly occurred between concrete and abstract mathematics, between practical science and engineering, and theoretical mathematics. The Resolution of the Paradox of Number The remedy for the problem of numbers is its expansion to include all Cauchy sequences of rational numbers, since these would be convergent so long as the point of their convergence existed.
In this way can be filled infor all primesand, in principle, all numbers that can be approximated to indefinite precision i. This new and much larger domain of numbers is no longer a countable infinity but an uncountable infinity of numbers, as shown by Cantor. Late Classical Mathematics Algebra, the science of equations, was already well developed in Babylonian and Egyptian times.Solver.
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Insert a math equation into a PowerPoint for Mac presentation. Write or change an equation. On the Insert tab, in the Symbols group, click Equation. A placeholder for your new equation is inserted, and you’ll see a set of tools in the ribbon for customizing and adding various elements to your equation. High school chemistry for teachers and students - Entire 11th grade course - Including labs, worksheets, handouts, notes, and PowerPoint lessons. Maths teaching resources for Key Stage 3/4 algebra topics.
first: she is not organized. 2nd: she makes a lot of mistakes in solving the equations and students correct her. The Purdue University Online Writing Lab serves writers from around the world and the Purdue University Writing Lab helps writers on Purdue's campus.
The concepts and ideas in the following write-up have been developed from a number of prominent and respected people in the field of OD, especially: Vol. 1 Making Good Looking Equations in PowerPoint Presentations. Starting the equation editor. Making Good Looking Equations in PowerPoint Presentations.
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