ed 327 391 title mathematics problem solving: a more ... · chapter 1 mathematics progra, as. this...

ED 327 391

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EDRS PRICEDESCRIPTORS

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SE 051 838

Mathematics Problem Solving: A More Advanced Skillfor Chapter 1. Workshop Leader's Guide.

Advanced Technology, Inc., Indianapolis, IN.; Chapter1 Technical Assistance Center, Indianapolis, IN.Region B.

Office of Elementary and Secondary Education (ED),Washington, DC. Compensatory Education Programs.Feb 89

94p.: Several pages contain type on gray backgroundand will reproduce poorly.Guides - Classroom Use - Guides (For Teachers) (052)--

MFOI/PC04 Plus Postage.*Elementary School Mathematics; Elementary SecondaryEducation; *Inservice Teacher Education;Instructional Materials; Learning Activities;Mathematics Education; Problem Sets; *ProblemSolving; *Secondary School Mathematics; TeachingGuides; *Workshops

This guide is designed to assist inservice providersin conducting successful workshops for teachers, administrators, andothers associated with Chapter 1 mathematics programs. It containsstep-by-step procedures for preparing, organizing, and presenting theworkshop. Included in this guide are: (1) an advanced planner, whichincludes a detailed checklist for materials and equipment needed toconduct a successful problem-solving workshops, and a workshopoutline, which includes detailed instructions with the goals of thewcsrkshop, specific problem-solving activities, and recommendationsfor using the wcrkshop materials; (2) blackline masters for theparticipant handouts and overhead transparencies; (3) a briefoverview of problem-solving topics; (4) resource materials andsupport materials including additional problems and suggestions; and(5) a bibliography of 12 additional resources. (CW)

* Reproductions supplied by EDRS are the best that can be made* from the original document.

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ED ATIONAL RESOURCES INFORMATIONCENTER IERIC)

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V T

MathematicsProblem Solving

A more advanced skillfor Chapter 1

Advanced Technology, Inc.2601 Fortune Circle East, Suite 300A

Indianapolis, IN 46241(317) 244-8160 (800) 456-2380

2

TableofContents

IIrnicmol

Wtot@hc)p

Section 1: IntroductionGetting StartedWorkshop Outline

Section 2: Participant Handouts

Section 3: Transparencies

Section 4: Background Paper

Section 5: Resource Articles

Section 6: Support Materials/Alternative Activities

Section 7: Bibliography

Mathematics Problem Solving

What's in this Guide?This Workshop Leader's Guide on MathematicsProblem Solving is designed to assist Technical As-sistance Cemer staff members and other in serviceproviders in conducting successful workshops forteachers, administrators, and others associated withChapter 1 mathematics progra, as. This guide con-tains step-by-step procedures for preparing, organ-izing, and presenting the Mathematics Problem-Solving Workshop. --,

Contents:

Section 1 includes the Introduction, Getting Started, and the Workshop Outline.The Getting Started section includes an Advance Planner and a detailed checklist for materials andequipment needed to conduct a successful problem solving workshop.The Workshop Outline includes detailed instructions for presenting a one-hour workshop includingthe goals of the workshop, specific problem-solving activities, and recommendations for using theoverhead transparencies and participant handouts . The outline for the one-hour workshop can beexpanded to provide the basis for a workshop up to three or more hours in length.

Sections 2 and 3 contain the blackline masters for the participant handouts and overhead transparenciesreferred to in the Workshop Outline.

Section 4, the Background section, gives a brief overview of problem-t.olving topics with which youshould become familiar.

Sections 5 and 6 contain Resource Articles and Support Materials iacluding additional problemsand suggestions for presenting a workshop on problem solving.

Section 7 includes a Bibliography of additional sources of information.

How to Use Th;s Guide

This Guide contains the planning and presentation materials neceaary to conduct a successfulworkshop on problem solving. The materials were developed to allow a great deal of flexibility.Suggestions for workshop variations and a variety of problems are included so Mat the workshop canbe adjusted to tit the needs and mathematical backgrounds of your audience. A wide range of prob-lem-solving approaches, types of prcblems, and specific research can be used in the actual workshoppresentation. You may choose to change, add, or eliminate a problem, activity, or transparency. Thesections in the Workshop Outline titled "Gaining an Understanding" and "A Model for TeachingProblem Solving" arc well suited for expansion or contraction, depending upon your presentationneeds.

Mathematics Problem Solving Workshop Page 1

,,1

INTRODUCTION

Getting Started

Begin your advance planning for the workshop by establishing some of the initial details, such items asdate, place, and type of audience. (See Advance Planner, a simple checklist for planning a workshop,p.4.) Then begin studying the contents of this guide by following the G-U-I-D-E steps outlined below:Glance, Understand, Investigate, Develop, and Edit.

G-11-1-D-EGlance through the entire set of materials,

This will give you a feel for the types ofmaterials contained in the Guide (and theirlocation) when you study the details latv.

Understand as many of the materials containedin the Guide as possible, Plan enough time

to develop a full grasp of the materials in orderto make more informed decisions aboutyour workshop presentation.

Investigate further. You may watu to doaddidonal research, try different problems,or experiment with various activities.

Develop additional materiak These may beworkshop notes, transparencies, handout pagesactivities, or any item resulting from your"investigating" activities.

Edit. Look carefully at the total picture,then elaborate or eliminate, if necessary.

Begin planning as soon as possible. Even if you use only the materials in this Guide, the G-U-I-D-E stepswill take time and should be included in your planning. It is especially important to allow yourself theopportunity to thoroughly explore and solve the problems in the Workshop Outline so the purpose andstrategies for each problem will become more apparent to you. In addition, as you engage in the prob-lem-solving experience, you are likely to discover that you need or want to try additional problems andactivities from several sources. (See Section 6, Support Materials.) You may find that these materialsare better suited to your parti,:ular workshop and, therefore, you may want to substitute them for otherworkshop materials in earlier sections of this guide.


r74.)

( SUPPORT MATERIAL!)

