Problem Solving: Getting to the Heart of Mathematics …academic.sun.ac.za/mathed/174WG/Problem...

PROBLEM SOLVING: GETTING TOTHE HEART OF MATHEMATICSIDEAS CREATE A WEB OF UNDERSTANDING EMBRACING CHANGE: INVOLVING

STUDENTS IN CHALLENGING MATH HOW TO WIN PARENTS’ TRUST

TECHNOLOGY AND OTHER RESOURCE TIPS

NWnorthwest

A MATHANDSCIENCEJOURNAL

A MATHANDSCIENCEJOURNALDEVOTED TO

RIGOROUS ANDIMAGINATIVE

LEARNING

TEACHERVOLUME 1 NUMBER 1 SPRING 2000

Northwest Regional Educational Laboratory

REFLECTING ROWBOAT PAINTED BY LIZETH RODRIGUEZ, 9, OF PORTLAND, OREGON.

TEACHERNORTHWEST

A Math and Science Journal Devoted to Rigorous

and Imaginative Learning

ON THE COVER:This painting of a moored row-boat, with its reflection spread-ing across blue waters, is byLizeth Rodriguez, 9, a studentat Woodlawn ElementarySchool in Portland, Oregon.It reveals the artist’s thinkingabout the concept of reflectionand her attempt to solve theproblem of representing thisconcept visually. Lizeth’s row-boat was exhibited in theYoung Artist Project of PacificNorthwest College of Art andPortland Public Schools.

VOLUME 1 NUMBER 1

SPRING 2000

contentsP R O B L E M S O LV I N G: G E T T I N G TO T H E H E A R T O F M AT H E M AT I C S

1 Editor’s Note D E N I S E JA R R E T T

2 A Word from the Director K I T P E I X O T T O

F E AT U R E S

3 Open-Ended Problem Solving: Weaving a Web of Ideas D E N I S E JA R R E T T

8 Teenager or Tyke, Students Learn Best by Tackling Challenging Math S U Z I E B O S S

14 Want to Gain Parents’ Trust? Let Them in on the Action ROSEMARY SHINOHARA

D E PA R T M E N T S

7 Reflections: Confessions of a Former Mathphobe J U D I T H C A R T E R

18 Connected Corner: Technology Tips J U D I M AT H I S J O H N S O N

19 Classroom Resources: A Library of Materials A M Y S U T TO N

20 Discourse: Integrating Problem Solving and Projects L O U I S N A D E L S O N

21 Inquiry: A Way of Seeing C H R I S S A N C H E Z

NORTHWEST TEACHER IS A PUBLICATION OF THE NORTHWEST REGIONAL EDUCATIONAL LABORATORY’SMATHEMATICS AND SCIENCE EDUCATION CENTER.

NWREL EXECUTIVE DIRECTOR/CEO Dr. Ethel Simon-McWilliams CENTER DIRECTOR

Kit Peixotto EDITOR Denise Jarrett CONTRIBUTING EDITORS Suzie Boss, Lee Sherman,Jennifer Stepanek CONSULTING EDITOR Andrew Giarelli, Portland State University CONTENT

SPECIALIST Robert McIntosh GRAPHIC DESIGN Shawna McKeown Design GRAPHIC

PRODUCTION Denise Crabtree PROOFREADING Catherine Paglin

MATHEMATICS AND SCIENCE EDUCATION CENTER Kit Peixotto, DIRECTOR Jolene Hinrichsen, SCIENCE

ASSOCIATE Robert McIntosh, MATHEMATICS ASSOCIATE Denise Jarrett, WRITER Jennifer Stepanek,WRITER Amy Sutton, RESOURCE SPECIALIST Amy Coyle, RESOURCE ASSISTANT Cathy Clark, MANAGEMENT

ASSISTANT Evelyn Lockhart, EXECUTIVE SECRETARY

NORTHWEST TEACHER is published three times a year by the Northwest Regional Educational Laboratory’s Mathematicsand Science Education Center, 101 S.W. Main Street, Suite 500, Portland, Oregon 97204, (503) 275-9575. Single copies of thepublication are available free of charge while supplies last to educators within the Center’s region of Alaska, Idaho, Oregon,Montana, and Washington. To request a free copy, contact the Center by e-mail at [email protected], by telephoneat (503) 275-0457, or place an order from the Center’s Web site at www.nwrel.org/msec/. An online version of the publicationis also available for downloading from this Web site. Multiple copies, and copies to individuals outside the Center’s region, maybe purchased through NWREL’s Document Reproduction Service at the above address, or direct orders by e-mail to [email protected]; by fax, (503) 275-0458, or by telephone, (503) 275-9519.

This publication is based on work supported wholly or in part both by a grant and contract number RJ96006501 from the U.S.Department of Education. The content of this document does not necessarily reflect the view of the department or any otheragency of the United States government. The practice of the Northwest Regional Educational Laboratory is to grant permissionto reproduce this publication in whole or in part for nonprofit educational use, with the acknowledgement of the Northwest Regional Educational Laboratory as the source on all copies.

http://www.nwrel.org/msec/

mailto:[email protected]



THE VASTNESS OF THE

Pacific Northwest can in-crease teachers’ feelings of isolation. By sharing storiesfrom Northwest classrooms,this journal aims to bridge thedistances, fostering a commu-nity of teaching professionals.

Welcome to the premiere issueof Northwest Teacher. We’re verypleased to be launching this freejournal for teachers in the PacificNorthwest. It’s intended asa place where teachers fromall grade levels who teachmath or science can readabout stories and issues ineducation that mattermost to them.

Published threetimes a year, eachedition of the journalwill focus on a keytopic in mathematicsand science instruc-tion, beginning inthis issue with math-ematics problemsolving and, in up-coming issues, theimpact of the stan-dards movement, and teachingscience through inquiry.

We know the teaching lifeleaves little extra time, so we’rekeeping the articles concise withpractical and, we hope, insight-ful information.You’ll find somegood ideas for classroom activi-ties and content here, but, ratherthan presenting “ready-made”lessons, Northwest Teacher strives toaddress the long view, taking alook at best practices and theirtheoretical underpinningsthrough the lens of real-lifeclassrooms.

We recognize that, in this eraof education reform and scien-tific discoveries about how peo-ple learn, teachers want succinct,readable, and accurate informa-tion about complex topics tohelp them make the most of new developments and growprofessionally.

The Pacific Northwest, like otherregions of the country, has a dis-tinctive character that influenceswhat takes place in classrooms.As a regional publication, this

journal aims to speak to the in-terests and needs of teachers andstudents in Northwest classrooms.

Each journal provides anoverview of the main theme tohelp readers reflect on the linkbetween theory and the realworld.Through interviews andanecdotes in feature articles,teachers, students, and parentsshare their concrete experiencesabout what it takes to teach andlearn mathematics and sciencewell. In columns, educators offersuggestions on using technologyin the classroom and finding in-

structional resources. Guest edi-torials and letters from readers aretwo more ways that teachers’ voicesare heard in this publication.

We urge you to write to uswith your ideas, tips on wheregood things are happening inNorthwest schools, feedback onthe magazine, and letters for theDiscourse column.

If you are a Northwest educatorand would like a free subscrip-tion to the journal, please con-tact us. Single copies are available

free of charge to educatorsin Alaska, Idaho, Montana,Oregon, and Washington.To request a single copy,

contact NWREL’sMathematics and Sci-ence Education Cen-ter by phone, (503)275-0457; e-mail,math_ [email protected]; or visit ourWeb site, www.nwrel.org/msec/pub.html.

Multiple copies orcopies to individualsoutside the regionmay be purchasedthrough NWREL’s

Document Reproduction Service,101 S.W. Main Street, Suite 500,Portland, OR 97204-3297;e-mail, [email protected];fax (503) 275-0458; phone(503) 275-9519.

Let us hear from you!

northwest regional educational laboratory SPRING 2000 northwest teacher 1

E D I TO R ’S N OT E

EDITOR’S NOTEDENISE JARRETT [email protected]

NWT

Drawing by Julia Rudinsky, 7, courtesy of Innerscape Art Center, Portland.



http://www.nwrel.org/msec/pub.html

http://www.nwrel.org/msec/pub.html


PROBLEM SOLVING IS AS

challenging for teachers asit is for students. But you don’thave to go it alone. NWREL’sMathematics and Science Edu-cation Center offers resourcesand technical assistance tohelp you.

In a recent position paper on thedevelopment of students’ numer-ical power, the National Councilof Supervisors of Mathematicsnotes that “all instruction shouldbe couched in an environmentof sense making, even if thatmeans taking longer to solveproblems …. [W]hen the major-ity of a child’s time in mathe-matics is spent memorizing whatthe child considers to be non-sense, he or she soon abandonsaltogether his or her efforts tomake sense of mathematics.”

The Northwest Regional Educa-tional Laboratory’s Mathematicsand Science Education Centerembraces this recommendationand that of the National Councilof Teachers of Mathematics toplace problem solving at the cen-ter of mathematics education. Itis for this reason that mathemat-ics problem solving was selectedas the theme of the inauguralissue of Northwest Teacher.

