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Non-Routine Word Problems: One Part of a Problem-SolvingProgram in the Elementary School

Peter Kloosterman Department of Curriculum and InstructionIndiana UniversityBloomington, Indiana 47405

It’s Sunday evening and Mr. Spector is thinking aboutteaching on Monday morning. He has been trying to put morechallenge in all of his mathematics lessons and now wants tohave a lesson involving word problems that require some realthinking. Where can he find some good problems? How canthis lesson tie into students* previous experiences with wordproblems? How should he introduce challenging problems andhow will he know if they are too easy or too hard? Andfinally, how is he going to have the lesson ready to teach byMonday morning?

Unfortunately, the Sunday evening panic is not unique toMr. Spector. TheNational Council ofTeachers ofMathematics(NCTM)recommendsteaching all mathematics usingaproblem-solving orientation (NCTM, 1989) but this is not easy to do. Inan effort to get students to be better problem solvers, manyteachers have good intentions about using non-routine wordproblems as one part of their problem-solving instruction;however, when they realize how much time it will take togenerate a good lesson involving non-routine word problems,they decide to put off the lesson for yet another week. Can thehurdle of having to do extensive planning for every lesson onnon-routine problems be overcome? Like Mr. Spector, mostteachers make their lessons more problem-solving orientedwhenever they can. Unfortunately, theyjustdon’t have the timeto do extensive planning for every new lesson.

Students’ textbooks usually contain a number of routine 1-and 2-step word problems. Unfortunately, most contain onlylimited examples of non-routine problems; that is, problemswhich cannot easily be solved by choosing an arithmeticoperation and applying it once or twice to the appropriatenumbers in the problem. One possibility for making instructionusing non-routine word problems easier is to use commerciallyavailablesupplementalproblem-solving materials. To determinethe feasibility of this option, supplemental materials were usedto teach lessons involving non-routine word problems on aregular basis. The primary objectives of the instruction were toget students to be better problem solvers and to see if lessons onnon-routinewordproblems could beprepared with little, if any,

Theprojectdescribed herewas supported by GrantTEl 8751478from the National Science Foundation. Opinions, conclusions,and recommendations expressed here are those of the authorand do not necessarily reflect the views of the Foundation.

more preparation time than is needed for traditional mathematicslessons. Overall, planning and teaching lessons involving non-routine problems did not turn out to be the labor intensiveendeavor that Mr. Spector envisioned. The purpose of thisarticle is to share insights about problem-solving instructionwith others who would like to teach using non-routineproblemsbut, like Mr. Spector, do not have the time to do extensive lessonplanning.

As background, it should be pointed out that the teachingdescribed in this article was done by a university instructorrather than regularclassroom teachers. In a teaching experiment,the instructor took over two fourth-grade classes about twice amonth for one school year. This instruction was a supplementto, rather than a replacement for, the regular mathematicscurriculum which was taught by the classroom teachers. Theschool in which the instruction took place served a workingclass area. Ability levels of students in the school were quitevaried, although the majorityofstudents in the classes describedin this article were relatively low achievers. Thecomments andinsights that follow are based on the university instructor’sexperiences teaching these students.

Materials

A variety of supplemental materials are available to aid inpreparing lessons using non-routine word problems. Mostinclude a variety ofchallenging problems and questions such asthose suggested by Charles and Lester (1982) and Cemen(1989). Make it Simpler (Meyer&S^lee, 1983) contains manythought-provoking problems for the upper elementary andmiddle school levels. The Lane County Mathematics Projectbooks (Problem Solving in Mathematics, 1983) have manyproblem-solving oriented mathematics activities, some ofwhich make excellent word problems for grades four througheight. Reasoning with Rabbit Counters (Goodnow &Hoogeboom, 1989) provides problems that can be modelledwithcounters for theprimary grades. TheCreativePublications*Problem Solver (1987) series, the Scott Foresman Problem-Solving Sourcebooks (1987), and the Addison-WesleyProblem-SolvingExperiencesinMathematics(19S5) all provideproblem books for grades one though eight. In addition toproblems, most of these materials provide suggestions fororganizing and teaching problem solving and sample questions

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a teacher can ask of students when presenting a problem. All ofthese materials are intended to help teachers teach problemsolving well without spending countless hours preparing eachweek’s lessons.

