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Achievement and self-efficacy of students with English as a second language based on problem type in.. Pel, Amanda Jean 2008

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ACHIEVEMENT AM) SELF-EFFICACY OF STUDENTS WITH ENGLISH AS A SECONDLANGUAGE BASED ON PROBLEM TYPE iN AN ENGLISH LANGUAGE-BASEDMATHEMATICS CURRICULUMbyAMAM)A JEAN PELB.Sc., The University ofBritish Columbia, 2003B.Ed., The University of British Columbia, 2004A THESIS SUBMITTED iN PARTIAL FULHLLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF ARTSinTHE FACULTY OF GRADUATE STUDIES(Curriculum Studies)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2008© Amanda Jean PdAbstractStudents who are learning English as a second language (ESL) have lower performance onmathematics problems based in language than students who are fully fluent in English.Students’ performance on word-based mathematics problems is directly related to theirEnglish reading comprehension and language fluency (Abedi & Lord, 2001; Brown, 2005;Hofstetter, 2003). This places students who are not fully fluent in Englishat a disadvantagein the mathematics classroom. Students’ self-efficacy beliefs also impacts their mathematicsperformance and motivation. The self-efficacy of students who are not fluent in English maybe negatively impacted by their struggle with language. For this exploratory study, image-based mathematics problems were created to communicate problem solving questions withpictures instead of language or computational symbols. This problem format wasinvestigated as a potential alternative to word-based or computation-based problems. Grade6 students registered in ESL level 2, ESL level 4, and not registered in ESL, completedamathematics task with four computation problems, four language-based problems, and fourimage-based problems. During a follow-up interview, students’ solution strategies andthought processes were explored further. The results ofthis study indicated thattheinclusion of wordless mathematics problems, such as image-based problems, assisted someofthe students who were learning basic English interpersonal communication skills. Asnonroutine problems, image-based mathematics also encouraged complex thought andmathematics understanding. Students in ESL Level 2 demonstrated higher self-efficacybeliefs on image-based problems than word problems.11Table of ContentsAbstract.iiTable of ContentsiiiList of TablesViList of FiguresviiAcknowledgementsViiiDedicationixResearch Motivation1Literature Review8Methods26Problem Selection26ESL Participant Group Selection33Mathematics Task Trials37School Selection and Approach38Subjects39Materials40Scoring the Mathematics Task42Procedure43Mathematics Task43Interview44Results48English in the Mathematics Classroom48ESL Level 2 Responses to the Useof English in the MathematicsClassroom49ESL Level 4 Responses to the Useof English in the MathematicsClassroom48Responses of Students Not Registeredin ESL to the Use of English in theMathematics Classroom50General Overview of All Three ProblemFormats 50ESL Level 2 Performance and Responseon Mathematics Task byProblem Type51ESL Level 4 Performance and Responseon Mathematics Task byProblem Type52Performance and Response on MathematicsTask by Problem Type ofStudents Not Registered in ESL54Percentage of Errors Made by Each ParticipantGroup by Problem Type 56Self-Efficacy by Problem Type60111ESL Level 2 Students’ Self-Efficacy by Problem Type 60ESL Level 4 Students’ Self-Efficacy by Problem Type 62Self-Efficacy by Problem Type of Students Not Registered in ESL 66Written and Oral Responses to Individual Picture Problems 69ESL Level 2 Students’ Responses to Question #1 69ESL Level 2 Students’ Responses to Question #2 70ESL Level 2 Students’ Responses to Question #3 72ESL Level 2 Students’ Responses to Question #4 73ESL Level 4 Students’ Responses to Question #1 75ESL Level 4 Students’ Responses to Question #2 76ESL Level 4 Students’ Responses to Question #3 78ESL Level 4 Students’ Responses to Question #4 79Responses to Question #1 of Students Fully Fluent in English 81Responses to Question #2 of Students Fully Fluent in English 82Responses to Question #3 of Students Fully Fluent in English 83Responses to Question #4 of Students Fully Fluent in English 85Clarity of Picture Problems 87Summary ofResults 90Discussion 92Picture Problems as Nonroutine Problems 93The Use of Active Learning in the Solution of Picture Problems 94The Role of Metacognition in the Completion of Picture Problems 99The Role of Will in the Completion ofNonroutine Problems 102Trends in Solution Strategies 104Trends in Solution Justification 104Trends in Solution Communication 106Meaningfhl Learning in Mathematics 106Conclusion 108Implications for Educators 108Implications for Research 110References 113Appendix A Computation Problems Used on the Mathematics Task 118Appendix B Word Problems Used on the Mathematics Task 119Appendix C Image-Based Problems Used on the Mathematics Task 120Appendix D Mathematics Task Question Booklet Cover Page123Appendix E Mathematics Task Work Booklet Cover Page 124Appendix F Simplified Chinese Interpretation of the Instructions for the MathematicsTask Problem Booklet and the Mathematics Task Work Booklet125ivAppendix G Sample Pagefor Written Work and QuestionResponse126Appendix H InterviewQuestions127Appendix I Behaviour ResearchEthics Board Certificate ofApproval128VList of TablesTable 1 Participant Gender and School Affiliation 39Table 2 Summary ofESL Level 2 Performance on the Mathematics Task by Problem 51Table 3 Summary ofESL Level 4 Performance on the Mathematics Task by Problem 53Table 4 Summary ofPerformance of Students Not Registered in ESL on the MathematicsTask by Problem 55Table 5 Percentage ofErrors Made by Each Participant Group by Problem Type 57Table 6 Percentage ofErrors Made by Each Participant Group by Problem Type AfterCalculation Errors are Removed 58Table 7 Methods of Improving Picture Problems Suggested by Students 87viList of FiguresFigure 1 Shelley’s Written Work for Question #1 70Figure 2 John’s Written Work for Question #2 71Figure 3 John’s Written Work for Question #3 73Figure 4 Sam’s Written Work for Question #1 75Figure 5 Ricky’s Written Work for Question #2 76Figure 6 Ricky’s Written Work for Question #3 78Figure 7 Sam’s Written Work for Question #4 80Figure 8 Mark’s Written Work for Question #2 82Figure 9 Jimmy’s Written Work for Question #2 84Figure 10 Steven’s Written Work for Question #3 83Figure 11 Chris’s Written Work for Question #3 85Figure 12 Jimmy’s Written Work for Question #4 86Figure 13 Flora’s Written Work for Question #4 86viiAcknowledgementsSpecial thanks to Dr. Ann Anderson forher enthusiasm, input, and constant support, and to Dr.Susan Gerofsky for her contribution, positivity,and knowledge. Thanks also to Zihao Chen forhis assistance developing the mathematicstask, and to all of the students who participated inthis study.viiiDedicationTo Maft, who always supports and encourages me.ixResearch MotivationWhen I look criticallyat my research motivation,I am acutely awarethat my earlyteaching experienceshave impacted my researchinterests more thanany other experience inmylife. During myenrolment in the TeacherEducation Program atthe University ofBritishColumbia in 2003,I completed my practicumin a grade 6 and7 classroom. Since it was amiddle-class school,I had little experience withstudents who spokeEnglish as a secondlanguage (ESL). Igraduated in the summerof 2004 with my BachelorDegree in Education anda concentrationin teaching the Intermediategrades.My first teachingposition was in a grade4 classroom, and because mypracticumexperience had beenin upper intermediate grades,I expected that I wouldhave no problem withthe grade 4 contentthat I would have to teach.I quickly learned that itwas not as easy as Iimagined. My primarydifficulty was with themathematics textbookthat the school had recentlyacquired. One of themost embarrassing thingsfor me to admit as a newteacher was that theword problemthat a grade 4 student had justshown me in the mathtextbook had me completelystumped. Althoughthis uncomfortable momentseemed to happen to me quiteoften in my firstyear of teaching, Ifelt that my knowledgeof mathematics was not atissue. As an elementarystudent, I had beenpart of a gifted mathematicsprograms and had experiencesolving complexcomputation and word-basedproblems. Most ofmy confusion, as a teacher, withthe grade 4mathematics textbook occurredbecause the word-basedmathematic problems were writtenwithvague or confusing language,and the intent ofthe problemwas unclear. The language usedinthe actual math problemswas the issue for me and mystudents; it was not our understandingofhow to interpret, applyand manipulate mathematicalterms. Sometimes I was able tofigure outa problem afterlooking the answer up inthe teacher handbook and workingbackwards. At othertimes, when even this strategycould not help me, I would justtell the students to move on to thenext question. ThoughI knew that it was the languageand the wording of the mathematics1problems that made them ambiguous and unclear,this situation made me feel that I wasincompetent as a teacher, and I began to have seriousquestions about my own abilities.However, I also knew that if! was experiencingthis level of frustration and anxiety, it was verypossible that my students, who were still only learning the mathconcepts that I fully understood,were also experiencing the same self-doubts. The frustration thatthey were dealing with wasworrisome to me, as it was enough to cause someofthem to label themselves “stupid” at math,and express this to me. It concerned me that all ofthe childrenin the classroom were fully fluentin English, yet they were having such problems in mathematics becauseof a current focus onlanguage in the mathematics curriculum. To help alleviate thesefeelings, so they would nothave this negative view of their abilities, I began rewriting the textbook problems tohelpincrease the clarity of the language and I added basic computation and manipulation questionsto help reinforce students’ learning. We increasedthe use of manipulatives and real worldapplication of mathematics through language-basedquestions developed with grocery flyers,school supplies, and other applicable situations.I wanted to help minimize the frustration in math, because the students’ negativeemotions and feelings did not just impact their willingness to learn math, but they also affectedthe students’ emotional state. This in turn altered their focus, concentration, openness tolearning, and willingness to be successful participants in the classroom. I was quite concernedthat some of the students were beginning to experience negative self-efficacy. The students’self-efficacy beliefs, their perceptions oftheir own capabilities, are linked to their motivation,feelings, thought-processes and behaviours (Bandura, 1994). The students in my classroom werebeginning to believe that their future attempts at problem solving with the textbook problemswould also be unsuccessful because oftheir past experiences. Thanlcfully, creating an alternativelanguage-based program for them helped to buoy their self-efficacy in the area of mathematicsproblem solving, and we finished the year using the textbook sparingly.2The following year, I taught in agrade 5 and 6 combined classroom in aninner-cityschool with a transient population, where approximatelyhalf of the students had an ESLdesignation. While the ESL studentswere not expected to have strong literacy skills because oftheir lack of language knowledge, many of thenative English speakers in the class were alsoreading below grade level and struggling withreading comprehension. The students in this classhad a significant number of languageand literacy difficulties, and the mathematics textbooksthat the school used typically assigned five to eightword problems per topic, very fewnumerical computation questions, andno picture-based questions. While I understood thebenefits of a language-based curriculum as a way to deepenstudents’ understanding ofmathematics and increase their ability to applyand transform their mathematics understandingto real situations, it was hard to justify inundatingstudents with written language-basedproblems when I was concerned that students werewithout the language and literacy skillsnecessary to participate in mathematics lessons, learning,and discussions, successfully. Howwas I meant to effectively teach these children mathematics conceptswhen they did not evenprocess enough ofthe language and vocabulary to understandwhat they were being asked? Howmany of them could have performed better or felt more successful ifthe mathematics programhad been based less on language and literacy, which were their areasofweakness, and insteadfocused more on numerical manipulation and mathematics explorationthrough hands-onlearning, relevant situations, or picture-based images? For howmany ofthe students did thelanguage component and their resulting difficulty have a negative effecton their own feelings ofmathematics capability and self-worth? These questions began to posean interesting dilemmafor me and made me passionate about fmding a way to allow students to achieve maximummathematics success in a culturally varied mathematics classroom whilestill experiencingmeaningful mathematics exploration.3One of the students inmy class that year was a refugeefrom a Middle Eastern countrywhere her dad had been employedas a professor with aPh.D. in mathematics. She had alwaysreceived very high marksin math. Not only did she come to Canadaand my classroom speakingvery limited English, but she wasalso used to a completely differentalphabet system. Thisimpeded her initial attempts notonly at acquiring the language but also atdeveloping literacyskills because she firsthad to acquire English letter recognitionskills. She had originallyexpected that when she came toCanada she would still be able to do well atmathematics, evenwith limited language skills. Likemany other immigrant students, she expectedmath to focusmore on numbers and computation and lesson language abilities. She was surprised anddismayed by the inclusion of a significantamount of language in mathematics, andher fathermet with me many times becausehe was so upset about seeing her receivefailing marks inmath. Her problem with math wasnot because she had difficulty understanding computationorapplying mathematics to problem-basedsituations; rather it was becauseof how heavily dousedin language and literacy the mathematicscurriculum has become. She was unable to answermany of the questions assigned toher in her math class because they were given in wordproblem format. In addition, her frustration aboutmath was affecting her performance andattitude in my classroom during the rest ofthe day. She began to withdraw, and her motivationdecreased. Though I did not teach grade6 mathematics, I was able to negotiate with hermathematics teacher to let me giveher an individualized program that minimized the need forEnglish and reading comprehension, butstill allowed her to investigate and explore the samemathematics as the rest of her peer group.This student’s personal struggle really affected meand made me begin to take a critical look atthe mathematics program as it is currentlystructured, and the successes and challengesthat a language-based mathematics curriculumbrings to a multicultural classroom, especiallyin areas with a high refugee or ESL population.4I feel that the educationsystem should allow childrento discover knowledge andteachthem to realize their fullpotential. Though I was ableto be successfil in the educationsystem asit was structured, I realizethat one form of educationis not necessarily best forall ofthestudents in a classroom.These experiences haveprompted me to look critically at shiftsin thestructure of educationin the area of mathematicsand how mathematics can be furtherdevelopedto allow maximum successof the widest range of learners.As an educator, my students haveopened my eyes to the importantissues in a classroom, andI have come to believe that highself-efficacy and perceivedsuccess are very importantqualities for students to possess. Throughmy research, I want to learnmore about how I can make eachday as successful as possible forthem so that they will be passionateabout education andlearning. In this particular study, I willinvestigate a potential alternative tolanguage-based mathematics whichmay allow students whoare not fluent in English tointerpret, apply and manipulatemathematics in a meaningful andsignificant way. Mathematicsquestions based purely on computationdo not teach students tothink deeply about mathematics asthey can just follow a learnedprocedure to recite memorizedfacts in order to find the correctanswer. These questions do not encouragestudents to discoverhow and why mathematics is used.Also, computation problems are not usuallyof comparabledifficulty to language-based problems.Other areas ofthe curriculum elicitinsight and discovery on the part ofthe student.Literacy learning asks students tomake inferences by “reading between the lines” bylooking atthe information presented, anddeveloping ideas and judgements based ontheir perceptions.Science and social studies demandcomplex, investigative thought by the studentsin order todelve into the “whys” of the world.This same principle of discovery in learning shouldalsoapply to mathematics even whendoing work in a textbook with a pencil and paper. Pictureproblems force students to be likedetectives in mathematics, trying to find clues to determinethe information given in the image and discover whatis missing.5For this exploratorystudy, I createdthe image-based problemsas a unique tool designedto remove languageand minimize thecomputational components,such as numbersormathematical symbols.Image-based mathematicsproblems are derivedfrom word problemswith similar problemoutcomes and complexitybut without the necessityof language. Similar toword problems, image-basedproblems have to be translatedby the students into a mathematicalcontext, interpreted to determinea solution strategy,and completed with accuratecomputation.The image-based problemscreated for the mathematicstask involve multiple stepsin a solutionstrategy and encourageanalysis and independentthought on the part of thestudent.Could creating mathematicsquestions that are image-basedprovide a suitable alternativeto language-basedproblems while studentsare first learning English? Couldimage-basedproblems retain the meaningfuland interpretive qualitiesthat language-based problemsoftenhave? Image-based mathematicscould allow students whoare still in the process of learningEnglish to have an opportunityto develop their mathematicsknowledge while still meetingtherequirements of the word-basedcurriculum (i.e. to provide studentswith situations in whichthey can interpret, apply,and manipulate mathematics).How would questions ofthis sort, whichcould allow students tocontinue their acquisition ofmathematics knowledge,impact students’self-efficacy?The research questionsthat guided this exploratorystudy are:1. According to ESLstudents’ self-reports, how istheir mathematics ability and selfefficacy affected bythe English used in the language-basedmathematics curriculum?2. How is mathematicsachievement and self-efficacy differentwhen mathematicsproblems are asked in a computation-basedformat, word-based format,or picturebased format for studentsin ESL level 2, ESL level 4, and those studentsnotregistered in ESL?63. What is the impact on students’ performance and self-efficacy when using themathematics modification of image-based problems, instead of language- orcomputation-based mathematics problems?7Literature ReviewMathematics educationfor elementary school childrenin British Columbia has shiftedrecently to include ahigher proportion ofword-basedproblems instead of focusing primarilyoncomputation and numericproblems. These word-based problemsrequire a certain level ofEnglish proficiency, literacy,and reading comprehension (Hofstetter,2003). According toBrenner et al. (1999), answering aword-based problem involves threesteps: translation,interpretation, and planning.Translation refers to comprehensionof the mathematics terms andtheir meaning, interpretationinvolves the application ofthe correctoperations, and planninginvolves combining theoperations with the numbersgiven and completing the question. I wouldsuggest that another importantfactor in students’ ability to translatethe problem is alsodependent on the proper interpretationofthe wording and vocabulary usedin the problem.Language-based problems whichallow multiple interpretations, manipulation,and a variety ofacceptable final answerscan minimize the confusion that canarise from languageunderstanding, and are considered tobe best practice (National Council of TeachersofMathematics, 1989; Baroody,1998). However, these questions are not oftenthe type of wordproblems found in mathematicstextbooks being used in the classroom. Inthe averagemathematics textbooks, studentsare expected to read the word-based problem,interpret what isbeing asked, translate the words into an equation, andsolve the problem to find one correctanswer. Having dealt with four differentEnglish word-based mathematics textbooks in the pastthree years, I have observed many studentsstruggle in mathematics, not because of a lack ofmathematics skill or comprehension, but dueto inadequate English, literacy abilities, or readingcomprehension skills. This has led me toquestion the validity of having a high proportion ofword-based problems for all students in the currentmath curriculum, and I wonder how thisexpectation affects students who donot have the language or literacy skills necessary to succeedin this form of mathematics curriculum.8For the 2005-2006 schoolyear, 10% of the studentsin the British Columbia publicschool system were designatedas having English as aSecond Language (ESL)(BritishColumbia Ministry ofEducation,2006). The British Columbia MinistryofEducation providesfunding for ESL studentsfor the first five years that theyare registered in the provincialpublicschool system (BritishColumbia Ministry ofEducation,2002). That money is granted directlyto the students’enrolling district and is intendedto be used for specificEnglish instruction forthose students still acquiringthe language. During the fiveyears of funding, students areexpected to move throughfive levels ofESL acquisition.Ideally, each level takes oneyear tocomplete so that the studentsare fimded for every yearthat they are registered in ESL classes.Due to individual studentlearning and variationin ability, some students may take lesstime tocomplete the five levelsand some students may need totake longer than the five years,however, any studentsrequiring more time aboveand beyond their first five yearsof enrolmentin the British Columbiaschool system receive no extrafunding for language support (BritishColumbia Ministry ofEducation, 1999).The Ministry ofEducation policystates that though the moneyis given directly to theschool districts in order to provideESL services, it is up to the individualdistricts to determineexactly how the moneywill be used (British Columbia Ministryof Education, 1999). In theHargrove School District,in a suburb of a large city in British Columbia,25% ofthe students inthe public school system areregistered as funded ESL learners and60% of the students in thedistrict do not haveEnglish as their first language (Carrigan,2005). These students come from avariety of language backgrounds, eachofwhich is attached to its own culture, traditionsandbeliefs. The district has organized acontinuum of ESL levels1 through 5 which provide variedamounts of support to ESL students basedon what the district feels the language needsofstudents in each ofthe five levels shouldbe. ESL levels 1 and 2 are designed to introducestudents to the fundamentals ofthe English language and provide them with comprehensionand9fluency in basic conversationalEnglish. These studentsgenerally receiveout-of-class supportwith a specialist teacherfor approximately 2 to3 hours a week (Carrigan,2005). Students inESL levels 3 through5, which are intended toteach academic English,often go without anyextra pullout support,or only sporadic meetingsor limited in-class assistancewith a specialistteacher during theschool year (Carrigan,2005). This lack of intensivesupport for ESL learnersin the upper three levelsofthe ESL systemis blamed primarily ona cut in funding whichoccurred approximately6 years ago and dramaticallydecreased the specializedsupport that thedistrict could provideto ESL students. Insteadof acquiring language throughfocusedinstruction, studentsare expected to learnmuch oftheir Englishknowledge from interactionswith peers, teachers andclassroom material(Carrigan, 2005).The focus ofthe presentstudy investigates howstudents who speak Englishas a secondlanguage (ESL) areimpacted by a language-basedmathematics curriculum. Word-basedproblems are a usefultool in the mathematicsclassroom to help studentsinvestigatemathematics and deepentheir understanding (Baroody,1998). However, I am interestedininvestigating how Englishability impacts a student’sperception of his orher own mathematicsability and self-efficacywhen she or he is placedin a language-based mathematicscurriculumthat is not taught inhis or her first language, orlanguage of fluency. Howis mathematics abilityand self-efficacy differentwhen questions are asked ina specific format, such asin straightcomputation problemformat, image-based problemformat or word-based problemformat? Inwhat ways does mathematicsability and self-efficacyincrease or decrease with increasedEnglish ability on each problemtype, or are they independentofEnglish ability?This study investigatedthe experience of ChineseESL students in British Columbiawithin the current problem- andlanguage-based mathematicscurriculum. For the purpose ofthisstudy I have limited myfocus group to studentsfrom China. This is not to say thatESL studentsfrom other cultural backgroundsdo not experience difficultywithin the English mathematics10curriculum, butbecause each culturemay experiencethese difficulties differently,massing themall together into asingle research projectmight not adequatelyconvey their experiences.Bylimiting this study toChinese immigrant students,I attempted to controlone variable byinvestigating theexperiences of a groupof students who sharea similar cultural background,community structure,and comparable experiencesin mathematics instruction.In the case ofFEPstudents, studies havebeen carried out to investigatetheirperformance on language-basedmathematics questions.Kiplinger, Hang andAbedi (2000)point out that averageFEP children’s performanceon language-based mathematicsquestions isbetween lO% and30% lower than theirperformance on similarquestions when they arepresented in a numericformat. Ifthis is the trendin the English-speakingpopulation, what is theperformance deficitof ESL students oncomparable word andcomputation problems? Ifit ishigher than the amountobserved in Englishspeaking children, thenthe impact of a language-based curriculum onESL students is even moredetrimental than theimpact on FEP students. Ahigher deficit for ESLstudents than FEP studentsshould cause educators toquestion ifwordproblems are a suitableassessment of ESLstudents’ mathematicsabilities.For the purposesof comparison, and dueto limited studies investigatingCanadianstudents’ mathematicsabilities compared with studentsin East Asian countries, I havechosen toinclude information aboutthe United States as a representationof English-speaking Westernclassrooms (Klassen,2004). The United Statesis comparable to Canadain its current focus onword-based mathematicsproblems and movement awayfrom rote computational proceduresand memorizationof math facts (Mayer, Tajika, &Stanley, 1991). In studiesofmathematicsperformance comparingEast Asian students (thosefrom China, Taiwan, and Japan)withstudents in the United States,computational skillsand problem solving abilities wereanalyzed.In tests of mathematics achievement,students from China,Taiwan and Japan consistentlyoutperform Americanstudents in computationand problem solving (Leung,2005).11Studies such asMayer, Tajika andStanley (1991), Brenner,Herman and Ho, (1999) andGeary, Liu and Chen(1999) break down mathematicsachievement into subgroupssuch ascomputation, representation,and problem solvingabilities. Mayer et al.