What You Need for the Workshop:

(EQUIPMENT

Overhead Transparency Projectorextension cord3-way plug adaptorextra bulb or spare projector

Blank Overhead Transparencies

Screen

Microphone (if needed)

MATERIAMnS

Workshop Outline

Supporting Notes

Participant Handouts(one for each participantfrom Section 2)

Overhead Transparencies(prepared from masters inSection 3)

Chart paper

Poster board

Marker and tape (or chalkboard and chalk)

Before You BeginMake copies of the overhead transparencies You pla'n to use in the workshop, and be sure

you have one copy of the handouts for each partkipant4 if you we preienting the workshopin a kvation with whkh you are unfamiliar, ask the tool contact penon to be sure the equipment listed above is available and in working order on the scheduled day and timeof the work-shop. If you will be supplying your own equipment, make arrangements for obtaining it well inadvance of the workshop and make sure everything is in working ortier,

\*

Mathematics Problem Solving Workshop

6

Page 3

c

rWorkshop Advance Planner

Presentation Information

Title

Date Day Time

Place

Audience Type Number

Purpose

\ontact Person Phone

Planning Task Date Completed

Contact Person(s) for PlanningConfirm Date, Time & PlaceMake Travel and Hotel PlansArrange for EquipmentSend Workshop Agenda to ContactPersonalize Workshop OutlineOther

Mathen:atics Problem Solving Workshop

7

Page 4

MATHEMATICS PROBLEM SOLVINGOne-Hour Workshop Outline

Setting the Tone 10 minutes

Goals and Background 10 minutes

Gaining an Understanding 20 minutes

A Model for TeachingProblem Solving 10 minutes

Summary and Conclusion 10 minutes


Setting the Tone

(10 minutes)


A More AdvancedSkill

For Chapter 1

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Display T-1: Workshop Title Slide:"Mathematics Problem Solving"


Activity 1

If a group of six participantswere to introduce themselvesto one another, how manyhandshakes would it utke

(It would take 15 handshakes)

N Read Activity 1:"Shaking Hands"

\. }

Introduce yourself and describe yourassociation with Chapter 1. Then followthe procedure below for Activity 1 whichinvolves participants meeting one another.

Procedure:

Encourage participants to work on the problemsin groups, sharing methods, answers, etc. Groupsshould write their answers on paper, not callthem out. Rather than discuss only the correctanswer(s), participants should discuss strategiesthey used to arrive at their answers. While thegroups are working, circulate around the roomlooking for different strategies.

After a minute or two, announce that 30 and 36are not correct solutions. This will encouragethose who "jumped" to incorrect solutions to co:- -tinue working toward a correct solution.

In the total group, discuss different strategiesused by participants. By observing variousgroups, you will see some very differentapproaches.

Use strategies shared by participants to start a"Problem Solving Strategies" list. Use chalk-board, chart paper, or poster board to modelthe process for panicipants.

Strategies will be added to this list throughoutthe workshop. Please refer to Background Paper(Section 4) for further information aboutproblem-solving strategies. Strategies used forthe handshake problem might include:

draw a picturemake a chartact it out

The list should develop from discussion andmay include different names like "make a sketch"instead of "draw a picture" or may have more orfewer strategies.

Extend the activity above by asking "What if...?"questions such as "What if there were 10, 20. or100 people in the group?" You might alsodiscuss ways to adapt the probiem to youngerchildren, fewer children, or a role-playing situ-ation.


-41 6

Page 7

Goals and Background(10 minutes)

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(These goals are also listed on ParticipantHandout, H-1.)

Display T.2: "Workshop Goals"

Present the goals of the workshop. Brieflydescribe how you plan to achieve these goalsduring the workshop.

Experiencing the workshop:Encourage participants to think about their workshop experiences in three ways, as a:

( Participant )experiencing the feelingof satisfaction duringproblem solving

Teacher

learning about the process ofproblem solving, choosing goodproblems, and modelingproblem-solving behavior

developing an awareness ofproblem-solving strategiesand the feeling of successthat accompanies it


-J.

To imprcwe the educational oppodu-maks of educationally deptived chil-

I Men by helping them:

Succeedin theregularprogram

Aitain grade !cvelproficiency

improve acNevement inbasic and more advancedskills

Reasoning

MoreAthancedSkillsIncluding:

A

Anal)si.

eProblemSolIng

Interpretation

DecisionMaking

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Display T-3:"Purposes of Chapter 1"

Discuss the purposes of Chapter 1 and the empha-sis on more advanced skills in the Chapter I law.

Display T-4:"More Advanced Skills"

Point out the acronym RAPID. Mention thatother (=kills such as organizing, integrating, andevaluating may be included as more advancedskills.

Mathematics Problem Solwng Iiilrkshop Page 9

Afto

MIINION

Why Only LowerOrder Skills?

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Display T-5: "Why Only LowerOrder Skills?"

Discuss reasons for the traditional emphasis onlower order skills.

Display T-6: "Why StudentsDon't Catch Up"

Discuss some of the reasons students don't"catch up" and suggestions for changing thesituation. Mention items such as the use ofan almost endless number of worksheetsfor computational drill and practice and thefocus on mechanics or algorithms t the ex-clusion of concepts and problem solving.


WHAT ISNEEDED

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Display T-7: "What is Needed"

Discuss what changes may help in providingdisadvantaged children with a more effectivecurriculum that would help them succeedat higher-order tasks when given theopportunity.

Display T-8: "NCTM Standards"Discuss the importance of problem solving intoday's mathematics curriculum, i.e., it is thefirst curriculum standard in each grade-levelgrouping.

Point out that problem solving is the reasonfor studying mathematics and that, as such,improved student problem-solving abilitymust be a goal for all mathematics instruction.

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Gaining An Understandingof the Problem-Solving Process

(20 minutes)

aActivity 2

Find the number ofcombinations of coinsthat could be used to buya can of soft drink for 54from a vending machine.The machine does not givechange and does not takefifty-cent pieces or dollarbills.

(There are 70 possible combinations.)

Read Activity 2:"Making Change"

Present the problem in Activity 2;then follow the suggested procedurebelow.

Procedure:

A good way to "involve" students in this problem-solving activity is to present the problem as a storyor relate it to a real-life situation. You may wantto model this with a story similar to the following:

(Using afifty cent piece as a prop for this problemmay add interest to the story and prompt a discus-sion of why it k: an unpopular coin.)

I was standing in front of a vending machinethat disper.sed SOO soft drinks. A woman camerunning up to me waving a large coin and act-ing very disturbed. When I asked if I couldhelp, she told me she desperately needed a softdrink, but had only a fifty-cent piece. I told herI would give her change for the vending ma-chine if she could tell me how many possiblecombinations of coins would work in the ma-chine.

Again, encourage participants to work in groups,sharing strategies and writing their answers.Circulate, answering questions and looking fordifferent strategies used. As a group, discussstrategies and add them to the list.

Possible strategies include:make a listmake an organized list(a refinement of the first)act it outuse real or play moneyuse maniptilatives

Extend the problem by asking "what if" ques-tions, like, "What if we could use pennies'?" or'What if the machine charged 600" Discussadapting the problem to younger children,perhaps having them use play or real coins.


1

Page 12

Definition

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PROBLEMS

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Display T-9: "Problem Solving"

Define problem solving. Emphasize that it is aprocess. Refer to the previously solved problems.Discuss what was involved in the solution.This discussion will help participants to betteridentify with and understand Polya's 4-stepmethod.

Display T-10: "Problems"

Explain that students need experience in solvingall types of problems, including both the tradition-al word problems typically found in textbooksand non-routine or process problems.


