Sunday, June 28, 2009

What Is Mathematics?

Posted by MJSC Pasir Salak




Mathematics is the study of quantity, structure, space, relation, change, and various topics of pattern, form and entity. Mathematicians seek out patterns and other quantitative aspects of the entities they study, whether these entities are numbers, spaces, natural sciences, computers, or abstract concepts.Mathematicians formulate new conjectures and establish truth by rigorous deduction from appropriately chosen axioms and definitions.



There is debate over whether mathematical objects exist objectively by nature of their logical purity, or whether they are manmade and detached from reality. The mathematician Benjamin Peirce called mathematics "the science that draws necessary conclusions. Albert Einstein, on the other hand, stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.



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. Knowledge and use of basic mathematics have always been an inherent and integral part of individual and group life. Refinements of the basic ideas are visible in mathematical texts originating in the ancient Egyptian, Mesopotamian, Indian, Chinese, Greek and Islamic worlds. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. The development continued in fitful bursts until the Renaissance period of the 16th century, when mathematical innovations interacted with new scientific discoveries, leading to an acceleration in research that continues to the present day.


Today, mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences such as economics and psychology. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new disciplines. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered later.

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Creators Of Mathematics?

Posted by MJSC Pasir Salak

Pythagoras, Euler, and G.H. Hardy,

If you’re interested in math or if you paid attention in school, chances are you’ve heard of these famous names and may be familiar with their work and accomplishments. But how about Agnesi, Cartwright or Goldwasser? These three names are amongst ten that we’ve selected as a means of highlighting some of the best and brightest female mathematicians of all time; women who unlike their male counterparts, have not always received the same level of recognition even though their achievements and contributions to the world of mathematics are just as important. These women were often groundbreakers, highly determined and very dedicated. They are shining examples of the fact that mathematics is not a “boys only” club, even if at many points in time it’s appeared that way on the surface. Today their work is recognized and appreciated, and they stand as fantastic sources of inspiration for a new generation of students and math enthusiasts – both female and male.

Hypatia of Alexandria (AD 350 to 370 – 415)

Born nearly 17 centuries ago, Hypatia of Alexandria was a brazen, highly intelligent woman who excelled in the fields of science, math and philosophy, which at the time (and for hundreds upon hundreds of years further) were seen squarely as the domain of men. Hypatia’s foremost teacher was her father, Theon Alexandricus, a mathematician and philosopher, who she would later go on to contribute to several mathematical works with. Hypatia herself was a teacher, as well as being the inventor of the hydrometer. Though she forged ahead in a time when women were all but ignored in the realm of mathematics, this bright Greek woman eventually met with a tragic death when her chariot was attacked and she was brutally murdered by a gang of Christians. Though her life was cut short, while she was alive, through her accomplishments, Hypatia was able lay the groundwork for future female pioneers f mathematics.


Gabrielle Émilie Le Tonnelier de Breteuil, marquise du Châtelet (December 17, 1706 – September 10, 1749):

A woman of many intellectual interests, Émilie was a mathematician, author, and physicist who hailed from France. Born into a well-to-do family, Châtelet was a gifted child with a natural penchant for linguistics. Given her family’s high social status, Émilie was able to receive a degree of education far above the vast majority of French women at the time. Her place in society also put her in a position wherein she was able to mingle with some of the leading minds of her time (such as Voltarie, who would go onto become one of her lovers). In 1740, Châtelet published a book entitled Institutions de Physique, which put forth some of her knowledge regarding both science and philosophy. In her last year of life, Émilie translated Newton’s well-known Principia Mathematica. In her early forties she became pregnant, and though she initially survived the pregnancy, a few days later both she and her newborn child passed away. Émilie was an independent, articulate and highly intelligent woman, who was somehow able to hold down both her role as a leading lady in French high society and as a mathematician, an equation which deserves respect in its own right.


Maria Gaetana Agnesi (May 16, 1718 – January 9, 1799):

A woman of many skills, Agnesi was an Italian mathematician, linguist, and philosopher whose profound intelligence was evident from an early age. Born into a wealthy and large family (due in part to siblings which sprang from her father’s two subsequent marriages after Maria’s mother passed away), Agnesi was a devoted and studious woman who would go onto publish the first book that dealt with both integral and differential calculus. In 1750, Maria was appointed as chair of mathematics and natural philosophy at the Bologna Academy of Sciences, an incredible accomplishment for any woman in the mid eighteenth century, when exceptionally few universities in Europe allowed women to study, let alone hold teaching positions. Later in life, Agnesi, a deeply religious woman, joined a nunnery and ended her days tending to the less fortunate




Marie-Sophie Germain (April 1, 1776 – June 27, 1831):

