The Heart of Mathematics: An Invitation to Effective Thinking, Third Edition

2010·40.18 MB·italian

by  Edward B. Burger

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Instructor’s Edition
of
Th e Heart of Mathematics:
An invitation to eff ective thinking
by
EDWARD B. BURGER AND MICHAEL STARBIRD
Contents of Instructor’s Edition opening sections:
I. A course that changes lives: aiming high
II. Courses for which Th e Heart of Mathematics is
appropriate
III. Goals of this textbook and course
IV. Th e HoM approach: Fundamental features of Th e
Heart of Mathematics: An invitation to eff ective
thinking
V. Overall suggestions for use
VI. Features of the Th ird Edition
VII. Annotated Table of Contents
VIII. Th e Heart of Mathematics supplements
Th e Instructor’s Resource book
Th e kit
Th e Web site
IX. Acknowledgments and thanks
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I. A course that changes lives
Aiming high
A general education mathematics class has the potential to be one of the most
signifi cant courses in a student’s education. In such a course, students can learn
great ideas and practical methods of effective thinking that can change their
lives and how they view and understand the world around them.
AIMING AT GREAT IDEAS. In their other courses, students study the great works
of literature, philosophy, art, and music. But frequently students’ mathematics
courses do not present correspondingly signifi cant ideas. Precalculus or college
algebra should not be the fi nal mathematics course for any student. Precalculus
and algebra provide essential techniques for a student who requires those skills,
but we should not have students struggle up the fi rst two rungs of a 100-rung
ladder that they will never climb.
MATHEMATICS COURSES FOR HUMANITIES STUDENTS CAN PRESENT FASCINATING TRIUMPHS
OF THOUGHT. Embracing the ideas of infi nity, facing quantitatively the uncertain
and unknown, exploring geometrical refl ections of our world and the geometry of
the abstract worlds of the fourth dimension and topology, appreciating the nuances
of number, and discovering the implications and beauty of chaos and fractals are
just a few imaginative journeys that resonate with the human spirit. After taking
a mathematics course that celebrates such intriguing topics,
I can see in a student’s face that moment students view mathematics as containing some of the greatest
when something puzzling suddenly ideas of human history.
makes perfect sense. Th e Heart of
TEACH POWERFUL METHODS OF THINKING. Nonscience students
Mathematics daily sets the stage for
will quickly forget technical terminology, notation, and
this kind of intellectual growth.
details of proofs. What can we realistically hope these
—TINA CARTER, BUFFALO STATE COLLEGE students will retain from their mathematics course after ten
years? Answer: powerful methods of analysis, reasoning, and
thought that transcend mathematics.
MATHEMATICIANS HAVE DEVELOPED POWERFUL TECHNIQUES FOR EXPLORING THE
UNKNOWN AND ANALYZING COMPLEX SITUATIONS. Teaching effective thinking is an
essential feature of any course adopting the methods and philosophy of this text.
Mathematics clearly illustrates potent methods of thought, and teaching those
methods can be an overt feature of the everyday content of the course.
EMPHASIZE DISCOVERY. To illustrate techniques for analyzing the unknown, we can
demonstrate how various habits of thought lead to rich intellectual discoveries.
Defi nitions and formal statements are not where the quest for discovery begins.
They evolve through a process of exploration and analysis. Students can learn
methods for developing insight such as: fi rst formulating questions arising from
observations; then looking for patterns, analogies, generalizations, examples, and
beauty; and, fi nally, making and verifying conjectures. These habits of discovery
help students understand how abstract concepts are created and how such
techniques are of value in their everyday lives. To assimilate these modes of thought,
students need to experience them repeatedly throughout the course rather than
just meeting them once in an introductory section. In a very real sense, this focus
invites students to see mathematics as useful and important in their everyday lives.
These are the true “applications” of mathematics for the nonscientifi c population.
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FOSTER POSITIVE ATTITUDES. Knowledge comes and goes, but hatred lasts forever.
Many students enter their required mathematics courses with apathy, dislike,
fear, and low expectations. For many, mathematics is problems. A critical goal
for this course is for students to appreciate and enjoy mathematics more at the
end of the term than at its beginning. If students enjoy mathematics and enjoy
thinking in a mathematical way, they will do so for the rest of their lives.
To make this course enjoyable and intelligible to real students, The
Heart of Mathematics employs a whole suite of enticements. The most basic
feature of the text is that it is designed to be read by students. Every page is
designed to encourage students to read on. The mathematical tendency to use
cryptic notation and sparse prose does not engage student readers.
HUMOR IS NO JOKE. Mathematical standards of proof and elegance are not
diminished by an entertaining treatment. Attractive art, humor, lively writing,
intriguing questions and surprising discoveries all make the substance more
inviting, even for skeptical students.