We believe that mathematicalproblem solving involves muchmore than the routine use of al-gorithms. Instead, problem solv-ing should engage students inrigorous and complex tasks thatrequire them to think, reason,communicate, and apply theirunderstanding of number con-cepts and operations.

This view of mathematicalproblem solving has guided ourresearch and development workover the past two years.The Cen-ter has developed the NWRELMathematics Problem-SolvingModel to provide resources andassistance in support of effectiveproblem-solving instruction andassessment. Like the Six-TraitWritingTM Assessment Model (andother models in the NWRELfamily of trait-based assessments),the mathematics problem-solving

model focuses on student learn-ing by helping teachers to useinformation from formative eval-uation activities to improve andadjust their instruction based onstudents’ performance.

It is this approach to problemsolving that the stories and

columns in this issue of NorthwestTeacher explore.

An overview of the Mathemat-ics Problem-Solving Model isavailable at the Web site www.nwrel.org/msec/mpm. In addi-tion to the model, NWREL’sMathematics and Science Educa-tion Center will offer a summerinstitute on problem solving,open by application to individu-als who are interested in becom-ing “certified” trainers. For moreinformation about the model andinstitute, contact MathematicsAssociate Robert McIntosh at(503) 275-0689.

Two additional resources onproblem solving will be availablein fall 2000. A video will showwhat effective problem solving“looks like” and the roles studentsand teachers play. An upcomingmonograph will synthesize theresearch on effective problem-solving strategies.

Watch our Web site for infor-mation on how to obtain theseupcoming resources.

Our goal for this publication iscaptured in its subtitle: a mathand science journal devoted torigorous and imaginative learn-ing.The articles illustrate howseveral Northwest teachers areusing problem solving to achieverigorous and imaginative learningin their classrooms.The columnsoffer resource ideas and infor-mation to support others in cre-ating similar opportunities forlearning.

We are excited about this newventure and look forward tohearing readers’ reactions.

2 northwest teacher SPRING 2000 northwest regional educational laboratory

A WO R D F R O M T H E D I R E C TO R

A WORDFROM THE DIRECTOR

KIT PEIXOTTO

[email protected]

NWT

Art courtesy of Innerscape Art Center.


http://www.nwrel.org/msec/mpm/

http://www.nwrel.org/msec/mpm/

THE ALLURE OF PROBLEM

solving isn’t just for greatmathematicians. Being absorbedin a difficult task is common toall of us, and, often, these mo-ments prove to be some of ourmost rewarding experiences.

Take the case of Andrew Wiles.At 33, the Princeton Universityprofessor set out to prove Fer-mat’s Last Theorem, joining a350-year-old contest. Pierre deFermat posed thischallenge to theworld in the 17thcentury by writing anote in the margin ofhis favorite book ofancient mathematics.He wrote that he’ddeveloped a proof fora particularly knottyproblem, but therewasn’t enough roomin the margin to writeit all out.

Fermat’s theoremstated that there areno whole number solutions for ngreater than 2 for the equationxn + yn = zn (an extrapolation ofthe Pythagorean Theorem, x2 +y2 = z2).

In the centuries since Fermatprovoked the contest, legions of mathematicians have tried tobuild on the scanty hints he leftbehind. One of the first great

breakthroughs came in the 19thcentury by French mathemati-cian Sophie Germain. Like Wiles,she was transfixed by the prob-lem, working on it single-mind-edly for several years. Germain’swork paved the way for morebreakthroughs. But it remainedfor Wiles to solve at the end ofthe 20th century, drawing on allthe power of modern mathemat-ics (Singh, 1997).

The story of Wiles’ pursuit ofthe elusive proof is dramatic and

emotional. He worked on it foreight years, experiencing allmanner of ups and downs. Hisstruggle to find a solution epito-mizes the power and allure ofopen-ended problem solving.His triumph in 1993-94 madefor exultant headlines and a riv-eting BBC Horizon documentaryfilm—the theorem even foundits way into Arcadia,Tom Stop-

pard’s play of that year aboutlove, mathematics, and the na-ture of scientific discovery.

Wiles’ story appeals, even tothose who don’t see themselvesas mathematically inclined, be-cause solving problems is a basichuman drive.We may not under-stand the mathematics involvedin Wiles’ proof, indeed few math-ematicians do, but we understandhis enormous capacity for curios-ity, perseverance, and resiliency.

We understand, perhaps, be-cause we’ve all experi-enced that state of“flow,” as psychologistMihaly Csikszentmi-halyi (say Chick-SENT-me-high) terms it, inwhich we’re so fo-cused on doing some-thing that we losetrack of time.Thesemoments of immer-sion in a meaningfulchallenge are not onlysome of our most sat-isfying experiences inlife, they are also the

richest for learning.

The concept of flow is a usefulanalogy to describe the level ofengagement Wiles must have ex-perienced while grappling withFermat’s problem.This level ofabsorption is the kind of learn-ing experience students are meantto attain through open-endedproblem solving.


OPEN-ENDED PROBLEM SOLVINGWeaving a Web of Ideas

STORY BY Denise Jarrett PHOTOS BY Suzie Boss

Young problem solvers buddy up at Clarendon Elementary School inTacoma, Washington.

F E AT U R E

Czikszentmihalyi (1990) writesthat flow occurs when a person’sabilities are fully engaged inovercoming a challenge that isinteresting and “just about man-ageable.” During flow, the personis controlling the direction of hisapproach to the task, constructinghis own learning as he stretcheshis abilities to master the activity,whether it be building a shed,quilting, writing a poem, orsolving a math problem.

Freedom to make one’s owndecisions about how to approacha problem and what strategies toemploy is also a key to open-ended problem solving.This re-quires a fundamental shift awayfrom traditional methods thatemphasize teacher-directed rotelearning.

While the phraseopen-ended problemsolving may soundforbidding, it ba-sically describesthis heightenedlearning experi-ence of being fullyabsorbed in a dif-ficult and interest-ing task. Research shows open-ended problem solving to beparticularly effective in promot-ing deep mathematical under-standing (Hiebert, Carpenter,Fennema et al., 1996; Schoen-feld, 1992). A teacher’s role is to make classroom conditions favorable for this kind of learn-ing to take place.

weaving ideas Czikszentmihalyi notes that“playing with ideas is extremelyexhilarating” and when made alifelong habit, it weaves a web ofconnected ideas, giving resiliency

and snap to intelligence.Transferthis concept to the mathematicsclassroom, and the writings ofJames Hiebert, Professor of Edu-cation at University of Delaware,come to mind. Hiebert’s researchon problem solving has donemuch in the past 20 years to helpidentify and articulate the featuresand benefits of problem solvingin the teaching and learning ofmathematics.The real value ofproblem solving, he concludes,is in the ideas it produces.

When a student learns math bygrappling with difficult and ab-sorbing problems—rather thanby simply memorizing and prac-ticing predetermined procedures—she is free to “wonder whythings are, to inquire, to search

for solutions, and to resolve in-congruities,” he says. “This ap-proach yields deep understand-ings of the kinds that we value”(Hiebert, Carpenter, Fennema,et al., 1996).

Thinking abstractly about ideasincreases the flexibility of one’sthinking capacity. (Though ideaswithin a “real-world” context arealso valuable.) The National Re-search Council explains the im-portance of developing flexiblethinking in its influential 1989publication Everybody Counts:A Re-port to the Nation on the Future of Math-ematics Education:

“Experience with mathematicalmodes of thought builds mathe-matical power—a capacity ofmind of increasing value in thistechnological age that enablesone to read critically, to identifyfallacies, to detect bias, to assessrisk, and to suggest alternatives.”

To help young people to bebetter problem solvers is to pre-pare them not only to thinkmathematically, but to approachlife’s ever-changing challengeswith confidence in their prob-lem-solving ability.

open-ended problem solvingOpen-ended problem solving in-volves problems that have multi-ple solution methods and answers.

Teachers shouldchoose problemsthat are just be-yond the solver’sskill level.The dif-ficulty should be an intellectualimpasse, notesSchoenfeld(1992), ratherthan a computa-

tional one. In fact, students whohaven’t yet mastered computa-tions should be allowed to doopen-ended problem solving.Nonroutine problems can pro-vide ample opportunity to buildcomputation skills while engagingthe student in more challengingmathematics and higher-orderthinking. All students deserve theopportunity to develop theirproblem-solving ability.

Working collaboratively is a keyfeature of open-ended problemsolving, though students will alsowork individually. To solve a prob-lem, students draw on their pre-


F E AT U R E

“THINGS LEARNED WITH UNDERSTANDING ARE THE

MOST USEFUL THINGS TO KNOW IN A CHANGING

AND UNPREDICTABLE WORLD.”—JAMES HIEBERT AND COLLEAGUES (1996)

vious knowledge and experiencewith related problems.They de-cide which solution method tofollow, perhaps even constructingtheir own method, trying this andthat, before arriving at a solution.They then reflect on the experi-ence, tracing their own thinkingprocesses and reviewing thestrategies they attempted, deter-mining why some worked andothers didn’t. Students will dis-cuss the problem, identifying itsfeatures, considering possible so-lution methods, conjecturing,and explaining their thinking.