Parisian born Germain was a passionate mathematician with a love of number theory and differential geometry. During her lifetime (which, in the context of both France and Europe in general, was a highly tumultuous era) Germain often corresponded under a pseudonym (Monsieur Le Blanc) as a means of hiding her gender when writing to leading male mathematicians of the time such as Lagrange and Gauss. In 1816 Sophie won a contest that was held by the French Academy of Science which dealt with the area of vibrations on elastic surfaces, that in turn lead her to become the first woman (short of some of the staffs’ wives) to attend classes at the Academy. In 1831, the University of Gottengen bestowed an honorary degree to Germain, however she died as a result of breast cancer before she was able to receive the degree. A self-taught mathematician who came of age during a truly unstable period in French history, Sophie will long be remembered for her mathematical contributions in the field of number theory.


Augusta Ada Byron King, Countess of Lovelace (December 10, 1815 – November 27, 1852):

English born Ada was the daughter of famed poet Lord Byron, though he was not active in his daughter’s life. Aside from her famous father, Ada is primarily known for her programming work regarding Charles Babbage’s invention of the analytical engine, a very early mechanical general-purpose computer. Lovelace was ahead of her time in this field, as she believed that computers held the capacity to do more than just simply act as calculators. Like many of the women in this list, Ada met with an early death; she was only 36 when she died due to uterine cancer. Today Lovelace is remembered fondly as the first female computer programmer (in era before the modern computer came into existence), and the programming language Ada was named in her honor.


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Mathematics Curiculum In USA and England

Posted by MJSC Pasir Salak

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
  • The teaching of heuristics and other problem-solving strategies to solve non routine problems.
    Methods of teaching mathematics have varied in line with changing objectives.
Standards


Throughout most of history, standards for mathematics education were set locally, by individual schools or teachers, depending on the levels of achievement that were relevant to and realistic for their pupils.
In modern times there has been a move towards regional or national standards, usually under the umbrella of a wider standard school curriculum. In England, for example, standards for mathematics education are set as part of the National Curriculum for England, while Scotland maintains its own educational system.
Ma (2000) summarized the research of others who found, based on nationwide data, that students with higher scores on standardized math tests had taken more mathematics courses in high school. This led some states to require three years of math instead of two. But because this requirement was often met by taking another lower level math course, the additional courses had a “diluted” effect in raising achievement levels. [2]
In North America, the National Council of Teachers of Mathematics (NCTM) has published the Principles and Standards for School Mathematics. In 2006, they released the Curriculum Focal Points, which recommend the most important mathematical topics for each grade level through grade 8. However, these standards are not nationally enforced in US schools.

Content and age levels

Different levels of mathematics are taught at different ages. Sometimes a class may be taught at an earlier age as a special or "honors" class. A rough guide to the ages at which the certain topics of arithmetic are taught in the United States is as follows:

  • Addition: ages 5-7; more digits ages 8-9
  • Subtraction: ages 5-7; more digits ages 8-9
  • Multiplication: ages 7-8; more digits ages 9-10
  • Division: age 8; more digits ages 9-10

The ages at which other math subjects (rational numbers, geometry, measurement, problem solving, logic, algebraic thinking, probability, statistics, reasoning skills and so on) are taught vary considerably from state to state.
Elementary mathematics in other countries is similar, though fractions (typically taught from 1st grade in the United States) are often taught later, since the metric system does not require young children to be familiar with them. Most countries tend to cover fewer topics in greater depth than in the United States.[3]

A typical pre-college sequence of mathematics courses in the United States would include some of the following, especially Geometry and Algebra I and II:

  • Pre-algebra: ages 11-13 (Pre-Algebra taught in schools as early as 6th grade as an honor course. Algebraic reasoning can be taught in elementary school, though)
  • Algebra I (basic algebra): ages 12+ (Algebra I is taught at 9th grade on average, or as early as 7th or 8th grade for an honors course)
  • Geometry: ages 13+ (Geometry taught at 10th grade on average, or as early as 8th grade as an honors course)
  • Algebra II: ages 14+; usually includes powers and roots, polynomials, quadratic functions, coordinate geometry, exponential and logarithmic functions, probability, matrices, and basic trigonometry
  • Trigonometry or Algebra 3 or Pre-Calculus: ages 15+
    Statistics: ages 15+ (Probability and statistics topics are taught throughout the curriculum from early elementary grades, but may form a special course in high school.)
  • Calculus: ages 16+ (usually seen in 12th grade, if at all; some honors students may see it earlier)
Mathematics in most other countries and in a few US states is integrated, with topics of algebra, geometry and analysis (pre-calculus and calculus) studied every year. Students in many countries choose an option or pre-defined course of study rather than choosing courses à la carte as in North America. Students in science-oriented curricula typically study differential calculus and trigonometry at age 16-17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions and infinite series their final year of high school

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