BY WAY OF INTRODUCTION: Empowering Students through ConnectionsAuthor(s): Carol A. Thornton and Randall I. CharlesSource: The Arithmetic Teacher, Vol. 40, No. 6, FOCUS ISSUE: EMPOWERING STUDENTSTHROUGH CONNECTIONS (FEBRUARY 1993), pp. 298-299Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/41195583 .

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^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^н BY WAY OF INTRODUCHON

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Empowering Students through Connections

Mathematics is a way of thinking that

involves understanding, recogniz- ing, valuing, and using relationships in many different contexts. The connections between and among mathematical ideas are more

significant than any isolated mathematical encounter, task, or activity. In a real sense, the whole of mathematics is greater than the sum of its parts. For this reason, the Curricu- lum and Evaluation Standards for School Mathematics (NCTM 1989) emphasizes the need "to connect ideas and procedures both among different mathematical topics and with other content areas" (p. 11).

A holistic view of the nature and power of mathematics can and should be nurtured by emphasizing mathematical connections. This view can be developed in many ways. The curriculum and evaluation standards document presents a dynamic vision of a classroom in which making connections is

emphasized (NCTM 1989, 32):

Ideas flow naturally from one lesson to another, rather than each lesson being restricted to a narrow objective. Lessons frequently extend over several days so that connections can be explored, discussed, and generalized. Once introduced, a topic is used throughout the mathematics program. Teachers seize opportunities that arise from classroom situations to relate different areas and uses of mathematics. Chil- dren are asked to compare and contrast concepts and procedures. They are helped to construct bridges between the concrete and the abstract and between different ways of representing a problem or concept. Learning and using mathematics are important as- pects of the entire school curriculum.

This characterization of a classroom sug- gests that a broad range of important con- nections can and should be made when

teaching mathematics. The underlying as- sumption is that emphasizing connections will nurture unified views of mathematics that will give rise to mathematical power as students recognize, explore, and use con- nections within and across mathematics strands; among what is spoken, modeled, and recorded in different ways; between

mathematics and real-world problem con- texts; between their mathematics experi- ences in school and their mathematics experiences at home; and between math- ematics and other disciplines.

The overriding goal is for students to develop a broad understanding of math- ematics. Limited mind sets and under- standings stem from students' having lim- ited experience in mathematics. Over time as mathematical connections are made through various enriching experiences, stu- dents can view mathematics in a way that allows them to glimpse the whole picture rather than just a part. This view lets stu- dents recognize that many aspects of math- ematics are linked in important ways, that mathematics is a relevant part of other activities, and that mathematical thinking is useful for solving many problems in and out of school.

In This Issue This issue contains various articles that address the need to emphasize mathemati- cal connections as a natural part of teaching and learning. In "One Point of View" (p. 300), Heidi Jacobs' s comments focus on the relation to mathematics of the interdis- ciplinary curriculum, which is based on making important links and explaining re- lationships with mathematics during teach- ing. In discussing "Mathematics Integra- tion: A Common-Sense Approach to Curriculum Development," Jacobs draws from her experience with teachers and stu- dents in classrooms to make three practical suggestions for coordinating mathematics with other subject areas.

In "Stepping Stones to Mathematical Understanding" (p. 303), Ruby Midkiff and Mary Cramer respond to Jacobs' s sug- gestion by illustrating how literature fur- nishes a natural link between mathematics

and the language of young students. Spe- cific books are discussed that can enhance learning many mathematical topics in the early school curriculum. The authors en- courage teachers to review children's lit- erature from a "mathematical point of view," that is, to search for and use appropriate books to lend an interesting setting for learning.

A different slant on emphasizing important connections for mathematics instruction is taken by Sheryl Orman in "Mathematics Backpacks: Making the Home-School Connection" (p. 306). The spirit of her message is that the open- ended, manipulative-based, cooperative- problem-solving nature of the backpack activities can help parents understand first- hand the revised methods and content of mathematics programs based on NCTM' s curriculum standards. Orman outlines her basic goals, presents ready-to-use samples, and shares guidelines that have worked well for creating mathematics backpacks at both the primary and the intermediate level.

