The Teaching and Assessing of Mathematical Problem Solving by Randall I. Charles; Edward A.SilverReview by: David J. WhitinThe Arithmetic Teacher, Vol. 37, No. 7 (MARCH 1990), pp. 56-57Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41193865 .

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that leads to a castle. As the student travels the maze, questions must be answered correctly before the student can pass through the closed gates. Although I have usually considered this type of format to be a distraction from the math- ematics practice, the fact that the student can control movements with the mouse and the fact that the Macintosh graphics are brisk and re- sponsive lead me to overlook that negative fea- ture in this program. The mathematics practice is the prime goal, and the mazes simply furnish a context and a motivational challenge for the students to do the mathematics.

Students' responses to the questions can be entered in a variety of ways. The student can enter digits from the keypad by pointing with the mouse and clicking on the appropriate digit or by dragging the digit into the proper place in the answer. A feature that I appreciated is the flexibility that is allowed in entering the an- swers. The flexibility becomes important for the student who can quickly get the answer men- tally. Rather than enter each part of the arith- metic layout, the student can drag digits into the answer portion, increasing the intensity of the practice and eliminating some of the boredom found in those drill programs that accept an- swers in a very strict sequence. This flexibility is further demonstrated in arithmetic exercises. Many programs require the student to enter an- swers from right to left, following the paper- and-pencil algorithm, whereas this program al- lows the student to enter digits from either direction. Entering numbers from left to right is more natural in exercises in which mental arith- metic yields a quick response.

The management system allows the teacher to set up grade-level or custom practice sets for individual students. It is well documented in the manual and easy to accomplish. The student reports are presented as tables and graphs, giv- ing clear profiles of students' progress.

If your classroom has access to a Macintosh, I believe this program warrants a second look as a possible addition to your software li- brary. - Vincent F. O'Connor.

WEST CHESTER UNIVERSITY DEPT. OF MATHEMATICS &

COMPUTER SCIENCE WEST CHESTER, PA 19383

A tenure-track position in Math. Educ. at the assistant or associate professor rank is avail- able beginning September 1990. Applicants should hold a doctorate in math, and/or math,

educ, have a strong background in math., and demonstrate a capability for research in K-8 math. educ. Teaching load includes pre- service elementary and undergraduate math, courses. Send transcripts, vita, and three let- ters of reference to Mark Wiener, Dept. of Math, and Computer Science, WEST CHES- TER UNIVERSITY, West Chester, PA 1 9383.

Application deadline has been extended to March 20, 1990. AA/EOE. Women and minorities are encouraged to apply.

REVIEWING AND VIEWING

New Books

For Teachers

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

New Directions for Elementary School Mathematics, 1989 Year- book, Paul R. Trafton and Albert P, Shulte, eds. 1989, viii + 245 pp., $18 cloth. ISBN 0- 87353-272-4. National Council of Teachers of Mathematics, 1906 Association Dr., Reston, VA 22091.

The 1989 Yearbook is a valuable resource for elementary school teachers and mathematics educators. The theory and activities share the vision of the Standards. The yearbook focuses on the importance of developing a solid foun- dation in mathematical understanding and ap- preciation during the elementary school experi- ence.

The book is organized into five parts. The balance between theoretical considerations and classroom applications makes the book useful.

The opening chapter presents a discussion on the five areas of needed change in elementary school: the view of mathematics and mathemat- ics learning, curriculum, instruction, evalua- tion, and support. Other chapters in this part support this discussion by examining the roles of communication, reasoning, and problem solving. Of particular importance is the chapter on the role of computation.

The two articles in the second part examine the implications on instruction of evidence gathered from recent research on children's thinking and learning. Teacher-pupil conversa- tions illustrate strategies that guide meaningful instruction.

Part 3 presents approaches for developing the content of the changed curriculum. Interesting considerations include connections between language experience and mathematics and some very practical suggestions for instructional ac- tivities on data analysis, measurement, frac- tions, and calculator use. Part 4 continues this theme in the three chapters on mathematical exploration and active involvement of students.

The concluding section focuses on factors that influence the way mathematics is taught

Edited by Grace M. Burton University of North Carolina at

Wilmington Wilmington, NC 28403 Hilde Howden Mathematics Consultant Albuquerque, N M 87114

and learned. Of particular interest are the chap- ters on cooperative learning and staff develop- ment.

Samples of students' work appear between sections. These are refreshing examples of what mathematics papers can look like.

This yearbook does provide "new direc- tions." Teachers could lend their copy to their superintendent, building principal, and curricu- lum director as one way to answer their ques- tions, "What are the Standards about?" and "What can our school do?" Chapter 1 and chapter 20 (staff development) would be a good place to start.- Margaret M. Scott, Oklahoma State University, Stillwater, OK 74078.

