TIPS TO HELP IMPROVE MATH SKILLS TO YOUR KIDS

Tips to Help Improve Math Skills to Your Kids



• Help your child master the basic facts.

It is saddening that the ASER reports shows that more than 70% of students of class 8 cannot do two digit multiplication. The onus falls on all of us as the RTE 2009 has taken the toll and students are doing drastic in Mathematics. If you ask a class 10 students to tell you the table of 15 or square of 34, cube of 14 etc. in maximum number of time you will get a wrong answer. Mastery of the basic facts means that your child can give an answer to a math fact in three seconds or less. Drills are recommended to be able to give quick and accurate responses. My teacher used to say that Drill Mathematics until you get mastery over it.

• Make sure your child understands mathematical concepts.



Mathematics need to visualize. If you fail to make mathematics a lovable subject it is obviously your fault. You need to co-relate mathematics with daily needs or daily uses. You have to put the notion in the head of students that life without mathematics is nothing and they should have a proper attention to master the basic facts, then only they can be master in Mathematics. Explain and review the concept with your child and check for understanding.

• Teach your child to write numbers neatly.


Mathematical errors may occur because of messy number writing or not lining numbers up correctly. Let your child understand the types of numbers with its property as well. Let them differentiate between prime and rational, even and odd, natural and whole. They should understand the place value system and face value of a number and while noting numbers in their copy, they should be aware of the facts which number should be written where and at what place. Improve your child's number writing skills by having them trace over numbers you have written, or by using graph paper to write the numbers neatly and more organized.

• Help correct skill difficulty before your child moves to the next skill level.



Math is a subject in which everything builds upon what has been previously learned. If the foundation stone is missing you can’t expect a building to last long. If your child has a problem in doing addition and subtraction he/she will have difficulty in multiplication and division too. If one is not good in fraction he will not be able to arrange them properly and will fail in doing percentage problem.

• Review skill technique with their math homework.


Homework pays an important role in building the math concept in a child. Let your child be in a habit of doing homework regularly. Check your kids’ copy and give suggestion if a mistake is done. Let your child practice with worksheet or sample papers. Doing math homework reinforces the skills your child is learning in class.

• Encourage your child to do more than the assigned work.


Only the text book is not sufficient. Let your child practice with some other books and sample papers so that he/she can encounter a varieties of problems to solve. It is well said -Practice makes perfect and this is true with your lovely subject mathematics too. Practice is necessary for your child to become comfortable with their math skills.

• Explain how to solve word problems.


Solving word problems is a tough job even for a teenagers. I have seen the students of Higher class in trouble. Let your child be prepared to face such problem boldly and for that you need to Teach your child to read a word problem at least two times, if not more, in order to fully comprehend what is being asked of them. Relate the problem with a real life situation to understand it fully. You can make a chart in order to describe it. Ask them what is being asked, what type of operation is necessary, and discuss the steps necessary to solve the problem. Operation dealing is an area of concern and let your child understand the facts the sign changes when you change the side.

• Help your child learn the vocabulary of mathematics.



Understanding Mathematics words and learning it is important. Let your child understand the word with proper definition. Buy a small dictionary for your child and help them to learn it. You can teach the meaning with examples and proper definition.

• Teach them how to do math in their head- "Mental Math".



This is an era of doing fast calculation. You can read book such as – VEDIC MATHEMATICS, Trenchberg system of fast mathematics and let your child do faster calculation mentally. ABACUS also plays handy in doing fast calculation. Help them to understand that if they know 2+3=5, then they can solve 20+30=50. Similarly if 2 x 3 = 6, so as 200 x 300 = 60000.

• Make math a part of your child's daily life.



The sad parts of Mathematics is that it has the stigma of being unpopular or insipid. The reason behind it that teacher hardly relate mathematics with the daily life. Math will become more meaningful when your child sees how important it is in real life situations. Encourage them to use math in practical ways. I remember that when my son Nilabh was 3 years I used to teach them 3 digits numbers or 4 digits numbers while driving with the help of car or bike number plate. I also use to give him a 10 rupee note and tell him to buy chips or something and teach him the concept of minus, plus or addition.


DR. RAJESH KUMAR THAKUR

युद्ध और गणित (war and mathematics)

आप सोच रहे होंगे की गणित को युद्ध में कैसे प्रयोग किया जा सकता है पर चौंकिए नहीं युद्ध में गणित का प्रयोग प्राचीन समय से होता आ रहा है . ईसा पूर्व 18 वी शदी के बेबीलोनियन पट्टी में युद्ध में पकडे सैनिको के बारे में जानकारी मिलती है साथ ही युद्ध की योजना जैसे- बंकर बनाने में कितने ईंटो की आवश्यकता होगी , छिपने के लिए जमीन को कितना गहरा कर खोदना पड़ेगा और युद्ध बंदियों को रखने के लिए बनाये जाने वाले जेल में कमरे की संख्या का आकलन करने के लिए प्राचीन समय में भी गणना के लिए गणित का प्रयोग किया जाता रहा है .

ईसा पूर्व 18 वी शदी के बेबीलोनियन पट्टी त्रेता युग में जब सीता का अपहरण कर रावण लंका ले गए और श्री राम के लाख प्रयासों के बाबजूद युद्ध अवश्यम्भावी हो गया तो श्री राम की सेना ने लंका की ओर कूच किया और इसके लिए समुद्र पर 100 योजन लम्बा और 10 योजन चौड़ा पुल का निर्माण किया गया. श्री राम की विशाल सेना देख रावण आशंकित हो गया और अपने गुप्तचर शुक को श्री राम की सेन्य बल का पता लगाने के लिए भेजा . शुक बानर का रूप धारण कर श्री राम के सेना में शामिल हो गया और वहां के युद्ध तैयारी का पता लगाया और फिर लंका पहुंचकर श्री राम की सेना के वारे में जो सुचना दी उसका विवरण श्री वाल्मीकि रामायण में इस प्रकार दिया है – इस श्लोक में 1060 तक की संख्या जिसे महौघ कहा जाता था का उल्लेख दिखता है .

