I submitted my book, One Mathematical Cat, Please!, to Brooks/Cole Publishers, and they loved it. Unfortunately, however, it is not a textbook for any particular course—instead, it is a supplement to many courses, from junior high through early college—and, after many deliberations, the underlying concern that “supplements don't sell” won out (sigh). Anyways... I will always cherish the line in this review: “If you can find the right place to put it, this book will do a great service to the mathematical educational world.” Maybe one of you, out there in cyberspace, can help me to locate “the right place to put it”!

Most of the items mentioned in this review that ask for changes have already been incorporated in the version of the text that is on the web, so you usually won't be able to see what the reviewer was “complaining” about. (I benefited tremendously from this review; my book is better because of it.)

ONE MATHEMATICAL CAT, PLEASE!

by Dr. Carol J.V. Fisher Burns

This review consists of the answers to your questions and the enclosed manuscript with numerous comments and suggestions written directly on them.

**1.** *Please describe the course in which you would consider using this manuscript. What is the greatest
teaching problem in the course?* This manuscript is not a textbook for a course. It is supplementary
material for students to help them understand mathematical language, and in particular the distinction
between a mathematical *thing* (an expression or number) and a mathematical *sentence* (an
assertion, often that two things are the same). I would __love__ for my students to understand this,
in lots of courses. I could imagine using this book as an optional (or maybe even required) supplement
for remedial courses (Elementary Algebra, Intermediate Algebra), the "mathematics for elementary education
majors" course, the precalculus course, and the finite math and baby calculus course for business majors.

But if you want to single out one course in which I'd use it, it is MTH 100, a special course we offer
in the summer before the freshman year to disadvantaged students who need help getting to the college
level in mathematics and writing. The content of this course is basically Elementary Algebra, but I would
use this book at the beginning before we got into the algebra per se. The enrollment is usually in the
50–80 range, with small sections, some taught by regular faculty and some by part-timers. The
greatest teaching problem in this course is the poor mathematical background of the students and various
social and intellectual dynamics that get in the way of their proper attitude toward learning. In answering
the rest of the questions, I'll keep this course in mind.

**2.** *Do you agree with the rationale for the contents and approach of the book? Does the
author succeed in his/her goal for the book?* Yes and yes. There is a real need for students to
understand the ideas in this book. Some students pick it up easily as they study mathematics in
high school and beyond. Others are so used to thinking of mathematics as pushing symbols around
rather than communicating ideas in a special language that they never understand these ideas. This
hampers their success. Thus I am convinced that the need for this book is immense. Your hardest job
will be to convince unthinking faculty members that the need is there. The author does a good job of
getting her points across—moving very gently and deliberately, saying things just right for the most part.

**3.** *Which topics to you think should be added? Deleted?* I might add some more examples
of story problems, but I don't think I'd add any more *topics*. (Well, maybe a discussion of the
order of operations, picking up on the problems raised on pages 44–46.) I might also like to see more
of a discussion of typical errors that students make (like writing 2+3 = 5·2 = 10), integrated throughout the
book. In Section 10 I would add the word "function" to the general discussion, rather than just as a sidebar
on page 144 (even elementary school students are told about "function machines"). One page 112, I would
stress this distinction between declarative and imperative modes more. I looked at the web site
mentioned there and see that the present manuscript is part of a wider conspiracy!

A few things don't seem to work as well as the rest, and I would consider dropping these (they are
marked in the manuscript—specifically the material on page 19 and some or all of the last section).

**4.** *Is the level of detail in the explanations of the manuscript appropriate for your
students? Are the tone and style of the manuscript suitable for your students? Should any changes
be made? Are the organization and level of problems suitable for your students?* Except where
I've marked it, the level of detail, tone, style, organization, and level of problems seem about
right for my students. It's not clear that the reading level isn't too high for some of the intended
audience, however. (I'm not an expert in education, but it seemed to me that seventh grade students
might not have the intellectual sophistication or the reading skills needed to appreciate the discussion
in some places.) Various suggested changes are marked throughout the manuscript.