The Teaching and Assessing of Mathematical Problem Solving, Ran- dall I. Charles and Edward A. Silver, eds. 1988, 282 pp., $15 paper. ISBN 0-87353-267-8. Na- tional Council of Teachers of Mathematics, 1906 Association Dr., Reston, VA 22091.

This volume contains an excellent set of articles that focus on the salient issues in the area of research on teaching and assessing mathemati- cal problem solving. It addresses the concern that problem solving not be perceived as a list of skills or techniques but as a way of thinking and understanding. Schools must consider problem solving from a broader perspective in which students tackle a wide variety of prob- lems in many different contexts.

Many of the authors stress the importance of collaborative mathematical thinking. Jean Love, Steve Smith, and Michael Butler propose an apprenticeship model for school mathemat- ics that is derived from the field of anthropol- ogy. This interdisciplinary approach, which uses the theoretical perspectives from other fields of study, is a common theme throughout this volume. This apprenticeship model views learners as active constructors of their own knowledge, collaborating with peers on real problems that are often ill defined. Important goals that people gain from this apprenticeship are "recognizing opportunities or problem find- ing, knowing when and how to apply skills that have been learned in other contexts, and ex- ploiting properties of the present sitation" (p. 63). Lauren Resnick also urges that mathemat- ics be viewed as a language that learners dis- cover, not receive; she advocates treating mathematics as an ill-structured discipline that invites argumentation, debate, and multiple in- terpretations. The socializing of mathematics learning that she proposes calls for students to invent, justify, and critique solutions to prob- lems.

In another article Alan Schoenfeld reiterates the concern for context in mathematical prob- lem solving by discussing the impact of the cul- tural milieu of the classroom on students' atti- tudes and beliefs. He argues that a classroom culture must encourage the art of sense-making, since mathematicians spend most of their time making sense of things, figuring out how things fit together in a particular way.

The role of metacognition in problem solving is addressed in articles by Frank Lester and by

** ARITHMETIC TEACHER

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Joseph Campione, Ann Brown, and Michael Connell. After discussing the limitations of tra- ditional instructional practice, Campione, Brown, and Connell advocate the use of recip- rocal teaching, a guided cooperative-learning strategy that employs questioning, clarifying, summarizing, and predicting. Students work in small groups to solve problems, with each member, in turn, serving as the discussion leader as well as a supportive critic. Lester de- scribes a metacognitive strategy that encom- passes three components: (1) teacher as exter- nal monitor, such as a discussion leader; (2) teacher as a facilitator of students' metacogni- tive awareness; and (3) teacher as a model of metacognitively aware problem solver.

Other articles address the history of problem solving in the mathematics curriculum; testing problem solving; and implications of this cur- rent research for programs of teacher educa- tion. The volume is an excellent summary of current studies, as well as an agenda for prom- ising new research. It is an important book for those interested in conducting research in math- ematical problem solving. - David J. Whitin, University of South Carolina, Columbia, S С 29208.

From Other Publishers

Connections: Linking Manipulatives to Mathematics, Grades 1 through 5, Linda Holden Charles and Micaelia Ran- dolph-Brummett. 1989, Grades 1-4, 92 pp. ea.; Grade 5, 89 pp.; $17.50 ea., paper. Grade 1, ISBN 0-88488-768-5; Grade 2, 0-88488-769-3; Grade 3, 0-88488-770-7; Grade 4, 0-88488-771-5; Grade 5, 0-88488-772-3. Creative Publications, 788 Palomar Ave., Sunnyvale, CA 94086.

Connections is a series of SVz-inch-by-ll-inch spiral-bound resource books of manipulative- based activities, one for each of grades 1-6 (number 6 was not sent for review).

Several features distinguish Connections from most other manipulative resource materi- als. The twenty activities in each book are de- signed to correlate with the mathematics con- cepts taught at a specific grade level and are organized according to these concepts, thus fa- cilitating the integration of active learning into any curriculum. Each activity incorporates the philosophy of Connections, "active learning is the foundation of understanding," by connect- ing hands-on experiences with mathematics and language. To insure these connections, each four-page lesson includes specific teaching sug- gestions for introducing the problem, exploring with the identified manipulative, recording the connection, reporting and displaying results, and extending the concept.

Lessons are structured for instruction with a whole class. Each lesson suggests the class- room organization to be used (size and number of groups of students and whether they should work together or independently) and specifies the kind and number of materials needed by each group. Since Connections was designed to accompany the Connections manipulative kits,

the materials used at each grade level are con- tained in the corresponding kit. This feature should not be a deterrent, however, since Pat- tern Blocks, base-ten blocks, Unifix cubes, tan- grams, plastic coins, and geoboards are com- mon to most kits of manipulatives. In addition, black-line masters of paper versions of Pattern Blocks, base-ten blocks, coins, teddy-bear counters, and geoboard dot paper are supplied in the books that specify their use. Rainbow Cubes, which are centimeter cubes of various colors, and Fraction Factory Pieces, which are rectangular color-coded tiles that represent halves, thirds, fourths, . . . , twelfths of a given area, can either be purchased individually or easily made.