Mathematics does not exist for children. It must be created in meaningful ways from the language of their experience. This the- sis is advanced by L. Diane Miller with her suggestions for "Making the Connection with Language"(p. 311). Miller makes the point that mathematics has a vocabulary of its own, which is used but often not mean- ingfully understood by students. She distin- guishes between "receptive" and "expres- sive" aspects of formal mathematics language in classroom discussion. She also makes the case for offering experiences that will allow students to understand, pro- cess, or otherwise act on what is read or heard so they can effectively use, speak, and write about the mathematics of these situations.

Marlene Kliman asks us to go with fourth- grade students on an imaginative journey, which challenges the students to identify patterns and relationships, solve problems, and communicate their findings. In "Inte-

grating Mathematics and Literature in the Elementary Classroom" (p. 318), Kliman

reports how students read My Travels with Gulliver and use mathematics both to en- hance their understanding of the story and to create their own stories and drawings. She includes guidelines for integrating lit- erature and mathematics in ways that make mathematics "meaningful and relevant to the exploration and understanding of dif-

296 ARITHMETIC TEACHER

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ferent worlds, real and imaginary." What are the mathematical characteris-

tics of proportional situations? What differ- ent ways can students apply their math- ematical knowledge to solve these kinds of problems? Kathleen Cramer and Tom Post address these questions and emphasize important connections students might make in relation both to the mathematics of pro- portional situations and to solving related problems. Nurturing an understanding of underlying mathematical concepts with confident, flexible, problem-solving ap- proaches serves to empower students.

Joanne Lobato raises important issues related to "Making Connections with Esti- mation" (p. 348). Concerned that some students seem unaware that the power of estimating lies in the speed with which a quantity or numeric result "close enough for purposes at hand" is derived, she em- phasizes the role and nature of connecting learning with other curricular areas, among mathematical topics, and with real-world- problem situations.

Through "Angle Sense: A Valuable Connector" (p. 353), Rheta Rubenstein, Glenda Lappan, Elizabeth Phillips, and William Fitzgerald collaborate to present a rich example for helping middle school students recognize that mathematics is an "integrated whole" rather than a cluster of disjointed concepts. The authors report how students use patterning blocks and a goni- ometer with manipulatives and in game settings to explore angles, rational num- bers, coordinate graphing, number- theory concepts, and other areas of mathematics.

Mathematics is a language for analyz- ing, understanding, quantifying, compar- ing, reasoning, identifying patterns, and communicating precisely. The authors of this focus issue suggest that making natural connections to different topics within math- ematics or to other relevant aspects of students' lives, in and out of school, is an important way to empower students. It is hoped that readers will want to use and extend the ideas presented in their class- rooms and other educational forums.

Reference National Council of Teachers of Mathematics. Cur-

riculum and Evaluation Standards for School Math- ematics. Reston, Va.: The Council, 1989.

Carol A. Thornton Randall I. Charles For the Editorial Panel W

Г- Z] The ARITHMETIC TEACHER: MATHEMATICS EDUCA- ШВ TION THROUGH THE MIDDLE GRADES (ARITHMETIC

^*^^ ЩШ TEACHER) ¡s an official journal of the National Council of ^*^^ Teachers of Mathematics. It is a forum for the exchange of ideas and a source of techniques for mathematics education through the middle grades. It presents new developments in curricu- lum, instruction, learning, and teacher education; interprets the results of research; and in general provides information on any aspect of the broad spectrum of mathematics education appropri- ate Council

for preservice and in-service teachers. The publications of the Council present a variety of viewpoints. The views expressed or implied in this publication, unless otherwise noted, should not be interpreted as official positions of the Council.

