THE RELATIONSHIP BETWEEN ATTITUDES TOWARDS SPECIFICMATHEMATICS TOPICS AND ACHIEVEMENT IN THOSE DOMAINSbyCARMEL FRANCES WALSHB.A., The University of British Columbia, 1975A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF ARTSinTHE FACULTY OF GRADUATE STUDIES(Department of Mathematics and Science Education)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIADecember, 1991© Carmel Frances Walsh, 1991In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(SignatureDepartment of 'IVA-al^6k--14-a -Ae-L-e-zArubThe University of British ColumbiaVancouver, CanadaDate^ , I c DE-6 (2/88)ABSTRACTThe purpose of this study was to investigate the nature of the relationship betweenhigh school students' attitudes towards particular mathematics topics and their achievementin those areas. In order to examine this relationship, data collected by the 1990 BritishColumbia Mathematics Assessment concerning students in Grade 7 and Grade 10 wereanalyzed. This assessment involved over 37 000 students at the Grade 7 level and over31 000 Grade 10 students. Data concerning students' perceptions as to the importance,difficulty, and likeability of various mathematics topics were collected by the assessment.Achievement scores based on student performance on a 40-item, multiple-choice test ofmathematical ability were also obtained. Each of the domains of the British Columbiamathematics curriculum were represented on the achievement test. Achievement items werealso constructed at three cognitive behaviour levels: computation, comprehension, andapplication/problem solving.Geometry and data analysis were the two domains of the mathematics curriculumwhich were the focus of this work. The data relating to these topics were analyzed throughthe use of chi-square analysis. Matrices were designed which compared students'perceptions of geometry and data analysis with their achievement in those domains. Eachof the three components of attitude—difficulty, importance, and likeability—were treated asindependent variables. Chi-square values were determined for each matrix and an analysisof the patterns exhibited by the cells was also undertaken. With one exception, each matrixhad a chi-square value which was significant at the 0.0001 level. The remaining matrixwas significant at the 0.001 level.An examination of the Grade 10 data indicated that a significant relationshipbetween students' attitudes towards geometry and data analysis and students' achievementin those domains existed. The number of students who considered geometry or dataanalysis to be important, easy, or likeable and who also obtained good scores on theachievement portion of the assessment was greater than the expected value for those cells ofthe matrices. Likewise, the number of students who indicated that geometry or dataanalysis was not important, was difficult, or disliked the topic and who also obtained lowachievement scores was greater than the expected value. Similar patterns were observedwhen students' overall achievement in mathematics was compared with their attitudestowards data analysis and geometry.Students in Grade 7 generally achieved higher scores in the mathematics assessmentand held more favourable views towards data analysis and geometry than did students inGrade 10. However, results showed that the relationship between each of the componentsof attitude and achievement in geometry and data analysis followed trends similar to, butnot as strong as, those found for students in Grade 10.For the purposes of this study, the Grade 10 data were also separated into twogroups. The data concerning students enrolled in the more challenging Mathematics 10course were compared with the data relating to students enrolled in the less difficultMathematics 10A course. Students enrolled in the Mathematics 10 course achieved higherscores and held more positive views towards data analysis and geometry than did thestudents enrolled in Mathematics 10A. The relationships between each of the componentsof attitude studied and achievement within each of the domains, however, were similar forboth groups of students.CONTENTSPageABSTRACT ^iiLIST OF TABLES ^viiLIST OF FIGURES ^ixACKNOWLEDGEMENT ^xCHAPTER1. INTRODUCTION ^1Background ^1Statement of the Problem ^32. REVIEW OF THE LITERATURE ^5Use of the term "attitude" ^6Early Studies ^8Measurement Scales ^10Mid-1970's to the Present ^11Components of Attitude ^11Age and Ability Levels ^14Large—Scale Assessment Studies ^17National Assessment of Educational Progress ^17Second International Mathematics Study ^18British Columbia Mathematics Assessment ^20The Use of Interviews ^21Geometry and Data Analysis ^21Summary ^22i vPage3 . METHOD^ 24The 1990 British Columbia Mathematics Assessment ^24Participants in the Assessment ^24Structure of the Assessment Materials ^25Content of the Assessment Materials ^28Sample Selection ^30Description of the Sample ^30Data Collection Instruments ^32Data Analysis ^34Coding of Sample Materials ^35Chi—Square Analysis ^36Summary ^374. FINDINGS ^38Description of the Variables ^38Student Attitudes ^38Student Achievement ^40Data Analysis and Geometry ^41Grade 10—Geometry and Data Analysis ^42Grade 10—Geometry ^42Grade 10—Data Analysis ^45Overall Achievement in Mathematics ^48Overall Achievement and Geometry ^48Overall Achievement and Data Analysis ^51PageGrade Levels—Grade 7 and Grade 10 ^53Grade 7—Geometry ^54Grade 7—Data Analysis ^57Streaming—Mathematics 10 and Mathematics 10A ^59Mathematics 10 and Mathematics 10A—Geometry ^60Mathematics 10 and Mathematics 10A—Data Analysis ^64Grade 10—Non-participation of Students ^695. SUMMARY AND CONCLUSIONS^ 71Findings and Conclusions ^71Implications ^76Limitations of the Study ^77Suggestions for Further Research ^78REFERENCES ^80APPENDIXA. British Columbia Mathematics Assessment Forms ^83Grade 10: Forms C and DGrade 7: Form AVILIST OF TABLESPageChapter 31. Statistical Properties of the Assessment Booklets ^262. Content Strands for Achievement Items ^293. Distribution of Subjects by Course Enrollment ^314. Strands and Item Assignments ^335. Classification of Achievement Scores ^35Chapter 41. Student Responses to Attitude Items ^392. Number of Items Used to Assess Achievement ^413. Grade 10: Chi-Square Analysis—Importance of Geometry andAchievement in Geometry ^434. Grade 10: Chi-Square Analysis—Difficulty of Geometry andAchievement in Geometry ^435. Grade 10: Chi-Square Analysis—Likeability of Geometry andAchievement in Geometry ^446. Grade 10: Chi-Square Analysis—Importance of Data Analysis andAchievement in Data Analysis ^467. Grade 10: Chi-Square Analysis—Difficulty of Data Analysis andAchievement in Data Analysis ^468. Grade 10: Chi-Square Analysis—Likeability of Data Analysis andAchivement in Data Analysis ^479. Grade 10: Chi-Square Analysis—Importance of Geometry andOverall Achievement in Mathematics ^4810. Grade 10: Chi-Square Analysis—Difficulty of Geometry andOverall Achievement in Mathematics ^4911. Grade 10: Chi-Square Analysis—Likeability of Geometry andOverall Achievement in Mathematics ^4912. Grade 10: Chi-Square Analysis Importance of Data Analysis andOverall Achievement in Mathematics ^5113. Grade 10: Chi-Square Analysis—Difficulty of Data Analysis andOverall Achievement in Mathematics ^52vii14. Grade 10: Chi-Square Analysis—Likeability of Data Analysis and OverallAchievement in Mathematics ^5215. Grade 7: Chi-Square Analysis—Importance of Geometry and Achievement inGeometry ^5416. Grade 7: Chi-Square Analysis—Difficulty of Geometry and Achievement inGeometry ^5517. Grade 7: Chi-Square Analysis—Likeability of Geometry and Achievement inGeometry ^5518. Grade 7: Chi-Square Analysis—Importance of Data Analysis and Achievement inData Analysis ^5719. Grade 7:Chi-Square Analysis—Difficulty of Data Analysis and Achievement inData Analysis ^5820. Grade 7: Chi-Square Analysis—Likeability of Data Analysis and Achievement inData Analysis ^5821. Mathematics 10 & 10A: Chi-Square Analysis—Importance of Geometry andAchievement in Geometry ^6122. Mathematics 10 & 10A: Chi-Square Analysis—Difficulty of Geometry andAchievement in Geometry ^6223. Mathematics 10 & 10A: Chi-Square Analysis—Likeability of Geometry andAchievement in Geometry ^6324. Mathematics 10 & 10A: Chi-Square Analysis—Importance of Data Analysis andAchievement in Data Analysis ^6625. Mathematics 10 & 10A: Chi-Square Analysis—Difficulty of Data Analysis andAchievement in Data Analysis ^6726. Mathematics 10 & 10A: Chi-Square Analysis—Likeability of Data Analysis andAchievement in Data Analysis ^68vii iLIST OF FIGURESPageChapter 31. Grade 10 background items: Geometry and data analysis ^33i xACKNOWLEDGEMENTI wish to thank the members of my thesis committee, Dr. David Robitaille, Dr.Mike Marshall, and Dr. Walter Szetela for their guidance. It has been a privilege for me towork with them.I would also like to thank my family for their support, patience, andencouragement.xCHAPTER 1INTRODUCTIONBackgroundThe teaching of mathematics involves an awareness of both the affective and thecognitive domains. An element of the affective domain which may have a significantrelationship to the learning of mathematics is attitude. Although attitude has been defined inmany different ways, a common theme in most definitions of the term is that an attitude caninfluence and determine the direction of an individual's behaviour. Thus, attitudes, whichmay predispose an individual towards certain behaviours, can play an important role in theprocess of learning mathematics. It has been speculated that attitudes may affect theamount of effort an individual is willing to make in order to learn mathematics, mayinfluence the selection of specific mathematics courses, and may be linked to individualdifferences in learning mathematics (Fennema & Sherman, 1976). Attitudes have also beenviewed from the reverse perspective where they have been seen to be an outcome of theprocess of learning mathematics (Jackson, 1990; Newman, 1984).Numerous investigations have been undertaken to determine the nature and strengthof the relationship between the learning of mathematics and the attitudes of studentstowards the subject. In 1970 and 1976, Aiken conducted surveys of the literatureconcerned with attitudes and performance in mathematics. Based on the results cited in thisliterature, he concluded that there is a low to moderate correlation between these variables(Aiken, 1976). More recent research has suggested that there might be a strongerassociation than was previously indicated and that this relationship between attitude andmathematics achievement is a complex one involving many different factors. Two studiesconducted since the mid-1970's found that attitude, when treated as a single entity, had a1significant, moderate correlation with student progress in mathematics (Campbell &Schoen, 1977; Tsai & Walberg, 1983).However, with the development of new instruments for measuring specificdimensions of attitude (Fennema & Sherman, 1976; Sandman, 1980), the emphasis inrecent studies has been on investigating particular aspects of attitude rather than on workingwith attitude as a single variable (Bassarear, 1987; Brassell, Petry & Brooks, 1980;Cheung, 1988; Kifer & Robitaille, 1989; Newman, 1984; Taylor & Robitaille, 1987).Anxiety, difficulty in learning mathematics, enjoyment of mathematics, students' self-concept of their ability to learn mathematics, and the value of mathematics in society are allattributes of attitude which have been linked to achievement in mathematics (Brassell,Petry & Brooks, 1980; Cheung, 1988; Hembree, 1990; Reavis, 1989; Taylor & Robitaille,1987). However, the strength of the relationship to achievement was not found to be thesame for each of these components. Although the findings were not consistent across allstudies, in general, the component which accounted for the greatest variance in achievementwas self-concept of ability (Brassell, Petry & Brooks, 1980; Cheung, 1988).Research has also been undertaken to determine if the link between attitude andmathematics achievement varies depending upon specific characteristics of the studentsinvolved. Grade level (Hembree, 1990; Newman, 1984; Taylor & Robitaille, 1987),intelligence (Minato & Yanase, 1984) and mathematical ability (Brassell, Petry & Brooks,1980; Hembree, 1990) have all been found to be associated with the degree to whichattitudes interact with mathematics performance.The relationship between attitudes and mathematics achievement appears to be anintricate one. Although the research tends to support the view that attitudes are linked to thelearning of mathematics, it also suggests that to make generalizations about all students andthe impact of attitude on their learning of mathematics may be inappropriate. While it isimportant to recognize that attitude may be a factor in the learning of mathematics, it mustalso be recognized that its influence varies depending upon the circumstances. As noted2previously, the individual student and the specific aspect of attitude under considerationboth appear to be important elements in this relationship.The relationship between attitudes and mathematics performance may also dependupon the specific mathematical topic under study. However, little is known about thispossible relationship. Several large scale studies of mathematics achievement have shownthat student performance varies over different topic areas. The 1985 British ColumbiaMathematics Assessment (Robitaille & O'Shea, 1985), the 1986 Fourth NationalAssessment of Educational Progress in the United States (Brown, Carpenter, Kouba,Lindquist, Silver & Swafford, 1988a, 1988b) and the Second International MathematicsStudy (Robitaille & Garden, 1989) all report that student achievement in mathematics is notthe same for all areas of the curriculum.There has also been some descriptive statistical evidence that suggests that studentshave different attitudes toward different parts of the mathematics curriculum (Hogan, 1977;Kifer & Robitaille, 1989; Robitaille and O'Shea, 1985). However, there appears to havebeen little research done to determine the nature of the relationship between attitudes and thelearning of these specific mathematical topics. Increased knowledge in this area would leadto a greater understanding of the mechanism by which student achievement and studentattitudes interact.Statement of the ProblemDuring the 1989-90 school year, the Fourth British Columbia MathematicsAssessment was conducted. Approximately 108,000 students and 4500 teachers fromGrades 4, 7, and 10 took part. The major focus of the assessment was to investigate thelevel of student achievement in mathematics and the perceptions of students and teacherstowards topics relating to the learning of mathematics (Robitaille, in press). Thus, thisassessment contained data which could be used to investigate the connection between3achievement in mathematics and attitudes towards specific areas of the mathematicscurriculum.The contents of the achievement portion of the British Columbia MathematicsAssessment reflected the domains and objectives of the mathematics curriculum in thatprovince. The mathematical content for each grade level was partitioned into major strandswhich were then divided into several topics. For Grades 4 and 7 these strands wereAlgebra, Data Analysis, Geometry, Measurement, Rational Numbers and Whole Numbers.The strands for Grade 10 were the same except that the Whole Numbers and RationalNumbers sections were combined under the single heading entitled Numbers & Operations.For each of these major strands, questions were developed at three cognitive levels:computation, comprehension, and application.In the section dealing with background, attitudes, and opinions, students andteachers were asked to indicate their views on many of the strands evaluated in theachievement section of the study. They were questioned on their perceptions of theimportance, enjoyment, and difficulty of learning or teaching such topics as geometry,trigonometry, data