To Students from University of Arizona Students
David Brokl, Jennifer Lowe, Ian Scott, and Michael Van Zeeland
This book is different! Most math books and math classes follow the
simple formula (1) present an abstract definition or theorem,
(2) attempt to clarify this definition or theorem with examples and
dialogue, and then (3) assign exercises which can be done by
repeating the examples in the book with changed numbers. This book
often works contrary to that formula in that graphical and numerical
examples are presented first, and from them some general conclusions
are drawn. Sometimes sections start with a data set or real world
example, often about situations we have encountered in our other
science and engineering classes. This is great, as it makes the book
much more interesting to read.
As you are working through these examples, keep your eyes open for
connections to past results. These are there, and when you finish,
there are some new conclusions which await you. Sometimes there are
clues as to what to look for in the ''Where We Are Going—and Why''
at the start of each chapter.
To help understand these general conclusions, we would first read
through a new section as if reading a story, looking for main ideas,
but not dwelling on them. Then we would go back through the section
carefully with pencil and paper (and often a computer or graphing
calculator) at hand to make sure the details made sense. After each
example, we stopped to examine the logic and details of the analysis.
In many places of the text there were questions asking why something was
true (these are contained in parentheses). We found it very worthwhile
to figure out the answer before going on. The words ''Comments about''
appeared at various places in the book. These were not simply idle
comments, but either emphasize an important point or made useful remarks a
bout a prior result. Be sure to read and think about these "Comments''!
As you work through each section, pay close attention to every single
graph. In other math books, graphs may seem to be incidental, but in
this book the graphs are central to understanding the ideas. A
computer, or graphing calculator, was also crucial to understanding
the material. We would question what we saw on the screen and guess
what would happen if we changed one of the inputs. We would also make
sure that what we saw agreed with our analytical work. Computers can
be misleading, so we questioned things which did not make sense. We
found that answering these questions led to a much better understanding of
the material. It was a great feeling to discover that we could explain
why the computer results were misleading.
Most of the exercises cannot be done by simply looking at the examples
in the book without reading the section. However by working through
the material in the book you will be able to recognize similarities
between the exercises and the examples and see the proper approach for
the exercise. If part of a section is confusing, look ahead to a ''How
to ... '' box—which contains concise explanations—or to
related definitions, or maybe to the ''What Have We Learned?'' summary
at the end of the chapter.
Finally, find a friend (or friends) to study with. These people should
be able to trade ideas with you and discuss alternative approaches to
solving problems. Having several points of view is often a big help in
solving some of the more challenging problems. Someone else can often
easily spot the place where your thinking or analysis is incorrect. Of
course explaining things to each other is an effective way to really
understand the material. We really enjoyed using this book, and think
you will also.