After considering each of these sources, the fourth-gradeversion of the teacher sourcebook and blackline masters forProblem-Solving Experiences in Mathematics (1985) was usedas the primary reference for the lessons. In addition to teachingstrategies, the book contains 30 sets of problem-solvingactivities. Each setcontains routine problems in the form ofskillactivities and 1-step wordproblems (see Figure 1). Each set alsocontains one non-routine multiple step and two non-routineprocess problems (see Figure 1). Like the other sources noted,this program is designed as a supplement to any mathematicstextbook and, theoretically, can be used with about the sameamount of preparation as would be needed for a regularmathematics lesson.

Figure 1. Sample problems.

Skill Activities1. Identify the extra information in this problem:

Monica played 6 games and scored a total of 67 points. Bridgetplayed 5 games and scored a total of45 points. How many morepoints did Monica score than Bridget?

2. Write a story problem that can you can solve using thenumber sentence 24 x 5 = ?

1-Step Problems1. 86 boys and 114 girls attended the soccer game. How

many more girls than boys attended the game?2. A table is 4 feet long. How many inches long is the

table?

Multiple-Step Problems1. Mr. Lewis drives to and from work each day. He lives 6

miles from his office. If he works 4 days this week, what is histotal mileage driving to work and back this week?

2. Oranges are priced at 2 for 380. Apples are priced at 3 for450. How much would 6 oranges and 6 apples cost?

Process Problems1. Jeffrey asked his mom to tell him where the oatmeal

cookie recipe was in the cookbook. She said "The recipe is ontwo pages next to each other. When you add the two pagenumbers, the sum is 113." On what pages will Jeffrey find thecookie recipe?

2. Every ice cream sundae at the Dairy Delite has one flavorofice cream and one flavoroftopping. The choices for ice creamare chocolate, vanilla, and strawberry. The choices for toppingare fudge, blucberry, and butterscotch. For example, you canorder a sundae with vanilla ice cream and butterscotch topping.How many different sundaes can be ordered?

Using the Materials

One set of problems (a skill activity, 1-step problem,multiple-step problem, and two process problems) is designedto be used over a 1-week period. Because the universityinstructor was in each classroom only about twice a month,most of a problem set was covered each day. On some days. askill activity was first, followedby a 1-step problem. Althoughthese routine problems were intended to help students buildbasic problem-solving skills, they were also good confidencebuilders. The students were given one problem at a time in aneffort to overcome their desires to rush through problems asfast as they could. When they only had one problem to workon, there was no incentive to put down a poorly thought outanswerbecause they could not moveahead. Although studentscould usually complete the skill and 1-step problems fairlyeasily, one or two similar problems were always available ifit appeared more practice was needed. For example, whenstudents had trouble with the first 1-step problem shown inFigure 1, the problem, "There were 108 dogs and 67 cats in theanimal show. How many more dogs than cats were in theshow?" was written on the chalkboard. On some days, two orthree additional examples were needed before all childrenappeared to understand.

Although 1-stepproblemsare important, mostofeach classperiod was spent having the students work on multiple-stepand process problems because these are the problems thatrequire the most thinking and are the ones on which studentshave the most difficulty (Dossey, Mullis, Lindquist, &Chambers. 1989). After a problem was passed out, a studentwould read thatproblem out loud. Although this was the fourthgrade, some ofthe students in the class were reading at a secondgrade level and having the problem read out loud ensured theyall knew what the problem said. The students then respondedto a series of questions for understanding to make sure theyknew what the words in the problem meant, what the problemwas asking, what extraneous information was in the problem,etc. Figure2 shows sample questions for understanding for theproblems given in Figure 1. To add variation, students in thetwo classes were sometimes asked to generate their ownquestions for understanding. Note that some of the problembooks contain examples of questions for understanding thatprovide a good basis from which to start. In most cases,however, sample questions needed to be supplemented withothers. By thinking through the problems ahead of time, thiswas easy to do.