C

PP Qc 13

Word/Story Problems

Follow develcpment cot an arithmeticoperation or other slgonthm

Can be solved by direct apphcatioi, of one ormore previously lesnied algorithms

Task is to identify wluch operation(s) oralgonthm(s) to use

Emphasis is on getting the correct answer

WO

To Reinforcerelationship between the

operations and theirapplication in realworld situations

11.

Display T-11: "Standard TextbookProblems"

Ask participants where story problems are usu-ally found in math textbooks. (Most often, theyare at the beginning or end of a page or chpter.)Identify the attributes of standard textbook prob-lems listed on the transparency. This should helpbring to life many of the attributes of standardtextbook problems.

(Note: An algorithm is a procedure that frequently in-volves repetition of an operation or a series of steps. Oneexample is the long division algorithm.)

Display T-12: "Why TeachTraditional Story Problems"

Point out that textbook problems arc important.They can help improve the recall of basic facts,strengthen skills with the basic operations andalgorithms, and reinforce relationships betweenthe operations and their application in real worldsituations.


1 7

Page 14

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Process Problems

Stress the process of obtaining solution.not just numerical answer

Require the use of one or more strategies

Frequently can be solved in several ways

Often have more than one answer

Usually take longer to solve

Provide opportunities for cooperativelearning

I.

Display T-13:"Non-routine Problems"

Review the six points on the trans-parency in light of participant in-volvement with the two problemspresented in the workshop. Pointout that most of the problems inthe handout are considered "pro-cess problems".

Example;Ask participants if the following question is a traditional story problem or aprocess problem:

There are 32 turkeys that are to be shipped in Crates that hold 4 turkeys each.How many orates will be needed?

(The problem is from a fourth-grade mathematics text)

Discuss whether this question would involve fourth-grade students in realproblem solving. Compare this to the other problems in the workshop. Re-mind participants that problems of this type are important, but that studentsneed experience in solving process problems. Process problems will improvetheir ability to solve both non-routine and textbook type problems.


isPage 15

Display T-14: "Problem Solving-The Reason for LearningMathematics"

Stress that problem solving should beincluded throughout the mathematics cur-riculum. Problem soiving should not betaught as a separate unit, or only at "specialtimes," then forgotten. Teachers need to makeproblem solving the focus of mathematicsinstruction and take advantage of the manyproblem-solving opportunities that arise inthe classroom.

Mention that additional value is placed onother skills such as computation and measure-ment when students realize the value of theseskills for solving problems.

The ReasonFor LearningMathematks

9 la


_1.)

A Model For Teaching Problem Solving

(10 minutes)

Display T-15: "Polya's Model for ProblemSolving"

Present the problem-solving model from GeorgePolya's landmark 1945 book, How to Solve It.Mention that most current models of problemsolving are adapted from this heuristic (or guide)for the problem-solving process originally pro-posed by Polya, who is often referred to as the"father of modem problem solving." Follow theprocedure below to expand upon Polya's model.(See Handout H-11.)

Procedure:Discuss the four steps 0 the problem-solving process in rdation to the problems participantsworked on during the workattop. Emphasize that learning to use a systematic, problem-solvingprocess takes time, bat once the process is discovemd, it can be used over and over again. En-courage participants to model the four steps in their teat, "'lather than have students merelymemorize the steps.

Consider displaying the model on a large piece of paperor poster board and taping it to the wallby the strategies list. Encourage participants to display the model permanently in the classroomfor easy reference and use by the teachers and students.


Activity 3

A farmer has some pigs andchickens. He sent his son anddaughter to count how manyof each he had. The son count-ed 70 heads and the daughtercounted 200 legs. How manyof each does the fanner have?

(Ans.: 30 pigs and 40 chickens)

Read Activity 3:"Pigs and Chickens"

Participants will again work ingroups. Many times groups try tosolve the problem (Step 3: Carryout the plan) before they thor-oughly understand the problem orcarefully devise a plan. As youcirculate, suggest to groups hav-ing difficulty to use the steps inPolya's model.

Procedure:"Trial and error" or "guess and check" is the most common strategy for solving thisproblem. Participants may feel that setting up algebraic equations is the only legitimateway to soive the problem. Note that trial and error is a legitimate problem-solvingstrategy and is commonly used in "real-world" mathematics. Point out that more thancorrect answers should be valued in the classroom. In this problem, the "mistakes" arenot only useful, but essential to successfully solving the problem.

Point out that the strategies list is a valuable tool for problem solving in the classroom.

Teachers can:

Create a strategies list with their students(add to the list as modeled in the workshop)

Display the list openly for easy reference

Encourage students to create their own strategies when needed

Refer students to the list of strategies whet.' "Devising a Plan"(Step 2 of Polya's Model)

Note: The One-Hour Workshop allows participants to use a variety of strategiesbut does not give them experience with re-using those methods previously listed.


Summary/Conclusion(10 minutes)

Distribute the Participant Handouts and Review Their Contents:Take a few minutes to review the contents of the handout. Much of the handout is notspecifically covered in the workshop but is meant as i reference for future use byparticipants. Reviewing the handout content will encourage its later use.

H-1 Goals and Experiencing the Workshop

Review the workshop goals and how each wasdemonstrated during the presentation.

H-2 through H-5 Reduced Copies ofTransparencies

Remind participants that they have reducedcopies of the transparencies, so they will notneed to copy them when they are displayed.

H-6 through H-10 Effective MathematicsTeaching

Review and discuss findings from research,myths to overcome, typical problem-solvinEactivities in the classroom, and proceduresfor encouraging more effective problemsolving.

H-11 Polya's Problem-Solving Model

Review Polya's problem-solving heuristic andits use as demonstrated on H-12.

H-I3 through H-23

These pages contain a variety of process problemsthat car. oe used or adapted for Chapter I students.To promote the process of "how-to-solve" ratherthan that of "finding-the-answer," answers aredeliberately not included.

H-13, Visiting Friends, is not easily adaptablefor early elementary but is an excellent activityfor participants. It can be included in a longerworkshop format or substituted for one of thegiven activities in the one-hour workshop.

H-21 and H-22 des ribe problem solving as usedin cooperative groups. The problem example forolder children, Number Bracelets , successfullymodels problem solving as a group activityand would be an excellent inclusion in a longerworkshop.

H-23, Sample Chapter I Desired Outcome...includes a rating scale as well as an exampleof desired outcomes for problem solving.

H-24 gives a Bibliography listing of suggestedreadings. Remind participants that titles can beprocured by requests on institutional letterhead.


4.

Page 19

?

TEACHER'S ROLEIN PROBLEM

SOLVING

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2. Create NI atmosphere whew working OM

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4. Lead peetsolutlo discuseles la whichtt mdeloto share their methods.