(1991) claim that eachof these subgroups shouldbe investigatedseparately and formulatethe theory that whenthis isdone, American studentsactually outperformJapanese studentsin the area of problem solvingtechniques. Thisclaim of an Americanadvantage is misleading,because it is only evidentwhenthe proportion of correctresponses is examinedat each ability levelseparately. For example,Japanese students whoscored 11 or higheron a 15-question achievementtest were also given an18-question testoftheir problem solvingabilities. This secondtest did not ask for the questionsto be answered;rather, it investigatedthe ideas that studentshad about the steps theywould taketowards solving thisproblem. Whenthe American studentswho received a scoreof 11 or higheron the achievementtest took the problemsolving test, they answeredproportionally morequestions correctly thantheir Japanese counterparts.This finding doesnot account for the factthat in the relativelysimilar sample sizesof American and Japanesechildren who completed themathematics achievementtest, only 5 Americanchildren achieved ascore of 11 or above,while77 Japanese childrenachieved this score.Geary et al. (1999)point out that this disparitybetween the groupsmight favour the Americanstudents taking the problemsolving test in termsof IQ. The majorityof American childrenin the achievementsample scored below 7 on theirmathematics achievementtests. Therefore, the few whohad scores of 11 or above significantlyoutperformed theirpeers. It could be argued thatthese students’high deviation from the normisdue to highly superiorIQs and thinking abilities.Since the majority ofJapanese children hadscores of 11 or above,that sample is more likely tobe indicative of a broadrange of IQs. Itcould be argued that theAmerican students takingthe second test were ahigher IQ populationand were being inequitablycompared to a sample ofJapanesestudents with an average IQ base.The problem solving testmay also be considered tobe an inaccurate representationof the12capabilities of the two samplegroups, as it only asks students to devise aplan for how to solve aword-based problem and notfor the actual computation to be carried out.However, the questionneeds to be raised: Can problemsolving really be assessed without studentscompleting theproblem or is problem completionan integral step in problem solving? When carried out tocompletion, critical problem solvinginvolves students assessing the validity oftheir answer andpossibly adapting their approach tothe question accordingly, which may result in studentsaltering their problem solving methods(Baroody, 1998).Brenner et al. (1999) conducted an investigationinto problem solving abilities, butincluded full completion ofthe problem aspart of the experimental design. Contrary to Mayer etal. (1991), Brenner et al. (1999)found that on representational forms of problem solving, wherea problem had to be interpreted,modeled around previous knowledge, and strategies forsolvingthe problem put in place with the questionactually being completed, East Asian students scored3 to 5 times higher than their American counterparts.Brenner et al. (1999) suggests that this isdue to East Asian students’ strong conceptualand abstract mathematics skills, which allow themto interpret and solve more difficult problems with higheraccuracy. Also contrary to thefindings ofMayer et al. (1991), Geary et al. (1999) findsthat when questions involving problemsolving are carried out to completion, American studentsofthe same IQ and computationalfluency as East Asian students have no advantage on theproblem solving component of aquestion. Geary et al.(1999) further confirms that East Asian studentsare superior to Westernstudents in computational abilities, and that thisstrength, coupled with the demonstrated equalability in problem solving, allows them to have strongeroverall mathematics achievement thantheir Western peers, regardless of problem type.The National Council of Teachers ofMathematics reacted to the finding that Americanstudents did not perform as well in mathematics skill assessments as studentsin China,Singapore, and Japan by changing the Western mathematics curriculum so that it teaches13mathematics through alanguage-based structure(Hook, Bishop, & Hook,2007; Leung, 2005).They feel that this willassist students by helpingthem “become mathematicalproblem solversand learn to communicatemathematically” (NationalCouncil of Teachers ofMathematics,1989). This focus movesaway from rote memorizationof facts and procedures,and attempts toprovide a fuller understandingof the derivation and manipulationofthe strategies and systemsused in mathematics.The desired outcome ofthe curriculum changeis to provide students witha deeper andmore complete understandingofthe mathematics process,using language toinvestigate, discuss,and transform mathematical ideas(Salend & Hofstetter,1996).In studies that investigatethe impact of languageon mathematics performancein aWestern English-speakingclassroom, there is a strongconsensus that students’performance onword-based problemsis directly related to theirEnglish reading proficiency(Abedi & Lord,2001; Brown, 2005;Hofstetter, 2003; Kiplinger,Haug, & Abedi, 2000).Students who strugglein reading, such as FEPstudents with poor readingcomprehension skills orESL students withlimited English languageknowledge, demonstrate a decreasein mathematics problem solvingability when the problemis in a language-based format(Kiplinger et al., 2000). Brown(2005)reveals an importantcorrelation between ESL students’scores in reading and their achievementin mathematics dueto the significant language componentsinvolved in both subjects as a resultof the NCTM’s curriculumchange. ESL students’ literacyabilities are greatly impacted bytheirdeveloping English skills andtheir lack of vocabulary (Hofstetter,2003; Brown, 2005). Theygenerally are slowerthan FEP students at reading andreading comprehension because of thisdifficulty. Studentswho struggle with reading comprehensionskills and vocabulary knowledgeneed extra time to complete word-basedmathematics problems and oftenscore lower on tests ofthis sort than their peers whohave a thorough grasp of the languageand stronger readingcomprehension skills (Brown,2005). This is not to say that whencompared to FEP students,ESL students will always have lowermathematics scores in testsoftheir problem solving ability14and lower readingcomprehension scoresin the English languageclassroom, but this is certainlythe case as their Englishlanguage fluency isdeveloping.Language modificationscan be made to mathematicsword problems in order toslightlybenefit both ESL andFEP students who strugglewith reading comprehension.One suchmodification is simplifyingthe language level byusing familiar or simplelanguage, active verbsinstead of passive verbs,replacing conditional and relativeclauses, and simplifying questionphrases and long nominalsto make them concise (Hofstetter,2003). While both ESL studentsand FEP students areable to benefit fromthis modification, ESL students’mathematicsachievement scoresshow a greater improvementthan the mathematics achievementscores ofFEP students who strugglewith reading comprehension(Abedi & Lord, 2001).This mayindicate that unfamiliarvocabulary and languagedifficulties are a largercomponent ofESLstudents’ struggle inword-based mathematicsthan their ability to understandor complete theactual mathematics.Ifthis were not true, simplifyinglanguage to increase readingcomprehension should showsimilar increases in the mathematicsachievement scores ofbothESL and FEP students. Itshould be noted, however,that the improvements to mathematicsproblem solving achievementfor either group is not consideredto be statistically significantwith language modification.An alternative languagemodification is the inclusionof a glossary of terms which maybe uncommon or unknownto students. This alternativealso does not produce a statisticallysignificant increasein ESL or FEP students’ mathematicsachievement scores. However, whenprovided with a glossary,ESL students do have slightlyhigher mathematics achievement scoresthan in situations when no modificationsare provided or when mathematicsis modified bylanguage simplification (Kiplinger etal., 2000). When students were providedwith a glossaryon achievement-based tests, it actuallyincreased the amount of readingthat had to be done bythe student in the same allotted time framebecause now they were expectedto read the15mathematics problemand the extra informationin the glossary. Withoutproviding extra timefor the students tocomplete the mathematicsachievement test,glossary modificationmay bemore difficult forESL students thanFEP students becausetheir reading and comprehensionrates are generallyslower than FEPstudents (Kiplinger etal., 2000). Since the modificationsofmathematics word-basedproblems using simplifiedlanguage or a glossaryonly allow formodest increasesin achievement, thesealternatives are not theanswer to the problem ofanachievement disparityon mathematics wordproblems betweenESL students and FEP students.Regardless ofthetype of modificationmade to a mathematicsproblem, it is important toensure that thereis no change to the complexityor difficulty of the question,as was guaranteedby Kiplinger etal. (2000) and Hofstetter(2003), When language issimplified or a glossaryisprovided, if the mathematicsdifficulty remains unchangedbetween tests, FEP studentswith asatisfactory level ofreading comprehensionshould perform equallywell on modified andnon-modified versions ofthe tests (Kiplinger etal., 2000).For language-basedmathematics to be assuccessful as possiblein classrooms thatcombine studentswho are fully fluentin English with thosewho have limited Englishabilities,math problemsmust use English which studentsare familiar with (Brenner,Herman, & Ho,1999). Key mathematicsterms and academic levelEnglish should only be includedinassessment situationsafter they have beenclearly defined and used multipletimes within theclassroom and instructionalsetting through teacher-directedsituations such as modelingandstudent-directed situationssuch as problem solving.Regardless of a student’s first language,heor she is best ableto answer mathematicsword problems which usemathematics languagewhich is in the sameform as he or she has beenintroduced to in theclassroom (Brenner,Herman, & Ho, 1999). Forexample, if the mathematicsterms difference or howmany are usedin instruction, the same termsshould be used in a similarcontext in word-based mathematics16problems to allow studentsto use their existinglanguage knowledge tointerpret, translate andanswer the question(Brenner, Herman, &Ho, 1999).Due to students’ existinglanguage limitations, thevocabulary used in mathematicsproblems is of theutmost importance to themathematics success ofESL students. When astudent is presentedwith a word problem, heor she must interpret theEnglish words used,deduce the intendedmeaning behind thewords within the frameworkof the question,understand the mathematicsterms used, apply thecorrect mathematical proceduresandoperations in the appropriatecontext, set up thequestion to integrate theoperations andnumerals, andcarry out the mathematicsto completion.The first two steps in thisprocessinvolve language skills,while the last threefocus primarily on thestudents’ mathematics abilityand knowledgewithin the context of theestablished mathematicsproblem. Mathematicseducators have attemptedto blur the distinctionbetween manipulation and applicationofmathematics andlanguage, or the communicationand discussionof math, so that these twostrands are not separateand distinct realmsof education (Anderson, personalcommunication,January 21, 2008).For students who havelimited languageabilities, presenting mathematicsproblems in an image-basedformat, devoid of language,may be a way to allow studentstounderstand and communicatemathematics knowledgewithout filly developed languagecapabilities.Brown (2005) notes thathigher socioeconomic status(SES) benefits ESL studentsattempting to learnEnglish by allowing them tobe exposed to a print rich environmentandproviding them with more pertinentEnglish experiences,especially in the area of academicEnglish. ConversationalEnglish, also called basic interpersonalcommunication skills (BISC),isoften acquired by ESL studentswithin two to three yearsof immersion, whereas academicEnglish, or cognitive academicEnglish proficiency (CALP),which includes the technicallanguage necessary toread textbooks and technicalwriting with subject-specific vocabulary,17often takes betweenfive and seven yearsof immersion and specificinstruction to attain(Cummins, 1997;Hofstetter, 2003).Brown (2005) findsthat high SES ESL studentsscore muchlower on mathematicsword-based problemsthan their high SESFEP counterparts, a findingthat he attributes toESL students’ limitedknowledge of English.Brown (2005) and AbediandLord (2001) furtherinvestigated the impactof SES on reading abilityand emphasize thatforboth ESL and FEPstudents, high SESstudents have higherreading proficiency thantheir lowSES counterparts.Because many mathematicsproblems in thecurrent curriculum are word-based, this findinghas an impact onmathematics also.Neither one of these studiesdelvedspecifically into theconnection between self-efficacyand performance, butBrown (2005) doespoint out that for ESLstudents to continueto be motivated and succeedin mathematics, theymust be given theopportunity to demonstratethe full extent oftheirabilities. The discrepancybetween socioeconomicstatus, mathematicalability and literacy skillsallows Brown (2005) toclaim that a language-basedassessment ofmathematics ability is unfairfor ESL students. A fairassessment system wouldallow high SESESL students to performon par with high SESFEPstudents, allowingfor some lag in the scoresof ESL students due tomathematics instructionwhich takes placein a language with whichthey are not yet fully proficient.Brown (2005)argues that, ideally,equal treatment ofESLstudents would allowthem to show theirmathematics ability, ratherthan be restrained by theirincomplete English languageproficiency.Presently, no matter howcompetent a student may bein mathematics, if his or her Englishknowledge is not advancedenough to complete wordproblems, his or her scores onmathematics assessmentin a language-based curriculumwill be lower than expectedand thestudent’s potential mathematicsunderstanding and ability cannotbe accurately assessed(Brown, 2005).It is worth exploring howthe language-based format usedin the current mathematicscurriculum impactsESL students’ assessed performance,since their performance appears to18drop with the inclusion ahigh proportion of word-based problems.All cultures experiencemathematics learning in a differentfashion, and the way students are taught and assessedandthe importance placed on specificskills may also differ (Gutstein, 2003). The effectthat thisshift of focus has on studentswho are new to the Western classroom is a topic thatshould beinvestigated in an effort to make the transition asuncomplicated and smooth as possible.Self-efficacy, which is an individual’s self-assessmentof his or her likelihood of successon an upcoming event or situation, has its basis inBandura’s social cognitive theory. Thistheory argues that an individual makes judgmentson his or her own ability to achieve a givenresult in relation to a specific task before that taskis carried out (Bandura, 1986). Anindividual’s judgment of self-efficacy is linked to his or hermotivation, and is a predictor offuture behaviour in the given area (Schunk & Gunn, 1986). Self-efficacy measuresare not staticthroughout a person’s life and are impacted positively and negatively by past performance,vicarious experiences, social persuasion, and emotional arousal (Klassen, 2004). For each event,individuals form an efficacy judgment as they undertake and complete further related tasks(Pajares & Kranzler, 1995). The outcomes ofthese efforts are then used to positively ornegatively influence their self-efficacy in future tasks.Bandura (1986) asserts that self-efficacy is task-specific; meaning that self-efficacy inword problems in mathematics will not necessarily be relevant or interchangeable with self-efficacy in relation to computation problems or image-based problems, thus they each need tobe investigated separately. Pajares and IVliller (1995) argue convincingly that with clearlydefined self-efficacy measures, which are closely related to the math problems being solved, thepredictive outcome of that relationship is increased. This makes a self-efficacy measure that isbased specifically on the task being investigated (ex. image-based problems) a more reliablemeasure than a self-efficacy scale loosely based on the general topic being studied (ex.mathematics). A student’s confidence in mathematics of his or her ability to solve given19problems is incorporated aspart of a self-efficacy measurement,along with the belief that he orshe possesses the skills necessaryto answer the questions(Pajares & Miller, 1994). This dualityof self-efficacy is importantwhen investigating the self-efficacy ofESLstudents, because itaddresses their perceivedability when solving a language-basedmathematics problem and theirfeelings about whether ornot they possess the pertinentmathematics skills for completion. Self-efficacy is different fromself-concept in that self-conceptis a measure of an individual’sperceived self-worth based onjudgments of competence. Self-efficacyis only a measure of astudent’s reports ofhis or her ownability to be successful in a certain taskand is not tied to anyjudgments of self-worth (Pajares &Kranzler, 1995).This begs the question of how the inabilityto show the full extent of one’s mathematicsknowledge impacts a student’sself-efficacy and perceptions of mathematics. Whenstudentshave high self-efficacy in mathematics,they are more likely to be motivated to investmore timeand energy into solving mathproblems in the future because their self-perceivedmeasure ofability is higher (Pajares & Kranzler,1995; Baroody, 1993). Students who experience problemsin mathematics or have low self-efficacy tend to developmathematics anxiety and a fear ofmathematics-related situations(Beasley, Long, & Natali, 2001). Contrary to this, students whohave higher self-efficacy are lesslikely to experience anxiety about their abilities inmathematics (Pajares & Kranzler, 1995).Math anxiety begins to develop as early as the primarygrades in a student’s mathematics educationexperience and, without early intervention gearedspecifically towards the anxiety and its root cause,it will continue to mount as a student is facedwith mathematics situations (Baroody, 1993;Beasley et al., 2001). In an attempt to decreasediscomfort, over time individuals begin toavoid situations which increase their anxiety, such asoptional mathematics courses in high schooland college courses or professions which involvemathematics. This avoidance of advanced or upper level mathematicsmay have a detrimental20effect on a student’sfuture, as it seriously limitsemployment options (Beasley etal., 2001;Baroody, 1993).Many of the studies carried outin the area of mathematicsand language are donethrough quantitativemethods, where mathematics abilitiesare measured with tests.However,previous studies havefound a connection between self-efficacyand mathematics performance(Pajares & Miller, 1994;Pajares & Miller, 1995) and self-efficacyis shown to have as muchimpact on performance as astudent’s actual ability to solve a mathematicsquestion (Pajares &Kranzler, 1995). Because ofthis, self-efficacy in mathematics should beof interest toresearchers, in addition towritten assessments of a students’ability. A shift in focus alsonecessitates a shift in researchmethodologies from primarily quantitativeto a mix ofquantitative and qualitativemethodologies. In order to accuratelyinvestigate the impact ofmathematics and languageon self-efficacy, it is importantto also investigate students’ opinionsand feelings on the subjectfrom a qualitative perspective. Thestudent’s own perceptions whichlead to his or her judgmentsof self-efficacy are very personal andcannot be uncovered througha written mathematics assessment.The researcher needs to providethe student with the time andopportunity to explain how and whyhe or she feels a certain way about a mathematicsconceptor problem type.Many ofthe studies discussed inthis paper assess students’ self-efficacy using a five oreight point scale to measure theirperceived ability on mathematics word-based problems.Todelve further into the issue ofhow to positively influence a student’s self-efficacy, weshouldexamine the topic with moredepth than simply interpreting a number as a comprehensiveexplanation for students’ feelings and perceptions.Supporting a scaled measure with aqualitative analysis of students’ self-efficacywould be beneficial in order to better understandwhy they might have higher or lower self-efficacybeliefs and how they feel it affects them.Abedi and Lord (2001) used some qualitativeanalysis in their study ofthe importance of21language and languagemodification in mathematicstesting for ESL andFEP students. Theyconducted a seriesof interviews with asmall sample of the participantswhere the researchersasked students abouttheir perceptions ofcertain math word problemsand their ability tocomplete them. Thisinterview step is importantto allow students toinform the researcher oftheir preferences andwhy they have madecertain selections and choices.Without theinformation abouttheir feelings and opinionscoming straight from thestudents, a researchercan only formulate an educatedguess as to why studentshave specific preferences.Interviewsalso give students the opportunityto discuss how they feel theirself-efficacy has developed andchanged over time and howself-efficacy influencestheir mathematics achievement (Pajares &Kranzler, 1995).In a quantitative analysisof self-efficacy measuresin mathematics, Pajares and Kranzler(1995) discovered that86% of students are overconfidentin their mathematics abilities. Thesestudents demonstratehigher self-efficacy ratingsthan their actual abilities justif’.Pajares andKranzler (1995) suggestthat this is beneficial because amarginally inflated self-efficacymeasure actually motivatesthe student in the area ofmathematics. A positive view ofone’sability to performwell on a mathematics task alsoimpacts the apparent effortthat is required fora successful outcome(Chen, 2002). As students becomemore confident in their abilities,theyreport that their perceivedeffort on a mathematics taskdecreases even if the actual effortuseddoes not. Perceived effortand motivation are negativelycorrelated so that a decrease inperceived effort actually increasesstudents’ motivation andwillingness to learn furthermathematics (Chen, 2002).In a study comparing the self-efficacyof Asian-Canadian immigrantstudents and AngloCanadian students, Klassen(2004) reports that individuals fromEast Asian cultures have lowerself-efficacy beliefs thantheir Western peers regardless oftheirsuperior success in assessmentsof mathematics ability (Whang & Hancock1994). There has been no study carried out to22specifically investigatethe proportion ofEastAsian students who overestimateor underestimatetheir abilities in comparisonto their actual assessedperformance. This is anarea that should bestudied because offindings that link positiveself-efficacy with increasesin a student’sperformance, willingnessto put forth greater effort,and level of persistenceon difficult or novelmath concepts (Brown,2005). If East Asian students’self-efficacy in mathematicsis alreadylower than their Westernpeers, how does this impacttheir motivation whentheir assessment ina Western classroomdoes not reflect their capabilitiesand is lower than they wouldexpect?Kiassen (2004) investigatedthe impact that culturehas on self-efficacy beliefs. Heidentifies that thereare differences between individualistcultures, such as Canadian orAmerican cultures,and collectivist cultures, suchas those found in manyAsian countries. In anindividualist culture, theemphasis is on the individual,and his or her personal goalsand ideals.In a collectivist culture, the focusis on the group as a wholeand its members’ duty within andfor the group. Klassen’s(2004) study of cross-cultural self-efficacycompared Indo-Canadianimmigrant students (a collectivistculture) to Anglo-Canadian students(an individualist culture).For both collectivist culturesand individualist cultures, pastperformance on similar tasks hasthe strongest effect on students’mathematics self-efficacy beliefs.For Indo-Canadian immigrantstudents, the next mostinfluential contributors to self-efficacyare students observing theabilities of individualssimilar to themselves and the degreethat their own parents value successin mathematics. In an individualisticculture, students’ self-efficacyis influenced more stronglyby emotional arousal, such asthe fear of failure, thanthe influence ofthose around them(Klassen, 2004).Future studies into cross-culturalself-efficacy could provideresearchers and educatorswith beneficial information abouthow self-efficacy is formed and experiencedwithin particularcultures (Kiassen, 2004). Itis necessary for self-efficacy to be lookedat within a specificcultural group because it wouldbe naive to believe that the self-efficacyexperiences of all23students enrolled ina Western mathematicsclassroom would beindicative of all of thepopulations and culturesthat come together tocomprise the classroom community.Klassen(2004) also points out thatthe factors which have thegreatest influence on self-efficacymaychange over time regardlessof cultural background, asstudents are acculturated intoWesternindividualistic culture andmove away from adhering tothe expectations of a collectivistculture.This is another areawhich would be worthwhile to studyas the knowledge couldincrease ESLstudents’ self-efficacy inthe area of mathematics atparticular stages of their education.The purpose ofthis studyis to investigate how the self-efficacyof students who areEnglish language learnersis impacted by curriculum andassessment practices which focus onlearning through language-basedword problems. Can a modification,such as using image-basedmathematics, increasethe mathematics performanceofESL students as they learn English?How does thismodification impact an ESL student’sself-efficacy belief?I compared themathematics self-efficacyof students in ESL level 2with that of students in ESL level4 and ofstudents who are notregistered in ESL, who are consideredby the British ColumbiaMinistry ofEducation and the HargroveSchool District as fluent inEnglish. Investigating potentialalternatives to the currentlanguage-based mathematics programis important in order to increasethe mathematics knowledgeand learning of students who arelimited by a word-basedcurriculum due to a lack ofEnglish language knowledge. Duringthe transition to the Englishlanguage classroom, educatorsshould endeavour to provideESL students with programs thatwill allow them to acquire thelanguage but also as muchof the curriculum knowledge aspossible. If image-basedmathematics can still provide meaningfulmathematics experiences thatallow students to manipulate, investigate,and explore mathematics without languagebarriers, itmay be a useable alternative learningapproach to allow ESL student to acquiremuch of thesame mathematics learning astheir peers who are fluent in English.The option would helpminimize any gap in learning dueto a period of language acquisition whichmight result in ESL24students falling behind their FEP classmates.It is also important to investigate using image-based mathematics problems as apotential alternative for ESL students to increase their self-efficacy beliefs, because modificationswhich increase self-efficacy also help to increasestudents’ present performance. The link between self-efficacyand motivation cannot beminimized because of the role that motivation plays in educational performancein theclassroom. Increased motivation leads to increased effort and perseverance,which both help tocontribute to increased performance. If motivation and, in turn, performancecan be increased,this may allow the student to have a more positive view of school and learning (Pajares andMiller, 1995). The fIiture increase in the likelihood that the student will pursue college classes ora career that is