analysis and fractions. As a result, information regarding students'attitudes towards certain domains of the mathematics curriculum and their achievement inthose domains was obtained by this assessment.The general purpose of this study, then, was to use data from the 1990 BritishColumbia Mathematics Assessment to investigate the nature of the relationship betweenhigh school students' attitudes towards particular mathematics topics and their achievementin those areas. Information from the geometry and data analysis domains in Grade 7 and10 was used to generalize about the nature of this relationship for students at these agelevels. These particular topics were chosen because they are areas common to the course ofstudy of both Grades 7 and 10 and within each of these topics there are many objectiveswhich are common to both grades. Also, the 1985 British Columbia MathematicsAssessment (Robitaille & O'Shea, 1985) determined that students' performance in4geometry was poor at all three grade levels and that students and teachers consideredgeometry to be the least important of a group of ten mathematical topics. This weakperformance and the suggestion of a possible connection to attitude made geometry a topicof particular interest.The specific questions addressed by the study are as follows:1.What relationships exist between students' attitudes towards the geometry and dataanalysis domains of the mathematics curriculum and students' achievement in those domains?2. What relationships exist among students' overall mathematics ability, their achievementin the geometry and data analysis domains, and their attitudes towards these topics?3. What differences, if any, exist in the nature of the relationships in questions 1 and 2among students at different grade levels?4. What differences, if any, exist in the relationships in questions 1 and 2 among studentsin the same grade who are enrolled in different mathematics courses?5CHAPTER 2REVIEW OF THE LITERATUREResearch into the relationship between attitudes and mathematics education has beenconducted from a variety of perspectives. This review of the literature focuses on thosestudies which were concerned with students' attitudes towards mathematics and how theyrelate to achievement in mathematics. The emphasis is on those works which dealt withstudents learning mathematics at the high school level.The first part of the review discusses the term "attitude" and its usage in theliterature. The next section is organized on a chronological basis. Changes in this field ofstudy have been made during the past few decades and these changes are reflected in theliterature. The latter portion of the review focuses on the results of recent major studies,and on the work done in some specific content areas.Use of the term "attitude"There is no universally accepted definition of the term "attitude" and, in theliterature cited in this review, little attempt was made by the authors to explain its meaning.A formal definition was given in only two research reports. One definition, included in afootnote and prefaced with the explanation that "... there is no standard definition of theterm attitude...", stated that, in general, attitude "...refers to a learned predisposition ortendency on the part of an individual to respond positively or negatively to some object,situation, concept, or another person" (Aiken, 1970). In another work, attitude wasreferred to as "...affectively toned perceptions of situations in which mathematics is learnedas well as to views of mathematics as a subject" (Cheung, 1988). Although there has beenconsiderable work done towards the development of a definition of attitude (Allport, 1967;Anderson, 1988; Hart, 1989; Shaw & Wright, 1967), the discrepancies between the twodefinitions mentioned previously indicate that there is no consensus amongst researchers inmathematics education as to the precise meaning of this term.6In most of the studies cited in this review, it seems to have been assumed that,when the word attitude is used, there is a general, but unstated, understanding of itsmeaning. Other terms related to the affective domain, such as opinions, beliefs, emotions,and values, have also escaped definition.A structure for distinguishing between these various terms has been developed byMcLeod (1989). McLeod has indicated that beliefs, attitudes, and emotions are terms thatdescribe different aspects of the affective domain. Each of these terms represents differentlevels of cognitive involvement. Beliefs, attitudes, and emotions represent a range ofresponses with "...increasing affective involvement, decreasing cognitive involvement,increasing intensity, and decreasing stability" (p.246).Beliefs about mathematics are considered by McLeod to be the most stable of thethree terms and he divides beliefs into two general categories. One set of beliefs concernthose about mathematics as a discipline. A second set of beliefs deals with how studentsview their own relationship towards mathematics. This second group of beliefs involves agreater affective response than the first group, and it includes such concepts as self-confidence and self-concept of ability.McLeod suggests that attitudes represent those `...affective responses that involvepositive or negative feelings of moderate intensity and reasonable stability" (p.249) and areresistant to change. Whether or not a student likes a particular topic would be an exampleof an attitude towards mathematics. McLeod considers attitudes to be stable enough to beable to be measured through the use of a questionnaire.Emotions, however, are the least stable of the three terms and cannot be measuredin this traditional manner. Emotions involve feelings that are more intense than thoserepresented by attitudes and beliefs and these feelings may change quickly. As an example,McLeod suggests that a student might feel frustrated while trying to solve a problem. Thisfeeling will often disappear once the problem is solved.7The situation described above suggests that, because the term attitude has not beenexplicitly defined in all of the research, the studies concerning its relationship toachievement in mathematics may lack validity. However, in most works attitude has beendefined, in an indirect manner, by the instruments used to measure the term. Self-reportingscales, which require subjects in a study to report on a variety of items, were the mostcommonly used means of assessing attitude. In any particular study, the items selected forthis purpose would then represent the defmition of attitude for that piece of research. Thefindings of these works, however, could only be related to attitude as defined in thiscontext.In more recent studies, some researchers have chosen to separate attitude into avariety of components, such as self-concept of ability or motivation in mathematics, andhave then defined these individual characterisics. However, the defintion of these terms forthe purposes of the research are, again, dependent upon the instruments used to evaluatethem. As Kifer (1990) explained:[T]hose who do attitude research ... must face the fact that there is no one rightdefinition of an attitude and no one right way to measure [it]. ... In a strict sense there is adifference between problems of defining attitudes and those of trying to know what hasbeen measured (p. 4). ... Instead one should provide justifications and strongspecifications for what is done regardless of the labels that are attached to our measures(p. 6).Early StudiesIn his book, Life in Classrooms, which was first published in 1968, P. W. Jackson(1990) presented the "common sense" arguments which are sometimes given to linkacademic success with positive attitudes towards school. He referred to the view held bysome psychologists that positive and negative reinforcement will produce correspondingpositive and negative feelings. The good grades and positive feedback received by somestudents should elicit a different view of schooling than that held by students who obtainlow grades and negative personal feedback. Thus, achievement is a predictor of attitude.8Jackson also discussed this relationship from the reverse direction. In order to dowell in school, or in any job, one must want to do the necessary work. Therefore,accomplishment in school depends upon one's attitude towards the task.These same arguments could also be applied to the study of the specific content areaof mathematics. However, despite the apparent logical basis to the above reasoning, earlyresearch into attitudes and mathematics achievement did not produce consistent significantlinks between these variables.In 1970, a major survey by Aiken of the research into attitudes towardsmathematics was published. He also presented an updated study in 1976. Both of theseworks contained sections discussing attitude and achievement. The research cited produceda group of results ranging from no apparent relationship to significant correlations. Ingeneral, however, Aiken reported that low to moderate correlations between attitude andmathematics achievement were found.In the conclusions of his works on attitude, Aiken expressed concern over theattitude measurement instruments and the statistical methodology being used at the time.He stated that:There are too many "home-grown", unstandardized attitude scales, and too manyresearchers have not bothered to become informed about the uses of ...methodology. ...Designers of attitude instruments should begin to provideevidence of the extent to which an instrument is a precise measure of attitude and issensitive to changes in attitude (Aiken, 1976, p. 302).An example of the difficulty involved in investigating attitudes at that time can befound in the doctoral work of the following individuals who were not referred to in Aiken'smaterial. Burbank (1970) and Caezza (1970) each used the Dutton Mathematics AttitudeScale to study the relationship between students' attitudes toward mathematics andstudents' achievement in mathematics. Burbank worked with 411 seventh grade pupilsfrom a single junior high school. The Pearson correlation was used to test four nullhypotheses regarding students' attitudes toward mathematics as related to their achievementin mathematical reasoning, achievement in mathematical concepts, achievement in9mathematical computations and overall mathematical achievement. Statistically significantresults were found in all four cases.In contrast, Caezza did not find any correlations of a practical significance betweenstudent achievement and teacher or student attitude. The subjects in his study consisted of2765 pupils from Grades 2 through 6, and all of their teachers, from a single district inNew York State. In addition to presenting the results of his research, Caezza concludedthat the Dutton Arithmetic Attitude Scale was inadequate as an instrument for linkingattitude to academic variables. While the difference in the fmdings in the work of Caezzaand Burbank could be due to a variety of factors, it does reflect the uncertainty of theresearch at that time.Measurement ScalesIn the mid-1970's, two attitude scales were developed which have since gainedwide-spread acceptance by researchers. Richard Sandman (1980) designed theMathematics Attitude Inventory (MAI) to measure the attitudes of students in Grades 7through 12 towards mathematics. The MAI consists of 48 statements about mathematicswhich are divided into six groups measuring different constructs of mathematics attitude.These categories are: perception of the mathematics teacher, anxiety toward mathematics,value of mathematics in society, self-concept in mathematics, enjoyment of mathematics,and motivation in mathematics.In order to gain information regarding the learning of mathematics by females,Elizabeth Fennema and Julia Sherman (1976) developed the Fennema-ShermanMathematics Attitude Scales. They wanted an instrument which would go beyond themeasure of global attitudes, one which would define specific dimensions related to thelearning of mathematics. Nine affective domains were established. Some of these, such asthe Teacher Scale, the Mathematics Anxiety Scale, the Effectance Motivation Scale inMathematics and the Mathematics Usefulness Scale, are similar to ones presented by10Sandman, whereas others, such as the Mathematics as a Male Domain Scale and theMother/Father Scale, were designed to study the relationship of gender to attitude andachievement. The remaining domains were the Attitude toward Success in MathematicsScale and the Confidence in Learning Mathematics Scale.Both of the above instruments use a Likert-type response format where studentschoose from a four- or five-point scale ranging from strongly agree to strongly disagree.Statistical evidence supporting the validity of the instrument has been supplied by theauthors of each of these scales (Fennema & Sherman, 1976; Sandman 1980).Mid-1970's to the PresentThe research done since the mid-1970's has been influenced by the development ofthe previously mentioned scales. These instruments have frequently been used or theyappear to have been the model for the design of other attitude scales. While someresearchers continued to treat attitude as a single variable, many have instead worked withspecific dimensions of attitude.In addition to separating attitude into its various components, studies have also beendone which divide subjects into different groups. Grade level and mathematical ability aresome of the classifications which have been used. Since the development of the Fennema-Sherman scale there has also been increased interest in gender and its relationship toattitudes and achievement. However, it is not the intent of this review to survey that lattercollection of material.Components of AttitudeRecent research has focused more on studying the components of attitude than onworking with attitude as a unitary concept. Two aspects of attitude which have sometimesbeen associated with achievement are self-concept and anxiety. While these terms have notbeen defined in the same manner throughout the literature, there does seem to be a common11understanding of the meaning of the terms. In the MAI scale, Sandman defined self-concept in mathematics as "a student's perception of his or her own competence inmathematics" and anxiety as "the uneasiness a student feels in situations involvingmathematics" (p. 149). None of the scales in the Fennema-Sherman attitude inventory arespecifically entitled "self-concept", but this idea is incorporated into the other dimensions.Mathematics anxiety is defined in a manner similar to Sandman. Some of the findingsrelating to anxiety and self-concept are discussed in the following material.The role of anxiety was investigated by Hembree (1990) who used meta-analysis toexamine the fmdings of 151 studies concerned with mathematics anxiety. Based on thissurvey, Hembree concluded that mathematics anxiety contributes to lower performance inmathematics. However, this relationship was not found to be reciprocal. Lack of successin mathematics did not appear to cause mathematics anxiety.Using the Mathematics Attitude Inventory, Brassell, Petry, and Brooks (1980)undertook an investigation of mathematics achievement as it relates to ability grouping andstudent attitudes towards the subject. Their research also included a study of therelationship of anxiety and self-concept to the learning of mathematics. Their fmdings onthe role of grouping will be discussed later in this review in the section entitled Age andAbility Levels.Brassell, Petry, and Brooks worked with 714 Grade 7 pupils representative of fivejunior high schools in a suburban community in the United States. Scores fromcomponents of the California Test of Basic Skills and its total score were used to measureachievement. The findings on anxiety indicated weak, negative, correlations ranging from-0.27 to -0.30 for the relationship between mathematics anxiety and mathematicsapplications, concepts, computations, and total