19th Edition

by Warren Esty

Most math courses concentrate on computational skills for particular types of problems, but they don't teach you how to learn on your own, nor do they teach logical reasoning. This course is much different from those courses.

Mathematical language skills include the
abilities to read with
comprehension, to express mathematical thoughts clearly, to reason
logically, and to recognize and employ common patterns of mathematical
thought.

This text is appropriate for a broad range
of students from
elementary-education students (it is almost critical for them) and
liberal arts majors to math majors.

This site updated December 28, 2016

For information about ordering a copy.

For information about ordering a copy.

You live in a world which is highly
mathematical, even if you don't personally do math (Most adults don't).
But if you are in school more math lies ahead of you and if you are, or
expect to be, a parent, a lot more math lies ahead of you when your
kids take it. If you will be an elementary-education teacher, you will be doing math every day. Why not learn how it works?

Here is a parallel. Suppose you were going to go to Germany. Would you learn German for the trip?

Probably not, if the trip were only going to be a week long. Someone can translate for you or maybe you don't need to know what is going on during a short trip. But, if you were going to be there a year it would be worthwhile to learn German. In regular school the teacher translates math for you, sentence by sentence and method by method, and most students do not figure out what is going on. They never learn the language. No one asks them to, and no one requires them to. They never learn "German" and every new topic requires new translation by a teacher. This text is the mathematical equivalent of learning German. You won't need a translator again.

College faculty will be interested in a discussion about "added value" further down the page.

Here are some unsolicited comments:

Here is a parallel. Suppose you were going to go to Germany. Would you learn German for the trip?

Probably not, if the trip were only going to be a week long. Someone can translate for you or maybe you don't need to know what is going on during a short trip. But, if you were going to be there a year it would be worthwhile to learn German. In regular school the teacher translates math for you, sentence by sentence and method by method, and most students do not figure out what is going on. They never learn the language. No one asks them to, and no one requires them to. They never learn "German" and every new topic requires new translation by a teacher. This text is the mathematical equivalent of learning German. You won't need a translator again.

College faculty will be interested in a discussion about "added value" further down the page.

Here are some unsolicited comments:

I ordered your book in June and it has been a real benefit for the time I invested studying it. I am now teaching precalculus (both plain jane and goosed-up versions) and calculus and I am able to use more precise language and to express concepts that I otherwise would have used improvised language. The $42 price of your self-published text was a real investment.

-- a teacher

I've found your book to be a wonderful map! I definitely feel like it's been aiding my adventure into the mathematical region of my imagination. Thank you for your earnest concern about the subject; your enthusiasm is as infectious as it is appreciated!

-- an on-line buyer

My experience tells me that a very small portion of the population really understands what math is. Your work is one of few that brings the real process to an accessible level.-- a high school teacher

My perspective [of mathematics] has definitely changed. It's opened my mind to what it's all about. Now it's one of those things where I respect it in a different way. [This class] taught me to think more. Before, I never even thought or cared about why it happens. You just did it and you were done with it.

-- an anonymous student

I recently purchased your Language of Mathematics text, and found it to be more than I could have possibly hoped for, as such texts go! Never have I seen so many salient components of this beautiful language exposed and discussed in the comprehensive way that you do in your text.

-- a graduate student in mathematics education

I am finding your book, The Language of Mathematics, very pleasant to study. I don't know if you remember me. I am the one studying Mathematics at the University of [xx]. I strongly believe that every math major and high-school student should have a copy of your text! I agree with your articles; Mathematics is a language on its own and it is essential that students understand the true meaning of the subject. I wish I had a copy before entering University!!What's it like? Here is the first section in pdf format.

Even as an experienced mathematics professor, teaching the "Language of Mathematics" allowed me to learn new ways to explain to students the nature of mathematics, its interpretation as a language, and its notation. Esty's text provides a wonderful exploration of the deep issues involved in understanding and teaching even the most elementary mathematical concepts. Working with the text has and will influence, expand, and change what I emphasize when teaching all other courses, whether aimed at general education students, mathematics, science, and education majors, or graduate students.

-- a Professor of Mathematics

What is is in it? Here is the Instructor's Manual, which is designed to for new instructors. It will help you grasp what the text does.

This text has been so

is a core-course (Math 147) at Montana State University and elsewhere.

The text, also entitled

is now available in its sixteenth (!) edition.

Many individuals, serious about improving their grasp of mathematics,

have studied from the text on their own.

The extensive (42 small-print pages crammed with solutions) solution manual gives them feedback they need.

For information about ordering a copy.

The Language of Mathematics

- Students' attitudes about math improve dramatically. [Many students changed from avoiding math and being "math anxious" to high enjoyment of and involvement in math.]
- Students' perceptions of their ability to tackle another math course go way up.
- Students' involvement in and enjoyment of math increase significantly.

**Prerequisite**: The math prerequisite is near the
level of
completion of Algebra I. Much more important is the English
prerequisite: students must be willing and able to read at the college
level. Many students who enroll do not have anywhere near
Algebra II-level skills, and many are extremely "math-anxious" by their
own admission.

Most math courses are filters, not pumps,

but this one is different -- it is designed to promote **success**

For information
about
ordering
a copy.

19th Editon

Mathematical methods and results are expressed in a foreign language.

That language, like other languages, has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, abbreviations, sentence structure, and paragraph structure. It has certain language features unparalleled in other languages (for example, theorems expressed using the letter "x" also apply to "b" and "2x-5").

**Purpose**: To teach essential language concepts which
have
been underemphasized in the usual mathematics curriculum. To emphasize
the basic patterns of mathematical expression and thought. This is what
college "core" mathematics should be.

