PREFACE TO ‘HOW THE OTHER HALF THINKS: ADVENTURES IN MATHEMATICAL REASONING’
A
Occasionally, in some difficult musical compositions, there are beautiful, but easy parts - parts so simple a beginner could play them. So it is with mathematics as well. There are some discoveries in advanced mathematics that do not depend on specialized knowledge, not even on algebra, geometry, or trigonometry. Instead, they may involve, at most, a little arithmetic, such as ‘the sum of two odd numbers is even’, and common sense. Each of the eight chapters in this book illustrates this phenomenon. Anyone can understand every step in the reasoning. The thinking in each chapter uses at most only elementary arithmetic, and sometimes not even that. Thus all readers will have the chance to participate in a mathematical experience, to appreciate the beauty of mathematics, and to become familiar with its logical, yet intuitive, style of thinking.
B
One of my purposes in writing this book is to give readers who haven’t had the opportunity to see and enjoy real mathematics the chance to appreciate the mathematical way of thinking. I want to reveal not only some of the fascinating discoveries, but, more importantly, the reasoning behind them. In that respect, this book differs from most books on mathematics written for the general public. Some present the lives of colorful mathematicians. Others describe important applications of mathematics. Yet others go into mathematical procedures, but assume that the reader is adept in using algebra.
C
I hope this book will help bridge that notorious gap that separates the two cultures: the humanities and the sciences, or should I say the right brain (intuitive) and the left brain (analytical, numerical). As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role. The alleged gap can be narrowed or completely overcome by anyone, in part because each of us is far from using the full capacity of either side of the brain. To illustrate our human potential, I cite a structural engineer who is an artist, an electrical engineer who is an opera singer, an opera singer who published mathematical research, and a mathematician who publishes short stories.
D
Other scientists have written books to explain their fields to non-scientists, but have necessarily had to omit the mathematics, although it provides the foundation of their theories. The reader must remain a tantalized spectator rather than an involved participant, since the appropriate language for describing the details in much of science is mathematics, whether the subject is expanding universe, subatomic particles, or chromosomes. Though the broad.outline of a scientific theory can be sketched intuitively, when a part of the physical universe is finally understood, its description often looks like a page in a mathematics text.
E
Still, the non-mathematical reader can go far in understanding mathematical reasoning. This book presents the details that illustrate the mathematical style of thinking, which involves sustained, step-by-step analysis, experiments, and insights. You will turn these pages much more slowly than when reading a novel or a newspaper. It may help to have a pencil and paper ready to check claims and carry out experiments.
F
As I wrote, I kept in mind two types of readers: those who enjoyed mathematics until they were turned off by an unpleasant episode, usually around fifth grade, and mathematics aficionados, who will find much that is new throughout the book. This book also serves readers who simply want to sharpen their analytical skills. Many careers, such as law and medicine, require extended, precise analysis. Each chapter offers practice in following a sustained and closely argued line of thought. That mathematics can develop this skill is shown by these two testimonials:
G
A physician wrote, The discipline of analytical thought processes [in mathematics] prepared me extremely well for medical school. In medicine one is faced with a problem which must be thoroughly analyzed before a solution can be found. The process is similar to doing mathematics.’
A lawyer made the same point, “Although I had no background in law - not even one political science course — I did well at one of the best law schools. I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles. Lawyers who have studied mathematics can master the legal principles in a way that most others cannot.’
Questions 27-34
Reading Passage 196 has seven sections, A-G. Which section contains the following information?
Write the correct letter, A— G, in boxes 27 — 34 on your answer sheet.
NB. You may use any letter more than once.
27. a reference to books that assume a lack of mathematical knowledge
28. the way in which this is not a typical book about mathematics
29. personal examples of being helped by mathematics
30. examples of people who each had abilities that seemed incompatible
31. mention of different focuses of books about mathematics
32. a contrast between reading this book and reading other kinds of publication
33. a claim that the whole of the book is accessible to everybody
34. a reference to different categories of intended readers of this book
Questions 35-40
Complete the sentences below. Choose ONE WORD ONLY from the passage for each answer.