mathematics score. Moderate, positivecorrelations of 0.35 to 0.40 were found for mathematics self-concept and the same fourmeasures of achievement. In general, as self-concept decreased, anxiety increased.12A major study by Newman (1984), however, reached a different conclusion. In1971, a longitudinal study initially involving 255 children who were enrolled inkindergarten in four Minneapolis schools was begun. These students were againmonitored in Grades 2, 5, and 10. By Grade 10, only 103 students remained in the study.Newman began the report on his work with a detailed discussion of the research that hadbeen done into the relationship between academic achievement and self-concept of ability.He concluded that the "...findings support both directions of causality between children'sachievement and self-perceptions of ability" (p. 858). That is, self-concept of ability couldaffect a child's academic achievement and, in turn, academic achievement could affect aperson's self-concept. These conclusions lend support to the "common-sense" argumentsregarding attitude and achievement presented by Jackson.The purpose of Newman's study was to examine this connection within the area ofmathematics and to do so using an improved longitudinal design. By using more than twotime periods, the intention was to avoid any hidden curvilinear relationships. Achievementwas measured by the score on a series of tests, and the self-concept measure was basedupon student Likert-type ratings of mathematics performance.Using path-analysis, Newman concluded that, between Grades 2 and 5,mathematics achievement influenced students' perceptions of their ability. However, thisinfluence did not operate in the reverse direction. The effect of self-concept on achievementwas almost zero. Between Grades 5 and 10, self-concept again did not have anysignificant causal relationship to achievement. Depending upon the model he constructed,Newman found that between these two grades achievement had either a low or no causaleffect on self-concept. It was also determined that the accuracy of these student self-ratingsrelative to actual ability increased between Grades 2 and 5 but did not do so in later years,whereas achievement remained stable throughout these years. Newman's work is one ofthe few studies to investigate if there is a two-way relationship between attitudes towardsmathematics and the learning of the material.13Campbell and Schoen (1977), in their investigation of teacher behaviours, alsostudied the relationship between attitudes and mathematics achievement. The subjects were1602 pre-algebra students attending school in Oklahoma. Using a 3-point scale, studentsreported how much they liked mathematics and this response was used to assess theirattitude towards mathematics. Achievement was determined by the student's grade in thesubject. A correlation of 0.34 between these grades and attitude was found. However, asattitude was defined in a narrow context in this study, these results apply only to theconcept of 'liking' mathematics rather than to a broader definition of attitude.The results of major assessments which have also considered the role of specificcomponents of attitude in their studies will be discussed in a subsequent section entitledLarge-Scale Assessment Studies. However, the overall body of work done since the mid-1970's which has focused on the particular components of attitude has been moreconsistent in its findings than the work done prior to that time. Although the findings arenot the same across all studies, in general, the research has indicated that there is asignificant, moderate correlation between various dimensions of students' attitudes towardsmathematics and students' achievement in mathematics.Age and Ability Leyels In addition to looking at different attitude components, researchers have also beeninterested in determining if the relationship between attitude and mathematics achievementvaries with the characteristics of the subjects concerned. The grade level of the studentsand their level of academic ability are two areas which have been studied.Newman, for example, in the study cited previously in this review, indicated thatthe strength of this relationship between performance in mathematics and students' attitudesis linked to the grade level of the students. The Third British Columbia MathematicsAssessment was a major study which also worked with students in different grade levels.Its findings will be discussed in the section dealing with large-scale studies.14Other researchers have grouped subjects according to academic ability. In a study ofover 800 Grade 8 pupils attending three schools in Japan, Minato and Yanase (1984)placed students into three categories based on their scores on an intelligence test which hasbeen used extensively in Japan. For statistical purposes, attitude was treated as a singleentity and its measurement was based on the results of two attitude scales developed inJapan. Achievement was measured by tests which covered the topics of numbers, linearequations, and inequalities.The purpose of this study was not to determine if attitudes affect achievement; butrather to determine what is the influence of ability on performance if attitude is assumed toinfluence learning. Based on their findings, Minato and Yanase concluded that the effect ofattitudes on mathematics achievement for students who ranked low on the intelligence testwas greater than for those who ranked in the middle range and greater still than for thosewith a high ranking.Brassell, Petry, and Brooks also examined the impact of ability grouping. In theirstudy, students had already been assigned by the school district to one of three differentlevels based on a measure of their mathematical ability. Within each of these district levels,teachers were asked to divide the pupils into three groups, (high, medium and low),according to their ability.Brassell, Petry and Brooks reported that, among each of the three district levels,there were significant differences on five out of six attitude scales. On the value ofmathematics in society and anxiety towards mathematics scales the differences were weakwhile moderate differences were reported on the enjoyment of mathematics and attitudetowards the teacher scales. Strong, significant differences were reported on the self-concept scale. Motivation was the only non-significant scale.For each district level, the mean scores on the attitude scales within the high,medium and low sub-groups were given. These figures indicate that, for each districtlevel, anxiety increased and self-concept decreased as the ranking of the student went from15high to low. It is interesting to note that the students with the lowest self-concept and thehighest anxiety level were the low-ranked students within the medium level classes. Incontrast to the findings of Minato and Yanase, which indicated that attitude had the leasteffect on students of high intelligence, Brassell, Petry, and Brooks reported that the high-ranked students in the high-level district group had the lowest anxiety level and the highestself-concept.Hembree's study also investigated the impact of anxiety on students working atdifferent levels and his findings were similar to those of Brassell, Petry, and Brooks.When subjects were compared based upon their ability, high ability students were found tohave lower levels of anxiety than average to low ability students. No difference was foundbetween the average and low ability groupings. It was also determined that the level ofanxiety for students increases throughout junior high school, peaking and levelling off nearGrades 9 and 10.In a longitudinal study on sex differences in mathematical reasoning ability,Benbow and Stanley (1982) worked with students in the United States who had scored inthe upper five percent on the national norms for a standardized achievement test. In thisresearch, attitude was defined as a composite score on a scale which consisted of studentratings of their views on the importance of mathematics for obtaining a job, whether or notthey liked mathematics, and how they ranked mathematics relative to other high schoolsubjects. For these high-ability students, Benbow and Stanley found that... there does not appear to be much relationship between attitudes towardmathematics and achievement in mathematics in a high-aptitude group, unless thevariables measured in this study were inadequate indicators of attitudes towardmathematics. ... For example, [it has been] demonstrated that attitude towardmathematics involves several distinct components (p. 617-618).These results, like those of Minato and Yanase, were based on the study of attitudeas a single variable and they do not coincide with the findings of Hembree or Brassell,Petry, and Brooks whose works were based on the study of the individual components ofattitude. As alluded to by Stanley and Benbow, in order to gain an accurate understanding16of the relationship between attitude and achievement in mathematics, the various aspects ofattitude should be taken into account. The studies cited also indicate that when consideringthe impact of attitude on the learning of mathematics it should not be assumed that it will bethe same for all students.Large—Scale Assessment StudiesData from three major assessments, each involving thousands of students, has alsobeen analyzed to determine the relationship between attitudes and achievement inmathematics. This research differs from the works mentioned previously, not only interms of the numbers of subjects involved, but also in that these subjects were selectedfrom broad geographical regions. The findings of the studies based on these assessmentsare discussed below.National Assessment of Educational ProgressA study involving high school students was conducted by Horn and Walberg(1984) using the data collected in 1977-78 by the second National Assessment ofEducational Progress (NAEP) in the United States. Representative communities fromthroughout the country were selected. Within each community, schools were chosen andthen within the schools a random sample of age-specific students was selected. Using thedata from the study of 17-year-old students, Horn and Walberg worked with a sample of1480 cases. Although attitudes were not investigated directly, the relationship betweeninterest in mathematics and achievement was analyzed.For the purpose of the study, interest was a composite variable determined by theresponses of the students to the following items: "How often did you work ahead in yourmathematics book?"; "How often did you do mathematics problems that were notassigned?"; and "How often did you study mathematics topics that were not in the17textbook?". Horn and Walberg reported a correlation value of 0.04 between interest andachievement and thus failed to find a link between the two variables.Over 2000 thirteen-year-old students who participated in the same NAEP studywere investigated by Tsai and Walberg (1983). An achievement test consisting of 74 itemswas used along with an attitude scale of 14 items. Analysis of variance was used toinvestigate the relationship between attitude and achievement. It was determined thatattitude was associated with significant differences in achievement and that achievementwas also associated with significant differences in attitude.Further statistical analysis indicated that seven factors (sex, ethnicity, father's education,mother's education, home environment, experience, and attitude) accounted for 32 percent of thevariance in achievement and this value was significant at the 0.001 level. Attitude was the factorwith the second strongest link to achievement. Ethnicity had the greatest link. However, althoughachievement was determined to be significantly associated with attitude, only eight percent of thevariance in attitude could be accounted for by a combination of seven factors, one of which wasachievement, and this amount of variance was insignificant at the 0.05 level. These findings,which indicate that attitude may influence achievement but that achievement may not have an effecton attitude, coincide with the work of Newman who, likewise, did not find a causal relationshiplinking achievement to attitude.Second International Mathematics StudyDuring the 1980-81 and 1981-82 school years, students from 20 educationsystems from around the world took part in the Second International Mathematics Study.Two populations of students, aged approximately 13 and 17, were sampled. A variety ofdata in both the attitude and achievement domains was collected.Hong Kong was one of the countries which participated and Cheung (1988)examined the data for the 5644 Grade 7 students who participated in that region. Five-point Likert scales were used to determine student views on 10 attitudinal dimensions.18Moderate correlations with achievement were reported for the dimensions Mathematics-Myself (SELF), Mathematics and Society (SOC), and Mathematics-Create (CREATE). Ofthe achievement variance, 22.6 percent was associated with these three items treated as asingle component. Of all the attitude dimensions, the SELF component, which isstudents' assessment of their own ability to do mathematics and is similar to the self-concept dimension discussed previously, had the greatest correlation with achievement.Another aspect of this study concerned the reciprocal relationship between theattitude variables mentioned above and mathematics achievement. The unique contributionsof achievement on SELF, SOC and CREATE were 0.047, 0.026 and 0.008 respectively.When combined with two of these three attitude components, achievement was found toexplain between 24 percent and 35 percent of the variance in the remaining thirdcomponent. Based on these findings, Cheung claimed that "the commonality analysesresults ... made clear that the relationship of mathematics achievement and attitudes towardsmathematics learning is reciprocal in nature" (p.218-219). However, the amount of thecontribution of achievement on SELF, SOC and CREATE as reported by Cheung is lowand does not appear to support his claim.In a report on the attitudes of students from all of the jurisdictions involved in thesame study, Kifer and Robitaille (1989) found large differences in the perception ofmathematics by students of different countries. Kifer and Robitaille hypothesized that thesevariations may be due to such factors as differing curricula and educational philosophies.Another suggested hypothesis was that social factors may also have an effect on studentattitudes. If this latter view is indeed correct, then the relationship between attitudes andmathematics performance may also be dependent upon the cultural context within which thestudents function.19British Columbia Mathematics AssessmentIn 1985, the third British Columbia Mathematics Assessment, involving moststudents in Grades 4, 7, and 10 in the province, took place. The assessment included itemson achievement, background, and attitude. The achievement section contained 150 itemsfor each grade. Attitude was assessed using three scales with a total of 35 items coveringthe areas of mathematics in school, gender, calculators, and computers. Approximately95,000 students took part in the assessment.Taylor and Robitaille (1987) conducted a study of this data in part to examine theeffect of students' opinions about mathematics on student achievement in this subject. Thestudents were questioned on their opinions about the importance of mathematics, thedifficulty in learning mathematics, and the enjoyment in learning mathematics. Forstudents in Grades 4 and 7, the correlation with achievement for each of these items wasweak and ranged in value from 0.09 to 0.27. At the Grade 10 level, moderate correlationsof 0.42 and 0.45 were found between achievement and importance and achievement anddifficulty respectively.A regression analysis of student opinions and their relationship to achievementdetermined that, at the Grade 4 