Precalculus -- Preparing Students for
by Warren Esty
(This page updated Sept. 25, 2012)
designed to produce a deep understanding of algebra and
trigonometry so that students will be comfortable with their
next math. Students will be well-prepared for calculus.
This text is designed to be appropriate for self-study, as
well as classroom use.
The content includes the usual precalculus
material (functions, powers, polynomials, logs, exponentials,
trig, etc.). Graphing calculators play an important role.
However, this text is unlike others because it does not just
use calculators to do old-style problems, but actually
incorporates calculators as a learning tool
and not just a "doing" tool.
This text has been used at Montana State University and
elsewhere by about a hundred different instructors and many
thousands of students. A great deal of experience has gone into
making this text an effective learning tool.
"A few months ago, I ordered your
Precalculus text, and thank goodness I did. ...." An
unsolicited testimonial continued
(with additional testimonials, too).
"As a home-schooled junior in high school
this year, I used your Precalculus book. [clip] I would just
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entire perspective on math. Up to this year, I considered math
to be a distasteful medicine [clip]. No text book on a language
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ideas, and now I love Math. Precalculus was my favorite subject
this year [clip. Read the whole letter
"I ordered your precalculus book, and thus far
it's been quite a learning experience. I've learned more in the
past 3 months than I learned in 4 years of high school. ....
Thanks." [an adult student]
"I'm taking Precalculus for graduate school,
and I have always hated math and been terrible at it, I was taught
only to memorize, never to understand. I'm starting to love math,
and it's because my dad (who loves math) has your book and we've
been working with it! So I wanted my own copy. Thank you for
writing such a great precalculus book!"
"I am the teacher who used your
Precalculus text with some homeschoolers over the internet the
year before last. I now have more students asking for the class.
I would like to use your text again. (I was very pleased that
the students who used your book and went on to take the SAT got
math scores very near 800. It'll be interesting to see what
happens with the new SAT exam.) ..."
"I love teaching the text and the other
instructor feels likewise. The course has been a challenge for my
students. For those who have buckled
down and done the requisite assigned work (and shift in conceptual
focus), well, the results are very, very evident." [continued here]
-- A high school teacher.
"Speaking as an
aerospace engineer, this was a great course, great professor [at
a different school], and great textbook. I would have never
found that textbook on my
own. I would not change anything with this course. It meets the
demands of the Math Analysis course in a previous era... the one
I grew up in. For the
most part, Calculus and Pre-Calculus classes have been
dramatically watered down since the 1980s. This course restores
the ancient paths.
"I think this is the best math
class she [his daughter] has had for her mathematical
understanding [continued here]"
-- A parent
Here is a link to information
ordering a copy.
The Table of Contents.
Of course, the presentation of most topics resembles that of
other precalculus texts. So does the organization, at least
1 (which is unique). But it is particularly effective because of
its numerous distinguishing features:
Here is a link to the AMATYC summary of an article I wrote on
"Teaching about Inverse Functions".
- An effective new approach that incorporates
a learning tool (not just a calculating tool).
This may be the first text to fully adjust to the fact that
calculators can do the calculations and therefore accelerate
learning about algebra, exponentials, logs, and trig, but students
must still learn and understand the math. Calculators
- Concentrate attention on essential points
- Increase the rate at which students
gather experience with the subject
- Focus on learning math that is valuable (essential!) even
though calculators can do all the calculations. There
is still a lot to learn about math, even though calculators
can do a lot. This text clearly distinguishes between learning
about calculators and learning about math with the help of
calculators. (Dr. Esty has given numerous conference talks
about learning with the help of calculators.)
- Emphasis on learning how the symbolic
language of math
is used. This is probably the most
distinguishing feature of the text. It is an explicit goal of
Chapter 1 that students learn how thoughts about methods are
written in modern mathematical notation. The goal of
Chapter 1 is to have students become able to learn math by
reading math. Let's face it, most students do not learn
math by reading it. This text has explicit reading lessons!
- Increase students' ability to generalize
- Increase students' ability to learn
outside of class. [There is so much more time outside of
class than in class. Wouldn't it be great if students could
(not just practice) outside class!]
homework in addition to the usual type of calculation
problems for practice.
- (For example, most "B1" problems ask for an
illustration, or explanation, rather than a computation.)
visual illustrations that generate correct concepts.
- Emphasis on connections to lower- and higher-level
- (Calculus-style applications of algebra are
devoted to how to do word problems. Students who
can't do word problems are missing something important about
algebra -- how symbolism is used to represent operations. The
emphasis on symbolism for expressing thoughts about operations
in Chapter 1 helps students learn how to do word problems, and
they would never be ready without it. The author's research on
word problems shows that students cannot do word problems just
by taking years of algebra. They need to study writing about
operations in symbols first.
- Emphasis on graphs and their interpretation, and
effective use of graphing calculators.
- Emphasis on mathematical concepts that will not
become obsolete when the next generation of calculators
- Instructor-friendly and student-friendly
(this text does not require new teaching techniques or
- Excellent for
teaching yourself. It is hard to learn math on your own.
You must have good (English) reading skills. In contrast to
all other texts, this one has lessons in Chapter 1 on how math
is written and how to read it. This should help you get the
most of the text even if you don't have a teacher.
- A solution manual
with solutions to odd-numbered problems so you can follow how
problems are done if you have questions.
Six articles by Dr. Esty on learning precalculus with the
aid of calculators have appeared in the recent proceedings of
the International Conference on Technology in Collegiate
Mathematics (some are on-line with links below).
The importance of conceptual development that is
specifically algebraic is discussed in
Language, and Word Problems," an article by Dr. Esty and
Dr. Anne Teppo in the 1996 Yearbook: Communication in
Mathematics, published by the National Council of Teachers
of Mathematics. They have also written related articles on
problem-solving and algebraic thinking in several issues of Psychology
in Mathematics Education.
Here is a pdf image of a published article for educators
about the philosophy of the text. "Learning Precalculus Concepts
using Graphing Calculators and Emphasizing Symbolic Language"
by Warren Esty. As far as I know, no other Precalculus text
comes close to recognizing the importance of learning to read
Mathematics Using Graphing Calculators." What makes
a graphing calculator exercise a good one? Find out in
this published article, imaged as a pdf.
"What do we need to
teach about algebra, now that 'Calculators can do it all'?"
When is it all right to use calculators? Find out in this
published article, imaged as a pdf.
The author. Warren Esty is a Professor in the Department of Mathematical
Sciences at Montana State University.
Here is a link to three paragraphs about the author, Warren Esty.
Warren Esty has written another text, The
Language of Mathematics, designed to constitute a
core course in mathematics,
and a third jointly with Norah Esty, Proof:
Introduction to Higher Mathematics.
Warren Esty, Dept. of Mathematical Sciences, Montana State
e-mail Warren Esty at