There are a limited number of frequently repeated patterns
of
expression and thought in Mathematics. This text identifies, isolates,
and emphasizes the essential patterns, illustrating them in several
subject areas of mathematics.

There are a limited number of key vocabulary words from logic ("and", "or", "not", "if... then", "if and only if", "for all", and "there exists") which are frequently used in mathematics.

One Goal: Students will learn to read math. The text teaches how to read math well enough in order to learn math by reading. It sounds like a tall order, but it works!

What's it like? Here is the first section in pdf format.

What is is in it? Here is the Instructor's Manual, which is designed to for new instructors. It will help you grasp what the text does.

This course adds value. Any good course should change you. Most math courses are intended to change (add to) your calculation skills and change (add to) your knowledge base. Those changes are good, but less useful than they were a decade ago. Now calculators and computers can do all the calculations you will ever need to do. Mathematical facts are on the web for the asking, if you know how to ask and can read the mathematical language on the web pages you find. Unlike years ago when the usual math curriculum was developed, calculations are now very cheap and facts are very cheap. So you have not added very much value to yourself if that is all you have learned!

I gave a talk about why the approach of The Language of Mathematics is important for colleges to adopt. Here are the bare PowerPoint slides which you could skim in a few minutes.

This course adds value by emphasizing how the language of mathematics works so you can think mathematically, reason logically, read mathematics with comprehension, and learn mathematical skills and facts by reading. Mathematics is a written language-- a foreign language. This course is the equivalent of language lessons that will help you get along when you visit the land of mathematics. In the future, if you take more math, you will be able to read the book and get a lot out of it. If you need to be able to understand or do some math which is new to you, you will be able to read how, even without an instructor. Long after this course and college are over, you will still be able to add to your own value.

Faculty at colleges and universities might consider adding this course to their school's offerings. It is particularly suitable for elementary-education majors who are often not comfortable with algebraic notation but really should be, given they will be teaching math! This course also makes a great "liberal arts" course because, compared to other liberal arts math courses, its emphasis on logic, reasoning, and thinking skills makes a much higher fraction of the course actually benefit students in their future lives.

College and university faculty who are intrigued by these arguments may contact me about obtaining a copy.

Write me, Warren, at:

I gave a talk at the Joint Math Meetings in New Orleans, January 2011, making these points. Here are my PowerPoint slides. (I said a lot that is not reproduced on the slides, but you can figure it out.)

What's it like? Here is the first section in pdf format.

What is is in it? Here is the Instructor's Manual, which is designed to for new instructors. It will help you grasp what the text does.

What is

- A lot!

- Constant emphasis of
**patterns**of thought and expression which recur throughout mathematics - Thorough explanation of what makes mathematics
"algebra" and how to think "in algebra."

- Emphasis on bringing the students
**up**to a mathematical, abstract, level of expression and understanding - Emphasis on mathematical examples of sentences and reasoning (not logic of this sort: "If it's raining, then I will get wet...")
- Emphasis on alternative ways to express the same
information until students are comfortable with all the ways
mathematical thoughts are expressed

- logical equivalences
- letter-switching
- theorems which use "iff"
- definitions
- English v. mathematical expression
- abbreviations, notation
- Making implicit usages explicit
- Little equation-solving until they have the ability to
read
the theorems which justify the steps (learning to
**read**in order to learn is a major thrust of the text). This is not a calculation-oriented text.

- Algebraic methods are justified (and students understand the justifications)
- Proofs are introduced near the end, after students have all the background they need.

- Math-anxious students love it!
- They can finally understand what's going on in a math course! Math majors love it!
- Math grad students love it!
- They don't take the course, but some get the book, read it, and come back to me saying they wish they had it before taking advanced calculus (or even regular calculus)
- School math teachers love it!
- Yes, Montana State has taught from the same text to school math teachers in our summer Master's degree program (with a somewhat more sophisticated emphasis) three times. They see many applications to their own teaching. (And, I am sure their increased comfort with reasoning and the meaning of symbolism makes them much better all-around mathematically.)

*The Language of Mathematics*, the book -- the Table of
Contents
and detailed
descriptions of each section.

For information about ordering a copy.

e-mail Warren Esty at

Department of Mathematical
Sciences

Montana State University

Bozeman, MT 59717

Warren Esty has written another text, *Precalculus*,
designed to prepare students for calculus.

**Articles on language and math**.
What are the language concepts of mathematics? See Warren Esty's
article, "Language Concepts of Mathematics" (pdf image), in *FOCUS -- On
Learning Problems in Mathematics*, volume 14.4, Fall 1992. His
(long) joint article with Anne Teppo, "A General-Education Course
Emphasizing Mathematical Language and Reasoning" (pdf image) in the same
journal, volume 16.1, Winter 1994, describes the research which
demonstrates the improvement of students' attitudes and abilities.

Jointly with Anne Teppo, Warren Esty published an article
in
the *Mathematics Teacher* (Nov. 1992, 616-618) entitled "Grade assignment based on
progressive improvement" (pdf image) which was reprinted in the NCTM's *Emphasis
on Assessment*. In a language course, you can expect continual
improvement. This article discusses why grading should not be based on
averages of unit-exam scores and how a course like "The Language of
Mathematics" can be graded.

More work of theirs on algebraic language was published
in
the 1996 NCTM Yearbook, *Communication in Mathematics*. Their
"Mathematical Contexts and the Perception of Meaning in Algebraic
Symbols" was published in 2002 in *The Future of the Teaching and
Learning of Algebra, Volume 2*, and many other articles of
Prof. Esty have appeared in other publications.

Go to a concise
description of The Language of
Mathematics.

What's it like? Here is the first section in pdf format.

For information about ordering a copy.