Teaching mathematicalproblem solving

/ november 1987 besöktes lärarhögskolan i Mölndal av envärldsberömd problemlösningsexpert, professor Frank Lester,Indiana, USA. Hans mycket uppskattade föreläsning publicerasnu i Nämnaren.

All mathematics educators agree thatproblem solving is a very important,if not the most important goal, ofmathematics instruction at everylevel. Indeed, some have even gone sofar as to insist that Problem solvingshould be the focus of school mathe-matics (National Council of Teachersof Mathematics, 1980, p. 1).1) Unfor-tunately, when this pronouncementwas made it was not accompanied byany suggestions as to how to makeproblem solving the focus of instruc-tion. Since about 1980 problem solv-ing has been the most written about,but possibly the least understood,topic in the mathematics curriculumin the United States. It probably issafe to say that most teachers agreethat the development of students'problem-solving abilities is a primaryobjective of instruction. It is equallyas evident that these same teacherswould admit that it is quite anothermatter to decide how this goal is to bereached (i.e., where to begin, whatproblems and problem-solving experi-ences to use, when to give problemsolving particular attention, etc.). Al-though acceptance of the notion thatproblem solving should play a promi-nent role in the curriculum has beenwidespread, there has been anythingbut widespread acceptance of how tomake it an integral part of the curric-

ulum. Indeed, it is common to hearteachers voice concerns like: As ifthere wasn't already enough contentto cover. Now the "experts" want usto add problem solving.Comments of this sort cannot bebrushed aside lightly. They point tothe fact that to date no mathematicsprogram has been developed that ad-equately addresses the issue of mak-ing problem solving the central focusof the curriculum. Instead of pro-grams with coherence and direction,what teachers have been given is awell-intentioned melange of storyproblems, lists of strategies to betaught, and suggestions for classroomactivities. If problem solving is tobecome a more prominent goal ofmathematics instruction, more seri-ous and thoughtful attention must begiven to what it means to make prob-lem solving the focus of school math-ematics. Before presenting my ideasfor teaching problem solving let meillustrate the severity of the problemfacing us as teachers by means of afew examples from my own experi-ence.

1) The National Council of Teachers ofMathematics (NCTM) is an American profes-sional organization for persons interested inthe teaching and learning of mathematics.Currently, the NCTM has approximately70,000 members.

Why Is Mathematical Problem SolvingDifficult for Students?

I have been a mathematics teacher formore than 22 years. During my careerI have taught students ranging in agefrom 6 years to adult (my oldest stu-dent was over 60 years old when Itaught her). As one would expect,some of these students have been ex-ceptionally talented in mathematicsand a few have found mathematics aparticularly troublesome subject tolearn. But, on the whole, most of mystudents have been of average abilityin mathematics. The examples thatfollow are taken from my experienceswith this large majority of averageability students. As is true of anyreasonably serious teacher, I havebeen puzzled from time to time by thebehavior of some of these students.My puzzlement has been nowheremore pronounced than in my obser-vations of students' problem-solvingbehavior. Consider the followingthree "problems" and the behaviorsof several students who have attempt-ed to solve them.

The Frog in the Well

A frog is at the bottom of a wellthat is 10 meters deep. During thedaytime the frog climbs up the sideof the well 4 meters, but at night itslides back 2 meters when it sleeps.At this rate, how many days will ittake the frog to reach the top of thewell?

It has been my experience that moststudents (and people in general) overthe age of 8 or so determine that itwill take the frog 5 days to reach thetop of the well. (Their reasoning goessomething like this: 4 m - 2 m = 2 mgain each day and 10 days - 2 m perday = 5 days.) By contrast, children8 years old or less draw some sort ofpicture and arrive at either 3 1/2 or 4days as their answer (each of whichcan be considered correct). What hap-pens to students that makes them lesssuccessful on this problem as theybecome older?

The Chickens and Pigs

Tom and Susan went to theirgrandparents' farm and saw somechickens and pigs in the barnyard.Tom said he saw 18 animals in thebarnyard. Susan agreed with himand added that she counted 52 legsin all. How many chickens and howmany pigs were in the barnyard?

