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Problem Solving: Tips for Teachers by Phares G. O'DafferReview by: Dana S. AdamsThe Arithmetic Teacher, Vol. 36, No. 8 (April 1989), p. 58Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193676 .Accessed: 12/06/2014 16:23

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Judith Pachciarz, who became a scientist and a physician, also has the strong support of a loving family from the beginning of her life. This support is an important piece of the pic- ture.

This reviewer would argue that all five books taken together present one explicit theme: you can make it, you can overcome the odds in spite of all handicaps, if you are highly motivated and work hard. But two implicit themes can also be detected. One is that you can make it if you are "smart." As I pointed out earlier, not everyone is, particularly in the specific ways that enabled these women to succeed; this theme might pose problems for this reason.

The second implicit theme is that you can make it if you have a caring family to give you extra support during your early years. It seems appropriate then to close with a question. What would happen if all of our nation's children were born into a larger caring family, a support- ive community, a country invested in insuring that all its children, one way or another, "make it"? These books would have more impact if they were treating community conscience, as well as individual initiative, as a real issue. This reviewer believes that the books in this series have value; they could be still more valuable if their impact on all children were reexamined from every possible perspective. - Margaret Parish, University of North Carolina at Wil- mington, Wilmington, NC 28403.

For Teachers

From NCTM

20-percent discount for individual NCTM members on NCTM publications

Problem Solving: Tips for Teach- ers, Phares G. O'Daffer, ed. 1988, 83 pp., $7 paper. ISBN 0-87353-264-3. National Council of Teachers of Mathematics, 1906 Association Dr.,Reston, VA 22091.

Problem Solving: Tips for Teachers contains selections from the monthly feature of the same title found in the Arithmetic Teacher over the past four years. With the current focus on the importance of developing problem-solving abil- ities, it is a timely publication that is designed to help teachers plan and implement problem- solving activities in the classroom on a regular basis.

To help students develop skills in each of the process steps of problem solving (under- standing the problem situation, dealing with data, planning a solution, finding a solution, and analyzing and evaluating the solution), seven goals have been identified. The student should learn to (1) use problem-solving thinking skills; (2) select and use problem-solving strategies; (3) build helpful attitudes and beliefs about problem solving; (4) use related knowledge; (5) monitor and evaluate their thinking and prog- ress while solving problems; (6) solve problems in cooperative-learning situations; and (7) find correct answers to a variety of problems.

The book is divided into four sections de-

signed to assist the teacher in planning an instructional program to meet the aforemen- tioned goals. The sections are "Developing Problem-solving Strategies," "Extending Prob- lem-solving Strategies," "Developing Problem- solving Skills," and "Implementing a Problem- solving Program." For example, the first section focuses on helping students improve their ability to plan a solution to a problem by working with such strategies as guess and check, working backward, and finding a pat- tern. Each strategy in the section includes a problem with specific information to help the teacher aid the students in focusing their think- ing. Included as well are valuable tidbits on the classroom climate, resources for additional ideas, extension ideas for various grade levels, and tips on developing problem-solving skills.

The format of the book is inviting, and most strategies are covered in two pages in a bulletin- board-type format. The tips are designed to be used by teachers in the primary through the intermediate grades, and some lists include resources for the preschool years. This book is a valuable resource for the teacher who is serious about helping students become better problem solvers. - Dana S. Adams, University of North Carolina at Wilmington, Wilmington, N 28403.

Readings for Enrichment in Sec- ondary School Mathematics, Max a. Sobel, ed. 1988, v + 258 pp., $10 paper. ISBN 0-87353-252-X. National Council of Teachers of Mathematics, 1906 Association Dr., Reston, VA 22091.

Readings for Enrichment in Secondary School Mathematics is addressed to teachers of the academically talented student in mathematics and includes articles from Topics for Mathe- matics Clubs, the Mathematics Teacher, NCTM' s twenty-eighth yearbook, and several new articles.

Articles from the twenty-eighth yearbook, Enrichment Mathematics for High School, are "Number Theory," "Mathematical Induc- tion," "A Permutation Group and Its Isomorphs," "Mathematical Models of Growth and Decay," "Problem Solving and Some Problems," "More Nonroutine Problems," and "Random Walks." The entries from Topics for Mathematics Clubs are "Fibonacci Se- quences," "Infinity and Transfinite Numbers," "Pascal's Triangle," "Experiments with Natu- ral Numbers," and "Non-Euclidean Geome- tries." From the Mathematics Teacher come the following articles: "Problem Solving in Ge- ometry - a Sequence of Reuleaux Triangles," "Reflective Paths to Minimum-Distance Solu- tions," "Logo and the Closed-Path Theorem," "Some Challenging Constructions," "The Shoemaker's Knife - an Approach of the Polya Type," "Using Magic Borders to Generate Magic Squares," "The Harmonic Triangle: Op- portunities for Pattern Identification and Gen- eralization," and "Geometric Probability - a Source of Interesting and Significant Applica- tions of High School Mathematics." Original contributions include "The Harmonic Mean and Its Place among Means," "Rotation Matri- ces, Complex Numbers, and Trigonometry,"

and "How Computers and Calculators Do Arithmetic."