mathematically based also encourages investigation into the increase of selfefficacy beliefs (Pajares & Miller, 1995).If an improvement can be made in the area ofmathematics self-efficacy, it would be wise to pursue it in hopes of increasing mathematicsperformance in elementary school and at all stages of life.25MethodsProblem SelectionAfter examining many ofthe problems foundin resources currently used by teachers ingrade 6 mathematics classrooms throughout the HargroveSchool District, I selected themathematics problems used on the mathematics taskadministered to grade 6 participants. Idetermined which resources, textbooks, and problemswere being used in classrooms throughobservation in a variety of schools and through informalconversations with teachers andstudents about the resources and materials they were using and assignmentsbeing given. I foundthat, while there are many recently published textbooks available which arebeing used tovarying degrees in classrooms, many teachers supplement these new resourceswith oldertextbooks and non-Ministry-approved resources, suchas Scholastic computation andmathematics drill books.Many ofthe resources found in classrooms contain mathematics problems that wouldnot be considered by mathematics educators to be a reflection of currently accepted best-practice problems, because current best-practice problems involve studentscompletingproblems which involve multiple steps, manipulation of numbers, and complex thought aboutthe mathematical processes involved (Baroody, 1998). Best-practice problems also involveanswers which are not fixed so that the problems are open to interpretation by the students, thusallowing a variety of acceptable solutions (Baroody, 1998). The computation and wordproblems found in the mathematics task used in this study include problems that may be moresimilar to the ones in the classroom resources than to current best practice problems, because itwas my intention that the mathematics task problems be an accurate reflection of currentclassroom practice in the Hargrove School District. In my attempt to include computation andword-based problems that are as close to best practice as possible, the problems that I selectedinvolve multiple steps for students to work through. However, unlike best-practice problems, the26problems included in the mathematicstask are only open to a few interpretations, sothat thereare only one or two correct solutionsto each problem. I made alterationsto the problemsdiscovered in classroom resourcesin order to maintain the language and problemformat used inthe original problems, but alteredthe questions enough so that they are differentthan thosefound in the classroom resources.This was done to ensure that therewould be no chance ofstudents having ever completedan identical mathematics problem before. The alterationsincluded, but are not limited to,numbers, names, locations, and otheridentifying details foundin the problems.The mathematics skills and knowledgeneeded to complete the problems asked on themathematics task do not directlycorrespond to the skills that are taught and acquired as apart ofthe grade 6 mathematics curriculum,as outlined by the Prescribed Learning Outcomes(PLOs)of the British Columbia Ministry of Education(British Columbia Ministry ofEducation, n.d.).The mathematics task used in this study, though administeredto grade 6 students, generallyincludes mathematics concepts that are coveredin the Learning Outcomes of earlier grade years.For example, Question #9, an image-based problemwhich involves the passengers on a bus,asks students to add and subtract single digit and twodigit numbers and count up to 11,mathematics skills that students should have masteredyears earlier in grades 2 to 3 andKindergarten to grade 1, respectively (British ColumbiaMinistry ofEducation, 1996a; BritishColumbia Ministry ofEducation, 1996b). Questionswith mathematics skills acquired in earliergrades are included in the mathematics task in computation,language-based and image-basedproblem formats because the mathematics task is not designed to assess students’ acquisition ofgrade 6 learning requirements. Rather, the purpose of the mathematicstask is to investigate howstudents process mathematics problems in computation, word-basedand image-based problemformats, and what effect the format itself has on students and their opinionof mathematics. TheBritish Columbia educational curriculum does not expect students to have mastered the learning27outcomes ofthe PLOs bythe end of the grade year,though they are expected to have agoodunderstanding of them.The curriculum as designed creates arepetitive, circular format so thateach ofthe learningoutcomes taught in the previousyears is revisited and built uponin latergrades. I took this learningpattern into account as I createdthe mathematics task, as well asthefact that because students donot all learn at the same speedmany of them might need theadditional years to fI.illygrasp a mathematical concept.I was concerned that theinclusion of a high proportionof problems that require studentsto use mathematics thatthey have only been introducedto recently might impact or skew theresults of the study. If only grade6 level problems had been included in themathematics task,when a student experiences difficultywith a specific problem, I would notknow if it is causedby the problem formator the mathematics skills requiredof the student. The inclusion ofmathematics problems that useskills that students should be adeptat and familiar with becauseof years of practice makesit more likely that any difficulty encountered isthe result oftheproblem itself.While computation and word-based problemswere created using the textbooks andclassroom resources as models (see AppendixA and Appendix B respectively), I developedimage-based problems as an original component,with the mathematics skills used in thecomputation and word-based problems as aguide (see Appendix C). The image-basedproblemsused in the mathematics task were developedto meet the goal of producing problems devoidoflanguage which minimize the useof numbers and computational symbols in the problem itself,while still clearly communicating the mathematics problemthat needs to be solved.Mathematical symbols and numerical digitswere generally avoided in an attempt to minimizethe likelihood that the image-based problems wouldtransform into computation questions thatare simply illustrated. The images used in the problemsare intended to be interpreted by theparticipant into a mathematics problem and carriedout to completion.28Question marks are usedin all of the image-basedproblems instead of an equalsymbolto indicate tostudents which value theyare meant to discover.Arrows are included in theproblems as a methodintended to direct students’attention to all areas oftheproblem, so thatthey would take into considerationall of the important visualinformation before attemptingtoprovide a solution strategyand final answer. Forexample, image-basedproblem Question #2,which involves bananas intendedto be evenly distributedbetween four monkeys, usesan arrowto show studentsthat all ofthe bananasshould be placed into the greenbucket before studentscontinue with the problemand distribute the bananas tothe monkeys. Unlike the other image-based problems, thisproblem presents the need for astandard mathematics symbol to be usedinorder to make it clear and understandableto students. The monkeysshown in the problem arenot all of equal size and asI was designing the problems,I was concerned that, because of thissize discrepancy, studentsmight not assign an equal numberof bananas to each monkey, asintended. My suspicions wereconfirmed through conversationsthat I had with trial participantsas they completed themathematics task. These participantsindicated that the monkeys’ sizedifference made them wonderif there is a ratio or other factor bywhich to distribute the bananasto monkeys according tosize. When the trial participants didnot find this information aboutdistribution, someof them asked whether they could randomlyassign bananas to monkeysbased on their assumptionsof the representative size discrepancyand the total number ofbananas available.In order to minimize this confusion, I placedan equal sign between thequestion marks above each monkey.I chose to use this representation because, withoutan equalsymbol, even ifthe monkeyswere the same size, the students could decide toallocate thebananas as they wished. Themathematical symbol placed betweenthe question marks on thearrows pointing to each monkeywas meant to indicate to students thatthe number of bananasallocated to each monkey shouldbe equal regardless of size. Trials ofthe mathematicstask thatwere conducted after an equal sign was addedyielded results that were more closely matched to29my desired outcomefor the problem. Studentsin the second mathematicstrial still took note ofthe size difference betweenthe monkeys but assignedan equal numberof bananas to eachmonkey when theynoticed the equal sign. Addinga standard mathematicssymbol in an effort tominimize students’ confusionand decrease the possibilityof multiple answers successfullydirected the trialparticipants to the desiredoperation of division. Thisdoes raise the issue offuture designs for image-basedproblems and the need to createimages that are easilyunderstandable to all students.This issue will be considered furtherwhen discussingimplications for future research.A question mark is alsoused to signify a combinationof missing values in Question #1,the image-based probleminvolving sporting equipment.In this instance, the question markisplaced in the middle ofthe problem to indicate the placementofthe unknown. Students need toinfer that more than oneofthe objects needs to be purchasedbecause the total cost is $82,shown on the cash registerdisplay, which is more than the costof any one single item shown onthe left of the shopping cart.This problem is presented tostudents in the form x1+ x2 + ... + x=y where x1,x2, and x7 are unknown andy isthe total paid. During trials ofthe mathematicstask, students used a seriesof addition and subtractionproblems to complete the problem. Theexpectations of this question were clearto participants, so its original format wasmaintained.Image-based problem Question#3, a comparative measure question involving ajellyfishand a whale, requires students tocomplete multiple steps in order to determinethe finalsolution. Students are asked to observethe number of segments for eachcreature and multiplyor add by the indicated amountfor two segments to find the total length of the creature.Thisquestion involves multiplicationof decimals, though it can also be solvedor verified throughrepeated addition of decimals, both ofwhich are learning outcomes assigned to grade 5 (BritishColumbia Ministry of Education,1996a). The whale length and jellyfish length are bothwithinthe acceptable size parameters for these oceancreatures in their natural habitats, which allows30students to question whether or not their answers are realistic and valid. A third question mark,placed in the unoccupied space that remains when the two creatures are placed side by side, asksstudents to determine the difference in lengths between the two creatures. The answer to the sizedifference can be found by subtracting the length ofthe jellyfish from the length of the whale orusing a guess-and-check addition strategy, though this would be more time-intensive. In trials ofthe mathematics task, all of the students understood the representative meaning ofthe threequestion marks.The image-based problem, Question #4, involves people moving on and off a bus andrequires students to follow the movement of passengers. This question involves simplecounting, moving both directions on a number line, and working with addition and subtraction.In trials of the mathematics task, some students observed that the bus driver was still on the busin the final frame, while others did not. I initially contemplated removing the bus driver fromthe image, but hesitated because image-based problems should still demand the concentrationand keen observation that is necessary for computation- and word-based mathematics. I wasinterested in investigating the information that students notice, the distinction needed in images,and the information that students believe to be extraneous. In trials ofthis problem, somestudents followed the scenes sequentially, adding and subtracting as passengers moved on andoff of the bus, and using guess-and-check to find an appropriate answer, while others took noteof all of the passengers moving offthe bus and subtracted this from the total number ofpassengers who boarded the bus to determine the number of passengers who were initially onthe bus.Mathematics problems based on language were created using textbooks currently used inclassrooms as a guide, but are tailored to ensure that students use all basic mathematicoperations such as subtraction, addition, division and multiplication, and that these operationsare carried out with the same difficulty in the computation and image-based problems. Some31number values used in the word-basedproblems are written in words as opposed tonumericalformat, because this limits students’ability to place all of the digits shown in the wordprobleminto an equation and end up findingthe correct solution without an understandingofthe reasonor rationale involved. This also forcesthe question to be Ilirther based on language knowledge,as opposed to digit recognition.The computation questions selected forinclusion in the mathematics task were chosen inan attempt to minimize the “easy” or straightforwardcomputation questions in which a series ofoperations are given that need to be completed inorder to find a final answer. It should be notedthat computation questions are usuallynot directly comparable to word-based problems orimage-based problems in mathematics difficultyor sophistication because, unlike the otherproblem formats, computation problems do not requirethat students interpret any information inorder to create a solution strategy or equation; itis given to them in the problem already. Due tothis, computation problems often do not require thatthe student possess the same level ofmathematics understanding, sophistication, or complexthought as the two other problemformats require.Recognizing this discrepancy and my desire for the difficulty of each of the problemformats to be as similar as possible, I modified the computation problemsI found in theclassroom resources in such a way that the altered problems requiredstudents to manipulate thenumbers and equation formats given in order to solve each problem. The computation problemsused in the mathematics task did not simply ask students to regurgitate a memorizedsolution toa series of operations; rather, most of the problems were presented in aform which askedstudents to find a missing value(s) to make the given solution to the problem true. Whentheyencountered a computation problem in a form which could not be directly solved, studentsneeded to determine the strategy they should use in order to answer the problem, because thisinformation was not provided for them. Optional strategies for many ofthese problems include32guess-and-check or reorderingthe question to create an equation tofind the missing value(s).The values that needed tobe found by students were indicatedon the mathematics task by anempty box inserted intothe place of the missing value. Havingthe participants solve to find amissing value in a computationproblem increased the difficulty of theproblem and minimizedthe incidence of studentsanswering in mechanical fashion byrecalling memorized mathematicsfacts.ESL Participant Group SelectionIn the Hargrove SchoolDistrict, the ESL program has five levelsthat students learningEnglish are registered in, based ontheir knowledge of the English language at each level.Afterconsideration of the requirementsof each level and the students’ knowledgeof English, I choseto include students in ESLlevel 2 and ESL level 4 in this study. Theselevels were selectedbecause the students in ESLlevel 2 and ESL level 4 represent beginnerand intermediate ESLlevels respectively. My choice was basedon my previous experience with ESL studentsand myunderstanding of students’ progression throughthe ESL levels.Students who are registeredin ESL level 1 are generally in their first year of instructionin the English language. These students haveusually arrived in Canada sometime after the startof the current school year and it is quite possiblethat some ESL level 1 students have onlyrecently arrived in Canada. Manyof the students in ESL level 1 come into the school systempossessing little or no knowledge of theEnglish language, and this might create manydifficulties in the completion of the study. A major factorin my choice not to include students inESL level 1 in this study was that, even thoughthe study results may be more dramatic withthese participants, I was unsure what the emotional effectof a mathematics task with languagebased problems and an English interview would be onESL level 1 participants. Also, due to thelimited nature ofthis study, for valid results with ESLlevel 1 students, I would need an33appropriate translatoralong for all interviewsand at all times when languagewould be used forinstruction or informationbetween myself and the participants,and this was not possible withinthe confines of my study.IfESL level 1 students were includedwithout a translator present,their English answerswould likely be limited and theirresponses, though they could be writtenin Cantonese or Mandarin,might cause miscommunicationor an information gap where keyknowledge might be missed or leftout. While it would belikely that students in ESL level 1would be able to complete the mathematicstask without a translator, their participationin theinterview component would belimited, and I would not be able tohave a complete grasp oftheir thoughts, feelings, and understandingof mathematics problems in language-based,computation-based and image-basedformats.Most ofthe students inESL level 2 have been in an English languageclassroom inCanada for one or twoyears. ESL level 2 students are usually in theirsecond year of languageacquisition, so they have acquiredsome basic conversationalEnglish abilities and are workingtowards mastery of conversationalEnglish. To further develop their Englishknowledge,students in ESL level 2 still receivefocused English instruction with a specialistteacher for aperiod oftime each week. These studentsmay experience difficulty with uncommon vocabularyor complex sentences in oral andwritten language, because they are still at a stage ofgenerallanguage acquisition. Students in ESL level2 were selected to participate in the mathematicsstudy because they have enoughEnglish knowledge to understand theEnglish mathematics taskinstruction and interview questions, if itis supplemented with a written translation of simplifiedChinese and a voice recording in both Cantoneseand Mandarin. These aids were used to allowall ESL level 2 participants to completethe task and interview without unreasonable stress ordiscomfort. The ESL level 2 students’ basicconversational English abilities enabled them toparticipate in the interview process withbasic fluency, and allowed me to understand their34statements, respond totheir answers naturally, and knowwhen to probe for further or relatedinformation.Students in ESL level 3 wereomitted from the study because,though they are beginningto learn academic English, thedifference in English ability betweenESL level 2 students andESL level 3 students mightnot be significant enough to includeboth groups in this study.Students in ESL level 4 wereincluded in this study because according tothe timestandards set by the BritishColumbia Ministry of Educationin their funding model, thesestudents have acquired fluency inconversational English and are workingtowards fluency inacademic English (BC Ministry ofEducation, 2002; Carrigan, 2005). ESLlevel 4 studentsgenerally have four years of instructionin the English language classroomwith the first threeyears supplemented by specialized Englishinstruction (Carrigan, 2005). The written instructionsand oral directions on the mathematicstask should not have been a challenge forstudents inESL level 4. These students wereable to participate fully in the English language interviewprocess because theycould understand questions posed in English and respond to the interviewquestions in English. Because every copyof the mathematics task had the instructions translatedinto Chinese, students in ESL 4also had access to this language aid, though they shouldnothave needed to use it. These studentswere also provided with a written and oral translation ofthe interview questions for extra support butnone of the ESL level 4 students used them tosupplement the oral interview.Students with no ESL background were alsoincluded in the study to function as acontrol group. Students with no ESL background were selected according tothe same criteria asthe ESL students. They had to be enrolled in a grade 6classroom, taking grade 6 math, and beof Chinese descent, having lived for at least some timein China. The cultural descent ofthestudents is important to keep constant between the participant groupsbecause self-efficacy inmathematics has been shown by Klassen (2004) to be culture-specific.This specificity is due to35the fluctuation between culturesregarding the importance of schooling,mathematics, and theirrole within an individualistor collectivist community structure.The students with noESL experience have been exposed tothe Western mathematicscurriculum for the entiretyof their time within the school systemand were considered by theHargrove School District and theMinistry ofEducation to be fluent inEnglish within the firstfew years of their entrance into theschool system. The impact that their culturalheritage mayhave on their judgments of schooland mathematics could vary betweenthe students based ontheir family’s time in Canada andacculturation (Kiassen, 2004). The departure from culturalheritage possible over time willlikely happen for ESL students as they are immersed inCanadian culture, society and expectations.Any gradual change in self-efficacy due toacculturation or cultural influencecannot be teased apart from the impact that themathematicsclassroom has on a student’s self-efficacy, becauseself-efficacy is based on both previous andpresent experiences. Furthermore,there should be no attempt to take these two factors apart,because doing so would not be authentic orpossible: a student’s self-efficacy is based on theentirety of a student’s mathematics experiences. Theimpact of cultural background and Westernimmersion does complicate my findings, because self-efficacycannot be attributed singularly tothe problem format of the mathematics task, and problemformat can only be a contributingfactor to a student’s measure of self-efficacy. The difficulty encounteredbecause of multiplefactors contributing to a student’s self-efficacy further justifies conductinginterviews with thestudy participants. The interviews give the childrenthe opportunity to share their opinions aboutthe impact of problem format on their ability to solve the problem,and this may be one of theonly ways to isolate the impact of problem format from othercontributing factors.In order to minimize the stress on the participants and allow all students to participate tothe full extent of their abilities, I attempted to provide the students oflimited English knowledgewith the appropriate language translations needed to complete the mathematics task and the36interview. As mentioned earlier, the instructions for the mathematics task weretranslated intowritten simplified Chinese on the second page of the task booklet, and were alsoprovided to thestudents through an audio recording of the instructions in both Cantonese and Mandarin.Theinstructions were provided both orally and in written Chinese because ofthe awarenessthatmany individuals who speak Mandarin or Cantonese are not fluent in the writtenforms of thelanguage, especially not at such a young age. Unlike most ofthe other materials inthis study,the language-based word problems on the mathematics task were not translated forstudents,because the goal ofthis study was to investigate the impact that mathematicsbased in Englishhas on ESL students. To provide them with a translation would negate thestudy and be anunrealistic reflection of the challenges that ESE students face in the English-languageclassroom. If desired, the interview, which included a series of questions asked orallyoftheparticipant, was provided to participants in either Cantonese and/or Mandarin throughthe use ofprerecorded questions in both ofthese languages. The question was asked firstby the researcherand then played in the appropriate language directly afterwards. Students wereencouraged torespond in either English or the language of their choice if they did not feel that theycould frillyexplain their thoughts or opinions in English. This flexibility was communicatedto allparticipants in all three languages at the beginning of each interview.Mathematics Task TrialsThree formal and three informal trials were conductedto identify any potential problemswith the mathematics tasks. The formal trials were carriedout one-on-one with participantsfrom grade six and seven classes. Participants providedfeedback to the researcher and sharedtheir solution strategies for each problem duringcompletion. These trials were conducted overthe course of two weeks, and changes were madeto the mathematics task based on the inputfrom trial participants.37Informal trials were conducted with children in grades 5, 6 and 7 who were available atschool to provide input after completing their assigned school work. The informal trials wereconducted after the formal trials and did not result in any changes being made to themathematics task.All of the participants in the formal and informal trials were of Chinese descent and werefluent in BISC.School Selection and ApproachEvery September, schools register their ESL students with the British Columbia Ministryof Education in order to receive fbnding for the students for the upcoming year. These numbersare reported on the BC Ministry ofEducation website as downloadable PDF files for publicviewing. I downloaded the reports for all ofthe schools in the Hargrove School District andcontacted the schools with 12 or more ESL students in grade 6. There were seven schools forthe 2007-2008 school year who met this criteria. I contacted all seven schools by telephone tointroduce myself and my study and arrange a meeting with the school principal to furtherexplain the study. Ofthe seven schools contacted, five accepted the offer for a meeting andtwodeclined. I contacted only one school with fewer than 12 ESL students in grade 6 to find moreparticipants for the study.The enthusiasm ofthe participating schools was mixed. One ofthe schools had noparticipant response, while another had only one student volunteer to participate. Two otherschools provided me with six participants each. The fifth school provided me with fiveparticipants and the remaining three participants came from the sixth school.38SubjectsAt the time of the study, all participants were enrolled in all grade 6 courses, includinggrade 6 mathematics. Table 1 shows the distribution of participants by gender and school.Table IParticipant Gender and School AffillationNumber of Participants Per SchoolStudent Group TotalA B C D EESL Level 2Female I 1 1 1 4Male 2 2ESL Level 4Female 2 2 1 5Male 2 1 3Not Registered in ESLFemale I 1 1 3Male 2 1 1 4Participants were between 11 and 12 years old and all of Chinese descent with varyingEnglish language abilities, having lived for at least some period oftime in China with theirfamilies. Six ofthe participants were registered in ESL level 2, designating them as studentswho were still acquiring basic interpersonal communication skills (BISC) in English (Cummins,1980; Carrigan, 2005). Eight participants were registered in ESL level 4 which designated themas students who had mastered BISC and were working towards mastery of academic English(cognitive academic English proficiency or CALP). Seven participants who were fiilly fluent inEnglish, and not registered in ESL, were included in the study as a control group. These students39had received at least