अतः श्री राम की विशाल सैन्यबल की कल्पना करना अपने आप में एक डर पैदा करता है . क्या आप जानते है की पृथ्वी का क्षेत्रफल 1016 वर्ग फीट से कम है तो इतनी बड़ी सेना के लिए धरती पर खड़ा होना और युद्ध करना भी असंभव प्रतीत होता है और शायद इसीलिए गोस्वामी तुलसीदास ने रामचरित मानस में लिखा – राम ते अधिक राम कर दासा. इसी प्रकार महाभारत की युद्ध में भी सेना के प्रकार और उसमे सेना की संख्या के बारे में गणना का विवरण दिखता है जी गणित के सार्थक ज्ञान के विना संभव नहीं था . इस श्लोक के अनुसार प्रत्येक रथ में 4 घोड़े को बांधा जाता था और 100 धनुष रखने की व्यवस्था थी . एक रथ के पीछे 10 हाथी और एक हाथी के पीछे 10 घोड़े और एक घोड़े के पीछे 10 पैदल सेना चलती थी . इसका मतलब 1 रथ के पीछे 10 हाथी , 100 घोड़े और 1000 सैन्यबल चलता था और यह सभी संख्या गुणोत्तर क्रम में दिखते हैं. यही नहीं 1 सेना में 500 हाथी और 500 रथ होते थे जिसका सीधा मतलब ये है की एक सेना में पैदल सैन्यबल = 500 x 1000 = 500000. महाभारत में अक्षोनी सेना के बारे में भी वर्णन मिलता है. अक्षोनी सेना का प्रयोग चतुरंग सेना के सन्दर्भ में होता है. एक चतुरंग सेना के चार अंग जिसमे – 1. रथ 2. हाथी 3. घोड़े और 4. पैदल सेना होते थे . एक अक्षोनी सेना के पास 21870 रथ , 21870 हाथी , 65610 घोड़े और 109350 पैदल सेना होते थे और मजे की बात ये है की इनका अनुपात 1 : 1 : 3 : 5 है . यदि एक रथ में सारथी और योद्धा मिलाकर 2 लोग मान लिया जाये तो इनके पीछे चलने वाले हाथी पर 2 लोग सवार हो तो 1 अक्षोनी में 183680 योद्धा होंगे. 2 × 21870 + 2× 21870 + 65610 + 109350 = 262440 उपरोक्त श्लोक के अनुसार पांडवो के पास 7 अक्षोनी सेना और कौरवों के पास 11 अक्षोनी सेना थे अर्थात दोनों के हिस्से में कुल 18 अक्षोनी सेना थे या यूँ कहा जाये तो 18 अक्षोनी में 4723920 सेना उनके तरफ से लड़ रहे थे . इस तरह के सेना किसी चिर परिचित राजा द्वारा उनके समर्थन में लड़ रहे हो सकते हैं पर पूरी महाभारत पढने के बाद आपको लगेगा की ये सेना की संख्या इससे कही अधिक थी . मसलन युद्ध समाप्ति के बाद जब धृतराष्ट्र ने युधिष्ठिर से पूछा की युद्ध के उपरांत कितने सैनिक जीवित बचे और कितने वीर गति को प्राप्त हुए तो युधिष्ठिर का जबाब कुछ और ही बाते बयां करते हैं . युधिष्ठिर के अनुसार युद्ध में मरने बाले सैनिको की कुल संख्या 1 अरब 66 करोड़ 20 हज़ार तथा जिन्दा बचे सैनिको की संख्या 2 लाख 40 हज़ार एक सौ पेशा पैसठ है . महाभारत की लड़ाई की सबसे खास बात चक्रब्यूह की है . चक्रव्यूह या पद्मव्यूह सेनाओं का ऐसा समूह है जिसमे सैनिक 7 सकेंद्रिय वृत के रूप में एक चक्र की रचना करते थे और इसमें घुसने के एक ही रास्ते थे और योद्धा के इस व्यूह में घुसते ही व्यूह दायीं और घूम जाता और निकलने का रास्ता बदल जाता था और व्यूह के योद्धा अन्दर प्रवेश करने वाले योद्धा को मार देते थे . विकिपीडिया के अनुसार चक्रव्यूह की संरचना सैनिकों की एक वहू परतीय रक्षात्मक गठन को दिखता है जो ऊपर से देखने से एक कमल फूल जैसा दिखता था . इसमें हर परत के साथ सैनिकों की संख्या और योग्यता का स्तर बढ़ता जाता था और कृष्ण, अर्जुन , प्रद्युम्न और अभिमन्यु को छोड़ कर किसी पांडव योद्धा को चक्रव्यूह को तोड़ना नहीं आता था,

चक्रव्यूह सैनिकों के युद्ध में वृताकार , स्तम्भ , सारिणी , वर्गाकार और कील के आकार के रचना का उल्लेख पुस्तकों में मिलता है जहाँ सेनापति अपने सेना को अलग अलग आकृति के व्यूह रचना में युद्ध के लिए तैयार करते थे. ग्रीक लोगों के द्वारा वर्गाकार रचना में युद्ध लड़ने का उल्लेख मिलता है. क्या आप जानते है की युद्ध के लिए बनाये जाने वाले अधिकांश उपकरण – जहाज, पनडुब्बी , हथियार, टेलिस्कोप, गोला –बारूद, कैमरा , दूरबीन बिना गणितीय ज्ञान के बनाना संभव ही नहीं है . सेराक्युस शहर की रक्षा के लिए आर्कमिडीज ने एक पंजा डिजाईन किया था जिसमे एक क्रेन जैसे भुजा थी जिससे एक बड़ा धातु का हुक लटका हुआ था .जब इस पंजे को एक आक्रमण करते हुए जहाज पर डाला जाता था तो इसकी भुजा ऊपर की ओर उठती थी और जहाज को उठाकर पानी में डूबा देती थी. यही नहीं सेराक्युस की घेराबंदी के दौरान आर्कमिडीज ने अपने द्वारा बनाये बड़े- बड़े दर्पण का उपयोग कर समुद्र में ही सेरिक्युस की ओर आने वाले जहाज को राजमहल से ही दर्पण द्वारा सूर्य के प्रकाश को फोकस कर आग के हवाले कर देते थे.

Archimedes Burning Mirror

15 वी शदी में तरताग्लिया (? 1500-1557) ने युद्ध प्रक्षेपकी के लिए एक नया सिधांत विकसित किया और इसका उपयोग गोली और गोला दागने में किया जाने लगा और सम्भबतः गणित का युद्ध के तकनीक में यह पहला प्रयोग हो.


राजेश कुमार ठाकुर

HOW TO READ FOR MATHEMATICS EXAMINATION

How to Study For Examinations?

Examination!! Isn’t it a word that makes you tense?

Let me quote a Sanskrit text first and then we shall start discussing how to prepare you for the examination?
तावद भयस्य भेतव्यं यवाद भयमनागतम
आगतं तु भयं विक्ष्यम प्रतिकुर्याधथोचितम
The meaning of the text is ---- Get afraid of fear as long as it doesn’t comes to you. The moment fear approaches you, come forward and embrace it.

Hey guys , don’t get tensed over the examination. The tension at little amount is essential as it motivates you to work harder. Prepare well in advance and keep the tension away. Here are few tips and strategies that will work well if you follow it as advised.

1. Exams are terrible and stressful things to study for, especially knowing that they can make or break your final mark. In the age of Technology, we have different types of books available in the market or e-book available on internet. You can buy a ten years question paper or sample paper to see the pattern of question paper. If you are net savvy, you can download the previous year question paper or sample papers from the CBSE /ICSE or respective Boards’ websites. Besides that, you can download the blue- print of the syllabus which will guide you to decide which chapter will fetch you maximum or minimum marks. The blue print will also guide you about the number of questions being asked in the examination and marking scheme. The most interesting part of the question paper available on the internet on the CBSE website (www.cbse.nic.in) is it has also a solved question paper with detail description of the step-wise marking scheme. If you go through this solved paper, you will get the knowledge about the marks being awarded in the Board Examination. Even this pattern is being followed in the school.


2. You can buy sample papers available in the market. The publishing company has now designed the sample paper on the CCE pattern for Secondary classes. For the senior secondary classes you can get books on sample papers. This book contains solved and unsolved question paper on the Board pattern. Now look for a peaceful space in your home where you can sit 3 hours continuously to solve the sample paper. Mind one thing, you won’t be able to study while there are distraction like ----- A) your little sister or brother is running screaming around the house. B) Your favourite program or film is running on TV. C) There is dim light in your room. D) You are in a mess. Statistics say that 75 % of guys that study in a better light will focus better on study. For girls it is indicated that 90% of time, they study and focus better in a brighter room with little noise, their chance of grasping the matter is better. So chose a peaceful place where you don’t get any distraction. Close the room and sit there for three hours. Don’t practice in installment. Though you can take 1-2 minute break in between but that too in emergency call.
3. After practicing 2-3 papers, reduce the time by 15-30 minutes so that you can keep these 15 minutes for revising the papers. 4. Once you complete the paper, get it examined by your class teacher or tutor so that you get to know the area where you need to put extra labour. 5. The ideal time to start sample paper is 2- month before the final examination takes place. A sample paper book of 20 odd papers should be finished in 30 days so that before the examination is scheduled to happen, you get to revise all the sample papers at least 2-3 times. 6. Solving sample paper will not only control the mistake you make in examination but it will help you to learn the effective way of writing in examination. Besides that, you will learn how to complete your paper within the stipulated time. Solving sample paper before examination will serve three purpose: - a) It will control the mistake b) manages your time effectively c) you learn how to write the paper effectively. What to do a night before examination?