MAKING IT PHYSICAL. Physical experiences bring mathematics
One of my greatest joys in teaching from
to life. The accompanying manipulative kit allows students
Th e Heart of Mathematics is seeing
to build the Platonic solids, physically try topological
students discover not only mathematical
contortions and puzzles, and experiment with strange dice.
ideas but also the mathematical
The accompanying website allows for otherwise impossible
abilities within themselves.
simulations, experimentation, and discovery—for example,
students can create their own fractal images. —TIM CHARTIER, DAVIDSON COLLEGE
PLEASURE FOR FACULTY. Conveying truly fascinating ideas
should be a pleasure for the teachers as well as the students.
Teaching inherently interesting ideas is a strong start. In addition, the Instructor’s
Resource companion contains detailed suggestions for course structure, individual
class activities, homework assignments, and tests, all to make conducting the class
a positive (and, ideally, “easy”) experience for all instructors.
Making liberal arts mathematics courses fun, important, and satisfying has
far-reaching benefi ts. Future societal leaders will view mathematics as a living
source of powerful ideas. We can improve the lives of every student. However,
we must meet people where they live and take that reality as the starting place.
The Heart of Mathematics: An invitation to effective thinking will help you tell
the story of the mathematical adventure. Your students can learn and appreciate
fascinating ideas and leave the course with habits of thought that can help them
along any path their lives take.
II. Courses for which The Heart of Mathematics is
appropriate
• Pure mathematical ideas for humanities students
• An applied introduction to mathematical ideas
• Quantitative literacy
• Math for future teachers
• Introduction to higher pure mathematics
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STUDENT AUDIENCES FOR WHOM THIS TEXT IS APPROPRIATE. This book has a bountiful
set of sections. Choosing various collections of sections and various levels of
mathematical rigor in treating those sections can make the course appropriate
for several different educational goals and student audiences. Educational goals
include introducing great pure mathematical ideas, introducing great applied
mathematical ideas, developing quantitative literacy, and presenting teaching
themes for future teachers. Appropriate student groups for courses using this text
include humanities students, students preparing to be teachers of any subject,
students who want a view of higher mathematics, and students who could be
enticed into further studies of higher-level mathematical ideas.
III. Goals of this textbook and course
Th e easy-to-read explanations of very
sophisticated, interesting advanced
One goal is to inspire students to be actively engaged in
mathematics topics in Th e Heart of
mathematical thought. We want students to discover ideas on
Mathematics have made many of my
their own, grapple with challenging new concepts, and learn
students see math in a whole new
various techniques of thought through repeated exposure
light—and curious to learn much more!
throughout the course. Many students entering this course
—KERIN KEYS, CITY COLLEGE OF SAN FRANCISCO do not arrive with a deep understanding of college-level
mathematics. So we have presented topics, many of which
are rather advanced, in such a manner as to make them
accessible, interesting, and enticing to students who do not have a fi rm knowledge
of the vocabulary and symbolic representation of mathematics. Our goal is not to
teach these students the vocabulary and symbols of math that they will never use
in their everyday lives. Instead the goal is to open the students’ minds to ideas and
to help them learn innovative modes of thought to empower them to approach
and conquer all types of issues within and outside mathematics.
IV. The HoM approach: Fundamental features of The
Heart of Mathematics: An invitation to effective thinking
The hallmark features of The Heart of Mathematics have made it the most widely
adopted new textbook in liberal arts mathematics and teacher preparation in
over ten years:
• A focus on the important ideas of mathematics and mathematical
methods of investigation.
• A style of writing designed to be read and enjoyed by students and
faculty, general-interest readers, or professionals.
• “Life Lessons,” offering effective methods of thinking that students will
retain and apply beyond their college years.
• Entertaining and stimulating end-of-section “Mindscape” exercises for the
development of application, problem-solving, and argumentation skills.
• Activities that encourage collaborative learning and group work.
• An integrated use of various visualization techniques, technology, and a
hands-on manipulative kit, which direct students to model their thinking
and to actively explore the world around them.
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Praised by reviewers, instructors, and students since its publication in 1999,
The Heart of Mathematics: An invitation to effective thinking has set a new
standard for liberal arts mathematics courses and become a perennial favorite
on campuses across the United States. Its review in the June–July 2001 issue of
The American Mathematical Monthly states:
This is very possibly the best “mathematics for non-mathematician” book
that I have seen—and that includes popular (non-textbook) books that one
would fi nd in a general bookstore.