Once most students understood a problem, additionalquestions were introduced to help students plan a solution.Cemen (1989) calls these "questions forgetting started." Sampleplanning/getting started questions are shown in Figure 3. Onsome problems, there was no need to use planning questionsbecause students in the class obviously all had good ideas forproceeding. Note that even though it is possible to distinguishbetween understanding and planning/getting started questions,

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many questions fall into both categories. Thinking about theexact type of question being asked sometimes makes writingquestions easier; however, the exact type of question is not asimportant as knowing that every question is designed to guidestudents to an understanding of the problem and its solution.

Figure 2. Questions/or understanding for the multiple-stepand process problems in Figure 1.

Multiple-Step 11. Does Mr. Lewis drive the same distance to and from

work?2. Do we need to know the speed at which Mr. Lewis drives

to answer the problem?

Multiple-Step 21. What is the price of one orange?2. What is the price of one apple?

Process 11. What does it mean when we say the sum is 113?2. Could the page numbers be 100 and 13?

Process 21. Is strawberry ice cream with chocolate topping a

possibility?2. Can you have two flavors of ice cream in a Dairy Delite

sundae?

Figure 3. Questions for planning/getting startedfor themultiple-step and process problems shown inFigure 1.

Multiple-Step 11. How many miles will Mr. Lewis drive in one day?2. How many miles would Mr. Lewis drive in two days?

Multiple-Step 21. What is the price of four oranges?2. What is the price of six oranges?

Process 11. Will the page numbers have to be more than 50?2. Try using 74 and 75 as the numbers. What do you know

from trying them?

Process 21. Besides butterscotch, what toppings can you have with

vanilla ice cream?2. Can you list the sundaes that can be made with chocolate

ice cream?

Once students came up with answers to the problem theywere given, they wanted to know immediately if their answers

were correct. Rather than telling them, the instructor askedstudents to explain verbally why they believed their answerswere correct. Because the students were used to gettingimmediate feedback on the correctness of their mathematicsproblems, they often complained when they were not told whatthe correct answer was. After a few class periods, however,they stopped taking wild guesses and stopped asking if theiranswers were correct until they could give a good reason whythey felt they were. Thus, the tactic offailing to give immediatefeedback, while initially frustrating to the students, eventuallymade them much more reflective about the answers they weregetting for problems.

Several oftheproblem books suggest thatprobicm solutionsbe discussed once a problem is completed. When planning thelessons, mental or written notes about follow-up questionswere made to guide this discussion. One personal guideline forthese questions was to makea minorchange in thecontext oftheproblem or in the numbers in the problem. Sample follow-upquestions arcshown in Figure4. On someoccasions, discussionleading up to a problem’s solution made it fairly obvious thatstudents did understand their answers and thus follow-upquestions were not necessary.

Figure 4. Follow-up questions for multiple-step and processproblems shown in Figure 1.

Multiple-Step 11. How many miles would Mr. Lewis drive in five days?2. Suppose Mr. Lewis went to the grocery store on the way

home on the fourth day. This added two miles to his trip. Howmany miles would he drive in four days?

Multiple-Step 21. Do you have to find the price of one orange to solve this

problem?2. Suppose, instead of the six oranges, you bought 3

grapefruit that cost 380 each. What is the total cost?

Process 11. Suppose the sum was 115. What would thepage numbers

be?2. Could the page numbers add up to 114? Why or why not?

Process 21. Suppose the Dairy Delite started offering mint ice cream

instead of chocolate? How many different sundaes could beordered?