If you are studying one night before examination, you must have some hours left to prepare. So don’t get frightened, you need time management and you can manage to get pass or get good marks depending upon your earlier preparation.


1. Eat some brain food:- A night before examination is a crucial moment for your life. The tension grows if you haven’t had studied much in past or you have examination phobia. Get some brain food. Brain food not only gives us energy and helps us to live longer but it can also help you to get good marks. Green tea is a best brain food as it contents polyphenols. The bitter tasting substance protects the brain from wear and tear plus it helps in mood enhancement. You can eat walnuts, eggs, dark chocolate etc. Brain food boosts your brain’s workability by giving it what it needs to function properly. Moreover, by eating something before study you will be less tempted to get hungry and quit study early.


2. Dress comfortably: - Hey, I am not joking. Go to the bathroom and dress comfortably so you don’t fall asleep. Prepare your body as much as you can so that when you start the study season you have no excuses to get up and go somewhere.

3. Organize your study materials:- Get all the materials such as class-notes, text-book, reference book out on your desk so you can see what you have to work with. This preparation should be done in advance so that you don’t have to break your study every now and then.


4. Don't cram. Cramming the night before is proven to be useless, because you're taking in so much information at once that it's impossible to memorize it at all -- in fact, you'll hardly retain anything. I know it's been preached to you many times before, but it's true: Studying before and going over it multiple times really is the best way to learn the material. This is especially true with things like history and subjects dealing with theory. Most of the math is not memory work but recognizing patterns and applying useful techniques to problems. The only way to learn mathematical skills is to practice.

5. Plan: Always create a plan before you start studying, not to mention that this plan has to be achievable. If out of 5 lessons 3 are easy and can be finished fast, finish them first, that way you can spend quality time on the difficult lessons without fretting. Small tricks like these will help you complete your portions quickly. Go through the previous year question paper to make a list of important questions so that you don’t have to run through all pages. Check whether you have gone through the solved examples or not.


6. Don’t read whole night: - Though Examination is a do or die situation but you can do well in exam if you are a regular reader. Remember one thing that mathematics can’t be read in overnight. There is nothing you can gain just by cramming few hours/days before examination. So make a habit of regular reading. Our brain works like a muscle. It must be exercised regularly to be strong. Don’t put much pressure a night before examination. Let your mind take a rest. Watch film, do some recreational work instead so that your mind is relaxed and can perform better. Study less, with more concentration, if possible. If you want to study for a long time then you can take breaks in between to regain your concentration after you return from the break. It would be a better choice to study in 1 hour slot followed by a 5 minute break. If you try to study indefinitely for hours and hours, your brain will overload and you will have to work to regain your focus on studying.


7. Sleep well: - Nothing will make you do worse on a test than pulling an all- nighter. You may be tempted to study whole night and cram in as much as is possible, but this practice will do no good to you. Sleep well before exams, so your mind will be alert before, after, and during the exam. Reading whole night and sleeping less than the usual hours of sleeping will get you feel tired. It's better to study for two hours in one day than to try and cram in that daily hour of studying at two in the morning. Excess amount of reading one night before exam will make you feel sleepy in the exam hall also. Remember this analogy: 'Study without ambition is a bird without wings'.


8. Don’t call your friend: - Many students keep calling his/ her friend for help or to inquire important questions. Avoid making any call especially one night before examination, because you may get confused. Don't depend heavily on the help of others. Avoid stress from other people, if at all possible. Don't hang around friends that worry and stress. This will rub off on you. Remember don’t call your competitor, although he may look like your good friend. I have seen my competitor calling me to tell the list of important questions one night before examination only to distract my study. Your competitor may make some plan to disturb you so that you get less mark and he gets chance to prove his excellence in class/ board. So better consult your friend a month ago not a night before examination to avoid any last minute mistake.



9. Overcome test anxiety: - Examination and anxiety are supplementary words. You will hardly find any persons on the earth who are not afraid of examination. To a certain extent it is normal to be nervous about taking test, even if you are a very good student. While mild anxiety can be a motivating factor, high anxiety triggers your flight response and makes you avoid the subject that is a dangerous issue. You can go through the chapter how to overcome the math anxiety in the book itself.


10. Get up early and REVISE: - This is one of the last steps I shall advise you before you go for test. Get up early and go through the important formula, HOTS, etc. STOP reading 3 hours before examination. Have breakfast so that your mind gets proper nutrients it needs to work at full capacity. What to do in Examination Just as it is important to think about how you spend your study time, it is important to think about what strategies you will use when you take a test. Good test taking strategies can make a big difference to your score. Strategies to write a math test • Read the instruction first. The instruction is provided to help you and make strategies accordingly.

In many tests there are provision of negative marks so if you haven’t gone through the instruction, you may lose your marks.

• CBSE provides extra 15 minutes to read question paper so look over the entire paper to get a sense of its length. Try to identify those problems you are sure about it and you are having some doubt.

• Write important formula in the margin you think you will need for the exam. Write them all out on your text the moment it gets handed to you. The answer sheet should be folded into three parts. The middle part should have more space and the first and last fold should have less space. Once you finish your paper, cross the rough area to avoid any confusion.

• Go through the question paper properly. Do the easiest one first and save the hardest for last. It is not a wiser idea to try with the hardest. Writing the easiest question first will build your confidence and help you focus your time and energy on the tough ones later. This is a good way to manage your time as well.


• Don’t get panic after reading the question paper. Give yourself time to adjust. Take a long breathe and let it out slowly. Don’t watch other students working as it may put pressure on mind. Cool down and see how many question you can solve. Attempt those question first you find easy. Once you start solving you will find you know some more problems to solve.

• Don’t lose point through careless errors. Check each step as you go, and make sure that it really follows correctly from the previous step. Watch for things like dropped minus signs or missed exponent or mistake you generally make.

• Work properly, writing all steps. Discipline yourself to sow all the steps in your solution and show them one after the other, not little bits of math written here and there. Jumping steps should be avoided. Give reasoning whereas necessary. Underline the important formula. Put your answer in a box. This gives a good look to your answer sheet and makes a better impression in the mind of examiner.


SEND YOUR VALUABLE COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com


Question: - Sum of angle of a triangle is 180 degree. Given: - A triangle ABC To prove :-

Proof: - Since BC // DE and AB is a transversal line. <2 = <4 ------------- (Alternate Angle)

(1) Again, BC // DE and AC is a transversal line. <3 = <5 ----------- (Alternate Angle)


(2) Adding equation 1 and 2 we get, <2 + <3 = <4 + <5 -----------


(3) Add <1 both side in equation (3), we get, <1 + <2 + <3 = <1+<4+<5 Since, <1+<4+<5 = 1800 (Angle on straight line) Hence, <1 + <2 + <3 = 1800 Q.E.D.
• In order to answer the geometric proof the order should be ---

a) Given b) Required to Proof c) Construction d) Proof The proof should have proper reasoning. If you are applying the theorem in order to proof a question, don’t forget to mention it.


• In graph, write the scale and also write the equation of the line. Let us now plot a velocity-time (v - t) graph for the following data. Velocity in m/s 0 10 20 30 40 50 Time in Seconds 0 2 4 6 8 10

• Leave at least two- three lines between two consecutive questions so that it makes a good impression to the examiner.


• Time is very crucial factor in exam. If there are 20 questions to be solved in 3 hours test, then one question will have on an average 9 minutes. Don’t spend much time in one question. If you are working on objective question give less time to it so that you have some extra time to work out in long type question. Keep checking the time so that you can manage the time well.

