DOCUMENT RESUMEED 280 700SE 047 862AUTHORTITLEMarks, RickProblem Solving with a Small "p": A Teacher'sPUB DATENOTEView.Apr 8769p.; Papel: presented at the Annual Meeting of thePUB TYPEEpRs PRICEDESCRIPTORSIDENTIFIERSAmerican Educational Research Association(Washington, DC, April, 1987).Speeches/Conference Papers (150) -- ReportsResearch/Technical (143)MF01/PC03 Plus Postage.*Case Studies; Educational Philosophy; EducationalResearch; *Mathematics Instruction; MathematicsTeachers; *Problem Solving; Secondary Eeucation;*Secondary School Mathematics; *Teaching Methods*Mathematics Education ResearchABSTRACTThis case study examined an experienced secondaryschool mathematics teacher's knowledge and teaching of problemsolving, using interviews, classroom observations, teachingdocuments, and experimental tasks. The informant revealed a broadinterpretation of problem solving, integrated with mathematics butwidely applicable. This interpretation appeared consistently in hisknowledge of problem-solving content, knowledge of pedagogy, andinstructional behavior. The informant's own background significantlyinfluenced his knowledge of problem solving, which in turn shaped histeaching of problem solving. This study extends recent casework inmathematics teaching and has important implications for research,teaching, and educational policy. Appendices indicate data sources,interview schedules, task schedules, and a problem-solving ctions supplied by EDRS are the best that can be madefrom the original ******************************
PROBLEM SOLVING WITH A SMALLA TEACHER'S VIEWU.S. DEPARTMENT OF EDUCATIONOffice of Educational Researchand ImprovementEDUCATIONAL RESOURCES INFORMATIONCENTER (ERIC)XThis document has been reproduced asreceived from tho person or organizationoriginating it.0 Minor changes have been made to improvereproduction quality.o Points of view or opinions stated in this docu-ment do not necessarily represent officialOERI position or policy.Rick MarksStanford University"PERMISSION TO REPRODUCE THISL HAS BEEN GRANTED BYMATETO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)."Paper presented at AERA annual conferenceWashington, D.C., April 19872
icsteacher's knowledg? and teaching of roblem solving, usinginterviews, classroom observations, teaching documents, andexperimental tasks.The informant revealed a broadinterpretation of problem solving, integrated with mathematicsbut widely applicable.This interpretation appearedconsistently in his knowledge of problem-solving content,knowledge of pedagogy, and instructional behavior.Theinforrnant's own background significantly influenced hisknowledge of problem solving, which in turn shaped histeaching of problem solving.This study extends recentcasework in mathematics teaching and has irr.,portantimWications for research, teaching, and educational policy.
PROBLEM SOLVING WITH A SMALL "p": A TEACHER'S VIEW(Math teacher, in interview, describing problem solving) Whenyou have a situation that derminds attention, either because ofthe positive things that the sOution might bring or thenegative situation that exists now that you want to alleviate,then that's a problem. And you seek to deal with that situation.At a certain point, you cal; what you have a solution, I guess.Sometimes it's a completely satisfying solution, such asbalancing a checkbook; other t;:ries it'S a temporary solution;and sometimes you give up, 2ind you say this is no longer aproblem. A very broad 3ense.I think students might not be making the split t 3t.ween problemsolving and non-problem solving; the teachers are. Forstudents, who don't divide up the curriculum like we do, I thinkit's all problem &lying.This teacher views problem solving in a distinctive way. It is notProblem Solving, the new find in math education, or Prohiem Solving, a topicto be squeezed in somewhere between Fractions and Quadratic Equations. Itis problem solving with a small "p", applicable across disciplines and evenoutside of school, woven into the fabric of mathematics rather thanstamped on top. But what does this approach to problem solving look like indetail? What alternative paradigms of probkm solving might other mathteachers hold? And how does the teacher's view of problem solving shapeher teaching of it? These questions motivate the present study.Problem solvthg as an issue in mathematics education is not newborn,but it has reached its adolescence in the 1980s: everyone knows it'saround, but no one is quite sure how to handle it. Policy makers (CaliforniaDepartment of Public Instruction, 1935) and educational researchers (Beg le,