About six years ago I gave the Chick-ens and Pigs problem to a class ofgrade three students (ages 8 and 9).Nearly half of them gave 70 as their

answer—not 70 animals or 70 legs,just 70. Some of these children point-ed out that the words "in all" in theproblem tell you to add. A few weekslater I asked a class of grade fivestudents to solve this same problem.Many of them wrote nothing on theirpapers. When questioned as to whythey did not give an answer, typicalresponses included: I don't know howto do it, and / think you have todivide, but 18 doesn't go into 52. Notone student in a class of 30 drew anysort of diagram, made any guesses, orotherwise used any "natural" prob-lem-solving strategies. Many of theyounger students had been taught afaulty strategy (viz., look for "keywords"), but even worse, the olderstudents had actually "learned" thatthey could not solve mathematicsproblems.

The Car Trip

A man drove his car from his hometo a friend's house at a speed of 64kph and it took him 20 minutes.When he returned to his home hetravelled along the same roads butat a speed of 80 kph. How long didthe return trip take?

I have not been particularly puzzledby the fact that many high school anduniversity students fail to solve theCar Trip problem successfully. Whathas puzzled me is that so many (near-ly 25 %) give 25 minutes as the answer(64/20 = 80/x). Doesn't it defy logicand common sense that if you gofaster, it would take longer to make atrip?

Examples such as these are all toocommon. In fact, most mathematics

teachers are quick to observe thatmany of their students are unable tosolve any but the most routine prob-lems despite the fact that their stu-dents seem to have "mastered" all ofthe requisite computational skills,facts, and algorithmic procedures.The first reason why students are of-ten unable to solve any but the mostroutine problems is that solving amathematics problem requires the in-dividual to engage in a variety ofcognitive actions, each of which re-quires some knowledge and skill, andsome of which are not routine. Fur-thermore, these cognitive actions areinfluenced by a number of noncogni-tive factors. That is to say, by its verynature problem solving is an extreme-ly complex form of human endeavourthat involves much more than thesimple recall of facts or the applica-tion of well-learned procedures.

Successful problem solving involvesthe process of coordinating previousexperiences, knowledge, and intuitionin an effort to determine an outcomeof a situation for which a procedurefor determining the outcome is notknown.

A second reason why so many stu-dents have trouble becoming profi-cient problem solvers is that they aregiven too few opportunities to engagein real problem solving. That is, theydo not develop expertise in problemsolving because they are neither guid-ed nor challenged to do so. Sinceproblem solving is so complex stu-dents need to be given carefully de-signed problem-solving instructionand they must have extensive experi-ences in solving a wide variety ofproblems and reflecting on their per-formance.

The Complex Natureof Mathematical Problem Solving

The ability to solve mathematicsproblems develops slowly over a verylong period of time because it requiresmuch more than merely the directapplication of some mathematicalcontent knowledge. Problem-solvingperformance seems to be a functionof at least five broad, interdependentcategories of factors:

(1) knowledge acquisition and utiliza-tion

(2) control(3) beliefs(4) affects(5) socio-cultural contexts.

Let me say a few words about each ofthese categories.

Knowledge Acquisitionand Utilization

It is safe to say that the overwhelmingmajority of research in mathematicseducation has been devoted to thestudy of how mathematical knowl-edge is acquired and utilized. By"knowledge" I mean both informaland intuitive knowledge as well asformal knowledge. Included in thiscategory are a wide range of resourcesthat can assist the individual's mathe-matical performance. Especially im-portant types of resources are thefollowing: facts and definitions (e.g.,7 is a prime number, a square is arectangle having 4 congruent sides),algorithms (e.g., the long division al-gorithm), heuristics (e.g., drawingpictures, looking for patterns, work-ing backwards), problem schemas

(i.e., packages of information aboutproblem types), and the host of rou-tine, but not algorithmic, proceduresthat an individual can bring to bearon a mathematical task (e.g., pro-cedures for solving equations, generaltechniques of integration). Of par-ticular significance to this discourse isthe way individuals organize, repre-sent, and ultimately utilize theirknowledge. There is not doubt butthat many problem-solving deficien-cies can be attributed to the existenceof "unstable conceptual systems"(Lesh, 1985). That is, when individ-uals are engaged in solving a problemit is likely that at least some of therelevant mathematical concepts are atintermediate stages of development.In such cases problem solvers mustadapt their concepts to fit the prob-lem situation. To the extent that theyare able to make appropriate adapta-tions, they are successful in solvingthe problem.