“Communication works to-gether with reflection to producenew relationships and connec-tions,” writes Hiebert (1996).“Students who reflect on whatthey do and communicate withothers about it are in the bestposition to build useful connec-tions in mathematics.”

(Even Andrew Wiles, whoworked in seclusion most ofthose eight years, needed the direct collaboration of others tohelp him finally solve Fermat’sLast Theorem.)

Through it all, the solver con-structs a mathematical under-standing of the problem that isboth deep and flexible. By con-necting her prior knowledge withnew concepts and skills, she gainsdepth of understanding.Theseconnections also create flexibilityin her thinking, enabling her toextend her knowledge to newmathematical situations and be-yond. Good problem solvers, saysRobert McIntosh, MathematicsAssociate for the Northwest Re-gional Educational Laboratory,can see past the surface featuresof problems to common under-lying structures.They can moni-

tor their own thinking strategies,recognizing when an approachor tactic is not being productiveand modifying it as necessary.Self-awareness and the ability toreflect are essential for improv-ing one’s problem-solving abil-ity. Furthermore, good problemsolvers are resourceful, confident,and willing to explore.They’repersistent and tolerate a measureof frustration.

Japanese students are some of the most tenacious problemsolvers. Indeed, when the U.S.members of the Third Interna-tional Mathematicsand Science Studyviewed videotapesof Japanese mathclasses, they wereamazed by stu-dents’ perseverance(Stigler & Hiebert,1999). Morepointedly, anothercomparative studyof first-grade stu-dents working adifficult task (in fact, the taskwas unsolvable, but the studentsdidn’t know that), reported thatU.S. students gave up in about 15seconds, while Japanese studentsdidn’t stop until the class cameto an end an hour later (Stigler,1999).

“To develop these abilities, stu-dents need ample opportunitiesto experience the frustration andexhilaration that comes fromstruggling with, and overcom-ing, a daunting intellectual ob-stacle,” says McIntosh.

The beauty and utility of open-ended problem solving is justthis: It leads to understandingthat is transferable. And in thisincreasingly complex world, the

ability to transfer knowledge andskills to meet changing condi-tions and challenges is essential.

standards: a statement of valuesStandards, says Hiebert (1999),are simply a statement aboutwhat we most value. From ourbest judgment, we create educa-tion standards based on past ex-periences, research, advice frompractitioners, and societal expec-tations. At least since the 1940swhen George Pólya identifiedproblem solving as an essential

math skill—the heart of mathe-matics, in fact—problem solvinghas been a stated education pri-ority.When the National Councilof Teachers of Mathematics beganissuing its standards for mathteaching and learning in the1980s, problem solving rose tothe fore of standards reform. Inits Principles and Standards for SchoolMathematics (NCTM, 2000), thecouncil continues to identifyproblem solving as a core strandof math learning for all gradelevels.

Across the country, states arewriting mathematics standardsand assessments to include prob-lem solving, acknowledging theimportance of assessing students’


F E AT U R E

Manipulatives help these students solve a geometric problem.

ability to reason, communicate,make connections, and applytheir knowledge to problem situ-ations.Thus, problem solving isan area teachers are increasinglyexpected to teach and assess.

making the changeTeachers are often caught be-tween daily pressure from col-leagues, parents, and communitymembers to uphold conventionin the classroom, and pressurefrom administrators and policy-makers to employ standards-basedpractices that show immediateand positive results on achieve-ment tests. One must considerthese opinions and mandates, butteachers who make meaningfulchanges are those who developtheir own inner voice of author-ity (Wilson & Lloyd, 2000).

Teaching problem solving is anart mastered over a long periodof time (Thompson, 1989). Byreflecting on their personal un-derstandings of teaching andlearning—as well as their stu-dents’ understandings—teachersdevelop inner authority for im-proving their instruction.Throughreflection, teachers determine forthemselves the value of particu-lar reform innovations (Wilson& Lloyd, 2000).

Many teachers do recognize thatnontraditional strategies are nec-essary to meet the learning needsof their increasingly diverse stu-dents. Embracing change can be unsettling, but these teachersventure into new territory, open-ing a world of discovery for them-selves and their students. For theyknow that a young mind care-fully nurtured may be the nextbig thinker to solve another ofthe world’s mysteries.

Denise Jarrett is an education writer andeditor of Northwest Teacher.

references Csikszentmihalyi, M. (1990). Flow:The psy-chology of optimal experience. New York, NY:Harper Collins.

Hiebert, J. (1999). Relationships betweenresearch and the NCTM standards. Journal forResearch in Mathematics Education, 30(1), 3-19.

Hiebert, J., Carpenter,T.P., Fennema, E.,Fuson, K., Human, P., Murray, H., Olivier,A., & Wearne, D. (1996). Problem solvingas a basis for reform in curriculum and in-struction:The case of mathematics. Educa-tional Researcher, 25(4), 12-21.

National Council of Teachers of Mathemat-ics. (2000). Principles and Standards for SchoolMathematics. Reston,VA: Author.

Schoenfeld, A.H. (1992). Learning to thinkmathematically: Problem solving, metacog-nition, and sense making in mathematics.In D.A. Grouws (Ed.), Handbook of research onmathematics teaching and learning (pp. 334-367).New York, NY: Macmillan.

Singh, S. (1997). Fermat’s enigma:The epic questto solve the world’s greatest mathematical problem.New York, NY:Walker.

Stigler, J.W. (1999, September). Improving thequality of teaching in mathematics and science. Paperpresented at the meeting of the NationalCommission on Mathematics and ScienceTeaching for the 21st Century,Washington,DC.

Stigler, J.W., & Hiebert, J. (1999). The teach-ing gap: Best ideas from the world’s teachers for im-proving education in the classroom. New York, NY:Free Press.

Thompson, A.G. (1989). Learning to teachmathematical problem solving: Changes inteachers’ conceptions and beliefs. In R.I.Charles & E.A. Silver (Eds.), The teaching andassessing of mathematical problem solving. (Vol. 3,pp. 232-243). Reston,VA: National Councilof Teachers of Mathematics.

Wilson, M.S., & Lloyd, G.M. (2000). Shar-ing mathematical authority with students:The challenge for high school teachers [Ed-itorial]. Journal of Curriculum and Supervision,15(2), 146-169.


F E AT U R E

If problem solving is at theheart of mathematics, then non-routine problems are at theheart of problem solving.

True problem solving involvesnonroutine, or open-ended,problems. Moving into the terri-tory of nonroutine problems isfull of unknowns. Solution meth-ods and answers are not madeexplicit. Decisionmaking is sharedbetween teachers and students.Teachers must predict what tac-tics and questions might comeup in class, preparing for themas best they can.

Teachers need mathematical expertise to anticipate students’approaches to a problem andhow promising those approachesmight be. They must choose taskswhich are appropriately difficultfor their particular students. Theymust decide when and how togive help so that students canbe successful but still retainownership of the solution. Teach-ers will sometimes find them-selves in the uncomfortableposition of not knowing the so-lution to a problem. Letting goof the “expert” role requires ex-perience, confidence, and self-awareness.

Teachers should choose taskswith nonroutine problems that(Hiebert et al., 1996):

• Make the subject problematicso students see the task as inter-esting and challenging

• Connect with students’ pres-ent level of understanding, al-lowing them to use theirknowledge and skills to developmethods for completing the task

• Allow students to reflect onimportant math ideas

NONROUTINE PROBLEMS

NWT


confessions OF A former mathphobe

R E F L E C T I O N S

TO CONFRONT HER FEAR OF MATH,elementary school teacher JudithCarter took a math course for edu-cators. For the first time, she waslearning math in an environmentthat encouraged independentthinking and problem solving.Below, she reflects on the changesthis experience prompted in herpersonal beliefs about teaching and learning.

W hen I entered a graduateprogram to become a

teacher seven years ago, I hadbeen aware for some time thatas a student of mathematics Ihad been injured by the peda-gogy of my teachers. In the classesI had taken from kindergarten tohigh school it was never accept-able to use pictures or objects tohelp me understand what thenumbers and letters represented.No one let me talk about what Ithought, or encouraged me to askquestions. In fact, to ask a ques-tion was taking a big risk that Imight be seen as someone who“didn’t get it.” Most of all, noone said it was okay to struggleand even more, no one viewedstruggling and confusion as partof problem solving. Had the ped-agogical use of manipulatives,social interaction, reflective pro-cessing from various perspectives,and honoring the perseverance it takes to solve complex tasksbeen taught by any of my teach-ers, my mathematical experienceand self-concept would havebeen tremendously different.

One of the main reasons I tookthis math class was to overcomemy fear of math and change theimage I had of myself as a mathstudent. As a student of any sub-ject I have always wanted to know“why” and math has been no ex-ception. This class was all aboutfinding “why.” But what was sodifferent was that we were en-couraged to find out “why” inour own way and in our own time.Everyone’s paths were exploredand honored and, though I foundthat my path to “why” was oftencircuitous, I learned that the longway is not the wrong way. I strug-gled and I was confused, but thecommitment that had led me totake the class in the first placekept reappearing in my persist-ence with each problem. Becauseperseverance was supported, bycollaboration, by honoring diverseapproaches, by being given timeto solve problems, I found thatmy perseverance increased expo-nentially. There were some awe-some moments of revelation inthis class and I could feel the childin me want to jump out of my seat(I think I did this a few times) andshout, “I get it!” There were alsooccasions when the injured mathstudent appeared, and I learnedhow to be patient with her. Per-severance and patience are nowpart of my new self-image.

