The Language of Mathematics
by Warren W. Esty
Copyright © 2007
This page updated July 28, 2007
Comments on what
is in the chapters.
Pace of the course.
Table of Contents, 2007
Edition
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CHAPTER I
ALGEBRA IS A LANGUAGE
1.1. Reexamining Mathematics
Mathematics as a Language
Problem-Solving Methods as Formulas
Problem-Solving Methods as Identities
1.2. Order matters!
Guide to Pronunciation
1.3 Reading Mathematics
1.4. Algebra and Arithmetic
1.5. Reconsidering Numbers
CHAPTER 2
SETS, FUNCTIONS, AND ALGEBRA
- 2.1. Sets
- Equality, Subset
- Intersection, Union
- Complement
- 2.2. Functions
- Why they are important
- Formulas
- Notation
- Composition of Functions
- 2.3. Solving Equations
- The Rules
- Reading the Rules
- The Theory of Equation-Solving
- RULES OF ALGEBRA
- 2.4. Word Problems
CHAPTER 3
LOGIC FOR MATHEMATICS
- 3.1. Connectives
- 3.2. Logical Equivalences
- 3.3. Logical Equivalences with a Negation
- 3.4. Tautologies and Proofs
- RESULTS FROM CHAPTER 3
CHAPTER 4
SENTENCES, VARIABLES, AND CONNECTIVES
- 4.1. Sentences with One Variable
- 4.2. Generalizations and Variables
- Generalizations and "True-False" Questions
- 4.3. Existence Statements and Negation
- 4.4. Ways to State Generalizations
- 4.5. Reading Theorems and Definitions
- Conditional Sentences as Defining Conditions
- Existence Statements as Defining Conditions
- 4.6. Different Appearance -- Same Meaning
- (A summary of all the ways sentences can look different and yet
have the
same meaning)
CHAPTER 5
PROOFS
- 5.0. Why Learn to do Proofs?
- 5.1. Proof
- "True" or "Proved"-- The List Approach
- 5.2. Proofs, Logic, and Absolute Values
- 5.3. Translation and Organization
- 5.4. The Theory of Proofs
- 5.5. Existence Statements and
Existence Proofs
- 5.6. Proofs by Contradiction or Contrapositive
- 5.7. Mathematical Induction
Students who finish Chapters 1
through 5 are prepared to move on to higher mathematics.
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Chapter 1 begins
with a section on the definition of abstraction
and discusses its importance in written mathematics. Then it discusses language and
how symbolic language can be used to express the
methods
of arithmetic. The order conventions of the written language are given.
The procedures of arithmetic are described as simple symbolic
sentences.
The student begins to practice with symbolism which is applied to
topics
he or she already knows. The context for the language lessons of
Chapter
1 is numbers. Students learn that symbolic sentences can express
methods.
In the last section examples of sentences include properties of
absolute
values and methods of solving inequalities.
- Give the order in which the operations are to be executed
in
the
expression
"-(8 + 4x² )";
- State the algebraic formulation of the method used
to
evaluate
these
expressions: "12 - (-8)" [ans: a - (-b) = a + b] , (3/4)/5 [ans:
(a/b)/c
= a/(bc)], etc.
Chapter 2 introduces
sets and functions as examples of mathematical
concepts that can be discussed in the language. Terms such as set
"intersection,"
"union," and "subset" help introduce the key vocabulary words "and,"
"or,"
and "if...,then...". Theorems for solving the most basic types of
equations
also introduce these vocabulary words from logic (these logical
connectives
become the subject of the next chapter, but you cannot learn about
connectives
without ideas to connect!)
Section 2.4 on "Word problems" shows how the algebraic
concepts
emphasized
so far (especially the conepts of operations and order = functions) are
essential to doing word problems.
From the section on functions here are typical questions of a
conceptual
nature.
- Give the mathematical expression in terms of "x" which
expresses the
rule
"Add 7 and then multiply by two";
- Give a descriptive imperative name for f (which does not
mention "x")
when
f(x) = (x + 5)²." [Answer: "add 5 and then square."]
- What is the difference between "f" and "f(x)"?
From the long section on solving equations:
- [E1 and E2 denote successive equations] If E1 --> E2,
how
do their
solution
sets compare? Can they be equal? If one has more solutions than the
other,
which is it?
- To solve the initial equation, would you prefer to have
your
sequence
of
equations connected by"-->" or "iff"? Why?
- What is an "extraneous" solution?
- When solving (simple) equations in Chapter 2, the
instructions are, "In
the following homework the solution is not the only goal. Exhibit every
step, exhibit the connective [iff for equivalence, sometimes "-->"
is necessary],
and cite a rule." A typical problem ranges from a factorable quadratic
to a harder problem where a square root must be eliminated: x - 1 =
sqrt(x
+ 11) [extraneous solutions may arise].
- Suppose you are asked to solve the equation "(x + 1)²
=
x(x + 2)."
After properly using some of our rules you would find it is equivalent
to "1 = 0. " That looks wrong. Did you make a mistake? Now what? What
is
its solution? Why?
- What would be the most likely next equation [when solving
these]? a)
h/g
= f; b) f² = fg. c) fgh = 0.
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Chapter 3 introduces
truth-table logic, emphasizing a dozen basic
logical equivalences and a few tautologies. The logical equivalences
selected
are those equivalences most often used in mathematics to provide
alternative
ways of expressing the same thought. The tautologies are those most
often
used in equation-solving and proofs. Chapter 3 emphasizes the primary
patterns
of mathematical reasoning. In contrast to truth-table logic as taught
in
"discrete" or "finite" math courses, the examples are nearly all
mathematical
and the study of logic continues immediately in Chapter 4 with its
application to the truth of sentences with variables. Open sentences,
generalizations,
existence statements, and negations are discussed in the context of
algebra.