V. Overall suggestions for use
TOO MUCH FOR ONE SEMESTER. This book contains many more topics than can be
treated in one semester (or even two semesters). Instructors are given the freedom
to craft the appropriate course by selecting those topics most interesting and suitable
for their students. The omitted sections are excellent sources for independent
projects such as term papers, oral presentations, or class poster sessions.
ONE SECTION (cid:1) ONE CLASS SESSION. In creating your syllabus, we wish to point out
an important fact: One section (cid:1) one class session. Some sections require more
than one day, while a few require less than a full class. Many sections can be
used at different levels to allow shorter or longer treatments. For each section,
the Instructor’s Resource (which we describe more fully later) includes our
estimate of the number of class days required, but you may well have a different
experience. Some instructors block out two 50-minute classes per section.
STUDENTS SHOULD READ THE BOOK. We have put a great deal of effort into making
the book actually readable by real students. We do not visualize this book being
used by assigning exercises at the end of the sections and asking the students to
fi nd and adapt model examples previously worked out in the text. We are aiming
for student thought that includes reading and understanding the material before
grappling with the exercises. A basic homework assignment is for the students
to read the text and discuss the ideas with their classmates, roommates, friends,
and family. We have found in our experience that there are suffi ciently many
intriguing ideas that many students actually do discuss (and debate) the ideas
outside of class. These discussions can be lively and interesting and should be
encouraged. Sometimes in exercises, we ask the students to explain a particular
notion to a friend not in the class and report on the experience.
EXERCISES, CALLED MINDSCAPES. The exercises at the end of each section fall into
four categories. The fi rst group of questions is designed to be straightforward
and often offer more traditional algebraic-style questions. They get students
to solidify their understanding of the basic ideas of the section and offer
quantitative practice if that is a component of the course. The second group of
exercises may require more synthesis of ideas. The third group of exercises is
more conceptual and challenging. Answers, hints, and extended hints to selected
questions appear in the back of the book; those questions are marked by (H),
(ExH), or (S). Solutions to all the exercises appear at the end of the Instructors’
Resource.
The last group of exercises in each section is a collection of open-ended
questions to have students think creatively and generally about the ideas of the
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section. These questions add a writing component to the course and encourage
the students to express mathematical ideas in both ordinary and creative
language.
TEACHING STYLES. Many formats of teaching are viable with this book. The
presentation of the material has an implicit discovery learning aspect in that we
present examples and scenarios that evoke ideas in the students’ minds before
we present worked-out notions. Any teaching style that follows the pattern of this
book will encourage students’ involvement in constructing and refi ning ideas for
themselves. The Instructor’s Resource suggests numerous opportunities for small
group or entire class involvement and participation. These various activities can
easily be converted to a more traditional lecture format if desired.
AN ENJOYABLE COURSE TO TEACH. We have tried to make this book fun for the
students, but we have also tried to make this course a pleasure for instructors.
We have taught this course many times ourselves and realize the challenge of
enticing nonquantitative students to the joys of mathematics. The Instructor’s
Resource offers many detailed suggestions to make it easier to teach this course.
We have found that the students respond well to the intellectually substantial
topics in this book and that the presentation suggested here makes those ideas
accessible to a large number of students. We hope you fi nd a surprising pleasure
in this course, just as the students fi nd a surprising pleasure in the mathematics.
Good luck and have fun.
VI. Features of the third edition
WITH THE THIRD EDITION COMES AN EVEN RICHER AND MORE REWARDING EXPERIENCE FOR
STUDENTS, INSTRUCTORS, AND GENERAL READERS.
• A new section (5.3) called “Circuit Training: From the Königsberg Bridge
Conundrum to graphs” demonstrates how a specifi c question can lead
to a whole new branch of mathematics. Important concepts arise from
isolating essential features from a specifi c conundrum.
• The new Section 5.3, “Circuit Training,” combines well with Section 5.4,
“Feeling Edgy? Exploring relationships among vertices, edges, and faces”
to give a nice introduction to graph theory.
• The second edition’s Chapter 6, “Chaos and Fractals,” has been reorganized
to create the new Chapter 6, “Fractals and Chaos.” The reorganization
allows the reader to more clearly appreciate the various features of fractals
separately from the conundrums of chaos.
• The second edition’s Chapter 7, “Taming Uncertainty,” which dealt with
probability and statistics, has been greatly expanded and split into two
new chapters: Chapter 7, “Taming Uncertainty,” focuses on probability
while Chapter 8, “Meaning from Data,” treats statistical reasoning.
• In Chapter 7, “Taming Uncertainty,” readers will fi nd intriguing
illustrations of what to expect from randomness. Visual examples and
results from coin fl ips illustrate the wisdom that from randomness, we
should learn to expect the unexpected.
