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Problem Solving as a Means Toward Mathematics for All: An Exploratory Look Through a Class Lens Sarah Theule Lubienski, Iowa State University As a researcher-teacher, I examined 7th-graders’ experiences with a problem-centered curriculum and pedagogy, focusing on SES differences in students’ reactions to learning mathematics through problem solving. Although higher SES students tended to display confidence and solve problems with an eye toward the intended mathematical ideas, the lower SES students preferred more external direction and sometimes approached problems in a way that caused them to miss their intended mathematical points. An examination of sociological literature revealed ways in which these patterns in the data could be related to more than individual differences in tempera- ment or achievement among the children. I suggest that class cultural differences could relate to students’ approaches to learning mathematics through solving open, contextualized problems. Key Words: Curriculum; Equity/diversity; Gender issues; Problem solving; Race/ethnicity/SES; Reform in mathematics education; Social and cultural issues; Teaching (role, style, methods) The National Council of Teachers of Mathematics (NCTM) argued that problem solving should become “the focus” of mathematics in school (1989, p. 6). According to NCTM (1989, 1991), centering mathematics instruction around problem solving can help all students learn key concepts and skills within motivating contexts. In contrast with previous emphases on problem solving as an end in itself (e.g., focusing on heuristics), the current emphasis is on problem solving as a means to learn mathematical content and processes. CLASS 1 DISMISSED? Because current reforms are intended to help all students gain mathematical power, studying how the reforms affect the least powerful in our society—lower Journal for Research in Mathematics Education 2000, Vol. 31, No. 4, 454–482 The research reported in this article was part of my doctoral dissertation, directed by Deborah Loewenberg Ball. The author wishes to thank Deborah, Suzanne Wilson, Daniel Chazan, and David Labaree as well as Glenda Lappan and other members of the Connected Mathematics Project for their support in this endeavor. I am grateful to Kathy Burgis, Angela Krebs, and Chris Lubienski for their help in interviewing students. Chris and Deborah have my gratitude for nurturing me and this work in every stage of the process. 1 I use the term SES when referring to particular students, but I use class when discussing larger, soci- etal structures and groups. SES can be thought of as an approximation for one's class, which connotes more permanence and shared beliefs about roles in society and in relation to power (Secada, 1992).

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  • Problem Solving as a Means Toward Mathematics for All:

    An Exploratory Look Through a Class Lens

    Sarah Theule Lubienski, Iowa State University

    As a researcher-teacher, I examined 7th-graders experiences with a problem-centered curriculumand pedagogy, focusing on SES differences in students reactions to learning mathematicsthrough problem solving. Although higher SES students tended to display confidence and solveproblems with an eye toward the intended mathematical ideas, the lower SES students preferredmore external direction and sometimes approached problems in a way that caused them to misstheir intended mathematical points. An examination of sociological literature revealed ways inwhich these patterns in the data could be related to more than individual differences in tempera-ment or achievement among the children. I suggest that class cultural differences could relateto students approaches to learning mathematics through solving open, contextualized problems.

    Key Words: Curriculum; Equity/diversity; Gender issues; Problem solving; Race/ethnicity/SES;Reform in mathematics education; Social and cultural issues; Teaching (role, style, methods)

    The National Council of Teachers of Mathematics (NCTM) argued that problemsolving should become the focus of mathematics in school (1989, p. 6). Accordingto NCTM (1989, 1991), centering mathematics instruction around problem solvingcan help all students learn key concepts and skills within motivating contexts. Incontrast with previous emphases on problem solving as an end in itself (e.g.,focusing on heuristics), the current emphasis is on problem solving as a means tolearn mathematical content and processes.

    CLASS1 DISMISSED?

    Because current reforms are intended to help all students gain mathematicalpower, studying how the reforms affect the least powerful in our societylower

    Journal for Research in Mathematics Education2000, Vol. 31, No. 4, 454482

    The research reported in this article was part of my doctoral dissertation, directed byDeborah Loewenberg Ball. The author wishes to thank Deborah, Suzanne Wilson, DanielChazan, and David Labaree as well as Glenda Lappan and other members of the ConnectedMathematics Project for their support in this endeavor. I am grateful to Kathy Burgis, AngelaKrebs, and Chris Lubienski for their help in interviewing students. Chris and Deborah havemy gratitude for nurturing me and this work in every stage of the process.

    1I use the term SES when referring to particular students, but I use class when discussing larger, soci-etal structures and groups. SES can be thought of as an approximation for one's class, which connotesmore permanence and shared beliefs about roles in society and in relation to power (Secada, 1992).

  • 455Sarah Theule Lubienski

    class childrenmakes sense. Considering class-related equity issues is particularlyimportant for mathematics educators, inasmuch as mathematics serves as a crit-ical filter (Campbell, 1989, p. 95), with successful students potentially rewardedwith high occupational status and pay.

    Yet, class is rarely focal in current educational studies. In a recent ERIC data-base survey of roughly 1,000 research articles published in U.S. mathematicseducation journals from 19821997, 89 gender-related articles, 32 ethnicity-relatedarticles, and only 6 class-related articles were found (Lubienski, 1999).2 Afterreviewing the literature on equity, Secada (1992) noted the lack of serious atten-tion given to social class in mathematics education:

    It is as if social class differences were inevitable or that, if we find them, the resultsare somehow explained. While the research literature and mathematics-educationreform documents (for example, NCTM, 1989; NRC, 1989) at least mention womenand minorities, issues of poverty and social class are absent from their discussions.(p. 640)Perhaps the educational research community overlooks class-based differences

    in mathematics achievement because many problems lower class families face seembeyond the schools control. Among the barriers that lower class students need toovercome, some, however, arise within schools, even within mathematics class-rooms. Changing mathematics curricula and pedagogy can remove or add barriersfor lower class students.

    As Reyes and Stanic (1988) argued, researchers need to look inside classroomsto study the relationship between SES and mathematics learning.

    Before quantifying classroom processes, more qualitative work is necessary to pointout categories that may better capture the wholeness and richness of classroom life.Within mathematics education research, relatively little work has been done on race-related differences and almost no work has been done on the relationship between SESand mathematics performance. (pp. 3940)

    Frankenstein (1987) agreed, arguing that issues previously examined with thegender lens should be examined with respect to race and class; these issues includeattitudes about mathematics and types of pedagogies and curricula that are helpfulor harmful for various groups.

    Yet, class is difficult to study. In this time of emphasizing the positive aspectsof diversity, discussing class differences and raising issues about potentially nega-tive aspects of lower class cultures are risky endeavors. Additionally, the lack ofagreement about definitions and indicators of class makes class particularly diffi-cult to study (Duberman, 1976). Despite these difficulties, in this study I focus onclass differences in students experiences in one problem-centered mathematicsclassroom.

    2Class fared only slightly better in a more general pool of mathematics education research articlesfrom 48 U.S. and international education research journals. Among the 3,000 articles examined, 323articles pertained to gender, 112 pertained to ethnicity, and 52 pertained to class (Lubienski & Bowen,in press).

  • 456 Examining Problem Solving Through a Class Lens

    Although in a larger study (Theule-Lubienski, 1996/1997) I explored lower andhigher SES students general experiences with a Standards-based (NCTM, 1989,1991) pedagogy and curriculum, in this article I discuss two aspects of the NCTM-inspired mathematics problems that seemed to play out differently for the lowerand higher SES students studied. The first aspect is openness. NCTM advocatedusing problems or tasks that have no obvious solution and that can be solved invarious ways. Some problems have no single right answer and can take hours, daysand even weeks to solve (NCTM, 1989, p. 6). The second aspect is context.According to NCTM (1989), problems should arise from a motivating situation thatis often based in the real world but occasionally is strictly mathematical or fictional.

    THE PROMISE OF OPEN, CONTEXTUALIZED PROBLEMS

    The use of open, contextualized problems seems sensible at many levels. Insteadof having students complete meaningless exercises and memorize what the teachertells them, why not have students learn key mathematical ideas while solving inter-esting problems? Not only do these ideas seem sensible for all students, but in someways instruction centered around open, contextualized problems might seemparticularly promising for lower SES students.