1979) agree with the National Council of Teachers of Mathematics (1980)that "problem solving [should] be the focus of school mathematics in the1980s" (p. 1). Yet despite consensus on the importance of problem solving,it is an elusive concept to def ine and a complicated one to study. Briars(1982) and Kilpatrick (1985) desc, ibe various research programs inmathematical problem solving: analysis of problems; cognitive processes,whether through verbal protocols or computer simulations; expert-novicecomparisons; and training sequences. Each of these approaches highlights adifferent facet of problem solvmg.This extensive body of problem-solving research has examined thecontent, the student, and the instruci.ion, but it has largely ignored theteacher (Grouws, 1985; Silver, 1985). Even when research has focused onteachers, it has typically concentrated on how they behave in the classroomrather than on how they think about the content, students, and instruction(Shulman & Elstein, 1975). Fortunately, a recent trend in mathematicseducation re' earch has begun to correct this oversight by delving intotachers knowleage and its rel-Ition to their teaching (Shulman, 1985).Thompson (1984) and Steinberg, Haymore, and ;/arks (1985) demonstrated insets of case studies that math teachers conceptions of mathematicsstrongly influence the way they teach. Cooney (1985), studying problemsolving in Fred, a beginning math teacher, accounted for a good deal of histeaching behavior in terms of his conceptions of mathematics, problemsolving, and teaching.The present case study extends this line of research by investigat;ng anexperienced mathematics teacher's knowledge and teaching of problemsolving. This study examines haw the informant's background helped form
his content knowledge of problem solving, how this in turn influenced hispedagogical knowledge, how these aspects of knowledge shaped his planningand teaching, and how features of the teaching context affected thisprocess. These substantive outcomes also suggest a more formal model forthe tc-aching of problem solving and similar topics.Conceptual FrameworkThe categories and relationships of interest in this case study appear inFigure 1. Following Shulman's (1986) recent work, knowledge of content isconsidered separately from knowledge of teaching that content. Knowledgeas used in this paper encomp.Asses not only factual knowledge but alsoskills, beliefs, attitudes, and feelings. Problem solving will be defined bythe teacher rather than by the researcher, since the chief purpose of thisstudy is to discover the teachers's own conceptions of problem solving,together with their origins and effects. I did, however, construct adescriptive hierarchy of problem-solving behavior (see Appendix D, Figure 4)to assist in generating examples of various problern-sdlving featuresthroughout the study. I also intended to use this hierarchy to analyzeinstances of proolem solving in the data but found it difficult to ?plyreliably.Knowledge of proble:11 solving comprises three major categories:sources, content knowledge, and pedagogical knowledge. Sources areinfluences on the development of problem-solving knowledge and mayinclude teaching experience, teacher education, other schooling, anlnon-academic factors. Content knowledge here connotes the teacher'sknowledge of problem solving per se, apart from teaching. Three46
ontent KnowledgePee.mimics! KnowledgecurriculumskillsO methodscontentconceptionsattitudesKTIOSILEOGE OF PRO8LE11 SOLOINGTERCHINO OF PROBLEM SOLUINGInstructionPlanning* incidenceincidence* ontextcoursetextbookstudentsotherFigure 1.Influence of teacher's knowledge en teaching.