I had always said I believed thatall children can learn but, becauseof my own fears, I think there wasa part of me that was still think-ing, “but not me.” So if I contin-

ued to believe this about myself,wouldn’t I also believe it aboutsome of my students? I know Ican learn difficult math conceptsand explain my thinking to oth-ers. This will have a profound ef-fect on how I view my studentsbecause I know now that all chil-dren can learn when they are al-lowed to work in an environmentthat is safe for taking risks.

To feel safe and gain confidencein math, students need to haveopportunities to process theirthinking in a variety of ways:using manipulatives, writing,working with others, and pre-senting their work to the class.They need to be able to assesstheir own work and to reflect onit and to redo the work as theylearn more. I hope that this year Ican create an environment wherestudents will see themselves asproblem solvers and value theirown path to “why.”

(For a related story featuringJudith Carter, see Page 10.)

JUDITH CARTER

is a second-grade teacher atClarendon Elementary in Portland.

confessions OF A former mathphobe


F E AT U R E

TACOMA, WASHINGTON—Whenshe joined the staff of Wilson HighSchool four years ago, Heidi Ewerwas the new kid on the faculty.

Then only 27 years old andshorter than most of her students,she had only two years of teach-ing experience behind her, nei-ther at the high school level. Ewerwas assigned her first couple of

years to teach pre- and remedialalgebra.

“I dreaded it,” she admits.“These kids had failed this sub-ject matter over and over. Mostof them had been through it threeor four times already.” She dis-covered that students were fail-ing math largely due to behaviors,not from an inability to under-stand concepts. “They had poorattendance.They were bored.And they didn’t care about grades.It was a struggle to find rewards

or even consequences that wouldmotivate them to learn.Whatwould drive them internally?”

Ewer decided to depart fromthe traditional, textbook-basedteaching methods. “Even I wasgetting bored with the routine of lecture and homework,” sheadmits. At the University of PugetSound in Tacoma, where sheearned an undergraduate degreein math and a master’s in teach-ing, she had studied the theorybehind research-based teachingpractices that actively engage stu-dents in learning. During the twoyears she spent as an elementaryteacher, she had seen the benefits

of hands-on learning. But she’dnever seen such concepts mod-eled with secondary students.

making ideas centralA little apprehensive, she decidedto add some new approaches toher repertoire. Rather than drill-and-practice desk work, she triedusing cooperative learning—al-lowing teenagers to socialize a bitin class, so long as they focused

on math. She created opportuni-ties for students to learn by hav-ing them investigate ideas ratherthan memorize formulas. She as-signed open-ended problemsthat could be solved in morethan one way, using hands-onmaterials. She attended confer-ences and a workshop on mathreform to build her confidenceand gather new resources. Andher students responded. “I couldsee the lights click on,” she says.

Ever since, Ewer has stuck withher decision to use student-cen-tered teaching methods that stillmake her a bit of a maverick atWilson High, a racially diverseschool of 1,800 students. Moretraditional teachers raise theireyebrows when she turns herstudents loose with tipsy-lookingglass vases and beakers of water

to investigate volumes. Colleagueswonder what’s up when animatedstudent conversations about howto harness a horse to maximizegrazing space spill out of herdoorway and echo down the hall.A few ask pointedly whether Ewerwill have time to cover all thematerial in the textbook.

Even new students sometimesmisinterpret what goes on in herclasses, Ewer admits. “I have a rep-utation for teaching fun classes,”in subjects such as geometry and

AFTER EMBARKING ON

A PROBLEM-SOLVING

APPROACH, TWO

TEACHERS CONCLUDE:

STORY AND PHOTOS BY Suzie Boss

If she sees a student struggling, Heidi Eweroffers support but doesn’t give away any answers.

HEIDI EWER

Teenager or Tyke,Students Learn Best

BY TACKLING CHALLENGING MATH

advanced algebra. “Some studentsthink that ‘fun’ means ‘easy,’” shesays with a laugh that meansthey’ll soon learn otherwise.

Teenagers living in Tacoma don’thave to look far to find examplesof mathematical problem solvingout in the real world.There’s theTacoma Dome rising on the hori-zon, its convex roof built to allowunobstructed views for concertgoers. Cargo ships docked at theharbor must have freight properlydistributed in order to stay afloatin Puget Sound. Ewer routinelybrings such real-world problemsright into her classroom, askingstudents to determine how muchcable would be needed to sup-port, say, the Tacoma NarrowsBridge.

And students seem to welcomethe challenge. Says a girl namedAmanda, “When you’re the onepouring the water and measuringthe rate of change in volume, it’smore real than reading somethingout of a book.You learn how todo it yourself. Eventually, whenwe have jobs, we’ll need to knowthe best way to solve problems.”Adds Matthew, “This way of

learning makes you more open-minded.There’s no one rightway to solve these kinds of prob-lems.That’s changed my perspec-tive—not just about math, butabout life.” And Evan, who hassought out Ewer’s classes for threeyears in a row, says her approach“makes you think, and be opento other people’s ideas. It’s notenough to figure out a problem.She expects you to be able to ex-plain your thinking in a writtenreport.That means you really haveto get the concept for yourself.”

fostering perseveranceAlthough buoyed by her students’enthusiasm, Ewer is quick toadmit that this way of teachingposes its own challenges. Stu-dents who are more sequentiallearners, for instance, often balkat open-ended problems.They’reused to learning one methodthat will lead them to the rightanswer, and get frustrated whenthe rules change. During a recentproblem-solving activity, Ewersaw tears welling up in a girl whocan blitz through more traditional

worksheets without batting aneye. Ewer came to her aid, coach-ing her to consider possible so-lutions.The teacher’s dilemma:“How can I lead her without justgiving her the answer?”

Letting students struggle is valu-able, up to a point. “Strugglinghelps them see this as an invest-ment of their own time and en-ergy. It makes them more willingto learn,” Ewer says. Struggling tosolve problems requires studentsto use their intuitive skills to in-vestigate concepts, she explains,and, in this way, they gain a deeperand more lasting understandingof the mathematics. “A studentwho has a good grasp of problem-solving strategies and conceptswill be so much stronger than onewith a knowledge of the materiallearned through memorization.They will know how to use theirskills and confidence to attacknew concepts and learn thoughtheir own method.” But Ewer alsopeppers her students with encour-agement: “Have confidence.” “Lis-ten to the wind (learn from whatothers are saying).” “Try—it’sbetter than guessing.”

The evolution of her own class-room practice reminds Ewer thatstudents need a variety of ap-proaches to help them learn. “It’slike teaching reading to elemen-tary students,” she says, harkingback to her student teaching days.“Should you use phonics or amore holistic approach? Somestudents will learn better eachway. You need to pull in a varietyof methods,” she asserts, in orderto serve all students well. Classdiscussions and collaboration alsohelp students learn to appreciateone another’s strengths and dif-ferent learning styles. Ewer’s

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LEFT: Problems that are based on real lifehold students’ interests.

ABOVE: Advanced algebra students put theirheads together to solve a problem involvingthe volume of a glass vase.

brown eyes widen with delight,for instance, when Amanda ob-serves, “Some days, I learn morefrom other students than I dofrom the teacher.”

meeting the challengesOccasionally, a student’s behaviorwill interrupt the dialogue andcollaboration that’s central to theproblem-solving process. “A lotof teachers will say they can’t dothis because of behavior prob-lems,” admits Ewer. “But I teachmath and behavior.” Rather thanstarting off with problems thatinvolve glass, water, or modelrace cars, she builds up frompaper and tape to more challeng-ing materials. “I ask them to beaccountable, and I have a solidbehavior plan.”When one boyacted edgy during a vase-and-water problem, Ewer kept himon track with good-humored re-minders, direct eye contact, anda promise to remove him if hecouldn’t manage himself. In fouryears, she’s never had to bouncea student out of class during aproblem-solving lesson.

Time poses another classroomchallenge.Wilson High used to

offer a block schedule.With 110minutes, Ewer could take her classfrom warm-up dialogue to hands-on work to wrap-up discussion.This year, the school is back to55-minute periods.That means a constant race against the clock.She had to cut short a discussionone morning, just as studentswere making some keen insights,to shift to a hands-on problem. Aclosing discussion would have towait for another day.

Although Ewer feels supportedin her teaching approach by dis-trict administrators, her principal,and cohorts from other schools,she remains a bit of a loner withinher own building. Eager to sharethe problem-solving materialsshe’s gathered at workshops, shegets few takers. “I get kind ofpreachy,” she admits, “like I’mon a pulpit.”

But the students keep her pas-sion burning bright. “I go intomy classroom and do the best Ican for my kids. I hope it poursout of my room so that otherteachers will see the sparks. I hopethey’ll wonder what in the worldwe’re doing in here and comefind out for themselves.”