    For example, some researchers have argued that lower SES students tend toreceive more than their share of rote instruction and low-level exercises fromteachers with low expectations (Means & Knapp, 1991). Anyon (1981) found thatstudents of lower SES received rote instruction whereas higher SES students wereactively involved in problem solving. This pattern would seem to perpetuate inequal-ities: Higher SES students are educated to be leaders whereas lower SES studentsare trained to be followers. In opposition to this trend, Frankenstein (1987) arguedthat lower SES and other disadvantaged students should learn mathematics throughanalyzing complex social issues, such as poverty and discrimination.

    Additionally, according to some scholars, lower class families tend to be moreoriented toward contextualized language and meanings. Bernstein (1975) arguedthat linguistic codes (or the underlying principles of speech) are affected by theclass system. The more privileged classes, who have power and tend to be indi-vidualistic, are socialized in a way that develops high-status knowledge and thelanguage of control and innovation. Because of the emphasis on the individual,meanings are not assumed to be shared with others. Hence, middle-class familiesuse what Bernstein called elaborated codes, language with meaning that is moreexplicit and less tied to local contexts. Lower status families use restricted codes,language with implicit and context-dependent meanings that make sense incontexts in which emphasis is placed on community and common knowledge andvalues are assumed to be shared. Additionally, Holland (1981) found that middle-class children tended to categorize pictures in terms of transsituational properties(e.g., grouping foods together that were made from milk or came from the sea),whereas working-class children tended to categorize pictures in terms of morepersonalized, context-dependent meanings (e.g., grouping foods they eat at

  • 457Sarah Theule Lubienski

    Grandmas house). Holland concluded not that children could not think differentlybut that they had been raised with a particular orientation. If Bernstein andHolland were correct, then contextualized mathematics problems seem to alignnicely with lower class students preferred ways of thinking.

    Hence, in theory, the use of open, contextualized mathematics problems appearspromising for promoting equity for lower SES students. Yet, in this exploratorystudy I raise questions about this theory as I examine differences in lower SES andhigher SES students experiences in one problem-centered mathematics classroom.

    METHOD

    In this study, I played a dual role as researcher and pilot teacher for a problem-centered curriculum-development project. Several aspects of the studys method-ology merit discussion, including characteristics of the school, the curriculum, theteacher-researcher role, my pedagogy, and the methods of data collection andanalysis.

    The School and Classroom

    This study was conducted in a socioeconomically diverse school located in amedium-sized midwestern city. The school was located in a distinct part of the citythat, largely because of the success of the local automobile industry, had previouslyenjoyed a solid tax base. Because of changes in the industry (e.g., downsizing),however, the economy had begun to decline in the past decade.

    The school had a socioeconomic mix of studentsa few upper-middle-classstudents (e.g., with professional parents holding graduate degrees), some middle-class students (e.g., with college-educated parents in professions such as teachingor engineering), some working-class students (e.g., with parents who worked infactories or held service jobs), and a few lower class students (e.g., with parentswith very limited educations, no steady jobs, and incomes below the poverty line).The schools 500 students were primarily Caucasian, with roughly 2% AsianAmerican, 3% Hispanic American, and 11% African American.

    The school was a pilot site for the Connected Mathematics Project (CMP), andthe students in this study had used the CMP trial materials during the year precedingthis study. As a year-long guest in the classroom of a seventh-grade mathematicsteacher, my role in the school included piloting the trial materials with one class whileserving as a CMP liaison and teaching model for other teachers in the school.

    There were roughly 30 students in the class (a few came and went across the year).The students were evenly balanced in terms of gender, and, with the exception oftwo African American males and one Mexican American female (whose familyhad lived in the United States for several generations), all the students wereCaucasian. These students remained together throughout most of each day, rotatingamong a team of four teachers, each of whom specialized in teaching mathe-matics, English, science, or social studies.

  • 458 Examining Problem Solving Through a Class Lens

    The Curriculum

    The CMP is a middle school curriculum-development project funded by theNational Science Foundation to create problem-centered materials aligned with theNCTM Standards. In this study, CMP trial materials provided the mathematicalfocus of each class period, with students working on the problems both in andoutside of the classroom each day. I refer to the materials as they appeared in a draftstage; I cannot and do not attempt to make any claims about the final materials.This study is not a test of one curriculum but an exploration of how currently popularideas being implemented with various curricula might differentially affect lowerSES and higher SES students.

    Both the CMP authors and NCTM share an emphasis on students learning math-ematical content and processes through solving open, contextualized problems. TheCMP authors described their vision as follows:

    The curriculum is organized around rich problem settings. Students solve problemsand in so doing they observe patterns and relationships; they conjecture, test, discuss,verbalize, and generalize these patterns and relationships. Through this process theydiscover the salient features of the pattern or relationship; construct understandings ofconcepts, processes, and relationships; develop a language to talk about the problem;and learn to integrate and discriminate among patterns or relationships. The studentsengage in making sense of the problems that are posed, and with the aid of the teacher,to abstract powerful mathematical ideas, problem solving strategies, and ways ofthinking that are made accessible by the investigations. (CMP, 1995, p. 24)

    Hence, the goal is not for students to simply learn about solving the individual prob-lems but for them to complete problem explorations having abstracted importantmathematical ideas or processes. This abstraction occurs both because problemsare carefully designed to require students to think about particular mathematicalideas and relationships and because the teacher plays an active role in highlightingthe intended ideas.

    The seventh-grade CMP trial materials consisted of eight units, each containingseveral investigations. The one to four problems in each investigation were care-fully constructed and sequenced to help students understand the intended, over-arching mathematical ideas of the unit. The problems were usually set in a real-worldcontext (e.g., analyzing disaster data or designing a house) and varied in the amountof time required to solve them from several minutes to several days. After many ofthe problems, there were follow-up questions to help students focus on key patternsand ideas. Each investigation concluded with a section of homework problems thatwere mathematically similar to the problems in the investigation but were gener-ally set in different contexts. Writing prompts to help students summarize the math-ematics in the unit were being added to the materials at this stage of development.

    The Teacher-Researcher Role

    Cochran-Smith and Lytle (1990) explained how teacher-researchers often askquestions that arise from working at the intersection between theory and practice.

  • 459Sarah Theule Lubienski

    This was true for me, inasmuch as it was during my CMP pilot teaching the yearprior to this study in a primarily middle-class, sixth-grade classroom that I beganto wonder about possible SES-related differences in students reactions to learningmathematics in ways aligned with current reforms. Although I felt generallysuccessful in promoting problem solving and deep discussions of mathematicalideas, I wondered why the few students who seemed to be of lower SES were alsothe students who said that they preferred a more traditional approach. When I hadthe opportunity to teach in a more socioeconomically diverse school (as describedabove) the following year, I decided to investigate my growing, yet still open andloosely focused, questions about this issue.

    These questions could have been addressed in various ways, but in this study,being the teacher as well as the researcher allowed me to design the case to bestudied (Ball, 1998). Because I played such a major role in this study, discussingthe context provided by my own background is important.

    Coming from a working-class family, I brought to this study an interest in thepotential of mathematics education to perpetuate or break the cycle of poverty. Ialso held expectations that lower SES students can succeed in mathematics, as Ihad succeeded. At the time of this study, I held a masters degree in mathematicsas well as a secondary mathematics teaching certificate. I was completing mydoctoral program, in which I had eagerly studied the theories underlying thereforms and worked with key figures in the NCTM reform movement. Withconcerns that lower SES students were too often taught to be followers instead ofcritically thinking leaders, I was enthusiastic about the reformers goals of devel-oping reasoning and problem-solving skills in all students. This was my third yearof working with the CMP in writing and piloting trial materials and preparingteachers to use the materials. When piloting the materials, I found that my math-ematics background and deep familiarity and agreement with the goals of NCTMand the CMP curriculum helped me guide students explorations and discussionsof the problems. Although my middle school teaching experience had been limited,3the CMP directors considered my year-long implementation of the sixth-gradecurriculum successful enough to merit employing me as a seventh-grade pilotteacher and teaching model the following year (the year of this study).

    My ability to implement a reasonable interpretation of a reformed classroomis an important contextual factor in this study. Although I certainly have more tolearn as a mathematics teacher, I contend that I had at least an average chance ofeffectively helping students learn mathematics through problem solving, as envi-sioned by the CMP authors and the NCTM Standards (1989, 1991).

