The Language of
Mathematics
Algebra is written in a symbolic language that is designed to
express mathematical thoughts, including how to do problems. This
website describes a text for a course that emphasizes how
mathematical methods are expressed in symbolism. Most math courses
concentrate on computational skills for particular types of
problems. This one concentrates on how mathematical
methods and facts are expressed, with many examples, so that if
you ever need to know how to do a problem you will be able to read
and learn how. Did you know that "x" is used in two radically different ways? This course emphasizes all the ways that
mathematics is used to express thoughts in algebra and
higher-level classes. It will make you far better at math, and at
learning math, regardless of your current level.
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.
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
What's it like? Here is the first section in pdf format.
This text has been so successful that it has been the
subject of two published (and several unpublished) research studies to identify why it works and what
it does to and for students.
The Language of Mathematics
is a core-course
(Math
147) at Montana State University and elsewhere.
The text, also entitled The Language of Mathematics,
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.
Here is a link to three paragraphs about the author, Warren Esty
For information
about
ordering
a
copy.
The Language of Mathematics, the book -- the Table of
Contents and descriptions of each section.
Research Results:
Reference: Warren Esty and Anne Teppo, "A General-Education Course
Emphasizing Mathematical Languge 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. 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.
The Language of
Mathematics
by Warren W. Esty
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.
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!
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. They 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.)
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.
What's it like? Here is the first section in pdf format.
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" 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," 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," 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"
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.
