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### BrownBag Seminar talk byProfessor B V Ramana on

“THE ART OF TEXT BOOK WRITING”

Who am I?

- I am B V Ramana .
- Ph.D. from Indian Institute of Technology,

(IIT) Bombay, India.

- Professor & Head of Mathematics for 20 years

at JN Technological University, Hyderabad, India.

- Wrote 8 text books published by McGraw-Hill and Pearson Education.
- 6 years of foreign assignments.
- Presently professor at SKKU .

Good News!!!

- Professor Sang-Gu Lee, Chairman, Department of Mathematics , SKKU, has initiated a project of writing text books in MATHEMATICS .
- I am happy to inform you that myself and

Professor Sang-Gu Lee have already started writing

a text book entitled “CALCULUS” in English for Korean students.

- We have completed few chapters so far.
- This book consisting of 18 chapters is comprehensive and self contained.
- This project work is in progress.
- God bless SKKU !

Preamble:

- Writing a text book is both an art and science. By being a text book writer, you will be liberated from the confines of the walls of a class room. You can project your innovative ideas and methods of teaching to a very large section of audience, both students and faculty (professors). You will be popular and will earn lot of money too.
- It is well known that a CALCULUS text book writer from USA is earning more than one million US dollars every year only through royalties!!!.

Preamble( contd)

- Your name will spread from your parent University to the rest of the temples of learning. In other areas of scientific work, your work and knowledge may be appreciated by a very few colleagues in that specialization only.
- But by writing a popular text book, your knowledge/presentation will be appreciated by many and you have self satisfaction and extra money worth earning year after year. So in this talk, let me share with you the one million dollar secret of text book writing.

Introduction

- The word mathematics is derived from the Greek word mathemasembodies the notions of knowledge, cognition, understanding, and perception. In the end, mathematics is about ideas. Putting mathemas on paper will require writing sentences and paragraphs in addition to the equations and formulas.
- Mathematicians actually spend a great deal of time writing. If a mathematician wants to contribute to the greater body of mathematical knowledge, he/she must be able to communicate his/her ideas in a way which is comprehensible to others. Thus, being able to write clearly is as important a mathematical skill as being able to solve equations.

Ideas:

- Learning how to communicate mathematical ideas clearly can help you explain your thinking process to another person like your boss, a co-worker, or an elected official who is likely knows fewer maths than you do.
- Hence, mathematics which is written clearly and carefully is more likely to be correct. However, a list of calculations without any explanations omits ideas. The ideas are the mathematics. So a page of computations without any writing or explanation contains no math.

Communication:

- Through a text book one can communicate mathematical reasoning and ideas clearly to another person.
- Writing is an essential form of communication, especially for mathematics. In mathematics all the tools of ordinary language plus the additional conventions of mathematical symbolism are used.
- Solutions consist of both words and symbols. In mathematics there are additional constraints and conventions.

Mathematical writing:

- Represent mathematical information graphically, symbolically, numerically, and verbally with clarity, accuracy, and precision. Learn to communicate mathematically. Writing mathematics is not the easiest thing to do. Writing mathematics is a skill which takes practice and experience to learn. Thus learning to write mathematics can only be done by actually by doing it.
- It may be frustrating at first, but it will get easier with time and you will get better at it. Being able to write mathematics well is a good skill to learn, and one which you will keep for a life time.

About the Reader:

- Although a text book is evaluated by (present) colleagues, the main aim is to make the text book useful, stimulating and interesting to the (future generation of) students.
- The level or standard of the text book depends on the reader’s previous knowledge, intelligence, interest in the subject.

Size of text book:

- Note that a physical lecture is interactive

(face -to- face) whereas a text book is a lecturer in absence. Text book is neither an encyclopedia to be elaborate nor a research paper to be brief.

- While writing a mathematics text book, remember that the author is well versed in the subject, while the reader/student may be a novice or just a beginner.