I

Display T-16:"Teacher's Role in Problem Solving"

Remind participants that teachers need to invite their students to think.Research indicates that students retain knowledge more effectivelywhen learning is experienced, when the student is actively involved .

Thus the effectiveness of problem solving translates to knowledge andprocess retention.

After going through the transparency, refer back to 11-6, EffectiveMathematics Teaching (What Does Research Say). Note that researchfindings indicate the success of utilizing problem solving in theclassroom.

Refer to H-7, Improving Problem Solving in Your Classroom, and toH-8, Self Assessment Questions for Teachers of Problem Solving.These materials have been developed for participant use. H-7 givespractical suggestions for expanding the use of problem solving in theparticular teaching situation. H-8 is a self-assessment tool designedto aid the individual in determining what teaching strategies they noweffectively use and to indicate which strategies may need greaterutilization.


Display T-17:

"Model the Problem-solving Process "

Teachers should be encouraged tothink aloud. Such process modelinghas gained popularity in the field, andresearch indicates that this instruct-ional approach of letting students "inon the secrets" is an effective wayof teaching not only problem solving,but reading comprehension and otheradvanced skills as well.

Model the problem-solvingprocess at every opportunity.

Let students listen to you think.

Maihematics Problem Solving Workshop Page 21

Problem Solving

Prot, lem solving is the

process by which thestudent resolves anunfamiliar situation ornon-routine problem.

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"Problem solving is whatyou do when you don't knowwhat to do"

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Display T-18:"Problem Solving"

Emphasize the term "process." Point out thatteaching and utilizing problem solving isn't justa matter of equipping students with a bag of tricksbut encompasses teaching students how toapproach, analyze and understand a situation;d wise a plan; implement a plan; assess theoutcome; and adjust where necessary. Polyaproposes strategies and a mind-set which empowersthose who use them.

Display T-19:"Problem Solving is what you do..."

Read the statement emphatically. This quoteencapsulatts the complexity and difficulty ofproblem solving in a statement that is easilyunderstood; it stresses for participants what prob-lem solving should be for their students.


25Page 22

fSUGGESTIONS

FORWORKSHOPS INVOLVING PROBLEM SOLVING

Leaders of workshops that involve problem solving should consider the following suggestions:

Model the process of problem solving.

Teach problem-solving heuristics -- drawing a picture, making a table, using trial and error,and so forth.

Work the same problen ...tral different ways, using different heuristics.

Obtain an abundant supply of interesting problems.

Have a variety of problems that are suited to the various mathematical backgrounds of theparticipants.

Create a friendly, nonthreatening atmosphere.

Have each participant work on problems that are individually challenging, but notoverwhelming.

Allot a large portion of the time for actually solving problems.

Give participants opportunities to work cooperatively in small groups.

Give participants ample opportunity to talk about solving problems.

Emphasize estimation and reasonableness of results.

Use calculators.

Have participants generalize results.

Encourage participants to create problems.

Give participants a variety of nontraditional problems to take with them.

Give participants a list of sources of good problems.

Source: Professional Development for Teachers of Mathematics: A Handbook. NationalCouncil of Teachers of Mathematics, National Council of Supervisors of Mathematics, 1936.


ZS

Mathematicsnioblem Solving


For Chapter 1

Transparencies

27



For Chapter 1

The One HourWorkshop

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RReasoning

MoreAdvancedSkillsIncluding:

ARAnalysis

PProblemSolving

IInterpretation

Source: Section 1471 of the August F.HawkinsRobert T. Stafford, Elementary andSecondary School Improvement Amendmentsof 1988

DDecisionMaking

1-4


Belief that introducingmore advanced skills iscounterproductive becausestudents need basics first'1,44Z&VI,k*\,,,:eZea-v**40iWAPCM,

Instructional staff(particularly aides) arenot well prepared toteach more advanced skills

Failure to considerthe idea of focusing onmore advanced skills

...................................................

State testing programsfocused on basic skillsdiscourage inclusionof more advanced skills

Source: Program Design Study Chap-ter 1, Nation ssessment 1985-86

V', ,," k, '"::.:`

2 1-5


Reduced Learning Expectationsby both students and educators

Slow, Plodding Pacethat reinforces low expectations

Emphasis on Repetitionthrough drill and practice

Crucial Learning Skills Omitted

Mechanics Stressed Over Content

...the Achievement Gap Widens

Source: Levin, H., New Schools for the Disadvantaged, 1987

T-6

WHAT ISNEEDED

An effective curriculum for thedisadvantaged should not only befaster paced and actively engagethe interests of such children toenhance their motivation, but mustalso include:

conceptsanalysisproblem solvinginteresting applications

T-7

Mathematicsas

Problem Solvingis the

FirstStandard

at each of thegrade groupings

K4, 5-8, 9-12

STANDARDS

C.`

Source: Curriculum and Evaluation,Standards for School Mathematics,NCTM, 1989

35T-8

problem solving problem solving problem solvingsolving problem solving problem solving problemproblem solving problem solving problem solvingsolving problem solving problem solving problemproblem solving problem solving problem solving

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so vmprosolvprosolvprosolvproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problem

Problem solving is the processby which the student resolves

an unfamiliar situation ornon-routine problem.

problem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problemproblem solvingsolving problem

3CT-9

Standard Textbook Problems

Word/Story Problems

Follow development of an arithmeticoperation or other algorithm

Can be solved by direct application of one ormore previously learned algorithms

Task is to identify which operation(s) oralgorithm(s) to use


37

Why TeachTraditional

Story Problems ?

To Improverecall of basic facts

To Strengthenskills with the basic operations

and algorithms



3 2 T- 1 2

=1.1=1=.

Nonsaoutine Problems

Process Problems

Stress the process of obtaining a solution,not just a numerical answer






3;) T- 13

--

ChartsAnd

Graphs

Geometry

Estimation

Problem

Solving

EquationsAnd

Inequalities

Computation

Ratioand

Percentage

Measurement

i

The ReasonFor LearningMathematics

,f u T- 14

POLYA'S CON1..lZPTUAL MODEL00

1

1. Understand the problem

Devise a plan 2.

T

3. Carry out the plan

Look Back 4.L_ _J

fr 1 T-15


SOLVING

Make sure the children understandthe problem.

Create an atmosphere where working onthe problem is more important thangetting the right answer.

Encourage the children to talk with eachother and to work together.

Lead a post-solution discussion in whichthe students share their methods.

/".

T-16



z, 3T-17

Problem Solving

Problem solving is theprocess by which thestudent resolves anunfamiliar situation ornon-routine problem.

44T- 18


(Wheatley, 1984)

45 T-19



For Chapter 1

The One HourWorkshop

1

40Transparency 1

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To improve the educationalopportunities of educationallydeprived children by helpingthem:

succeed in theregular program

attain gradelevelproficiency

improve achievementin basic and more advancedskills.