six years oftheir educational instruction in English and have hadfillcomprehension ofthe languagesince at least grade 3.MaterialsChildren were asked to complete a mathematicstask problem booklet consisting of 12problems similar to those found in thegrade 6 mathematics curriculum (see Appendix A,Appendix B, & Appendix C). Both the Mathematics TaskQuestion Booklet and theMathematics Task Work Booklet had covers with writteninstructions for the mathematics task(see Appendix D and AppendixE, respectively). The second pages of both of these booklets arethe completion instructions written in simplified Chinese (seeAppendix F). Four problems werecomputation-based; four were image-based problems thatmerge interpretation, computation andlogic while minimizing the need for language; and fourwere language-based problems typicalof the ones currently used in the provincially recommended textbook resources. Two questionsof each problem type were easy, one of each was of medium difficulty, and the remainingproblem was more difficult. The questions in each problem format were of comparabledifficulty to one another and designed to assess the use of mathematics skills involving addition,subtraction, multiplication and division knowledge within each problem format. The skillsneeded to complete the mathematics task were based on the British Columbia PLOs describingthe curriculum expectations for grade 6 students BC Ministry ofEducation, 1996).The order ofthe mathematics problems was randomized in each of the mathematics taskproblem booklets so that the computation, word-based and image-based problems were in noparticular order. Students in each focus group (ESL level 2, ESL level 4, and those notregistered ESL) were given one of eight problem booklets. No two students in the same focusgroup had a mathematics task problem booklet with the questions in the same order.Randomization of the problems was used to ensure that if success or difficulty is encountered40repeatedly on a question, the observedoutcome or pattern could not be attributed totheproblem’s placement in the mathematicstask problem booklet or on the influenceof a particularproblem preceding it. The problemswere not placed on the mathematics task in any orderrelated to their difficulty and were alsorandomized so that the difficulty oftheproblems cannotbe assumed by the student from theirplacement (cx. from easiest to hardest or viceversa). Thequestions were tested in trials on ESLand non-ESL students and amended before the actualmathematics task was administered to study participants.Students participating in the mathematicstask recorded their written work in a separatedesignated booklet. It provided onefull page for each problem and an adjoining page allocatedfor a series of opinion questions about the mathematicsproblems (see Appendix G). Aftercompleting the written work foreach question, students were asked to circle a word that coulddescribe their perceived level of difficulty for the question:very easy, somewhat easy,somewhat hard or very hard. I chose to use words, rather than a number-basedscale becausenumbers on a scale do not have a fixed meaning. For example, on a scaleof 1 to 10, where 1represents no difficulty and 10 represents extreme difficulty, onestudent may chose to assignmoderate difficulty to a 7, while another might chose to assign the exact same difficulty to a5.Giving the students words instead of numbers could help minimize this disparity. Taking thestudents’ possible language difficulties into consideration, the English words were translatedinto simplified Chinese directly below to further reduce any confusion. Translation ofterms inthis instance was beneficial because students’ responses to these questions of opinion should beas informed and complete as possible. Students were also asked in separate opinion questions torate their confidence in their answer and their ability to answer a similar question.41Scoring the MathematicsTaskWhen marking the responsesof participants on each ofthe three typesof questions, theywere assessed based on work shown,areas of error, and accurate answers.Before markingbegan, multiple possible solutionswere written for each problem (cx. asolution to a problemthat used repeat addition andan alternative solution to the same problemthat was equally validbut used multiplication). Possiblesolutions were developed by the researcherthroughcollaboration with the trial participantswho tested the mathematics task before it wasadministered to the study participants.Errors on the mathematics task werecoded as one of four types: no answergiven,computational error, solutionstrategy error or comprehension error. Those questionswhich haveno marks written in the solution areawere coded as “no answer”. Some students made a“calculation error” during the completionof a question. These were coded as such when thesolution strategy presented was viableand demonstrate that without the calculation error anaccurate answer would have been reached. In otherinstances, the solution strategy used toanswer the problem was not effective and couldnot result in an accurate answer. The coding forsuch responses was “solution strategyerror” and was only used when the strategy could notsatisfy the requirements of the problem. Alack of understanding based on the problem itselfwas also possible. In a “comprehension error”,the student exhibited confusion about what theproblem was asking, rather than what mathematicsshould be used. These errors were harder todeduce and had to be uncovered through considerationof the work shown and the oral responsesprovided by the participant.The incidence of correct and incorrect answers, perceiveddifficulty, assessment ofcorrect answers, and high or low self-efficacy reports were tabulated and recorded ingroupingsdetermined by English ability (ESL level 2, ESL level4, or no ESL background).42ProcedureMathematics TaskThe mathematics task was given to participants after school hours in a small groupsetting with other participants from the same school. ESL designation was not a considerationwhen grouping participants for the assessment. The task was completed in a small group settingto alleviate any stress that students may have experienced in a one-on-one assessment with theresearcher. Before beginning, the researcher explained to all ofthe participants that theproblems that they would be solving are at a grade 6 mathematics skill level or lower, and thatmany ofthe problems were very similar to those found in textbooks that had been used at somepoint in grade 6 classrooms in British Columbia. Each student was given one Mathematics TaskQuestion (Problem) Booklet, one Mathematics Task Work Booklet, and a pen. The directionsfor the Mathematics Task Problem Booklet and the Mathematics Task Work Booklet were givenverbally, but were also written and translated into Mandarin and Cantonese and provided to eachparticipant. The researcher explained that the Mathematics Task Question Booklet contained themathematics problems only, and should not be written in. Participants were also advised that theMathematics Task Work Booklet provided the space for them to record all of their work andanswer opinion questions. Students were asked to complete each question to the best of theirability. They were encouraged not to leave a blank space in the event that they were unsure ofan answer, but rather to make an educated guess as to what the problem was asking and howtoformulate their response. Participants were strongly encouraged to show all oftheir thinking onthe paper and asked not to erase any of their work. They were advised that the researcherwanted to see, not only the end result oftheir thinking, but also how they got to that answer. Ifparticipants thought that any of their work was incoffect, they were asked to simply cross outtheir mistake with a single line and make their corrections in the space beside their error. It wasexplained to them that these mistakes and the written work they complete before they finda43final solution often provides valuable information to the researcher along with the final answeritself. After answering each mathematics problem, students were asked to make three judgmentsabout it: to rate the difficulty of the questions; provide their beliefthat they have reached thecorrect answer; and their ability to successfiully complete more questions just like it. Their self-perceived ability to correctly answer similar problems in the fhture was a measure of their self-efficacy beliefs relating to tasks ofthe same nature.After the participants completed the mathematics task, they handed in both theMathematics Task Problem Booklet and the Mathematics Task Work Booklet to the researcherso that correct answers could be marked and their work could be investigated for completion,errors, omissions, and demonstration of mathematics understanding.InterviewThe individual student interview and the mathematics task were not administered on thesame day because twelve mathematics problems are enough to saturate most students after a dayof regular school work. Since the interview portion ofthe investigation was important tounderstanding students’ thought processes it was completed within one week of completion ofthe written mathematics test. Students met individually with myself as the researcher at thelocation oftheir parents’ choice, such as their house or the public library. This was done to tryto provide the students with the power of selecting an area that they would feel comfortable in,as suggested by Punch (2002). The interviews were audio taped and took approximately 0.5hours to 1 hour to complete. It was comprised of a set series of interview questions (seeAppendix H), discussion ofthe mathematics task, and time for participants to share theirmathematics experiences and thoughts with the researcher. I introduced myself as a studentatthe University of British Columbia in an attempt to minimize the powerimbalance that mighthave existed between me as an adult and the participant as a child (Punch, 2002).My teaching44background and years ofworking with children provided me with the experience that I neededto establish rapport with participants before the interview began. This was done to try toencourage students to feel comfortable sharing their real opinions, rather than providing theanswers that they felt that I wanted to hear, which has been reported as a problem in both childand adult research (Punch, 2002). I took time at the beginning ofthe interview to explain to eachof the participants that his or her honest thoughts, ideas and insights about mathematics andmathematics instruction were important, informative and appreciated. This was done to try to setan open tone with the participants so they felt like their opinions were valued.The interview was completed by the researcher asking the questions verbally, whileproviding students with an oral translation of the questions, so to minimize the language barrier.Students shared their responses verbally while the researcher audio taped their verbal responsesand recorded notes about their answers and their nonverbal responses. If students were unable tosuccessfully communicate their responses in English, they were given the opportunity torespond orally in the language of their choice, which could be translated later from the audiorecording. The researcher gave students the time to explain their feelings about mathematics ingeneral and compare their present math experiences with any they may have had in the past.Before discussing their responses to the mathematics task, students were asked todescribe their degree of success and enjoyment of computation problems, word problems, andpicture problems, and asked to explain their choices with some discussion. This was asked earlyin the interview so that any discussion and questions asked about specific problems on themathematics task would not impact students’ judgments or responses about self-efficacy orenjoyment. The researcher then asked students questions related to the mathematics taskthatthey had completed. Students were not told if their answers to the mathematics task wereaccurate because this may have swayed their initial responses regarding self-efficacy. Itwasintended that students’ self-efficacy judgments be based on their regular classroomexperience45and not overly influenced by the mathematics task. Theparticipants were shown an originalunmarked copy of the mathematics taskwith their written work and question responses. Theinterviewer asked students about all the picture problemsand only one or two computationproblems and word problems chosen at random. This wasdone because the focus ofthis studywas primarily on students’ responses to picture problems,not the other forms of mathematicsproblems. They were asked to identify the questioneach problem was asking in their own wordsand to describe their solution strategies. Students were asked to elaborate on howthe problemscould be made clearer and asked why they felt they would or would not be able to completesimilar questions in the fhture. The questions on the mathematics task were discussed withstudents in no particular order so that order would not influence students’ opinions, therebyimpacting the results of the interview. At the end of each interview, participants were asked ifthere was anything about the mathematics task that they wanted to share with the researcher.Each participant was thanked for his or her time.Not all ofthe responses to the interview questions directly answered the main researchquestions being investigated. Some questions allowed me to develop a sense of the student’slanguage background and his or her general view of mathematics. This paper discusses only theresponses to questions that address English use in the mathematics classroom, self-efficacy, andperformance on problem solving in different formats, as outlined earlier in the researchquestions for this study.46ResultsThe mathematics task and interview, administered to21 grade six students in one ofthree designated ESL levels during the latter part ofthe school year, demonstrated someinteresting trends and noteworthy observations. Due tolimited sample size, these findings arenot generalizable but serve as an indicationof performance for the selected group. Trendsobserved in this group can provide a baseline for furtherinvestigation. The reader is remindedthat the research questions being asked are:1. According to ESL students’ self-reports, how istheir mathematics ability and self-efficacy affected by the English used in the language-based mathematicscurriculum?2. How is mathematics achievement and self-efficacydifferent when mathematicsproblems are asked in a computation-based format, word-based format or picture-based format for students in ESL level 2, ESL level4, and those students notregistered in ESL?3. What is the impact on students’ performance and self-efficacy when using themathematics modification of image-based problems, instead of language- orcomputation-based mathematics problems?I will share student judgments about English in the mathematics classroom, and discusstrends and differences seen between student approaches to the problems and solution strategiesas individuals and groups. Finally, I will present the written and oral responses of students inESL level 2, ESL level 4, and not registered in ESL to each ofthe picture problems collectedthrough the mathematics task and subsequent interview. All students have been givenpseudonyms to conceal their identity.Due to the exploratory nature of the study, the results provide us with initial insights intothe role image-based problems may play, and provides us with directions for future research intothis issue.47English in the Mathematics ClassroomIn order to investigate student’sfeelings about the use of English in a language-basedmathematics curriculum, I askedeach participant for his or her input. The question was raisedwithin the first few minutes of eachstudent’s interview in order to minimize any influencethatthe interview itself might have on the student’s opinions.There were varied responses to thequestion “What is your opinion about the amountof English used in the mathematicsclassroom?”ESL Level 2 Response to the Use of English in the Mathematics ClassroomAmong students registered in ESL level 2, responsesshowed an acute awareness oftheprevalence ofEnglish in their mathematics classroom. Many ofthe students explained that theyfeel that as their English improves, the amount of difficulty caused by the languageinmathematics decreases. Emily mentioned that she has difficultywith both the words andsentences used in mathematics. “Some sentence is hard to understand... . Some, like, thosequestions, urn, some has a really difficult words, so I can’t understand.”Mikey mentioned that he has noticed the dominance of the English language in histextbook; “For the new mathematics textbook there are a lot of those word questions. There’sonly one or two question that [are] numbers or shapes and all the other questions are words, justwords. So [the amount ofEnglish in the math classroom is] pretty much like 90%.” Anotherstudent explained that the focus on words in the curriculum impacts her ability to participate inmathematics learning.Liz: I can’t understand completely for the questions. Oh, I can guess.Interviewer: What’s hard to understand?Liz: Words.Interviewer: Single words, when they’re put together [in sentences] or both?Liz: Both.48ESL Level 4 Response to the Use of English in the Mathematics ClassroomStudents in ESL level 4 occupy a unique position in the world of mathematics educationfor ESL learners. While they have acquired conversational English, permitting them access tothe discussion and instruction in the mathematics classroom, they still do not have all of thetechnical language necessary for true fluency. These students are also able to reflect on a recenttime when they were participants on the periphery in the mathematics class because oftheirlimited knowledge ofthe English language.When asked to respond regarding their opinion about the amount of English in themathematics classroom, two participants stated there is not a lot and they do not put muchthought into it. I was surprised by their comments, but both ofthese students later elaborated onthis point and mentioned that their mathematics teachers focus on the completion ofcomputation problems. This individual classroom focus could clarify why they are notconcerned about the amount of English in the mathematics classroom. This was different thanthe majority of the participants in this study registered in ESL level 4. While one participantstated that she finds the language in mathematics “pretty easy,” five ofthe respondentscommunicated that they still struggle with the language component of math, primarily due toword problems. Many also acknowledged that they struggle less with the vocabulary than theyhave in past years. “Like my first year in, in Canada was really hard ‘cause, ‘cause, like, someof the words I don’t know, but like, simple questions, like 20 times 10 then I can, like, withoutword questions, then I can answer. But now I can answer both ofthem except for explaining.‘Cause I know how to explain in Mandarin, but I have to translate it to English. That’s a little bithard.”In addition, some students shared coping strategies they have developed. One studentexplained that her strategy for learning new math concepts is to bring the worksheets home totranslate it into her first language and review the concepts with her mother so that she does not49miss any important information. Another student shared that his trick is to locate the digits in thewritten questions to ensure that they are all somehow placed into the number sentence that hederives from the word problem.Response of Students Not Registered in ESL to the Use of English in theMathematics ClassroomStudents in this study who are fluent in English recognized that mathematics instructionand problems are generally presented to students in a language-based format. Six of the sevenstudents explained that they are comfortable working within this format, while Jimmy statedthat he experiences some difficulty with these problems because he finds the wording hard tounderstand. “Some ofthe question they ask you are easy, urn, but they put really hard words tosort of confuse you.” Austin explained that the amount ofEnglish used in the mathematicsclassroom “doesn’t really matter to me ‘cause, like, I can understand it.” Tracy recognized thatESL students have difficulty with the language component of mathematics, “For the ESLstudents in our class, they can do the math but sometimes they have to ask, urn, fellowclassmates what it means. And they all have this electronic dictionary thing [to help them].”General Overview of All Three Problem FormatsStudents completed four computation problems, four word problems, and four pictureproblems on the mathematics task. The results ofthe mathematics task were very surprising. Ihad expected students with no ESL designation to perform equally well on all three forms ofmathematics, ESL level 4 students to perform slightly worse on word problems than the othertwo forms, and ESL level 2 students to struggle most dramatically with word problems. Not allof these assumptions were confirmed. I will provide basic information about the students’responses to computation problems and word problems so that these two mathematics formats,50with which students are familiar, can be compared to pictureproblems, a unique mathematicsformat. Because picture problems are the mathematics modification being investigated in thisstudy, I will elaborate on students’ responses to picture problemslater in detail as to ensure thatthe reader is able to understand the students’ thought processes and solution strategies. Pleasenote that Question #3 has three times the number of responses than the other problems becausestudents had to provide answers for three separate components to satisfy the requirements ofthispicture problem.ESL Level 2 Performance and Response on Mathematics Task by Problem TypeThe six students in ESL level 2 made errors on every problem except Question #7 andQuestion #12, and only one error was made on Question #1 and Question #11 (see Table 2).Table 2Summary of ESL Level 2 Performance on Mathematics Task by ProblemProblem TypePicture WordError Type #1 #2 #3 #4 #6 #7#5 #8 #9 #12NoError 5 4 15 3 4 2 6 3 3 3 5 6Calculation 0 0 2 0 1 0 0 2 0 0 1 0Solution Strategy 0 2 1 0 1 4 0 1 2 0 0 0NoAnswer 0 0 0 2 0 0 0 0 1 1 0 0Comprehension 1 0 0 1 0 0 0 0 0 1 0 0Unsolvablea0 0 0 0 0 0 0 0 0 1 0 0Note. The values represent the number of solutions coded with that error type. For all problems, except#3, n=6. For #3, n=18 because the problem asked for three separate values.aAprinting error occurred in the Mathematics Task Problem Booklet which made Problem #10unsolvable for some students. This outcome was not recorded as an error on the part of the student.Computation#10 #1151Out of all ofthe computation problems, studentshad the most difficulty on Question #9because two of the students felt thatthe question was “impossible” due to ineffective solutionstrategies. Another student felt that it would take too muchtime to solve it so he did not providean answer.Word problem Question #6 was difficult for two thirds of the participants.All four of thestudents with errors on this problem mentioned that they had difficulty with the language of thequestion, especially the last sentence “What number is the solution to the problem?” Shelleycommented, “I don’t really get this question so I couldn’t answer this question, like, for sure.”The students demonstrated attempts at placing all of the digits into a number sentence that usessubtraction and addition, though not necessarily in the order intended by the word problem.Out of all ofthe picture problems, ESL level 2 students only demonstrated considerabledifficulty on picture problem Question #4. During her interview, Shelley struggled to identifythe question being asked by the image. Consequently, she was not able to complete the questioneffectively. Two other students explained that they were unsure about the appropriate solutionstrategy to use and chose not to answer the problem altogether. Both of these students alsoexplained that if the question mark had been at the end of the problem they would have beenmore successful.ESL Level 4 Performance and Response on Mathematics Task by Problem TypeThe eight students registered in ESL level 4 presented many ofthe same difficulties andstrengths as the students in ESL level 2, though their strengths and weaknesses appear to bemore pronounced (see Table 3). Similar to ESL level 2 students, all ESL level 4 studentsaccurately answered one word problem (Question #6) and one computation problem (Question#10).52There were only four errors made on computationproblems. Two of the errors wereanswer omissions, one made because a student found Question#9 too difficult to answer, andthe other omission on Question #11 for no obviousreason. Question #12 has two errors, one dueto inaccurate subtraction and the other asolution strategy error because the student altered theorder of operations in the problem so that it no longerled to an accurate answer.Table 3Summary of ESL Level 4 Performance on MathematicsTask by ProblemProblem TypePicture WordError Type #1 #2 #3 #4 #6 #7#5 #8 #9#12NoError 5 4 14 7 2 8 6 4 7 47 6Calculation 0 1 3 0 0 0 1 3 0 0 01Solution Strategy 2 3 5 1 6 0 1 1 00 0 1NoAnswer 0 0 2 0 0 0 0 0 1 0 1 0Comprehension 1 0 0 0 0 0 0 0 0 0 0 0Unsolvablea0 0 0 0 0 0 0 0 0 4 0 0Note. The values represent the number of solutions coded with that error type. For allproblems, except#3, n=8. For #3, n=24 because the problem asked for three separate values.aAprinting error occurred in the Mathematics Task Problem Booklet which made Problem #10unsolvable for some students. This outcome was not recorded as an error on the part of the student.The students in ESL level 4 showed the most difficulty on word problem Question #5.Every error was the result of students subtracting the cost of supplies from the final profitinstead of adding to it. The question asked “How many customers did they have if their profitwas $84 after they paid $12 for their cleaning supplies?” In their interviews, many of thestudents who made this error explained that the fact that the people holding the carwash paidthis money meant that it needed to be subtracted. Students also explained that they choose tocarry out the subtraction of $12 before they determined the number of customers because of theComputation#10 #1153apparent direction given by the useof the word “after.” This indicated to them that thesubtraction of $12 needed to bedone before they could divide to find the number of customers.This particular error was made by ESLlevel 2 students in only a few instances. The presence ofthis error in responses provided byESL level 4 seems to demonstrate that these students stillexperience language difficulty in mathematicsword problems even though they are more fluentin English than ESL level 2 students.Some participants elaborated on their difficulties with thelanguage during their interview, thoughothers made no mention of it. Each of the students whoanswered Question #5 mistakenlywere sure that they could answer another question like this,and half ofthem were completely sureoftheir answer.Picture problems were more difficult for the participants in ESL level 4 than thoseinESL level 2. The most difficulty was experiencedon Question #2 and the least on Question #4.Most of the errors were made due to incorrectsolution strategies. Picture problems will bediscussed in more detail later in this chapter.Performance and Response on Mathematics Task by Problem Type of Students NotRegistered in ESLThe seven students not registered in ESL answered more computation problemscorrectly than word problems or picture problems (see Table 4). No errors were made oncomputation problems due to ineffective solution strategies. All seven participants were able toanswer Question #10 correctly and only one error was made on each ofthe other threecomputation problems, two of these due to answers omitted by the same student.The majority of the errors made on word problems were due to calculation errors withonly three errors because of ineffective solution strategies.Students not registered in ESL made twice as many errors on