Control

Control refers to the marshalling andsubsequent allocation of available re-sources to deal successfully withmathematical situations. More spe-cifically, it includes executive deci-sions about planning, evaluating,monitoring, and regulating. Two as-pects of control processes have be-come increasingly popular as objectsof research in recent years: knowledgeabout and regulation of cognition.The processes used to regulate one'sbehavior are often referred to as me-

tacognitive processes, and these haverecently become the focus of muchattention within the mathematics edu-cation research community. In fact,recent research suggests that an im-portant difference between successfuland unsuccessful problem solvers isthat successful problem solvers aremuch better at controlling (i.e., moni-toring and regulating) their activities.It is clear that a lack of control canhave disastrous effects on problem-solving performance.

Beliefs

Schoenfeld (1985) refers to beliefs, or"belief systems" to use his term, asthe individual's mathematical worldview; that is, " . . . the perspectivewith which one approaches mathe-matics and mathematical tasks" (p.45). Beliefs constitute the individual'ssubjective knowledge about self,mathematics, the environment, andthe topics dealt with in particularmathematical tasks. For example, mycolleagues and I have found thatmany elementary school children be-lieve that all mathematics story prob-lems can be solved by direct applica-tion of one or more arithmetic opera-tions, and which operation to use isdetermined by the "key words" in theproblem (Lester & Garofalo, 1982). Itseems apparent that beliefs shape atti-tudes and emotions and direct thedecisions made during mathematicalactivity. In my own research I havebeen particularly interested in stu-dents' beliefs about the nature ofproblem solving as well as about theirown capabilities and limitations (Les-ter, Garofalo & Kroll, in press).

Affects

This domain includes individual feel-

ings, attitudes and emotions. Mathe-matics education research in this areaoften has been limited to examina-tions of the correlation between atti-tudes and performance in mathemat-ics. Not surprisingly, attitudes thathave been shown to be related toperformance include: motivation, in-terest, confidence, perseverance, will-ingness to take risks, tolerance ofambiguity, and resistance to prema-ture closure.

To distinguish between attitudesand emotions I choose to regard atti-tudes as traits, albeit perhaps tran-sient ones, of the individual, whereasemotions are situation-specific states.An individual may have developed aparticular attitude toward some as-pect of mathematics which affects heror his performance (e.g., a studentmay greatly dislike problems involv-ing percents). At the same time, aparticular mathematics task may giverise to an unanticipated emotion(e.g., frustration may set in when astudent finds that he or she has madelittle progress toward solving a prob-lem after working diligently on it fora considerable amount of time). Thepoint is that an individual's perform-ance on a mathematics task is verymuch influenced by a host of affec-tive factors, at times to the point ofdominating the individual's thinkingand actions.

Socio-Cultural Contexts

In recent years, the point has beenraised within the cognitive psychologycommunity that human intellectualbehaviour must be studied in the con-text in which it takes place (Neisser,1976; Norman, 1981). That is to say,since human beings are immersed in areality that both affects and is affect-

ed by human behaviour, it is essentialto consider the ways in which socio-cultural factors influence cognition.In particular, the development, un-derstanding, and use of mathematicalideas and techniques grow out of so-cial and cultural situations. D'Am-brosio (1985) argues that childrenbring to school their own mathemat-ics which has developed within theirown socio-cultural environment. Thismathematics, which he calls "ethno-mathematics," provides the individ-ual with a wealth of intuitions andinformal procedures for dealing withmathematical phenomena. Further-more, one need not look outside theschool for evidence of social and cul-tural conditions that influence mathe-matical behavior. The interactionsthat students have among themselvesand with their teachers, as well as thevalues and expectations that are nur-

tured in school, shape not only whatmathematics is learned, but also howit is learned and how it is perceived(cf., Cobb, 1986). The point then isthat the wealth of socio-cultural con-ditions that make up an individual'sreality plays a prominent role in de-termining the individual's potentialfor success in doing mathematics bothin and out of school.

These five categories overlap (e.g.,it is not possible to completely sepa-rate affects, beliefs, and socio-cul-tural contexts) and they interact in avariety of ways too numerous toname in these few pages (e.g., beliefsinfluence affects, and they both influ-ence knowledge utilization and con-trol; socio-cultural contexts have animpact on all the other categories). Itis perhaps due to the interdependenceof these categories that problem solv-ing is so difficult for students.

