The Language of Mathematics, by Warren W. Esty

This page updated March 6, 2024.    (Return to the main page of this website.)
      Table of Contents, 19th Edition (for extended comments, see below)

To the student


        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


     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 (summarized)
    2.4. Word Problems


     3.1. Connectives
     3.2. Logical Equivalences
     3.3. Logical Equivalences with a Negation
     3.4. Tautologies and Proofs


     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
               Conditional Sentences as Defining Conditions
               Existence Statements as Defining Conditions
     4.5. Reading Theorems and Definitions
     4.6. Different Appearance -- Same Meaning
               (A summary of all the ways sentences can look different and yet have the same meaning)


     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.

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.

From the long section on solving equations:

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:

Chapter 5 examines proof. The roles of prior results, tautologies, reorganization, and definitions in mathematical proofs are discussed.
Some questions on proofs are:  


For instructors--comments on pace. 

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.


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e-mail to Warren Esty:    e-mal
Warren Esty has written another text, Precalculus, designed to prepare students for calculus.