picture problems due toineffective solution strategies than all other error types combined. Every student made a solution54strategy error on Question #2. Six calculation errors were made on Question #3 and no othercalculation errors were made on picture problems. Again, detailed responses to picture problemswill be discussed in more detail later in this chapter.Table 4Summary of Performance of Students Not Registered in ESL on Mathematics Task by ProblemProblem TypeWord Computation#6 #7 #9 #10 #11 #12PictureError Type #1 #2 #3 #4 #5 #8NoError 5 0 11 4 4 6 5 4 6 7 6 6Calculation 0 0 6 0 1 1 2 2 0 0 0 1Solution Strategy 0 7 3 3 2 0 0 1 0 0 0 0No Answer 0 0 1 0 0 0 0 0 1 0 1 0Comprehension 2 0 0 0 0 0 0 0 0 0 0 0Unsolvablea0 0 0 0 0 0 0 0 0 0 0 0Note. The values represent the number of solutions coded with that error type. For all problems, except#3, n=7. For #3, n=21 because the problem asked for three separate values.aAprinting error occurred in the Mathematics Task Problem Booklet which made Problem #10unsolvable for some students. This outcome was not recorded as an error on the part of the student.The high number of errors made on picture problems was not due to any obviousweakness in mathematics on the part ofthe non-ESL students. They demonstrated that theywere more than adept at computational mathematics and were successful at word problems.These participants were used to being successful within the current format of the mathematicsclassroom and might have been unable or unwilling to deviate from the format that they werecomfortable with: “We don’t really use pictures, we just use words and numbers and I’m notreally used to pictures.” However, it can be argued that if students possess mathematicalunderstanding, they should be able to apply their knowledge of mathematics to a wide variety ofsituations, including those problems presented without English language accompaniment.55Percentage of Errors Made by Each Participant Group by Problem TypeAll participant groups made errors on each problem type: picture problems, wordproblems, and computation problems (see Table 5). Although students had no previousexperience with picture problems, unlike computation and word problems, it seems “unfair” todirectly compare their performance on picture problems to computation- and word-basedproblems. However, trends between the participant groups were present. The percentage oferrors for each participant group and problem type is calculated by determining the number oferrors due to solution omission, inaccurate computation, ineffective solution strategies orerroneous interpretation of the problem itself, and dividing the total number of errors by thenumber ofeach type ofproblem attempted by students (see Equation 1).Error calculation for each problem type = Total number of errors (1)Total number of problems attemptedThe percentage of errors made by participants registered in ESL level 2 on pictureproblems and computation problems was the same, whereas the percentage of errors on wordproblems was slightly higher. The latter findings about word problems were expected as thesestudents are not fluent in English and could be expected to struggle with language-basedproblems. ESL level 2 students made the fewest number of errors on picture problems whencompared with the other two participant groups and made more errors on computation than theother two groups combined. It is possible that the ESL level 2 students had a mathematicsweakness in the area of computation so their success rate in that area was below the other twogroups. However, ifthis computation weakness did exist, it did not seem to have a large impacton ESL level 2 students’ ability to complete word problems because their results were the sameas ESL level 4 students and only slightly weaker than students not registered in ESL. The56difficulty demonstrated by ESL level 2 students on computationdid not affect their ability tocomplete picture problems when compared to the other twoparticipant groups because theirsuccess rate was much higher than either group, especially thosestudents not registered in ESL.Table 5Percentage of Errors Made by Each Participant Group by Problem TypeErrors by Problem Type (%)ParticipantGroupESL Level 2ESL Level 4Computation Word Picture25.0 37.5 25.014.3 37.5 41.6No ESL10.7 35.7 64.3Note. Errors types include: calculation, solution strategy, answer omission,and comprehension.The participants in ESL level 4 made roughly the same number of errors on wordproblems and picture problems, though the percentage of errors on picture problems wasslightly higher. They made considerably fewer errors on computation problems than eitherpicture or word problems. This finding suggests that ESL level 4 students’ difficulty on pictureproblems and word problems may not be linked to poor math skills but perhaps due to difficultywith other factors such as interpretation, comprehension, or solution strategy. When comparedwith participants who are not registered in ESL, the number of errors made on computationproblems was roughly similar. When compared to the performance ofESL level 2 students, ESLlevel 4 students performed equally well on word problems, and better on computation problems.However, they performed worse on picture problems than students in ESL level 2.Out of all three participant groups, students fully fluent in English performed the best on57computation problems. Their performanceon word problems was only slightly better than ESLlevel 2 and ESL level 4 students but their performanceon picture problems was drasticallybelow the performance of students in eitherESL level. Students not registered in ESL madeerrors in over half of the picture problems which wereapproximately double the number oferrors made by ESL level 2 students on these samepicture problems.Table 6Percentage of Errors Made by Each Participant Group by Problem Type AfterCalculation Errors are RemovedErrors by Problem Type without Calculation Errors (%)ParticipantGroupESL Level 2ESL Level 4No ESLComputation Word Picture20.8 25.0 19.410.7 25.0 31.37.1 14.3 50.0Note. Errors types include: solution strategy, answer omission, and comprehension.In order to try and understand the number of errors made on mathematics problems dueto misunderstanding the problem or misguided solutionstrategies, Table 6 shows percentage oferrors made with calculation errors removed. The percentage of errors made due to inaccuratesolution strategy or poor comprehension of the question is calculated by dividing the number ofthese errors by the total number of each problem attempted by students (see Equation 2).Omissions of a final answer are also included as this error category and not considered to be acalculation error.Error calculation for problem type = (Total number of errors — Number of calculation errors) (2)Total number of problems attempted58The same sorts of trends were present when calculationerrors were removed as whenthey were included. Students in ESL level 2 nowperformed slightly better on picture problemsthan computation problems and had the most difficultywith word problems. Students in ESLlevel 4 and those not registered in ESL still demonstratedthe same trends as when calculationerrors are included. Both ofthese groups performedbetter on computation problems than wordproblems and had the most difficulty on picture problems. Students registered inESL level 2and ESL level 4 still made the same number of errors on wordproblems as each other and feweron computation problems than before calculation errors were removed. Studentsin ESL level 2made double the number of errors on computation problems as ESL level 4 students andthreetimes as many computation errors as students not registered in ESL. Thepercentage of errorsmade by students not registered in ESL on word problems dropped noticeably when calculationerrors were removed. This suggests that their difficulty on word problems was superficial sincea large part of the errors they made were simplecalculation errors which could be easilyremedied and their performance on computation problems, using their basic mathskills, wasstrong.Students in ESL level 2 made roughly the same number of errors on each problem type(within 6%). Students in ESL level 4 made over twice as many errors on word problems thancomputation problems and three times as many errors on picture problems as computationproblems. Students who are fully fluent in English showed the most dramatic difference inability on each problem type. They made twice as many errors on word problems thancomputation problems and seven times the errors on picture problems than computationproblems. This suggests that their ability to work within multiple mathematics formats was notas well developed as those students registered in ESL level 4 and much less consistent thanthose students registered in ESL level 2.59Self-Efficacy by Problem TypeSelf-efficacy, a student’s judgment of his or her successon future tasks similar to onesafready completed, was investigatedfor each problem format on the mathematics task by askingstudents how they would perform on the same sortoftask in the future. Students were asked fortheir judgment of self-efficacy and to gauge theirenjoyment of computation problems, wordproblems, and image-based problems.ESL Level 2 Students’ Self-Efficacy by ProblemTypeWhen considering how well they would do on a pageof number problems, wordproblems, and picture problems, students in ESL level 2 unanimously felt that theywould beable to answer the highest percentage of computation problemscorrectly. The reason Aliciaproposed for this expectation is “cause it’s easy. Every questionis like the same way to do it.”Another reason given by multiple participants was the importanceof mathematics facts inChina. One student explained, “Urn, in China, math was morefocused about adding andsubtracting, dividing and timesing [sic].” Shelley alsovolunteered that her ability to succeed onnumber problems has decreased since her immersion in theWestern classroom. “Because, urn,like, in China the questions were more difficult, but when I came here for like more than oneyear I didn’t do much, like, old questions like that. I probably wouldn’t do as well as before.”All ofthe ESL level 2 students voiced concerns that their success on word problemswould be significantly lower than number problems. Shelley stated, “Um, some English mathquestions, like, I don’t get it, so yeah. Some sentence is hard to understand.” When asked for anexample, she elaborated that, “some, like those questions, urn, some has a really difficult wordsso I can’t understand.” Another student continued this idea by explaining his struggle with wordproblems. “Some words I don’t understand or I misunderstand some so it might get me wrong.Sometime the word combinations I don’t understand.” The other ESL level 2 students echoed60this concern. They indicated thatthey also experience difficulty determiningmissinginformation, creating accurate translations,and having the necessary mathematicsvocabularyavailable to them.When asked about their enjoymentofword problems, five ofthe studentsexplained thatthey do not enjoy working on wordproblems for a variety of reasons. Two students statedthattheir lack of enjoyment is simply because they donot enjoy mathematics. Three othersimplicated their incomplete understanding ofthe English language as their reason for notenjoying word problems. One student,Emily, stated that she enjoys word problems when theytell a story but does not enjoy themwhen they are used simply to elicit a series of mathematicaloperations.After the ESL level 2 students gainedexperience completing the four picture problemson the mathematics task, when asked toconsider how well they would do on future pictureproblems, their responses were mixed. Fourstudents stated that in some instances pictureproblems confused them. Yet, five ofthe sevenstudents thought that they would perform betteron picture problems than word problems. They attributedthis primarily to not having totranslate any words in order to be successful on theproblem. The two students who believedthat they would likely perform worse on picture problemsthan word problems were the onlytwo students who explained that theyare used to problems in word format and are more likely tosucceed in a form of mathematics more familiar tothem. One student stated, “I’m too used todoing numbers and words, so I would do, like,pretty bad on [picture problems].. . . In China, inevery country, you learn numbers andthen you come to Canada, you learn words but right nowyou’re learning a new kind of math that, like, new. And some symbols you don’t understand oryou don’t know how to think about it so it really confuses you.” The other student mentionedthat while she understood two ofthe picture problems, the two that she did not understand werehard for her because “the picture look(s) weird and in school I always do word problems so....”61These two students explainedthat they are more comfortable withproblems similar to theirprevious experiences, than newproblem formats. This desire forfamiliarity was seenrepeatedly, and to a greater extentin participants in ESL level 4, and even more so withparticipants not registered in ESL.All ofthe students in ESLlevel 2 felt that if they were taught how to answer pictureproblems they would be highly successfulon them relative to their level of successon wordproblems. “If I learned the pictureproblems compared to the word problems, I don’tthink Iwould misunderstand anything. Becauseevery country uses pictures and pictures areeasier toremember than words.” Another studentwent on to say, “I think I can do good because picturestell me, like, it doesn’t tell me thewords, but pictures tell me how many, like, what islike priceor we have to add something and thepicture tell me you have to get minus or times sothat youhave to just minus something.”Even though all of the ESLlevel 2 students felt they would be most successful atcomputation problems, they all rankedtheir enjoyment of picture problems on par or higher thanboth number and word-based problems. Threeparticipants explained that picture problems werefun and enjoyable due to the visuals and different colours, butalso warned that their enjoymentwas jeopardized when the pictureswere confusing.ESL Level 4 Students’ Self-Efficacy byProblem TypeSimilar to the responses provided by ESLlevel 2 students, all of the ESL level 4participants shared the same preference for computation-basedmathematics problems over wordor picture problems. All ofthe eight ESL level 4 students believedthat they would be highlysuccessful at number problems. Five students mentioned that theirsuccess would be amplifiedby the absence of words in computation problems.This demonstrates that even though ESLlevel 4 students are thought to be only one year away from English fluency, they still experience62nervousness when they encounter English in mathematics.When Candy was asked how wellshe would do on number problems she responded, “good ‘causethey’re not in English so Iwon’t have to worry if I understand it or not,”while Brandon explained that number problemsraise no concerns for him because there are “just numbersand no words to understand.[Djustlook at the numbers and just write it out.” All eightof the students in ESL level 4 shared thatwhen they first came to Canada they would have doneas well, or better than now, oncomputation problems due to the focus on computationproblems and recall of math facts inChina.Five of the eight students replied positivelythat they enjoy number problems. As onestudent elaborated, “the numbers are really clear to you and then it’s simple to do them, urn,because you can easily see what the numbersare and then you just need to know what numbersare there to add, subtract, divide, and multiply.” Most students agreed that their enjoyment ofthis problem format comes from a clear understanding of how to answerthe problem and simplyneeding to recall mathematical principles.ESL level 4 students demonstrated less apprehension than ESL level 2 students abouttheir perceived level of success on word problems. All ESL level 4 students stated that theirsuccess would be fine, okay, or good on word problems. Michelle explained that she would do“fine” on word problems because “I learned pretty much English, so I could understand simplequestions.” One student explained that she believes “word problems are good. They help theESL [students] learn words and then they, if you don’t read them clearly, they can make theanswer wrong for you, so it’s tricky. So you have to be careful of, urn, be careful of the numbersthere. And then sometimes the word problems, they hide the numbers by making the numbers inwords.” Two students mentioned that they have actually practiced word problems outside ofschool hours to improve their performance on this problem format.63During their interviews, many ESL level 4 students spontaneously described momentswhich illustrated the struggles they have had with word problems in the past. One student sharedthat when she was first learning English she “asked a lot of questions like what does this wordmean, what does this word mean? And sometimes, even though they explain it to me, I stillwon’t understand.” This same frustration was also touched upon by another participant as well.Brandon: I wouldn’t get like, if I got one I would be really lucky because, like,when I first came I didn’t know exactly English or what. Like, I didn’t know anyEnglish actually, and then I couldn’t understand anything except for ‘okay.’ Justone word. I don’t know anything when I first came to Canada. When I was givenword problems I didn’t know what to do so then I told another person who knew,like, Chinese to tell the teacher I didn’t know what to do and then the teachertook it away and gave me number problems instead.Interviewer: How did that make you feel?Brandon: Strange, because I was looking at this page with words on themand Ididn’t know what they meant, so yeah.Interviewer: And how did you feel about that [page ofwords] being math?Brandon: Pretty weird. It’s such a jumble of letters together.Compared to students registered in ESL level 2, students registeredin ESL level 4demonstrated increased self-efficacy on word-based problems andincreased enjoyment of suchproblems as well. Three of the eight students stated that they find wordproblems enjoyablebecause oftheir increased familiarity with this problem format andthe opportunity to “figureout new words for the definitions to improve. . . math skills” and grammarskills. Five oftheeight participants believed that they would“sort of’ enjoy word problems and explained thattheir enjoyment is hampered by difficultyunderstanding the vocabulary or sentence structureused in the problem.When asked how well they would performon picture problems, five of the eightparticipants in ESL level 4 believed that theywould not perform as well on picture-basedproblems as word problems, while the otherthree participants felt that they wouldperformbetter on picture-based problems. Somestudents’ hesitation regarding picture problemsstemmed from a desire to experience minimalconfusion in mathematics. As one studentstated,64“I would do average, because, urn, the picture problems, they’re, like, sometimes moreconfusing than the word problems because picture problems you have to, like, know whatthey’re trying to tell you and word problems, they just tell you right away.” Ricky was quick toquestion his understanding of picture problems and demonstrated discomfort when it waspossible that his answers were wrong.Despite ESL level 4 students feeling some confusion on picture problems, all of them,except Brandon, actually explained that they enjoyed picture problems more than wordproblems. The majority ofthe students expressed that they would enjoy completing pictureproblems because of the visual and mental stimulation, ease of comprehension, and assistance itgives to ESL learners. Nicole stated that when she first caine to Canada she would haveperformed “like the same [on picture problems] as number problems. Because, like, pictures,they don’t have any words and English to solve. I think the children in [Canada] will do betterthan me because the teachers in here taught in a more imaginative and creative way, but inChina, it’s just, like, textbooks.”Many students in ESL level 4 also explained that the multiplicity of question andanswerpossibilities, coupled with the onus on the student to determine the correct meaning, forcedthem to think logically and analytically, which they enjoyed. Matthewexplained that, “It’s morefun because picture problems actually give you those pictures andthen you [are] trying to findout what they mean.” Debby repeated many times that it is important for herto be activelyengaged in mathematics, not simply repeating tasks over and over, andenjoyed pictureproblems because they challenged her and they forced herto think during each problem. “Wellfirst of all you get to look at this pictures right. It’s kind of fundoing that... It won’t reallyboring me that much.”Some students did not share this sentiment however.Brandon’s reason for decreasedenjoyment of picture problems was directly related to hislack of exposure to them. He65demonstrated a fear ofbeing “incorrect” and a preference for the inclusion of language. Brandonhad no knowledge of whether his answers on the picture problems on the mathematics task were“correct” or “incorrect,” yet he afready insisted that he is unable to do them. He stated that hisenjoyment of picture problems was low because “I don’t usually do picture problems thatmuch.” However, he believed that if he were to practice, he would “do good at them.” Brandonactually grossly underestimated his performance on picture problems because he was able tointerpret all ofthe picture problems accurately and answer them effectively.Self-Efficacy by Problem Type of Students Not Registered in ESLStudents with no ESL designation demonstrated quite a different opinionabout the threemathematics problem formats than ESL level 2 or ESL level 4 students. All seven participantsfelt that they would be successful completing computation problemsbecause they knew themathematics rules and “you don’t have to read anything and it takes less time.”Most ofthestudents felt that they would get at least 90% ofthe problems “correct”.Chris commented that“I’ll get most of them right. A few wrong. Sometimes I forget somethings or I forget to carrythe one or something,” indicating that the only difficulties heexperienced are simple calculationerrors, and as a consequence of carelessness, not comprehension.When asked to explain their enjoyment of computation problemsthere was a myriad ofresponses. Two students enjoy computation problems,one commented that his enjoyment islinked to the fact that “they’re easyto do. Just adding.. . you don’t have to really thinktoomuch.” Tracy stated that her enjoyment isdependent on the variety of computationproblemsgiven because simple, repetitive questions are“boring.” One student did not know howmuch hewould enjoy computation problems and anothersaid that he would experience moderateenjoyment. Mark asserted that he would not enjoy numberproblems because they are math and66he does not enjoy math. This was consistentwith his responses for enjoyment ofword problemsand picture problems as well.The students fully fluent in English relayed thattheir ability to be successful on a pageof word problems is either ‘okay’ (four responses), ‘good,’ ‘prettygood,’ or ‘very good’. Themost striking difference between this group of participants andthose enrolled in ESL was thatonly one student who is fully fluent in English seems to haveconcerns about the vocabulary andsentence structure posing a difficulty for him. He would be “sad” becausehe does not “like[word problems] at all. Um, find, um, I find them hard to understand.” Two other studentsviewed the language used in word problems as a trick that they are usually capable of sussingout and not falling for.Though all ofthe students with no ESL designation ranked their success on wordproblems from moderate to high, there was no consensus among the group about theirenjoyment of such problems. Two students responded that they are happy receiving problems ofthis type, two felt that they would have moderate enjoyment with the potential for boredom, andthe other three believed that they would not enjoy completing such problems. Mark commentedthat his lack of enjoyment is directly related to the fact that word problems are harder thannumber problems which make them less enjoyable.Students with no ESL designation all had different opinions regarding their success onpicture problems. Two students believed that they would do “bad,” three felt they would do“okay” and two others responded with more optimistic judgments of “pretty well” and “prettygood.” Tracy stated in her interview, “I can understand stuff from pictures and it uses the samemath skills which I afready know.” The students who felt that they would have low or moderatesuccess blamed this on finding picture problems confusing. “Like, sometimes they just putpictures there and then question marks and then, yeah, I don’t get it.”67Brandon:This one,I thoughtit was justurn, likethis one,I didn’t evenknow howyoudo it butthen I usedmy fingerslike thisand went“One, two.”I times itby two. Ifigureout [thejellyfish length]first and Itimes it bytwo andthen that’s it.Interviewer:But youdidn’t writeit downbecause...?Brandon:Cause like Ithought, Iwouldn’t getit right sothen put somerandom thingthere andthen yeah.Interviewer:So you justput nothing?Brandon:Yeah, becauseI would havegotten itwrong anyway.Brandonstated thathis mathematicsabilitieson this pictureproblem were“really badbecauseit’s all thosepicturethings andthen I did notlike picturethings at all sothen I just didn’tunderstandthis one.”He seemedto havestrong negativefeelings towardspicture problemsandapplyingmathematicsto a differentproblem format,somethingalso demonstratedby some ofthe otherstudents, especiallythose not registeredin ESL.All ofthe studentswho thoughtthat they woulddo poorly on apage of pictureproblemsalso statedthat they wouldnot enjoycompletingthose problems.Conversely,all ofthe studentswho felt thatthey wouldperform atleast satisfactorilyon picture problemswould be “happy”todo pictureproblems, foundthem “fun,”and consideredtheir enjoymentof picture problemstobe “betterthan [that of]word problems.”For picture problems,a new problemformat, enjoymentfor this groupof students wasdirectly relatedto their self-perceivedlevel of successon future pictureproblems, or self-efficacy. This directlink was not obviousin participantsenrolled in ESLlevel 2 or level4. Withthe exceptionof a few studentswho were adamantthat they wereunable to understandpictureproblemsand consequentlywould not enjoythem, the studentsin ESL level 2or level 4 wereless likely to basetheir enjoymentof picture problemsdirectly on theirself-efficacy.Austin, a participantwho has not beenregistered inESL since grade1, demonstrated aresistance to applyingmathematics to anew format suchas picture problems.He was used tointerpreting andcompleting mathematicsproblems in a formatthat he is able toeasilyunderstand, suchas numbersor English language-basedproblems. Becauseofthis he seemed68conditioned to only functioning in mathematicswhere his English language fluency allowshimto succeed. Whenasked about how well he would do on pictureproblems he replied, “Bad!”Austin: I can’t anticipate the pictures.If they’re included with words then I willdo very well butifthey’re just pictures, like, it would bevery confusing.Interviewer: Why would it be confusing?Austin: There’s no, like, question andthen you, like, have to figure out thequestion on our own.During Chris’ interview, he communicatedhis frustration that picture problems had insufficientwritten information to determine theactual math questions from the picture alone. Even after heclaimed that he was mostly certain about hisanswers and felt that picture problems were sort ofeasy, he reverted, stating that a subsequentquestion was “another picture so I don’t know whatto do. I wouldn’t know what the correctanswer would be.”Written and Oral Response to Individual Picture ProblemsIn order to understand how the unique mathematicsformat of picture problems impactstudents in their mathematics problem solving, eachof the participant group’s responses to thefour picture problems will be discussedin some detail. Those students who provided the correctsolution strategy will be discussed more briefly. Where students erred,explanations oftheirwork and thought processes will be explored in greater detail.ESL Level 2 Students’ Responses to Question #1All of the participants in ESL level 2, except one, answeredQuestion #1 correctly. Whenasked to describe what was being asked in Question #1 withthe sporting goods, shopping cart,and cash register, Alicia replied that it was not immediately clear toher what the image wasasking.69Alicia: [The question is:] What did I buy to put it in the cart andget the amount of money that’s $82? But in the first time, Ithought maybe they’re asking you how much dollars that’s moredollars or less dollars than $82.Interviewer: How did you decide to do this? [Points to thewritten work on the mathematics task.]Alicia: Because, urn, because of the shopping cart. Ifthey ask you[for the change] it might be, like, a different picture I think.Shelley’s attempt to answer Question #1 demonstrated that her confusion about thequestion led directly to her inability to answer the problem intended by the image (see Figure 1).She stated that “I thought [the cash register] was showing [that all of the itemsadded up] was $82, so, urn, or this couldtalk about if they paid more than this7 _it6much, like they exchanged $82. This,0[öjconfuse me.” Shelley’s confusion about______what the problem desired stopped herFigure 1. Shelley’s written work for Question #1.from determining a definitive solutionstrategy. It was unclear from her work and her interview whether she intended to have $82 asthe change or as the amount difference (i.e. the difference between the cost of all ofthe itemsand the total displayed on the cash register). Rather than subtracting $82 from the five item totalof $140, as is suggested by the “minus” symbol in her number sentence, Shelley added the twonumbers resulting in a total of $222 which she stated she paid, placing herself into the positionof the customer. This unclear solution strategy mimicked her confhsion about what the questionappeared to be asking and her resulting work did not satisfy the question the researcher intendedas indicated by the placement ofthe question mark and arrows on the image.ESL Level 2 Students’ Responses to Question #2Students in all participant groups demonstrated metacognition, reflection on their own70thought processes, while completingQuestion #2. During the interviews, manyofthe studentsacross all participant groups explainedthat they debated two components ofQuestion #2: thefirst being whether to distributethe bananas individually or by bunches, and the secondbeingwhether the bananas should be distributed tomonkeys equally or distributed dependenton theirsize.Shelley debated whether to divideindividual bananas or bunches of bananas between themonkeys and she decided to answerwhat she considered the simpler of the two problemsbecause “Urn, I thought if you go tobunches, you can share too.” Four ofthe six studentsESLLevel 2 students chose to divide the bananasequally, while the others divided the bananasaccording to their assumptions aboutthe size difference between the monkeys.Figure 2. John’s written work for Question#2.ck2C411- nosoud&k.