Mathematics

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Mathematics education,teacher education,Great mathematicians

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  Mathematics - The abstract science of number, quantity, and space Definitions- Aristotle  defined mathematics as : The science of  quantity The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. A group of related sciences, including algebra, geometry, and calculus, concerned with the study of number, quantity, shape, and space and their interrelationships by using a specialized notation The study of the measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are branches of mathematics.

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Mathematics  is the study of  quantity ,  structure ,  space , and  change .  Mathematicians  seek out patterns , formulate new  conjectures , and establish truth by  rigorous   deduction  from appropriately chosen  axioms  and  definitions

Mathematics:

Mathematics Through the use of  abstraction  and  logical   reasoning , mathematics evolved from  counting , calculation ,  measurement , and the systematic study of the  shapes  and  motions  of physical objects. Mathematics arises from many different kinds of problems. At first these were found in  commerce , land measurement ,  architecture  and later  astronomy ; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself.

"Let none ignorant of geometry enter here." :

"Let none ignorant of geometry enter here."  Plato  (427-347 BC), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, Academy "Let none ignorant of geometry enter here." Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked  ruler  or a  protractor , because these were a workman’s tools, not worthy of a scholar. Aristotle  (384-322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see  Logic ) which was not substantially improved upon until the 19th century.

In contemporary education, mathematics education is the practice of teaching and learning mathematics, along with the associated scholarly research.:

In contemporary education,  mathematics education  is the practice of  teaching  and  learning   mathematics , along with the associated scholarly  research . In  mathematics education , the term word problem is often used to refer to any mathematical exercise where significant background information on the problem is presented as text rather than in  mathematical notation . As word problems often involve a  narrative  of some sort, they are occasionally also referred to as  story problems  and may vary in the amount of language used

Geometry :

Geometry   Geometry  ( Greek   γεωμετρία ; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre- modern mathematics , the other being the study of numbers. Classic geometry was focused in  compass and straightedge constructions . Geometry was revolutionized by  Euclid , who introduced  mathematical rigor  and the  axiomatic method  still in use today. His book,  The Elements  is widely considered the most influential textbook of all time

Objectives :

Objectives At different times and in different cultures and countries, mathematics education has attempted to achieve a variety of different objectives. These objectives have included: The teaching of basic  numeracy  skills to all pupils The teaching of practical mathematics ( arithmetic ,  elementary algebra , plane and solid  geometry ,  trigonometry ) to most pupils, to equip them to follow a trade or craft The teaching of abstract mathematical concepts (such as  set  and  function ) at an early age The teaching of selected areas of mathematics (such as  Euclidean geometry ) as an example of an  axiomatic system  and a model of deductive reasoning The teaching of selected areas of mathematics (such as  calculus ) as an example of the intellectual achievements of the  modern world The teaching of advanced mathematics to those pupils who wish to follow a career in  Science, Technology, Engineering, and Mathematics (STEM)  fields. The teaching of  heuristics  and other problem-solving strategies to solve non-routine problems.

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Egyptian geometry   Egyptian mathematics The ancient Egyptians knew that they could approximate the area of a circle as follows : Area of Circle ≈ [ (Diameter) x 8/9] 2

Thales and Pythagoras :

Thales and Pythagoras Thales  (635-543 BC) of  Miletus  (now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived.  Pythagoras  (582-496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to  Babylon  and  Egypt . The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. 

Euclid:

Euclid Euclid's most famous work is his treatise on mathematics  The Elements . The book was a compilation of knowledge that became the centre of mathematical teaching for 2000 years  Euclid  (c. 325-265 BC), of  Alexandria , probably a student of one of Plato’s students, wrote a treatise in 13 chapters, titled  The Elements of Geometry , in which he presented geometry in an ideal  axiomatic  form, which came to be known as  Euclidean geometry . Euclid himself wrote eight more advanced books on geometry

The Elements is divided into 13 books. Books one to six deal with plane geometry.:

The  Elements  is divided into 13 books. Books one to six deal with plane geometry. In particular Books 1 & 2 set out basic properties of triangles, parallels, parallelograms, rectangles and squares. Book three studies properties of the circle while book four deals with problems about circles  Book 6 looks at applications of the results of Book 5 to plane geometry. Books 7 to 9 deal with number theory. Book 10 deals with the theory of irrational numbers Books 11 to 13 deal with 3-dimensional geometry.

THE fundamental geometric principles ARE called axioms or postulates:

THE fundamental geometric principles ARE called  axioms  or postulates Euclid’s axioms / Euclid’s common notions 01 Given two points there is one straight line that joins them. 02 A straight line segment can be prolonged indefinitely. 03 A circle can be constructed when a point for its centre and a distance for its radius are given. 04 All right angles are equal. 05 If a straight line (Transversal) falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles. 06 Things equal to the same thing are equal. 07 If equals are added to equals, the wholes are equal. 08 If equals are subtracted from equals, the remainders are equal. 09 Things that coincide with one another are equal. 10 The whole is greater than a part.