level, student opinions, with a beta weight of 0.19, had alimited effect on achievement. Student background, problem solving processes, andclassroom organization were found to be more influential. For Grade 7 students, studentopinions had a beta weight of 0.23 and had the greatest effect of all factors studied onchange in achievement. In Grade 10, student opinion had a beta weight of 0.27 but thegreatest predictor of change at this level was student background with a beta weight of0.49.In 1987, a replication study of the 1985 British Columbia Assessment wasconducted. Pre-test and post-test information was gathered and used to do a comparison ofcross-sectional and longitudinal models. When student entry-level factors were controlled20in the longitudinal study, student opinion variables were considerably weaker predictors ofachievement. However, the relative effects of the different variables such as opinions,classroom processes and background were the same as for the cross-sectional model.Taylor and Robitaille cautioned that "findings, based on student background and opinionsof mathematics, should not be attributed solely to current classroom practices" (1987: p.55).The Use of InterviewsIn each of the research studies mentioned previously, the instrument used to assessattitude was a self-reporting scale in which the subjects involved in the study respondeddirectly to a questionnaire. Recent doctoral studies by Bassarear (1987) and Lucock (1989)have involved the use of interviews, essays, and observations in studies of attitudes andachievement in mathematics. The findings of these works are consistent with the results ofmost of the previous research. In a three-year longitudinal study which collected datathrough the use of a self-reporting scale and interviews, Lucock concluded that attitudes doaffect mathematics performance. Similar results were found by Bassarear. In a study ofability, performance, and attitude of 16 students in a college remedial class, data regardingattitudes was collected through the use of questionnaires, essays, and interviews. It wasfound that "... several attitudes significantly added to the amount of variance explained inthe exam average by the measures of ability" (1987: p. 2492-A).Geometry and Data AnalysisAlthough it was noted by Aiken in 1970 that there was a need for research intoattitudes towards specific mathematical topics, little work has been done in this area. Ingeneral, researchers have concentrated on segmenting attitude into its various componentsor on investigating the specific characteristics of students. Individual sections of thecurriculum have not received the same attention.21Hogan (1977) made reference to this deficiency in his report on student interest in avariety of mathematical topics. Approximately 13,000 students from 10 states, who wereenrolled in Grades 1 through 8, participated in a survey of student likes and dislikesregarding a list of mathematics items. A descriptive analysis of the findings was presentedwhich indicated that, in general, as students progressed from Grades 1 through 8 theirattitudes towards mathematics became less favourable. This situation, however, was notthe same for all topics. Interest in computation remained fairly stable while the greatestdecline in interest was in geometry.However, the descriptive analysis of the data from the Second InternationalMathematics Study presented by Kifer and Robitaille (1989) presented a different view.When students at the Grade 8 level were asked to rate different parts of the mathematicscurriculum using the following scales - important-unimportant, easy-difficult and like-dislike - the three geometry topics listed did not rank as being the least favoured ones.Depending upon the specific geometry topic, it was liked by approximately 40 percent to 60percent of the respondents and considered important by 50 percent to 65 percent of thestudents. While research has been done which focuses on attitude towards some areas ofmathematics such as problem solving and calculator use, there do not appear to have beenany further studies linking attitude to achievement in geometry or data analysis.SummaryThe results of early studies into the relationship between attitudes and mathematicsachievement were inconsistent. This research was limited in that the measurementinstruments and the statistical methodology used needed improvement. Since the mid-1970's, the relationship between attitude and achievement has generally been seen to be anintricate one, involving many different dimensions. Recent studies have focused onspecific aspects of attitudes and on certain attributes of the subjects. In general, theresearch has indicated that there is a significant relationship between some of the22components of attitude and mathematics performance and that this relationship variesdepending upon the grade and ability level of the students.The gender of the student, the particular mathematical topic under study, and theculture within which the student lives have also been suggested as additional factors to beconsidered when investigating the connection between attitude and achievement. Whilethere has been considerable research into the relationship between attitudes and gender,there has been little work done regarding the attitudes of students towards specific topicswithin the mathematics curriculum or the impact of culture on these attitudes. Althoughthere is evidence that students have different attitudes towards different mathematics topics,it is not yet known if attitude has different links to achievement depending upon theparticular curriculum domain under consideration. It has not been determined if themathematical topic under consideration is a significant factor in the relationship betweenattitudes and mathematics achievement.23CHAPTER 3METHODThe purpose of this study was to investigate the relationship between studentattitudes towards geometry and data analysis and student achievement in those domains ofthe mathematics curriculum. In order to examine this relationship, data from the 1990British Columbia Mathematics Assessment was used. A description of the pertinentassessment materials and their development are found in this chapter. The details of sampleselection and data analysis are also explained.The 1990 British Columbia Mathematics AssessmentParticipants in the AssessmentMost students in Grades 4, 7, and 10, who were enrolled in public and independentschools in British Columbia, were expected to take part in the 1990 MathematicsAssessment. Grades 4, 7, and 10 were selected because these grades represent criticaljunctures in the school system in that province. By Grade 4, students have completed theirprimary schooling and have the ability to participate in such a study. Grade 7 is the lastyear of elementary school and, as schooling in British Columbia is compulsory only to age16, Grade 10 is the last year that most students are legally required to attend school. BothFrench and English versions of the assessment materials were available for use and theonly students not expected to participate were those with moderate to severe mentalhandicaps.As province-wide student enrollment figures were not available for the actual datesof the assessment and, as school enrollment fluctuates throughout the year, it is notpossible to determine precise rates of participation by students in the assessment.However, as enrollment numbers are available for September 1989, it is possible toestimate these rates. Approximately 40,000 Grade 4 students and 37,000 Grade 7 students24took part in the assessment. Based on the enrollment numbers for September, 1989, thesefigures indicate that there was almost full participation by students in these two grades. InGrade 10, the participation rate was lower with about 31,000 students, or approximately 84percent of the Grade 10 student population as defined in September, 1989, taking part.This lower rate may have been due to students withdrawing from school during the year orto a higher absentee rate among students at the high school level than at the elementaryschool level. Some students of weaker mathematical ability may even have beendiscouraged from taking part in the study by their teachers.Teachers of mathematics at Grades 4, 7, and 10 were also requested to participate inthe assessment. Different teacher questionnaires were designed for each grade level and,for each of these grades, multiple versions of the questionnaire were developed. However,teachers were required to complete only one questionnaire, even if they taught more thanone mathematics class. The numbers of forms completed by teachers were 1980 for Grade4, 1692 for Grade 7, and 912 for Grade 10.Structure of the Assessment MaterialsThe 1990 Mathematics Assessment collected data from both students and teachers.Students provided information on their personal backgrounds, on their attitudes towardmathematics, and on their perceptions of classroom practices. They also completed anachievement test. Teachers reported on their background characterisics, the implementationof the new mathematics curriculum, their classroom practices, their attitudes towardsmathematics, and the content of the mathematics course which they were teaching.All students and teachers completed forms consisting entirely of multiple-choiceitems. Information on students' ability to solve problems was obtained through theadditional use of open-ended forms which were distributed to an eight percent randomsample of students. For each grade level, four multiple-choice student forms, called FormA, Form B, Form C, and Form D, were randomly distributed within each classroom.25These four forms were each divided into two sections entitled BackgroundInformation and Achievement Survey, respectively. In the Grade 7 forms, the first sevenof the 14 background items dealt with personal information and student perceptions ofmathematics in general. These items were the same on all four forms. The remainingquestions consisted of items concerned with student views on specific mathematics topicsand items relating to student perceptions of classroom practices. This latter section was notthe same on all forms although there were some individual items which were identical ontwo or more forms. The background section of the Grade 10 booklets, which contained 17items, was structured in a similar manner. There were eleven items common to all fourbooklets. The last six items, dealing with individual mathematics topics and classroompractices, were not the same on all forms.In the Achievement Survey section of the assessment, all students were asked 40questions. The content of this section was not the same on all forms but it was intendedthat the forms be "... parallel by content weighting, cognitive behavior level, and difficultyat each grade" (Taylor & Robitaille, in press). An analysis of the psychometric propertiesof the forms confirmed that the forms were parallel. For any one grade, the mean score foreach of the forms never differed by more than two percent and the standard deviationsabout the mean indicated the scores had similar variance on each form. The values of thereliability coefficients, which had a maximum range of 0.04 for any one grade, alsoindicated that the forms were consistent and stable as measures of student achievement. Adetailed list of the statistical properties of the assessment booklets is given in Table 1 on thefollowing page.26Table 1Statistical Properties of the Assessment BookletsMean StandardDeviationReliability(KR 20)Grade 4Form A 20.2 7.1 0.85Form B 20.6 7.6 0.86Form C 20.1 7.4 0.87Form D 20.6 7.1 0.84Grade 7Form A 21.7 7.1 0.85Form B 20.5 7.2 0.85Form C 21.6 7.8 0.87Form D 21.3 7.5 0.86Mathematics 10Form A 21.6 7.8 0.87Form B 20.5 7.9 0.87Form C 21.6 7.5 0.87Form D 20.3 7.6 0.87Mathematics 10AForm A 13.7 5.6 0.76Form B 13.4 5.9 0.79Form C 14.8 5.8 0.78Form D 14.5 5.9 0.80The achievement portion of the Grade 10 booklet was organized in a mannerdifferent from the Grades 4 and 7 booklets. In British Columbia, students have an optionof enrolling in Mathematics 10 or Mathematics 10A. Mathematics 10 is the prerequisite toMathematics 11 which is a course required for university entrance in that province.Completion of Mathematics 10A, however, does not entitle a student to enroll inMathematics 11. Students may enroll in Mathematics 11A or they may complete anintermediate course, Introductory Mathematics 11, and then enroll in Mathematics 11. Thecourse outlines for Mathematics 10 and Mathematics 10A have approximately 30 percent of27their content in common. Thus, the achievement sections of the Grade 10 booklets weredivided into three parts. Part 1 consisted of 20 items which were to be answered by allstudents and it contained items which were common to the curriculums of Mathematics 10and Mathematics 10A. The 20 items in Part 2 were to be answered only by studentsenrolled in Mathematics 10A and the Mathematics 10 students answered the 20 items inPart 3.Different versions of the teacher questionnaire were also developed. There werethree forms for Grade 4, three for Grade 7 and two for Grade 10. The questionnaires weredivided into five sections. The Background Information and Implementation Informationsections were common to all versions whereas the Classroom Practices, Mathematics inSchool and Opportunity to Learn sections were not the same on all forms. Full details ofall of the forms are available in the Technical Report of the assessment (Robitaille, inpress).All students were allocated one hour of time for the assessment. A pilot study hadindicated that one hour should provide enough time for almost all students to answer thequestions in their booklets. There was no limitation on the amount of time used forcompletion of the teacher questionnaires.Content of the Assessment Materials The initial development of the items for the assessment forms was the responsibilityof the Contract Team which consisted of individuals from throughout the mathematicseducation community in British Columbia. The work of the Contract Team was reviewedby the Advisory Committee and the Review Panel whose members had been selected so asto provide a cross-section of opinions regarding the learning of mathematics. TheAdvisory Committee advised the Contract Team and reviewed all items before they werepresented to the Review Panel. The Review Panel then considered each of the items in the28context of the original objectives of the assessment . This development process wasestablished so as to ensure the content validity of the assessment instruments.The achievement component of the Mathematics Assessment student forms wasintended to reflect the content of the British Columbia mathematics curriculum for studentsfrom Grades 1 through 10. The achievement items on the Grade 4 forms were based onthe mathematics curriculum for Grades 1 through 4, the Grade 7 items were based on thecurriculum for Grades 5 through 7, and the Grade 10 items were based on the curriculumfor Grades 8, 9 and 10.The content of the achievement section was divided into six strands for Grades 4and 7. These strands were whole numbers, rational numbers, data analysis, geometry,measurement and algebra. For Grade 10, the whole numbers and rational numbers strandswere combined into one section called numbers and operations. The other four strandswere the same as those used in Grades 4 and 7. This information is summarized in Table2.Table 2Content Strands for Achievement ItemsGrade 4^Grade 7^Grade 10Whole NumbersRational NumbersAlgebraData AnalysisGeometryMeasurementWhole NumbersRational NumbersAlgebraData AnalysisGeometryMeasurementNumbers & OperationsAlgebraData AnalysisGeometryMeasurementAt each grade level, each of the strands listed above was divided into topics. Thesetopics were then separated into three cognitive behaviour levels: computation,comprehension and application/problem solving. This structure was used as the basis forthe development and selection of