‘+mokese?+L++3-fL3L1.3L&2A:7.b’7During the interview, John explained that he believedQuestion #2 was the most difficultpicture problem primarily because ofthe confusion he experiencedwhile determining what thequestion was asking (see Figure 2). His first attempt at interpreting the problem led him tobelieve that it was asking “how many bananas are in the bucket?” He stated that he realized thatthe bananas were to be distributed to the monkeys and he chose to assign bananas to monkeysbased on their relative sizes. After he distributed the bananas to themonkeys, he found that he71had bananas left over. Inorder to deal with the remainingbananas, John changed hisinterpretation of the problem to“How many are left [in] the bucket?” Thisfit more accuratelywith his selected solution strategyand allowed him to accountfor them, without creating asolution containing remainders.When questioned about his choice to assignbananas based onthe size difference, John statedthat “if they were supposed to [get the] samenumber themonkeys would be the same size.”He explained that the visual cues givenabout size trump theinstructions indicated by themathematical symbols present.Alicia indicated that she was not aware thatthe equal symbols placed between thequestion marks above each monkeywere mathematical symbols. She explainedthat she simplyassumed that they were placedthere “as decoration.” She asserted thatthe equal symbols shouldbe placed in between the monkeysinstead ifthey were meant to indicatethat equal distributionshould be used.ESL Level 2 Students’ Responses toQuestion #3All of the students in ESL level 2 said theyunderstood the question being asked by theimages in Question #3. In herinterview, Emily had difficulty explaining what thequestion wasasking for with words, but wasable to determine a solution that satisfies all three components.She achieved this by using herfingers to indicate the values for which she was solving. Four ofthe six students solved all three componentsof this problem accurately. Each ofthemdetermined the length of the whaleand the jellyfish by multiplication and addition ofthesegments and subtracting the length ofthejellyfish from the whale to find the difference inlength. Ivlikey also used this solution strategybut made a multiplication error when determiningthe length of the whale resulting in an inaccurateanswer for the difference in length.John followed an effective solution strategy for thelength of the whale and jellyfish andwas able to accurately determine both ofthese values.72Figure 3. John’s written work for Question#3.AJekf41tc OrnA:v1teic. 35A:-ed1IeveflCe54°62)0--?2)390/ 1However, he made a solutionstrategy error when trying to determinethe difference in lengthbetween the whale and thejellyfish. Using a series of mathematicaloperations, in a mannersimilar to a guess and check strategy, Johnconcluded that the size difference is1.350 metres(see Figure 3). Johnhadpreviously determined thathalfofthewhale’s length isQ.f 21.95 metres and his0+ 9,.Lmeasurement for the lengthdifference was less than halfthe length of the whale.0:225However, the pictureshowsthe space left by thedifference oftheir lengths ismore than halfthe length of S/ o2the whale.o225b/5’OESL Level 2 Students’ Responses to Question #4Three ofthe six students completed Question #4with a correct solution strategy. All ofthese students understood that they mustview the question with people moving in reverse.Therefore, as passengers get onto thebus, they subtracted the equivalent number, and whenpassengers departed from the bus, they addedthe same amount, resulting in accurate answers.Of the other three students registered in ESL level2, one student used an erroneoussolution strategy and two students did not provide answers tothe problem. Shelley’s answer hada mistaken solution strategy. Viewing the movementof passengers on and off the bus enabledShelley to determine that there had to be at least 11passengers on the bus at some point in order73for 11 passengers to disembarkin the third to last frame. Sheselected 20 people as the numberof people on the first bus,demonstrating that she was awarethat this is a previously unknownvariable. After workingthrough the passenger movement,she stated that seven people would beleft on the bus. Even thoughher answer does not satisfy the problem,it took some thought forShelley, which, she stated, madeher feel good about her mathematics ability.Both Mikey and Emily chosenot to provide an answer. In the interview,Mikeyattributed this to a lack of understandingof the problem itself. He correctly identifiedthat “Ithink maybe it was asking howmany people on [the bus with the questionmark].” Likewise, hedescribed a viable strategy (guessand check), but was confi.ised because he believed thatsuch astrategy could lead to many possible outcomesdepending on the number selected for thevariable and he knew that he musthave zero passengers remaining.“Because if it asking me how manypeople are on the bus. I don’t know howmany people on the bus in the first picture. Andthere like six people gets off, Istill don’t know how many people on thebus. So... maybe if there was like 10people on the bus there will be only likefour left but if it’s a different number,there will be a different answer, so it’s,like, pretty confi.ising.”While Emily knew that she was to determine howmany people begin as passengers onthe bus, she was unsure of how to deal with11 people leaving the bus in the third to last frame.She originally followed the problem throughwith the idea that the first six people to come off ofthe bus were the original passengers. When 11 peoplehad to leave the bus, this number waslarger that the number of people she thought wason the bus. This created a situation involving anegative number of people, whichshe did not know how to resolve.All three students who made errors on Question#4 explained that if the direction of thequestion was reversed, so that the unknown numberof passengers was in the final frame afterthe passenger movement, the question would be mucheasier. Similar to many other studentswith difficulty on this problem, Emily found the directionof the question challenging. “Urn,because if you, if you like, draw it backwards, I would understand,like, it comes down and then74goes up, like, the last one is [the bus with the questionmark] and the first one is [the emptybus].” Before she reviewed her answer, sheoriginally assumed that picture problems would beeasy; however her difficulty on this problem ledEmily to feel that “picture math is not as easyas I think.”ESL Level 4 Students’ Responses to Question #1Sam determined that theFigure 4. Sam’s written work for Question cost of all ofthe items was$140 (see Figure 4). Rather thanusing guess and check toGdetermine the three itemspurchased, he found the cost oftheitems not bought and used guessand check to eliminate those items\(0that cost that amount, a faster c\t o4 c,ciktk\1process than the solution method4,&Lc \:\\.used by most students. Once hedetermined the items not bought,O)\-k \he was able to deduce which itemswere purchased.Similar to Shelley in ESL level 2, Nicole’s interpretation ofthe problem in Question #1did not agree with the image and demonstrated difficulty comprehending the intentions oftheimage. However, Nicole explained that she believed that the customer wanted to buy all fiveitems but only had $82 so she determined how much more money he or she needed in order topurchase all ofthe items. Such an interpretation does not seem to reflect the usual social75meaning of an amount shown on acash register in a purchasing situation. For thisinterpretationto be viable, the amountof $82 would have to be shown as belonging to apurchaser.Candy and Ricky both employedinaccurate solution strategies to answer Question #1.Candy interpreted the total onthe cash register as the change given, after all ofthe itemswerepurchased. While this solution strategy included thenumbers shown on the image, it did notactually account for the placement ofthe questionmark in the image. The question mark in thepicture problems was placed on theFigure 5. Ricky’s written work for Question cart to indicate that the solutiondesired from students should take the place93of the question mark, inside the shoppingcart. Candy’s answer did not provide asolution at this point in the problem so itdid not agree with the image represented.Picky determined which items must beremoved from the cart but only provided a—‘partial solution because he did notcommunicate with pictures, numbers, orwords which items were left in the cart which would satisfy the picture problem (seeFigure 5).ESL Level 4 Students’ Responses to Question #2Similar to the students in ESL level 2, students in ESL level 4 clearly reported duringtheir interviews that they experienced metacognition when answering Question#2 due to thepresence of bunches, and the size variance ofthe monkeys. They shared moments ofquestioning, rethinking, and wondering aloud. This was apparent in Matthew’s reasoning for hissolution to the problem, which did not contain an error. His interview demonstrated that he was76quite perceptive aboutthe visuals in the image such as thedifferent numbers of bananas in thebunches and the equalsymbols.Matthew: Um, I didn’t really getthis question. But now I think that it’s askingme that if you put all thesebananas into this bucket,how much bananas willthere be? And if you sharethem among 4 monkeys equally, howmuch will eachget?Interviewer: Why do thebananas go into bucket?Matthew: Because there’san arrow pointing to the bucket and there’s aquestionmark on it so I thought it meant,like, how much bananas are the total.Interviewer: Why equally?Matthew: Because each monkeythere is a question mark above it and thereis aequal size to every questionmark. That means all the numbers are the, must bethe same, and that basicallymeans like equally sharing between themonkeys.Interviewer: How did youdecide whether to do bunches or bananas?Matthew: Because I see some buncheshave four bananas instead. Some havethree so if you share with bunches,I don’t think that will be fair.Although Matthew explained inhis interview that he was dismayed by the presenceofremainders while completingthe problem, it did not stop him frommaking decisions about theproblem that were in agreement with theimages presented.Three students made solution strategyerrors, two ofwhom divided the bananasaccording to the size of the monkey. LikeMichelle, Ricky became very uncomfortable when herealized that some bunches had three bananasand others had four. This made him question hisdecision to divide the bananas by bunchesamong the monkeys. Although he blamed hisconfusion about the question on this visual,he admitted that he noticed the different numbers ofbananas during the interview only. Debby wasaware ofthe equal symbol but chose not to placemuch importance on it. “Cause like at firstI didn’t get the monkey part and then I was thinking,like, what does the equal sign mean and then why isthe monkey like, has, like, different sizesand then cause, like, I used guessing.” Inorder to account for the presence ofthe equal sign inher answer, she placed equal symbols betweenthe different amounts of bananas that themonkeys received. Candy allocated bananas based onan estimation of monkey size. Whenasked during the interview for more information abouther logic, she insisted that she did not77understand what the questionwanted her to do, despitehaving earlier stated that she had tomove the bananas to thebucket and then distribute them tothe monkeys.Though Sam determinedan effective solution strategy andused equal division, he madea calculation error andhad difficulty transferring the numberof leftover bananas to a decimal,resulting in an incorrect solution.ESL Level 4 Students’ Responsesto Question #3All eight ESL level 4 students correctlyidentified the objectives of Question#3, andwere able to formulatean effective solution strategy for solving atleast one of the whale andjellyfish lengths.Three students used inaccurate solutionstrategies to find the length ofthe whale orjellyfish. When Debby explainedher solution strategy, she stopped and gasped becauseshenoticed that her answer to thelength of the jellyfish was inaccurate. When completingthemathematics task, she did not noticethat the 0.4 meter measurement wasthe length oftwosegments out of five on the jellyfishand thought that it represented theentire length. Afterrealizing this error during the interview,she explained the correct solution strategy precisely.“I’ll put 0.4 times 0.2. 1 mean plus. I mean0.4 plus 0.4 plus 0.2. That would giveme the length of the jellyfish. And then I’lluse, and then I’ll use, the length ofthe shark subtract by the length ofthe jellyfishand I get a difference.”Ricky explained his understanding that since twosegments together are 0.6m in length, onesegment would beFigure 6. Riclcyswrittenhalf ofthis measure (see Figure 6). However,when he usedthese numbers to determine the total lengthof the whale, hedid the calculation in his head and ended upwith anincredibly inflated measurement of33 metres.Q‘78Six students struggledwith solving the difference in lengthbetween the whale andjellyfish. Three students used atypical subtraction solution strategywith calculation errors, twostudents devised alternatesolution strategies, and one student optednot to answer.Candy and Ricky both used estimationto solve for the difference in length.Candy usedher fingers to estimate how manytimes the length of the jellyfish fitsinside the empty space.She determined that 2.4 jellyfishwould fit. Ricky was unable toexplain the strategy he used toestimate the difference.He mentioned that he needed to find out“how long is the jellyfish and[the space] is” then quickly stated,“Actually, I don’t understand the question,so I just guesslike ‘cause there’s no like numberhelping us figure this out so I just go[3 metres].” In thisresponse, he stated that he didnot understand the question.However, by recording ameasurement for the length difference,he demonstrated the he was fullyaware ofwhat thequestion was asking, but wasunable to formulate the corresponding solutionstrategy.Brandon did not record an answerfor either the jellyfish length or the comparativedifference in length. He insistedthat he did determine the length of the jellyfishand simply didnot record it. He explained thathe chose not to record an answer forthe difference in lengthbetween the two creatures becausehe was unsure about his answer and solution strategy.“Iwould have gotten it wrong anyway.”ESL Level 4 Students’ Responses to Question#4All students, except one, were successful answeringQuestion #4. Matthew and Debbytook the opportunity to apply their early knowledgeof algebra to assist them in tackling thisproblem. They set the unknown numberof passengers in the first bus to a variable and solvedfor that variable. Other students used guessand check, grouping positive and negative values,and making a numerical list as solution strategies.When answering Question #4, Brandon demonstratedresistance to using mathematics in79unfamiliar situations. Hewas able to correctly interpretthe problem and he derived a correctsolution to the problem. However,his difficulty with having to dothe problem backwardsresulted in him asserting that“I just, urn, because I justdidn’t want to do another questionlikethis, but because, like, it justmade me all confused, like,make me mixed up and then I’mnot used to working backwards.”Sam followed through thevisual information given in the picture problemby writing alist of sentences describingthe movement ofthe passengers onand off the bus (see Figure 7).Figure 7. Sam’s written workfor Question #4.CL1(ev> ci‘(3’.‘?‘10 U. O-‘ec‘c’.çeo‘‘t.’4\\)cL.k .80He erroneously added all ofthe passengers together, rather than assigning one directionofmovement a positive value and the other a negativevalue. He did not verify his final solution byplacing it into the position ofthe unknownand checking that the answer was zero. He stated thatwhile he thought he knew “most ofwhatthe picture means,” he “still [didn’t] know how tosolve it.”Responses to Question #1 of Students Fully Fluentin EnglishOnly two students not registered in ESL made errorson Question #1, of which both werecomputational errors. Both students provided interpretationsthat were not in agreement with theimage. Mark stated, “I think the persons is buying the things,the items and then look at the cashregister and it says $82... Maybe it was save some money day.”Mark’s understanding of thesignificance ofthe price shown being less than the total ofall ofthe items was that there was adiscount provided to the purchaser. As a final answer, Mark found the differencebetween theitems total and the price displayed on the cash register. This answer was creativeand could fitwith the social meaning of a smaller than expected price due to the presenceof sale prices.However, similar to the answer given by Candy in ESL level 4, Mark’s solution did not agreewith the image shown in the picture problem. Chris, on the other hand, did not account for thetotal prices shown on the cash register. Chris believed that the question being asked was “Howmuch was the total at the end. How much did the person use to buy all the equipment?” As asolution, he calculated the total cost of all of the items. His answer was isolated to the first twoimage blocks shown in the problem, the assortment of items, and the shopping cart. His solutiondid not consider that there was a price already displayed on the cash register, and the cost wasnever factored into his answer.81Responses to Question #2 of StudentsFully Fluent in EnglishAll ofthe students not registeredin ESL made errors on Question#2. This was despitethe fact that all of them, except Tracy,chose to distribute bananas equally tothe monkeys.Austin made an error counting the numberofbananas and, consequently, his answer wasincorrect. Much like Johnin ESL level 2, Austin left the leftover bananas (ofwhich he had onetoo many) in the bucketin order to deal with the remaining bananas. Duringhis interview it wasdifficult to get Austin to state whathe thought the picture problem was asking.In his initialresponse to this inquiry he provided manypossible interpretations.“I had no idea what the question’sasking and then it said, I think it means, like,split the bananas equally amongthe four guys, or monkeys, and then I kept ongetting these other, like, wacko ideas so itwas very confusing. First, it’s like,how many, what the big guy getsmore and the little guys get less, and then theyshould all be split equally, andthen they’re just a family, and then they just eatwhat they get.”While this demonstrated an awarenessofthe multiplicity of possible questions and solutionswith picture problems, Austin hesitated toisolate which question he actually answered tocomplete the mathematics task.Interviewer: So what is [the picture problem]asking?Austin: I don’t know.Interviewer: But you came up with an answer, so whatdid you answer?Austin: How can you split the bananas amongthe four monkey dudes? Yeah, itdidn’t really work.During the interview,FigureS. Mark’s written work for Question #2.Mark clarified that whileJ(gIofn(completing the mathematicstask, he felt that the question£41being asked was how many+4bananas fit into the bucket (see/Figure 8). The four82recorded in his work indicatedthe number of monkeys shown and was not meant to beananswer to the question he stated.He explained that the question marks over the monkeys’headswere there to indicate that the monkeyswere “thinking” or “wondering about something.” Herelayed that at this point in the mathematicstask, he stopped writing anything down because hedid not know what to do.Jimmy chose to describe his solution strategy inwords rather than with numbers toanswer the problem (see Figure 9). Thestrategy that he described appears to be accurate but iscoded as incomplete becauseFigure 9. Jimmy’s written work for Question numerical solution isY’provided.-c CI)-*\Cn\x’cTracy was the only*\estudent who did not choose todistribute the bananas equally to the monkeys. She explained that she ignored the presence ofthe equal symbols in the image when she completed the mathematics task. Upon reflection,during her interview she spontaneously stated that it might be correct to divide the bananasequally amongst the monkeys instead ofthe strategy that she chose to employ. When askedabout her ability to complete another question of similar design, she explained that “maybeIwould still try again but using a different kind ofway ofunderstanding the question.” Herresponse indicates both reflection and metacognition during completion.Responses to Question #3 of Students Fully Fluent in EnglishFive out of eight participants not registered in ESL accurately answered all threecomponents of Question #3. They determined the answers ofthe two creatures’ lengths throughmultiplication and addition, and found the difference in length through subtraction.83Three students made errors dueFigure 10. Steven’s written work forQuestion a combination of solutionstrategy andcalculation challenges. Steven’serrorson Question #3 were basedon an .inaccurate use of decimal,unit length,c-L54 vV, t4-and solution strategy (seeFigure 10).He skip-counted by 0.3 metresfor the whale length and 0.2 metresfor the jellyfish length.While the digits recordedwere correct for both, the placementofthe decimals and hisconversion to centimetres wasnot accurate. Steven’s strategyto find the difference in lengthbetween the two creatures was toskip count each of the whole segmentsshown in the whaleimage and then add what heestimated the length of the partial remainingspace to be. Hementioned that Question #3 “mademe think about like what to do with the problem. Because,like, well, the one part I didn’tknow was the gap here. Maybe just becauseof the part wherethere was, urn, like, nolength here and you have to like look around tofind a clue, yeah.”Steven did not choose to usesubtraction as a viable strategy and instead used thepicture clues,like segment length and a visualestimation of segment size, to attempt to discover a reasonablesolution.Mark used an incorrect strategy todetermine the length ofthe jellyfish as he equated 0.4metres with one segment instead oftwo.His strategy for determining the length differencebetween the two creatures wascorrect and by chance his answer also ended up being correct.This was due to the same resulting differencebetween his two earlier erroneous calculations.Chris used a variety of strategies to answer Question#3 (see Figure 11). He wrotenumber sentences for all ofthe components basedon his interpretation of the visualinformation. He incorrectly multiplied during his attempt tofind the length of the whale.However, using the same solution strategy, he correctly determinedthe length of the jellyfish.84$,&ks LOct’°‘‘c‘, teO,M°i.oDuring his interview, Chris explained that heestimated the length of the space below the whaleand counted the equivalent number of segmentsshown on the whale. He then multiplied thenumber of segments by the measurementoftwo segments in the jellyfish. After explaining thisprocedure, he mentioned that he felt thathis answer for the length of the whale was incorrectFigure 11. Chris’s written work for Question#3.because of what he afreadyknew about these twocreatures. “Like, the jellyfishis 1 metre and the whale is 1.5metres so, urn, I learned aboutwhales before and they weresupposed to be a lot biggerthan a jellyfish.” He used hisbackground knowledge andillustrated his understanding of“a reasonable answer” toinvestigate his solution.Responses to Question #4 of Students Fully Fluent in EnglishFour students not registered in ESL, that answer Question #4 accurately, all mentionedduring their interviews that they struggled with the question. Chris explained that it took himtwo attempts to answer Question #4. On his version of the booklet, Question #4 was the firstpicture problem and thus, this was the first problem of this form that he had ever encountered.He described that he looked at the problem for a few minutes and moved on to complete theother questions on the task, coming back to it to answer it after completing some ofthe otherpicture problems first. Chris initially chose to add all of the passengers together, regardless of85their direction oftravel, but he then concluded that the direction oftravel should be included. Heproceeded to use guess and check to randomly assign values to the unknown bus and completedthe number sentence he had created. Chris commented that “once I figured out [how the get theanswer] it was pretty easy.” After explaining his solution strategy, Steven stated, “[Thisproblem] made me feel pretty good about [my mathematics ability] ‘cause at first I thought itwould be a bit hard but I got through it.”Austin was one ofthe few students who noticed the bus driver and the only student whoactually included the driver in his solution.Jimmy’s answer to Question #4 was the single sentence shown in Figure 12. In theinterview, he correctly identified the purpose of the picture problem. He stated that had therebeen a pattern in