PORTLAND, OREGON—Gatheredaround their teacher’s chair, the second-graders join in a spirited discussion abouttheir favorite flavors of birthday cake.Theirteacher, Judith Carter, uses the dialogue tointroduce a delicious-sounding problemthat the students will tackle in pairs.“Todayyou’re going to make a pretend cake, agourmet cake, the most special cake thereis,” says Carter.“It comes from the Tri-Cities Bakery. Can anyone guess whatshape their cakes are? What words do weknow that start with tri?” Hands shootinto the air as the children clamor toshare their ideas.

facing her fearsWhen she was the tender age ofher students, Judith Carter usedto avoid raising her hand duringmath lessons. Asking a questionwhen she felt confused wouldnot result in a different explana-tion or open an alternate routeto understanding. It would onlyreveal that she didn’t get it.

“My teachers had one way ofseeing math, one way of teach-ing,” says Carter. Now 45, withsandy brown hair and eyes alightwith intelligence, Carter doesn’tsound bitter when she describes


Collaborative work allows Tacoma High schoolers to discuss ideas together.

JUDITH CARTER

Judith Carter coaxes children to keep tryinguntil they “muddle through to understanding.”

F E AT U R E

her own classroom experiences.In a matter-of-fact tone, she re-lates a tale of missed opportuni-ties. Roles were clear-cut in themath classes of the ’50s and ’60s,she explains, not only in Vermontwhere she grew up, but all acrossthe country. The teacher’s jobwas to explain the concepts inthe textbook.The students’ job,to memorize and practice them.Answers were either right orwrong. It was a formula that left no room for uncertainty.

By middle school, Carter hadadopted a survival strategy: “Ikept quiet and did the minimum.”Math was something she couldn’tget away from fast enough. Andfor decades, she managed to keepthe subject at a safe distance, earn-ing a college degree in Englishand working in a variety of office jobs. She didwind up marrying a mathteacher, but math was his thing.Not hers. Or so she thought.

In her late thirties, living inWashington with her husbandand three children, Carter wasready for a career change.As partof her master’s in teaching pro-gram at Evergreen State Collegein Olympia, she walked backinto a mathematics class—onlyto discover that the universe hadshifted. “In my math methodsclass, they gave us hands-on ma-terials, manipulatives. That’s whenI realized there was another wayto learn math than how I wastaught.” For the first time in years,she felt intrigued by math. Butgetting comfortable with the sub-ject would take her a little longer.

Once the class has determined that Tri-Cities Bakery produces triangle-shapedcakes, Carter adds another dimension tothe problem: Each big cake is assembled

by fitting together smaller pieces that arecut into geometric shapes to identify dif-ferent flavors. Lemon cake is cut into hexa-gons; blueberry, parallelograms; chocolatemint, small triangles; and strawberry,trapezoids.The students’ task is to create abig cake containing at least two pieces eachof the four flavors.“What do we have inthe room that would help you solve thisproblem?” Carter asks.A boy points to thepattern blocks, which just happen to be inplain view.“Remember,”Carter adds,“there’sgoing to be more than one way to solvethis problem. How do you feel when youhave a problem to solve?” Hands wave.“Excited,” says one girl,“because it’ssomething I’ve never done before.”

When she started teaching withthe Tacoma School District sixyears ago, Carter spent a year as a Title I reading specialist. But ever

since, she’s been teaching all sub-jects, including mathematics. InTacoma she taught fourth- andfifth-graders in a high-povertyurban school.When Carter’s fam-ily relocated to Portland, she washired as a second-grade teacher.She’s now in her second year atClarendon Elementary, a modern-looking school built with openpods instead of traditional class-rooms that serves about 320children in a racially diverseneighborhood. For about a thirdof the students, English is a sec-ond language.

Both the Tacoma and Portlanddistricts have adopted math pro-grams that favor student-centeredlearning. Philosophically, Cartershares this commitment to im-proving how math is taught. Shewants to be the kind of teacherwho allows children to make dis-coveries and connect math to theworld beyond the classroom. Sheembraces the concept of open-ended problem solving, whichallows children to ask questions,develop strategies, and cementtheir own understandings.

echoes from the pastBut as Carter has discovered,moving from theory to practicetakes more than good intentions.“You can have a philosophy you

believe, you can have great train-ing in good practices, you canhave a sound curriculum. Butthen you’ll hear your own fearscoming from your own mouth,”she says.

“This is hard,” complains a girl who’sstruggling to fit her pattern blocks intothe shape of a large triangle. Her partner’sstuck, too. Carter bends down to their eyelevel, asking questions. She doesn’t want tospoon-feed them a solution, just start themthinking. She wants the youngsters to beginrecognizing patterns and relationships, anddeveloping their reasoning and communi-cation skills. They’ll have to use theircomputation skills, too, to figure out how

F E AT U R E


“REMEMBER,” Carter adds, “THERE’S GOING TO BE

MORE THAN ONE WAY TO SOLVE THIS PROBLEM. HOW

DO YOU FEEL WHEN YOU HAVE A PROBLEM TO SOLVE?”Hands wave. “EXCITED,” says one girl,

“BECAUSE IT’S SOMETHING I’VE NEVER DONE BEFORE.”

much their big cake will cost. By trialand error, the students discover that placingthe largest pieces—the hexagons—is thehardest step. Once they’re in place, the restfit more easily. Carter slips away to thenext table, leaving the two to finish ontheir own.

When she began teaching ele-mentary math, Carter was startledto hear echoes of her old mathteachers in her classroom. “I hadto keep fighting the way I wastaught,” she says. She’d go homeat night and debrief with hermath-teacher husband. He en-couraged her to keep going downthe path of problem solving, evenif it felt a little bumpy. “I had tostop myself from blurting outthat my students’ solutions wereright or wrong. Sometimes, I’dhave to take a student’s solutionhome and think about it. I had tobe willing to show my kids that I don’t always know the answer.If I was going to do my kids jus-tice,” she says, “I realized I’d haveto get over my own math phobia.And it was deep.”

Last summer, Carter faced herfears head-on. She enrolled in atwo-week, intensive seminar inalgebraic thinking offered by theMathematics Education Collabo-rative of Ferndale,Washington.Designed for educators, the courseoffered challenging content andpromised to model an optimal

learning environment. She wouldlearn the “big ideas” of algebrain the same kind of classroom shewanted to create for her ownstudents.

Carter had studied algebra inhigh school, of course, “but I re-membered none of it. As elemen-tary teachers, we need to knowalgebra so we can see the con-nections between what we teachand what our students will learnin middle school and high school.When we teach patterns duringcalendar time in second grade,we need to know that those pat-terns will come back later, whenour kids learn algebra.” Seeingthose connections, Carter says,“helps us see the value of whatwe’re doing.”

During the two-week class,Carter’s moods rose and fell likea stock market graph. “Some daysI’d go home wondering, whathave I gotten myself into? ThenI’d go struggle over my home-work.” Because this was her per-sonal quest, Carter refused herhusband’s help. “This was mystruggle. I had to figure it out on my own.”

The first time she was invitedto walk up to the overhead pro-jector and explain her solutionto the rest of the class, Carter felther old insecurities well up. “Toput my thinking out there was

scary,” she admits. But to heramazement, once she had actu-ally figured out a solution, shewas excited to share her thinking.

a new view of mathematicsThe biggest insight she gainedthat summer? “Being good atmath is not something you’reborn with. Most people who aregood at math work hard at it.They have a high tolerance forconfusion and uncertainty. Youhave to muddle through.That’swhat learning is—muddlingthrough to understanding.Theteacher’s challenge is to help stu-dents be okay in that space offeeling muddled. Don’t let themfeel defeated. Help them under-stand that it takes perseverance.The key work of teachers is todevelop that quality in our stu-dents so they’ll keep working atit, trying things, until they reachtheir own solution.”

Lunchtime is fast approaching whenCarter calls a halt. Her second-gradershave been working hard for more than anhour. There’s just time to regroup and in-vite students to explain their strategies.Carter moves her chair to the back in hopesof tempering her tendency to jump intothe student-led conversation too soon. Shesees so much rich learning going on duringproblem solving, and wants her students tosee it, too. But they’re only second-graders,


Carter provides just enough help tofurther her students’ thinking.

When children feel “stuck,” Judith Carter asksprobing questions to get them thinking.

Working in pairs, second-graders learn fromeach other’s strategies.

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F E AT U R E

after all. Sometimes they need a littleprompting.“Tell us,” Carter says, leaningforward,“what did you discover?”

When she finishes a problem-solving activity, Carter often feelsexhausted.This is high-energyteaching, requiring her to planeverything from which materialsshe’ll need to how she should pairstudents to maximize collabora-tion. It’s not an activity she canfit into every day, or even everyweek. But now that she’s devel-oped her confidence to teachthis way, she wouldn’t dream ofnot making time for open-endedproblem solving. She loves hear-ing the buzz of conversation thatwells up during these activities.“Children can develop theirmathematical power as early assecond grade,” she says. “If wedon’t give them a chance to beproblem solvers, we’re sellingthem short.”

Suzie Boss is associate editor of NWREL’squarterly magazine, Northwest Education.

