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Publisher:

Mathematical Association of America

Publication Date:

2005

Number of Pages:

260

Format:

Hardcover

Series:

Classroom Resource Materials

Price:

54.00

ISBN:

978-0-88385-739-7

Category:

General

[Reviewed by , on ]

Steve Maurer

08/13/2005

This is a very valuable and well written book that goes into depth about certain high school mathematics topics, connects them, and in the process illustrates how to think like a mathematician. Its primary audience is strong high school teachers, especially in workshops and study groups, and professors who teach current or future teachers. However, most professional mathematicians would enjoy the book too and find things they didn't know.

Now more detail.

In the introduction to this book, the author Cuoco writes:

For too long, professional development for high school teachers has meant either taking yet another university mathematics course (with no connection to what we teach) or taking a workshop about teaching (with no mathematics in it at all). But the foundations of high school mathematics go deep enough to involve sophisticated thinking, hard problems, and subtle connections, all the while staying connected to the kind of mathematics we talk about with our students every day.

In short, high school teachers need courses for themselves that provide a more advanced viewpoint on the math they teach, not just courses on college and university mathematics and courses on pedagogy.

This book is for such a course.

Such books are definitely needed. Indeed, others have recognized this too recently. See *Mathematics for High School Teachers: an Advanced Perspective* [5] by Usiskin et al. Perhaps Hartshorne's *Geometry: Euclid and Beyond* [2] should also be mentioned, though it was written for undergraduates. Of course, there are also old classics on elementary topics from an advanced standpoint by Klein [3] and Moise [4].

But Cuoco's book is very different from all of these. It has a second premise:

High school teachers need to see how to think about mathematics (how to explore, how to discover, how to generalize, what sort of issues to think about, how to prove) and this is missing from standard texts, for them or for most anybody.

To meet both needs, Cuoco has written a very personal book. It has a very personal choice of topics — it does not attempt to be comprehensive, even about a segment of high school mathematics. Second, it has a very personal (and personable) style. He talks to the reader about how to think about things, and cycles around, including false starts, until a deep understanding is reached.

To give an example, consider Chapter I. Its main question is: Can we fit a polynomial to a table of (*x*,*y*) values? At first the tables have consecutive *x* values. Cuoco introduces differences *f*(*n*+1) - *f*(*n*) and soon readers discover that you can find a formula for the original function if you can sum the differences. So you can find formulas if you can sum the first *n* *k*-th powers for various *k*. But this seems hard, so Cuoco tries another approach, involving higher order differences. This leads to Newton's Difference Formula, the discrete analog of Taylor polynomials. Then the problem is generalized to nonconsecutive *x* values, and Lagrange interpolation is developed.

Along the way, various important issues begin to emerge. One begins to get a feeling for the linearity of data fitting methods, that this makes bases important, and that {1, *x*, *x*^{2}, ...} is not the right polynomial basis for every problem. Indeed, the combinatorial polynomials are shown to be a much better basis for difference algebra.

The approach involving sums of *k*-th powers is not forgotten. It is completed in the final Chapter 5, using methods and sequences of numbers that in the meantime have already appeared in answer to a combinatorial lock problem in Chapter 4. It is such interplay between seemingly disparate problems that leads to the word Connections in the title. Other connections can be gleaned from the very nice Annotated Table of Contents. Here I just list the chapter Titles:

- Difference Tables and Polynomial Fits
- Form and Function: The Algebra of Polynomials
- Complex Numbers, Complex Maps, and Trigonometry
- Combinations and Locks
- Sums of Powers.

There is a danger with any book based on "how I think about it," that it will fall flat outside the hands of the author. Other people think differently, and they may have more trouble following your journey than following a straightforward, codifying approach, even if the latter is less interesting. Cuoco is aware of this danger and works hard to confront it. His method? He made this a problems book. There is text, but that's the tip of the iceberg. There are many problems, and you are asked to do them all, in order. Each problem has a point, and the points of the problems are what make the author's way of thinking about the material follow naturally. I could see the point of many problems by reading them, but there were other problems which I realized I too would have to solve in order to see the point.

Of course, not everyone, certainly not independent readers, will really do every problem and in order. But Cuoco helps in various ways. First, he groups problems on common themes, so at least you can work one theme at a time. Second, if you do skip an important problem — many problems are needed later — at that later point he tells you what problem to go back and do. Third, at the end of each chapter he has notes for selected problems that often clue you in on what you missed.

So will the book work? Probably Yes, at least in the hands of experienced independent readers, or good teachers who assign most of the problems and use the reading to inform their own classroom plan. Time will tell. It's certainly worth a try.

Readers need to understand: This is a very algebraic book. This does not mean *mechanical* algebra, as in old-fashioned high school algebra books that spend 90% of their time expanding or factoring expressions. In fact, one of the very nice features of this book is that Cuoco has just the right attitude toward Computer Algebra — he encourages the reader to leave carrying out the calculations to a CAS as appropriate. But the book is nonetheless algebraic in two senses. First, Cuoco makes no apologies for using and expecting users to be comfortable reading and manipulating long algebraic expressions. Indeed, he finds such expressions delightful.