subcategories are useful for analysis. Skills represent the teacher's ownproblem-solving experience and competence, which many math educatorsargue are indispensable in an effective teacher of problem solving (e.g.,House, 1980). Conceptions are knowleoge and beliefs about the subjectmatter itself (e.g., "problem solving is central to mathematics"), whileattitudes are affects and beliefs abo'it the teacher's own relation to thesubject matter ("I use problem solving all the time"). Pedagogicalknowledge describes the teacher's knowledge of teaching problem solving inparticular, not just of teaching in general. Again three types are important.Curriculum consists of general NiewE; of problem-solving instruction: itsaims, significance, scope and structure, and its proper place in the totalinstructional program. Methods include specific teaching techniques andmaterials--for example, the use of cooperative small groups or of computersoftware to teach problem solving. Content as a subcategory of pedagogyrefers to knowledge the teacher wants his students to acquire, such as alist of Polya's (1957) heuristics or a scheme for solving river-crossingproblems.Teaching of problem solving likewise includes three major categories:planning, instruction, and context. Though planning precedes instruction,the two activities correspond closely, so each contains the same foursubcategories. Incidence describes the frequency and emphasis of problemsolving and the way it is incorporated into the mathematics class. Contenthere is the knowledge and techniques of problem solving that the teacherteaches, explicitly or implicitly. Methods include the procedures andmaterials which the teacher uses to teach problem solving but which aremeans and not ends with regard to student learning (e.g., the use of small8
groups). Attitudes are subjective beliefs and feelings about problem solvingwhich the teacher communicates, again explicitly or implicitly. Contextconsists of those features of the particular teaching situation which callfor adapting the teacher's general knowledge specifically to that situation.Various aspects of context may be important, but three are related closelyto the content. Course refers to prescriptive factors such as curriculumoutlines, advice from colleagues, or department policy. The textbookprovides a particular treatment (or non-treatment) of problem solving,while the students bring a distinctive set of problem-solving skills,aptitudes, and attitudes to the class.The relationships in Figure I depict the five main questions in this casestudy. What skills, conceptions, and attitudes does the teacher haveregarding problem solving? How do these influence his ideas about teachingproblem solving? How did the teacher acquire this knowledge? How doesthis knowledge shape his actual planning and teaching of problem solving?How does teaching context affect this process? Though this conceptualmodel was designed to explore these questions in the domain of problemsolving, the same model could be applied to virtually any content area.Methods and ProceduresSince the purpose of this study was not to determine the typicality ofmathematical problem-solving instruction but to describe and explain itwhen it does occur, the informant had to be a math teacher who also taughtproblem solving in some form. One of my teacher education studentsrecommended her master teacher, Sandy, who happened to be a casual friendof mine. After a brief interview which confirmed his suitability and9
willingness, Sandy became my informant. He was extremely cooperative andinterested throughout the study and especially articulate in interviews, notsurprising since he also does educational research at a nearby university.Data for this study came from several different sources: nine structuredinterviews with the teacher, nine observations of classroom teaching,numerous documents, eight experimental tasks for the teacher, and onesummary debriefing interview. (See Appendix A for a complete list of datasources, Appendix B for the interview schedules, and Appendix C for the taskschedules.) At least two different types of data address each subcaterryin the conceptual framework (see Appendix A, Table 1). The variety ofsources serves both to increase the descriptive and explanatory power ofthe study and to increase construct validity through triangulation.One or two 45-minute interviews each addressed background, contentknowledge, pedagogical knowledge, and context, while shorter interviewsconducted before and after each ooservation unit covered planning andinstruction. The interviews included indirect as well as direct approaches:for example, "Suppose your math department organized a task force torecommend ways to include more problem solving in the curriculum, andthey chose you as the chair of the task force. What would you do?" Eachinterview was audiotaped and later transcribed in full.Three observation units of three days each explored instruction for oneclass on one topic, a different class on the same topic, and the same classon a different topic. Data in each class consisted of extensive handwrittennotes and a short, open-ended debriefing interview after class. Documentsincluded school course descriptions, unit and lesson plans, handouts andtests, reproductions of the textbook, and copies of other resources the810