Teachers are constantly making changes to meet the evolving needs oftheir students and to try new instructional practices. And they exercise con-siderable control over the kinds of changes they decide to make (Richard-son, 1990). Standards-reform measures, therefore, can’t determine teachers’decisions directly but can serve as a resource for developing inspired butpersonally tailored innovations (Ball, 1996). To be most effective, teachersalso need to keep abreast of developments in the field and educationalfindings from research (Richardson, 1990).

Research. Research is one basis for improving practice. With an under-standing of the theory behind standards-reform measures, and an opportu-nity to discuss with others how theories agree or disagree with teachers’own experiences, teachers are more likely to make informed and effectivechanges. Otherwise, teachers may accept or reject reform-based practiceson the basis of whether they fit one’s personality or accommodate the dailypressures of classroom management and content coverage (Richardson,1990).

Reflection. The role of reflection in improving practice is essential. Teach-ers need time and a context for reflecting on their own teaching experiencesin light of what research reveals about teaching and learning. They need tobe able to discuss their ideas with others; articulating ideas to colleagueshelps teachers clarify their own beliefs and practices.

Risk. Moving in the direction of reform practices, such as open-ended prob-lem solving, means confronting the uncertainties and complexities of learn-ing and understanding.

“When we ask students to voice their ideas, we run the risk of discoveringwhat they really do and do not know,” writes Deborah Loewenberg Ball(1996). “In pursuing reform goals, deep anxieties about one’s effectivenessand one’s knowledge are likely to surface.”

Teaching with an open-ended problem-solving approach requires more thanmathematical knowledge and skill. Personal qualities, such as patience, cu-riosity, generosity, trust, and imagination, are essential. So are an interest inseeing the world from another’s perspective, a sense of humor, tolerance forconfusion, and concern for the frustration and shame of others, says Ball.

“As teachers build their own understandings and relationships with mathe-matics, they chart new mathematical courses with their students,” she says,“and conversely, as they move on new paths with students, their own math-ematical understandings change.”

Ball, D.L. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn. Phi Delta Kappa, 77(7), 500-508.

Richardson, V. (1990, October). Significant and worthwhile change in teaching practice.Educational Researcher, 19(7), 10-18.


EMBRACING CHANGE

NWT

http://www.nwrel.org/nwedu/

http://www.nwrel.org/nwedu/

MAKING MOMS AND DADS

classroom collaboratorsis a good way to persuadethem of the value of open-ended problem solving andlearning math without a textbook.

PALMER, ALASKA—When JanaDePriest began focusing her mathclasses on problem solving aboutfive years ago, she had to workhard to overcome the fears ofparents.

“Parents weren’t seeing the maththey knew when they grew up,”recalls DePriest, a fourth- andfifth-grade teacher at Finger LakeElementary School. “Change ishard.”

Some moms and dads still ques-tion what DePriest is doing, butusually not for long. She has wonthem over with a reputation forgetting children engaged in math,and by a determined effort tokeep parents informed and to invite them into her classroom.

“Parents in her class love her,”says Dave Nufer, the Finger Lakeprincipal. “Other teachers lookto her as a mentor for ideas.”

a need for changeDePriest, 37, is in her 13th yearof teaching, and her sixth year at Finger Lake Elementary. Theschool serves rural subdivisions

in the Matanuska-Susitna Bor-ough, about 45 miles north ofAnchorage in a fertile valley cutby two glacier-fed rivers. FingerLake Elementary sits halfway be-tween Palmer and Wasilla—once

colony towns, now bedroomcommunities for Anchorage pro-fessionals. Most of the school’s500 pupils live in the new sub-divisions and apartments thathave bloomed like fireweed onformer farmland. But cows stilldot the slopes of remaining pas-tures, a reminder of the pioneerfarmers who settled here in the1930s to grow robust vegetablesand livestock under summer’sso-called midnight sun. In fact,DePriest and her husband, Ray,an educational technologist, liveon Ray’s childhood farm, grow-ing hay in summer—their farmis now surrounded by housingdevelopments and a golf course.

The shifting of the Mat-Su Val-ley landscape was gradual, overdecades. But sometimes whenchange occurs, it’s more like the

ground moves and suddenlyyou’re seeing it from a new per-spective.That’s what it was likefor DePriest when she decidedon a radical change in her teach-ing approach.

As a beginning teacher, she re-calls, “It was pretty much youfollowed a textbook page by page.I used manipulatives when thebook told me to.”

But when students got to harderproblems, such as multiplyingtwo digits by one digit, they re-lied on memorization, followinga process by rote rather than fullyunderstanding the concept ofmultiplication.

“I would ask, ‘Why are you re-grouping that one to the top?’They didn’t have the answers.Theydidn’t really understand whatthey were doing.”

So DePriest read the researchon using real-life problem solv-ing to help students learn math,and began building it into herprogram.


WANT TO GAIN PARENTS’ TRUST?Let Them in on the Action

STORY BY Rosemary Shinohara PHOTOS BY Stephen Nowers

Finger Lake Elementary School in Palmer, Alaska.

F E AT U R E

“Right when I read it, I knew itwould work. I still know it’s theright change,” she says.

DePriest is now one of a hand-ful of teachers at Finger Lake whoteach mathematics by involvingstudents in open-ended problemsolving, avoiding the district’stextbooks and focusing insteadon math problems that have areal-world context. She coversthe material set out in the Alaskamathematics standards, whichare similar to the national stan-dards, but she does so primarilythrough hands-on activities. Sheuses a balance of paper-and-pen-cil tests, oral questioning of eachstudent, and projects to evaluatewhere students stand.Whileshe’s questioning students,she’s prodding them tothink. Students reflect on their learning injournals which are turned inat the end of a unit of study.The projects and individualassessments allow DePriestto adapt her instruction tomeet the needs of studentsat different levels of learningin her multigrade classroom.

DePriest job-shares, teachingscience and math afternoonswhile teaching partner LyndaChud handles language arts andsocial studies mornings.When-ever possible, DePriest combinesher two subjects.This spring, theclass explored the principles offlight.The students set up scien-tific experiments to collect andexamine data on how differentvariables affect flight.Theylearned and applied distance,time, and rate formulas.Theymeasured in centimeters andlearned to use a stopwatch and a protractor.

winning parents’ trust In the beginning, many parentswere skeptical of the change, wor-ried that their children wouldn’tbe prepared for middle school.Most of the 21 teachers use Saxonmath texts, a much more struc-tured approach, says Nufer, theprincipal. Some parents whosechildren studied math from Saxontexts in earlier grades worriedabout the lack of apparent struc-ture in DePriest’s class. Saxon les-sons follow a script.They involverepetition and memorization. Stu-dents bring home worksheetsevery night.This is many parents’

image of what mathematicslearning is all about.

“When we were growing up,we had books we brought home,”says Pam Wooding, mother of10-year-old Mackenzie Deiman.When she learned Mackenziedidn’t have a math text, shethought, “Whoa, how can youteach math without a book?”

With no text, Mackenzie’s par-ents didn’t know how to helpher with her homework.Thatwas last year, when Mackenziewas in fourth grade.Then, at aparent-teacher conference, De-Priest explained to her that stu-dents must not simply memorize

how to do problems, but be ableto explain their thinking and de-scribe their strategies for solvingproblems. Now Wooding is a con-vert.Wooding says she hated mathas a child and figures it was tooabstract. She has watched DePriestand Chud make math more real-

world by incorporating mathinto learning throughout the day.

Lisa Greenwood, mother offourth-grader Emma, says shehas no concerns that her daugh-ter’s missing anything in her un-conventional math class. “I can

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ABOVE: Jesse Eshleman examines hercards during a math game in Jana DePriest’s class.

LEFT: Veronica Iversen counts money aftera lunchtime opening of the class store.

BOTTOM: Students play cribbage andother math-oriented games on a recentafternoon.

see her applying math in every-day situations,” Greenwood says.“Like how much dog food shewould need for her puppies—how many pounds are in the bag.”

math-filled daysTo provide real-world contextsfor problem solving, the teachershad students create a mini-societyin which they made productsand sold them in a market. Class

members also run a real store at lunchtime from a table in thefront lobby of the school.Theysell erasers that look like dollarbills, pens that smell fruity, rulers,and clear plastic balls with rep-tiles inside. Students eagerly vol-unteer to be sales people, givingup their lunch breaks. Fourth-grader Raymond Hajduk explains:“You have to get all your home-work done, and make sure your

name isn’t up onthe board for miss-ing assignments.”

The students de-cide what merchan-dise to order, and,in the end, how tospend the profits.Raymond suggeststhey use the moneyto save mountain

gorillas in danger of extinction.Someone else proposes that theycould buy school equipment andthrow a party. But all that will beafter they pay back a $100 loan,plus 8 percent interest, to theteacher. In the meantime, thereare sales to be made. “Thank youfor doing business at the schoolstore,” Raymond tells a customerpurchasing a pink, smelly pencil.

involving moms and dadsKathy Krell says she doesn’t mindthat her child doesn’t have a mathbook because DePriest sendshome detailed notes to parentsexplaining what’s going on inher classroom. In fact, DePriesthas made a conscious effort toinform parents through regularnewsletters and by sending homethorough descriptions of mathunits. Her description of the re-cent unit on flight was four pages

long. In it, DePriest laid out themath and science concepts andthe learning goals, the state con-tent standards that would be cov-ered, the questions studentswould consider, and the criteriafor student success. She includedwriting standards and a planningguide. She also gave parents achecklist for assisting their chil-dren in the final project that en-tailed building a flying machineand writing a paper about it.