    3With the exception of student teaching and NCTM pilot teaching, my teaching experience consistedof 2 years at the college level, at which I taught a variety of courses, including a Mathematics forElementary Teachers course intended to help prepare students to teach mathematics in ways alignedwith current reforms. It was during my teaching of this course that I won a teaching award.

  • 460 Examining Problem Solving Through a Class Lens

    Pedagogy

    As advocated by the CMP, lessons were structured using a Launch, Explore,Summarize model. In a typical lesson (which could take more or less than one classperiod), I launched a problem with the class, and then students worked in groupsto explore and solve the problem. Finally, I facilitated a whole-class discussionto summarize students explorations. During these discussions, I asked studentsto explain how they had solved the problems, encouraged students to compare ideasshared by their classmates, and posed questions to highlight key mathematical ideas.

    Overall, I viewed my pedagogy as reasonably consistent with the NCTMStandards: I facilitated students exploration, discussion, and sense making ofimportant mathematical ideas. Evidence from surveys and interviews indicated thatstudents shared my perspective about key aspects of my pedagogy. For example,I intended to guide students problem solving, prompting their explorations infruitful directions when they were stuck instead of giving them the rightanswer. On a multiple-choice survey question about my response to students whowere stuck on a mathematics problem, the majority of students chose Our teacherencourages us to figure it out for ourselves, and no student indicated Our teachertells us the answer. Similarly, students confirmed that I usually encouraged theconsideration of different problem-solving methods and that students worked ingroups and used calculators in our mathematics class at least half the time.

    Data Collection

    I began this study wanting to understand how lower SES and higher SES studentsin my classroom experienced the CMP trial curriculum and my pedagogy. I wasunsure what I would find, but I suspected that perhaps the contexts of the problemswould be more appealing to the middle-class students, inasmuch as a middle-classgroup of curriculum developers had chosen the settings. Or perhaps I would findthat the middle-class students would get more help at home on their nontraditionalhomework. I found little evidence to support either of these hypotheses, but my datacollection and analysis methods were open enough to allow other trends to surface.

    I used three sets of interviews, various surveys, student work, teaching-journalentries, and daily audio recordings to document students experiences, includingtheir struggles and successes with problems, what they thought they were learning,and what they found helpful or hindering.

    To get a sense of students SES backgrounds, I surveyed parents about their occu-pations, educations, incomes, numbers of books and computers in the homes, andnewspapers read regularly. These are commonly used SES indicators (e.g., seeDuberman, 1976; Kohr, Coldiron, Skiffington, Masters, & Blust, 1988). I used thesedata to place the students into two, admittedly rough, categories: lower SES andhigher SES. The lower SES students were primarily working class, but a few ofthem could be considered lower class. The higher SES students were what mostAmericans would call middle class, with some families bordering on upper-middleclass. I ultimately gained permission to include 22 of the 30 students, 18 of whom

  • 461Sarah Theule Lubienski

    provided clear SES data that allowed me to categorize them as lower SES orhigher SES. Lower SES students are referred to by one-syllable pseudonyms: Thefemales are Rose, Anne, Dawn, Sue, and Lynn; the males are Carl, James, Nick,and Mark. Higher SES students are referred to by three-syllable pseudonyms: Thefemales are Samantha, Rebecca, Guinevere, and Andrea; the males are Benjamin,Timothy, Christopher, Harrison, and Samuel. Two-syllable pseudonyms refer tostudents who were not categorizable because of missing or ambiguous SES data.

    I collected homework and survey data from all participating students and inter-viewed them at the end of the year. I also more closely followed a smaller groupof target students, interviewing them at the beginning of the year and havingcolleagues interview them later in the year. A low- and a high-achieving male andfemale from each SES group were selected to be target students to help me distin-guish SES from achievement (as measured by students initial performance in theclass, including the quality of their quizzes, homework, and participation).4

    Survey and interview questions focused on students experiences with and reac-tions to the CMP curriculum and my pedagogy. In later interviews students wereasked to consider how their opinions about the class and curriculum might bechanging and why.

    Students also completed CMP surveys regarding the pedagogy and environmentin our classroom. Additionally, in CMP-designed Show What You Knowsurveys, students were asked to describe their experiences with particular units; theywere asked, for example, which mathematical ideas were most interesting andimportant and what activities were most enlightening.

    Data Analysis

    In analyzing the data, I focused on similarities and differences of lower SES andhigher SES students experiences and also considered interactions with gender.(Scholars have pointed out the dangers of studying the variables of race, class, andgender in isolation [e.g., Campbell, 1989]. Elsewhere [Lubienski, 2000], I havegiven more attention to the gender interactions in this study.) With a primarilyCaucasian sample, I was able to focus on gender and class without much variationin race within those categories.

    After completing the school year, I began formal analysis with a systematic exam-ination of my teaching journal as well as of tables containing transcriptions of allsurvey and interview data organized by question. I referred to audio recordings andstudent work relevant to specific questions that arose from my analyses of thesurveys, interviews, and my journal. Through these initial analyses, I developed

    4For a variety of reasonsincluding fear of attrition, the desire of some students to be heard, and thedesire to fill gaps due to diversity within the categoriesI interviewed more than 8 students. For example,I added Sue to my pool of interviewees because she requested the opportunity to be interviewed, andshe added some dimensions that her fellow lower SES females did not (she was only midachieving,though she tried very hard, and she raised some important gender issues because she was often ridiculedby the boys in the class). Twelve students participated in the first round of interviews, 14 participatedin the second round, and 18 students participated in the final round.

  • 462 Examining Problem Solving Through a Class Lens

    roughly 60 questions or mini-themes, such as reactions to being frustrated onproblems, views on whether feelings are hurt in class discussions, and beliefs aboutwhich style of instruction helps students think and learn more. For each of thesethemes I created a table in which all relevant survey or interview data for eachstudent were recorded and categorized by SES and gender. I wrote summaries ofthe data pertaining to each theme as well as the data relating across themes, care-fully noting conflicting evidence for individual students and SES or gender groups.I paid particular attention to data on high-achieving, lower SES students and low-achieving, higher SES students because doing so helped me sort out differencesthat seemed more aligned with achievement than with SES.

    Because many recurring themes related to students experiences in whole-classdiscussion, I also systematically examined some of these discussions to comparestudents participation with their reported experiences in the discussions. I randomlyselected 2 days from each of the seven curricular units used throughout the year,and I analyzed the audiotapes of the whole-class discussions from those days. Icoded each student contribution to discussions (a total of almost 300 contributionsover the 14 days) using 20 categories designed to capture the content, problemcontext, social context, reasoning, visual/tactile references, tone, correctness,insightfulness, mathematical relevance, and difficulty level associated with eachcontribution. I looked across the results of each days coding for gender and SESpatterns in students participation.

    In addition to the analyses of the data for all participating students, I wrote casesof the female target students. Looking across the larger data set helped me see trendsin SES differences; looking closely at individuals helped me understand the trendsand how they interacted to shape the experiences of individual students.

    RESULTS

    During the year, I enjoyed watching some successes, such as the most seeminglyapathetic students becoming engaged in problem explorations, unfold. Yet, inconsidering which students seemed to experience and benefit from the curriculumin ways we authors intended, I noted some SES-related trends pertaining to twokey aspects of the curricular problems: their openness and their contextualizednature. In the first several sections below, I present data on students experiencesthat relate primarily to the open nature of the curricular problems, includingstudents appreciation of the open problems in relation to more computation-oriented curricula, students struggles and motivations to make sense of and solvethe problems, and students thoughts about what they were learning. In the finalsection, I focus directly on the contextualized nature of the problems and how thisaspect might have differentially affected students.

    General Feelings About Traditional Versus CMP CurriculaThe survey and interview evidence relating to students curricular preferences

    indicated that more of the higher SES students than the lower SES students

  • 463Sarah Theule Lubienski

    preferred the CMP trial materials to the more typical mathematics books they hadexperienced through fifth grade. Many students expressed a mixture of opinionsabout the curriculum, but the four students who consistently expressed preferencefor CMP were all higher SES. Four of the six students who consistently said thatthey preferred typical mathematics to CMP were lower SES.