Second, Cuoco is interested in the abstract algebra aspects of high school polynomial algebra: traditional theory of equations, but also the theory of differences and derivatives as very similar formal operators on the ring of polynomials.

He is also interested in complicated patterns among coefficients of polynomials. Indeed, if there is one theme that shows up in every chapter, patterns in polynomial coefficients is it. Bernoulli numbers, Stirling numbers, combinations, *k*th-differences, *k*th-derivatives — they all show up as coefficients of terms in polynomials written various ways. This sort of delight with algebraic rewritings reminded me of the college book *Concrete Mathematics* [1], in which Knuth et al revive Euler's attitude. Indeed, Cuoco makes several glowing references to *Concrete Mathematics*. Come to think of it, Cuoco's book could be described as a high school oriented *Concrete Mathematics*. I do feel that most readers will weary of polynomial algebra long before Cuoco — he pushes each idea in this book for all it's worth.

Now, is a highly algebraic book for teachers a good thing? Probably so. There has been a tendency to downplay algebra in recent years in reform curricula, and this book is a nice counterweight. I myself admit to having mixed feelings on this important issue — how much algebra should school curricula contain? In any event, teachers should know a fair amount, though they may need to be weaned from the turn-the-crank approach to it. This book may do a good job of weaning them.

This is a fine book, but every reader dislikes some things, perhaps unreasonably. Reviewers get to tell the world what they don't like. Here's my list.

**1. The Title.**

It doesn't really convey what the book is about. It should have been something like "Explorations into High School Mathematics: Mathematical Thinking for Teachers and Others." Since you're reading this review, you don't need a better title, but others do.

**2. Sidebars.**

This book makes heavy use of sidebar comments (again like *Concrete Mathematics*). The reason proffered is a good one: a major theme is how to think about mathematics, and such thinking is often nonlinear; hence it doesn't always fit into the linear flow of traditional text. However, I continue to be unimpressed with the sidebar "solution" to this problem. First, sidebars give only one level of commentary. Second, since writers abhor a vacuum (aka blank space), there is a tendency for authors to look for something to fill (clutter?) the wide margin. I think the real response to the nonlinearity issue will be hypertext electronic books, where clicks open new windows (no space need be reserved on the main pages) and the interlinking of comments can be quite complex. This is what Ted Nelson had in mind 45 years ago when he proposed the Xanadu system that eventually led to the Web.

**3. The Index.**

It's barely over 1 page and woefully incomplete. The first 5 things I tried to look up — none of them were there. For a book where you are supposed to go back and forth in an ever growing circle of ideas, a really good index is vital.

This is a very nice book, for high school teachers, for teacher educators, and for anyone curious to read something lively and learn more about high school algebra. Readers who commit themselves to doing a good proportion of the problems in the order presented won't be disappointed. It is an excellent book for summer institutes and term-time working groups.

On the other hand, if you are an educator looking for a text for a comprehensive course on high school mathematics from a deeper perspective, you are better off considering [5].

**References:**

1. Graham, Ronald L., Knuth, Donald E., and Patashnik, Oren, *Concrete Mathematics : a Foundation for Computer Science*, 2nd ed, Addison-Wesley, Reading MA, 1994.

2. Robin Hartshorne, *Geometry: Euclid and Beyond*, Springer-Verlag, New York, 2000.

3. Klein, Felix, *Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis*. Translated from the third German edition by E. R. Hedrick and C. A. Noble, Macmillan, 1932.

4. Moise, Edwin E., *Elementary Geometry from an Advanced Standpoint* 2nd ed, Addison-Wesley, Reading, MA, 1974.

5. Usiskin, Zalman, et al, *Mathematics for High School Teachers: an Advanced Perspective*, Pearson Education, Upper Saddle River, NJ, 2003.

**Note:**

Cuoco says "we" because he was a high school teacher for over 20 years. Since 1993 he has been fulltime at Educational Development Center, where he has become well known for many innovative projects and teaches teachers. Or really, collaborates with teachers. Some of the novel approaches in Mathematical Connections were devised in collaboration with teachers, for instance, a development of the geometry of complex multiplication that allows one to prove the trig addition formulas from the properties of complex multiplication rather than vice versa. Oh, Cuoco got a Ph.D. in algebraic number theory in the middle of his school teaching years.

Steve Maurer (smaurer1@swarthmore.edu) is Professor of Mathematics at Swarthmore College near Philadelphia, and currently Chair of Mathematics & Statistics there. He is coauthor of Discrete Algorithmic Mathematics and the MAA's Contest Problem Book V. In his spare time he tends his garden, writes emails or TeX macros, and wishes he were riding trains.

See the table of contents in pdf format.

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