Transparency 3

4-8

RReasoning

MoreAdvanced SkillsIncluding:

AAnalysis

PProblemSolving

IInterpretation

DDecisionMaking

Source:Section 1471 of the August F. Hawkins - Robert T. Stafford,Elementary and Secondary School Improvement Amendments of 1988

Transparency 4


Transparency 5

Belief that introducingmore advanced skills iscounterproductive becausestudents need basics first

Instructional staff(particularly aides) arenot well prepared toteach more advanced skills

P.Z.*it*SONIIM

Failure to considerthe idea of focusing onmore advanced skills

State testing programsfocused on basic skillsdiscourage inclusionof more advanced skills

-4410104WraMt't Aktm;rM-VONSIMIIII

50

.S4

Source: Program Design StudyChapter 1, National Assessment 1985-86


Reduced Learning Expectationsby both students and educators

Slow, Plodding Pacethat reinforces low expectations


Crucial Learning Skills Omitted

Mechnanics Stressed Over Content


Source: Levin, H., New Schools for the Disadvantaged,1987.

Transparency 6

WHAT ISNEEDED

An effective curriculum for thedisadvantaged should not only befaster paced and actively engagethe interests of such children toenhance their motivation, but mustalso include:

conceptsanalysisproblem solvinginteresting applications

Transparency 7

54,-;

Mathematicsas Problem Solving

is the

FIRSTstandardat each

of the grade groupings

K4, 5-8, 9-12

STANDARDSSource:Curriculum and Evaluation, Standards for School Mathematics, NCTM, 1989

Transparency 8

53

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54 Transparenc y 9

PROBLEMS

TraditionalWord/Story

Problems

I

Non-RoutineProblems

S5

1

Transparency 10

Standard Textbook Problems

Word/Story Problems


Can be solved by direct application of one ormore previously learned algorithms

Task is to identify which operation(s) oralgorithm(s) to use


Transparency 11

66


Story Problems ?


To Strengthenskills with the basic operations

and algorithms



,=

1 _ _ _ -

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Transparency 12

57

Non-Routine Problems

Process Problems

Stress the process of obtaining a solution,not just a numerical answer






Transparency 13

5F,

THEREASONFOR LEARNINGMATHEMATICS

Transparenc y 14

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SOLVING

1. Make sure the children understandthe problem

2. Create an atmosphere where working onthe problem is more important thangetting the right answer

3. Encourage the children to talk with eachother and to work together

4. Lead a post-solution discussion in whichthe students share their methods

Transparency 16

Ci



Twisparency 17

...

Problem Solving

Problem solving is theprocess by which thestudent resolves anunfamiliar situation ornon-routine problem.

Transparancy 18

e3


(Wheatley, 1984)

Transparancy 19

C4

Alternate Activity 1

Six teams are in a junior vollyballleague: The Spikers, Pounders,

Setters, Servers, Netters, and theVollys. If all six teams must playeach other one time, how many

games must be scheduled?

(It would take 15 Games)

Possible Strategies:


1. Draw A Picture

1 2 3 4 5 6

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r,5

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5

4

1-

2

15 Games

2. Draw A Diagram

1 I... ....- i

:I. . 4. / I

5i...1, 41.1. 1. / I '.1 / 0°I/ I4'

2

Number of games

1 plays 5

2 plays 4 (Since already played 1)

3 plays 3 (Since already played l& 2)

4 plays 2 (Since already played 1,2, &I)

5 plays 1 (Since already played 1,2,3,4)

15 Games

3. Act It Out

People will assume a team name or number, then "play games" with each other until all havePlayed each other once.

6 6


If you choose one food fromeach category on the menu,how many different combina-tions are possible?

(12 combinations)

(


Salad Main Course Dessert

\.

Ranch Dressing

French Dressing

Turkey

Beef

Ham

Ice Cream

Cake

Possible Strategies:1. (Tree) Diagram

Dressing Main Course Dessert

1 I II/C

T

it-"'..°6.a...icR H I

C

T I,..,...vC

B '' IH Cc..c.....ci

C

67'

(12 combinations)

2. Organized List

Dressing Main Course Dessert

Ranch Turkey Ice Cream

Ranch Turkey Cake

Ranch Beef Ice Cream

Ranch Beef Cake

Ranch Ham Ice Cream

Ranch Ham Cake

French Turkey Ice Cream

French Turkey Cake

French Beef Ice Cream

French Beef Cake

French Ham Ice Cream

French Ham Cake

(12 combinations)

6 8

=MO


A toy manufacturer makes carsand motorcycles. The supplierscan only send 22 headlights and44 tires. How many possiblecars and motorcycles can bemanufactured?Extension: How many combi-nations are possible?

(There are 12 combinations)

Possible Strategies:


1. Create a Chart

1

Cars = 2 Headlights and 4 TiresMotorcycles = 1 iicadlight and 2 Tires

Headlights Tires

0 Cars 0 022 Motorcycles 11 44

2 1 Car 2 420 Motorcycles 20 40

3 2 Cars 4 8

18 Motorcycles 18 36

4 3 Cars 6 1/16 Motorcycles 16 31

5 4 Cars11 Motorcycles

6 5 Cars14 Motorcycles

7-12Etc. (A total of 12 Combinations)

65

Other Possible Strategies:

2. Guess & Check

3. Create an Oganized List

4. Find a Pattern

5. Algebraic:

2 C + 1 M = 224 C+ 2 M= 44

Note: The algebra does notsolve the problem in a simplestep. Even algebra requires-

some additional work!

70

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41

POLYA'S CONCEPTUAL MODEL


.... ...

Devise a plan

r

L

3. Carry out the plan

Look Back 4.

1


SOLVING

1. Make sure the children enderstandthe probima.

2. Create en atmosphere where *or king enthe problem I. mere important thangetting the right answer.

3.

1

1 4.

1

I

Encourage the children to talk with eachether and us work toaethec

Lead a post-solution distuation in whkhthe students Mere their methods.


74

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EFFECTIVE MATHEMATICS TEACHING(What Does Research Say?)

Class Organization:

The more effective teachers devote about half of each class to a combination of explaining, demon-strating, and discussing. The less effective teachers use only about a fourth of each class for these ac-tivities and over half of the class for individual seatwort

Questioning:

The backbone for the teaching of problem solving is consistently asking questions that involvestudents and invite them to think.

Effective teachers ask more process questions (calling for explanations) and more product ques-tions (calling for short answers) than do less effective teachers. They also ask more new questionsafter correct answers have been given.

Effective teachers also allow more wait-time between questions and answers both the pause fol-lowing a teacher's question and the pause following a student's response.

Encouragement:

Effective teachers try to keep most of their students, low achievers as well as high achievers, fromslipping into passive learning. Effective teachers are more encouraging and receptive to student inputthan are their less effective colleagues. Effective teachers reinforce questions and requests for help.