SUPER HIGH WAY

- Imagine the text book as a highway with the teacher as the road signs ( road map) and with the student as a unfamiliar traveler.
- Keep in mind the mathematical knowledge of the reader. While writing the solution to a problem, imagine that you are writing for a student in your class.
- You are writing for someone who doesn\'t know the subject. (That someone may be you! )

Characteristics of a Good Text Book

- Communicate to the reader why and how the solution is arrived at. Convince that your particular reasons and your particular means to the solution are correct. A good text book not only should provide clear explanations, but should also be able to persuade a skeptical reader.
- Many times, if the same solution is arrived through alternate routes, make your writing more persuasive. Comparison of a graph with real-world information, pictures and graphical depictions can be very helpful for the reader.

General to specific:

- Specific examples will also help to make book more persuasive.
- Help a reader understand an abstract general argument by showing how the argument applies to a specific case.
- Also use “extreme" cases to show the limits of an argument.

Innovativeness:

- Thinking about the reader will help to decide which details need to be included and which details should be left out.
- To understand mathematics better, judge what is important and what is unimportant. Leave out what is unimportant.
- On the other hand, don\'t leave out anything which is critical to the key ideas you are trying to explain.

Myth or misconception?

- Neverassume that the reader is familiar with the problem being solved
- or that the reader is in the same mind set as that of the author.

Historical notes:

- Wherever necessary, provide historical back ground, anecdotes, and authors’ comments of the subject in general and of scientists, research papers in particular.

For example: Pierre Simon Marquis De Laplace

( 1749-1827), French mathematician, known as the Newton of France and teacher to the French emperor Napoleon Bonaparte.

Cultural notes:

- For instance:
- Probability distribution of t is also known as “Student t-distribution” which was published by

William Sealy Gosset ( 1876-1937) English statistician, who published under the pseudonym “student” .

Foreign words:

- Explain the significance of the foreign words and phrases.
- For example: Greek: “orthogonal” : right angle
- Latin : “trajectories” : cut across.
- For instance :

Wronski determinant or Wronskian:

I.M.Hone (1778-1853) Polish mathematician, who changed his name to Wronski.

Lecture plan:

- A text book may be planned for lecture lessons as follows:
- one lecture of 60 minutes ≈15 pages consisting of
- 5 pages of introduction of concepts, proof of theorems
- 5 pages of examples and
- 5 pages of graded exercises

( starting from the simple to moderate to difficult) , quiz, true-false questions, additional hard questions.

- Provide, wherever possible, the geometrical, physical and practical significance of the topic .

Theorems:

- Proofs of theorems should contain implications and the converse (with counter examples).
- For example: Differentiability ⇒ Continuity,
- But converse is not true.
- Counter Example: f(x) = I xI is continuous but not differentiable at x= 0.
- Keeping in view of the teacher (instructor), a text book may contain analysis of the lesson, additional information and concrete models on the topic.

Formulas:

- Write important or long formulas on separate lines. It is important to use words and symbols appropriately.
- Part of being able to write mathematics well is knowing when to use symbols and knowing when to use words.

Misuse of “=“

- Don\'t use mathematical symbols when you really mean something else.
- A common mistake is to misuse the “=" symbol.
- For instance :

IBI = IBTI ,IA –λI I = I(A- λ I)T I = IAT – (λ I)T I = IAT – λ ITI = IAT – λ I I

- Do not use the equal sign to really mean “the next step is" or “implies".
- A slight improvement is to use arrows instead of equal signs , but still not desirable:

IBI =IBTI,IA –λI I = I(A- λ I)T I ⇒IAT –(λ I)TI= IAT–λ ITI = IAT – λ I I

Descriptive:

- With a sequence of calculations, sometimes it is best to just place each equation on a separate line.
- For a difficult computation where the reader might not readily follow each step, include words to describe the steps taken.

Good presentation:

We know that IBI = IBTI .

In the above result with B = A –λI , we have

IA – λII = I(A- λI)T I

= IAT –(λI )TI applying transposition

= IAT – λITI since transpose of a scalar λ is λ itself

= IAT –λI I since transpose of I is I itself.

- Thus A and AT have the same characteristic polynomial.
- Hence the Eigen values of A and AT are same.