2. On Friday, the Dairy Delite ran out of fudge topping.How many sundae choices were lost?

After students have completed a multi-step or processproblem, the usual next step is to givethem aproblem extension.An extension is a new problem that can be solved using aprocess which is similar to the original problem (Charles &

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Lester; 1982). Problem books often provideproblem extensionswhich were used for many of the problems. Figure 5 givessample extensions for the multiple-step and process problemsshown in Figure 1. Students were generally not given a problemextension until the problem on which it was based had beencompleted and discussed by the whole class. By doing this, itwas possible to use questions for understanding and planning/getting started for the extensions. On someoccasions, however,a few students would finish long before the others. If the earlyfinishers appeared to understand their solutions, they wereallowed to go on with the extensions with the agreement thatthey would have to ask questions of each other rather than theteacher until all students in the class completed the initialproblem.

Figure 5. Problem extensions for the multiple-step andprocess problems shown in Figure 1.

Multiple-Step 1During January, Mr. Lewis went to work 4 days the firstweek, 5 days the second week, 5 days the third week, andonly 2 days the fourth week. How many miles did hedrive to and from work in January?

Multiple-Step 2Jenna and her friend bought 4 pears that were priced at 2for 290 and 4 peaches that were priced at 2 for 350. Howmuch did they each pay if they split the cost evenly?

Process Problem 1Andrew said *Tm thinking of three numbers whose sumis 72. The second number is 10 more than the first andthe third is 10 more than the second." What numbers wasAndrew thinking of?

Process Problem 2On the Fourth of July, the Dairy Delite was giving two-scoop ice cream cones for the price of a single scoopcone. The flavors were vanilla, chocolate, strawberry,and cherry. If you can get two scoops of the same flavoror two scoops of different flavors, how many choices doyou have for a two-scoop cone?

A major assumption behind assigning problem extensions isthat once students have completed and discussed a problem,they understand their solutions well enough to move on tosomething a little bit harder. Because so many questions wereasked duringthesolutionandfollow-upphasesofmostproblems,it was easy to determine how well a majority of the class wascomprehending. Many times they had grown frustrated andtired of the problem that the class had started with and thus werenot ready to move on to something harder. This was particularlytrue of the more difficult process problems. On these occasions,extensions which were very similar to the original problem were

given. Figure6 shows extensions for additional practice on theprocess problemsin Figure 1. There were somedays wheretheentire class period was spenton practice-oriented extensions ofthe same problem. While this took a lot of time, students didfinally understand the problem and were confident they coulddo problems similar to it. Thus, time spent with multipleexamples of the same type of problem was time well spent.

Figure 6. Problem extensions for additional practice onprocess problems shown in Figure 1.

Process Problem 11. When he finished baking the oatmeal cookies, Jeffrey

decided to make muffins. His father said "The recipe is on twopages next to each other and the sum ofthose pages is 159." Onwhat pages will Jeffrey find the muffin recipe?

2. Sarah’s teacher told her to open her math book so thatthe sum of the two pages she was looking at was 125. Whatpages was she looking at?

Process Problem 21. Next week, the Dairy Delite will alsohavechocolate chip

ice cream. How many choices for sundaes will be availablethen?

2. The Dairy Dream is across the street from the DairyDelite. The Dairy Dream has vanilla, cherry, and peach icecream. The toppings they have are blueberry, strawberry,boysenberry, and raspberry. If their sundaes also have onescoop of ice cream and one topping, how many differentsundaes can be ordered?

Insights

After a year of teaching lessons involving non-routineproblems, it would be unfair to claim that every lesson wassuccessful. There were days when it seemed that few of thestudents had understood theproblems thathad been workedon.On the whole, however, the lessons were a very worthwhileexperience for the students as well as for the instructor. Byusing published materials, the amount of time it took to planeach lesson was minimal in comparison to the amount it wouldhave taken to write new problems and questions. Specifically.for the first several lessons, it took close to an hour to workthrough the problems in each lesson and to write down a longlist of questions to ask. Planning time dropped to 10 to 20minutes per lesson as the year progressed.

The following are insights gained that should be useful toothers who want to use commercially produced problem-solving materials in a classroom setting. Many of thesesuggestions seem obvious now but they were things that werenot apparent when the year began. Many other teachers maynot have thought about them either.