Modeling:

More effective teachers engage in more problem-solving behavior. The teachers show the waythrough their own examples. Problem-solving instruction is most effective when students sense twothings:

1. that the teacher regards problem solving as an important activity;

2. that the teacher actively engages in solving problems as part of mathematics instruction.

Source: Research Within Reach: Secondary School Mathematics (NCTM, 1982)

H-6

76

IMPROVING PROBLEM SOLVING IN YOUR CLASSROOM

MYTHS TO OVERCOME:

We can't do that! These students don't know their basics yet!

These students can't think abstractly.

There is not enough time.

If it is not in the textbook, it isn't important.

POSITIVE STEPS:

Approach problem solving as a process, not as a series of specific skills or tricks to learnand apply.

Select problems that interest students.

Approach problems as "Here is a situation, let's think about it," rather than, "Here is aproblem, let's find the answer."

Take time with problem solving. Do not hurry the students into finding the answer.

Act as a "facilitator" of problem Solving. Careful modeling, questioning, and leading ismore effective than telling.

Encourage students to work together in problem-solving activities. Students should sharetheir thoughts.

Be patient. Be accepting of different approaches or methods. Encourage your students totell how they approached a problem, not just the answer.

Take time for the post-solution discussion. It is a very important part of the problem-solving process.

Create a successful problem-solving atmosphere. Problem-solving ability is bestimproved by solving many problems in an atmosphere of exploring and sharing ideas.

H-7

77

SELF-ASSESSMENT QUESTIONS

for teachers o f problem solvingRate each item using the following number s ystem

1= never 2=rarely 3= sometimes 4=often 5=always

I ask students to explain why they chose a particular method.

I ask students for more details after they have given a specificanswer

I ask students to give details of their reasoning process when theyanswer orally.

I give students the opportunity to explore a variety of methodswhen solving problems.

I allow students to explore and discover methods that work bestfor them.

I demonstrate a variety of strategies for solving problems.

I discuss with students how to choose the best strategy for, solvingproblems.

I show students how to solve a single problem using more than onemethod.

_ I explain the reasons for choosing a certain procedure ora roach when discussin problems with students.I compare and/or contrast methods when discussing the circum-stances under which a particular method is appropriate.

I allow students to listen to the thinking processes I use when Isolve problems.

I reward students for correct methods and good ideas even whenthe numerical answer is not correct.

H8

78

STANDARD TEXTBOOK PROBLEMS(Word/Story Problems)

Characteristics:

Follow development of an arithmetic operation or other algorithm

Can be solved by direct application of one or more previously learnedalgorithms

Task is to identify which operation(s) or algorithm(s) to use

Emphasis on getting the correct answer

Purposes:

Improve the recall of basic facts

Strengthen skills with the fundamental operations and algorithms

Reinforce the relationship between the operations and their applicationsin real-world situations

Improving.' Students' and Teachers' Attitudes Toward Story Problems:

Many teachers do not feel very successful in teaching story problems and many students find storyproblems one of the more difficult challenges in mathematics and, therefore, do not like them. Success leadsto positive attitudes, and so we must begin with success. Throw out any consideration of grade level andstart with very easy problems. Display or write simple problems that all the children in your group can solve.

Since long sets can be distasteful, give short sets two, three, or perhaps four problems are plenty!However, give these sets of exercises three or four times each week.

The treatment for changing students attitudes from negative to positive is to provide very frequent shortsets of problems on which the students experience absolute success.

And what about you, the teacher? Who will help you with your attitude toward story problems? Don'tworry about it. As you see your students succeeding in solving story problems and perhaps even liking them,your own attitude will improve; you will find it interesting and exciting to teach problem solving.

Source: Problem Solving in School Mathematics (NCTM, 1980 Yearbook)

H-9

79

NON-ROUTINE PROBLEMS

Non-routine problems are questions that cannot be answered with knowledgeimmediately available to the student.

Problem solvinq is the process by which the student resolves an unfamiliar situ-ation or non-routine problem.

PROCESS PROBLEMS AS THE VEHICLEFOR PROBLEM SOLVING

Process Problems:

Require use of strategies

Stress theot obtaining the solution rather than the solutionitself

Frequently have more than one answer

Are used to encourage thl development and practice of problem-solving strategies

Provide opportunity for cooperative learning

Build student confidence in problem solving

GOOD PROBLEMS USUALLY HAVE ONE OR M^REOF THESE CHARIMERISTICS:

Are in a context that interests childrenCannot be solved directly with a computational algorithm(at least, not one known to the children)Have multi* solutionsRequire finding several preliminary answers to get the final solution

Source. Problem Solving in School Mathematics (NCTM 1980 Yearbook)

H-10

go

POLYA'S CONCEPTUAL MODELFOR PROBLEM SOLVING

Ei Lit Understand The Problem

What do we know for sure?Anything else?What are we trying to find out?Is there any information we don't understand?Is there any information we don't need?Any guesses or estimates of what the solution might be?

Second: Devise A Plan

Does the problem remind you of any we have solved before?Would any of the following strategies help?

sketch or diagramlist, chart, or tableconcrete or manipulative aidstrial and error, guess and checklooking for pattemssimpler numberssimulation or experimentation, act it out

Can we determine what operation(s) we might use?

nal: Carry Out The Plan

(Do not overemphasize this step. Emphasize the process of problem solving and differentways of solving the problem.)

Use the plan you devised.

Fourth: Look Back/Extend

(Taking the time to look back is very important but often overlooked, especially when"getting the answer at any cost" is 'he approach )

Is your answer reasonable?Is it close to your estimate?Does it answer the question as stated in the problem?Did anyone E olve the problem another way?Could there to (Jther answers?Extend the problem:

ask 'vohat if..." questionsmake up other problems similar to this one

H-11

81

USING POLYA'S MODEL"The Dinner Party Problem"

Next month I am hosting a dinner party for thirty (30) people.1 want to borrow only enough card tables, the size that seatone person on each side, to arrange them in a long rowend-to-end and still seat all my guests. What is the fewestnumber of tables I should borrow?

Understanding the Problem

What do we know?

What are we trying to find out?

Devising a Plan

How could we go about solving this problem?

What kind(s) of sketch(s) wol:ld help us?

How could a partial sketch aid us?

Carrying out the Plan

What does our sketch(s) look like?

How many tables did we need in our solution(s)?

Looking Back

How can we be sure our solutions are accurate?

In what other way could we have solved the problem?

What if more people were invited? 42? 100? Fewer?

What if the tables were arranged differently?In rows? In a U-shape? In an L? In a square?

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82

VISITING FRIENDS .. .

Five of my friends ..ci I live in the sane neighborhood, which has a lot of short streets.