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• A new section 7.5 in Chapter 7 discusses several real-life issues involving
probability:
• the meaning of the surprisingly poorly understood phrase “There is
a 30% chance of rain tomorrow.”
• some game theory, including an application to decision making on
the football fi eld, and
• how a doctor can use probabilistic reasoning when new test results
alter the diagnosis of a disease.
• Probabilistic reasoning can help us all understand our world in a
more nuanced manner. The new features of Chapter 7 explore that
perspective.
• The new Chapter 8, “Meaning from Data,” begins by presenting statistical
ideas through statistical pitfalls. These examples of dubious reasoning
expose the themes that sound statistical reasoning develops.
• Chapter 8 emphasizes the challenge of taking a set of data and fi nding
strategies for extracting meaning from those data.
• The themes of organizing, describing, and summarizing a collection of
data lead naturally to the concepts of descriptive statistics.
• The goal of understanding what we can infer about an entire data set
when all we actually see is a sample from that data set leads us to concepts
of inferential statistics.
• Common sense is a fundamental tenet of sound statistical reasoning, and
common sense is front and center in our presentation.
• An entirely new section 8.5 presents statistical challenges in real life, from
estimating the numbers of Nazi tanks in World War II to understanding
how great baseball players of the past might have been.
• Throughout the book, new expanded hints illustrate for the student how
to apply sound methods of reasoning to selected Mindscapes.
• The expanded hints and many additional Mindscapes allow students to
translate ideas into action and effective learning.
• New Mindscapes that bolster mathematical techniques allow instructors
who wish to do so to include more quantitative skill development in the
course along with conceptual richness.
VII. Annotated Table of Contents
In what follows, we have described the mathematical content of each section,
together with the general educational themes that are included in that
section.
CHAPTER ONE Fun and Games: An introduction to rigorous thought
Chapter 1 introduces students to methods of critical thinking through puzzles
whose solutions foreshadow concepts in the future chapters.
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CHAPTER TWO Number Contemplation
SECTION 2.1 COUNTING. [Pigeonhole principle]. This section introduces the natural
numbers. The power of quantifi cation and estimation is illustrated, and the
Pigeonhole principle is introduced. An important goal of the section is to have
students learn the value of moving from qualitative to quantitative thinking and
reasoning, and to have them learn the power of estimation.
SECTION 2.2 NUMERICAL PATTERNS IN NATURE. [Fibonacci numbers]. Fibonacci
numbers are found in nature and art, and students are encouraged to fi nd
them in the spiral counts of various plants. The Golden Ratio is shown
F
to be lim n+1 . The idea that mathematical relationships are refl ected in nature
x→∞ F
n
may be a new and powerful idea to some students and an eye-opening concept
to some. There can be great value in looking at simple things deeply, fi nding a
pattern, and using the pattern to gain new insights.
SECTION 2.3 PRIME CUTS OF NUMBERS. [prime numbers]. One fundamental principle
of this section is that idea of seeking elementary building blocks as a technique
for understanding. The section contains proofs of the prime factorization
theorem and the theorem that there are infi nitely many primes. It concludes with
a discussion of the Prime Number Theorem involving the distribution of primes
and a discussion of some famous unsolved problems.
SECTION 2.4 CRAZY CLOCKS AND CHECKING-OUT BARS. [modular arithmetic]. This section
presents several opportunities for presenting general habits of thought. Starting
with the ordinary experience of clocks, students can be drawn to develop the
ideas of modular arithmetic. Check digits on universal product codes are good
everyday examples of mathematics in ordinary life.
SECTION 2.5 SECRET CODES AND HOW TO BECOME A SPY. [RSA public key cryptography].
This is probably the most notationally, algebraically, and computationally
challenging section in the book, which we often omit. It also is one of the most
modern and interesting applications of number theory. This dichotomy makes
the topic a great challenge for both students and instructors. This topic illustrates
that things that seem abstract and devoid of application today may be central in
our daily lives tomorrow.
SECTION 2.6 THE IRRATIONAL SIDE OF NUMBERS. [irrational numbers]. This section and
Section 2.7, “Get Real,” comprise an introduction to the real numbers. Section
2.6 begins by pointing out that between every two rational numbers there is
another rational number. The principal content of the section is a proof that 2
is irrational. The same proof is shown to apply to other cases such as 3 and
. These proofs provide excellent examples of the technique of following
2 + 3
the consequences of an assumption.
SECTION 2.7 GET REAL. [the real number line]. This section describes the real
numbers and how rationals and irrationals are characterized by their decimal
expansions. One challenging idea presented is that 0.9999 . . . (cid:2) 1. Another
challenging idea to students about the reals is that between every two reals there
are irrational and rational numbers. The chapter ends with the provocative idea
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