the achievement items to be used on the assessment.29Many of the individual achievement items were developed specifically for thisassessment. The mathematics curriculum in British Columbia has recently undergonerevision and the development of items was required for new topics which had beenintroduced into the curriculum. Other items were drawn from such sources as the 1985Provincial Mathematics Assessment, the National Assessment of Educational Progress inthe United States, and the Second International Mathematics Study.As noted previously, all of the achievement questions followed a multiple-choiceformat. Students selected a response from a range of five possible options. The fifthoption was always the statement "I don't know".A pool of achievement items was developed and pilot tested in October, 1989. TheGrade 4 material was administered to 46 Grade 5 classes, the Grade 7 material was given to42 Grade 8 classes, and the Grade 10 items were administered to 43 Grade 11 classes.Questions where not all of the response options were selected, or where more than 95percent or less than 10 percent of the students answered the item correctly, were eitherchanged or not used in the assessment. Items where the point-biserial correlation betweenthe correct answer and the total test score was less than 0.20 or less than the correspondingcorrelation between a distractor and the total test score were also altered or deleted.Information about students' perceptions of specific mathematics topics wasdetermined by asking participants to rate how important the topic was, how easy the topicwas, and how much the topic was liked. The Grade 10 forms referred to nine topics andtwelve topics were listed in the Grade 7 forms. Teachers were asked to rate the same topicsby indicating how important the topic was for the class, how easy it was to teach the topic,and how much they liked teaching the topic.Students and teachers responded to the above items on a five-point Likert scalewith a range of options from "not at all important" to "very important", "very difficult" to"very easy" and "dislike a lot" to "like a lot". This scale had been adapted from onedeveloped by the International Association for the Evaluation of Educational Achievement.30Sample SelectionThe sample used in this study was selected from the data collected by the 1990Mathematics Assessment. Authorization to use this data was obtained from the BritishColumbia Ministry of Education.Description of the Sample As the purpose of this study was to investigate students' attitudes towards geometryand data analysis, the participants in the sample consisted of all students enrolled inMathematics 7, 10, or 10A who answered the background questions in the assessmentforms relating to these concerns. While all forms contained items about students'perceptions of individual mathematics topics, not all forms asked students about geometryand data analysis. At the Grade 7 level, only students who received Form A were given theopportunity to state their views on these topics and in Grade 10 a similar opportunity wasgiven to the recipients of Forms C or D.Within every class which participated in the assessment, four different assessmentforms were distributed randomly to the students. The sample size of 9491 cases selected atthe Grade 7 level represents the number of students who received Form A and isapproximately 25 percent of the total enrollment in that grade. As the forms weredistributed randomly and there was almost a 100 percent participation rate for this grade,this sample is representative of the population of students enrolled in Grade 7 in BritishColumbia at that time.One of the items asked students who received the Grade 10 forms whichmathematics course they had been enrolled in during the 1989-1990 school year. A total of14,786 of the students who responded to Forms C or D indicated that they had beenenrolled in Mathematics 10 or 10A. Of these students, 10,907 had studied Mathematics 1031and 3879 had taken Mathematics 10A. The distribution of the participants by grade issummarized in Table 3.Table 3Distribution of Subjects by Course EnrollmentCourse Number of SubjectsMathematics 7 9491Mathematics 10 10907Mathematics 10A 3879The participation rate in the assessment at the Grade 10 level was approximately 84percent. It is not certain if the 16 percent who did not participate represented a randomselection of students or if they belonged to a particular sub-group of the population. As theGrade 10 forms were distributed randomly to the students, the sample cases arerepresentative of the population of students enrolled in Grade 10 mathematics courses whoparticipated in the assessment but they are not necessarily representative of all students whowere enrolled in Grade 10 at the time of the assessment.Data Collection InstrumentsIn order to assess students' perceptions of geometry and data analysis, studentresponses to the relevant Background Information items on the assessment forms wereanalyzed. The specific questions relating to geometry and data analysis which were askedof the Grade 10 students are displayed in Figure 1. The Grade 7 students' questions weresimilar in content, the only difference being that the headings "Geometry" and "Dataanalysis" were replaced with the titles "Learning geometry" and "Working with data andgraphs".32For each of the next three items, three answers are neededA) Tell how important you think the topic is.B) Tell how easy you think the topic is.C) Tell how much you like the topic.If you are not sure what a topic means, leave its three answersblank.GeometryA^B^Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lotData AnalysisA^B^Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant^easy likevery important^very easy^like a lotFigure 1. Grade 10 background items: Geometry and data analysis.Student achievement was evaluated based upon performance on the AchievementSurvey portion of the assessment. Those items which corresponded directly to thegeometry and data analysis strands of the mathematics curriculum were used to assessstudent performance in those areas. As Grade 7 Form A and Grade 10 Forms C and Dwere the only forms containing background items relating to geometry and data analysis,achievement assessment items were selected from those forms. A list of the item numbersused to evaluate achievement in geometry and data analysis at each grade level is given inTable 4. Table 4 also lists the intended learning outcome number from the mathematics33curriculum to which each item corresponds. The content of the geometry and data analysisquestions which were asked can be found in Appendix A.Table 4Strands and Item AssignmentsCourse^Strand^Form and^Intended LearningItem Number Outcome NumberMath 10^Geometry C: 9, 10, 11*, 1245, 46, 47, 48, 49,50, 51, 52, 60D: 9, 10, 11*, 12,45, 46, 47, 48, 49,50, 51, 528.52, 8.48, 10A.24, 10A.24,10.23, 10.26, 8.52, 9.15, 9.159.14, 10.19, 9.17, 8.549A.32, 8.51,10A.24, 10A.25,9.25, 8.56, 8.51, 9.14, 10.17,9A.37, 8.47, 10.20Data^C: 5*, 6, 7*,^10A.21, 10A.19, 9A.31,Analysis^8*, 43, 44* 9A.31, 8.38, 9.13D: 5*, 6, 7*,8*, 43,44*Math 10A Geometry C: 9, 10, 11*, 12,33*, 34, 35, 36D: 9, 10, 11*, 12,33*, 34, 35, 3610A.21, 10A.14, 9A.31,9A.31, 10A.19, 9.138.52, 8.48, 10A.24, 10A.24,9A.37,10A.29,10A.27,10A.329A.32, 8.51, 10A.24, 10A.25,9A.37,10A.26,10A.29, 10A.32Data C: 5*, 6, 7*, 8*, 10A.21, 10A.19, 9A.31, 9A.31Analysis 30, 31, 32 10A.19, 9A.24, 8.37D: 5*, 6, 7*, 8*, 10A.21, 10A.14, 9A.31, 9A.3130, 31, 32 8.40, 9A.24, 9A.24Math 7 Geometry A: 25, 26, 27, 7G33, 7G33, 7G5,28, 29, 30 7G23, 7G3, 7G47Data A: 19, 21, 22, 7D13, 7D12, 7D1,Analysis 23, 24 7D12, 7D1*items common to both forms34Data AnalysisThe mathematics assessment materials were sent to the British Columbia Ministryof Education for marking and coding. All responses to the background and achievementitems were coded. Responses to achievement items were also recorded as being eithercorrect or incorrect. Items to which there was no response were coded as blanks. Theresulting data file was used as the basis of the analysis for this study.Coding of Sample MaterialsThe items in the assessment which related to student perceptions of individualmathematics topics requested that students indicate their views on how important the topicwas, how easy the topic was, and how much the topic was liked. For the purposes of thisstudy, these three dimensions were considered to be separate components of attitude andthe data for each area analyzed independently.As indicated in Figure 1, there were five possible responses which students couldmake when answering the questions relating to geometry and data analysis. When the datawere coded for this study, these five options were replaced with just two categories.Student responses were recoded to indicate if the topic was important or not important,easy or difficult, or if it was liked or disliked. For example, responses which indicated thatgeometry was "not at all important" or "not important" were all coded under the heading"not important" while those characterizing it as being "important" or "very important" weregrouped together under the heading "important". All "undecided" responses were ignored.Similar coding was used to classify the "easy" and "like" dimensions of attitude.All responses to the achievement items were coded as being correct or incorrect. Ifthe question was not answered, it was considered to have been answered incorrectly.Achievement in geometry and data analysis was determined by calculating the percent of the35responses which were correct in each of these strands. These results for each strand werethen divided into five groups. The range of scores in each group is listed in Table 5.Table 5Classification of Achievement ScoresGroup n = achievement in percent1 0% < n <20%2 20% < n <40%3 40% < n 5.60%4 60% <n <80%5 80% < n < 100%Chi-Square Analysis The relationship between student attitudes towards geometry and data analysis andachievement in those areas was investigated through the use of chi-square analysis. Inorder to conduct this investigation, matrices which linked attitude and achievement weredeveloped. The scale on the vertical axis of these matrices consisted of one of thefollowing pairs of values: important/not important, easy/difficult, or like/dislike. Thehorizontal scale consisted of the five achievement groupings. Thus, six 2 X 5 matriceswere developed, one for each of the three attitude dimensions associated with geometry anddata analysis. The statistical program SPSS/PC+, Version 3.1, was used to construct thesematrices and calculate the chi-square value. The results were then analyzed.The relationship between overall mathematical ability and attitude was also studiedthrough the use of this procedure. Mathematical ability was determined by the results onthe entire achievement part of the forms. In order to investigate the relationship amongattitudes, achievement and the grade of the student, data from both Grades 7 and 10 wasanalyzed. Students in Grade 10 were also classified according to which Grade 10mathematics course they had taken. This collection of data was then examined to determineif there was an association among streaming, attitudes and achievement.36As approximately 16 percent of the Grade 10 student population in BritishColumbia did not participate in the mathematics assessment, the Grade 10 data wereinvestigated further in order to study the possible impact of this situation. It is not knownif the non-participants were a representative sample of the entire student population or ifthey represented a particular sub-group. Individual cells in the matrices were adjusted toreflect this latter "worst-case" scenario and the chi-square value was then calculated. Theseresults were then analyzed.SummaryThis chapter has described the 1990 British Columbia Assessment materials fromwhich the sample data for this study was drawn. The sampling procedure and the methodsof data analysis were also discussed. The results of the study are reported in the nextchapter. In the final chapter, the conclusions based on these findings and the implicationsof the results are presented.37CHAPTER 4FINDINGSDiscussed in this chapter are the findings of the study. A description of thevariables is given as well as the results of the chi-square analysis of the Grade 10 data foreach of the components of attitude which were investigated. Possible effects of overallachievement in mathematics, grade level, and streaming are also explored. The chapterconcludes with a discussion of the impact on the research of the absence of 16% of theGrade 10 students from participation in the study.Description of the VariablesTwo variables relating to the learning of mathematics were investigated in thisstudy. These were attitude and achievement. The relationship between these variableswithin the domains of geometry and data analysis was the central focus of the research.Student AttitudesAs indicated in the previous chapter, the 1990 British Columbia MathematicsAssessment requested that students indicate their views regarding certain topics inmathematics. Students were asked to respond in three separate categories. They were tostate how important the topic was, how easy the topic was, and how much the topic wasliked. For the purpose of this study, these three dimensions of attitude were analyzedindependently of each other. Attitude was not defined as a single variable. Therefore, thestatistical analysis of students' attitudes involved working with the data regarding theirviews on the importance, difficulty, and likeability of geometry and data analysis.Students in Grade 7 who received Form A were asked about their perceptions ofgeometry and data analysis. The number of students who completed this form was 9491.Forms C and D were the Grade 10 assessment materials which contained questionsregarding geometry and data analysis and 15,602 of these forms were completed. Only38those students who indicated that they were enrolled in Mathematics 10 or Mathematics10A were included in the sample.Not all students who received these forms responded to every question. Studentswho did not respond to a question were not included in the statistical analysis for thatparticular item but their responses to other items were included in other analyses.Similarly, students who selected the "undecided" option when responding to an attitudeitem were also omitted from the sample. For each item, the remaining responses were thengrouped into two categories. The topics were consided to be important or not important,easy or difficult, and liked or disliked. The level of response to each of the attitude items islisted below in Table 1.Table 1Student Responses to Attitude ItemsformscompletedGrade 7^9491itemanswered"undecided"option selectedsamplesizeGeometry: importance 9315 1759 7556Geometry: difficulty 9206 1812 7394Geometry: likeability 9215 1735 7480Data Analysis: importance 9235 1886 7349Data Analysis: difficulty 9131 2268 6863Data Analysis: likeability 9099 2187 6912Mathematics 10 & 10A 14786Geometry: importance 14502 3824 10678Geometry: difficulty 14443 2890 11553Geometry: likeability 14461 3149 11312Data Analysis: importance 13718 5018 8700Data Analysis: difficulty 13667 6821 6846Data Analysis: likeability 13657 7059 659839In Grade 10, the response rate for the data analysis items was lower than for thegeometry items. This difference may have been due to the fact that, as reported in thetechnical report on the mathematics assessment (Robitaille, in press), at the time of theassessment a large number of teachers of Grade 10 mathematics had not yet taught thematerial in the data analysis strand to their students. Students may not have been familiarenough with the material to enable them to respond. Also, some students may not havebeen certain as to what was meant by the phrase "data analysis". The data analysis strandhas been introduced recently into the British Columbia mathematics curriculum. Teachersand students