    In using the CMP, students are asked to think harder than and differently frombefore, so it is not surprising that many students reacted negatively to the new expec-tations. Yet, their complaints were not all the same. The predominant theme amongthe lower SES students was that the problems were frustrating because they weretoo confusing or hard (a theme for eight of the nine lower SES students). Forexample, Sue said, The books are confusing [because the] questions are too longand complicated. Dawn declared, I dont like this math book because it doesntexplain EXACTLY! Moreover, six lower SES students specifically said thatthey were better at mathematics as it had been taught previously. For example, Suesaid, I used to do really good in math, and James explained, Im worse [now],cause I used to could do the work, but here I dont understand it. When pushedto explain what exactly was hard or confusing, the lower SES students would talkprimarily about difficulty in determining what they were supposed to do with theproblems, difficulty they attributed to the vocabulary and general sentence struc-tures used as well as to the lack of specific directions for how to solve the prob-lems. Because SES correlated with achievement, it is important to give attentionto those students who were of low SES and high achieving as well as to studentswho were of high SES and low achieving. Although both Rose and Lynn were high-achieving students in all school subjects, they shared their lower SES peers atti-tudes about the difficulty of the curriculum. Rose consistently said that the CMPwas harder for her because of the lack of specific direction. She explained, Thesebooks are bad because they are so confusing. We are told to do a page as home-work, and the page gives directions but it doesnt explain how to do it. Lynn said,Im better at number problems than problem solving.

    Some higher SES students also complained about the books being confusing,but their complaints were usually voiced toward the beginning of the year (whenit was cool to complain about this) and were not so passionate or personalized;they often offered a specific suggestion or pointed to one problem or word that wasunclear. For example, Guinevere suggested that the CMP authors include a glos-sary, and Samantha recalled that the CMPs use of the word correspondingconfused her until she learned what it meant. These specific complaints were quitedifferent from those of the lower SES students, who conveyed the feeling ofhaving no idea how to proceed on the problems.

    Although no lower SES students said that the CMP curriculum was easier forthem than typical curricula, several higher SES students made comments aboutCMP problems being easier than computation. Guinevere explained that CMPproblems are a lot easier [for me because] I guess our familys justwe are word-problem kind of people. Similarly, Christopher explained that CMP is easierbecause Im pretty good at problem solving. Benjamin encouraged the CMP

  • 464 Examining Problem Solving Through a Class Lens

    authors to make their problems more challenging. Additionally, Rebeccaexplained her views to an interviewer (I):R: This year were doing stuff that I like. Two years ago we did 20 times 4; I hated it

    because I wasnt good at it and Im good at this. [CMP is] easier. [Before] we justsat there with hundreds of problems on a page.

    I: Why do you think its easier for you (when others say it is harder)?R: I dont know; its just my abilities. I: Would you rather have the teacher tell you rules, or would you rather figure them out?R: No. Learning about it like exploring how to do things is easier for me than sitting down

    and learning the rule.

    I suspected that part of the lower SES students frustration might be due to theirfeeling less than confident with the mathematical ideas that were prerequisites forthe curriculum. James, who had just transferred from another school, said that allthe material was new. Nick and Lynn said that most of it was new. Yet all otherlower SES students said that at least half of it was familiar. Apparently, at least frommost students points of view, if they were struggling, something other than themathematics posed the difficulty.

    Internal/External Direction

    As one might guess from their comments about wanting clearer directions in theCMP trial materials, the lower SES students preferred a more directive teacher role.When asked to rank various modes of working on problems, the lower SESstudents, especially the girls, ranked having specific teacher direction highly.Also, the lower SES girls more often asked me, Is this right? Only lower SESstudents said that they preferred having the teacher tell them the rules. Three ofthese lower SES students said that trying to explore the problems and figure outthe rules confused them, and two of them said that being told rules allowed themmore time to work with and understand them. When the lower SES students talkedabout aspects of my teaching that were good, they focused on my ability to explainthings well, particularly the CMP questions.

    Teacher direction seemed less important to higher SES students. For example,Andrea liked my not too strict style, and Benjamin said, Ms. Lubienski is a goodteacher because she doesnt give answers, but helps and is nice.

    In the face of the uncertainty the open problems presented, the lower SESstudents seemed more passive, less confident about how to proceed. On the varioussurveys and interviews, lower SES students tended to say that they would becomefrustrated and give up when stuck, whereas higher SES students often said thatthey would think harder about the problem or just interpret it in a sensible way andget on with it. For example, Christopher explained his reaction to confusing prob-lems as follows:

    I just, like, try to figure out what I was trying to do, and just go and do what I think itstrying to say. I just try my best to figure out the directions, so if I get it wrong, itsjust because of directions and not because I did the problem wrong.

  • 465Sarah Theule Lubienski

    The higher SES students more than the lower SES students displayed intrinsicmotivation to solve the problems and grapple with mathematical ideas. The fourstudents who said that they liked to figure problems out and really understand ideaswere higher SES. Although there were several examples in my field notes ofhigher SES students showing intellectual curiosity and excitement about chal-lenging mathematical problems, there were no such examples for lower SESstudents. For example, Benjamin once became totally engaged in a context-freeproblem that required him to find the dimensions of a cylinder with a volume of1,000 cubic units. Although he asked me a few questions as he excitedly usedvarious methods to get successively closer to 1,000, he made clear that he did notwant me to tell him too much. When finished, he proudly said that solving thisproblem was the hardest thing he had ever done. Christopher also became veryengaged in the same problem and, on his own, created an extension of the problem,which he showed me the following day.

    I do not mean to say that the lower SES students never engaged in or enjoyedthe problems but that they seemed to be motivated more by the activities involvingfun, games, and contexts of interest to them (such as sports for Nick or dream housesfor Rose and Sue); the mathematics did not tend to draw them in or to receive focus(I return to this later). Audiotapes of lower SES students group work revealedinstances of their hurrying through problems but not wanting to appear to befinished for fear I might push them to rethink or discuss their solutions. In my journalI recorded several instances in which the lower SES students (especially the high-achieving females) seemed more concerned about getting the algorithm that wouldallow them to complete assigned work than about grappling with or understandingmathematical ideas for their own sake.

    Overall, more of the higher SES students than the lower SES students seemedto possess orientations and skills that allowed them to actively interpret the openproblems, believe their interpretations were sensible, and follow their instincts infinding a solution. The lower SES students seemed more concerned with havingclear direction that enabled them to complete their work and were less apt tocreatively venture toward a solution.

    Thinking/Learning More

    Students of both lower and higher SES agreed that in their former mathematicsclasses they had learned more basic skills, such as multiplication and division.Several students of both SES groups said that the CMP made them think more,and most students said that CMP helped them see how mathematics is connectedto real life.

    One difference in the way students talked about the benefits of CMP was thatthe higher SES students clearly stated that they were personally helped by the CMP,whereas the lower SES students talked about the CMPs benefits from a moreexternal standpointeither saying what they thought they should be learning orwhat others said they were learning.

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    For example, Samantha explained, [In CMP] Ive learned new stuff and Im ableto think more stuff that will help in real life in the future. I can probably figureout a problem now, without having someone just tell me the rulelike if weredoing integers again, I could probably figure out a rule. Additionally, Andreaexplained in her final interview that in her former mathematics classes, she wouldforget everything over the summer, but with CMP she remembers what she learnslonger: [CMP] sticks with youit just stays with you.

    As with the higher SES students, most lower SES students could articulatemuch of what the CMP was intending. For example, Rose explained, CMP is kindof like real-life stuff, so I think that maybe they want you to extend your brain. Theyarent going to say, here are the rules; they want you to figure it out yourself.When pushed to say whether the CMP worked for them as the authors had intended,all higher SES students (except two who gave mixed reviews) said yes. But thelower SES students, especially the girls, tended to be hesitant, saying I dont know or kind of. The majority of lower SES students talked about their difficultiescaused by the curriculums interfering with their learning. Lynn (known by herteachers as a very motivated, high-achieving student) articulated views shared bymany of her lower SES (especially female) peers, as indicated by their survey andinterviews throughout the year:L: I usually just try to remember it [the rule] and do the problems. I dont ask how or why

    it works. I think sometimes it confuses me more, and sometimes Im not interested; Ijust want to know how to do the problems or something. Like when we were doingthe, um, the negatives and positives, adding and all that stuff, for a while there you triedto have us figure out the rules, and then when Miss Mattel [their regular classroomteacher who substituted for me while I was away] taught us, she just told us the divi-sion and multiplication laws, and I liked it when she told us so we could just do em,and I didnt have to sit and try to figure them out. Cause I think when I didnt knowthe rules, I got em wrong, and now that I know the rules, I get most of em right now.