1. Work through each of the problems when planning alesson. Most teachers can look at textbook word problems and

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know immediately how to do them. With process problems,this is not always the case. It only took a few minutes to gothrough each of the problems which were to be used in alesson, yet those few minutes gave insights into the types ofquestions students would have and into the types of questionsfor understanding, planning, and follow-up that would beappropriate. Having a good knowledge of the problems alsomade it easier to think of standard extensions and practiceextensions while planning and, when necessary, in the middleof a class.

2. Prepare lots of questions. The Professional StandardsforTeaching Mathematicsprepared byNCTM (1991) stress theimportanceofmathematical dialog in the classroom. Examplesof three types of questions were given in this article, notbecause the types are so important, butbecause questions are anexcellent vehicle for helping students understand challengingproblems. Some problem-solving books provide examples ofgood questions but there are usually not enough. It wasextremely difficult to predict how many questions would beneeded to get students started on a problem and thus it was bestto have lots of questions prepared. It was surprising how mucheasier it became to write good questions once it had been donea few times.

3. Select problems carefully. An assumption was made thatthe problems in the problem book were sequential andapproximately equal in difficulty. As a general rule, this wastrue; however, there were times when a multi-step problem wasmuch harder than a process problem in the same set. At othertimes, process problems were more difficult than theirextensions. Figure7 gives a process problem where this provedto be the case. Working through the problem, it seemed thatstarting on the middle rung, as stated in the original problem,would be harder for students than starting on the fourth rung, asstated in the extension. When this problem was used, theextension was done first, and the students found it to bechallenging but reasonable. Then they moved on to the originalproblem and found it to be very difficult. Ifthe original problemhad been used first, many would have been so frustrated thatthey would not have been able to attempt the extension eventhough it was easier. There were other instances where processproblems made little sense and thus they were omitted. Whenlooking for problems that required the use ofspecific strategies,problems were used in a different order than in the problembook. In brief, although the materials were an excellent sourceof problems, it was important to look over the problems aheadof time to make sure they were at an appropriate level ofdifficulty for the students.

Figure 7. Example where the original problem is moredifficult than the extension.

ProblemBen stood on the middle rung of a ladder when painting ahouse. He stepped up 3 rungs to paint one spot and thenstepped down 5 rungs to get a spot he had missed. Afterpainting that spot, he climbed up 7 rungs to finishpainting and then climbed the remaining 4 rungs to thetop of the ladder. How many rungs did the ladder have?

ExtensionIf Ben had started on the fourth rung of the ladder, howmany rungs would there be?

4. Be flexible in the strategies used to solve a problem. Forprocess problems, there are often several appropriate solutionstrategies. Most problem books provide hints or suggestedsolution strategies for each problem. Students, however,sometimes used other strategies that seemed more appropriate.When using commercially available materials, recommendedstrategies and questions for understanding and planning mustbe looked at as suggestions to be used when appropriate. Withexperience it becomes easier to take recommended solutiontechniques and questions and adapt them for a specific group ofstudents.

5. Be prepared to use manipulatives or pictures to explaina problem. In general, the more non-routine problems studentsare exposed to, the better they get at constructing their owndiagrams to help them find solutions. During the course of theyear, a number of coin problems (e.g.. How many ways canVanessa make change for a half dollar without using pennies?)were done. Some students were very good at making lists ofhow many nickels, dimes, and quarters they planned to use butothers had a very hard time. A few of the students in the classesneeded to use actual play money to find any combinations at allthat added to 500. Another problem involved a pattern ofboxesof different sizes. Some students actually made boxes usingpaper and tape to help visualize the problem.