may not have put the term "data analysis" into frequent use. These reasonsmay also explain why a large number of the students chose the "undecided" option whenresponding to the data analysis items.On the Grade 7 form, the question on data analysis was worded differently than onthe Grade 10 forms and it gave more information as to the content of the material inquestion. The Grade 7 students were asked for there perceptions on "working with dataand graphs". This difference may partially explain why their was a higher response rate tothe data analysis questions at the Grade 7 level than at the Grade 10 level.Student AchievementIn this study, student achievement values were calculated for both the geometry anddata analysis strands and for the entire achievement portion of the assessment. Studentachievement in geometry and data analysis was determined by the number of items whichwere correct in each of these domains of the assessment. Blank responses to an item wereconsidered to be incorrect. The total number correct was then converted to a percent. Thepercents were then divided into five groups each with a range of 20%. Table 5 in Chapter4 summarizes the percentage values assigned to each of these groups.A list of the individual items which were used to assess achievement in dataanalysis and geometry can be found in Table 4 of Chapter 3. Listed in Table 2 below is thenumber of items which were used to determine achievement in each domain. As indicatedin the table, the number of items in the geometry and data analysis strands was not the same40for Mathematics 10 as for Mathematics 10A. This situation reflects the fact that there is adifferent emphasis placed on data analysis and geometry in the curriculums of each of thesecourses.Table 2Number of Items Used to Assess AchievementData Analysis GeometryMathematics 7: Form A 5 6Mathematics 10: Form C 6 13Form D 6 12Mathematics 10A: Form C 7 8Form D 7 8The same procedure was used to calculate overall student achievement inmathematics. All students were asked 40 questions in the achievement portion of theassessment. The score, which was determined by the number of items answered correctly,was then converted to a percent.Data Analysis and GeometryAs reported earlier, the items which were used to assess achievement in the areas ofdata analysis and geometry were selected from the achievement section of the 1990 BritishColumbia Mathematics Assessment. These items were selected for inclusion in theassessment based upon their correspondence to the items in the mathematics curriculumguide of the province of British Columbia. Thus, for the purposes of this study, thedefinitions of the terms data analysis and geometry correspond to the definitions implied inthe use of these terms in the British Columbia mathematics curriculum guides for Grade 7and Grade 10.41Grade 10—Geometry and Data AnalysisThe analysis of the data involved the use of chi-square to determine if there was asignificant relationship between achievement in mathematics and attitudes toward certainmathematical topics. These analyses were concerned with three different components ofattitude and were performed on data from two grade levels. Presented below is adiscussion of the findings concerning the Grade 10 geometry and data analysis material.Grade 10—Geometry The analysis of the Grade 10 data was organized in the following manner. For eachof the three dimensions of attitude—importance, difficulty, and likeability—a 2 X 5 matrixwas designed. One axis of each matrix represented attitude and its scale was divided intotwo parts: not important/important, difficult/easy, or dislike/like. The other axisrepresented achievement in the geometry domain and its scale ranged in value from one tofive. These values corresponded to the five groupings which had been established for theachievement scores. A chi-square analysis was then performed on each of these matrices.The results of the analyses of the geometry data are given in Tables 3, 4, and 5 on thefollowing pages.As the chi-square values for all three matrices were significant at 0.0000, it can beconcluded that each of the three components of attitude towards geometry is related toachievement in geometry. However, given the large sample sizes involved in this study,these results do not necessarily indicate that a strong relationship exists.Of more interest are the trends that are revealed in the individual cells of thematrices. All three matrices indicate that a student who does well is more likely to considergeometry to be important, easy, and likeable than one who does poorly. In all three areas,the percentage of students who responded positively towards geometry rose steadily as theachievement levels increased. This situation is reflected in the relationship between the42Table 3Grade 10: Chi-Square AnalysisImportance of Geometry and Achievement in GeometryAchievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 6.4% 11.5% 8.9% 6.5% 2.0%ExpectedValue 4.7% 9.9% 9.1% 8.1% 3.6%Important 6.9% 16.5% 16.8% 16.4% 8.1%ExpectedValue 8.6% 18.1% 16.6% 14.8% 6.5%43Row Total35.3%64.7%N=10678Total^13.3%^28.0%^25.7%^22.9%^10.1%^100%Chi-square = 309.9; D.F. = 4; significance level = 0.0000Table 4Grade 10: Chi-Square AnalysisDifficulty of Geometry and Achievement in GeometryAchievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 7.8% 14.0% 9.7% 6.0% 1.7%ExpectedValue 5.3% 10.9% 10.0% 9.0% 4.0%Easy 5.7% 13.8% 15.9% 17.0% 8.5%ExpectedValue 8.2% 16.9% 15.6% 14.0% 6.2%Row Total39.2%60.8%N=11553Total^13.5%^27.8%^25.6%^23.0%^10.1%^100%Chi-square = 832.0; D.F. = 4; significance level = 0.00000 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 8.9% 15.8% 11.3% 7.6% 2.4%ExpectedValue 6.3% 12.9% 11.7% 10.5% 4.5%Like 5.0% 12.2% 14.2% 15.3% 7.3%ExpectedValue 7.5% 15.2% 13.8% 12.3% 5.3%Lkeab1tyRow Total46.0%54.0%Table 5Grade 10: Chi-Square AnalysisLikeability of Geometry and Achievement in GeometryAchievement (percent)44N=11312Total 13.8% 28.1% 25.5% 22.9% 9.8% 100%Chi-square = 719.3; D.F. = 4; significance level = 0.0000 actual numbers in each of the cells as compared to the expected values for each cell. In thecells which indicated a negative attitude towards geometry, the number of studentsachieving a score of less than or equal to 40% was greater than the expected value and thenumber achieving greater than 40% was less than the expected value. The reverse situationoccurred with students who had a positive attitude towards geometry. Fewer students thanthe expected value did poorly and the number of students who achieved greater than 40%was greater than the expected value.Having a positive attitude towards geometry did not necessarily mean that a studentobtained good results in the geometry portion of the assessment. For example, 65% of thestudents indicated that geometry was important. Of these students, only 33% had a resultof more than 60% on the geometry achievement section. However, those students whohad a positive attitude did tend to achieve better results than those who had the oppositeview. For example, only 24% of the students who stated that geometry was not importantachieved more than 60% on this section whereas 38% of those who thought it wasimportant obtained a score greater than 60%.Fewer students thought geometry was easy than those who considered it to beimportant, and even fewer still liked the topic. Only 54% of the students indicated that theyliked geometry whereas 61% stated that it was easy and 65% considered it to be important.While a majority of the students who did poorly found geometry to be difficult and dislikedit, more than half of the students who did poorly considered geometry to be important. Atall five achievement levels, more students considered geometry to be important thanconsidered it to be not important.Grade 10—Data Analysis The same method of analysis that was applied to the geometry strand was used toinvestigate the data analysis strand. Students' attitudes towards the importance, difficulty,and likeability of data analysis were examined to determine how they related to achievementin this area. The results of this analysis are given in Tables 6, 7, and 8 on the followingpages.Similar results were found for the data analysis strand as were found for thegeometry strand. The chi-square values were significant at the 0.0000 level in all threecases which suggests that achievement in data analysis is related to each of the componentsof attitude examined in the matrices. The number of students who achieved poor resultsand who also considered data analysis to be unimportant or difficult, or who disliked itexceeded the expected value for those cells. The number of students who did well and whohad negative views towards data analysis was less than the expected value. Conversely,less than the expected number of students with positive attitudes towards data analysis didpoorly and more than the expected value did well.45Table 6Grade 10: Chi-Square AnalysisImportance of Data Analysis and Achievement in Data Analysis46Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 3.9% 4.3% 5.0% 2.3% 1.5%ExpectedValue 2.5% 3.5% 5.2% 3.5% 2.4%Important 10.6% 16.0% 25.5% 18.2% 12.7%ExpectedValue 12.0% 16.9% 2.5% 17.0% 11.7%Row Total17.0%83.0%N=8700Total^14.5%^20.3%^30.5%^20.5%^14.1%^100%Chi-square = 191.4; D.F. = 4; significance level = 0.0000Table 7Grade 10: Chi-Square AnalysisDifficulty of Data Analysis and Achievement in Data AnalysisAchievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 8.3% 9.0% 11.9% 6.1% 3.0%ExpectedValue 5.9% 7.4% 11.7% 7.7% 5.5%Easy 7.2% 10.3% 18.7% 14.1% 11.4%ExpectedValue 9.6% 11.9% 18.9% 12.5% 8.9%Row Total38.3%61.7%N=6846Total^15.5%^19.3%^30.6%^20.2%^14.4%^100%Chi-square = 305.5; D.F. = 4; significance level = 0.00000 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 9.1% 9.9% 13.5% 7.4% 4.9%ExpectedValue 7.1% 8.8% 13.9% 8.8% 6.3%Like 6.7% 9.7% 17.4% 12.2% 9.1%ExpectedValue 8.7% 10.8% 17.0% 10.8% 7.7%Likeabi1itYRow Total44.8%55.2%Table 8Grade 10: Chi-Square AnalysisLikeability of Data Analysis and Achievement in Data AnalysisAchievement (percent)47N=6598Total^15.8%^19.7%^30.9%^19.6%^14.0%^100%Chi-square = 148.1; D.F. = 4; significance level = 0.0000 The percentage of students who liked data analysis or considered it to be easy waswithin one percentage point of the percentage of students who indicated those views aboutgeometry. Likewise, the distribution of the percentages of students who held these viewsover the five achievement levels was similar for both curriculum domains.However, more students indicated that data analysis was important than stated thatgeometry was important. Approximately 65% of the students responded that geometry wasimportant whereas 83% stated that data analysis was important. This high rate offavourable responses may explain why the gap between the expected value and the actualvalue of the number of students who considered data analysis to be important and whoachieved high scores was only 0.9%. However, the difference between expected value andactual value for low achievers who considered data analysis to be unimportant (37%)matched the patterns found for the other data.0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 3.2% 13.5% 12.0% 5.5% 1.0%ExpectedValue 2.3% 11.2% 12.5% 7.3% 2.0%Important 3.4% 18.2% 23.4% 15.2% 4.6%ExpectedValue 4.3% 20.5% 22.9% 13.4% 3.7%ImP0rtanceRow Total35.3%64.7%Overall Achievement in MathematicsIn order to gain an understanding of the relationship between students' overallmathematical achievement and their attitudes towards specific topics in the mathematicscurriculum, the Grade 10 assessment data were explored further. The results of this workwith geometry and data analysis are given below.Overall Achievement and GeometryThree matrices, similar to those used to study achievement in geometry, weredesigned to investigate the relationship between overall achievement in mathematics andattitudes towards geometry. The findings concerning these data are given in Tables 9, 10,and 11 below.Table 9Grade 10: Chi-Square AnalysisImportance of Geometry and Overall Achievement in MathematicsAchievement (percent)N=10678Total^6.6%^31.7%^35.4%^20.7%^5.6%^100%Chi-square = 289.7; D.F. = 4; significance level = 0.000048Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 4.1% 16.0% 13.5% 4.7% 0.8%ExpectedValue 2.6% 12.5% 13.8% 8.1% 2.3%Easy 2.5% 15.8% 21.6% 16.0% 5.0%ExpectedValue 4.0% 19.3% 21.3% 12.6% 3.5%Row Total39.2%60.8%Table 10Grade 10: Chi-Square AnalysisDifficulty of Geometry and Overall Achievement in MathematicsN=11553Total^6.6%^31.8%^35.1%^20.7%^5.8%^100%Chi-square = 806.5; D.F. = 4; significance level = 0.0000Table 11Grade 10: Chi-Square AnalysisLikeability of Geometry and Overall Achievement in Mathematics49Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 4.5% 18.3% 15.3% 6.5% 1.2%ExpectedValue 3.1% 15.0% 15.9% 9.5% 2.5%Like 2.1% 14.3% 19.3% 14.2% 4.1%ExpectedValue 3.6% 17.6% 18.7% 11.2% 2.9%Row Total46.0%54.0%N=11312Total^6.7%^32.7%^34.6%^20.7%^5.3%^100%Chi-square = 631.1; D.F. = 4; significance level = 0.0000As was the case with achievement in geometry, the chi-square values for thematrices dealing with the relationship between achievement in mathematics and attitudestowards geometry were significant at the 0.0000 level. These findings suggests that thereis a significant association between students' views as to the importance, difficulty, andlikeability of geometry and their overall performance in mathematics.However, the proportion of students scoring in each of the five achievementcategories was not the same for both achievement in geometry and achievement inmathematics. The distribution of geometry scores was more consistent across the fivecategories than the distribution of overall mathematics scores. In the geometry portion ofthe assessment, approximately 10% of the students scored higher than 80% and about 13%scored less than 20% whereas only 6% of the students achieved total assessment scores ofabove 80% and approximately 7% obtained scores of less than 20%. In general, studentsscored higher on the geometry portion of the assessment than they did on the overallassessment. Approximately 27% of the students obtained a score of 60% or better on theassessment whereas approximately 33% of the students achieved in this range of scores onthe geometry unit.In contrast, the percentage of students in each achievement category whoconsidered geometry to be important, easy, or likeable was similar for performance both ingeometry and in mathematics. A comparison of the matrices which examined performancein geometry with those which examined performance in mathematics reveals that thedifference between the percentage of students at any one achievement level who consideredgeometry favourably was never more than 4%. In eleven out of fifteen cases, thisdifference was 2% or less. For example, geometry was liked by approximately 75% of thestudents who obtained high scores in this topic and it was also liked by about 77% of thestudents who scored well on the overall assessment. It was disliked by approximately 64%of the students who did poorly on the geometry unit and by approximately 68% of thestudents who obtained low scores on the assessment.500 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 2.1% 7.2% 5.2% 2.0% 0.5%ExpectedValue 1.0% 5.1% 6.0% 3.8% 1.1%Important 4.0% 22.7% 29.9% 20.3% 6.1%ExpectedValue 5.1% 24.8% 29.2% 18.5% 5.4%ImP0rtanceRow Total17.0%83.0%Overall Achievement and Data AnalysisThe findings concerning overall achievement in mathematics and attitudes towardsdata analysis were determined in a manner similar to those regarding geometry. Threematrices were designed to examine these relationships. The results of this analysis can befound in Tables 12 through 14.The chi-square values for each of these matrices were significant at the 0.0000level. These results parallel the findings regarding attitudes towards data analysis andachievement in this domain and they suggest that achievement in mathematics