Teaching Problem Solving

For students who are struggling tobecome better problem solvers thedifficulty caused by the complexity ofproblem solving is compounded bythe fact that most of them do notreceive adequate instruction, either inquality or quantity. Since problemsolving is so complex, it is difficult toteach. Unfortunately, we do not yethave fool-proof, easily followed andimplemented methods of helping stu-dents to improve their problem-solv-ing ability.

In recent years there has been much

research conducted on various ap-proaches to mathematical problem-solving instruction. I will not discussthis research here except to say thatdetailed discussions of the researchconducted in the United States duringthe past 15 years can be found inLester (1980, 1983), Schoenfeld(1985) and Silver (1985). In the nextsection of this paper I present myanalysis of what this research suggeststo me about how mathematical prob-lem solving should be taught.

Fundamental Principles AboutTeaching Problem Solving

In my study of the research literatureI was able to isolate four basic princi-ples that stood out as common resultsof all of the research. These principlesare as follows.I. Students must solve many prob-

lem in order to improve theirproblem-solving ability.

II. Problem-solving ability developsslowly over a prolonged period oftime.

III. Students must believe that theirteacher thinks problem solving isimportant in order for them tobenefit from instruction.

IV. Most students benefit greatlyfrom systematically plannedproblem-solving instruction.

I will not elaborate on the first threeof these principles except to mentionthat although many factors arenecessary ingredients for a successfulproblem-solving program, perhapsnone is more important than principleIII. Teachers must demonstrate en-thusiasm for problem solving andcommunicate through their actionsand words the importance of problemsolving in mathematics. When teach-ers make a sincere commitment todeveloping students' problem-solvingskills, students will make a similarcommitment. In the remainder of thispaper I will discuss what is involvedin "systematically planned problem-solving instruction."

Essential Components ofa Systematically PlannedProblem-solving Program

Most problem-solving programs will

seem to work for a while in the class-room. However, for a program to besuccessful all year and year after year,it should be made up of three compo-nents: (a) appropriate content, (b) ateaching strategy, and (c) guidelinesfor managing the program. Let uslook at each component in turn.

A. Appropriate ContentFirst and foremost, a good problem-solving program must include appro-priate content. The content must beof suitable difficulty and must includeat least three types of experiences de-signed to improve problem-solvingperformance. These types of experi-ences are the following: (1) regularsessions devoted to solving a varietyof kinds of problems; (2) instructionin the use of various problem-solvingstrategies; and (3) practice aimed atthe development of specific problem-solving thinking processes and skills.

The focus of instruction should beon the solution of "process" prob-lems, but routine one-step verbalproblems and multiple-step verbalproblems should also be included.Briefly, a process problem is onewhose solution cannot be obtainedsimply by performing computations.Such problems are included becausethey exemplify the processes inherentin thinking through and solving a trueproblem (i.e., a situation for which aprocedure for solving the problem isnot readily at hand). These types ofproblems serve to develop generalstrategies for understanding, plan-ning, and solving problems, as well asevaluating attempts at solutions.

One reason why students have dif-ficulty with problem solving is thatmany of them have not been taughthow to use specific problem-solvingstrategies. Traditionally, most stu-

dents are taught only one strategy—choose an operation or operations toperform, then do the computations.Several of the strategies that shouldbe included in instruction in grades1-8 are the following:

choose an operation or operationsto performdraw a picturemake an organized listwrite an equationact out the situationmake a table or chartguess and checkwork backwardssolve a simpler problemuse objects or models

Instruction aimed at developing stu-dents' ability to use strategies such asthese needs to include two phases.During the first phase students aretaught how to use a particular strat-egy. This phase emphasizes the me-aning of a strategy and the techniquesinvolved in implementing it. Afterstudents are introduced to a strategythey are given practice using it tosolve problems. The second phase iswhere students are taught to decidewhen to use a strategy. Here studentsare given problems to solve but theyare not told which strategy to use.They must select from among thestrategies they have learned the one(s)that are appropriate for solving agiven problem. Both phases must beincluded in instruction. Unfortunate-ly, it is beyond the scope of this paperto elucidate effective ways to coordi-nate these two phases.