    I: Why do you think we [the other CMP authors and I] want you to figure out the rules?L: So you can, like, learn how they do it. Or try to, um, so you can just likeso you can,

    um, I dont know why really, but I know its a good reason.I: Why do you think its a good reason if you dont know it?L: Cause if you can figure em out, then maybe youre learning more; if you can figure

    out how they do it or how the rules go, maybe you understand it more or something.I: But you prefer not to do that?L: Yeah, cause it confuses me cause when I try to figure things out, I like the rule I

    get, and I stick to it even though its not right, and when we get the real one, it confusesme.

    Lynn provided a summary of many of the issues discussed above. Most lowerSES students (especially the females) preferred having the teacher tell them therule so they could get the right answer to problems. Lynn did not seem to viewlearning to work through ambiguities of problems as an important curricular goal;the ambiguities were simply confusing obstacles to knowing the right rule to useto obtain the correct answer. Like many other lower SES students, Lynn talked inthe third person about the curriculums being helpful. She thought that maybe the

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    other CMP authors and I had a good reason for wanting students to figure outthe rules themselves, but the lack of clear direction and rules confused her. In theend, Lynn did state that she thought that her experience with CMP had improvedover time, and she attributed this improvement to having a teacher who understoodthe problems and could explain things (or simply restate the problem) in plainEnglish. For example, she once asked me, Why dont they [CMP authors] writeit like you say it? When I figure out what Im supposed to do, I can do it; it justtakes me longer.

    In the data presented thus far, I have focused on the open nature of the problemsin the CMP trial materials. The other aspect to be discussed is the problemscontextualized nature.

    Contextualization

    One might expect that if lower SES students tend to have a contextualizedorientation to ideas, they would benefit from contextualized problems. Yet, the datafrom this study indicate that this supposition should be questioned.

    The data suggest that the lower SES students were, indeed, more contextualizedin their orientation toward ideas. I (Lubienski, 1997) interviewed the students abouttheir interpretations of mathematical claims presented in the media (e.g., magazineadvertisements, newspaper articles). The lower SES students were more likely thanhigher SES students to mistrust and ignore the data provided and to reason in person-alized and common sense ways, such as by referring to pictures or offering storiesabout friends and family members who had tried various products. Higher SESstudents tended to scrutinize given mathematical information to find a loophole.Their reasoning was not necessarily more correct than that of the lower SESstudents, but it was more mathematically focused and impersonal.

    Additionally, the type of language and proof used by students in their contribu-tions to class discussions (on the 14 randomly selected days) suggest at least amoderately more contextualized orientation among the lower SES students. Icoded whether students used generalized or contextualized language. For example,in a problem about differences in stores prices, I recorded as contextualized thelanguage of someone who referred to the objects, the dollars, and the stores, andas generalized the language of one who referred only to the numbers without contex-tual attachment. To avoid exaggerating or masking differences, I coded contribu-tions that were not clearly generalized or contextualized as in-between. Accordingto the data, roughly two thirds of each groups contributions were in-between. Still,there were patterns in how the remaining contributions fell. Contextualized languagewas used more by the lower SES males (16% of their contributions) and females(23%) than by the higher SES males (3%) and females (10%). Generalized languagewas used more by the higher SES males (38%) and females (24%) than by the lowerSES males (11%) and females (6%).

    I also developed six categories of proof: proof by pattern, common-sense proof,deference to rule, proof by one example, general proof, or in-between proof. A proof

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    by pattern involves arguing that something is true because it follows a predictablesequence. A common-sense proof is a nonmathematical (in the traditional sense)rationale that relates to everyday living, whereas deference to rules involvesarguing that something is true because of a rule learned previously. A proof by oneexample is an argument that a general statement is true because it holds in one case,and a general proof is what might be considered a typical, deductive proof: anabstract argument through which one logically derives a general (i.e., not tied tothe problem at hand) conclusion from previous conclusions. An in-between proofinvolves giving a general explanation of some sort using the problem context athand, hence, a proof that is somewhere between a general proof and a proof by oneexample. Again, when in doubt, I coded the contribution as in-between. Themajority of students contributions were in-between, ranging from 52% (lower SESfemales) to 76% (higher SES females). The higher SES students contributed 8 ofthe 10 general proofs offered. Proof by one example was used twice, both timesby lower SES girls. The lower SES girls gave 7 of the 9 common-sense proofs. Thelower SES students deferred to rules more often than did higher SES students,providing 16 of 24 instances (about 25% of lower SES students contributions werein this categorytwice the percentage for the higher SES students).

    Might these differences be attributable simply to achievement differencesbetween the SES groups? On the basis of the evidence, I suggest otherwise. Forexample, Rose was a very bright, high-achieving student, and yet she sometimesdrew conclusions from one example and seemed to have difficulty distinguishingbetween that and a more general proof. I noted the following incident in myjournal:

    Benjamin was very insistent about his (correct) theory that the area increases by thesquare of the scale factor. I was asking for explanations as to why this happens. Again,Rose chimed in with, Well, we know that the area is 18, so it has to be multiplied by3, and Benjamin said, But WHY does it happen?If you didnt know what itwould be, how could you figure it out? Rose has a very difficult time with thisthisis the second or third time this exact situation has come up.

    Rose also had difficulty distinguishing between an opinion-based question and onethat could be mathematically analyzed. For example, on a probability question aboutwhere a spinner is most likely to land, she said that because different people havedifferent guesses, it is just an opinion question. I also recorded several examplesin which Roses real-world, common sense reasoning seemed to take precedenceover her finding or realizing the usefulness of the intended mathematical solutionsto problems. For example, in response to a question about data stating that men arekilled in car accidents more often than women, Rose said, Maybe men were inthe wrong place at the wrong time. As another example, in a CMP problemstudents were asked to compare the volumes and prices of three sizes of containersof popcorn sold at a movie theater. Rose had no trouble finding the volumes fromthe given dimensions, and then she used solid common sense reasoning to arguethat, inasmuch as the prices went roughly in order of size, the choice should be deter-mined by need: It depends on how much popcorn you want. Although Roses

  • 469Sarah Theule Lubienski

    reasoning was completely sensible because she used the context to determine thedegree of accuracy needed, in using this approach, she did not have the intendedexperience of working with volumes and comparing unit prices.

    Rose was typical of many of the lower SES students who approached ideas in acontextualized manner, an orientation that could sometimes hinder understandingof relevant mathematical ideas. As an extended example, I offer the following datasurrounding a problem about sharing pizza. This is an example in which both lowerSES and higher SES students were actively engaged with the curriculum anddiscussion. But in looking beyond that similarity, one can see differences in thethinking and reasoning of the higher SES and lower SES students.

    Imagine that you entered a pizza parlor and saw pizzas being served at two tablesof your friends, one table with 10 people and the other with 8 people. Friends ateach of the tables call out to you to pull up a chair and share their pizzas. You couldchoose the table with fewer friends or the one with more friends. Or you mightchoose the table with the most pizza. But is this the best way to make the deci-sion? Are there other more helpful ways that we can use mathematics to comparethe two situations?

    Table 1 Table 2

    Problem 1.2: If you like the two groups of friends equally well, which table would you join and why?

    Figure 1. Problem from trial version of Comparing and Scaling.

    The authors intended the problem presented in Figure 1 to help students learnto create and compare ratios. As I launched this problem, I clarified that studentsshould assume that they want to go to the table at which they can eat the most pizza.Students worked on this problem in groups for about 15 minutes, and then Rosebegan our whole-class discussion by sharing the work she had done in her group.(T refers to me, the teacher.)Rose: Um, youd think they would be equal if they were divided up into 4; I dont

    anyway, if they were divided up into 4.

