# The Craft of Word Problems

## Using Algebra Word Problems to Explore Problem Space

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After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

## Bibliography for Teachers and Students

Berger, Debra Wolk,
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Getting Ready to Take the Math 10 CAPT Test
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. Brewster, New York: Kingsbridge, 2003.

The preparatory workbook many high schools in Connecticut are providing for students to cram for the tenth grade CAPT test in mathematics. A wide range of algebra and geometry problems is included and samples are provided with solutions for each section of the book. The nature of this and any test-prep book is that it emphasizes getting the answer more than understanding the nature of the mathematics. However, some interesting problems are included which could lead students to other investigations. In the short run, we are going to use these books, so mine them for valuable problems. The tenth grade CAPT test always has some great problems that stretch student thinking. All the “released” math questions from previous years of the test are available on the Connecticut State Department of Education’s website.

Carpenter, Thomas, Megan Loef Franke, and Linda Levi.
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Thinking Mathematically: Integrating Arithmetic & Algebra in Elementary School
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. Portsmouth: Heinemann, 2003.

Discussion by teachers actively engaged in classrooms about what it means to do and learn math. Filled with examples that illustrate the algebraic thinking processes, which are (or should be) a part of the learning of arithmetic at the earliest grades. Of all books read in our seminar, this one opened up the most discussion among teachers from grade one to high school. Also includes a CD featuring classroom videos of students engaged in the problem soling dialogues described by the authors.

Charles, Randall, and Carne Barnett,
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Problem-Solving Experiences in Pre-Algebra
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. New York: Addison-Wesley, 1992.

A workbook of “thinking activities, problem solving strategies, challenge problems and more--designed to support and encourage student problem solving in math classes. Covers a wide range of math problems but requires careful selecting by teacher to develop focused problem sets around a key concept..

Harnadek, Anita,
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Algebra Word Problems
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. Pacific Grove, CA: Midwest, 1988.

A Set of twelve slim paperback booklets, each with a theme or concept (though three of the booklets re “miscellaneous” problems) around which the 40 to 50 problems are built. The explanations are spare and drive mostly toward variable definition and equation solving. There is no reason why students could not use my “5-part problem template” for any of the problems. This resource is an inexpensive way to obtain problem sets for some algebra topics and the problems provided generally progress from less to more difficult.

Kordemsky, B. The Moscow puzzles. Edited by Martin Gardner. New York: Dover, 1992.

Mathematical puzzles and problems often appearing simple, but proving quite tricky. Several of these nicely demonstrate common pitfalls of algebraic problem solving and could be well worth the time spent working through with students.

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Lampert, Magdalene.
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Teaching Problems and the Problem of Teaching
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. New Haven: Yale University Press, 2001.

Detailed reflections by an experienced classroom teacher and university professor who spends a year teaching a fifth-grade math class as part of her research. All teachers of math can benefit from this thoughtful and well-written examination of the dynamics of teaching and teacher-student relationships. Particularly helpful is Lampert’s careful attention to student problem solving by individuals and her whole class.

Loyd, Sam
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. Mathematical Puzzles
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. Selected and Edited by Martin Gardner. New York: Dover, 1959.

Puzzles originally published in magazines and newspapers by Sam Loyd, “America’s greatest puzzlist” have been organized in this volume by mathematical category. Many of these puzzles are classic word problems that can fascinate curious students or drive them crazy trying to solve them. Fortunately, solutions are provided in the appendix. With some guidance, many of the puzzles might be launching pads for exploration of mathematical big ideas.

Olas, Carla,
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Problem Solving and Algebra Too
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. Boston: Northeastern University Custom, 1992.

An algebra course workbook/practice book, this text has hundreds of problems with answers provided in the appendix, but little to no explanations. The author does provide problems sets with instructions to read them and sort into groups of related problems--a key step in developing the ability to explore “problem space.” This book may serve as a source of problems for a particular type of problem, but it is a bit overloaded with tedious black and white graphics and unhelpful details.

Polya, George,
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How To Solve It
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. Princeton: Princeton U. Press, 1973.

One of the classics in the writing about problem solving, this book offers a problem analysis that boils down to four steps: understand the problem, devise a plan, carry out the plan, and examine the solution. Vee-charts of word problems as I use them are graphic organizers based roughly on Polya’s steps.

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Primary Mathematics Texts 6A and 6B
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, U.S. Edition. Curriculum Planning and Development Division, Singapore Ministry of Education. Singapore: Federal Publications, 2003.

As mentioned in my paper, this set of texts is a very accessible presentation of problems for pre-algebra to early algebra students. the arithmetic understandings developed will give a foundation for more advanced algebra. The “concrete--pictorial--abstractct’’ approach is helpful for students who struggle with too much perceived abstractness in mathematics.

Reimer, Wilbert, and Luetta Reimer,
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Historical Connections In Mathematics: Resources for Using History of Mathematics in the Classroom
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. Fresno: AIMS Educational Foundation, 1992.

A 3 volume series of books that use great mathematical thinkers to present historical connections as well as the key concepts the pioneer thinkers developed. Very useful for exploring the roots of some of the more interesting problem solving techniques (Napier’s Bones, for example).

Santi, Terri,
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Math Ties: Problem Solving, Logic Teasers, and Math Puzzles
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. Pacific Grove: Critical thinking books, 1998.

A great source of quick puzzles and problems to introduce new units or build into problem sets--simpler and more accessible than the Sam Loyd or Moscow Puzzles..

Smart, Margaret, and Mary Laycock,
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Hands-On Math for Secondary Teachers
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. Hayward, CA: Activity Resources, 1984.

Though somewhat older, this guide is an interesting resource for teachers who are seeking alternatives and different strategies for teaching fundamentals of math to junior and senior high school students. Of great potential are the use of graphic solutions to problems and use of simple drawings and manipulatives to make concepts real.

### Websites:

www.japanese-online.com/math

As mentioned in my paper, this site provides 225 selections of 20 different types of algebra word problems along with illustrations, hints, explanations, and suggested time expectations for solving by 12-year-old students in Japan. Teachers and students can use this site as a learning or tutoring tool.

www.mathpower.com

Author and math professor Ellen Freedman has created this site to offer basic math and algebra tutorials, word problems and sets of problems for students and teachers. The site uses humor and lively graphics to address concerns about math anxiety and to provide practice in solving problems. Links to nearly one hundred math and other educational websites are included.