• Never waste time in erasing. If you got some steps wrong just put it in a box and draw a line to show you have cut that part. Not only does erasing waste precious time, but you may erase something useful and for that you need to repent later.

• Write the dimension in the answer. If you are asked to find the volume of a solid objects the answer will contain the cubic unit (m3 ,cm3 ,--). If you are asked to find the area of any objects never forget to write the unit in square unit such as m2 , cm2 etc. Does the problem ask how old the woman was? Make sure that your x really did stand for the woman’s age and not her daughter’s age. Students lose lots of points on every test because they didn’t answer the question that was actually asked, but answered some other question instead. Make sure you have worked the problem to the end and that your answer is in the right form, and that you’ve answered all parts of the question. • Especially in problem of Trigonometry, Geometry, Co-ordinate geometry, Mensuration etc, make clean diagram. The diagram is essential in such question and the visual representation will clear your mind and help you figure out a way to solve it.


• Don’t copy or use unfair means in exam. Never match your answer with your friend Students sometimes ask their friend for help and sometimes cross check the answer. This is a dangerous practice. I still remember one incident, when I was a student of B.Sc. (Math Hons). In a Real analysis paper I answered a question that was wrong but I was confident of my answer. A friend of mine sitting back asked me the answer of that question and I told him the answer I got. Since I was a brilliant student and most of my friend used to trust me blindly so he crossed his answer and solved it again with the method I used. When I went home and checked my answer I found that I had applied wrong concept thus got wrong result. This made me sad because I managed to get 80 marks in that paper but my friend who could have scored 70 got 55 in that paper.


• Don’t leave the paper in hurry. Have you heard the story of Tortoise and Rabbit? Doing your paper fast and submitting it without re-checking it will not bear fruit. Many students like to leave a test paper early to show their fellow friend that how good he is to finish the paper early. Regardless of your motive, leaving the exam room early is a mistake. If you have finished the paper well in advance, re- checks it two to three times. Make sure you have actually answered each question correctly. Write your variables clearly so that you don’t confuse them with other ones. Ask yourself if your answer makes sense. Are the units correct? Is your answer consistent with the parameters of the question? Check your formulas etc before submitting it to the teacher. • The last but the important point is that your writing should not be illegible. Good handwriting makes a good impression in mind of teacher and illegible writing brings a wrong impression and your illegible writing sometime may be the reason of your poor marks in exam. Hope these tips will help you to prepare for your class test or board exam.


Always remember a good strategies and well planned preparation will help you to score good in exam.

Issac Newton

Isaac Newton With the possible exception of Albert Einstein no scientist, so altered the human perception of the Universe as Isaac Newton. His cosmos of absolute space and time drew upon the work of others such as Kepler and Galileo but it was Newton who brought together Galileo’s mechanics and Kapler’s law of planetary motion to fashion a universe which could run without the benefit of continual divine intervention. His successor have pointed Newton as the supreme intellect that the human race has produced—he who in genius surpassed the human kind. The English Physicist, Mathematician Isaac Newton was born on 25th December 1642 in Woolsthrope in England. His father Isaac died at the age of 37, before the birth of his son. Newton was a premature child. His mother Hannah Ayscough married the next door neighbour Barnabas Smith and left the 3 years old Newton to the care of his grandmother. As a child Newton was not robust and was forced to shun the rough games of boys of his age. Instead of playing with them he invented his own diversions in which his genius first showed up. Newton received his early education in the common village school of his vicinity. His maternal uncle William Ayscough was the first able man to recognise the talent of Newton. He persuaded Newton’s mother to send his son to Cambridge instead of keeping at home. On his uncle’s advice he had been sent to the Grantham Grammar School. While preparing for Cambridge Newton lodged with a Mr. Clarke and fell in love with his step daughter Miss Storey to whom he became engaged before leaving for Cambridge. But in Cambridge his growing absorption in work put his romance in the back seat and Newton never married. Newton attributed his success to three greatest mathematicians - Descartes, Kepler, and Galileo. Newton writes- If I have seen a little further than others it is because I have stood on the shoulders of giant. In June 1661, Newton entered Trinity College Cambridge where his maths teacher was Dr. Isaac Barrow. Newton passed his BA degree in January 1664. Till 1664, Newton did nothing great. Though a manuscript dated May 20, 1665 shows that Newton had scientifically developed the principles of Calculus at the age of 23. He called his method Fluxions. He had also discovered the Integration and Binomial Theorem in 1665. The second of his great discovery is the law of universal gravitation in 1666 at Woolsthrope’s apple garden. Newton always preferred to view the Universe as a gigantic watch spring wound up by the hand of God and left to run down on its own. He believed that it was impossible for science to transcend the barrier between our world and the spiritual world and was quite content to leave such speculations to other natural philosophers. He wrote:- We have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power... I have not been able to discover the cause of those properties of gravity from phenomena , and I frame no hypothesis; for whatever is not deduced from the phenomena is to be called an hypothesis; and hypothesises---- have no place in experimental philosophy. In 1667, Newton was elected a Fellow of Trinity and in 1669, at the age of 26, he succeeded Barrow as Lucasian Professor of mathematics. It is said the Isaac Barrow resigned from the Lucasian Professorship of mathematics in favour of his incomparable pupil Newton. His first lecture was on Optics. In 1668, he constructed a reflection telescope with his own hand and used it to observe the satellite of Jupiter. His master piece Principia was published at the cost of his friend Edmund Halley in 1687. It contained the Dynamics, Laws of Gravitation and the three laws of motion, which dominated the scientific view of the physical universe for the next three centuries. Newton showed that the motions of objects on Earth and of celestial bodies are governed by the same set of natural laws, by demonstrating the consistency between Kepler's laws of planetary motion and his theory of gravitation. The Principia had no mention of his Calculus. From 1670 to 1672, Newton lectured on optics. During this period he investigated the refraction of light, demonstrating that a prism could decompose white light into a spectrum of colours, and that a lens and a second prism could recompose the multi-coloured spectrum into white light. He also showed that the coloured light does not change its properties by separating out a coloured beam and shining it on various objects. This is known as Newton's theory of colour. Newton was also a member of the Parliament of England from 1689 to 1690. In 1696, at the age of 54, Newton became warden of the Royal Mint. His job was to reform the coinage. In the same year Johann Bernoulli and Leibniz concocted two challenges to the mathematician of Europe. The problem baffled the mathematician for next six months and none of the great mathematician of that very time could come up with the exact solution. Newton heard of the problem when a friend communicated him. He was at that time returned from his Mint office. Newton took his dinner and sat to solve the problem and finally got the solution of the two challenging problems. He communicated the solution to the Royal Society anonymously. On seeing the solution Bernoulli at once exclaimed, “Ah! I recognize the lion by his paw.” In 1701-02, Newton represented Cambridge University in Parliament and in 1703 was elected President of Royal Society. In 1705, he was knighted by Queen Anne. Newton had done a great service to mankind by his discovery but he was so humble that he once said: - I don’t know what I may appear to the world; but to myself I seem to have only like a boy playing on the seashore and diverting a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. Towards the end of his life, Newton took up residence at Cranbury Park, near Winchester with his niece and her husband, until his death in 1727.Newton died in his sleep in London on 31 March 1727 and was buried in Westminster Abbey. Let’s close the biography of Greatest mathematician of the world with the line of English Poet Alexander Pope:- Nature and nature's laws lay hid in night; God said "Let Newton be" and all was light Rajesh Kumar Thakur rkthakur1974@gmail.com