EDITORIAL PANEL Carol A. Thornton, Illinois State University, Chair; Anne Battei/ Minneapolis Public Schools; Randall I. Charles, San Jose State University; Linda Gojak, Hawken School, Lyndhurst, Ohio; Maxie Jenkins, Madison Public Schools, Wisconsin; Johnny Loft, University of Montana; Ian C. deOroot, Sutherland Secondary School, North Vancouver, Board of Directors Liaison; Harry B. Tunis, NCTM, Director of Publications

NCTM BOARD OF DIRECTORS Mary M. Lindquist, Columbus College, Georgia, President; Iris M. Cari, Houston, Texas, Past President, James D. Gates, NCTM, Executive Director; Gail F. Burnii, Whitnall High School, Wisconsin; Frances R. Curcio, Queens Col- lege of the City University of New York; Ian С deGroot, Sutherland Secondary School, British Columbia; Dan Dolan, Project to Increase Mastery of Mathematics and Science, Connecticut; Eariene K. Hemmer, Martha Fox Heck Elemen- tary School, Montana; David R. Johnson, Waupaca, Wisconsin; Miriam A. Leiva, University of North Carolina at Charlotte; Richard D. Lodholz, Parkway School District, Missouri; William J. Masalski, University of Massachusetts; Beverly W. Nichols, Shawnee Mission Public Schools, Kansas; Lee V. Stiff, North Carolina State University; Irvin E. Vance, Michigan State University

JOURNAL STAFF James D. Gates, Publisher; Harry B. Tunis, Managing Editor; Joan Armistead, Editorial Coordinator; Ann M. Butterfield, Editorial Associate; Pamela Grainger Halonen, Editorial Aide; Beth Hahn, Production Secretary; Lynn S. Gates, Program Assistant; Shannon Epperson, Journal Secretary; Betsy Maynard, Journal Secretary; Cynthia C. Rosso, Publications Business Manager; Rowena G. Martelino, Advertising Manager; Tom Pearson, Advertising Assistant; Sandra S. Belsley, Adver- tising Secretary. Art direction by William J. Kircher & Associates, Inc.

Correspondence should be addressed to the Arithmetic Teacher: Mathematics Education through the Middle Grades, 1 906 Association Drive, Reston, VA 22091-1593. Manuscripts should not exceed 10 pages of text and should be typewritten on one side only, double-spaced with wide margins. Figures should appear on separate sheets. Manu- scripts should be prepared according to the Chicago Manual of Style and the American National Metric Council's Metric Editorial Guide. No author identification should appear on the manuscript; the journal uses a blind review process. Five copies are required. Priority areas are listed in the editorial in the October 1 992 issue. Three returnable disks for computer programs with more than 15 statements should be furnished. Permission to photocopy material from the Arithmetic Teacher is granted to classroom teachers or their copy services for instructional use when the material is to be distributed free or at cost only, to authors of scholarly papers, and to librarians for limited numbers of reserve copies. Permission must be sought for quotation of journal content in advertising, for inclusion of articles in commercial books of readings, or when profit is intended. Use of material from the Arithmetic Teacher, other than described, should be brought to the attention of the National Council of Teachers of Mathematics. The journal will not participate in the unautho- rized reproduction of computerized courseware that bears an explicit or implicit copyright claim. For information on reprints or back issues write to Data Processing in the Headquarters Office. Information is available from the Headquarters Office regarding the two other official journals, the Mathematics Teacher and the Journal for Research in Mathematics Education. Special rates for students, institutions, bulk subscribers, and life and retired members are also available. The index for each volume appears in the May issue. The Arithmetic Teacher is indexed in Biography Index, Current Index to Journals in Education, Education Index, Exceptional Child Education Resources, Media Review Digest, and Zentralblatt für Didaktik der Mathematik.

The Arithmetic Teacher: Mathematics Education through the Middle Grades (ISSN 0004-1 36X) is published monthly except June, July, and August at 1906 Association Drive, Reston, VA 22091-1593. Dues for individual membership in the Council are $40, which includes $15 for an Arithmetic Teacher subscription and $3 for a five-issue NCTM News Bulletin subscription. Second-class postage paid at Reston, Virginia, and at additional mailing offices. POSTMASTER: Send address changes to the Arithmetic Teacher, 1906 Association Drive, Reston, VA 22091- 1593. Telephone: (703) 620-9840, Order Only (800) 235-7566; Fax: (703) 476-2970; CompuServe: 75445, 1161. Copyright© 1 993 by The National Coun- cil of Teachers of Mathematics, Inc. Printed in the U.S.A.

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FEBRUARY 1993 299

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