A key point is that sentences of different types may appear
symbolically
similar (e.g. "2(x + 3) = 2x + 6" and "2(x + 3) = 3x"), but require
radically
different mathematical interpretations. Chapter 4 culminates with a
section
summarizing four distinct mathematical reasons why two sentences may
appear
different yet express the same meaning.
In addition to problems which require students to construct
truth tables,
some questions on logical equivalences and their applications to
sentences
are:
- Are the following nouns, pronouns, or statements? a) 3(4 +
2)
= 18, b)
3(4 + 2) = 20, c) 3(4 + x), d) 3(4 + 2) [Note that b) is a statement a
false statement];
- [Use DeMorgan's Law to] Give the negation of "-5 < x
<
8."
- [Use a stated logical equivalence to] Give another sentence
logically
equivalent
to "If c > 0, then, if c² > 25, then c > 5." [One answer:
"If c >
0 and c² > 25, then c > 5." Another: "If c > 0 and c
<= 5, then
c² <= 25."]
- Here is a sentence: "|x - 5| > 2 --> x > 7." Give
its
contrapositive. Give
its negation. Is it true?; [Because it is an implicit generalization,
its
negation is an existence statement: "There exists x such that |x - 5|
>
2 and x <= 7. " Thus the counterexample x = 0 (one of many) proves
the
negation true and the original statement false.]
- Suppose each of the following sentences is true. Which
express
mathematical
facts, and which express facts which depend upon the particular things
represented by the letters? a) 3(x + 5) = 3x + 15, b) 3x = 12, c) |x|
<=
0, d) |x| < |x + 1|, e) (A and B) --> B, e) (A or B) --> B
["A" and
"B" represent statements];
- Explain the difference between a sentence with a variable
and
a
statement;
- Explain the difference between an equation and an identity;
- Express this fact using different letters: ab = 0 iff a = 0
or b = 0.
- Use the quadratic formula to solve for x in ax +3x² =
bc;
- Give the [logical] form of the sentence: If |x| > 5,
then
x > 5
or x <
-5. [Answer: A --> (B or C)]; Restate the assertion in a logically
equivalent
form.
- State the fact expressed in English as "The product of
positive numbers
is positive" using proper mathematical notation and connectives."
[Answer:
If a> 0 and b > 0, then ab > 0.]
- True or false? a) bc > 25 is equivalent to b > 5 or c
> 5; b)
b < c
is equivalent to b + d < c + d; c) a = b is equivalent to a² =
b².
- Discuss this conjecture: The hypothesis of a theorem can be
interpreted
as describing the cases for which the conclusion is true. [No. It
describes
some of the cases, but possibly not all.]
Chapter 5 examines
proof. The roles of prior results, tautologies,
reorganization, and definitions in mathematical proofs are discussed.
Some questions on proofs are:
- What is the difference between "prove" and "deduce"?
- Suppose "A --> B" is true. Discuss whether A "proves" B.
- Determine whether the steps are sufficient to deduce the
conclusion.
Steps:
H --> A. H --> B. (A and B) --> C. Conclusion: H --> C.
- Many examples of, and exercises on, proofs
********************
In practice, in forty class periods (50
minutes each) with many math-anxious
students I could get through Section 5.3 or so, which completes a good
introduction to proofs. When I taught school math teachers it was in
the
summer with longer, but fewer, class periods. Surprisingly, they were
unable
to go much faster. It seems that this material is accessible to almost
everyone, but unavailable elsewhere in the curriculum, so even math
teachers
had a lot to learn.
Pace. The material has not been artificially subdivided
into
single-day chunks. The inherent unity of some substantial topics
produces
some sections that are quite long (for example, functions, 1.5;
algebraic
methods, 4.2, introduction to proofs, 5.2). Here is the pace I use in a
Freshman-level class at Montana State University:
Section and number of days (excluding exams)
Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5
1.1: 1 2.1: 3 3.1: 2 4.1: 1 5.0: O+
1.2: 1 2.2: 2 3.2: 2 4.2: 1 5.1: 1
1.3: 1 2.3: 3-4 3.3: 1-2 4.3: 1 5.2: 3
1.4: 2 2.4: 2 3.4: 1 4.4: 1 remainder of
1.5: 2 4.5: 2 Chapter 5:
4.6: 1 1 day each
---------------------------------------------------------
7 10-11 6-7 7 4* (totals)
* Chapter 1 through Section 5.2 can be regarded as a complete course
which
provides some exposure to the theory of proofs.
Prerequisites: The prerequisite is some high-school algebra
because many of the examples in the text use algebraic symbolism. At
Montana
State University I have found that many older students have forgotten
most
of their algebra but do well anyway. Experience shows that many
strongly
"math anxious" students who cannot complete traditional courses can do
well -- even very well -- if they do the work.
A research study showed that this class provides a wide range
of
students
with the opportunity to reenter and succeed in mathematics learning
regardless
of their entry levels of manipulative skills and conceptual
development.
Conclusion. It has been traditional to offer abstract
courses
only to those advanced college students who have shown by their
continued
success that they are already "good at math." This course has
demonstrated
that many students who would normally be classified as "math rejects"
can
succeed and blossom when the reasoning and abstract methods of
expression
underlying the mathematical material are made explicit. The sequencing
and pace of the material provide an opportunity for success with
significant mathematics to weak
and strong students alike.
********************
For information about ordering a copy.
e-mail to Warren Esty:
wwesty AT theglobal.net
Warren Esty has written another text, Precalculus,
designed to prepare students for calculus.