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

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