How to Evaluate Progress in Problem Solving by Randall Charles; Frank Lester; PharesO'DafferReview by: Diane ThiessenThe Arithmetic Teacher, Vol. 36, No. 4 (December 1988), pp. 55-56Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41194393 .

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letter of the boxes in the order that Iggy opens the boxes. The graphic inch worm is cute, but its use increases the time it takes to run the program. As with the other games, the pupil can continually respond incorrectly without the computer's assuming control. In this situation, the message "You just used that" appears on the screen. On the basis of my review and classroom experience, I would say that the program works well at the K-l grade levels, but older students lose interest rapidly. Thus, this program could be a nice addition to the K-l resource library, but one program that provides chaining practice should be adequate. - John F. Martin, Jr.

Terrapin Logo. 1986, Apple Macintosh, 1 disk, 140-page tutorial, and reference guide; $79.95. Purchaser permitted to make one backup copy. Terrapin, 222 Third St., Cambridge, MA 02142. Terrapin Logo is an implementation of the MIT syntax of the Logo computer language for the Apple Macintosh computer. It maintains the clarity and simplicity of previous Terrapin Logos (e.g., for the Apple and Commodore computers) yet incorporates several new fea- tures. It can be used by students of all ages.

This review will emphasize specifics, because this package can be seen as an extension of previous versions of Terrapin Logo. Thus, the capabilities and reliability of previous versions can be assumed.

Several changes in syntax have been made, primarily in the direction of LCSI Logos. For example, several primitives have been renamed (e.g., READLIST, AND, OR, CLEAN, LOAD); the IF procedure takes lists as inputs; procedures can be buried; special characters are backslashed for printing; and primitives such as ED, for edit, evaluate their inputs - thus one must type ED "PROC. Terrapin Logo also contains several other conventions similar to those of the LCSI' s version of Logo for the Macintosh, such as a configuration program that allows adjustment of Logo's memory re- quirements, font, and number of turtles; win- dow characteristics; and the use of the ENTER instead of the RETURN key to signal execution of a command - not as convenient on the newer keyboards that have the ENTER key placed far to the right. Not all conventions have been changed; for example, I was pleased that Ter- rapin retained the use of a question mark in- stead of a P to denote predicates (e.g., MEM- BER? or NUMBER? as opposed to LSCI's MEMBERP or NUMBERP).

As an informal check of compatibility with other Terrapin versions of Logo, I loaded sev- eral programs that previously ran on the Apple II. Although differences were great enough to discourage the translation of more than a hand- ful of large programs, only minor editing -

mostly of syntax as described previously - was necessary to make the programs run smoothly.

In addition, Terrapin Logo includes several features not present in earlier versions. Graphic support includes DOT?, FILLSH (fill shape), STAMP, and SNAP primitives; multiple tur- tles; user-definable turtle shapes; and access to many Macintosh QuickDraw routines. The pro-

gramming features include extended CATCH and THROW primitives, input and output prim- itives (e.g., those related to the mouse), multi- dimensional arrays, strings, streams, and auto- matic load-on-call. Editing and debugging are enhanced by formatting of procedures and in- denting of command lines; online help; a FIND command available from a menu or command key; WATCH, TRACE, STEP, WHOCALLS, WHOUSES, and ERRORPAUSE primitives; and explicit error messages.

In comparison with some other Logos for the Macintosh, Terrapin's Logo does not refresh a graphic window that has been temporarily cov- ered by another window. Instead, the section that was covered is erased. It uses traditional workspace-based editing, instead of file-based editing, which is superior for larger programs. However, such editing can be simulated by using Logo in tandem with either a word- processing desk accessory or a word processor (the latter would require SWITCHER or MultiFinder, which allow more than one pro- gram to reside in the computer simultaneously). This version also lacks some features that are available in other Macintosh Logos, such as the multiplicity of numerical types, LISP-oriented primitives, object-oriented programming fea- tures, sound and speech primitives, and access to such Macintosh objects as menus.

On the positive side, the turtle wraps around any size screen, a feature not implemented in most other versions. It can also save and load pictures created by Logo or MacPaint; win- dows are resized to fit any loaded picture.

The documentation, which is split into two sections, is complete and organized. A feature not found in many other programs is that the tutorial section of the documentation attempts to take the beginner through all steps of learning to use the Macintosh. Most other programming concepts are introduced, explained, and sum- marized clearly and carefully, although recur- sion rears its somewhat complex head early. The second section, a reference guide, de- scribes advanced topics and includes a glossary and appendixes.

Terrapin Logo is recommended as a fine Macintosh implementation. Although not quite as innovative or powerful as other available versions, it nevertheless represents a clean and effective computer programming language with a couple of unique features. One can hope that the existence of this and other Logos will increase the effective use of the Macintosh in education. - Douglas Clements.