One stylistic point I'm not too enthusiastic about is the constant talk about what "mathematicians"
do. The author should think carefully about the down side of this approach—a student says to
himself, "I'm not going to be a mathematician, so this is irrelevant to my learning mathematics."
I'm not sure how to change this approach, though—it pervades the entire book. On a related note,
look at the use of "result" on page 77; we mathematicians forget sometimes that this is *not*
common English usage. (On the same page is another item I'd want to change—the lie that "theorem"
means "true statement" rather than "proved statement". It's a subtle point, not likely to matter to
this audience, but we should try to avoid telling white lies when possible, even at this level.)

I would not be quite so forgiving about using calculators for trivial arithmetic as this author is (e.g., averaging 0.1 and 0.2 on page 18), but that's a minor point and she has good reason for doing it her way (one being that she doesn't want the anxiety of doing the calculations to get in the way).

Since this is not a textbook, you can't really view the "problems" in the normal way—students are expected to do them all, and they're an integral part of the presentation, not a separate section of drill and practice. They are excellent.

In the later sections, more motivation would be useful (e.g., pages 149 and 151).

**5.** *How well does this book stand up to its competitors?* That's easy to answer: There are no
competitors. It is not a textbook for a course but a supplement, and I've never seen any similar thing
before. One marketing idea you might think about is stressing the NCTM *Standards* and their
emphasis on proper mathematical communication.

**6.** *If you were the sole decision-maker, would you adopt this book for your course? Why?* Yes,
I would require it as a supplement for MTH 100 (in addition to an elementary algebra text), and cover
it first, returning to it as needed throughout the course. The reason is that I want these students
to get the point this little book sets out to make: that they need to communicate mathematics
correctly by understanding the rationale and structure of its language.

**7.** *Any other comments?* Let me say that I really enjoyed reading this book. It is wonderfully
written and crafted with a care you rarely see. (I think I could find only two typographical errors.)
If you can find the right place to put it, this book will do a great service to the mathematical
educational world.

One point I jotted down to include as I was reading the manuscript was to tell you
*not to let the copy-editors ruin the presentation!* The author has been extremely careful
and thoughtful with how she wants to present the material, from layout to font to wording,
and there are many very subtle points that copy-editors, used to editing textbooks, will
not appreciate. Please give her wide latitude to dictate against any company policies that
will interfere with what she is trying to do. (On the other hand, I personally would take
out a few of the exclamation points. And I would use "whether" in many places where she
uses "if". Also, the word "any" needs to be changed in several places; I've marked many
such places.)

Another note to myself said that this little book seems to be just one of a series in "truth and language in mathematics"; you need to look at the series as a whole when deciding how to market these materials.

I loved the margin note approach to layout—more books should do this! And a good index is also a plus, even in such a small book.

There are a few errors that the author makes, such as confusing an object and the
name for an object (the person Carol versus the word she writes as 'Carol'), and in a book
with the emphasis of this one, that distinction *must* be maintained. See page 12
for one problem. (As someone who grew up with the "new math" of the early 1960s, I recall
that this was one of the first main points we were taught, and it's important!) The author
uses the word "digit" to mean "number" (e.g., page 6); she shouldn't. I don't know whether
she will want to add a comment or not, but the statement on page 8 that sentences are
never expressions in their own right is not true—just ask any mathematical logician
(and in fact she uses this idea herself later on, when she puts the verb "is equivalent to"
between sentences).

Another thing I would strongly suggest changing is the statement that the symbol for the empty set looks exactly like a zero with a slash through it (see page 31). I prefer to write the former as Ø [correct symbol not available on web] and the latter as ø [correct symbol not available on web] and make the distinction. Perhaps the author should make even a bigger point that the empty set and zero are two totally different animals. (She needn't tell these readers that in mathematical logic, zero is defined to be the empty set in some settings!)