*The Language of Mathematics*

19th Edition

by Warren Esty

This page updated Oct. 12, 2024

Algebra (and all higher math) is written in a symbolic language that is designed to express mathematical thoughts. English is a great language, but it is not designed for doing mathematics. This website describes a text for a course that simultaneously emphasizes mathematical thoughts, methods, facts, and reasoning and how they are expressed in symbolism. Students will learn how to read, write, and think mathematics.

Most math courses concentrate on computational skills for particular types of problems, but now computers can do all that for you. This course teaches you how to learn math on your own and how to reason logically. It is not a computation course.

Wikipedia is a vast storehouse of knowledge, but to access it you must be able to read. Clearly you can read sentences in *English*. This course gives you the ability to read sentences in *mathematics*.

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.

Here is **the Table of Contents**: http://estymath.com/toc.html

**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 or two 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!

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 and understand 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. (Reasoning properly is a very valuable skill in daily life, too. You may find you use skills from this course more outside mathematical contexts than in them.)

After many years of teaching and reflecting on the value of what students learned in various courses, I still think this is the most valuable math course of all.

**Who Should Take it**? This text is appropriate for a broad range of students from elementary-education students (it is almost critical for them because elementary-education teachers teach math every day; they need to understand it) and liberal arts majors to math majors. 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), gratis, solution manual gives them feedback they need. *The Language of Mathematics *has been a core-course at Montana State University and elsewhere.

What's it like? Here is the first section in pdf format. If you read it and want the rest of Chapter 1, write to tell me that you have finished 1.1 and would like to see the rest of Chapter 1. (It is an image file, not a pdf, and more than 20mb. I hope you can get such a large file in your e-mail.)

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:

I ordered your book in June [some years ago] 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. -- a teacher

I ordered The Language of Mathematics from you years ago, and I don't know that I ever followed up!

Your book helped me immensely. I struggled with math all the way through college, managed to scrape by and get a BS. After working for about 10 years I found I was really interested in statistics and wanted to go back to school, but first I had to pass calculus 1 & 2, which to me was a seemingly impossible task.

Your book allowed me to finally understand things, such as f(x) having multiple meanings (p. 91). I don't know why that was so hard for me before...no one ever explained it in a way that made sense.

I did go on to pass calculus, go to school, and earn my MS in Applied Statistics. I'm very proud of that achievement! Maybe pride isn't the right word, I try to be humble, but it was a huge deal for me, something I previously thought I could never do.

I'm not sure what I'm going to do professionally, I have about 20 years of experience in the food industry (R&D, manufacturing), and I've been using my stats skills every day!

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.-- a 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

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 Table of Contents and descriptions of each section.

Here is the Instructor's Manual (there is a separate solutions manual) which is designed for new instructors. It will help you grasp what the text does.

Here is a link to three paragraphs about the author, Warren Esty

For information about ordering a copy.

*The Language of Mathematics*, the book.

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.

Reference: Warren Esty and Anne Teppo, "A General-Education Course Emphasizing Mathematical Language and Reasoning," *FOCUS -- on Learning Problems in Mathematics*, volume 16.1, Winter 1994.

**Prerequisite**: The math prerequisite is near the level of completion of Algebra I (not Algebra II). Most 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.**

Mathematical methods and results are expressed in a foreign language. That language, like other languages, has its own grammar, syntax, vocabulary, word order (e.g. in "2 + 3*x*" multiplication is first), 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 "2*x*-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.

**This course ****adds value**. Most math courses are intended to add to your calculation skills and add to your knowledge base. However, unlike decades 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 at the Joint Math Meetings in New Orleans, January 2011, making these points about why the approach of *The Language of Mathematics* is **important for colleges to adopt**. Then I modified them for major talk I gave when I retired. Here are the bare PowerPoint slides: RetirementTalkWhatShouldWeTeachAboutMath? (in pdf form) which you could skim in a few minutes. (I said a lot that is not reproduced on the slides, but you can figure it out.)

**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 desk copy.

Write me, Warren, at:

What is **different** about *The Language of Mathematics*?

**A lot!**- Constant emphasis of
**patterns**of thought and expression which recur throughout mathematics - Thorough explanation of what makes mathematics "algebra" (as opposed to arithmetic) 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.)

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

For information about ordering a copy.

e-mail me, Warren Esty, at

**Books and Articles**. Warren Esty has written another text, *Precalculus*, designed to prepare students for calculus. With Norah Esty he has also written *Proof: Introduction to Higher Mathematics*, for college sophomores.

**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.

PowerPoint slides: RetirementTalkWhatShouldWeTeachAboutMath? (in pdf form) which you could skim in a few minutes. (I said a lot that is not reproduced on the slides, but you can figure it out.)