and attitudestowards data analysis are related.Table 12Grade 10: Chi-Square AnalysisImportance of Data Analysis and Overall Achievement in MathematicsAchievement (percent)51N=8700Total^6.1%^29.9%^35.1%^22.3%^6.5%^100%Chi-square =352.9; D.F. = 4; significance level = 0.0000Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 4.2% 15.1% 12.5% 5.6% 0.8%ExpectedValue 2.6% 11.8% 12.8% 8.4% 2.6%Easy 2.7% 15.8% 21.0% 16.3% 6.0%ExpectedValue 3.3% 15.0% 16.3% 10.6% 3.3%Row Total38.3%61.7%N=6846Table 13Grade 10: Chi-Square AnalysisDifficulty of Data Analysis and Overall Achievement in MathematicsTotal^4.3%^19.1%^20.7%^13.5%^4.2%^100%Chi-square = 451.9; D.F. = 4; signiicance level = 0.0000Table 14Grade 10: Chi-Square AnalysisLikeability of Data Analysis and Overall Achievement in Mathematics52Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 4.4% 16.8% 14.5% 7.1% 2.0%ExpectedValue 3.2% 14.1% 15.3% 9.5% 2.7%Like 2.7% 14.6% 19.6% 14.2% 4.1%ExpectedValue 3.9% 17.3% 18.9% 11.8% 3.4%Row Total44.8%55.2%N=6598Total^7.1%^31.3%^34.2%^21.2%^6.1%^100%Chi-square = 227.1; D.F. = 4; significance level = 0.0000The data analysis results were similar to the geometry results in that the achievementscores for data analysis were more consistent across the five achievement levels than werethe scores for the overall assessment. Approximately 14% of the students achieved morethan 80% on the data analysis section of the assessment while only 7% of the studentsachieved in this range on the overall assessment. Likewise, 7% of the students scoredbelow 20% on the assessment whereas 15% of the students had low scores on the dataanalysis unit. Also, more students scored 60% or higher on the data analysis unit than onthe overall assessment. The percentage of students obtaining marks in this range was 35%and 29% respectively.However, unlike the findings for geometry, the percentages of students at eachachievement level who viewed data analysis favourably were not the same for the matriceswhich dealt with performance in mathematics as they were for the matrices which wereconcerned with performance only in the area of data analysis. For these two sets ofmatrices, the difference between the percentages of students who had positive attitudestowards data analysis was at times as high as 9%. For example, 53% of the students whoobtained scores less than 20% on the data analysis unit considered data analysis to bedifficult whereas 61% of the students who had low overall scores on the assessment heldthese views. Data analysis was considered easy by 79% of the students who scored wellon the data analysis unit while 88% of the students who obtained high scores on theassessment considered data analysis easy.Grade Levels—Grade 7 and Grade 10The assessment materials for students in Grades 7 and 10 were examined in orderto determine if the grade level of the student was a factor in the relationship betweenattitudes towards geometry and data analysis and achievement in those areas. The Grade 7data was analyzed in the same manner as the Grade 10 data. For both geometry and dataanalysis, matrices which dealt with each of the three components of attitude were produced.530 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 2.1% 2.4% 3.2% 2.7% 1.7%ExpectedValue 1.8% 2.4% 3.0% 2.7% 2.3%Important 12.5% 17.3% 21.8% 19.4% 16.8%ExpectedValue 12.8% 17.3% 22.0% 19.4% 16.3%ImP0rtanceRow Total12.1%87.9%Grade 7-GeometryThe results of the work with the Grade 7 geometry data are reported in Tables 15through 17 and the data analysis results are presented in the next section. Each of the tablesgiven below illustrates the findings for the chi-square analysis of the relationship betweenachievement in geometry and a particular component of attitude.The matrices which were concerned with how much students liked geometry orwith how difficult they found it contained findings similar to the corresponding matrices forstudents in Grade 10. The chi-square values for each of these matrices were significant atTable 15Grade 7: Chi-Square AnalysisImportance of Geometry and Achievement in GeometryAchievement (percent)N=7556Total^14.6%^19.7%^25.0%^22.1%^18.6%^100%Chi-square = 17.8; D.F. = 4; significance level = 0.001354Table 16Grade 7: Chi-Square AnalysisDifficulty of Geometry and Achievement in Geometry55Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 5.5% 6.3% 6.9% 4.4% 2.5%ExpectedValue 3.6% 4.8% 6.5% 5.7% 4.9%Easy 8.6% 12.7% 18.4% 17.9% 16.8%ExpectedValue 10.5% 14.1% 18.8% 16.6% 14.4%Row Total25.6%74.4%N=7394Total^14.1%^19.0%^25.3%^22.3%^19.3%^100%Chi-square = 297.4; D.F. = 4; significance level = 0.0000Table 17Grade 7: Chi-Square AnalysisLikeability of Geometry and Achievement in GeometryAchievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 5.6% 6.6% 7.6% 6.1% 4.2%ExpectedValue 4.4% 5.9% 7.5% 6.7% 5.6%Like 9.2% 12.9% 17.2% 16.1% 14.5%ExpectedValue 10.3% 13.6% 17.4% 15.5% 13.1%Row Total30.1%69.9%N=7480Total^14.7%^19.5%^24.8%^22.2%^18.7%^100%Chi-square = 86.1; D.F. = 4; significance level = 0.0000the 0.0000 level which suggests that, for students in Grade 7, there is a significantrelationship between these two perceptions of geometry and achievement in geometry. Aswas the case for the Grade 10 students, the number of Grade 7 students who dislikedgeometry or found it difficult and who did poorly on the achievement section was greaterthan the expected value. Also, the number of students who liked geometry, or who foundit easy, and who achieved good scores on the geometry unit was greater than the expectedvalue.The relationship between the Grade 7 students' views of the importance ofgeometry and their performance in geometry, however, was different than that for Grade10 students. The chi-square value for the matrix dealing with this relationship had asignificance level of 0.0013. In all five achievement categories in this matrix, a minimumof 85% of the students considered geometry to be important with approximately 88% of thetotal Grade 7 sample holding this view. There was only a 5 point difference between thepercentage of students in the highest achievement category who considered geometry to beimportant and the percentage of students in the lowest category who indicated the sameview. In contrast, 65% of the Grade 10 students stated that geometry was important andthere was a 28 point spread between the percentage of low-achieving and high-achievingstudents who held this view.An examination of the matrices dealing with the other attitude components indicatessimilar results. Of those Grade 10 students who scored in the highest achievementcategory, approximately 83% considered geometry to be easy and about 75% indicated thatthey liked it. These values are 41 and 39 percentage points, respectively, higher than thepercentage of students in the lowest achievement categories who found geometry easy orwho liked it. The difference between the high and low scoring Grade 7 students in each ofthese categories, however, was only 26 and 16 percentage points, respectively.Overall, students in Grade 7 appear to view geometry more favourably and toachieve higher scores on the geometry unit than students in Grade 10 . As previously560 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 2.3% 2.3% 2.7% 1.8% 1.0%ExpectedValue 1.5% 2.0% 2.5% 2.4% 1.6%Important 12.9% 17.9% 22.5% 22.0% 14.7%ExpectedValue 13.7% 18.2% 22.6% 21.4% 14.1%Imp0rtanceRow Total10.0%90.0%noted, a greater percentage of Grade 7 than Grade 10 students considered geometry to beimportant. Likewise, about 74% of the Grade 7 students found geometry to be easy and70% liked it whereas the corresponding percentages for Grade 10 were 61% and 54%.Also, approximately 41% of the Grade 7 students obtained scores of 60% or more on thegeometry unit as compared to the 33% of Grade 10 students who achieved scores in thisrange.Grade 7-Data Analysis The data analysis strand of the assessment was examined for Grade 7 students inthe same manner as the other material. Tables 18 through 20 illustrate the findings of thiswork.Table 18Grade 7: Chi-Square AnalysisImportance of Data Analysis and Achievement in Data AnalysisAchievement (percent)57N=7349Total^15.2%^20.2%^25.1%^23.8%^15.7%^100%Chi-square = 69.8; D.F. = 4; significance level = 0.0000Table 19Grade 7: Chi-Square AnalysisDifficulty of Data Analysis and Achievement in Data Analysis58Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 6.1% 5.3% 5.4% 4.0% 1.6%ExpectedValue 3.5% 4.5% 5.6% 5.4% 3.5%Easy 9.3% 14.8% 19.6% 19.8% 14.1%ExpectedValue 11.9% 15.6% 19.4% 18.5% 12.1%Row Total22.5%77.5%N=6863Total^15.4%^20.1%^25.0%^23.8%^15.7%^100%Chi-square = 315.3; D.F. = 4; significance level = 0.0000Table 20Grade 7: Chi-Square AnalysisLikeability of Data Analysis and Achievement in Data AnalysisAchievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 5.4% 5.2% 5.7% 5.1% 3.1%ExpectedValue 3.9% 4.9% 6.1% 5.8% 3.8%Like 10.5% 14.9% 19.3% 18.6% 12.2%ExpectedValue 12.0% 15.2% 18.8% 17.8% 11.6%Row Total24.6%75.4%N=6912Total^16.0%^20.1%^24.9%^23.6%^15.4%^100%Chi-square = 76.6; D.F. = 4; significance level = 0.0000The Grade 7 data analysis results were similar to the Grade 10 findings in that allthree matrices had a chi-square value which was significant at the 0.0000 level. Thus,there appears to be a relationship between the attitudes of Grade 7 students towards dataanalysis and their achievement in this area. The Grade 7 results also parallelled the Grade10 findings in that there were more low achievers who had negative views towards dataanalysis than the expected value and there were more high achievers with positive viewsthan expected.As was the case with the geometry results, more Grade 7 students held favourableviews towards data analysis and achieved better results than did Grade 10 students. Also,the percentage of Grade 7 students who liked data analysis, found it easy, and considered itimportant was more consistent across the five achievement levels than it was for the Grade10 students. There was a greater gap between the percentage of high and low achievers inGrade 10 who held favourable views towards data analysis than between the high and lowachievers in Grade 7.Streaming—Mathematics 10 and Mathematics 10AThis study also investigated if the relationship between achievement and attitudetowards data analysis and geometry was different for students taking different mathematicscourses. As reported earlier, the students in Grade 10 who participated in this study wereenrolled in two different courses—Mathematics 10 and Mathematics 10A. These twocourses share approximately 30% of their curriculum content in common. Mathematics 10is the Grade 10 mathematics course which is a requirement for acceptance into university.Mathematics 10A is the mathematics course which is often chosen by students who do notintend to pursue post-secondary school studies. Data concerning these two groups ofstudents were used as the basis for this investigation.59Mathematics 10 and Mathematics 10A—GeometryThe data pertaining to students in Grade 10 were separated into two groups—Mathematics 10 and Mathematics 10A—and an analysis of the material, similar to thatundertaken for all of the Grade 10 data, was done. Tables 21 through 23 on the followingpages contain the findings of the analysis of the Mathematics 10 and Mathematics 10A dataconcerning geometry.For Mathematics 10, all three matrices linking achievement in geometry to attitudetowards geometry had chi-square values with levels of significance of 0.0000. ForMathematics 10A, the matrices concerned with the difficulty and likeability of learninggeometry were also significant at 0.0000. The matrix concerned with the importance oflearning geometry had a significance level of 0.0001. Thus, it seems that, for studentsenrolled in either Mathematics 10 or Mathematics 10A, achievement in geometry is relatedto each of the three components of attitude.The nature of this relationship between performance and attitude appears to besimilar, though not identical, for students enrolled in either course. Overall, students in theMathematics 10A classes had less favourable attitudes towards geometry than students inthe Mathematics 10 classes. They also achieved poorer results on the geometry unit. Only9% of the Mathematics 10A students had a geometry score of higher than 60% whereasapproximately 41% of the Mathematics 10 students scored in this range.The number of students in Mathematics 10 and Mathematics 10A who achieved lessthan or equal to 20% on the geometry section and who also had negative views towards thesubject was greater than the expected value for this cell in the matrices. Likewise, for bothcourses, the number of students who had favourable views concerning geometry and whoachieved scores higher than 80% was greater than the expected value. However, therelationship between the number of students performing in the 40% to 60% range and theexpected value for these cells was not the same for Mathematics 10 and Mathematics 10A.60Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 4.1% 9.1% 10.0% 7.7% 2.6%ExpectedValue 2.8% 7.2% 9.8% 9.3% 4.4%Important 4.4% 12.3% 19.2% 20.0% 10.6%ExpectedValue 5.6% 14.3% 19.4% 18.4% 8.8%Row Total33.5%66.5%Table 21Mathematics 10 & 10A: Chi-Square AnalysisImportance of Geometry and Achievement in GeometryMathematics 10N=8025Total^8.5%^21.4%^29.2%^27.7%^13.2%^100%Chi-square = 253.7; D.F. = 4; significance level = 0.0000Mathematics 10A61Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 13.3% 18.8% 5.5% 2.8% 0.2%ExpectedValue 11.4% 19.5% 6.1% 3.3% 0.3%Important 14.7% 29.1% 9.6% 5.4% 0.6%ExpectedValue 16.7% 28.5% 9.0% 4.8% 0.4%Row Total40.6%59.4%N=2653Total^28.0%^47.9%^15.1%^8.1%^0.8%^100%Chi-square = 24.9; D.F. = 4; significance level = 0.0001Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 5.0% 10.9% 11.1% 7.2% 2.2%ExpectedValue 3.1% 7.8% 10.5% 10.1% 4.8%Easy 3.5% 10.6% 17.8% 20.7% 11.0%ExpectedValue 5.4% 13.7% 18.5% 17.8% 8.4%Row Total36.3%63.7%Table 22Mathematics 10 & 10A: Chi-Square AnalysisDifficulty of Geometry and Achievement in GeometryMathematics 10N=8734Total^8.5%^21.5%^29.0%^27.9%^13.2%^100%Chi-square = 649.3; D.F. = 4; significance level = 0.0000Mathematics 10A62Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 16.4% 23.7% 5.3% 2.6% 0.2%ExpectedValue 13.9% 22.8% 7.3% 3.8% 0.3%Easy 12.5% 23.8% 9.8% 5.3% 0.5%ExpectedValue 15.0% 24.6% 8.0% 4.0% 0.4%Row Total48.1%51.9%N=2819Total^28.9%^47.5%^15.1%^7.8%^0.7%^100%Chi-square = 80.1; D.F. = 4; significance level = 0.0000Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 5.4% 12.3% 12.9% 9.0% 3.1%ExpectedValue 3.6% 9.4% 12.4% 11.9% 5.4%Like 3.1% 9.7% 16.0% 18.8% 9.6%ExpectedValue 4.9% 12.6% 16.5% 15.9% 7.3%Row Total42.7%57.3%Table 23Mathematics 10 & 10A: Chi-Square AnalysisLikeability of Geometry and Achievement in GeometryMathematics 10N=8501Total^8.5%^22.0%^23.9%^27.8%^12.7%^100%Chi-square = 508.8; D.F. = 4; significance level = 0.0000Mathematics 10A63Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 19.3% 26.5% 6.5% 3.2% 0.3%ExpectedValue 16.6% 25.9% 8.5% 4.4% 0.4%Like 10.5% 19.9% 8.8% 4.7% 0.5%ExpectedValue 13.2% 20.5% 6.8% 3.5% 0.3%Row Total55.8%44.2%N=2811Total^29.3%^46.4%^15.3%^7.8%^0.7%^100%Chi-square = 82.0; D.F. = 4; significance level = 0.0000There were fewer students than expected in Mathematics 10 who had favourable views ofgeometry and who were also achieving in the 40% to 60% range whereas in Mathematics10A the reverse situation occurred. There were a greater number of students than expectedachieving scores in this range.In general, for students in either course, as the level of achievement increased sotoo did the percentage of students who held favourable views regarding geometry.Students in Mathematics 10 and Mathematics 10A who obtained scores of less than orequal to 20% seem to share comparable views as to the importance, difficulty, andlikeability of geometry. For both courses, approximately 52% of the students in thiscategory stated that geometry was important. Approximately 37% of these