Archimedes:

Archimedes Archimedes  (287-212 BC), of  Syracuse ,  Sicily , when it was a  Greek city-state , is often considered to be the greatest of the Greek mathematicians, remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus.

Famous Quote: "Eureka“ Apparently when taking a bath, he discovered the buoyancy principle and jumped up and ran through the streets naked shouting 'Eureka' which means – I have found it. :

Famous Quote: "Eureka“ Apparently when taking a bath, he discovered the buoyancy principle and jumped up and ran through the streets naked shouting 'Eureka' which means – I have found it. Contributions: Discovered how to find the volume of a sphere and determined the exact value of  Pi . Principle of Buoyancy. It is believed that he was actually the first to have invented integral calculus , 2000 years before Newton and Leibniz. Powers of Ten , a way of counting that refers to the number of 0's in a number which eliminated the use of the Greek alphabet in the counting system. ( Scientific Notation ) A formula to find the area under a curve, the amount of space that is enclosed by a curve.

Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi. He also defined the spiral bearing his name, formulae for the volumes of solids of revolution, and an ingenious system for expressing very large numbers.  The Syracusia is said to have been the largest ship built in classical antiquity.[23] According to Athenaeus, it was capable of carrying 600 people and included garden decorations, a gymnasium and a temple dedicated to the goddess Aphrodite among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the Archimedes screw was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw described in Roman times by Vitruvius may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon:

Archimedes is generally considered to be the greatest  mathematician  of antiquity and one of the greatest of all time. He used the  method of exhaustion  to calculate the  area  under the arc of a  parabola  with the  summation of an infinite series , and gave a remarkably accurate approximation of  pi . He also defined the  spiral  bearing his name, formulae for the  volumes  of  solids of revolution , and an ingenious system for expressing very large numbers.  The  Syracusia  is said to have been the largest ship built in classical antiquity. [23]  According to Athenaeus , it was capable of carrying 600 people and included garden decorations, a  gymnasium  and a temple dedicated to the goddess  Aphrodite  among its facilities. Since a ship of this size would leak a considerable amount of water through the hull, the  Archimedes screw  was purportedly developed in order to remove the bilge water. Archimedes' machine was a device with a revolving screw-shaped blade inside a cylinder. It was turned by hand, and could also be used to transfer water from a low-lying body of water into irrigation canals. The Archimedes screw is still in use today for pumping liquids and granulated solids such as coal and grain. The Archimedes screw described in Roman times by  Vitruvius  may have been an improvement on a screw pump that was used to irrigate the Hanging Gardens of Babylon

While Archimedes did not invent the lever, he gave an explanation of the principle involved in his work On the Equilibrium of Planes. Earlier descriptions of the lever are found in the Peripatetic school of the followers of Aristotle, and are sometimes attributed to Archytas. According to Pappus of Alexandria, Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth.":

While Archimedes did not invent the  lever , he gave an explanation of the principle involved in his work  On the Equilibrium of Planes . Earlier descriptions of the lever are found in the  Peripatetic school  of the followers of  Aristotle , and are sometimes attributed to  Archytas . According to  Pappus of Alexandria , Archimedes' work on levers caused him to remark: "Give me a place to stand on, and I will move the Earth."

The last words attributed to Archimedes are "Do not disturb my circles”. Archimedes died c. 212 BC during the Second Punic War, when Roman forces under General Marcus Claudius Marcellus captured the city of Syracuse after a two-year-long siege. According to the popular account given by Plutarch, Archimedes was contemplating a mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword. :

The last words attributed to Archimedes are "Do not disturb my circles”. Archimedes died  c . 212 BC during the  Second Punic War , when Roman forces under General  Marcus Claudius Marcellus  captured the city of Syracuse after a two-year-long  siege . According to the popular account given by  Plutarch , Archimedes was contemplating a  mathematical diagram when the city was captured. A Roman soldier commanded him to come and meet General Marcellus but he declined, saying that he had to finish working on the problem. The soldier was enraged by this, and killed Archimedes with his sword.

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a sphere and a cylinder of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.:

The tomb of Archimedes carried a sculpture illustrating his favorite mathematical proof, consisting of a  sphere  and a  cylinder  of the same height and diameter. Archimedes had proven that the volume and surface area of the sphere are two thirds that of the cylinder including its bases. A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A  sphere  and  cylinder  were placed on the tomb of Archimedes at his request.

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