teacher used. The experimental tasks probed Sandy's knowledge of problemsolving in forms oiher than a direct verbal report, in an attempt to reducethe level of inference through multiple sources of evidence. Examples of thetasks are to solve a given math problem while thinking aloud, to classify agiven set of math problems according to their perceived degree of problemsolving, and to rank a set of 40 possible objectives for high school mathaccording to their importance for the students. Five of t. le tasks were alsoaudiotaped and transcribed.Data analysis was a variation of Glaser and Strauss's (1967) constantcomparative method. Analysis began during data collection. First 1 reducedeach interview, observation, document, and task to a set of brief numberedn3tes, placing each instance relevant to one of the framework's conceptualcategories into a single note. This stage involved selection hut almost nocommentary; the notes consisted of low-inference summaries, paraphrases,and quotations of the data. This technique not only reduced the mass ofverbiage but also suggested themes to guide the remainder of the datacollection. After reducing all the data to notes and studying them, I wrote aseries of short memos delineating substantive themes, both descriptive andexplanatory, and noted questions, contradictions, and conjectures. Afterorganizing these themes! constructed coding categories, then returned tocode some of the notes. I repeated this process twice, each time refiningthe memos, reducing the categories, and coding more of the notes forverification. Formal themes emerged eventually from the substantivememos. Finally, by the end of the third iteration, the themes formed acoherent structure and the coded data provided the supporting detail for thecase study report.9 11
Informant and Setting(Sandy, interview) Mostly I studied the social sciences andhumanities, with a heavy emphasis of psych, sociology,anthropology. Some literature, history. Not a great amount ofmath. I always liked math, yeah. like it more since I'mIteaching it. And I think there are some complementarythings about thinking in terms of mathematics that is a nicebalance to other parts of my life. I'm not one to believe thatsubject matter knowledge is the great decider of theeffectiveness of the teacher. My goal was to load up mycredential as much as possible, sort of like a utility infielderin baseball, who can serve different roles in a school.Sandy is an experienced and highly educated high school teacher withmulti-disciplinary interests and abilities. He majored in psychology butalso liked math and was good at it. In high school he went through theaccelerated math program, including calculus, and in college he took coursesin statistics, finite math, and computer science. Sandy holds an M.A. ineducation and a teaching credential in both math and social science. Inaddition, he has just completed a doctoral dissertation on teacherself-evaluation in math and social studies teachers. After eight years offull-time teaching, both math and social studies, he is still very happy withhis career and plans to continue teaching high school, at least for the nearfuture.The quotation above illustrates a personal characteristic germane to thepresent study: Sandy's tendency to integrate and qualify, to reject extremesand dichotomies. This trait appears both in his simultaneous interest insocial studies and math and in his minimizing of the importance of subject1012
matter knowledge. He speaks of the varied disciplinary perspectives asbalancing rather than competing. This characteristic manifests itselfthroughout the data and will enter the analysis later.Sandy teaches Algebra 2, Algebra 1, and Algebra .5 in a medium-sized,integrated suburban high school. The courses are not tracked, and all of hisclasses span an ordinary range of student abilities. He describes the twohigher-level courses as "college prep," "fast-paced," "challenging," and"difficult"; Algebra 2 in particular is "real solid." Algebra .5 is also "aserious math class" but moves more s;owly. These terms are indicative ofSandy's serious, no-nonsense attitude toward his teaching. The presentstudy focuses on Algebra 2 because Sandy's two sections of that coursepermitted more intensive observation and because preliminary datasuggested that cross-course comparisons would add little insight.Knowledge of Problem SolvingContent Knowledge(Sandy, interview) Elf I'm driving home from work and I noticeI'm low on cash] that's problem solving to me. Figuring out howam I going to get over to the bank, what's the best way to do it,is there any way I can put this off 'til tomorrow, do I have themoney in the bank, how can I avoid traffic, can I do this and getwhere I have to be. That to me is a problem to solve.I would say most math problems would involve problem solving.Conceptions. For Sandy, problem solving occurs whenever someone ismotivated to perform a task which is not strictly mechanical. The actualcarrying out of mechanical operations, such as stapling some papers orii13
performing arithmetic operations, is usually not problem solving, but thetask of having to get some papers stapled or of having to come up with asolution to an arithmetic problem does entail problem solving. However,what is mechanical for one person may not be for another, and so the act ofstapling papers or performing arithmetic can also constitute problemsolving. The designation depends on the performer as well as the task.Some of the types of problems Sandy describes are especially relevant tohis teaching. Learning to differentiate concepts and apply them is a f orm ofproblem solving: for example, "Identify which of these equations arequadratic equations," and "I don't have to think what a red light means anymore, but that, when I was learning, that was a problem to be solved."Another form of problem solving closely related to this is learning to selectand apply procedures, in a sot of heuristic rather than mechanical fashion:"If I give them a set of quadratic equations not written in standard form,and asked them to solve them using a variety of methods. Some that canbe factored, some that aren't quadratic, some that they get imaginary rootsusing the quadratic formula." Interactional and organizational tasks, suchas working together in groups or dealing with the teacher or taking notes,are another important type of problem solving. One of the most significantforms of problem solving for Sandy himself is his teaching, "this greatintellectual challenge, to teach well." He distinguishes interpersonal andprofessional problems from academic problems in that, with the former, "asan individual you can't necessarily control the outcome; one's own actionsare not enough to arrive at a solution that might be desired." .Most"important life tasks" are of this sort.
