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2.1 Aesthetic Sense in Mathematics
The major important changes in mathematics education over the past century. It evolved in response to perceptions as conceived of and practiced by research mathematicians and school mathematics, which is imbodies in textbooks and classrooms.
Meaning of Aesthetic:
‘Aesthetics’ or the ‘Philosophy of Art’ is the study of beauty and test. The term aesthetics is used to designate a particular style.
Example: Japanese aesthetics
- Aesthetic design principles include ornamentation texture, symmetry, color and harmony.
- Working in mathematics gives aesthetic pleasure to the mathematician.
- Desi and experience bunty duty in certain aspects of mathematics an art or creative work.
- An example of beauty in methods of mathematics is an elegant and simple proof of the Pythagorean theorem.
Russell one of the greatest mathematicians and philosopher describes the beauty of mathematics in the following words:
- Mathematics rightly possesses not only truth but supreme beauty.
- The true spirit of delight, the excellence, the sense of being more than man who is touchstone of the highest excellence.
2.1.1 Beauty in Mathematical Methods
Mathematical proofs have beauty on their own we describe a very pleasing method of proof as elegant. An proof has the following characteristics:
- Uses minimum number of assumptions songs for previous results.
- Devices a result from apparently unrelated results.
- Is best on now and original insights.
- is concise and clear.
- Can be adopted in solving problems of similar kind.
in the search for an allergen proof mathematicians of unlock for different independent ways to prove a result the first prove that is found may not be the best the theorem for which the greatest number of different floors have been discovered it possible the Pythagorean theorem with hundreds of rows having been published.
2.1.2 Beauty in Mathematical Results
Sometimes mathematical results establish a relation between two areas of specialization in mathematics who is seen to be totally on related South relations are usually referred to as results GS Hardy who is responsible for organisation of Ramanujan observed that beauty of mathematical results arise from an element of surprise.
Euler’s identify is often referred as d result and is called as one of the most remarkable formula in mathematics.
Euler’s identity is the equality
e+1 =0
Han yaar is the ulysse number the best of natural logarithms why is the complex number whose square is -1 -5 is the ratio of the circumference of a circle to its diameter.
what is so remarkable about its mathematical beauty in this equality is of the basic mathematical operations of addition multiplication and expansion occur exactly once father The Identity give the relation between:
The additive Identity zero The multiplicative Identity the number by which is predominant in trigonometry euclidean geometry and analytical mathematics the number is the base of natural logarithm the number I unit of complex numbers whose study leads to Deepa insights into many areas of logarithm length and integral calculus.
all the five quantities which sim to be totally unrelated are closely related by this identity.
2.1.3 Beauty in the Mathematical Language
the language of mathematics the way of expressing mathematical results and proof also known as mathematician the language of mathematical a great scientist Galileo said mathematics in the language with who is the god wrote the universe.
2.1.4 Beauty in Mathematical Experience
most mathematicians enter the mathematical beauty queen actively and wealth in mathematics it is particularly in number theory that while manipulating with numbers the mathematical experience becomes a delight it is true with many other branches also.
2.1.5 Three Aesthetic Experience Variable Identifies by Brikhoof and Their Relation
According to George David the typical experience of an object is a function of three variables namely:
Complexity of the object
The feeling value or aesthetic measure m and
Harmony symmetry order of the object.
in terms of these variables on experience is a sequence of the following phases:
1 preliminary report of attention which increases in proportion to complexity of the object feeling of aesthetic measure which hours this about a realisation that the objectives characterized by a certain harmony symmetry or order which is concealed in object.
an analysis of the aesthetic experience leads cost to believe that the aesthetic feelings are because of an unusual degree of harmonium interrelationship within the object these elements of harmonious interrelationship or order are referred ISM similarity equality symmetry balance and sequencing. These elements have positive effect on aesthetic measure but the complexity of the object makes it more difficult to experience the activity of the object the Arctic measure is given by the formula
M @ o/c
2.3 Recreational Mathematics
Recreational method is a term for mathematics carried for recreation rather than strictly research and application.
Mathematical Puzzles, Games and Riddles:
Any topic in this field requires no knowledge of advanced Mathematics and recreational method is open against to children and untrained adults.
The well-known topics in recreational mathematics are magic squares, logic puzzles and mathematical problems.
2.3.1 Importance of Recreational Mathematics
- Recreational math broadly include immensely popular puzzles as Sudoku and Ken Ken.
- No advanced mathematical knowledge like calculus be required for this puzzle.
- Sudoku type puzzles so addictive and easily generated by computers.
2.3.1 Role of Recreational Activities in Mathematics Learning
- Good curriculum and textbook are prepared mathematics education its success and the learning of mathematics depend on the teacher.
- Recreational activities plays a vital role in developing and understanding a mathematical concept in an enjoyable way by relating to the everyday life activities using fun and magic.
- They enable students to experience success and satisfaction for build and self confidence.
The recreational activities in mathematics help students to:
- Understand mathematical concept.
- Develop mathematical skills.
- Know mathematical facts.
- Learn the language of mathematics.
- Develop ability in mental thinking and reasoning.
Many teachers can be created in mathematics learning in the classroom through various activities using teaching learning materials like matchsticks, seeds, leaves, string blocks, games etc.
2.3.2 Recreational Activities
Mathematics interesting by tactful presentation of the subject matter. This can be done mainly:
- By focusing on the development of real understanding.
- By including applications of mathematics in other school subjects.
- by introducing the elements of recreation in the forms of patterns, games, magic, squares, riddles, puzzles etc.
Recreational mathematics depends on the teacher how he/she makes their use in enriching learning mathematics.
They encourage an alert and open attitude in youngsters and help them in developing high degree of logical thinking.
Example:
0x9+1=1
01×9+2=11
12×9+3=111
123×9+4=1111
1234×9+5=11111 and so on.
Example: Magic square order 3 constant of magic square is 15.
| 8 | 1 | 6 |
| 3 | 5 | 4 |
| 4 | 9 | 2 |
Even shorter ways of calculations create interest in mathematics.
2.3.4 Mathematical Games
- Many number games have been developed in mathematics which apart from killing the leisure time.
- It develops positive attitude towards the subject of mathematics.
- To clearly understand or play a number games to win one has to go to the minute details in the process of the game and thus this indirectly helps in developing the power of logical thinking and reasoning.
Example: Tell your friend to assume a number then tell to multiply 2 then tell for adding 2 then again tell for multiply 5 and adding of 5 after that repeat it for multiply 10 and add 10.
Now ask the final result from your friend and just subtract 1 from its 1’s place, the result will say the hopping number.
2.3.5 Mathematical Puzzles
Puzzle in Mathematics means statement apparently leading to a particular answer or result.
Puzzles are also called brain teasers.
By solving mathematical puzzles, it enrich and develop power of thinking and reasoning in students.
Example: in the diagram given here many rules are there from A to B if you are not allowed to go North or west.
| A | North | |||
| East | ||||
| South | B |
Solution: 126[√5+√4÷√5√4] routes since its root contains 5 horizontals and 4 verticals.
2.3.6 Mathematical Riddles
- Questions with clever or surprising answer are popularly called riddles.
- a true riddle always asks a question that can be insert reasonable riddles are world’s oldest guessing games.
- riddles creates interest in learning of mathematics and help in the development of logical thinking and reasoning.
Example: 9 dots are arranged in a square formation in 3 rows of 3 draw 4 straight line segments 2nd beginning where the 1st ends, the 3rd beginning where the 2nd ends and 4th beginning ends and 4th beginning where the 3rd ends so that eachdot is on at least one line segment.
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