Top principles in teaching mathematics

Relying upon math experts and enthusiasts to define the subject is likely to lead to a variety of conflicting and wide-ranging answers.

Top principles in teaching mathematics

See Article History Analysis, a branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiationand integration. Since the discovery of the differential and integral calculus by Isaac Newton and Gottfried Wilhelm Leibniz at the end of the 17th century, analysis has grown into an enormous and central field of mathematical research, with applications throughout the sciences and in areas such as finance, economics, and sociology.

The historical origins Top principles in teaching mathematics analysis can be found in attempts to calculate spatial quantities such as the length of a curved line or the area enclosed by a curve. These problems can be stated purely as questions of mathematical technique, but they have a far wider importance because they possess a broad variety of interpretations in the physical world.

The area inside a curve, for instance, is of direct interest in land measurement: But the same technique also determines the mass of a uniform sheet of material bounded by some chosen curve or the quantity of paint needed to cover an irregularly shaped surface.

Less obviously, these techniques can be used to find the total distance traveled by a vehicle moving at varying speeds, the depth at which a ship will float when placed in the sea, or the total fuel consumption of a rocket. Similarly, the mathematical technique for finding a tangent line to a curve at a given point can also be used to calculate the steepness of a curved hill or the angle through which a moving boat must turn to avoid a collision.

Less directly, it is related to the extremely important question of the calculation of instantaneous velocity or other instantaneous rates of change, such as the cooling of a warm object in a cold room or the propagation of a disease organism through a human population.

This article begins with a brief introduction to the historical background of analysis and to basic concepts such as number systems, functions, continuityinfinite series, and limits, all of which are necessary for an understanding of analysis.

Following this introduction is a full technical review, from calculus to nonstandard analysis, and then the article concludes with a complete history.

Top principles in teaching mathematics

Historical background Bridging the gap between arithmetic and geometry Mathematics divides phenomena into two broad classes, discrete and continuoushistorically corresponding to the division between arithmetic and geometry.

Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3, …. Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.

Understanding the true nature of such infinite decimals lies at the heart of analysis. The distinction between discrete mathematics and continuous mathematics is a central issue for mathematical modeling, the art of representing features of the natural world in mathematical form.

The universe does not contain or consist of actual mathematical objects, but many aspects of the universe closely resemble mathematical concepts. For example, the number two does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars.

In a similar manner, the real numbers provide satisfactory models for a variety of phenomena, even though no physical quantity can be measured accurately to more than a dozen or so decimal places.

It is not the values of infinitely many decimal places that apply to the real world but the deductive structures that they embody and enable. Analysis came into being because many aspects of the natural world can profitably be considered as being continuous—at least, to an excellent degree of approximation.

Again, this is a question of modeling, not of reality. Matter is not truly continuous; if matter is subdivided into sufficiently small pieces, then indivisible components, or atoms, will appear. But atoms are extremely small, and, for most applications, treating matter as though it were a continuum introduces negligible error while greatly simplifying the computations.

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For example, continuum modeling is standard engineering practice when studying the flow of fluids such as air or water, the bending of elastic materials, the distribution or flow of electric currentand the flow of heat. Discovery of the calculus and the search for foundations Two major steps led to the creation of analysis.

The first was the discovery of the surprising relationship, known as the fundamental theorem of calculusbetween spatial problems involving the calculation of some total size or value, such as length, area, or volume integrationand problems involving rates of change, such as slopes of tangents and velocities differentiation.

Credit for the independent discovery, aboutof the fundamental theorem of calculus together with the invention of techniques to apply this theorem goes jointly to Gottfried Wilhelm Leibniz and Isaac Newton. While the utility of calculus in explaining physical phenomena was immediately apparent, its use of infinity in calculations through the decomposition of curves, geometric bodies, and physical motions into infinitely many small parts generated widespread unease.

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In particular, the Anglican bishop George Berkeley published a famous pamphlet, The Analyst; or, A Discourse Addressed to an Infidel Mathematicianpointing out that calculus—at least, as presented by Newton and Leibniz—possessed serious logical flaws. Analysis grew out of the resulting painstakingly close examination of previously loosely defined concepts such as function and limit.

Although these developments were not entirely satisfactory from a foundational standpoint, they were fundamental to the eventual refinement of a rigorous basis for calculus by the Frenchman Augustin-Louis Cauchythe Bohemian Bernhard Bolzanoand above all the German Karl Weierstrass in the 19th century.rutadeltambor.com: Principles and Standards for School Mathematics: An Overview (): National Council of Teachers of Mathematics, NCTM: Books.

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Apply today. Analysis: Analysis, a branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration.

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Top principles in teaching mathematics

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