The black-line student recording sheet in- cluded with each activity helps students to ex- plore, organize their data, and summarize their results. In no activity is it a worksheet to be completed by supplying the "right" answers.

Teachers with extensive experience working with manipulatives, as well as novices, will ap- preciate these carefully planned activities that assist them in implementing the NCTM's Stan- dards into any mathematics program they may be using. - Hilde How den.

Cooperative Learning in Mathemat- ics: A Handbook for Teachers, Neil Davidson, ed. 1989, 409 pp., $25.60 paper. ISBN 0-201-23299-5. Addison-Wesley Publish- ing Co., 2725 Sand Hill Rd., Menlo Park, С А 94025.

This book is an anthology of twelve articles about various aspects and techniques for using small-group and cooperative learning in the mathematics classroom. The articles have been written by various authors and cover elemen- tary, secondary, and college levels, as well as general issues relevant to the subject. In addi- tion, appendixes contain further resources and results of surveys of teachers. An introduction to the book lists some helpful key questions to ask regarding such topics as assessment, class- room management, and student differences.

The first article is "The Math Solution: Using Groups of Four," by Marilyn Burns. The au- thor shares insights into how children learn mathematics better if they are given opportuni- ties to develop mathematical understanding through social interactions and the handling of physical objects. This kind of information is helpful to teachers who must give administra- tors and parents a rationale for what often is a significant change in teaching style. The same article also describes some sample elementary school lessons with good descriptions of stu- dent-student and teacher-student interactions.

Another article, "The Small-Group Discov- ery Method in Secondary and College Level Mathematics," by Neil Davidson, focuses on small-group learning in calculus. The author cites research from the field of educational phi- losophy that furnishes a basis for developing guidelines to facilitate group problem solving. This article also describes actual classroom sit- uations and students' explorations of such top- ics as computing integrals of simple polynomi-

als and the mean- value theorem. The blend of research, theory, and practical

suggestions makes this book helpful both to teachers who are just beginning to use cooper- ative learning in mathematics and to those who have had much experience. The publication of a book on this topic is especially timely in light of recent calls for reform in mathematics educa- tion. - Claire J. Fenton, New Mexico Depart- ment of Education, Santa Fe, N M 87501-2786.

Hands On Logic: Kindergarten through Grade ř4ine, Linda Sue Brisby, Andy Heideman, Natalie Hernandez, Jeanette Leng er, Ron Long, Petti Pfau, Scott Purdy, and Sharon Rodgers. 1989, 167 pp., $15.95 pa- per. Hands On, 2121 Rebild Dr., Solvang, С А 93463.

Since the time of the Greek philosophers, the education of the young in the logic of reasoning individuals has been a subject of considerable concern. It is most refreshing to find a book that treats the teaching of logic as a delightful inter- action with materials rather than a stodgy train- ing of the mind. This is the third volume in a series whose intent is to help teachers teach mathematics more effectively by using manip- ulative materials. The authors designed more than 150 lessons to go beyond logic puzzles and assist students to use organized data to draw new conclusions. Having done a task analysis of logic and found thirty-one components, the authors then present several activities for each, labeling them primary, middle, middle-upper, and upper and coding them to the analysis. Some of the lessons are quite innovative, whereas others are as traditional as syllogisms. And although all the lessons do not really in- volve hands-on materials, at least all do use a process approach.

The authors acknowledge that logical thought is not necessarily a step-by-step linear proce- dure in which one skill is mastered before mov- ing on to the next level. The activities build on the students' mathematical background but present computational proficiency not as an end in itself but as a means to understanding the practical situation. Some of the lessons are seemingly outside the area of mathematics, such as one that examines common tools of the 1800s and another that explores and/or/not statements. Others seem better placed in a sci- ence or music class, which should be viewed as a strength rather than a weakness, for the ap- proach presents problem solving as a matter in- volving integration of the total curriculum rather than the exclusive domain of the mathe- matician.

This book will appeal to the practical-minded educator. The materials suggested are easily obtainable, and the sketches, examples, and de- scriptions are surprisingly precise. Lessons can augment and enrich any mathematics program or series being used by the reader, since activity pages are organized by task from the primary to the upper level, allowing teachers to select for whatever level is needed. - Ann Lockledge, University of North Carolina at Wilmington, Wilmington, NC 28403.

MARCH 1990 SI

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