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    T: Into 4 pieces each you mean?Rose: Yeah, at Table 1 there would be 16 pieces.T: Okay, 16 pieces, all right. [I write her information on the board as she talks. My

    okays signal when I have it written.]Rose: And on Table 2 thered be 12. And so, um, thered be 4 people who wouldnt

    have seconds on Table 2 [pause as she waits for me to record her informationon the board] because that would be 8 and 8 is 16 (pause). And then the sameon Table 1.

    T: Four people without seconds, and four people without seconds. (pause) Doesanyone want to ask them anything? (pause) Sue?

    Sue: Okay um, you dont know how much the, the, the people at the table who alreadyate, and (pauseI prompted her to continue). You dont knowlike say youcame 5 minutes after they served everyone and everything, and say like twopizzas were gone at like both tables, and then theres only two on both of them.

    T: Okay, so if, so you dontokay, you dont know how much is left at each table,is that ?

    Sue: Yeah! You dont, youI dont know! [She mumbles something I cant hear.]T: (Pause) Okay, lets just assumethey dont say this explicitlybut lets assume

    they just got the pizzas and they havent started eating yet.Student: It says that.T: It does say that? Okay. [I checkedthe problem did say that.] Um, Samantha,

    youve had your hand up a long time.Samantha: Um, I want to take the peopleum, theres four people without seconds at each

    thing, but the Table 1 has more people, so, um, um, those people withoutseconds on Table 2, thats half the people, but the people without seconds onTable 1, thats, um, less than one half, so you have a better chance of gettingseconds at Table 1 than at Table 2. [We had recently finished a probability unit,and Samantha was able to use those ideas here.]

    T: Okay, so you would say Table 1 is better. So you like this idea of gettingseconds.

    Samantha: I have another idea.T: Go ahead.Samantha: Okay, I got the answer by dividing by, um, dividing how many pizzas there are

    by the number of people, and. [Getting seconds was not the original way shehad solved the problem. But she had listened to the argument and understoodits mathematical flaw and tried to address the flaw.]

    T: Okay. [Im writing on the board again.]Samantha: So I took 4 divided by 11 because you have to add more people if youre going

    to join the table.T: Because youre going to be there too, so youre saying theres 11 people and.Samantha: And you get 36% of the pizza.

    At this point, I pushed a bit on what the 36% was a portion of, one pizza or allthe pizza. Although many students seemed to understand Samanthas approach andanswer, the class period ended before I could pursue the differences in Roses andSamanthas approaches.

    While the bell was ringing and students were leaving, Rose approached me forhelp in understanding how to proceed on the first homework problem about which

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    of three parks was most crowded, given the areas and the number of children playingin each park. This problem is mathematically similar to but contextually verydifferent from the pizza problem.

    In looking closely at their homework from that night, I found the SES patternsstriking. Most of the higher SES students made good sense of the park problemand their solutions to it. Their answers involved ratios such as .087 kids per squaremeter or 11.4 square meters per person, quite abstract ideas. Most lower SESstudents performed some computation with the given numbers to arrive at ananswer, but they could not make sense of their answers when they finished. Forexample, Rose divided some numbers but could not interpret what they meant and,therefore, could not determine which park was most crowded. She wrote, Idivided the area by the number of children playing. This found the percent of peopleon each playground.

    Although stronger teacher intervention might have helped students pull out theintended mathematics after their group work on the problems, the issue remainsthat the lower SES students seemed to have more difficulty than the higher SESstudents in learning the intended mathematical ideas from the pizza problem.Instead of exploring the problem in a way that allowed them to learn about gener-alizable methods of comparing numbers using division or ratios, Sue and Rose usedapproaches that seemed heavily influenced by real-world concerns, such as gettingseconds on pizza or arriving late to dinner. Hence, these students did not solve theproblem in a way that helped them learn the powerful, generalizable methods ofusing division or ratios to make comparisons. As Samantha demonstrated, Rosesapproach with getting seconds could have allowed her to reach the right answer,in which case she might have left class confident that she understood what wasexpected. But Rose would not have learned the ideas that would have helped hersolve the mathematically similar homework problem that night.

    Other evidence indicates that the lower SES students tended to focus on indi-vidual problems without seeing the mathematical ideas connecting various prob-lems. Dawn said that she could never figure out what she was supposed to belearning until she took the test. Rose complained that she had difficulty seeing howthe problems we did in class related to the assigned homework problems. Rose alsodid not see connections between units; according to her, once we finished with anidea, we never revisited it. In contrast, several higher SES students (but no lowerSES students) noted that we encountered the same mathematical ideas over andover in various contexts.

    A Simple Difference in Previous Achievement, Ability, or Motivation?Some differences in students experiences and views were probably due to

    previous achievement differences, which correlated with SES. But various formsof data show that an achievement-differences explanation is incomplete. Inasmuchas lower SES students such as Rose or Lynn were known as very bright, highachievers in their other, more typical classes (as I learned from their teachers and

  • 472 Examining Problem Solving Through a Class Lens

    parents), why were they struggling with particular aspects of my class? Why didthe lower SES students (especially females) tend to say that they understood math-ematics better before CMP? Why did some higher SES students but no lower SESstudents say that for them traditional mathematics was more difficult than CMPmathematics?

    Many differences in the data might seem attributable to simple motivationaldifferences, but my analyses of the data do not support that hypothesis, particu-larly when I examine gender and SES together. For example, Table 1 shows thestudents average grades for the year. The first number for each student is thepercentage of assignments (homework and classwork) completed. This number isa rough measure of effort. The second number is the students quizzes/test average,which indicates the degree to which a student understood the intended mathemat-ical ideas in a way that allowed her or him to apply the ideas to new situations.

    Table 1Students Average Grades for the Year

    Assignment completion Quiz and test averageStudent (effort) (understanding)

    Higher SES malesChristopher 96 95Samuel 89 90Timothy 89 81Benjamin 77 92Harrison 70 69Average 84 85

    Higher SES femalesRebecca 99 95Samantha 98 96Andrea 91 86Guinevere 81 90Average 92 92

    Lower SES malesMark 82 84Nick 78 87Carl 69 61James 21 42Average 63 (76)a 69 (77)a

    Lower SES femalesRose 97 87Anne 94 88Lynn 94 86Sue 90 77Dawn 80 63Average 91 80aBecause James is an outlier for the lower SES males, I calculated the overall average both with and(without) his grades.

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    Gender differences were found in effort put forth on homework: Girls were morediligent than boys, completing more than 90% of the assignments. The amount ofeffort put forth on assignments correlated closely with quiz and test results for higherSES males and females and for lower SES males. But the story is quite differentfor the lower SES females, who showed the most frustration with the classthroughout the year. These girls made consistent effort, but they still did not under-stand the mathematics in a way that allowed them to do well on tests. In other words,effort did not pay off for them in the way it did for the higher SES girls.

    In students perceptions of their own mathematical abilities were patterns thatseemed to go beyond actual differences in achievement or ability. In a survey atthe end of the year, I asked students to name the top three mathematics students inthe class. Table 2 contains the tallies of those named. Notice that no lower SESstudent ranked herself among the top three, not even Rose, who was consideredby nine other students to be a top student. Meanwhile, every higher SES studentmentioned by anyone also mentioned himself or herself. Even Samuel andGuinevere, who were not mentioned by anyone else, named themselves. Accordingto their homework and test grades, Rose and Anne had as much cause to feel math-ematically confident as Timothy, Guinevere, or Samuel. And yet, they did not.

    Table 2Numbers of Times Students Were Mentioned in Response to Who Are the Best ThreeMath Students?

    Higher SES studentsMale student Times mentioned Female student Times mentionedBenjamina 18 Samanthaa 20Timothya 2 Rebeccaa 4Christophera 2 Guineverea 1Samuela 1 Andrea 0Harrison 0

    Lower SES studentsMale student Times mentioned Female student Times mentionedCarl 0 Rose 9James 0 Anne 2Nick 0 Dawn 0Mark 0 Sue 0

    Lynn 0aStudents who named themselves as one of the top three mathematics students.