Odd manners:

- Don\'t start a sentence with a formula. While it may be grammatically correct, it looks strange.
- t = 10 then w = 5000,
- f max at x = 6.
- Adding just a word or two makes these examples look better.
- When t = 10 then w = 5000, so we have
- The function f attains maximum at x =6.

Clarity:

- Use precise and correct language. Make sure that the words you use really have meaning.
- Mathematics requires very precise use of language.
- Avoid overuse of the word “it" or “that “. Mathematical papers with a lot of pronouns like “it" and ”that" tend to be hard to read. It is often hard for the reader to see what “it" is referring to.
- Try to write as simply and directly as possible.
- No one likes to read ponderous pretentious prose.

“An introduction”

- In the introduction, first, state the problem and its significance using “real-world” terms. The introduction should “capture" the reader.
- Make sure that you explain the problem to the reader. Assume that the reader is unfamiliar with the problem.
- The introduction should also try to indicate to the reader why the problem is interesting and give some indication of what will follow in the text.

Defining variables and formulas.

- Explicitly state all letters in formulas.
- Common phrases used in mathematics are “Let” ; “we get "; “ we have” ; “therefore”; “hence”;

“ since”; “because” , “it follows”, etc.

Double usage:

- When a symbol is used to represent two different things (even, or perhaps especially, if those things are similar), the reader (and the writer!) can become confused.
- A symbol used in two different ways is not only confusing, but often results in incorrect mathematics!

Using pictures in mathematics.

- A picture can really be worth a thousand words. However fully explain how a picture, a diagram, a graph, or some other visual mathematical representation included fits into the mathematical argument.
- All diagrams should be carefully labeled.

Requirements For a Good Author / Teacher :

- Disciplined character
- Creativity
- Analytical reasoning
- Respectability
- Morality

Final word:

- A person’s

(i) intelligence

(ii) knowledge

(iii) character

(iv) ability

(v) confidence

cannot be stolen by anybody.

So go ahead.

The world is before you.

Get name, fame and money by becoming a good author and teacher.

Checklist:

- Begin with an introduction if possible.
- To write technical equations use TEX or LATEX, professional mathematical typesetting languages.
- Use complete sentences, correct grammar, and correct spelling.
- Start each sentence with a word, not with a mathematical symbol.
- Two mathematical expressions or formulas in a sentence should be separated by more than just a space or by punctuation; use at least one word.

Checklist:

- Don\'t end a line with an equal sign or an inequality sign.
- Insure that every statement is mathematically correct.
- Strive for a good balance between words and symbols.
- Honor the equal sign.
- Use different letters for different things.
- Remember mathematics is case sensitive.

Checklist:

. Define and describe adequately all terms or variables,

in units if any.

. Label all the diagrams, tables, graphs and other pictures.

. Once a variable has been assigned a meaning, do not re-use it with a different meaning in the same context.

. There is a distinction between the definite article ( "the" ) and the indefinite articles ( "a" and "an" ).

. Be sure that the use of a term agrees with the definition

of that term.

. State all your assumptions.

Checklist:

- Answer the problem.
- Conclude the solution of a problem with a clear and complete statement of the conclusion in “real-world” terms.
- It is incorrect to write two operation symbols next to each other: For example: 3- - 4 is incorrect; Correct syntax: 3-(-4).
- Distinguish the words equation, expression, and function. (An expression is an algebraic combination of terms containing no verb, an equation is a mathematical statement which contains an equal sign , a function consists of a domain, a range, and a rule).
- Finally proof read everything including spell.ing and punctuation.

Do not use:

- Incorrect English.
- Incorrect mathematics.
- Too many words and too few symbols, or vice versa.
- Hard-to-read format.
- Frequently pronouns such as “it”, “that”.
- The same symbol for different quantities.
- Abbreviations.

Contents:

1. Functions

2. Limits

3.Differentaition

4. Application of differentiation

5. Integration

6. Applications of integration

7.Inverse functions

8. Techniques of integration

9. Further applications of integration.

10. Differential Equations

11.Parametric equations

12. Infinite sequences and series

13. Vector algebra

14.Partial derivatives

15. Multiple integrals

16. Vector Differential calculus

17 Vector Integral calculus

18. Ordinary Differential Equations of 2nd and higher order.

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