In Sandy's view virtually all math problems entail problem solving, butsome more so than others. In one of the experimental tasks (see Appendix C,Task 2) Sandy read a set of 23 problems appropriate to high school algebra,varying in content, conceptual difficulty, and conventionality. Examples are"Graph the equation x2 4y216y20 0" and "Would you and your familybe comfortable living in a 440 square foot house? Why or why not?" Hedecided that every one would involve problem solving for his students andfor himself. In classifying the problems according to their probable degreeof pro'clem solving, however, he rated all the more conceptual and lessstandard problems (such as the house example above) as excellent or goodand virtually all the more standard problems (such as the graphing example)as fair or poor. In another task (see Appendix C, Task 4) Sandy chose atopic, then generated various problems which he thought represented eitherlow or high degrees of problem solving within that topic. Again he selectedproblems with less standard forms as involving more problem solving,because "there may be more tasks layered in, there's options that they mayhave to explore, there may be more alternatives. It may take moresticking-to-it power for them . so it may take a higher degree ofmotivation." As for word problems, Sandy refers to them as "typical" or"traditional" problem solving. He believes they involve maximal problemsolving and are important, but they don't define problem solving for him asthey do for some teachers and in some textbooks.Problem solving refers to solution methods as well as problem types.Mthough Sandy never used the word "heuristics," several times he listedvarious Polya-style heuristics as typical problem-solving skills: forinstance, "Figure out what skills you need, what is the problem asking, lay1315
out a plan of attack, evaluate your answer at the end, work cooperatively,. what resources are necessary, do you have the abiiity to even begin thisproblem, . is this problem solvable?" These "broader, overarching skills"apply across content areas: "I don't see mathematical problem solving asbeing much different than other types of problem solving." What does varyacross disciplines or applications is the underlying content skills, whichare different from but necessary for successful problem solving: "There's adistinction between the mathematics involved and the mathematics problemsolving." "In order to solve a problem you need the background content." Atthe human level, "There is a generalized problem-solving ability that peoplehave," though "I wouldn't generalize it to all parts of that person's life.""People may get 'hung up on the skills involved." For example, SandyconsH-Ts himself good at solving many kinds of problems, but he lacksskillsplumbing. Consequently he could not solve a plumbing problemdirectly but would do so via a more general heuristic, seeking help--that is,calling a plumber.Skills and attitudes.(Sandy, interview) I'd say I'm a good problem solver. Ifdon't understand something, I will either stop and ask for help,or find a way to go through it. For example, if you gave me acalculus problem to solve now, I think I could solve it, eventhough I haven't seen calculus for fifteen years. I'm not sayingI could do it on my own, but if you said, okay, you know, comeback in X amount of time with the solution, if it was areasonable amount of time, I think I could f ind the solution.Because I would go backwards to the point, I'd be willing to putin the effort and time to figure out what background I needed,and then I would . rebuild my understanding from there.I1416
In arithmetic, where he has the skills, Sandy seems justified in callinghimself a good problem solver. He solved two non-standard problemscompetently, using a variety of heuristics in a deliberate and organizedfashion. For instance, in a "how many ways can you make change for a dollargiven certain coins" problem (see Appendix C, Task IF), he stated hissimplifying assumption that "all the pennies are the same"; made asimplifying substitution, reducing eight pennies to five; chose a search-treestrategy based on size of coin, and carried it out systematically; noted analternative strategy along the way; worked backward as well as forward;checked his work when finished, discovering a missed combination and anunnoticed relationship; and restated his assumption with the final solution,which was correct.Sandy considers himself skilled at academic and educational problemsbut not so good with physical problems. Not surprisingly, his attitudes runin the same direction. He enjoys his dissertation and the challenge of howto teach well but doesn't like problems involving his car or income tax. Asfor mathematics, Sandy seems to like problems more because they relate tohis teaching than for their own sake. One of his tasks (see Appendix C, Task3) was to rate 21 math problems, varying widely in topic and difficulty,according to how much he would enjoy solving them. He consistently ratedteaching-related and conceptual problems high (e.g., "How are complexnumbers like real numbers, and how are they different?"), number theoryproblems low (e.g., 'find MAID if 03 DAD and (IM)2 MOM"), and logicproblems and strategy games intermediate; algebraic problems ranked lowif simple and higher if harder. Sandy's own use of problem solving parallels