the number of peopleFigure 12. Jimmy’s written work for Question #4.moving on and offthe bus, he wouldhave been more likely to answer thequestion correctly. As it stands, hecorrectly judged that his answer did not satisfy the problem. “Cause it was very complicated.Ididn’t understand the question.” However, due to his clarity of understanding ofthe problem, hedemonstrated and explained that his confusion was not with the problem itself, but with whatsolution strategy he should employ.Unfortunately, Flora showed little work for Question #4, which wouldhelp clarify hersolution strategy (see Figure 13). In herFigure 13. Flora’s written work for Question #4.interview she explained that she adds all ofthe people shown getting on or off ofthe bustwereregardless oftheir direction, an erroneoussolution strategy.86In his interview, Mark explained an accurate solution but in his work sample he did theintermediate steps in his head, instead of writing it on the mathematics task. Because of this, Iam unaware whether his inaccurate final solution was due to an error in calculation or hissolution strategy.Clarity of Picture ProblemsStudent responses to the interview question “How can this picture problem be madeclearer?” are shown in Table 7. This was asked after discussing a picture problem with thestudent. This question helped to discover more about how the modification to image-basedproblems impacted the students and their preference for certain forms of problem solving.Students’ answers revealed the methods that they would choose to use to change the problems toincrease their perceived performance and self-efficacy.Table 7Methods of Improving Picture Problems Suggested by StudentsParticipantImprovement (%)GroupPicture Numbers Words NoneESL Level 2 26.3 31.6 15.8 26.3ESL Level 4 14.2 28.6 23.8 33.3No ESL 5.3 21.1 63.2 10.5The answers given by the participants fall into four general categories: clarificationthrough pictures, clarification through numbers, clarification through words or no clarification isnecessary. Students who believed that clarification would be best through pictures gaveexamples such as taking out unnecessary images, or adding or altering the images present inthepicture problems. Some students commented that numbers would help to increase clarity.Onestudent gave the example of having the number of passengers boarding and exiting thebus on87Question #4 written in each of the boxes sothat counting was unnecessary. Another exampleprovided to illustrate how clarity could be found throughnumbers would be to change themeasurement given in the segments on Question #3 to show thelength of one segment insteadof two. The use ofwords was also suggested so that the questionintended by the pictureproblem would be stated to supplement the image. Students also suggestedthat the entireproblem could simply be transformed into a word problem. When asked how pictureproblemscould be made clearer, some students also felt that picture problems needed noclarification as itwas already clear to them.Each group responded to this interview questionfor at least three ofthe pictureproblems. To get the percentage of improvement responses of each type, the number ofresponses by group to each specific form of clarification is divided by the total numberofresponses to picture problem clarification.The student responses were again more similar between the groups of students stillacquiring English and substantially different from the responses ofthose fluent in English.Students registered in either ofthe ESL levels were much more likely to say that there nothingneeds to be done to make a picture problem clearer than those students not registered in ESL.Students with no ESL designation responded that their main method for making the pictureproblems clearer would be to add more words; the approach the ESL level 2 group leastconsidered an option.ESL level 2 students demonstrated no obvious preference for form of clarification. Anequal number of students requested clarification through picture adjustments as stated that noclarification was needed. There were slightly more responses that favoured clarification throughthe addition of numbers or mathematical symbols and the fewest number of requests were forclarification through the addition ofwords or language. No one student requested the addition ofwords more than once in all of their responses to problem clarity. Since ESL level 2 students88indicated during their interviews thatthey often find that the language in mathematics problemsolving makes it difficult for them, itfollows that they would be least likely to suggest this as animprovement.ESL level 4 students showed a slight avoidance of clarification throughimage alteration,since fewer students requested this formof clarification over other forms. One third of the time,students in ESL level 4, with increased English comprehension, indicated thatno improvementsneeded to be made to make picture problems more coherent.There did seem to be an increase inthe number of times participants thought that language should be added over ESLlevel 2students. However, this increase is only slight and is in agreementwith the fact that thesestudents have also acquired more English and may be increasinglymore comfortable workingwithin this educational format.Students with no ESL designation overwhelmingly desired clarification through the useofwords (73.9% of responses). This desire to include language in picture problems is quadrupleand triple the preference of ESL level 2 and 4 students respectively. Only one responsesuggested that a picture-based method should be used for clarification. In only two instancesstudents believed that the information shown in the image was sufficient.Jimmy commented that it is important for picture problems to have words “so that[students] won’t have a hard time comprehending the question so then they might be hard tounderstand if they, like, yeah. I only understand the words ones.” When answering pictureproblems, students not registered in ESL exhibited a lack of confidence as evidenced by thesmall number of students who commented that picture problems were clear. On the other hand,students in ESL level 2 and ESL level 4 both felt that at least one quarter of the time theinformation presented in picture problems was clear enough.Through their interviews, it was apparent that much ofthe reason students who are not inESL desired clarification through language stemmed from wanting to be correct, and their desire89to work with familiar mathematicsproblem solving formats. Tracy explained that while shethought that picture problems werefun, she also felt that when they were even a little bitunclearthat she did not want to completethem. When I asked her why that was the case she toldme,“Um, it’s like you have risk. You get two answers for thequestions. I mean, there would be twokinds of questions for this picture and, urn, you could fallon one side or you could fall on theother side and you could get it right or you could get itwrong depending on what they’reasking.” Other students made statements similar to this.Tracy also explained that she neededthere to be an “answer sheet” for the work that she completedbecause “if I did a whole sheet ofmath problems, I would like toknow if I got them right or wrong.” Language was the mostguaranteed way for her to ensure that her answers were“correct.”Summary of ResultsImage-based problems provided the participating students with a unique mathematicsformat which they had not previously encountered. Thestudents in ESL level 2 were able to bethe most successful completing problems ofthis format,though they did not have the strongestperformance in either computation problems or word problems. The students in ESL level 4were less successful than ESL level 2 students when completing picture problems, and thestudents not registered in ESL struggled significantly with this problem format. The students’ability to notice details, think logically, and apply their mathematics knowledge to this newformat greatly impacted their rate of success when completing picture problems. The problemsolving self-efficacy of some students in ESL level 2 was positively impacted by pictureproblems. Many ofthe ESL level 2 students explained that the English used in mathematics isdifficult for them. Students in ESL level 4 also made comments about difficulty they have hadwith English in mathematics, though their self-efficacy on word problems is higher than thestudents in ESL level 2. Only one student not registered in ESL expressed any difficulty with90the language component in mathematics.These students had much higher self-efficacy whencompleting word problems than pictureproblems.91DiscussionThe present study endeavoured todetermine the effects of image-based, wordlessproblems on students’ perceptions oftheir ability to be successful when problem solving inmathematics and their performanceon problems ofthis type. During interviews, ESL studentsrepeatedly commented that theyexperience difficulty with the use of language in mathematics.This reaffirmed the need to investigateforms of problem solving that provide alternatives tolanguage-based word problems. Overall,the results of this study indicate that picture problemsmay be valuable in creating more positive self-efficacybeliefs for some ESL learners and alsoincrease their rate of problem solving success.Because the current study had a very small samplesize, the results found are notgeneralizable, but are indicative of the abilities of thisparticular participant group. Futureresearch should be carried out with a larger sample size tostrengthen the findings ofthis study.Students in ESL were expected to struggle with word problems to some extent becauseof language difficulties, but none of the participant groupswere expected to strugglesignificantly with picture problems. This surprise finding warrants attentionand I will discussthis in greater detail to attempt to explain these results. Picture problems did not requirelanguage knowledge, though there was some degree of cultural fluency necessary. For example,picture problem Question #1 showed a shopping cart, cash register, and numbers on tagsattached to sporting goods. In order to interpret this problem, students had to recognize theseitems, know how they work, and what the significance of each ofthese items would be in ashopping purchase. It was surprising that there is such a dramatic decrease in the percentage ofpicture problems successfully answered by students with more English knowledge. Therelationship between language acquisition and decreased performance on picture problems,suggests that there were factors other than language which inhibited students’ success on pictureproblems. This warrants further investigation.92Some students in ESL level 2 expressed a positive view of picture problems andprovided comments about possible future experiences with picture problems which seemed tosuggest that they would welcome this problem type. Many of these students demonstrated anacceptance of picture problems as an alternative problem solving format in mathematics. Thisacceptance was not as apparent in the responses of students in ESL level 4 and was essentiallynonexistent in the responses of students who are fully fluent in English. Their rationale for theseresponses will be discussed further, and the implications for future research will be outlined.Picture Problems as Nonroutine ProblemsAs a new problem format, picture problems presented students with situations in whichthey needed to apply their understanding of mathematics in new ways. Students were not taughtany solution strategies before they encountered picture problems on the mathematics task. Thisensured that students’ responses to picture problems were a genuine reflection oftheir ability totransfer their learned mathematics knowledge to new situations (Mayer, 1998).There are two types of problems that students can encounter when problem solving inmathematics: routine problems and nonroutine problems (Mayer, 1998). Routine problems areproblems that are similar to ones that students have previously encountered. Students havebackground knowledge of possible solution methods which can be applied to provide success onthe completion of routine problems (Mayer, 1998). For students in this study, computationproblems and word problems were examples of routine problems (ESL students may not findword problems completely routine due to their lack of language fluency). Routineproblemstypically allow students to use passive learning behaviours because they do not havetoconstruct or apply new knowledge (Mayer, 1998; Anthony, 1996). Passive learning includes theabsorption of information, completion of single, fixed answer problems, and recycling ofpreviously learned mathematics (Anthony, 1996). This format does not encourage meaning93making or the application of mathematics in a way thatwould increase and deepen students’understanding of the material (Anthony, 1996).Alternatively, nonroutine problems are questiontypes which students have not yetlearned how to solve (Mayer, 1998). The completion of nonroutine problemsdemands anunderstanding of mathematics application and knowledgeofhow, and when, to apply theirmathematics skills to problems (Anthony, 1996; Mayer,1998). Comments of students in thisstudy indicated that picture problems were nonroutine forthem and thus, confidence in theirsolution strategies or interpretations ofthe problem was affected.The Use of Active Learning in the Solution of Picture ProblemsNonroutine problems, such as picture problems, encourage students to engage in activelearning (Mayer, 1998). Active learning is the process of constructing new knowledge throughthe application of previous mathematics knowledge and skills (Anthony, 1996). This processengages students in their own learning and forces them to expend mental energy during theconstruction of a solution strategy (Anthony, 1996). Since deeper understanding of mathematicsgrows from students’ construction of mathematics knowledge and active engagement inlearning, nonroutine problems should be used to encourage this (Mayer, 1998).The use of active learning was present in the problem solving skills demonstrated by theESL students who participated in this study. Students in ESL level 2 and ESL level 4 explainedthat their overall expectation oftheir ability to succeed in mathematics has decreased sincecoming to Canada. Though they believe that they are becoming more comfortable with theEnglish language, students explained that they still struggle with the language used in wordproblems and felt that it impacts their success on word problems. ESL students’ beliefs aresupported by the results of the mathematics task since ESL students demonstrated weakerperformance on word problems than students who are fully fluent in English.94As difficult as word problemsare for ESL students, their struggle with these problemsmay actually prove beneficial whenstudents encounter nonroutine problems, such aspicture-based problems. When ESL studentscomplete word problems, they are forced to be moreactively engaged in problem solvingthan students who are fully fluent in English. This activelearning is the result of their limitedsuccessful experience with similar English language wordproblems. The likelihood that theESL students are familiar with the word problems that theyencounter is lower than for studentswho are fully fluent in English. ESL students cannot followpreviously used strategies for completion andneed to construct solution strategies for each wordproblem. Until word problems become routine or similarto ones they have seen in the past, ESLstudents need to develop solution strategies; this forcesthem to transfer previously learnedmathematics to the new situation (Mayer, 1998). Asthese problems become more familiar tothem, ESL students do not need to use active learning tothe same extent and they may begin torely more on passive learning (Anthony, 1996).When students encountered picture problemson the mathematics task, they wereintroduced to problems that facilitated active learning. That is, toanswer picture problems,students needed to analyse and investigate the problem,their solution, and their thinking. SinceESL level 2 students reported that they are still unable to fully understand word problems, theyremain in a state ofknowledge construction and active learning when completing such wordproblems. ESL level 2 students’ recent experience with problems that are nonroutine may makethem better equipped to apply active learning to picture problems as well. One possibleexplanation for their similar rates of errors on picture problems and word problems is thattheyapply active learning to both problem formats. Neither format is routine, as of yet.ESL level 2 students demonstrated a quick reversion to active learning when completingpicture problems. This strategy also suggests that their mathematics abilities are not being fullysupported by, or accurately assessed with, problems solving questions that are posed in a word95based format (Kiplinger et al., 2000, Brown, 2005). Studentsshould be able to completeproblems which enable them to develop the deepest understandingpossible (Anthony, 1996),but this cannot happen if ESL students are primarily encounteringword problems when problemsolving. This is supported by the comments made by the students during their interviews.Students’ responses in all groups suggested that the use of picture problems may allowmathematics problem solving to be more equitable for ESL learners, especially those who havenot yet mastered basic conversational English or acquired academic fluency.Though ESL level 4 students have had more experience with word problems than ESLlevel 2 students, they reported that they still struggle with language. However, they alsoconsistently mentioned that this problem solving format is becoming more familiar to them.These students have acquired the beginning of academic English, which includes mathematicslanguage, with its specific usage and meaning in mathematics word problems. Theirunderstanding ofthe nuances ofthe English used in word problems, and the practice that theyhave had applying this knowledge, are two factors stated by ESL level 4 students as beneficialto their success while solving mathematics word problems.ESL level 4 students were starting to demonstrate a methodical approach to solving wordproblems that Puchalska and Semadeni (1987) call playing the “word problem game.” Thisapproach uses an awareness of solution methods for past word problems which can be applied topresent problems to bring success. Previous knowledge ofthe routines and expectations ofwordproblems can allow ESL level 4 students to participate less in active learning and knowledgeconstruction and rely more on passive learning (Anthony, 1996). Consequently, when ESL level4 students encountered nonroutine problems, they may have been less inclined to revert back tousing active learning, thus completing picture problems with lower levels of success than ESLlevel 2 students.96Anthony (1996) explains that providingstudents with active learning situations andnonroutine problems does not guaranteethat they will make a transition frompassive learning toactive learning. Instead, this transitionis a decision that each student must make to take controlofthe construction of his or her ownlearning. Similar to Anthony (1996), Hegarty, Mayer,andMonk (1995) assert that students need a reason tomake the shift from a problem solvingstrategy that takes less mental effortand engagement, to one that insists that they become activelearners who accept mental challenges.According to Hegarty et al. (1995), students who engagein passive learning may use a direct translationapproach to problem solving. With thisapproach, students look at the problem,glean the basic information such as key words ornumbers, and place them into an equation.The errors made by students not registered in ESLsuggest that they used such an approach.Cobb, Wood, Yackel, and McNeal (ascited in Anthony, 1996) argue that directtranslation and passive learning strategies, suchas learning through the absorption ofinformation, routine practice, and memorizationare often sufficient for success in the typicalmathematics classroom. However, this strategyoften leads to these students becomingunsuccessful problem solvers when they are presentedwith more complex mathematicsproblems that require mental effort and the transferof mathematics learning (Hegarty, Mayer, &Monk, 1995, Anthony, 1996). The students in the study who are fullyfluent in English havelikely had direct translation, a weaker mathematical approach, reinforced through past successin the classroom (Hegarty et al., 1995). During their interviews, students stated that they havehad success on known problem formats in the past because they have been able to followlearned procedures for completion. Thus when completing tasks similar to previously learnedproblems, they were able to recycle this knowledge from previous tasks. The high rate ofsuccess on word problems and computation problems of students who are fully fluent in English97may be explained by their successfuluse ofthis approach to problemsolving on routineproblems.Conversely, students who arefully fluent in English had considerably more difficultywith picture problems than studentsin either of the other two participant groups. This coulddemonstrate reliance on the use of previouslylearned mathematics strategies, and may possiblyindicate that students in this study whoare not registered in ESL have a dependence on passivelearning. I would hypothesize that theirchoice to use direct translation on word problems andcomputation problems may be a signthat they have rarely encountered situations where theyhave had to shift to active learningin order to be successful. This is a claim that would needfurther research to substantiate, butwould be worth investigating, because few active learningsituations may result in students havingpoorly developed problem solving strategies that usediscovery, construction, and develop a deeperunderstanding of mathematics (Hegarty et al.,1995).In this study, students who arenot registered in ESL explained that they struggled whenthey were challenged with picture problems becauseit was a nonroutine problem format. Theyexpressed difficulty constructing meaning for theproblems and applying the appropriatemathematics. Their high rate of errors and interview statements stressedthat they attributed themajority oftheir difficulty with picture problems to inexperience with this problemformat. Thisresponse to picture problems suggests that the studentsin this study who are fully fluent inEnglish did not make the transition to engaging in active learning when completing nonroutineproblems. According to Hegarty et al. (1995), studentswho are used to experiencing high ratesof success on familiar problem formats are less willing to solve nonroutine problems. Theiravoidance of these tasks usually occurs because they are not able to employ a known algorithmto ensure their success (Hegarty et a!., 1995). The students in this studyalso explained thatmathematics problems should only have one “correct” answer. A single answer approach to98problem solving is further indicative of the applicationof passive learning (Anthony, 1996). Thelack of active learning and the desire to avoid failurethat is demonstrated by the students whoare filly fluent in English may account for their poor performanceon picture problems.The difficulty that students who are fully fluentin English demonstrated when applyingtheir knowledge of mathematics to picture problems suggests that, likemost students who areused to problem solving success, these students struggled withtransferring their knowledge ofmathematics to nonroutine problem solving tasks (Anthony, 1996; Kramarski,Mevarech, &Arami, 2002). When completing the mathematics task, these students did not demonstrate the“complex and flexible thought processes” necessary for successfully solving nonroutineproblems (Hegarty et al., 1995). The poor performance of students who are fully fluent inEnglish suggests that their ability to solve nonroutine problems was negatively impacted bytheir difficulty with knowledge transfer and also their use of passive learning.The Role of Metacognition in the Completion of Picture ProblemsThe majority of the students in all participant groups in the present study madecomments which indicate that picture problems encouraged them to mull over how to solve theproblems, what strategies they should use, and when they should be applied. Many students alsoexplained that they initially believed that picture problems could be solved with one method, butwhen considering their solution strategy, they reassessed those assumptions, and some of themaltered their approach. The evidence provided through interviews about students’ considerationof their own thought processes during the completion of picture problems suggests that theseproblems encouraged students to engage in consideration and analysis of their own thinking andlearning, a process also known as metacognition (Swanson, 1990; Kramarski et al., 2002). Somestudents in this study commented that they would work through a problem, come to a new pieceof information, and revaluate their initial procedure, adding in their new observations. Swanson99(1990) identifies that, regardless of their basic mathematics ability levels, those students whoengage in metacognition are more likely to be successful in mathematics than those who do notengage in metacognition. For example, one student in this study commented that she wouldsimply stop when she encountered information that was contrary to her initial completionstrategy and might have indicated that she had made an error. On such problems, when sheavoided delving into further thought in order to minimize her confusion or develop herunderstanding ofthe problem and possible solution strategies, her answers were incorrect.Cognitive factors such as the knowledge of mathematics skills or the ability to accuratelycomplete computation are not as important to successful problem solving completion as anunderstanding of one’s own learning and knowledge construction (Mayer, 1998; Swanson,1990). Following this argument, one might assume that all students who used metacognition aremore able to be successful on picture problems, but this was not the result shown in the study.Rather, students in ESL level 2, ESL level 4, and those not registered in ESL, who usedmetacognition while completing the mathematics task, still showed a vast difference in theirability to correctly answer picture problems, which may be due to the level of metacognitionthat they engaged in (Swanson, 1990).While metacogrntion was used more often during the completion of nonroutine problemsthan routine problems, the degree to which metacognition is used also seems to make adifference in the problem solving abilities of students in this study (Swanson, 1990). Duringtheir interviews, many ofthe students explained that they used different levels of metacognitionwhile completing picture problems. Many of the students who are fully fluent in English orregistered in ESL level 4 explained during their interviews that they used metacognitionsparingly. They conveyed that they had an initial idea ofthe goals ofthe problem and a possiblesolution strategy but when they realized that there could also be another option for the problemand its solution, they second-guessed themselves. The possible multiplicity of solution strategies100confused many of the students and created cognitivedissonance, where they were not sure whatto think. Rather than working throughthis issue with mental effort, students who did noteffectively use metacognition chose to ignore their thought-patterndiscrepancy and answer theproblem with their initial assumptions, or provide noanswer at all. Because the individualresolution of internal metacognitive conflicts in mathematics helps to developknowledge andfacilitate the transfer of mathematics knowledge to new situations (Kramarski etal., 2002),these students’ avoidance of cognitive dissonance may have been detrimental totheir ability tocomplete nonroutine problems (Mayer, 1998). Many ofthese students chose tocontinue withtheir first assumptions ofthe problem even though they claimed to know thattheir solutionstrategy was incorrect, or they simply avoided completing the problem. Had these students beenwilling to engage more in their own process of knowledge construction, they would haveinvested more time and effort into accurate completion ofthe problem to the best oftheirabilities (Mayer, 1998; Anthony, 1996).Conversely, when faced with a multiplicity of possiblequestions and solutions for asingle picture problem, other students in the study, especiallythose in ESL level 2, chose to usetheir cognitive dissonance as motivation to reexamine the problem. In order to make theirinterpretation of the problem and their resulting solution strategies acceptable to themselves,these students used higher levels of metacognition. Many ofthese students commented on theamount ofthought that they needed to use to complete picture problems and explained some ofthe inconsistencies that they encountered and their methods for solving these challenges. Forexample, one student explained that the first time she looked at the shopping cart problem(Question #1) she felt that all ofthe items should go into the cart, but realized that thisinterpretation was inconsistent with the information provided in the image, particularly theplacement of the shopping cart in the image, and she changed her interpretation and solutionstrategy accordingly. Increased metacognition on the part of the student such as this is seen in101the performance of these studentsin this study, results in increased mathematics achievement,especially on nonroutine problem types(Kramarski et al., 2002; Mayer, 1998).The Role of Will in the Completionof Nonroutine ProblemsAccording to Anthony (1996), metacogrntion is notthe only factor which impactsstudents’ success on nonroutine problems.A student’s will also impacts his or her performance.