    DISCUSSION

    Other potentially confounding factors in addition to achievement need to beaddressed. First, one might wonder if the CMP trial materials or my pedagogy weresimply ineffective. Yet, the point is that lower SES and higher SES students were

  • 474 Examining Problem Solving Through a Class Lens

    differentially affected by the open, contextualized nature of the problems (and theless directive nature of my pedagogy). Others might wonder if differences instudents reactions were rooted in their previous school experiences, but inasmuchas the students were all in the same classroom and the majority had been in thesame school system since kindergarten, this explanation does not seem plausible.

    Because of the nature of this qualitative study, I cannot draw conclusions aboutthe likelihood of differential experiences for higher and lower SES students in otherclassrooms aligned with current reforms, and I cannot say what would havehappened if the students had experienced a reform-inspired curriculum and peda-gogy throughout their school years. Additionally, the patterns in the data for thissmall sample could be due to individual differences that happened to fall alongclass lines or to other confounding factors, such as teacher/researcher bias.

    These concerns prompted me to review the literature on class cultures to shedlight on how the patterns in the data might or might not relate to issues of class. Inexploring this literature, I identified ways in which the differences in my studentsexperiences seem to link with class cultural differences found by other researchers.I now discuss these aspects, organized around two themes: (a) students persever-ance and desire for direction when solving problems and (b) issues of abstractionand contextualization.

    Perseverance and Direction in Problem Solving

    The open nature of the CMP trial curriculum required students to take initia-tive in interpreting and exploring problems instead of to follow step-by-step rulesgiven at the top of a page. The higher SES students generally felt able to explorethe open problems in this way, without becoming overly frustrated. In fact, manyhigher SES students voiced appreciation for their increasing mathematicalproblem-solving abilities. But the lower SES students, especially the females, moreconsistently complained of feeling confused about what to do with the problemsand asked (often passionately) for more teacher direction and a return to typicaldrill-and-practice problems.

    In Heaths (1983) famous study of middle-class and Black and White working-class communities, she found differences in parenting practices. In teaching theirchildren, the middle-class parents emphasized reasoning and discussing. Throughinteractions with their parents, the children developed ways of decontextualizingand surrounding with explanatory prose the knowledge gained from selectiveattention to objects (p. 56). Meanwhile, the White working-class parents empha-sized conformity, giving their children follow-the-number coloring books, forexample.5 They told their children, Do it like this, while demonstrating a skill

    5 In contrast, the Black working-class families emphasized creative story telling. Like Delpit (1986),Heath described the children in these families as linguistically fluent and argued that they need to learnmore decontextualized, factual, and explanatory means of reasoning and communicating for successin school.

  • 475Sarah Theule Lubienski

    (such as swinging a baseball bat), instead of discussing or explaining the featuresof the skill or the principles behind it. The White working-class children tried tomimic their parents actions. When frustrated, the children often tried to find away of diverting attention from the task (p. 62). These children became passiveknowledge receivers and did not learn to decontextualize knowledge and then shiftit into other contexts or frames. These students did well in early grades, but whenthey encountered more advanced activities that required more creativity and inde-pendence, they frequently had difficulty and asked the teacher, What do I do here?

    Similarly, Hess and Shipman (1965) found that middle-class mothers asked ques-tions to help focus childrens attention on key features of a problem, therebyteaching the children generalizable problem-solving strategies. Working-classmothers tended to explicitly tell their children how to solve a problem, oftensolving it for them.

    Other studies have described middle-class families as emphasizing initiative,curiosity, reflection, and self-confidence; they described working-class families asemphasizing obedience to authority and believing they have little control over theirown fate. Hence, middle-class children are taught to take initiative and to enjoylearning for learnings sake, whereas working-class children are more orientedtoward receiving knowledge from those in authority (Banks, 1988; Duberman,1976; Terrell, Durken, & Wiesley, 1959; Zigler & De Labry, 1962).

    These comparisons are likely to make many people (including myself) nervous.Bruner (1975), after surveying the literature on class cultural differences, concluded,Its not a simple matter of deficit (p. 41). But although he could see strengthsand weaknesses in the cultures of both the middle and lower classes, Bruner wasparticularly concerned about feelings of helplessness and hopelessness that seemmore pervasive in the culture of the lower classes. He concluded

    I am not arguing that middle-class culture is good for all or even good for the middle-class. Indeed, its denial of the problems of dispossession, poverty, and privilege makeit contemptible in the eyes of even compassionate critics. But, in effect, insofar asa subculture represents a reaction to defeat and insofar as it is caught by a sense ofpowerlessness, it [lower class culture] suppresses the potential of those who grow upunder its sway by discouraging problem solving. The source of powerlessness that sucha subculture generates, no matter how moving its by-products, produces instability inthe society and unfulfilled promise in human beings. (p. 42, emphasis added)Although class is the primary focus here, several researchers have suggested ways

    in which gender could be an interactive variable and might explain why the lowerSES females seemed most adamant in their preference for more traditional instruc-tion. A recent study on elementary school children showed that when solving prob-lems, girls were more likely than boys to use the given, standard algorithms asopposed to invented strategies (Fennema, Carpenter, Jacobs, Franke, & Levi,1998). Also, researchers have found that girls, especially those with low self-esteem,are more likely than boys to internalize their frustrations and failures as evidenceof their incompetence (American Association of University Women, 1992;Covington & Omelich, 1979). Moreover, in a case study of gender and ethnicity

  • 476 Examining Problem Solving Through a Class Lens

    in a seventh-grade classroom, Stanic and Hart (1995) found that White femalesexhibited the least amount of mathematical confidence.

    Taken together, these findings might explain why the lower SES students in thisstudy, especially the females, voiced such strong preferences for a traditionalcurriculum and teacher role. The lower SES students seemed to have lower self-esteem in the area of their mathematics performance (e.g., recall their rankings).6The lower SES females made strong, consistent efforts to succeed, and yet theystruggled when faced with problems requiring more than a standard algorithm. Thelower SES females became the most frustrated with the open nature of the prob-lems and my pedagogy, and this finding makes sense, according to the literature,because these are the students who were most likely to internalize their failure.

    Hence, in contrast with the reformers rhetoric of mathematical empower-ment, some of my students reacted to the more open, challenging mathematicsproblems by becoming overly frustrated and feeling increasingly mathematicallydisempowered.7 The lower SES students seemed to prefer more external directionfrom the textbook and the teacher. The lower SES students, particularly the females,seemed to internalize their struggles and shut down, preferring a more traditional,directive role from the teacher and text. These students longed to return to the daysin which they could see more direct results for their efforts (e.g., 48 out of 50 correcton the days worksheet).

    Abstraction From Contexts

    Because the open problems could be explored in various ways, students couldtake unexpected approaches. Although this openness is not necessarily problem-atic, in this study I raise questions about complications that can arise if, as Bernstein(1975) and Holland (1981) argued, lower SES students have a contextualizedorientation toward meanings.

    Many of the CMP problems were set in realistic situations, and these contextswere often effective motivators, prompting students to delve into the problems.Whereas the higher SES students seemed able and inclined to pull back from thecontext and analyze the intended mathematical ideas involved, the lower SESstudents seemed to focus more on the real-world constraints involved with thecontexts as they solved the problems, thereby missing some of the intended math-ematical ideas. Usually the lower SES students thinking about these real-worldconstraints was very sensible, and I (and the other CMP authors) did want students

    6Additionally, Kohr et al. (1988) found that SES correlates with students' general self-esteem in school(more so than race or gender).

    7Again, in this study I say nothing about the published versions of the CMP curriculum. The CMPproblems gained clarity with each revision, and, therefore, all students' struggles with the open natureof the problems would probably have been lessened if the published CMP materials had been used. Theissue raised in this study is SES-related differences in students' reactions to open, contextualizedcurricular problems, in general.

  • 477Sarah Theule Lubienski

    to be able to think about mathematical ideas in the messy, real-world contexts inwhich they are often embedded. But these contextualized problems were primarilyintended to be a means of learning more general mathematical ideas that could thenbe applied in other contexts. The tendency for the lower SES students to focus onthe immediate context, as well as their desire for more specific teacher directionfor their mathematical thinking and less intrinsic interest in understanding the math-ematics for its own sake, seemed to contribute to the lower SES students tenden-cies to become stuck8 in the contexts.