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New Books

Edited by Grace M. Burton University of North Carolina Wilmington, NC 28403 Diane Thiessen University of Northern Iowa Cedar Falls, 1A 50614

For Pupils

Counting Up tO 10, Jenny Tyler and Robyn Gee. 1987, 24 pp., $2.95 paper. ISBN 0-7460-0217-3. Usborne Publishing, 20 Garrick St., London WC2E 9BJ, England This book offers practice in counting and writ- ing the numbers 1 through 10. The authors recommend that an adult and child use the book together to enhance the child's counting expe- rience. Such skills as using manipulatives, un- derstanding 0, sets, and one-to-one correspon- dence are briefly discussed to give parents some idea of mathematics-related concepts that may help their child understand counting.

The pages are colorfully illustrated and invite the child to get involved with the picture in several ways. For example, on page 3 the child counts an animal once, writes the numeral 1 several times, and colors certain animals once.

The authors mention the importance of manipulatives, but they don't adequately em- phasize using these concrete teaching devices before using pictorial teaching methods. The pictures that represent numbers of higher value are very complicated, possibly interfering with the child's learning to count. The last several pages offer activities related to subtraction and ordering. These activities are clear and simple, but some children may have trouble grasping the concepts without adequate concrete in- struction. However, I think that this book can be a beneficial teaching tool if used to review rather than introduce the mathematics con- cepts.- Toi K. Creech, East Arcadia School, Riegelwood, NC 28456.

For Teachers

From NCTM 20-percent discount for individual NCTM members on NCTM publications

How to Evaluate Progress in Prob- lem Solving, Randall Charles, Frank Lester, and Phares O'Dajfer. 1987, v + 85 pp., $5.60 paper. ISBN 0-87353-241-4. NCTM, 1906 Association Dr., Reston, VA 22091.

Teachers' concerns about problem solving in- clude such questions as "What are you trying to evaluate? What are some evaluation tech- niques? How do you organize and manage an evaluation program? and How do you use eval- uation results?" A chapter is devoted to each of these questions in this excellent handbook on evaluating problem solving.

Seven goals for teaching problem solving are discussed; they include developing students' problem-solving skills, their ability to select and use problem-solving strategies, and their ability to monitor and evaluate their thinking and prog- ress while doing problems.

Evaluation techniques include observing and questioning, using students' self-assessments, holistic scoring, and multiple-choice and com- pletion testing. Each technique is defined and described; the advantages, disadvantages, and proper use of each technique are discussed.

December 1988 55

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Examples of teacher questions, surveys, and scales are given throughout. Sample multiple- choice test items are given for one-step, multi- ple-step, and process problems for each of seven thinking skills. Thinking skills include understanding the question, understanding the conditions and variables, and determining the reasonableness of results. Throughout the chapter on evaluation techniques, the reader is encouraged to work or write sample items for each technique as it is discussed.

To help the reader organize and implement an evaluation plan, guidelines and sample pro- grams are discussed in the third chapter. The focus of the fourth chapter is using evaluation results to help make instructional decisions on the basis of the strengths and weaknesses of students' problem-solving skills. This topic is discussed with regard to classroom climate, curriculum, teaching methods, and assigning grades. The appendixes include guidelines for in-service education and a list of resources on problem solving.

This book is part of the "How to . . ." series that is designed to be directly applicable to the classroom. The book definitely fits the criteri- on. The guidelines and suggestions are thought- fully presented in a very readable style. This handbook should be extremely helpful to any teacher - whether at the elementary, second- ary, or university level. - Diane Thiessen.

How to Teach Mathematics Using a Calculator, Terrence Coburn. 1987, v + 58 pp., $6.90 paper. ISBN 0-87353-245-7. NCTM, 1906 Association Dr., Reston, VA 22091. This collection of calculator activities for pri- mary, upper elementary, and middle school students is definitely what the title indicates - a resource for teaching mathematics to students. The introduction describes the calculator as an instructional tool to assist in the development of concepts, to help reinforce skills, to promote higher-level thinking, and to enhance problem- solving instruction.

The chapters in the collection cover numera- tion and counting, addition and subtraction, multiplication and division, decimals, problems and applications, prealgebra, measurement ge- ometry, and percent. Each activity is contained on a single page that can be reproduced for activity sheets or transparencies. Some activi- ties are games to be played by two people; others are appropriate for individual or small- group work; others are for whole-class instruc- tion.

One estimation activity uses the constant function to count by l' s to 50; the child is asked to predict and then check how long it would take to count by l's to 100, by 2's to 100, and by 5's to 100. A more advanced counting activity involves decimal fractions and explains how numbers such as forty hundredths are displayed on the calculator. A computational estimation activity includes deciding whether the sum of two numbers is more than 500, about 600, or more than 600; students use a calculator to check their estimate. A more advanced estima- tion activity includes problems such as .8 times what number is a product in the range of 125-150? The students estimate and then check.