In another note home to parentson multiplying and dividing largenumbers, DePriest told parentsfive ways a child might divide245 by eight. She included ques-tions a parent could ask to checktheir child’s understanding of aproblem:Why does your answermake sense? How are thesestrategies similar?

“I felt like parents and kidsthought I was making this stuffup off the top of my head,” De-Priest says. “I wanted them toknow that just because I was notusing a traditional textbook, Iwas still teaching math and haddone my research.”

Parents feel welcome to cometo class, sit at their child’s table,and dive into the work. Dad SteveMiller arrives as his daughter,Lindsey, and classmate Emma arepuzzling over a graph that has fivevertical bars, but no title, and nonumbers.

“What’s this a graph of?” he asks.

“We don’t know, we’re tryingto figure it out,” says Lindsey.

As clues, they have a story abouta variety of events that affectedearnings at the school store lastyear between January and May.Total earnings were about $300.The students’ task is to figure out

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Teacher Jana DePriest plays a math cardgame with her students.

Parent volunteer Kathy Krell advises studentsin the operation of the class store.

F E AT U R E


what the bar graph might be rep-resenting. DePriest wasn’t lookingfor just one answer. Any reason-able solution would be okay.Miller picks up one of their cal-culators and starts helping. Afteradding up all the bars and divid-ing them into $300, his groupdecides the monthly revenuescould be graphed in incrementsof $9. Others decide it makesmore sense to count by 10s.Either answer could be right.

“I’m looking for their strategiesand how they’re thinking,” De-Priest says.

On another afternoon, studentsuse their math problem-solvingskills to determine the mean,mode, and median of data col-lected during a science experi-ment. Blowing up tubular bal-loons to different lengths, thestudents and a couple of parentstape the inflated balloons to slid-ing devices that are attached tostrings stretching from wall towall. Releasing the trapped air,they let the balloons fly, gather-ing data that will allow them toevaluate Newton’s third law: Forevery action there is an equal butopposite reaction.

“This is fun,” says Greenwood.“I wish they taught math like thiswhen I was a kid.”

DePriest also invites parents formath games, including cribbageand other card games, every cou-ple of weeks on Friday afternoons.In one of her messages home,DePriest tells them:

“Math games develop students’sense of number and math oper-ations. Math games provide stu-dents with many opportunitiesto develop their own strategiesto solve arithmetic problems.

Creating strategies to solve prob-lems is an intellectually demand-ing endeavor because the studentsare combining mental math,arithmetic, and thinking aheadfor game strategy all at once.”

DePriest believes that some students can succeed at maththrough more traditional teach-ing methods, but that many morestudents will get the concepts ifthey see practical application. Andif they’re consistently asked toexplain their thinking, they’llconstruct a better understandingof why a solution works.Withparents largely onboard and stu-dents thriving in her problem-based classroom, DePriest is nowdeveloping ways to measure anddocument her students’ progress.

“I know it’s the right change,I can see the progress with thekids. But now I must documentand be able to prove to parentsjust how much their kids arelearning,” she says.

Creating a student-centeredclassroom, in which studentstackle challenging math problemsfor which there are no clear an-swers, is a time-consuming affair.So is communicating and inter-acting regularly with parents. Butthe payoff is big when strugglingstudents develop into skilled prob-lem solvers and proud parentsexpress their gratitude. Changingone’s practice can be a long anddifficult process, but, says De-Priest, “It’s worth it.”

Rosemary Shinohara is an education reporter and Stephen Nowers is a photog-rapher with the Anchorage DailyNews.

Young children are naturalproblem solvers, and early child-hood settings—where childreninteract with one another andparticipate in decisionmaking—offer countless opportunities forchildren to grow in their prob-lem-solving abilities. These im-portant experiences help childrenlearn to value different kinds ofthinking, think logically and cre-atively, and take an active role in their world.

Here are a few suggestions forparents on encouraging childrento be creative problem solvers:

Brainstorm. Invite children to be fluent thinkers by askingthem to respond to questionsthat have many right answers.

Reflect. Help children to beflexible thinkers by asking themto comment on specific objectsor situations.

Challenge. Encourage childrento practice critical and logicalthinking by asking them open-ended questions.

Listen. Encourage children toexpress their ideas. Asking ques-tions about things that don’tmake sense is another way chil-dren express critical thinking.

From Scholastic Early ChildhoodToday, April 1999. Copyright © 1999 by Scholastic Inc. Reprinted by permission.

TIPS FOR PARENTS

NWT

TO CONFRONT HER FEARS OF

MATHEMATICS AND BUILD HER

CONTENT KNOWLEDGE, elementaryteacher Judith Carter took a mathcontent course for educatorsthrough the Mathematics Educa-tion Collaborative of Ferndale,Washington. For the first time inher life, she was learning math inan environment that encouragedindependent thinking and problemsolving. In her class journal, ex-cerpted below, she reflects on theprofound changes this experienceprompted in her personal beliefsabout teaching and learning.

W hen I entered a graduateprogram to become a

teacher seven years ago, I hadbeen aware for some time thatas a student of mathematics Ihad been injured by the peda-gogy of my teachers. In theclasses I had taken from kinder-garten to high school it wasnever acceptable to use picturesor objects to help me understandwhat the numbers and lettersrepresented. No one let me talkabout what I thought or encour-aged me to ask questions. Infact, to ask a question was tak-ing a big risk that I might beseen as someone who “didn’tget it.” Most of all, no one said itwas okay to struggle and evenmore, no one viewed strugglingand confusion as part of problemsolving. Had the pedagogical useof manipulatives, social interac-tion, reflective processing fromvarious perspectives, and honor-

ing the perseverance it takes tosolve complex tasks been taughtby any of my teachers, my math-ematical experience and self-concept would have beentremendously different.

One of the main reasons I tookthis math class was to overcomemy fear of math and change theimage I had of myself as a mathstudent. As a student of any sub-ject I have always wanted toknow “why” and math has beenno exception. This class was allabout finding “why.” But whatwas so different was that wewere encouraged to find out“why” in our own way and inour own time. Everyone’s pathswere explored and honored and,though I found that my path to“why” was often circuitous, Ilearned that the long way is notthe wrong way. I struggled and Iwas confused but the commit-ment that had led me to take theclass in the first place kept reap-pearing in my persistence witheach problem. Because persever-ance was supported, by collabo-ration, by honoring diverseapproaches, by being given timeto solve problems, I found thatmy perseverance increased expo-nentially. There were some awe-some moments of revelation inthis class and I could feel thechild in me want to jump out ofmy seat (I think I did this a fewtimes) and shout, “I get it!”There were also occasions whenthe injured math student ap-

peared, and I learned how to bepatient with her. Perseveranceand patience are now part of mynew self-image.

I had always said I believedthat all children can learn but,because of my own fears, I thinkthere was a part of me that wasstill thinking, “but not me.” So ifI continued to believe this aboutmyself, wouldn’t I also believe itabout some of my students? Iknow I can learn difficult mathconcepts and explain my thinkingto others. This will have a pro-found effect on how I view mystudents because I know nowthat all children can learn whenthey are allowed to work in anenvironment that is safe for tak-ing risks.

To feel safe and gain confi-dence in math, students need tohave opportunities to processtheir thinking in a variety ofways: using manipulatives, writ-ing, working with others andpresenting their work to theclass. They need to be able to as-sess their own work and to re-flect on it and to redo the workas they learn more. I hope thatthis year I can create an environ-ment where students will seethemselves as problem solversand value their own path to“why.”

JUDITH CARTERsecond-grade teacher, ClarendenElementary, Portland, Oregon


C O N N E C T E D C O R N E R

BELOW ARE TECHNOLOGY RE-SOURCES and Internet sites thatcan facilitate learning throughmathematical problem solving.

Aspreadsheet program thatstores data so the numbers

can be crunched, graphed, andanalyzed is a practical classroomresource. However, teachingyoung students how to use aspreadsheet program can takevaluable class time. Now stu-dents can learn the value of aspreadsheet while the focus ofthe lesson is on problem solvingand decisionmaking.

Ice Cream Truck is a simulationprogram from Sunburst Technologyfor grades two through six (sun-burstdirect. sunburst.com/cgi-bin/SunburstDirect.storefront/).Student groups decorate an icecream truck, buy and sell icecream, and travel to different lo-cations. Daily and weekly salesresults are stored in summarysheets. Students can create cus-tom spreadsheets to compare,for example, each group’s salesresults or to organize and sortdata. Spreadsheet data can beexported to a more powerfulspreadsheet for graphing or sharing.