Hyptia of Alexandria

Hyptia of Alexandria Hypatia of Alexandria was the first woman to make a substantial contribution to the development of mathematics. She was a Greek Neoplatonist philosopher in Roman Egypt and the head of the Platonist school at Alexandria who taught philosophy and astronomy. The life of Hypatia was one enriched with a passion for knowledge. Historians are uncertain of different aspects of Hypatia's life. Her date of birth is one that is highly debated. Some historians believe that Hypatia was born in the year 370 AD. On the other hand, others argue that she was an older woman (around 60) at the time of her death, thus making her birth in the year 355 AD. She was the daughter of Theon of Alexandria who was a teacher of mathematics with the Museum of Alexandria in Egypt. Throughout her childhood, Theon raised Hypatia in an environment of thought. Historians believe that Theon tried to raise the perfect human in her . Theon himself was a well known scholar and a professor of mathematics at the University of Alexandria. Theon and Hypatia formed a strong bond as he taught Hypatia his own knowledge and shared his passion in the search for answers to the unknown. As Hypatia grew older, she began to develop an enthusiasm for mathematics and the sciences especially in astronomy and astrology. References in letters by Synesius, one of Hypatia's students, credit Hypatia with the invention of the astrolabe, a device used in studying astronomy. Most historians believe that Hypatia surpassed her father's knowledge at a young age. However, while Hypatia was still under her father's discipline, he also developed for her a physical routine to ensure for her a healthy body as well as a highly functional mind. In her education, Theon instructed Hypatia on the different religions of the world and taught her how to influence people with the power of words. He taught her the fundamentals of teaching, so that Hypatia became a profound orator. People from other cities came to study and learn from her. Hypatia became head of the Platonist school at Alexandria in about 400 AD. There she lectured on mathematics and philosophy, in particular teaching the philosophy of Neoplatonism. Hypatia based her teachings on those of Plotinus, the founder of Neoplatonism, and Iamblichus who was a developer of Neoplatonism around 300 AD. Hypatia dressed in the clothing of a scholar or teacher, rather than in women's clothing. She moved about freely, driving her own chariot, contrary to the norm for women's public behaviour. She exerted considerable political influence in the city. Hypatia was known more for the work she did in mathematics than in astronomy, primarily for her work on the ideas of conic sections introduced by Apollonius. She edited the work On the Conics of Apollonius, which divided cones into different parts by a plane. This concept developed the ideas of hyperbolas, parabolas, and ellipses. With Hypatia's work on this important book, she made the concepts easier to understand, thus making the work survive through many centuries. Hypatia was the first woman to have such a profound impact on the survival of early thought in mathematics. According to the Suda Lexicon, a 10th-century encyclopaedia, Hypatia wrote commentaries on the Arithmetica of Diophantus of Alexandria, on the Conics of Apollonius of Perga, and on an astronomical canon .The known titles of her works, combined with the letters of Synesius who consulted her about the construction of an astrolabe and a hydroscope. The existence of any strictly philosophical works by her is unknown. Hypatia lived in Alexandria when Christianity started to dominate over the other religions. In the early 390's, riots broke out frequently between the different religions. Cyril, a leader among the Christians, and Orestes, the civil governor, opposed each other. Hypatia was a friend of Orestes and it is believed that Cyril spread virulent rumours about her. In 415 AD, on Hypatia's way home, a mob attacked her, stripped her and killed her with pieces of broken pottery. Hypatia's life ended tragically, however her life's work remained. Later, Descartes, Newton, and Leibniz expanded on her work. Hypatia made extraordinary accomplishments for a woman in her time. Philosophers considered her a woman of great knowledge and an excellent teacher. Send your valuable comments to RAJESH KUMAR THAKUR rkthakur1974@gmail.com

George Cantor

George Cantor George Cantor, the founder of transfinite set theory revolutionized mathematical thinking in this area. For centuries, the concept of infinity had been a highly controversial one both mathematically and philosophically and his ideas could not get the full impact in his life time. George Ferdinand Ludwing Philipp Cantor was born in St. Petersburg, Russia. His father George Woldemar Cantor was a stockbroker while his mother Maria Anna came from a musical family. Cantor was raised in an intensely religious atmosphere. In 1856, when young Cantor was 11, the family moved from St. Petersburg to Germany where he attended the Gymnasium school. Cantor showed all round ability there but he had special affinity with mathematics and science and his father decided that he should train as an engineer. He was therefore enrolled in Polytechnikum at Zurich but seeing his special interest in mathematics his father later decided that he should study mathematics instead of engineering and he was admitted to University of Berlin, where he proved to be a good but not excellent student. Cantor moved to the University of Berlin where he became friends with Hermann Schwarz who was a fellow student. Cantor attended lectures by Weierstrass, Kummerand Kronecker. He spent the summer term of 1866 at the University of Göttingen, returning to Berlin to complete his dissertation on number theory De aequationibus secundi gradus indeterminatis in 1867. After obtaining his doctorate in 1867, he became a school teacher for a short period and later joined University of Halle where he remained for the rest of his entire carrier. He married Vally Guttmam in 1874 and they had 6 children. While he was in Berlin, he became interested in Number Theory and came to the notice of Kronecker. His first achievement was in traditional field of trigonometric series, especially the question of uniqueness for the Fourier representation of a given function. He gradually showed that the 19th Century idea in the connection between the dimension of a set and the number of its elements were dangerously fallacious. He showed that the concepts of Cardinal and Ordinal Number could be defined mathematically in such a way that it made a good sense to talk of infinite or transfinite numbers. In 1873 Cantor proved the rational numbers countable, i.e. they may be placed in one-one correspondence with the natural numbers. He also showed that the algebraic numbers, i.e. the numbers which are roots of polynomial equations with integer coefficients, were countable. However his attempts to decide whether the real numbers were countable proved harder. He had proved that the real numbers were not countable by December 1873 and published this in a paper in 1874. It is in this paper that the idea of a one-one correspondence appears for the first time, but it is only implicit in this work. Cantor went on to develop his revolutionary ideas in a series of paper published between 1879 to 1884. Cantor got the first part of his major work Beitrage zur Begrundung der transfiniten Mengenlehre (Contributions to the foundation of Transfinite set theroy) published in 1895. This book led to the circulation of Cantor’s idea to the world. Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space. Cantor was surprised at his own discovery and wrote:- I see it, but I don't believe it! In 1910 Cantor received an invitation from the University of St Andrews in Scotland to attend the 500t anniversary of the founding of the University as a distinguished foreign scholar .Cantor had hoped to meet with Russell who had just published the Principia Mathematica. However ill health and the news that his son had taken ill made Cantor return to Germany without seeing Russell. The following year Cantor was awarded the honorary degree of Doctor of Laws by the University of St Andrews but he was too ill to receive the degree in person. The second half of his life, his ideas was attacked by many mathematicians. Leopold Kronecker launched a particular vitriolic attack on Cantor that led to the decline in Cantor’s mental age and he died in a mental hospital on 6th January 1918. SEND YOUR VALUABLE COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com