Write your answers in boxes 35- 40 on your answer sheet.
35. Some areas of both music and mathematics are suitable for someone who is a ....................
36. It is sometimes possible to understand advanced mathematics using no more than a limited knowledge of ....................
37. The writer intends to show that mathematics requires .................... thinking, as well as analytical skills.
38. Some books written by .................... have had to leave out the mathematics that is central to their theories.
39. The writer advises non-mathematical readers to perform .................... while reading
40. A lawyer found that studying .................... helped even more than other areas of mathematics in the study of law.
LỜI GIẢI CHI TIẾT
27. A reference to books that assume a lack of mathematical knowledge.
Answer: D
Supporting sentence: Other scientists have written books to explain their fields to non-scientists, but have necessarily had to omit the mathematics, although it provides the foundation of their theories.
Keyword: non-scientists, omit, assume, mathematical knowledge
Key Location: Section D, 1st line
Explanation: This is the correct answer because when other scientists intend to write a book for people from a non-science background, they omit the mathematics as they rationally judge that these readers wouldn’t be able to comprehend the books otherwise. This is clearly mentioned in Section D.
28. The way in which this is not a typical book about mathematics.
Answer: B
Supporting sentence: I want to reveal not only some of the fascinating discoveries, but, more importantly, the reasoning behind them.
Keyword: differ, general public, reasoning, typical book
Key Location: Section B, 3rd line
Explanation: This is the correct answer because a typical math related book either talks about discoveries or mathematical procedures in detail but the author’s book intends to give a taste of mathematics to those who didn’t have the chance to enjoy it. He does so by revealing the reasoning behind various such thinking. In this way, the book becomes unique in itself. All this is explained in detail in Section B.
29. Personal examples of being helped by mathematics
Answer: G
Supporting sentence: Many careers, such as law and medicine, require extended, precise analysis.
Keyword: testimonials, analyze, many careers, physician, lawyers
Key Location: Section G, (1st & 2nd examples)
Explanation:This is the correct answer because in Section G, there are 2 testimonials one by a physician and the other by a lawyer where they mention how mathematics helped them in their respective fields.
30. Examples of people who each had abilities that seemed incompatible
Answer: C
Supporting sentence: To illustrate our human potential, I cite a structural engineer who is an artist, an electrical engineer who is an opera singer, an opera singer who published mathematical research, and a mathematician who publishes short stories.
Keyword: gap, not restricted, incompatible
Key Location: Section C, last line
Explanation: This is the correct answer because in section C, the author gives a lot of instances where people have excelled in both humanities and science to show that the gap doesn’t really exist and can be overcome by anyone.
31. Mention of different focuses of books about mathematics.
Answer: B
Supporting sentence:Some present the lives of colorful mathematicians. Others describe important applications of mathematics. Yet others go into mathematical procedures, but assume that the reader is adept in using algebra.
Keyword: different focuses, books, readers, general public
Key Location: Section B, last line
Explanation: This is the correct answer because in section B, the author mentions the various typical focuses of books on mathematics by other scientists.
32. A contrast between reading this book and reading other kinds of publication.
Answer: E
Supporting sentence: You will turn these pages much more slowly than when reading a novel or a newspaper.
Keyword: non-mathematical readers, contrast, other publications
Key Location: Section E, 3rd line
Explanation:This is the correct answer because in Section E, the author talks about how he has presented his book differently to the readers of all categories, thus drawing a contrast between his work and others.
33. A claim that the whole of the book is accessible to everybody
Answer: A
Supporting sentence: Each of the eight chapters in this book illustrates this phenomenon. Anyone can understand every step in the reasoning.