6. In a class with a wide range of ability levels, it is notpossible to make every problem appropriate for every student.In most schools, the mathematical skills of the students varyconsiderably. In classes with a wide range of ability levels, itis difficult, if not impossible, to find problems that arechallenging but reasonable for all students. Extensions arealways available for students who are the first to complete aproblem but one must teach problem solving knowing that it is

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impossible to keep all students involved 100% ofthetime. Makeproductive use of class time but be aware that when teachingproblem solving, like teaching mostother things, there are timeswhen the better students will have to sit through explanationsthey do not really need and there are times the slower studentswill not comprehend as much as you would like.

7. Problems withmultiple ans\vers are goodfor getting totalclass involvement. At various times during the school year,process problems were used that required finding patterns,drawing diagrams, working backwards, etc. While the interestofthe students in a lesson was hard to determine, problems withmultiple solutions seemed to be best for getting whole classinvolvement. Weaker students knew they had a good chance offinding at least one answer even if they couldn’t find them all.On problems like Process Problem 2 (see Figure 1), students gotbetter at systematic guessing when they did notknow if they hadcorrectly identified all combinations. This prompted the betterstudents to really think about their answers.

8. Communicate the difficulty of problems to students.Unfortunately, some students interpret making a lot ofmistakeson mutiple-step and process problems as indicative of lack ofmathematical ability. Students need to know when a problem ishard so that they realize that notknowing how to do theproblemcould be due to the problem, not to themselves (Kloosterman &German, 1990). It is important for students to be challenged butit is also important that they know that non-routine problems areoften difficult, and thus making mistakes on a problem is part ofthe learning process. Making mistakes does not imply that oneis incapable of doing mathematics.

9. Cooperative groupings are useful but not essential forteaching problem solving. As a general rule, cooperativegroupings area good organizational system for teaching problemsolving (Johnson & Johnson, 1989); however, it can take anumber ofclass periods for students to leam to work together incooperativegroups. Ifstudents areused to working cooperatively,then have them solve non-routine problems working in groups.If they are not, do not spend all of your problem-solving timeteaching them to be good group workers. Good problem-solving materials can be used for both cooperative group andwhole-class instruction. It was difficult to predict which daysstudents would be congenial enough to work cooperatively andwhich days they were better off working alone or in pairs. Onseveral occasions, they started work in groups but were told toreturn to individual work when the groups failed to functioneffectively.

10. There is a limit to the time that should be spent on anygiven problem. Persistence is usually a virtue in problemsolving and students were encouraged to keep working untilthey solved theproblem they were given. There were instances,however, where it was clear the frustration level ofa majority ofthe class was so high that it was pointless to continue. In thesecases, a very different type ofproblem was introduced. Teachersshould not feel guilty if a well-planned part of a lesson turns outto be too difficult.

11. Published grade levels are approximate for problem-solving materials. While specific grade levels are often givenfor problem-solving materials, many students have not hadextensive experience at problem solving, and thus it issometimes appropriate to begin instruction using non-routineproblems with materials that are a grade level below the levelyou are teaching. In general, the types of problems presentedand the strategies taught (guess and check, make a table, etc.)are similar across grade levels for a given publisher. Manythird-grade problems were found to be more difficult thanfourth-grade problems. Teachers are the best judges of whichproblems are appropriate for their students.

12. Personalize problemswhenpossible. To keep planningtime to a minimum, each lesson was started with problemsfrom the already-prepared blackline masters; however, whenextensions were needed the names of students in the class orother teachers in the school were used. This technique helpedto keep students’ interest on problems that they might nototherwise have been highly motivated to try. Overall, studentsenjoyed the non-routine problems because they were uniqueand challenging, not because they saw extensive real-lifeapplications in them.

Summary

This article describes experiences with teaching using non-routine word problems to dispel the myth that getting studentsinvolved in problem solving, by its nature, is harder and moretime consuming than traditional drill methods of instruction.This is not to say that teaching students to solve challengingproblems is easy, but it is not as hard as many teachers fear.There are inexpensive, good quality commercial materialsavailable, and it is much easier to use them as a framework forteaching than it is to create problem-solving lessons fromscratch. The strong mathematics background of the universityinstructor made it fairly easy for him to see the underlyingstructure of many of the problems presented to students andthus fairly easy to develop extensions andadditional questions.Any teacher, however, should be able to develop these skillswith a little practice. Onceone really understands how to do themultiple-step and process problems in the materials beingused, planning a problem-solving lesson takes no more timethan planning a traditional mathematics lesson.