All of us have strict parents who only allow us to visit friends who live on the same streetas us. Yesterday we all sat together in the school cafeteria and figured out who couldvisit who.

Beth and Carol can visit.Eva and Faye can visit.Deb and Eva can visit.Beth and Deb can visit.Deb and Faye can visit.Carol and Deb can visit.Carol and Faye can visit.Ann and Beth can visit.

Place each of uc at our correct home.

First St Second St. Third St. Fourth St.

Fifth St Sixth St.

AMIIM...=1

5%

Fifth St.

sbo"4'

Seventh St. Eighth St.

( IThird St. 4

83

H13

A

GETTING STUDENTS INVOLVEDIN PROBLEM DEVELOPMENT

Problems without numbers

[Bon Jovi's latest release is availab,e in both cassette tape and LP.How much could I save tii buying the album rather than the cassette?

QuestionsWhich costs more?What would you need to know to solve this problem?How would you solve the problem?

ActivitiesStudents put in their own pricesand solve the problem.

Extend the problem by addinginformation such as, "I have a!t 20 bill..." or "Why do pricesvary by label and artist?

Problems without questions

Beverly and Sheila go to the store. Beverly has $5.80 andlucky Sheila has $6.20. T ,eir intent is to buy a frisbee thatcosts $9.20.

Questions

Do the two have enough money?If Sheila uses all her money, how much will Beverly pay?What if Beverly uses all her money?If each pays the same amount, how much would each pay?Could they buy two frisbees?How much more money would they need to buy two?

H-14

84

GETTING STUDENTS INVOLVEDIN PROBLEM DEVELOPMENT

SCOREBOARD

HOME

QUARTER

VISITORS

ElTIME

=pgrFmmappimpFwimmipm...miimmeepuepuipmft,

Questions

Is the game over?How many total points were scored?If a quarter is 8 minutes long, how many minuteshave been played in the game so far?

Adapted from: "Problem Solving in School Mathematics," NCTM1980 Yearbook.

R5H-15

TARGET PROBLEMS

Both problems can be adapted by changingthe numbers of darts thrown, point values,complexity of the target, and questions asked.

If 4 darts are thrownand all hit the target,which of these scoresare possible?

100 28 2

17 36 93

If you throw two darts,what are all of yourpossible scores?

Adapted from: Jesse A. Ruanck, presentation, Minneapofis, 1987.

H-15

ISC

EASILY ADAPTED PROBLEMS

I have 5 pencils and want to give them to Chris, Joe, and Willie. How many ways can I give out the pencilsif each boy gets at least one pencil?

There are 7 people at a party. If each person shakes hands with everyone else, how many handshakes arethere in air

There were 10 handshakes performed at a party. You know each person shook everyone else's hand once.How many people were at the party?

At Tom's school there are 6 basketball teams. They want to plan an afternoon tournament so that each teamwill play every other team once. How many games must be played?

Some children are seated at a large table. They pass around a box of candy containing 25 pieces. Ted takesthe first piece. Each child takes one piece of candy as the box is passed around. Ted also gets the last pieceof candy, and he may have more than the first and last pieces. How many children could be seated aroundthe table?

Source: Problem Solving in School Mathematics (NCTM, 1980 Yearbook)

lio,mmaiicicici.i.q=fiaiaa.Laag=ommagica

A farmer has some pigs and some chickens. He sent his son and daughter to count how many of each hehad. "I counted 70 heads," said his son. "And I counted 200 legs," said his daughter. How many pigs andhow many chickens does the farmer actually have? (For younger children, use smaller numbers, perhaps6 heads and 16 legs.)

Last night I watched a little league baseball game. I noticed some boys and dogs playing in the grass. I

heard a noise and turned to see the boys and dogs running past me. I decided to count them in a differentway. I counted the legs and found there were 20. Now, what I want to know is how many boys and howmany dogs ran past me?

This problem . as a number of solutions. In looking back, you maywant to add a condition such as "I counted4 tails," or "I counted 8 heads." This will make a single solution.

Source: Adapted from a presentation by Jesse A. Rudnick, Temple Unwersrty, 1987.

H-17

87

I I have nickels and dimes in my pocket that total 30 cents.I have have 4 coins in all.How many nickels and how many dimes do I have?

MMMMMMMMMMMMMMWHAT IF . . .

I had 5 coins?

I had 7 coins totaling 60 centsand

there are nickels, dimes and quarters??

$8 Niwmwtak

$9

$4

What is the total cost of the five items?What 2 items could you buy for $13?What 3 items could you buy for $13?Could you buy 4 items for exactly $13?If you spend $15, what could you buy?

EXTEND the problem - - -

Students develop their own problemsfrom !he r,(lureStudents., develop original problemsand illustrations.Share either in small group or class-wide.

Adapted from: John F. Firkins, presentation, Gonzaga University, 1987.

H-18

88

cna-AsLE

TD©h]

Place the digits I through 9 in the small triangles so thatthe sums of the four digits in the three medium trianglesADF, BGI, and CHJ are the same.

V small

1'Po! sible sums:

17 19 20

21 23medium

Adapted from: Mathematics Teacher,, May /989.

H-19

89

FIND

THE

AVERAGE

Five students have test scores of 62, 75 , 80, 86, and 92.

Find the average.

How much is the average score increased IF

each student's score is increased by --

1 point 5 points

7 a01: ,0# : : 0/0"-

8 points n points

How much is the average score increased IFeach individual score is increased n points?

Convince another student that the statement is true.

Note: Some students may need more than three trials to make a generaliza;ion.U;ing a calculator or computer to explore various averages and their changeswill allow students to use more or greater numbers.

Adapted from: CU ffiCUIJITI and Evaluation Standards, NCTM,1989.

H-20

90

Problam 5oluingTip) For Taachar)

AN AGENDA FOR ACTION

Strategy Spotlight

30

Organizing the Classroomfor Problem Solving

A growing body of research points to the benefits ofhaving students learn in small cooperative groups.When students work in cooperative groups, the ac-tive participation of each student is maximized. Morestudents have the chance to speak than in whole-class discussions, resulting in more opportunities forstudents to clarify their thinking ,eIso many studentsfeel more comfortable in small-groue. "Ts and aretherefore more willing to explain their ideas, specu-late, question, and respond to the ideas of others. Insmall cooperative groups, students' opportunities tolearn with understanding are supported andenhanced.

Choosing Problems Suitable forCooperative Groups1. Choose problems for which a collatwxative effortwill benefit students, both in sharing ideas and in ac-complishing the ink.