A person who is learning to play amusical instrument, say the piano,does not learn to play simply byplaying musical scores. In addition,considerable time must be devoted to

activities designed to help her or himmaster certain skills and techniques.Such skill activities (e.g., finger exer-cises) are an essential part of beco-ming an accomplished pianist. In a si-milar manner, the novice problemsolver must be asked to do more thansimply solve problems. The third ofthe three types of experiences involvesactivities designed to develop certainproblem-solving thinking processesand skills. In my own work I haveidentified 8 thinking processes thatare involved in the solution of mathe-matics problems. A good problem-solving program includes numerousactivities that focus on these thinkingprocesses and the skills associatedwith them. A list of the thinking pro-cesses is given in Table 1.

Table 1: MathematicalProblem-solving Thinking

Processes(taken from Charles, Lester

& O'Daffer, 1987)

1. Understanding/formulating thequestion in the problem/situa-tion.

2. Understanding the conditionsand variables in the problem.

3. Selecting/finding data neededto solve the problem.

4. Formulating subproblems andselecting appropriate solutionstrategies to pursue.

5. Correctly implementing the so-lution strategy and attainingsubgoals.

6. Giving an answer in terms of thedata given in the problem.

7. Evaluating the reasonablenessof the answer.

8. Making appropriate generaliza-tions.

Finally, in addition to the three typesof experiences discussed above, thiscontent should be integrated through-out the entire mathematics program.(Occasional attention to problemsolving or including a short unit ofinstruction on problem solving simplyis not enough).

B. A Specific Teaching StrategyTeachers must have a specific strategyfor teaching the content. A teacherwho is not aware of specific ways toteach problem solving often resorts togeneral admonitions for students todo better when they need assistance.Comments like: "Read the problemagain," "Use your head," and"Think harder" are commonly madeby such a teacher. Teacher commentssuch as these may encourage studentsto try harder, but they are of littlehelp to students who are truly in needof help. Indeed, very few students canbecome successful problem solverswithout the aid of their teachers. Thesingle most challenging task for theteacher is to decide what kind ofguidance to provide and when to pro-vide it. The teacher must play anactive part during classroom prob-lem-solving activities by observing,

questioning, and, if necessary, byproviding direction. During the pastfifteen years I have collaborated withseveral colleagues in the developmentof a teaching strategy for problemsolving. This strategy has been testedand shown to be successful in quite alarge number of classrooms in a var-iety of schools throughout the UnitedStates (see Lester, 1983 and Charles &Lester, 1984 for discussions of theresults of our research). These"teaching actions" as we call themcan be used in a teacher-directed ac-tivity, beginning with a whole-classdiscussion of the problem, followedby individual or small-group work onthe problem, and ending with anotherwhole-class discussion of the prob-lem. A list of the teaching actions andthe purpose of each action are shownin the following table. These teachingactions cannot be used as a "formu-la" that will guarantee success for allstudents. Rather, they provide teach-ers with a means to guide studentssystematically through the process ofsolving problems in order to buildconfidence as well as competence. Asstudents become more capable asproblem solvers the teacher's overtrole diminishes.

Table 2: Teaching Actions for Problem Solving(taken from Charles & Lester, 1982)

Teaching Action PurposeClass Discussion BEFORE

1 Read the problem to the class orhave a student read it. Discuss vo-cabulary and the setting of thestory as needed.

Illustrate the importance ofreading problems carefully forgood understanding.

2 Ask questions related to under-standing the problem. Focus onwhat the problem is asking andwhich data are needed to solve theproblem.

Focus attention on important datain the problem and clarify confus-ing parts of the problem.

3 Have students suggest possiblesolution strategies. Do not censoror evaluate ideas at this time. (Asstudents become more successful,this action may be eliminated.)

Elicit ideas for possible ways tosolve problems. Encourage flexiblethinking and exploration.

DURING Students' Solution Efforts

4 Observe students as they solve theproblem. Ask them questionsabout their work.

Diagnose students' strengths andweaknesses in problem solving.Develop reflectiveness about theirwork.

5 Provide hints for students who arehopelessly stuck or who are be-coming too frustrated. Repeat un-derstanding questions as needed.

Help students past insurmountableobstacles in solving a problem andreduce the chances of the develop-ment of negative attitudes. Helpthem learn how to use particularstrategies.