George Boole

George Boole George Boole was one of the most brilliant mathematicians England has produced. He was born on Nov 2, 1815 at Lincoln England to John Boole and Mary Ann Joyce. He was born into the lowest economic stratum of society. The lower classes into whose ranks Boole had been born simply didn’t exist in the eye of the upper class. It is said that he was born in the wrong time, in the wrong place, and definitely in the wrong class. His father John used to make shoes but he was much interested in science and in particular the application of mathematics to scientific instruments. The family was not well off, partly because John’s love of science and mathematics meant that he didn’t devote the energy to developing his business in the way he might have done. Boole was lucky enough to have a father who passed along his own love of math. Young George took to learning like a politician to a pay rise and, by the age of eight, had outgrown his father's self-taught limits. Due to poverty and having been born in lower strata Boole could not get himself enrolled in a school of good repute. At that very time knowledge of Latin was considered to be important but no Latin was taught in the school where Boole was permitted to attend. Boole decided that he would learn Latin and a friend of his father helped him a little bit but Boole took the rest journey of learning Latin all alone. By 12, he had mastered enough Latin to translate a given paragraph into English properly. Boole learned his early lesson in mathematics from his father, an amateur mathematician and optical instrument maker. His father wanted him to join the business but after finishing his schooling Boole took a commercial course. By 16, he became a school teacher. This was rather forced on him since his father's business collapsed and he found himself having to support financially his parents, brothers and sister. He spent four years teaching elementary school. Boole later decided to become a clergy man so as to support his family in a better way and thus trained himself in the language of French, German and Italian and got mastery over it. At the age of 20, Boole opened up a school to prepare his pupils in mathematics. Over the next few years, Boole prepared himself with the tough mathematical courses with the help of mathematical journals borrowed from the local Mechanic's Institute. Boole struggled with Isaac Newton's 'Principia' and the works of 18th and 19th century French mathematicians Pierre-Simon Laplace and Joseph-Louis Lagrange. He later mastered himself the Mecanuque Cleste of Laplace and Mecanique Analytique of Lagrange by his own unaided efforts. In 1837, Boole submitted some of his work to the Cambridge Mathematical Journal. The style of presentation and its originality impressed Gregory and they become a good friend for life. David Gregory advised him to take a formal course of mathematics at Cambridge but he was unable to take Duncan Gregory's advice and study courses at Cambridge as he required the income from his school to look after his parents. In the summer of 1840 he had opened a boarding school in Lincoln and again the whole family had moved with him. He began publishing regularly in the Cambridge Mathematical Journal and his interests were influenced by Duncan Gregory as he began to study algebra. By 1844 he was concentrating on the uses of combined algebra and calculus to process infinitely small and large figures, and, in that same year, received a Royal Society medal for his contributions to analysis. In 1847, Boole published a pamphlet The Mathematical Analysis of Logic which bridged the gap previously separating mathematics from formal logic. This development was crucial in advancing the potential powers of the analytic engines of Babbage. It was this paper that won him, not only the admiration of the distinguished logician Augustus de Morgan but a place on the faculty of Ireland's Queen's College. This was his first public contribution to the vast subject which his work inaugurated and in which he was to win enduring fame for the boldness and perspicacity of his vision. Boole reduced logic to an extremely easy and simple type of algebra. This book laid the foundation of new branch of mathematics called Boolean algebra. In 1849, he was appointed Professor of mathematics there. He taught there for the rest of his life, gaining a reputation as an outstanding and dedicated teacher. This post made him financially independent and he made excellent use of his comparative freedom from financial worry. Without a school to run, Boole began to delve deeper into his own work, concentrating on refining his 'Mathematical Analysis', and determined to find a way to encode logical arguments into an indicative language that could be manipulated and solved mathematically. He came up with a type of linguistic algebra, the three most basic operations of which were (and still are) AND, OR and NOT. It was these three functions that formed the basis of his premise, and were the only operations necessary to perform comparisons or basic mathematical functions. In May 1851 Boole was elected as Dean of Science, a role he carried out conscientiously. By this time he had already met Mary Everest his would be wife with whom he married on 11 September 1855. It proved a very happy marriage with five daughters. In 1854, he published his work on logic An investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities. This research was based on a binary approach processing only two objects- the yes-no, true- false, on-off, zero-one approach. In 1859, he published his Treatise on the Calculus of Finite Differences. He published around 50 papers and was one of the first to investigate the basic properties of numbers, such as the distributive property, that underlie the subject of algebra. An American logician Charles Sanders Pierce spend more than 20 years modifying and expanding Boole’s treatise realising the potential for use in electronic circuitry and eventually designing a fundamental electrical logic circuit. He did introduce Boolean algebra into his university logic philosophy courses. The development of Boolean algebra was fundamental to mathematical logic, and is the basic logical tool in designing modern computer. Unfortunately, Boole's life was cut short when he died of a 'feverish cold' at the age of 49, after walking 2 miles through the rain to get to class and then lecturing in wet clothes. He died on December 8, 1864 in Ballintemple in Ireland. After his death his wife Mary Boole applied some of the ideas which she had acquired from him to rationalizing the education of young children. SEND YOUR COMMENTS TO RAJESH KUMAR THAKUR rkthakur1974@gmail.com

How to read mathematics

How to Read Mathematics?

Mathematics is a language that can neither be read nor understood without initiation.

You can’t read a math book the way you read other books. It takes a special approach to read maths. Mathematics is not like a novel reading. It can never be understood if you go through it starting page 1 to the last page in one breath. In order to understand mathematics you must develop a reading protocol to get benefit. Poetry calls for a different set of strategies than fiction, and fiction requires another set of strategies than nonfiction. Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics.  Always adopt pen and paper for doing mathematics otherwise you are simply deceiving yourself. You will never be able to understand mathematics unless or until you have done it in writing.

Reading mathematics is difficult because mathematics is difficult. It is considered the fiercest subject in the world and this is due to our pre notion approach. Mathematics is indeed the success gate of science and that’s why the great mathematician C F Gauss said-- Mathematics is the queen of all subjects.

Mathematics has the distinction of having the dense writing style in the text book which is not easy to understand. It is not true. Mathematics text requires more involvement from the reader than most texts in other subject. Hence, reading mathematics takes longer time and attempting to read mathematics too fast results to frustration. According to Adler- if a text is worth reading at all it is worth three readings at least. Mathematics is not a novel where you become absorbed in the plots and characters. The scene depicted is exaggerated but mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. Both a mathematics article and a novel are telling a story and developing complex ideas, but a math article does the job with a tiny fraction of the words and symbols of those used in a novel. Let us learn the technique to read mathematics.

1. Synthetic Reading of the Book: - In the synthetic reading, the reader proceed from the parts to the whole. The parts are the building blocks; you can’t jump on the second until you finish the first. Learn the basics and make a strong base. Reading mathematics is not at all a linear experience. Understanding the text requires cross references, scanning, pausing and revisiting. You can get a clear picture of next chapter if you have prepared and done well in the previous chapter. Nobody understands something complex on first reading. If after devoting for an hour you understand only one or two main points of the chapter don’t be sad about this. It doesn’t mean that there is anything wrong with you. The first reading just lays the framework for you to fill in later with details.

            A) Read the chapter introduction and each section summary.                                             B) Skim the chapter; circle the new words that you don’t understand. Consult the      dictionary for those words. Once you understand the concept erase the circle.                 For better clarification, if need be, consult your teacher.                          

C) Read with concentration. While reading the textbook, highlight the important result, formula, theorem etc.                                                                                    D) Before moving to the main topic, go through the example with noticing each and every step properly. In the example you might find many steps missing. Don’t jump to the conclusion that you will do the same in the examination. Try to find the missing steps with proper reasoning. Write those steps in text book for better understanding. Remember you learn math by doing, not by reading. If you don’t understand anything, refer to another math textbook, computer software program or consult your instructor, teacher for better understanding.                                  E) Once you find yourself equipped with the content of chapter, explain it to your friend.  If there is no one else available, can you explain it aloud, without stumbling? If you can do that, you probably understand it.

