The Language of Mathematics
19th Edition
by Warren Esty

    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 March 15, 2016

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. 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:

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!!

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

Research Results:

• 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.

For information about ordering a copy.
 

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.

• 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.

Results:

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.)

This is remarkable:

Even "math-anxious" students can do well in an abstract math course when the language is thoroughly explained.

 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.