Some of the prealgebra activities include ex- ploring properties and order of operations, solv- ing equations and proportions, and exploring exponents and integers. The presentation of these ideas is motivating and interesting. My only question concerns the use of the percent key. One activity involved finding 50, 10, 1, and 90 percent of a number; for the first three percents, it is faster to use common fractions and mental computation. If the purpose of the problems is to become familiar with the percent key, they are appropriate; otherwise, a calcula- tor isn't needed.

To help the teacher select a calculator, the introduction includes a brief discussion of dif- ferent types of calculators and their functions. Teacher notes are placed at the back of the book. The notes describe the purpose of each activity and identify which functions are prac- ticed. Each activity has an answer key and a grade-level classification. Sample tests for pri- mary, upper elementary, and middle school students are included.

As noted in the introduction, problem solving is emphasized throughout the collection. Stu- dents are involved in such activities as finding patterns, exploring numbers and operations, estimating results, making conjectures, and forming generalizations. The book presents an excellent set of activities that you and your students should enjoy exploring. - Diane Thies- sen.

The Ideas of Algebra K-12, Arthur f. Coxford, ed. 1988, viii + 248 pp., $16 cloth. ISBN 0-87353-250-3. NCTM, 1906 Association Dr., Reston, VA 22091. The Ideas of Algebra is a textbook for every mathematics teacher from elementary school through college. It begins with a chapter on the reshaping of school algebra. This chapter causes even the most experienced teachers to question and possibly reevaluate the traditional topics and teaching methods. Beginning teach- ers and teachers of elementary mathematics can gain insight from the chapters pertaining to the use of variables, conceptual difficulties that students have with algebra, and algebra as "generalized arithmetic." Other chapters that are valuable for experienced and preservice algebra teachers discuss teaching algebraic ex- pressions in a meaningful way and focusing on the difficulties that students have with the func- tion concept.

The needs of younger students are empha- sized in the chapters about establishing stu- dents' readiness for algebraic concepts. Even third graders can use manipulatives and picto- rial representations to solve problems, and they can manipulate numbers and operations before they are introduced to algebraic expressions. Applications and graphic interpretations of pro- portional reasoning can help students under- stand the relationship between algebra and arithmetic.

Algebraic equations and expressions are dis- cussed in depth by examining different ap- proaches students take to solving equations and by integrating equation-solving methods in a developmental learning sequence. These dis- cussions focus on the students' abilities to

understand the mathematics that they are trying to master. Although I am not sure that the chapter on polynomials in the school curricu- lum is best fitted to this section of the book, it is a very interesting chapter that negates the idea that factoring and polynomials are becoming topics of algebraic history.

Using diagrams to solve problems and stress- ing the importance of translating from words to algebra are incorporated in a chapter on teach- ing algebra with a word-problem focus. Also discussed is teachers' and students' use of computers and calculators to solve problems ranging from elementary arithmetic to interme- diate algebra. Whether programming the com- puter to learn algebraic concepts, finding roots of polynomials, generating tables, or using spreadsheets to teach algebra, excellent mate- rial is available to algebra instructors at all levels. The calculator is presented as a tool for discovery rather than as simply a mechanical device to facilitate the use of algorithms.

For the teacher who is always ready to try something new and interesting in the algebra class, the section pertaining to teaching ideas is perfect. Although these chapters are geared to algebra classes, they are worthwhile reading for everyone who teaches mathematics. The twen- ty-four special problems titled "Can your alge- bra class solve this?" (with answers) are incen- tives for teachers to make certain that their classes do know how to solve those types of problems. - Carol W. Brewer, Westover Junior High School, Fayetteville, NC 28303.

Preparing Elementary School Math- ematics Teachers: Readings from the Arithmetic Teacher, Joan worth, ed. 1988, vi + 169 pp., $10 paper. ISBN 0- 87353-251-1. NCTM, 1906 Association Dr., Reston, VA 22091. This publication consolidates forty-eight arti- cles on preservice elementary school mathe- matics teacher education that have appeared in the Arithmetic Teacher from 1962 to 1986. One purpose of the book is to remind teacher edu- cators of the broader issues involved in training elementary school teachers. The second pur- pose of the book is to serve as a resource of practical ideas to use with classes of preservice teachers.

The book is divided into six sections. The first three sections deal with the teacher- educator as teacher, program planning, and attitudes. The last three sections deal with methodological topics, such as integrating mathematical content and methods, modeling manipulatives in a laboratory setting, and using writing exercises in a mathematics education class.

Two features of the book were found to be slightly disappointing. First, the original publi- cation date of the most current reprinted article is 1986, which means that the work reported in the articles was done no later than 1985. Yes, historical perspectives are certainly valuable, but some of the information offered in several articles is already outdated. For example, a very comprehensive list of books recommended by outstanding mathematics educators is pre-

56 Arithmetic Teacher

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