“Will,” as defined by Mayer (1998),is the culmination of a student’s interest, self-efficacy, andmotivation when solving nonroutineproblems, such as picture problems. A more positive will tolearn in problem solving situations, tends to lead toincreased success during completion(Mayer, 1998). This aspect of problem solving is pertinent tothis study because students in allparticipant groups made comments about the effort, determination,and desire that they had tocomplete the problems on the mathematics task.The superior performance demonstrated by ESL level2 students in this study on pictureproblems, may be positively impacted by their willingness toanswer picture problems asfrequently mentioned during their interviews. The studentsin ESL level 2 were most likely toview picture problems as having both individual interest andsituational interest. Mayer (1998)defines individual interest as an individual’s preference for,or positive feelings towards, anactivity in general. Situational interest is more dependent on the task itself. Situational interestoccurs when a student judges a specific task to be interesting (Mayer,1998). ESL level 2students’ high opinion of both types of interest on picture problems predisposed them to puttingmore effort into thinking about, fully understanding, and working through the problems (Mayer,1998). Mayer (1998) also suggests that there is a link between interest and the transfer ofproblem solving abilities. This is supported by the findings of this study, although ESL 2students’ high success on picture problems cannot solely be attributed to their higher levels ofinterest.102ESL level2 studentsalso demonstratedmore positiveself-efficacybeliefsabout pictureproblemcompletionthan studentsin the twoother participantgroups. Moreofthe ESLlevel 2studentswere optimisticabouttheir abilityto be successfulon this problemformat,than ineitherthe ESLlevel4 or non-ESLgroups.Pintrichand DeGroot’sstudy (ascited inMayer,1998) foundthat studentswith morepositiveself-efficacyjudgementsare alsoincreasinglymore likelyto engagein activelearningduringmathematicsproblemsolving.Most importantly,however,students’self-efficacybeliefsabout successon similarproblemsin the future,directlyimpactstheir presentperformance(Mayer,1998).Studentswho expectsuccesson problemsbeforethey solvethem arelikely toexperiencesuccess.Conversely,studentswho expecttostrugglewith problemsoften experiencegreaterdifficulty(Mayer,1998).This resultingdifficultyarises becausestudentswho expectdifficultywith problemsolving tendto use onlypart ofthe informationgiven,demonstrateuncertaintyin theirwork anddo notunderstandthatmultipleanswersmay bepossible(Kramarskiet al.,2002;Mayer,1998).Coupledtogether,the elementsof will;namely interestand self-efficacy,played alargepart inthe problemsolvingsuccessof students(Mayer,1998). Duringtheir interviews,ESLlevel 2 studentsdemonstratedhigherlevels ofwill thanstudentsin the othertwo participantgroups.The increasedsuccessofESLlevel 2studentson pictureproblemssuggeststhat theirpositivewill for suchtasks didplay a rolein their success.Throughtheir interviews,studentswho arefully fluentin Englishdemonstratedthelowestlevels ofinterestand self-efficacyon pictureproblems,producinglow overallwill tocompleteproblemsofthis type.Thesestudentsalso demonstratedthe lowestperformanceonpicture problems.Similarto the studycarried outby Mayer(1998), thisstudy seemsto supportthe findingof a connectionbetweenstudents’will on problemsolving andtheir success.103Trendsin SolutionStrategiesWhileprobleminterpretationallowedfor multiplemeaningsfor pictureproblems,theyalsoallowedstudentsto usea varietyof strategiesduringcompletion.Whenansweringwordproblemsandcomputationproblemson themathematicstask,studentsused(i) logicalreasoning,(ii) writingnumbersentences,(iii)workingbackwardsand (iv)guessandchecka....solutionstrategies.Whencompletingpictureproblemson themathematicstask,studentsinacuse of(i)logicalreasoning(ii)writingnumbersentences,(iii)workingbackwards,(iv) guess—andcheck,(v) makinga list,(vi)findinga pattern,and(vii)estimating.Whilethevarietyofsolutionstrategiesmayhaveto dowiththe questionsplacedon themathematicstask,thereno reasonthatthe samerangeof strategiescouldnot havebeenusedfor computationand w—problemsas alternatesolutionstrategies.Pictureproblemson themathematicstask allowedstudentsto manipulatea varietyof strategies,especiallyin situationswheretheywereunsu---how toanswerthequestions.Whencompletingcomputationproblemson themathematics——-------studentswhodid notknowhow tofindanansweroftenleftit blank.Thougha fewstudent________chosenotto providea solutionto apictureproblemtheyfoundchallenging,manyalsochc-—-———usea previouslyunusedstrategy,such asestimation,toattemptto answerthe problem.Students’commentsindicatedthattheirviewsaboutthe “openness”of pictureproblemsthemfeelthattherewas arangeof acceptablesolutionstrategies.Thoughstudents’alternstrategiesdidnotalwaysproduceaccuratesolutions,the factthattheywerewillingto apz—----.-..otherstrategiessuggeststhatpictureproblemsmayelicitmoreof avarietyof solutionstr—— - -- - -thanothertypesof problems.Futureresearchin thisareais needed.Trendsin SolutionJustificationStudents’justificationof theirinterpretationsfor thepictureproblemsandtheirsubsequentsolutionstrategiesdemonstratedan interesting,andunexpected,trend.TlirC----..interviews, it was apparent that many students created situations or stories to justifytheirinterpretations of the image-based problems. The use ofjustification during nonroutine problemsolving is intended to increase students’ acceptance oftheir solution strategies(Kramarski et al.,2002). In this study, the instances of students’ elaboration of picture problems suggest that thesestudents were actively involved in sense making (Anthony, 1996). Their attempts to add details,explanations, and reasons to explain the problem situations, often led them to insert informationinto the picture problem that was not previously there.For some ofthe students in this study, their envisioned scenario agreed with all ofthedetails ofthe image, while others created situations that only accounted for part of the image.The situations that did not accurately account for all the information containedin the images,often resulted in solutions with errors because these situations omitted information that wasimportant for a complete understanding ofthe problem. This was similar to the findings ofHegarty et al. (1995), that successful problem solvers use all ofthe details given in the problemto develop an understanding of the relation of all of the items (Kramarski et al., 2002). On theother hand, unsuccessful problem solvers tended to incorporate only the key features oftheproblems, or the information that can be easily observed, into their solution strategy (Hegarty etal., 1995; Kramarski et al., 2002). In this instance, much of the important, but not obvious,information about the problem was lost from the solution strategy.In their explanations oftheir thought processes during the completion of pictureproblems, students in this study tended to place themselves inside ofthe problem as an activeparticipant. For example, they may have included themselves as the customer buying thesporting goods in picture problem Question #1 or the individual who was in charge ofbananadistribution in picture problem Question #3. According to Anthony (1996), such a procedure ofself-inclusion may encourage, but does not necessarily lead to, reflective thinking and105metacognition to minimize cognitive dissonance and create a new, conflict free understanding ofthe mathematics problem.Trends in Solution CommunicationStudents’ written communication of their understanding and solutions to pictureproblems demonstrated a tendency to show their solutions with a wider variety of techniquesthan they would during the completion of computation problems or word problems. Students’written answers to computation and word problems on the mathematics task were limited to theuse of words and numbers to communicate their solutions. When answering picture problems,students used words and numbers, but they also used illustrations to communicate theirsolutions. The use of illustrations was not present in any students’ solutions for the other twoproblem types. This suggests that the picture problem format may allow for an increased use ofvisual representation in solution communication. This may have to do with the students’ havinga sense of openness in problem interpretation and transferring this same sense of openness totheir solutions. Problems which allowed or encouraged the use of multiple solutionrepresentations such as numbers, words, and illustrations, may have allowed students to makeuse oftheir individual learning styles to help them better understand mathematics (Tang &Ginsberg, 1999). The use of illustrations by students were not isolated to students in anyparticular participant group, so it would seem that this preference for a visual descriptor of theirsolution was more based on an individual’s learning style than their language ability.Meaningful Learning in MathematicsActive learning, metacognition, will, individual story development, and using rationaleto justify picture problems, can all help to promote active student interaction with mathematicsproblems. This engagement in the process of mathematics problem solving helps to combat the106lack of connection that students feel exists between themselves and mathematics (Puchaiska &Semadeni, 1987; Baroody, 1993). As Baroody (1993) states, “Meaningful learning involvesseeing or making connections. Unlike rote learned knowledge, meaningful knowledge cannot beimposed from without but must be constructed from within. Meaningful learning, then, is notsimply a matter of passively absorbing information but entails actively making sense of theworld.” Many of the students in this study demonstrated that picture problems allowed them toexplore mathematics through their creation of appropriate questions for the images,determination and application of a solution strategy, and rationalization oftheir chosen solutionstrategy. Indeed, many of these students readily rationalized their solutions aloud during theinterview process.107Conclusion“Maximizing mathematical learning involves fosteringconceptual knowledge as well asprocedural knowledge, encouraging the development ofstrategies and metacognitive knowledgeand promoting a positive disposition” (Baroody, 1993).Image-based problems are wordlessproblems requiring interpretation ofthe problem andapplication ofthe appropriate mathematicsusing the visual information provided. As this study hasbegun to demonstrate, picture problemsmay provide students with the opportunities they need toengage in constructive mathematicsknowledge acquisition. Image-based mathematics problemsprovided some ofthe students whostruggle with language-based mathematics problems anopportunity to explore mathematics,while exposing them to mathematics that encouragesactive learning and engagement with theproblems. The ESL level 2 students in this study, with limited English language knowledge,expressed difficulty with the use of Englishin language-based problem solving. Many of themalso demonstrated more positive views about mathematics problem solving when completingpicture problems. It is believed that such positive views ofthese problems and the students’ability to complete such problems could lead to increased judgements of self-efficacy, whichwould help encourage further mathematics learning and motivation (Pajares & Miller, 1995).Implications for EducatorsMost educators continue to use word problems primarily as the vehicle for problemsolving in the classroom. Gerofsky (1996) suggests that this is because of our own inherentbelief in the “good” ofword problems and the established tradition of their use in themathematics classroom. However, though word problems can be beneficial because of thediscussion they create and the application of mathematics to new situations (Baroody, 1998),such problems do not have to be used in the classroom at the exclusion of all other forms of108problem solving. Rather, opportunities for problemsolving should be as plentiful as possible(Baroody, 1998).One overarching goal of mathematics is to equipstudents with the skills necessary to usemathematics successfully in their daily lives as theyexperience the world around them(Baroody, 1993). The mathematics they will have to do spontaneouslywhen shopping forgroceries, calculating a measurement or estimating a difference will arise primarilythroughsituations they encounter. It is likely that these problemswill not be written in digits or wordsbut will develop from each student’s interpretationofthe situation presented by his or herenvironment. The importance of students’ being able to apply the appropriatemathematics skillsand concepts to these situations as nonroutine problem solving cannot be understated. It seemsplausible that picture problems, as a nonroutine form of problem solving, may be a problemsolving method that can also be used to practice such thinking.In light ofthe findings ofthis study, it is suggested that image-based problems might beused by practitioners as an alternative assessment tool to assess students’acquisition andinternalization of mathematical concepts. The absence of words in picture problems may beginto permit this assessment regardless of the language of the educator or student. The use ofimage-based problems as a supplement to number problems and word problems seems to permitstudents to apply their knowledge of mathematics in new ways. For instance, rather thanstudents knowing to subtract because of a computational symbol or learned word such as“difference,” presenting students with picture problems encourages them to understand themathematics involved and construct an appropriate solution without such cues.As many students identified, there was more than one possible question andinterpretation for each of the picture problems which would agree with the image presented.This variety of approaches and outcomes can be useful in the mathematics classroom to createdialogue among students about mathematics, as desired by the NCTM (1989). Also, the109inclusion of images for interpretation allows those studentsnot adept at reading comprehensionor fluent in the language of instruction to participatein mathematics regardless of theirlimitations (Brown, 2005; Hofstetter, 2003). Whileit is desirable that students also be able tocomprehend language-based problems, image-basedproblems can be used to provide problemsolving experiences which do not demand language and allow students, especially thosewithlimited English knowledge, to explore mathematics. The allowance created by multiplepossiblequestions and solution strategies can help to encouragerisk taking, as some students in thisstudy explained (Baroody, 1993).Implications for ResearchIn order to investigate the validity of the findings of this study, a similar study should becarried out on a larger scale with more participants and revised image-based mathematicsproblems. A study this size would allow the trends seen in this participant group to be furtherexamined and would help to give more insight into the role that image-based mathematicsshould have in the mathematics classroom. One could investigate if there is a gender gap in theability to interpret and solve picture problems or if there is an ideal age for the introduction ofsuch problems. The ESL ability levels of the students needs to be determined as this appeared tohave an impact on their performance on picture problems with ESL level 2 students performingbetter than ESL level 4 students on such problems. The ability of students in ESL level 1, ESLlevel 3, and ESL level 5 to successfully complete picture problems could also be investigated todetermine which groups such problems are most beneficial for.As was encountered with some ofthe questions on the mathematics task, the intention ofthe image is not always clear to students. While image-based problems should encourage activelearning and discussion, some students found them confusing or frustrating in that they wereunsure of the requirements of the problem. Further development of picture problems should110attempt to make the problems as clear as possible, without unnecessary or confounding images.For example, Question #2, with the monkeys, could be redesigned to include monkeys of thesame size and individual bananas to minimize students’ confusion about the intention oftheproblem. Research should be done in this area to further understand students’ “image-basedliteracy” to aid in the development of problems ofthe appropriate mathematics level usingsuitable images. However, while image-based problems are visual problems that students couldwork on individually, they are intended to be a form of dynamic assessment. Picture problemsshould encourage dialogue and allow students to develop mathematics understanding andknowledge transfer from working with problems which challenge them. In thecontext ofmulticultural classrooms, research also should be conducted to determine how to use images in across-cultural environment so that such problems can be as effective and applicable as possibleto a wide variety of students.Further research should also be done to delve into the unexpected response ofFEPstudents to image-based problems. These students commented repeatedly that they were used toword problems and computation problems and that image-based problems were not clear tothem. Their poor performance on image-based problems was unexpected as it was assumed thatall participant groups would perform equally well on image-based problems which simplyremove the necessity for language, providing all students with an equitable foundation. Whywould students who are able to be very successful at computation problems and word-basedproblems not be able to transfer their understanding of mathematics to problems asked throughvisual images? Is their decreased performance impacted more by resistance to unfamiliarmathematics formats or a weakness due to an inability to apply their knowledge of mathematicsto alternative situations?I±i summary, this exploratory study demonstrated that picture problems provided thegreatest benefit to those students registered in ESL level 2. These students were able to perform111comparably well on picture problems as they were on computation- or language-basedproblems. Many ofthese same students also expressed more positive self-efficacy judgementswhen completing picture problems than language-based problems, which make up a large partofthe cuent mathematics curriculum. Further research needs to be done to investigate thetransferability of these resi.ilts to classroom practice and the ideal extent ofthe inclusion ofimage-based problems into elementary classrooms.112ReferencesAbedi, J., & Lord, C. (2001). The language factorin mathematics tests. 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Research with children: The sameor different from research with adults?Childhood, 9(3), 321-341.Salend, S. 3., & Hofstetter, E. (1996). Adapting a problem-solvingapproach to teachingmathematics to students with mild disabilities.Intervention in School & Clinic, 31(4), 209.Schunk, D. H., & Gunn, T. P. (1986). Self-efficacy and skill development: Influence oftaskstrategies and attributions. The Journal ofEducational Research (Washington, D. C.,), 79,238-244.Swanson, H. L. (1990). Influence of metacognitive knowledge and aptitude on problem solving.Journal ofEducational Psychology, 82(2), 306-314.Tang, E. P., & Ginsberg, H. P. (1999). Young children’s mathematical reasoning: Apsychological view. In L. Stifl & F. Curcio (Eds.), Developing mathematical reasoning ingrades K-12 (pp. 45-65). Reston, VA: NCTM.Whang, P. A., & Hancock, G. R. (1994). Motivation and mathematics achievement:Comparisons between Asian-American and non-Asian students. Contemporary EducationalPsychology, 19, 302-322.117Appendix A. Computationproblems used on themathematics task.Question 9LIXLI1078Question 1011.3 + 24.2LI61.7LI42.4LI55584.lQuestion 115873140263+571Question 1248/6+8-3LI118Appendix B. Word problems usedon the mathematics task.Question 5To raise money for theirupcoming performance, the drama club decided tohave a car wash. Theycharged $4 per car. How many customers did they haveif their profit was $84 after they paid $12for their cleaning supplies?Question 6To solve a problem, Jade needs to add426 to the difference of 732 and 929.What number is the solution tothe problem?Question 7How much change would you receivefrom a twenty dollar bill if you boughtone book for $3.98, one for $9.98 andtwo for $2.99?Question 8Mitchell Elementary School donated ninecases of 20 cans of soup to the foodbank. Byng Elementary School donated eightcases of 36 cans of soup to thefood bank. The food bank would like togive out the soup to needy families inboxes that fit 12 cans each. How many boxes will they needto use?1190CD 0 t’)C CD —. 0O Cl) CD CD C Cl) CD 0 I Cl)8CQuestion 3— — — — — — — — —— ? — — — — — — — — — —— — — — — — — — — — — —4—?121Question 4122Appendix D. Mathematics Task QuestionBooklet cover page.Mathematics TaskQuestion Bookletv3INSTRUCTIONS1. Please do all of your writing in the Mathematics Task Work Booklet, notin this book.2. Do your best to answer every question.3. Show all of your work. Do not erase any of your work. If you havemade a mistake, please just cross it out with one single line and keepgoing.4. After each question, write a note about any words, ideas or picturesthat you did not know or understand.5. After each question, answer the three questions at the end of thatpage, “How hard was this question?”, “How sure are you that youhave the best answer?” and “Could you answer another question likethis one?” by circling the word that best describes how you feel.Page 1123Appendix E. Mathematics Task Work Bookletcover page.Mathematics TaskWork BookletStudent Number_____Problem Book V___INSTRUCTIONS1. Please do all of your willing in the mathematics task work booklet.2. Do your best to answer every question.3. Show all of your work. Do not erase any of your work. If you havemade a mistake, please just cross it out with one single line and keepgoing.4. After each question, write a note about any words, ideas orpicturesthat you did not know or understand.5. After each question, answer the three questions at the end ofthatpage, “How hard was this question?”, “How sure areyou that youhave the best answer?” and “Could you answeranother question likethis one?” by circling the word that best describeshow you feel.124Appendix F. Simplified Chinese interpretation oftheinstructions for the Mathematics TaskProblem Booklet and the Mathematics Task Work Booklet.1. F#*, fl4th*W Mathematics Work Booklet.2. fl-fl3. flWfl+O Lt4nJ’tigT-±ej *ØJ<RTW**-W4a4. LBe1m , ,5. 2+P1E, fl*ii*:3tiüJ$1J?J*-±iLI#-1i?Page 2125Question#iQuestion#1contShowallofyourwork.Donoteraseanyofyourwork.IfyouhavemadeaWriteanoteaboutanywords,ideasorpicturesmistake,pleasejustcrossitoutwithonesinglelineandkeepgoing.thatyoudonotknoworunderstand.flffIf1.*T--’i’4jL#TICDHowhardwasthisquestion?leyouranswerCDVeryEasySortofEasySortofHardVeryHard0HowsureareyouthatyouleyouranswerCDhavethebestanswer?CDNotSureatAllSortofSurePrettySureCompletelyoSure Couldyouansweranother(eyranswquestionlikethisone?Yes,EasilylMaybe...No,ProbablyNotAppendix H. Interview Questions.1. Tell me about which languages you speak and where you speak them?2. How comfortable are you speaking English with me? In the classroom?3. How important do you think your parents feel math is? Can you tell me about that?4. Can you explain your school history to me? So, where you were to school, for how long andwhat languages you used there?5. Tell me a little about how your math experiences in China and in Canada have been the sameand different. How you feel about that?6. What do you think about the amount ofEnglish used in a math classroom?7. Now please think about math this year. How sure are you that you can do well in math? Why?8a. Tell me about how well you think you would you do if you were given a page of numberproblems? Why? What would your enjoyment be?8b. Tell me about how well you think you would you do if you were given a page ofword-basedmath problems? Why? What would your enjoyment be?8c. Tell me about how well you think you would do on a page of image-based math problems?Why? What would your enjoyment be?9. Which ofthese question types would be good or bad for ESL learners and why?10. What do you think it is important for me to know about the image-based problems beforegiving them to other students?For random computation and word problems, and all picture problems, ask:a. Can you tell me about how you solved this problem?b. Why was this question (easy/hard/etc) for you?c. I noticed on your math questions you said that your confidence on problem____is____Can you tell me a little about why you said that?127Appendix L Behaviour Research Ethics Board Certificate ofApproval.[C The Unwersity of British ColumbiaIOffice ofResearch Services\ JBehavioural Research Ethics BoardSuite 102, 6l9oAgronomy Road, Vancouver, B.C. V6T 1Z3CERTIFICATE OF APPROVAL- FULL BOARDRINCIPAL INVESTIGATOR: INS11TUTION I DEPARTMENT:PJBC BREB NUMBER:nn G. AndersonJBC/EducalioniCurncuIum Studies p108-00074NSTITU110N(S) WHERE RESEARCH WILL BE CARRIEDOUT:Institution ISiteI/AN/AIther locations where the researchwill be conducted:School DistriCt elementary schoolsO-lNVESTIGATOR(SJ:manda Jean PcI•PONSORING AGENCIES:ROJECT TiTLE:chievement and Self-Efficacy of Students with English as a SecondLanguage Based on Problem Type In anEnglish LanguageBased Mathematics CurriculumEB MEETiNG DATE: ICERTIFICATEEXPIRY DATE:lenuary 24, 2008 January 24, 2009)OCUMENTS INCLUDED IN THIS APPROVAL: IDATE APPROVED:February 15,2008)octzneiEName IVersionI Dateonsent Forms:‘rincipal Consent Form2 August 2, 2008‘arental Consent Form2 August 2, 2008ssent Forms:tudent Assent FormNIA November 1,2008tuestlonnaire. Questionnaire Cover Letter.Tests:ntervlew Questions N/A Noventher 1,20083ample Mathematics Task Problems N/A November 1, 2008Ehe application for ethical review and the document(s) listed above have been reviewed and the procedures wereound to be acceptable on ethical grounds for research involving human subjects.Approval is issued on behaitof the Behavioural Research Ethics Boardand signed electronically by one of the following:Dr. M. Jucith Lynam, ChairDr. Ken Craig, ChairDr. Jim Rupert, Associate ChairDr. Laurie Ford, Associate ChairDr. Daniel Salhani, Associate ChairDr. Anita Ho, Associate Chair128


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