The program includes a journal.Students can record why theychose a particular location, whythey purchased one item overanother, or how they decidedwhat to charge for the ice cream.Student journal reflections may

be used again and again as stu-dents explore the simulation, de-veloping an understanding ofrelationships between variablesby practicing math in a real-world scenario. Students can, forexample, determine how weather,marketing, location, and productvariety affect profits.

Group interaction and journalwriting foster students’ verbalskills. In their journals, studentscan write about their sales plan,and, later, reflect on the resultsof their plan.

Lessons and copyable handoutsare provided.

other resourcesThe Tenth Planet series of soft-ware products from Sunburst(www.tenthplanet.com) providean entrance into problem solvingfor elementary students. Assess-ment is an integral part of eachprogram. Electronic journals arealso a key feature of this resource.As students solve problems re-lated to the real world, they cansave their results and write com-ments beside their work.

The spreadsheet program,Cruncher 2.0 from KnowledgeAdventure (www.knowledge adventure.com) is the latest ver-sion of a very successful class-room product. It is designed formiddle-schoolers. Colorful organ-ization and larger fonts immedi-ately make this an inviting numberplayground. The teacher’s guidehas copyable pages linked to

prepared spreadsheets.

Geometer’s Sketchpad from KeyCurriculum Press (www.keypress.com/sketchpad) provides muchmore than just the technologytools for exploring geometric con-cepts. The built-in text windowsenable students, from grades five through 12, to describe theirproblem-solving processes as wellas their conclusions. The ability toexport data to a spreadsheet al-lows additional data analysis.

internet resourcesLOGAL Software offers mathe-matics, life science, and physicalscience simulation software pro-grams, available by subscription.To sign up for a free trial, visitwww.riverdeep.net.

To practice buying and sellingstocks, students can use the freesimulation at the Web site, www.mainXchange.com.

Teachers and students can access information and data onchanging global conditions andprint maps from National Geo-graphic Society’s Web site, www.nationalgeographic.com/education.

JUDI MATHIS JOHNSON

is a nationally published softwarereviewer.

TECHNOLOGY TIPS to facilitate learning

http://sunburstdirect.sunburst.com/cgi-bin/SunburstDirect.storefront/

http://sunburstdirect.sunburst.com/cgi-bin/SunburstDirect.storefront/

http://www.tenthplanet.com/

http://www.knowledgeadventure.com/

http://www.knowledgeadventure.com/

http://www.keypress.com/sketchpad/

http://www.keypress.com/sketchpad/

http://www.riverdeep.net/

http://www.mainXchange.com/

http://www.mainXchange.com/

http://magma.nationalgeographic.com/education/index.cfm

http://magma.nationalgeographic.com/education/index.cfm


C L A S S R O O M R E S O U R C E S

THE NWREL MATHEMATICS AND

SCIENCE EDUCATION CENTER’S RE-SOURCE COLLECTION is a lending library of teacher-support material.Search the collection and requestitems from the Web site www.nwrel.org/msec/resources/ or call (503)275-0457. The only cost is to mailitems back at library rate.

Following are titles on problemsolving for all grades:

A Collection of Performance Tasksand RubricsCharlotte Danielson & others (1997)

The four books in this series (pri-mary, upper elementary, middle,and high school) provide perform-ance tasks and scoring rubrics forkey math topics. Includes sam-ples of student work to clarifytasks and anchor the points ofthe scoring rubrics.Posing Open-Ended Questions inthe Primary ClassroomChristina Myren (1995)

Open-ended math questions forK-2 children, with children’s liter-ature serving as a springboardfor many of the lessons. Lessonscontain an overview, materialslist, description, student responsesand assessment discussion, andreferences.Projects to Enrich School Mathe-matics Level INational Council of Teachers of Math-ematics (1990)

Independent study projects forgrades four through six. Topicsencourage student research, in-vestigation and communicationof mathematics through report-ing and writing. Projects supply

teachers with materials adapt-able to a variety of settings.Elementary and Middle SchoolMathematics: Teaching Develop-mentallyJohn Van de Walle (1994)

A resource for major topics in theK-8 mathematics curriculum, thisguide and instructional resourceshows teachers how to help allchildren become confident “doersof mathematics.” Key elementsare the nature of mathematics as a science of pattern and order,an understanding of how childrenlearn mathematics, a view ofteaching mathematics as a prob-lem-solving endeavor, and spe-cific methods for integratingassessment with instruction.United We Solve: 116 Problemsfor Groups Grades 5-10Tim Erickson (1996)

Tasks provide a structure forlearning to work together andassume high standards for studentperformance, communication,and understanding of mathemat-ical ideas. Problems focus onproportional reasoning, general-izing from patterns, spatial rea-soning, and visualization.101 Short ProblemsJean Kerr Stenmark (1995)

Open-ended questions for gradesfour through nine. Leading ques-tions to motivate students to gobeyond the first answer. All prob-lems are in English and Spanish.

The New Sourcebook for Teaching

Reasoning and Problem Solving in

Junior and Senior High School

Stephen Krulik & Jesse Rudnick (1996)

A resource that educates teachersabout teaching and evaluatingreasoning and problem solving,and offers reasoning and prob-lem-solving challenges for stu-dents. Includes performance tasks,problem-specific rubrics, and rec-ommendations for basing instruc-tional decisions on accurateassessment of student problemsolving, reasoning, and commu-nication skills.High School Mathematics at WorkMathematical Sciences EducationBoard & National Research Council(1998)

Essays accompanied by illustra-tive tasks from workplace andeveryday contexts for high schoolstudents. How to develop curric-ula so that students learn to solveproblems they are likely to en-counter in life.Get It Together: Math Problemsfor Groups Grades 4-12Tim Erickson (1989)

More than 100 mathematicsproblems that three through sixpeople can solve together, witheach person holding informationneeded to find the solution. In-cludes advice on how to intro-duce, facilitate, and assesscooperative learning in the classroom.

AMY SUTTON

is a math and science resource specialist for NWREL.

books and materials available fromTHE CENTER’S LENDING RESOURCE COLLECTION

http://www.nwrel.org/msec/resource/

http://www.nwrel.org/msec/resource/


D I S C O U R S E

In my teaching experience,I’ve found that project-based

learning is an ideal context forproblem solving and construc-tivist learning, the foundation ofmy pedagogical philosophy. Tomeet my students’ learning needs,I’ve adopted a project-based cur-riculum that integrates subjects,creates opportunities to applyskills, and allows for higher-orderthinking and problem solving.

With project-based learning, stu-dents are able to apply their skillsand understanding of conceptsover an extended period that in-volves them in a variety of real-world problem-solving situations.

A wider range of student learn-ing styles and multiple intelli-gences can be accommodatedwith a project-based approach.For many students, the hands-onand applied manner of extendedprojects helps them to find mean-ing and justification for learning,often engaging those studentswho struggle most in academicsettings.

Building a project-based cur-riculum that emphasizes open-ended problem solving is noteasy. It requires one to shift pri-mary focus from the textbookand skill learning to a curriculumthat emphasizes active learning,application, synthesizing knowl-edge, and transferring strategiesand skills to new situations. Onemust be willing to experiment

with teaching and learning, to becreative in curricular approaches,using strategies that may not betraditional in school mathemat-ics.

One must be willing to acceptfailure, learn from mistakes, andrealize that change takes time,committing to a view of teachingand learning that is different fromthe traditional paradigm.

The rewards can be significant.By facilitating students’ learningof content knowledge, as well as their reasoning and problem-solving abilities, this approachcan help prepare them to meetstate standards and assessments.

In Washington, project-basedlearning can help students mas-

ter the Washington State Essen-tial Learnings, the state’s summaryof important skills and contentthat include communication,reasoning, making connections,problem solving, and conceptualand procedural knowledge.

After examining the Washing-ton Assessment of StudentLearning (WASL), I was not at allconcerned by how well my stu-dents might perform (see www.osd.wednet.edu/chsscimath/default.html). By combining skillbuilding with opportunities to ac-tively apply knowledge in a proj-ect-based curriculum, my studentsmet the Essential Learnings andhave performed well on theWASLs.

Project-based learning energizesand challenges me as a teacher.It motivates me to examine cur-riculum critically to ensure that Idevelop meaningful and effectiveprojects. I continually explore myrole as teacher and curriculumdeveloper, and this involves mein the excitement of research aswell as engaging me and my stu-dents in new activities.

LOUIS NADELSON

is a math and science teacher atCapital High School and EvergreenState College in Olympia.

problem solving AND project-basedlearning IN high school mathematics

LET USHEAR

FROM YOUSUBMIT LETTERS TO:

The EditorNorthwest Teacher


Mathematics and Science Education Center

101 S.W. Main StreetSuite 500

Portland, OR 97204or

[email protected]


CHRIS SANCHEZ, 14, A STUDENT AT KELLOGG MIDDLE SCHOOL IN PORTLAND, OREGON, employs the tools of the artist in this non-representational painting, using geometric shapes, colors, asymmetry, and perspective to explore relationships between di-verse elements. Chris’ painting was included in an exhibit by the Young Artist Project of Pacific Northwest College of Art andPortland Public Schools.The scholarship-based program provides time, materials, and encouragement to talented young artists,enabling them to exercise their creativity and build their skills in after-school art classes.

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