1. Learn Mathematical Symbols and vocabulary: - Before you attempt to understand the mathematical terms, you should learn the mathematical symbols commonly used in your textbook. In Algebra, you may see symbols like Σ (sigma) for addition, Π for multiplication. In trigonometry you may see the symbols like α (Alpha), β (Beta), γ (Gamma), δ (Delta), θ (Theta), φ (Phi) etc. Make a list of symbol and learn it. If possible, copy the list of math symbol from your math book or math dictionary and paste it on the wall so that you can revise the list every now and then. Moreover, it is also advisable to consult math dictionary for new words. Mathematics obtains much of its power by constructing a very precise vocabulary. A strong vocabulary can help you to understand the content of the book more easily and effectively. The most fundamental issue involved in reading mathematics for meaning is to get some sense of the mathematical words, phrases and applications. Mathematical text is a combination of jargon. Therefore we should not attempt to read mathematics as we would other type of reading. Reading mathematically is more than reading the printed words on the page. It requires linking the words with the mathematical ideas that are involved. So , making sense of mathematical prose is a complex process involving understanding mathematical terms and this can be done only when you learn the math vocabulary.
2. Learn Mathematical statement, formula: - In order to understand the mathematical concepts, you must learn the formula, statement of theorems, axioms etc. so that you may have the better understanding of the chapter. Try to learn the way how the formulas are generalized. It is better to learn the way formula is constructed rather than remembering the bags of formula. Make a poster of formula and paste it on the wall so that you can have a look of it every morning. Spend at least 5-10 minutes daily in revising these. You can develop your own way of remembering the formula. It is advisable to consult your teacher and ask him/ her to teach you the innovative ways of making and learning formula. You can also enquire about Pascal Triangle that is very helpful in understanding the binomial expansion of any positive power. Likewise you can also learn AFTER SCHOOL TO COLLEGE to remember the mathematical formula for sin(90+ө) , sin (180 ± ө), sin (270 ±ө) and sin (360±ө) etc. I still remember my first day in Trigonometry class when teacher wrote PANDIT BADRI PRASAD, HARI HARI BOLE to define the different Trigonometric ratios such as sine, cos, tan etc. I shall therefore put much emphasis on learning different techniques to learn formula rather than just learning the hundreds of formula.
3. Don’t be a passive Reader: - Mathematics is all about putting an extra effort. Go the extra mile and do some research works to find out what the particular page you are struck into want to tell you. Always keep a pen and paper with you while you read math text book. Write steps and solve the step by your own rather than seeing what is written in the book. Many results must be given in your book of which the details are suppressed. It is expected from the readers to fill the gap. If you are doing the work without pen and paper, you may probably not be able to understand certain thing. Always remember—to a great extent, people think mathematically through writing. It is hard to do in your head. Read the chapter introduction and each section summary. Skim the reading material that will allow you to see that if problems presented in one chapter is being explained in next chapter or not. As you skim the chapter, circle the new words that you don’t understand and ask these words to your parents first and then to teacher in the next day if you don’t understand these new words after reading the assignment .Mathematics says a lot with a little. Math uses special words to mean specific things. Sometimes, words are used differently in math than in regular language. For example PRIME, SET, VOLUME, COMPLEX, REAL, RATIONAL etc have different meaning in maths than they usually do. Understanding maths term will help you understand topic. Remember a math text book is very difficult. It might take you half an hour to read and understand just one page so don’t get impatient in such situation. You must be an active partner. At every stage you must decide whether or not the idea being presented is clear. Ask yourself these questions: ---                                                                                                                          Why is this idea true?                                                                                                          Is there any better option to challenge the idea?                                                              Could I convince someone else that it is true?                                                                       Why did the author use a different argument?                                                                         Why this particular formula or theorem was used here?
4. Study the examples and figures: - Most textbooks include examples with detailed solutions. The author of textbook design the exercise on the basis of examples discussed. Paying attention to the examples discussed provide an excellent opportunity for you to assess your readiness to begin the assigned exercises at the end of the section. If you go through the introduction of the chapter and read the example and correlate the matter, you may find the exercise handling is a cake walk for you.
5. Take a break when you are struck on a problem: - Many a time when you are trying to solve a math problem, you may force yourself to keep working at it until you find the solution. This may not be as useful as you think. If you get struck on one example, put it aside for a while.                                                                                                                                Take a break of 5-10 minutes.                                                                                                 Drink a glass of water or gat a snack.                                                                                          Do something completely different that will get your mind on something else. When you feel refreshed, go to the problem and read it with a fresh eye. This time you will have the solution. If problem still persists, ask your parents for immediate help.
6. Don’t make too much selection: - I have seen students making selection in reading. In the age of technology, students are aware of the pattern of the questions, blue prints of the syllabus and they make choices. A mediocre student selects the list of chapters that are easy to understand and estimate the necessary marks to pass by reading those few selected chapters. There is no harm in making selection of chapters but never thing that your ultimate goal is to pass in the examination with minimum passing marks. Go through all the chapters thoroughly and do all types of questions. Don’t make selection of important question at the very beginning. Remember                                                                                                                  Better know everything of something                                                                                    rather than something of everything                                                                                                       AND                                                                              Nothing like if you know everything of everything.                                              Selection of questions and chapters at the beginning will limited your knowledge and though you may feel good that this is the smartest way to pass the examination but it will do much harm to you in later phase of your life.
7. Put a special mark in your book: - Your teacher tells you to do some questions 5-10 times because that question is important. Your teacher with his experience knows those particular questions are important for the examination and lets you know in advance. My advice is in order to simplify the reading and also to select the important questions for your revision work before examination you make a regular habit of marking such questions with red ink to identify it and ease your learning during examination. You may mark with the initial letter of your name. Moreover, you can give special mark to those questions which you feel are difficult or require some special attention. You may write the special character at the first page of your book such as—R = Revision, I = Important, R*** = Revise it three  times etc. don’t include in this list those question which are based on the direct application of formula and which you think you can do.
8. Be a regular reader: - If you love watching cartoons/ serials on the Television and you miss a episode or two, you will still be in a position to judge what had happened in the last episode. But the same is not true for mathematics. Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. A mathematics article and a serial/ cartoon shown on TV are telling a story but a math article does the job with a tiny fraction of the words and symbols of those used in a serial/ cartoons. The TV serial uses language to evoke emotions and present themes which defy precise definition but the beauty in a mathematical article is in the elegant efficient way it concisely describe precise ideas of great complexity.                                                                         So if you are jumping one chapter or a single step in mathematics than there is greater chances that you fail to understand the whole concept and deflect the way. Always remember, mathematics can’t be mastered in a single day. You have to read mathematics regularly for at least 1 hour a day if you can’t put maximum in a particular day. Keeping yourself away for a week or two from math book will bring you to the mark zero and you have to pad up again to regain the knowledge because it is sure that you have forgotten almost 90% of what you have learnt a week or 15 days ago.  As you watch the television serial whether it is an entertainment channel or informative one, you must have seen that before the serial begins it recast the glimpses of the previous day. In the same manner, you must have to revise the odd words, formulas, important facts of the previous chapter before switching to the next.
9. Get your problem solved instantly: - Mathematics can be learnt if you resolve the problem instantly. Keep the phone number of your class teacher and your tutor and call him or her when you feel helpless in understanding a particular step or problem. It is better to consult the doctor when you feel uneasy rather than wait for the problem to grow. You can’t wait for a week to consult your class teacher to resolve the problem you face a week ago. Suppose you are reading a chapter from Algebra and while solving a particular sum you faced a problem. My simple advice is you first consult your parents. If they are good at mathematics then you will get the answer instantly. If you can’t get the answer, look at the example given in that chapter. If you have kept some reference book than you can consult the book too. If your labour fails in vain then don’t get hesitate in calling your teacher or tutor after all they are to help you. But don’t call them late night.
10. Keep the Reference Book/ Journal with you: - Reading mathematics is not at all a linear experience. Understanding the text requires cross references, scanning, pausing and revisiting. When you read a math book, you may or may not encounter several problems. Don’t assume that understanding each phrase, word, symbol etc. will enable you to understand the whole idea. This is like trying to see a portrait painting by staring at each square inch of it from the distance of your nose. You will see the detail, texture and colour but miss the portrait completely. A math article tells a story. Try to see what the story is before you delve into the details. You may not get the detail in your text book itself so you have to consult some other book or journals. A class text book has a limited number of problems and you may not get varieties of problem in it but a good reference book will help you to solve varieties of problem boosting your confidence. A good reference book will help you to understand the definition, properties of numbers, geometrical shapes etc and also the algorithm to solve problem of particular type.
11. Pay attention to your anxious feelings:- Some people feel like they are simply not able to learn math. They may have been unsuccessful in learning math earlier or may have been told that they could not do math. This is called math anxiety. Math anxiety has nothing to do with abilities. If you feel that you can’t do math simply doesn’t mean that you are unable to do math. The feeling can get in the way. If you see a problem that is difficult for you, you may unknowingly tell yourself that you can’t do it. A key to getting over math anxiety is to figure out what is going on and manage them until the problem over power you. Make a journal and write your good/ bad feeling in it. If you feel happy after solving a particular problem, write it down. In the similar way if you feel distressed or nervous when you fail to solve a problem, also write it. As you know thinking in math is related to doing. This exercise of writing your feeling allows your mind to critically analyse the problem. You analyse to anticipate problem areas when you write down your feeling and this way you are helping yourself in another way.
12. Don’t read too fast: - In the very beginning, I have told you that mathematics book should not be read as novel. You may read 10-20 pages in half hour if you are a avid novel reader but the same may not be true for mathematics. The same amount of hour in a math book may finishes with 1-2 pages depending on the chapter you are reading and how experienced you are at reading mathematics. There is no substitute of work and time. You can speed up your math reading skill by practicing. In novel reading, you may skip the unwanted paragraph but still you can understand what novel is trying to say. This may not be true in case you are reading math. A single paragraph may have hundreds of hidden facts that will be essential to understand the next paragraph. So, please be patience and hold your breath. Mathematics is called the queen of all subjects and you can’t make please queen in your zig-zag style of reading.
13. Practice, practice and practice: -       Mathematics is not a subject you learn in a single attempt. Keep pen and paper and do as many problems as is required to ensure that you understand the concept. The amount of practice may vary from person to person but you can’t skip practicing. You will want to practice a concept until it makes sense and until you are fluent at finding solutions to various problems within the concept readily. When you complete a set of questions in a row, you are probably to the point of understanding. Re- visit the same problem after a month to check whether you are still capable of doing the same problem with the same amount of easiness or not. Think of math the way one thinks about a musical instrument. Most of us don’t just sit down and play an instrument. We first take lessons, practice it several times before moving to the next lesson. A good musician takes out time to review and never stops practicing. Mathematics is like the same. You need to practice more and more. Do extra exercise. Go beyond what is asked for. If you are asked to do 20 odd questions, do it but never put yourself in a particular boundary. Buy another book and practice more and more until you reach to the point of fluency with the concept. Doing the extra practice questions will help you to grasp the concept more readily. Be sure to re- visit the exercise a few months later to ensure that you still have a grasp of it.
14. Discuss what you learnt with your friends:-  The best way to get good at math is to discuss what you have learnt to your friend. It is said- Two heads are better than one. When you discuss the problem in a group, you are clarifying the concept for you by looking at it in a different way. When you learn something new from your parents, tutor or book you read that is completely different from what you have learnt in class, never forget to share it with your friend. Remember what William Glasser says-                               10% of what we READ, 20% of what we HEAR, 30% of what we SEE , 50% of what we SEE and HEAR,70% of what is DISCUSSED with OTHERS ,80% of what is EXPERIENCED PERSONALLY , 95% of what we TEACH TO SOMEONE ELSE.                       
15. Read backwards and forwards: - Mathematical knowledge can’t be gained by straight reading. You have to move in all the directions. You may not fully learn something in chapter 1 until you are halfway through chapter 2 or chapter 3. Hence it is a good idea to look back once in a while over previous sections so that you can have the control of what you have learnt previously and your knowledge don’t get outdated.  