his attitudes for the most part. Problem solving is "a crucial ability" inSandy's life, one he uses "all the time": in planning lessons, responding tostudents, writing his university dissertation, figuring his income tax,juggling his.schedule. Mathematical problem solving, however, he uses agreat deal in his teaching but little if at all outside of teaching.Pedagogical KnowledgeCurriculum and content.(Experimental task--see Appendix C, Task 6) Given a list of 40things that students might learn in high school math, Sandyrated each on a five-point scale according to its importance.The items included skills, concepts, real-world applications,problem solving, and affects; they also covered arithmetic,algebra, geometry, and '.inspecified topic areas. Examplesinclude "Use a calculator efficiently to perform arithmetic";"Explain the relationship between area and linear measure";"Use functions to represent real-life relationships"; "Detectunreasonable results"; "Show enthusiasm for doingmathematics."Sandy spontaneously created two columns, for college- andnon-college-bound students. He consistently rated concept,application, skill, and problem-solving items high for thecollege group, but reduced the rankings of concepts,applications, and advanced skills for the non-college group,leaving problem solving and basic skills at the top. He rankedaffective items consistently low for both groups.Sandy sees problem solving as a critical component of mathematicsteaching; once he even described the content as a vehicle for problemsolving. He is aware of the current educational climate: "I think there is apublic emphasis on problem solving; it's somehow seen as a better thing to1618
be teaching." Charact2risU ally, however, he qualifies this view: "I don'tthink its quite that neat. I think its much more linked than the popularconception of problem solving." That is, he also values math content skills,without which mathematical problem solving is impossible, and believesthat problem solving must be fully integrated with the math curriculum. "Inorder to solve a problem you need the background content; even tounderstand content. hopefully it would be good to put in the context of aproblem where the content arose." Better understanding of the content isone of Sandy's goals for teaching problem solving.Another of his goals for problem-solving instructton is the developmentof generalized skills and affective qualities such as confidence andpersistence. This is one of Sandy's rare references to affect and at firstglance appears to contradict his low evaluation of affective objectives inthe task described above. However, four of the five affects in the task werespecifically related to mathematics--for example, "express appreciation forthe beauty of mathematics," or "plan to continue studying mathematics"-whereas the affective attributes Sandy values are generic across subjectareas. This is another instance of his cross-disciplinary propensity and isperfectly consistent with his views of problem solving in general.The problem-solving content that Sandy wants his students to learnconsists of general heuristics: defining the problem, determining the goal,developing a strategy, using previously learned skills, getting help if stuck,working through the problem, interpreting the answer, and using alternativemethods. The vehicle he cites for teaching these heuristics is all themathematical problems encountered in the standard curriculum, includingbut by no means limited to word problems. Sandy sometimes supplements911
the textbook's problems with algebra problems from other sources but neverwith problems unrelated to the current mathematical topic. This isconsonant with his own relative preference for problems
Problem Solving, the new find in math education, or Prohiem Solving, a topic, to be squeezed in somewhere between Fractions and Quadratic Equations. It, is problem solving with a small "p", applicable across disciplines and even outside of school, woven into the fabric of mathematics rather than stamped on top.