2. Select problems that allow for differentapproaches.

3. Se sure students understand both the problemand how they are to present their results.

4. When appropriate, have studentv post theirfindings for the class.

5. Allow for discussion time to summarize, during

Edited by him A. Vas de WelkVirginia Commonwealth UniversityRichmond, VA 2328440)1

Prepared by Marilyn SoresMarilyn Burns Education AssociatesSausalito, CA 94965

which groups present their findings and respond toother groups' results.

lin Early Grade Example "Making Change"ioung children need opportunities foc continuedpractice both with basic facts and with money. "Ma-king change" is a problem-solving activity that offersboth kinds of practice and also lends itself to cooper-ative group work.

Each group of students works to find all the waysthey could be (Pen 80.50. For example, they couldreceive two qualers, or one quarter and five nickels,and so on. They make a group record of their find-ings to present to the class. During a follow-up dis-cussion, they also need to tell why they think they'vefound all the ways.

An Example for Older Chlldren"Number Bracelets""Number bracelets" appeared in the May 1980 issueof the Mthmetic Teacher. In this activity, studentslook for patterns while practicing basic addition facts.Several benefits emerge as students tackle this prob-lem in groups. They share the task, thus making the

91 Arithmetic Teacher

work manageable, and they are able to check theirconjectures and evidence with each other.

Students start by writing any two numbers from 0to 9 and applying the following procedure: Add thetwo numbers and record just the digit that appears inthe ones place in the sum. For example, If you startwith "8 9," then the number that comes next is "7."Then add the last two numbers, the "9" and the "7,"and record "6 " Continue in this way, "8 9 7 6 3 92 ... ," until the pattern begins to repeat.

0 Tip Board0

0

aa

Each group of students is asked to answer the fol-lowing questions:

1. How many different possible pairs of numberscan you use to start?

2. What is the shortest bracelet you can find?3. What is the longest bracelet?4. Investigate the odd-even patterns in all your

bracelets.

5. Make up one more question and try to answerit.

_111 Guidelines for Coopt. itive Group MembersThree rules are useful when students work il cooper-ative groups. The rules need to be explained to thestudents and discussed, as they are only as usefulas they are understood and practiced.

1. You are responsible for your own work and be-havior.

2. You must be willing to help any group memberwho asks.

3. You may ask the teacher for help only when ev-eryone in your group has the same question.

The third rule often puts the greatest demand onteachers when first implementing cooperative groups.Children typically ask for individual help. Asking chil-dren to check with their group, rather than givingthem help at that time, is not a usual teacherresponse. However, it is an invaluable response forencouraging students to become more independentand to rely on each other. Assure students that youwill come and discuss whatever problem the entiregroup faces.

1111 NoviLarge Should WOOS Be?

.

A cooperativegroup requires

no Magic numberoi

children to work. At some times, studentswork best

in pairs, althoughgroups

of three to six stuientsare

successfulin other situations.

What is importantis

that groupsbe small

gh for all studentsto par

enou

-

tcipate.

IHow to Group StudentsIt is importantfor students

to be willpig to work andlearn with all their classmates.

Grouping studentsrandomly accomplishes

this objective.Students

can move their desks into ch _;ters offour each. The teacher labels each cluster withthe number of a playing card--ace,

two, three,and so on. The correspondingcards are shuffled

and distributed;children who hold aces go to the

ace cluster, childrenwith twos to the cluster la-

beled twi.), and so forth.Numbered

slips of paper can be drawn from athree or four,

"hat" to determine random groupings of either

811A CautionSeating

studentsin small groups

does not rnagicplly

produceinstantly

successfulcooperative

groupvvork

Practice,

and scssionare

required,but m

it is well worth the

III ReferenceTrotter, Terrel, Jr., and Mark D. Myers. "NumberBracelets: A Study in Patterns." Arithmetic Teacher27 IMay 1980):14-17.

kPart of the Tici Soled is reserved for techniques that you've found useful in teaching probiern solving In your class Send your dem lo theedttor ot the section

May 198892 31

Desired Outcome for Proble -Solving -

Definition

A desired outcome is a goal statementor measurable objective which focuseson what children will learn and accom-plish as a result of their participation inthe Chapter 1 program.

Goals

* Demonstrate a willingness toengage in problem solving.

* Be able to identify the importantinformation in a problem statement.

* Be able to use the followingstrategies:

choose the operafion,draw a picture,use objects,guess and check, andlook for a pattern.

Desired Outcome

80% of Chapter 1 mathematics studentswill show continuous growth in the threeproblem-solving goal areas as measuredby a Problem-Solving Performance RatingScale completed during September,January, and May.

Sample Rating Scales

S ales and their uses (as well as othermethods of evaluating more advancedmathematical skills) can be found inHow to Evaluate Progress in ProblemSolving (1987) available 4-om:

National Council of Teachersof Mathematics (NCTM)1906 Association DriveReston, Virginia 22091(703) 620 9840

Student:

4>"

Shows a willingness to try problemsDemonstrates self-confidenceSelects all important informationUses strategies appropriately

choose the operationdraw a picturelook for a patternguess and checkuses objects

Date:

frequently sometimes never

H-23

93

Additional Sources

Bazik, E.; Bolster, C.; Brannan, R.; Butts, T; Fair, J.; Nibbelink, W. Robitaille, D.;Schultz, J.; Shepardson, R.; Souviney, R.; Tucker, B.; & Wisner, R. (1985). ProblemSolving Experiences in Mathematics Grades 1-8. Addison-Wesley Publishing Co.

Burns, M. (1976). The Book of Think. Little Brown & Co.

Charles, R.; Garner, D.; Lester, F.; Martin, L.; Mason, R.; Moffat E.; Nofsinger, J.; &White, C. (1985). Invita!ion to Mathematics: Probk;m Solving Sourcebook (Grades 1-8).Scott, Foresman & Co.

Downie, D.; Slesnick, T.; & Kerr Stenmark, J. (1981). Math for Girls and Other ProblemSolvers. Math/Science Network, Lawrence Hall of Science, University of California-Berkeley.

Driscoll, M. (1982). Research Withfn Reach: Elementary School Mathematics. TheNational Council of Teachers of Mathematics.

Driscoll, M. (1982). Research Within Reach: Secondary School Mathematics. The Na-tional Council of Teachers of Mathematics.

Krulik, S. & Rudnick, J. (1987). Problem Solving: A Handbook for Teachers. Allyn andBacon, Inc.

Nelson, D. & Worth, J. (1983). How to Choose and Create Good Problems for PrimaryChildren. The National Council of Teachers of Mathematics.

Problem Solving in School Mathematics : NCTMI980 Yearbook. The National Councilof Teachers of Mathematics.

Raths. L.E.; Wasserman, S.; Jonas, A.; & Rothstein, A. (1986). Teaching for Thinking:Theory, Strategies, & Activities for the classroom. Columbia Teachers College Press.

Souviney, R. (1981). Solving Problems Kids Care About.. Scott, Foresman & Co.

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ed 327 391 title mathematics problem solving: a more ... · chapter 1 mathematics progra, as. this...

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