I hope my advice will help you to read mathematics in a planned manner and you know when any work is done with a full proof planning an outstanding result is sure to come.    

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Rajesh kumar Thakur

rkthakur1974@gmail.com

Mathematics and Hindu Religion lecture series

Fermat

Pierre De Fermat

Born: - August 17, 1601                     Died: - January 12, 1665

Fermat was one of the leading mathematicians of early 17^th century. Pierre Fermat's father was a wealthy leather merchant and second consul of Beaumont- de- Lomagne. Although there is little evidence concerning his school education it must have been at the local Franciscan monastery. He was an amateur mathematician.

 He attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. In Bordeaux he began his first serious mathematical researches and in 1629 he gave a copy of his restoration of Apollonius Plane loci to one of the mathematicians there. From Bordeaux Fermat went to Orléans where he studied law at the University and spent his working life as a magistrate in the small provincial town of Castres.

After he moved to Toulouse , he gained a new mathematical friend Carcavi.  In 1636 Carcavi went to Paris as royal librarian and made contact with Mersenne and his group. Mersenne's interest was aroused by Carcavi's descriptions of Fermat's discoveries on falling bodies, and he wrote to Fermat. Fermat replied on 26 April 1636 and, in addition to telling Mersenne about errors which he believed that Galileo had made in his description of free fall. Fermat had little interest in physical applications of mathematics. Even with his results on free fall he was much more interested in proving geometrical theorems than in their relation to the real world. Fermat sent a letter to Mersenne containing two problems on maxima which Fermat asked Mersenne to pass on to the Paris mathematicians and this was to be the typical style of Fermat's letters, he would challenge others to find results which he had already obtained. Roberval and Mersenne found that Fermat's problems in this first, and subsequent, letters were extremely difficult and usually not soluble using current techniques. They asked him to divulge his methods and Fermat sent Method for determining Maxima and Minima and Tangents to Curved Lines.

His reputation as one of the leading mathematicians in the world came quickly but attempts to get his work published failed mainly because Fermat never really wanted to put his work into a polished form. However some of his methods were published in Cursus mathematicus a work by Herigone. With Pascal, Fermat stands as one of the founder of mathematical theory of probability. Pierre de Fermat independently founded the new branch of mathematics called Analytical Geometry. This work led to violent controversies over question of priority with Rene Descartes. Fermat's pioneering work in analytic geometry was circulated in manuscript form in 1636, predating the publication of Descartes' famous La géométrie. This manuscript was published posthumously in 1679 in "Varia opera mathematica", as Ad Locos Planos et Solidos Isagoge, ("Introduction to Plane and Solid Loci").^[

In Methodus ad disquirendam maximam et minima and in De tangentibus linearum curvarum, Fermat developed a method for determining maxima, minima, and tangents to various curves that was equivalent to differentiation.

He is probably best known for his work on number theory. He also left one of the famous unsolved problems in maths called- Fermat’s last theorem. This theorem states that xⁿ + yⁿ = zⁿ has no non-zero integer solutions for x, y and z when n > 2. in the margin of Bachet's translation of Diophantus's Arithmetica , Fermat worte “ I have discovered a truly remarkable proof which this margin is too small to contain.” These marginal notes only became known after Fermat's son Samuel published an edition of Bachet's translation of Diophantus's Arithmetica with his father's notes in 1670. Unsuccessful attempts to prove the theorem over a 300 year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries. The truth of Fermat's assertion was proved in June 1993 by the British mathematician Andrew Wiles.

                                          

 

The second stamp was released after it was proved by Andrew Wiles.

In 1656 Fermat had started a correspondence with Huygens. This grew out of Huygens interest in probability and the correspondence was soon manipulated by Fermat onto topics of number theory. This topic did not interest Huygens but Fermat tried hard and in New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others. Fermat described his method of infinite descent and gave an example on how it could be used to prove that every prime of the form 4k + 1 could be written as the sum of two squares. 

He died at Castres, Tarn on January 12, 1655. The oldest and most prestigious high school in Toulouse is named after him: 

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rkthakur1974@gmail.com