While teaching the two fourth-grade classes, frequentdiscussions with the students’ regular teachers were held. Theteachers provided feedback about the teaching of problemsolving and often used similar questioning techniques in theirdaily mathematics lessons. All teachers at the school wereinvolved in aprojecton teaching mathematics from aproblem-solving orientation, and thus students were often involved inmanipulative activities and other projects designed to improveunderstanding of mathematical concepts. The fourth gradeteachers noted that they felt the better students in their classesgot the most out of the non-routine problem lessons, but they

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also felt it was an excellent experience for all of their students.Other teachers in the school said that students with weakcomputational skills were often the best at solving processproblems. Giving weak students a chance to do non-routineproblems can often be a key to improving poor mathematicalskills.

On a closing note, there are examples of teachers whoapproached the teaching of non-routine word problems liketheir students approached the actual problems-thcy tried it,found it was hard, and gave up. Few things that are reallyworthwhile are easy to attain. It will take some time andexperimentation to become good at teaching students to solvenon-routine problems. The main point of this article is thatusing good materials makes teaching problem solving mucheasier than many teachers realize. With persistence and the rightresources, non-routine word problems can be relatively easy,enjoyable, and worthwhile for both teachers and students.

References

Cemen, P. B. (1989). Developing a problem-solving lesson.Arithmetic Teacher, 37(2), 14-19.

Charles. R., & Lester, F. (1982). Teaching problem solving:What, why, and how. Palo Alto, CA: Dale SeymourPublications.

Dossey.J.A., Mullis,I.V.S., Lindquist.M.M.. & Chambers,D.L. (1989). What can students do? (Level ofmathematicsproficiency for the nation and demographic subgroups). InM. M. Lindquist (Ed.), Resultsfrom the fourthmathematicsassessment of the national assessment of educationalprogress (pp. 117-134). Reston, VA: National Council ofTeachers of Mathematics.

Goodnow,J., & Hoogeboom, S. (1989). Reasoning withrabbitcounters. Sunnyvale, CA: Creative Publications.

Johnson,D.W.,&Johnson,R.T.(1989). Cooperative learningin mathematics education. In P. R. Trafton & A. P. Shulte(Eds.), New directions for elementary school fnathematics(pp. 234-245). Reston, VA: National Council of Teachersof Mathematics.

Kloosterman, P., & Gorman. J. (1990). Building motivation inthe elementary school. School Science and Mathematics,

90. 375-382.Meyer, C., & Sallee. T. (1983). Make it simpler: A practical

guide to problem solving in mathematics. Menio Park, CA:Addison-Wesley.

National Council of Teachers of Mathematics. (1989).Curriculum and evaluation standards for schoolmathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991).Professional standards for teaching mathematics. Reston,VA: Author.

Problem-solving experiences in mathematics. (1985). Menio

Park, CA: Addison-Wesley.Problemsolving inmathematics: The Lane County mathematics

project. (1983). PaloAlto.CA: Dale SeymourPublications.Problem-solving sour-cebook:Invitation to mathematics. (1987).

Glenview, IL: Scott Foresman.The problem solver: Activities for learning problem-solving

strategies. (1987). Sunnyvale, CA: Creative Publications.

EMERITUS MEMBER FUND

The Association solicits donationsto its Emeritus Fund. Income fromthe invested gift is used to offset theexpenses of providing School Sci-ence and Mathematics to the Emeri-tus members. Retirees who havebeen SSMA members for 25 consec-utive years are eligible for Emeritusmembership.. Increases in the pub-lishing costs and numbers of Emeri-tus members require additionalfunding for this purpose. Tax de-ductible gifts may be sent to the Ex-ecutive Secretary.

Volume 92(1), January 1992