Each of the articles is accompanied by a bibliography. Some of the articles include hints or solutions.

Although this book is directed toward the gifted high school student, Sobol perceives the need to enrich all students in mathematics through historical topics, challenging problems, class experiments, "mathemagic," selected topics, applications, and mathematical recre- ations.

Especially noteworthy, because of the em- phasis of the Atlantic Region Mathematics League (ARML) and the Olympiad, are the articles on number theory, problem solving, and geometric probability. The article on math- ematical induction gives many good examples and an interesting approach to obtaining formu- las (the method of least differences is hidden in the article), and the classic "tower of Hanoi" problem is included. The article on mathemati- cal models of growth and decay does an inter- esting job of using an algebraic solution of the problem as opposed to the traditional calculus approach.

The articles on problem solving are especially fun, giving the students some new problems to solve and making intriguing suggestions regard- ing the benefits of problem solving. One strong suggestion is that problem solving should lead to further study of mathematics, not just to further problem solving.

The article on natural numbers is written for students and combines uses of the computer with a number of innovative approaches to primes, superprimes, Goldbach' s conjecture, and Euclid's primes. A short article on reflec- tive paths to minimum-distance solutions should whet any student's appetite. In addition to pure mathematics, it addresses the mathe- matics of two types of billiard games.

I purchased this book at the NCTM meeting in Chicago and would recommend it as a valu- able addition to any high school teacher's class- room library. Some of the topics will be of interest to any advanced class. I was a little frustrated by the fact that some of the articles are directed to the teacher and some to the student. I would have preferred for all articles to be directed to the student. - Jean Taylor, Hoggard High School, Wilmington, NC 28403.

From Other Publishers

Build YOUr Own Polyhedra, Peter Hil- ton and Jean Pedersen. 1988, 175 pp., $27.50 paper. ISBN 0-201-22060-1. Addison-Wesley Publishing Co., 2725 Sand Hill Rd., Menlo Park, 94025.

All the lovers of geometry will delight in the paper-folding adventures that Hilton and Peder- son have in store for us in Build Your Own Polyhedra. However, this book is much more than a geometry supplement to teachers' librar- ies. From beginning to end, you will find a manuscript written with enthusiasm and with apparent enjoyment, both of which are conta- gious. The book is on and about investigating problems in mathematics that lead to inventing

58 Arithmetic Teacher

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Article Contentsp. 58

Issue Table of ContentsThe Arithmetic Teacher, Vol. 36, No. 8 (April 1989), pp. 1-60Front MatterReaders' Dialogue [pp. 2-5]One Point of View: Tracking Is Inconsistent with the Standards [pp. 6-6]"Look Ahead" Activities Spark Success in Addition and Subtraction Number-Fact Learning [pp. 8-11]Project Hands-On Math: Making a Difference in K2 Classrooms [pp. 14-16]Facilitating Understandings of Geometry [pp. 17-20]Algebra Can Be Elementary... When It's Concrete [pp. 21-24]Principles For Principals: Steps toward Building a Successful Problem-solving Program [pp. 25-26]Ideas [pp. 27-32]Research into PracticeIssues in Problem-solving Instruction [pp. 34-35]

"Equals" Means "Is the Same As" [pp. 36-40]Basic Facts DrillCard Games [pp. 41-43]Magic with Magic Squares [pp. 44-49]Two Technological Fables [pp. 50-51]Teaching Mathematics with Technology: Mathematics and Spreadsheets [pp. 52-53]Reviewing and ViewingComputer MaterialsReview: untitled [pp. 56-56]

New BooksFor PupilsReview: untitled [pp. 56-58]

For TeachersFrom NCTMReview: untitled [pp. 58-58]Review: untitled [pp. 58-58]

From Other PublishersReview: untitled [pp. 58-59]Review: untitled [pp. 59-59]Review: untitled [pp. 59-60]Review: untitled [pp. 60-60]

EtceteraReview: untitled [pp. 60-60]Review: untitled [pp. 60, 54]

Back Matter