    Hence, although students with more of a contextualized orientation towardmeaning might be expected to benefit from contextualized mathematics prob-lems, in this study I raise questions about possible drawbacks of contextualizedproblems in the curriculum. Other researchers recently raised similar questionsabout the fairness of contextualized mathematics assessment items. In comparingBritish working- and middle-class students approaches to standardized test items,Cooper, Dunne, and Rodgers (1997) found that class differences were greater onquestions involving realistic situations. In examining these differences, they discov-ered that the contexts were obstacles for working-class students, who approachedthe problems in heavily context-laden ways unintended by test writers.

    Ball (1995) raised additional equity issues surrounding the use of realisticcontexts. She suggested that teaching mathematics through real-world problemscan pose difficulties because of (a) differences in students interpretations andapproaches and (b) uneven access to relevant contextual knowledge. In her class-room experience, Ball found that abstract mathematical contexts often seemed moreinclusive, giving students a sense of common understanding and purpose. Whenher students explored real-world problems, they were distracted, or confused, orthe differences among them were accentuated in ways that diminished the senseof collective purpose and joint work (p. 672).

    Although students need to become able and inclined to critically analyze real-world problems, particularly those involving inequities, I question whether theseproblems are equitable means of building the mathematical understandings neces-sary for such critical analyses.

    IMPLICATIONS

    The data and the literature reviewed above could suggest that lower SES studentshave the most to gain from problem-solving instruction, inasmuch as it might helpthese students gain knowledge and skills they are less likely to gain elsewhere. Yet,

    8Although some might take issue with the negative connotation the word stuck carries, I think it fitshere. Stuck indicates that the students' thinking could become so heavily immersed in the real-worldconstraints of the problem that they missed some important mathematical ideas. Again, this is not tosay that the lower SES students' thinking was less sophisticated than the higher SES students' thinking,but the lower SES students' approaches did not always allow them to reap the intended mathematicalbenefits of the problems.

  • 478 Examining Problem Solving Through a Class Lens

    in this study I raise questions about problem solving, not as an end in itself but asa means of learning other mathematical concepts and skills.

    From this study of one classroom, one cannot conclude that lower SES studentsare incapable of functioning in a reformed classroom, and one cannot concludethat lower SES students will learn less from problem-centered teaching than frommore typical teaching. In fact, some studies have indicated that teaching with anemphasis on critical thinking, discussion, and problem solving can, indeed, beimplemented in lower SES classrooms (e.g., Silver, Smith, & Nelson, 1995) andthat teaching for meaning can increase poor students problem-solving and compu-tational abilities when compared with teaching via rote memorization (Knapp,Shields, & Turnbull, 1995).

    Yet, two issues remain. First, few scholars give attention to the gap between lowerSES and higher SES students mathematics performance, a gap that is of partic-ular concern given the gatekeeping function of mathematical knowledge. This studyhighlights the need to consider the possibility that reform-oriented teaching couldimprove both lower SES and higher SES students understanding of mathematicswhile also increasing the gap in their mathematics performance. In other words,although repetition of abstract algorithms given by the teacher or text might be dulland ineffective in promoting true understanding, compared with reform-orientedteaching, the algorithmic mode of instruction might provide a relatively levelplaying field by having clear rules and being equally disconnected from all studentsrealities.

    Still, there is some encouraging evidence that suggests otherwise. In a large-scaleschool-restructuring study in which Newmann and Wehlage (1995) examinedinstruction in mathematics and other content areas, they found that authentic peda-gogy (which they defined in terms of an emphasis on student thinking and under-standing) was at least as effective for lower SES students as for higher SESstudents. However, when Newmann and Wehlage did not control for prior achieve-ment, the results were less encouraging. Sigurdson and Olson (1992) did not studySES specifically, but they also found that teaching mathematics with meaningincreased the gap between low-achieving and other students. Although separatingachievement and SES is important analytically, from a practical standpoint, theseresults raise questions about whom authentic pedagogy benefits, given the corre-lation between SES and achievement. Researchers must continue to investigate theeffects of various teaching methods on the gap between lower SES and higher SESstudents performances on achievement tests, particularly those used as gate-keepers to high-status opportunities.

    A second issue intertwined in the above discussion is that great care must be givento interpreting the implications of studies that report on the effects of teaching formeaning or authentic pedagogy. When one looks closely at how various researchersdefine the meaning-oriented teaching they contrast with other, more typical typesof teaching, there are often differences between the researchers definitions andsome aspects of the vision outlined in the NCTM Standards (1991). For example,Knapp et al. (1995) observed few teachers who were implementing the Standards

  • 479Sarah Theule Lubienski

    fully (p. 771), and they looked instead for teachers who emphasized conceptualunderstanding in some way and who broadened the range of mathematical topicsbeyond arithmetic. Although Newmann and Wehlages (1995) criteria for authenticpedagogy sound much like what NCTM outlined, note that their conception ofproblem solving centers around application (as opposed to learning) of ideas andthat their view of an authentic teachers role included a more traditional, teacher-centered form than some interpret the Standards as advocating.9

    Hence, we, as researchers, should not be too quick to conclude that the reformswork or do not work for lower SES students. When attempting to understandthe effects of various reform-oriented teaching practices, one should avoiddichotomizing traditional and reformed teaching (Chazan & Ball, 1995). In thisarticle, concerns have been raised about one particular aspect of instructiontheuse of open, contextualized problems. The reader should note that the data fromthis study did not indicate cause for concern about numerous other instructionalmethods aligned with current reforms (that were also utilized in this classroom),such as the use of technology, hands-on activities, heterogeneous-group work,or teaching with an emphasis on the meaning of mathematical concepts. If our goalis to develop mathematical power for all students, then we may need to disentanglevarious aspects of the instructional vision outlined by NCTM (1989, 1991) toconsider which elements might be more or less effective in helping us reach thisgoal with lower SES students.

    Two mathematics projects known to be successful with disadvantaged studentssuggest some instructional variations that relate to issues raised in this study. First,Bob Mosess (Moses, Kamii, Swap, & Howard, 1989) Algebra Project is knownto be successful in helping disadvantaged middle school students learn algebra.Moses developed a five-step plan to help students avert frustration when movingfrom concrete events to abstract representations. He advocated teaching studentsproblem-solving skills explicitly and helping them learn, in nonthreatening ways,how to be more self-reliant learners of mathematics, including how to generalizefrom concrete situations. Additionally, through Project SEED, Phillips andEbrahimi (1993) aimed to help low-income and minority elementary and middleschool students learn abstract mathematics in order to promote the study of moreadvanced mathematics later. Project SEED students actively engage in mathe-matical thinking and problem exploration, yet SEED students do not explore openproblems independently; instead the teacher leads the entire class through theexploration, using focusing questions. Also, contextualized problems are notemphasized; abstract ideas are taught in the abstract.

    Although in this study I explored various issues that arose in one classroom, morefocused studies with larger numbers of students are needed to address questions

    9Newmann and Wehlage (1995) summarized their criteria for authentic pedagogy as follows: Ourstandards emphasize teaching that requires students to think, to develop in-depth understanding, andto apply academic learning to important, realistic problems (p. 3, emphasis added). They explain thatauthentic pedagogy can be teacher-centered or student-centered (p. 16).

  • 480 Examining Problem Solving Through a Class Lens

    raised here (such as the differential effect of abstract versus contextualized prob-lems). With NCTMs release of Principles and Standards (2000), continuedresearch is needed so educators can understand potential gaps between aspects ofthe classroom culture envisioned by reformers and the cultures of lower SESstudents (including interactions with ethnicity and gender) and identify ways inwhich exemplary teachers are able to bridge such gaps. However, considerationsof how to adapt instruction to meet lower SES students needs might containinherent dilemmas about what types of instruction and goals are intrinsically morevaluable yet also could increase the gap between lower SES and higher SESstudents mathematics performances.

    I hope that the issues raised here can deepen researchers thinking aboutcomplexities involved with the mathematical power for all agenda and remindthem to keep equity at the forefront of discussions regarding curriculum and peda-gogy; methods that are promising for many students could pose unexpected diffi-culties for students who most need mathematical empowerment.

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    Author

    Sarah Theule Lubienski, Assistant Professor, Curriculum and Instruction Department, Iowa StateUniversity, Ames, IA 50011; [email protected]