Keyword: all readers, accessible, chance to participate
Key Location: Section A, 4th line
Explanation:This is the correct answer because in Section A, the author indicates his intention that he wants to give everyone a chance to participate in the mathematical experience, thus making the whole book accessible to everyone.
34. A reference to different categories of intended readers of this book
Answer: F
Supporting sentence: As I wrote, I kept in mind two types of readers: those who enjoyed mathematics until they were turned off by an unpleasant episode…This book also serves readers who simply want to sharpen their analytical skills.
Keyword: types of readers, precise analysis, aficionados
Key Location: Section F, 1st & 2nd line
Explanation:This is the correct answer because in section F, the author mentions how he has 2 types of readers in mind while writing the book.
Questions 35-40
Complete the sentences below. Choose ONE WORD ONLY from the passage for each answer.
Write your answers in boxes 35 - 40 on your answer sheet.
35. Some areas of both music and mathematics are suitable for someone who is a Beginner
Answer: : Beginner
Supporting sentence: Occasionally, in some difficult musical compositions, there are beautiful, but easy parts - parts so simple a beginner could play them. So it is with mathematics as well.
Keyword: easy parts, simple, suitable, beginner, music, mathematics
Key Location: Section A, 1st line
Explanation: This is the correct answer because the author talks about how both music and math can be really complex but even in those complexity, there will be parts which even a beginner can process and play which means that both the domains are suitable for people just starting out.Hence “beginner” fits the blank perfectly.
36. It is sometimes possible to understand advanced mathematics using no more than a limited knowledge of Arithmetic
Answer: Arithmetic
Supporting sentence: There are some discoveries in advanced mathematics that do not depend on specialized knowledge, not even on algebra, geometry, or trigonometry. Instead, they may involve, at most, a little arithmetic, such as ‘the sum of two odd numbers is even’, and common sense.
Keyword: do not depend, little arithmetic, advanced mathematics
Key Location: Section A, 3rd and 4th line
Explanation: This is the correct answer because the author gives instances where complex mathematical discoveries often require a person to know just the basic arithmetic and some common sense to grasp the concept.
37. The writer intends to show that mathematics requires intuitive thinking, as well as analytical skills.
Answer: Intuitive
Supporting sentence: As the chapters will illustrate, mathematics is not restricted to the analytical and numerical; intuition plays a significant role.
Keyword: humanities, science, intuition, analytical, numerical
Key Location: Section C, 2nd line
Explanation:This is the correct answer because the author is trying to prove there isn’t any gap in the way science and humanities people think, and math too requires intuition along with analytical thinking.
38. Some books written by scientists have had to leave out the mathematics that is central to their theories.
Answer: Scientists
Supporting sentence: Other scientists have written books to explain their fields to non-scientists, but have necessarily had to omit the mathematics, although it provides the foundation of their theories.
Keyword: omit, foundation, other scientists
Key Location: Section D, 1st line
Explanation: This is the correct answer because the author explicitly mentions how other scientists omit mathematics to explain their field to non-scientists even though that’s the foundation of their theories .
39. The writer advises non-mathematical readers to perform experiments while reading
Answer: Experiments
Supporting sentence: It may help to have a pencil and paper ready to check claims and carry out experiments.
Keyword: carry out experiments, check claims
Key Location: Section E, last line
Explanation: This is the correct answer because while describing how differently the book is written, the author elaborates the step-wise method which may compel the readers to perform it on their own to verify. Hence experiment is the correct one word.
40. A lawyer found that studying theorems helped even more than other areas of mathematics in the study of law.
Answer: Theorems
Supporting sentence: I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles.
Keyword: success, in particular, theorems, analyze
Key Location: Section G, 2nd para, 2nd line
Explanation: This is the correct answer because the lawyer in his testimonial mentions how mathematics helped him in excelling in Law, but he particularly mentions, “theorems” which helped in his success than any other domain. Hence the word “theorems” is befitting.
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