6 When students get an answer, re-quire them to check their workagainst data in the problem.

Develop students' ability to evalu-ate their work.

7 Give students who finish early avariation of the original (do thiswith all students as time permits).

Help students learn to generalizetheir solutions.

Class Discussion AFTER

8 Discuss students' solutions to theproblem. Identify different waysthe problem might be solved.

Help students learn when to use aparticular strategy and how to useit efficiently. Encourage flexibilityin solving problems.

9 Compare the problem just solvedwith problems solved previouslyand discuss any variations thatmay have been solved.

Foster transfer of learning.

10 Discuss any special features of theproblem such as misleading infor-mation.

Help students recognize problemfeatures that influence how prob-lems are solved.

C. Guidelines for Managingthe Program

In addition to knowing what and howto teach, the teacher should be pro-vided specific suggestions regardingthe management of the program. Inparticular, the teacher must knowhow to deal with issues such as:

1) the amount of time to devote toproblem solving

2) ways to group students for instruc-tion (allowing students to work onproblems in groups of three orfour has been shown to be quitesuccessful)

3) adjusting instruction for high andlow achievers in the same class-room

4) how to evaluate students' perfor-mance

5) how to create and maintain a posi-tive classroom climate for problemsolving. There is so much involvedin this final component that itcould easily be the topic of a sep-arate paper. In fact, recently I col-laborated with two colleagues inthe preparation of a book on thesingle topic of evaluation of stu-dents' problem-solving perfor-mance (Charles, Lester &O'Daffer, 1987). More informa-tion about this component can befound in the evaluation book or ina book written earlier by R.Charles and me (Charles & Lester,1982).

Closing Comments

In this short paper I have attemptedto provide some perspective about thenature of mathematical problem solv-ing and I have offered several sugges-tions as to how to begin to implementa mathematics program that hasproblem solving at its core. Yes,mathematical problem solving is dif-ficult for children to do and it isdifficult for teachers to teach. How-ever, helping children to be betterproblem solvers in mathematics is notonly an extremely important goal, it is

also the most challenging and excitingone that a teacher can have. If I wereallowed to give only one bit of adviceto a teacher who was planning tobegin to make problem solving thefocus of instruction, it would be toremember that children are naturalproblem solvers. The teacher's job isto try to develop this natural ability toits maximum extent and to add to thealready extensive repertoire of prob-lem-solving techniques that childrenhave at their disposal.

References

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

Charles, R. & Lester, F. K. (1984).An evaluation of a process-orientedmathematical problem-solving in-structional program in grades fiveand seven. Journal for Research inMathematics Education, 15, pp. 15-34.

Charles, R., Lester, F. & O'Daffer,P. (1987). How to evaluate progressin problem. Reston, VA: NationalCouncil of Teachers of Mathematics.

Cobb, P. (1986). Contexts, goals, be-liefs, and learning mathematics. Forthe Learning of Mathematics, 6, 2-9.

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Lester, F. K. & Garofalo, J. (1982,April). Metacognitive aspects of ele-mentary school students' perfor-mance on arithmetic tasks. Paper pre-sented at the annual meeting of theAmerican Educational Research As-sociation, New York.

Lester, F. K., Garofalo, J. & Kroll,D. L. (in press). Self-confidence, in-terest, beliefs, and metacognition:Key influences on problem-solvingbehavior. In D. McLeod (Ed.), Affectand mathematical problem solving: Anew perspective. New York: Springer-Verlag.

National Council of Teachers ofMathematics (1980). An agenda foraction: Recommendations for schoolmathematics for the 1980s. Reston,V. A.: NCTM.

Neisser, U. (1976). Cognition and re-ality. San Francisco: Freeman & Co.

Norman, D. A. (1981). Twelve issuesfor cognitive science. In D. A. Nor-man (ed.), Perspectives in cognitivescience. Norwood: N. J.: Ablex.

Schoenfeld, A. H. (1985). Mathemat-ical problem solving. Orlando, FL:Academic Press.

Silver, E. A. (1985). Research onteaching mathematical problem solv-ing: Some underrepresented themesand needed directions. In E. A. Silver(Ed.), Teaching and learning mathe-matical problem solving: Mutiple re-search perspectives. Hillsdale, N. J.:LEA.