Mathematics 10students liked geometry which was only 2% more than the percentage of Mathematics 10Astudents who liked it. There was also only 2 points difference between the percentage oflow-achieving Mathematics 10A students who found geometry easy as compared to thepercentage of low-achieving Mathematics 10 students who expressed this attitude.As the performance level of the students improved, however, greater differences inthe views of the students appeared. Among students who achieved scores higher than60%, a larger percentage of the Mathematics 10 students than the Mathematics 10Astudents considered geometry favourably. There was an 8—point to 10—point differencebetween the percentage of Mathematics 10 students who stated that geometry wasimportant, easy, or likeable as compared to the percentage of Mathematics 10A studentswho held these views.Mathematics 10 and Mathematics 10A Data Analysis All six matrices concerning Mathematics 10 and Mathematics 10A students and theirwork with data analysis had chi-square values which were significant at the 0.0000 level.Once again it can be concluded that, for students enrolled in Mathematics 10 orMathematics 10A, there is a relationship between performance in data analysis and attitudes64towards data analysis. The results of the analysis of this data can be found in Tables 24through 26 on the following pages.As was the case with geometry, the Mathematics 10A students considered dataanalysis to be less important, more difficult, and less likeable than did the Mathematics 10students. The students in the Mathematics 10A courses also achieved lower scores on thedata analysis achievement section than did the students in Mathematics 10. Approximately43% of the Mathematics 10 students scored over 60% on this unit whereas only 11% of theMathematics 10A students scored in this range. In general, for both courses, the higher theachievement level, the greater the percentage of students who viewed data analysisfavourably.However, data analysis results were unlike the geometry results in that at allachievement levels there were differences between the percentages of students who vieweddata analysis positively. At all five levels, a greater percentage of Mathematics 10 studentsthan Mathematics 10A students considered data analyis to be important, easy, andenjoyable.The data analysis results parallelled the geometry results in that there were morethan the expected number of students in both courses who obtained less than 40% on theachievement portion and who held unfavourable views towards data analysis. Also, for the"easy" and "like" matrices, there were more than the expected number of students who hadscores greater than 60% and who indicated favourable views towards data analysis. Thematrix describing Mathematics 10A students' attitudes towards the importance of dataanalysis and their achievement in this area differed from all other matrices which describeGrade 10 data. In this matrix, the number of students achieving scores higher than 80%who also considered data analysis to be important was less than the expected value for thiscell.65Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 2.7% 3.3% 3.3% 2.6% 1.7%ExpectedValue 1.7% 2.7% 3.5% 3.4% 2.5%Important 9.7% 16.2% 22.2% 21.8% 16.4%ExpectedValue 10.7% 16.8% 22.0% 21.1% 15.6%Row Total13.7%86.3%Table 24Mathematics 10 & 10A: Chi-Square AnalysisImportance of Data Analysis and Achievement in Data AnalysisMathematics 10N=6532Total^12.4%^19.5%^25.5%^24.5%^18.1%^100%Chi-square = 87.9; D.F. = 4; significance level = 0.0000Mathematics 10A66Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100NotImportant 7.5% 7.3% 10.1% 1.4% 0.6%ExpectedValue 5.6% 6.1% 12.3% 2.3% 0.6%Important 13.2% 15.5% 35.5% 7.3% 1.6%ExpectedValue 15.1% 16.7% 33.3% 6.3% 1.6%Row Total26.9%73.1%N=2168Total^20.7%^22.8%^45.6%^8.7%^2.2%^100%Chi-square = 48.4; D.F. = 4; significance level = 0.0000Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 6.7% 8.1% 9.7% 7.3% 3.8%ExpectedValue 4.7% 6.6% 8.9% 8.7% 6.7%Easy 6.4% 10.3% 15.4% 17.2% 15.1%ExpectedValue 8.4% 11.9% 16.1% 15.8% 12.2%Row Total35.6%64.4%Table 25Mathematics 10 & 10A: Chi-Square AnalysisDifficulty of Data Analysis and Achievement in Data AnalysisMathematics 10N=5042Total^13.1%^18.4%^25.0%^24.5%^18.9%^100%Chi-square = 220.3 D.F. = 4; significance level = 0.0000Mathematics 10A67Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Difficult 12.6% 11.4% 18.1% 2.9% 0.7%ExpectedValue 10.1% 9.8% 21.0% 3.7% 0.9%Easy 9.6% 10.2% 28.1% 5.3% 1.3%ExpectedValue 12.1% 11.7% 25.1% 4.4% 1.1%Row Total45.6%54.4%N=1804Total^22.2%^21.6%^46.2%^8.1%^1.9%^100%Chi-square = 50.0; D.F. = 4; significance level = 0.0000Table 26Mathematics 10 & 10A: Chi-Square AnalysisLikeability of Data Analysis and Achievement in Data AnalysisMathematics 1068Achievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 7.7% 9.0% 10.5% 9.0% 6.4%ExpectedValue 5.9% 8.0% 10.8% 10.1% 7.9%Like 6.1% 9.7% 14.7% 14.8% 12.1%ExpectedValue 7.9% 10.7% 14.5% 13.7% 10.6%Row Total42.6%57.4%N=4856Total^13.7%^18.7%^25.3%^23.8%^18.5%^100%Chi-square = 94.2; D.F. = 4; significance level = 0.0000 Mathematics 10AAchievement (percent)0 - 20 20.1 - 40 40.1 - 60 60.1 - 80 80.1 - 100Dislike 13.0% 12.5% 21.8% 3.1% 0.7%ExpectedValue 10.9% 11.4% 23.8% 4.1% 0.8%Like 8.4% 9.9% 24.9% 4.9% 0.9%ExpectedValue 10.4% 10.9% 22.8% 3.9% 0.8%Row Total51.1%48.9%N=1742Total^21.4%^22.4%^46.6%^8.0%^1.6%^100%Chi-square = 32.4; D.F. = 4; significance level = 0.0000Grade 10—Non-participation of StudentsIn the 1990, there was almost full participation by students in Grades 4 and 7 in theBritish Columbia Mathematics Assessment. However, only 84% of the Grade 10 studentsin the province participated in the assessment. There is no information available as to thereasons why certain students did not participate, nor is there any data which describes themathematical ability of these absentee students or their attitudes towards mathematics. Asthe sample for this study was drawn from the assessment data, the statistical significance ofhaving 16% of the students not participate was investigated to determine if it held anyimplications for this study.The students who did not participate in the assessment could have been arepresentative sample of the entire Grade 10 student population. If this was the case, thenthe non-participation of the these students would not have an impact on this study. Thesmallest Grade 10 sample analyzed contained 6598 subjects. Even though this value is lessthan half the number of students who were enrolled in Grade 10 in British Columbia in1989, it is still a large sample size.However, as discussed in Chapter 3, it is more likely that the individuals who didnot take part in the assessment had either withdrawn from school, were absent on the dayof the assessment, or were discouraged from participating. If these assumptions arecorrect, then these students would not form a representative sample of the general studentpopulation. Instead, many of the Grade 10 students who did not take part in theassessment might have been students of weak mathematical ability or students with littleinterest in school or in mathematics.In order to explore the potential impact on the findings of this missing information,the following points were taken into consideration. For each of the matrices which dealtwith the Grade 10 data, a chi-square analysis was performed. A chi-square value isobtained by subtracting the expected number (E) from the observed number in each cell of a69matrix. The square of (0 - E) is then divided by E. The subsequent values for each cell arethen added to produce the final chi-square value for the matrix.The chi-square value can be reduced in size by increasing the number of cells where(0 - E) is equal to zero. The participation of more Grade 10 students in the assessmentcould reduce the size of a chi-square value of a matrix and, thus, its subsequentsignificance level. If the missing students were placed in cells where 0 was less than E soas to make 0 equal to E, then the chi-square value would lessen in size.In each of the matrices dealing with Grade 10 data, there were five cells with 0 lessthan E and all of the cells, except one, had a chi- square value which was significant at the0.0000 level. In order to examine the potential impact of the missing Grade 10information, the matrix where the chi-square value was 0.0001 was selected for study.This matrix dealt with the Mathematics 10A data and can be found in Table 21. The valuesin the matrix were manipulated in the following manner. Each of the five cells where 0was less than E was changed so that 0 was equal to E. A new chi-square value was thencalculated and was found to be 12.97. This value has a significance level between 0.01and 0.02. This resulting level of significance is different than the original value but, for thepurposes of this study, it still represents a reasonable value.70CHAPTER 5SUMMARY AND CONCLUSIONSThis study was undertaken to determine the nature of the relationship between highschool students' attitudes towards particular mathematics topics and their achievement inthose areas. The sample used in this work was drawn from the data collected by the 1990British Columbia Mathematics Assessment of students in Grades 7 and 10 in the province.These data were then investigated through the use of chi-square analysis. The specificmathematical topics which were the focus of this research were geometry and data analysis.The study of the relationship between attitudes and achievement included an examination ofthe role of grade level and student ability.Presented in the following sections are conclusions based on the findings of thiswork. Implications of the results and the limitations of the study are then discussed. Thechapter concludes with proposed suggestions for further research.Findings and ConclusionsThe four research questions listed in Chapter 1 were addressed through the analysisof the data obtained from the 1990 British Columbia Mathematics Assessment. Each ofthese questions is stated below and a discussion of the relevant findings and conclusions ispresented immediately after each question. The first question reads as follows:1. What relationships exist between students' attitudes towards the geometry anddata analysis domains of the mathematics curriculum and students' achievement inthose domains?The results of this study support the findings of many researchers that there is aconnection between attitude and performance in mathematics (Brassell, Petry, & Brooks,1980; Campbell & Schoen, 1972; Hembree, 1990; Tsai & Walberg, 1983). In order toanswer the above question, the assessment data for students in Grade 10 were examined.The chi-square analysis of these data showed that there was a significant relationship71between students' attitudes towards geometry and data analysis and their performance inthose strands. This relationship existed for each of the three investigated components ofattitude.The chi-square value for each of the six matrices which were designed to studythese relationships was 0.0000. As this value was the same in all six cases, it did notprovide sufficient information to determine if the nature of the relationship between attitudeand achievement was different for each of the components of attitude and each of themathematics topics. Likewise, as the sample size for the study was large, a chi-squarevalue of 0.0000 is not an unexpected result and does not necessarily indicate the strength ofthe relationship.A further examination of the patterns present in the data, however, suggests thatthere is a strong connection between performance in geometry and data analysis andstudents' perceptions of these topics. Grade 10 students who were achieving good resultsin geometry were likely to view it as being important, easy, and enjoyable. Students whowere obtaining poor results were likely to hold the opposite viewpoint. Similar resultswere found for the easy/difficult and like/dislike components of attitude and theirrelationship to achievement in data analysis. Approximately the same percentage ofstudents considered data analysis to be easy and likeable as held these views aboutgeometry.The relationship between students' views as to the importance of data analysis andtheir performance in this area differed from the other findings in that approximately 85% ofthe students considered data analysis to be important. There was little difference betweenthe expected value and the actual number of high-scoring students who thought that dataanalysis was important. The majority of the students who did not consider data analysis tobe important achieved scores of less than 60%.Earlier work determined that students may have different attitudes towards differentmathematical topics and that they do not necessarily achieve the same results on all domains72of the mathematics curriculum (Hogan, 1977; Kifer & Robitaille, 1989). This research,however, suggests that the relationship between achievement and the 'easy/difficult' and`like/dislike' components of attitude is similar for both the data analysis and geometrydomains of the mathematics curriculum.As was noted previously, the relationship between the 'importance' component ofattitude and students' performance in mathematics is not the same for all topics. It is notclear from this initial analysis if these differing results are due to the particular domains ofthe curriculum under study or if they are a function of the 'importance' component ofattitude.The second question which was researched is stated below:2. What relationships exist among students' overall mathematics ability, theirachievement in the geometry and data analysis domains, and their attitudes towardsthese topics?The response to this question was based on a further analysis of the Grade 10 data.On both the geometry and data analysis units, Grade 10 students achieved higher scoresthan on the overall assessment and the distribution of the assessment scores was differentthan the distribution of the scores for specific topics. However, significant relationshipsbetween attitudes towards data analysis and geometry and overall achievement inmathematics were determined for each of the three components of attitude studied. The chi-square value for each of the matrices which examined these relationships was significant atthe 0.0000 level.The patterns which emerged within the data concerning geometry and overallachievement were similar to those which occurred with the data relating to achievementwithin the geometry domain. The percentage of students who viewed geometry favourablyand achieved high scores was similar in both cases as was the percentage of students whohad low scores and negative views.73Different results were found for data analysis. The relationship betweenperformance and attitude within the matrices was not the same for overall achievement as itwas for achievement within the data analysis unit. At each achievement level, thepercentage of students who found data analysis to be important, easy, or likeable was notthe same for each set of matrices.Therefore, it can be concluded that, as stated previously, there is an associationbetween attitudes towards specific topics within the mathematics curriculum and overallachievement in mathematics. However, this relationship will not necessarily be the same asthe relationship that exists between performance within a particular domain of thecurriculum and attitudes towards that domain. Similar patterns existed for the geometryunit but not for the data analysis unit.The third research question focused on the impact of the student's grade level andreads as follows:3. What differences, if any, exist in the nature of the relationships in questions 1and 2 among students at different grade levels?The assessment data concerned with Grade 7 and Grade 10 students provided thebasis for the study of this question. Five of the six matrices designed to illustrate theconnection between performance by Grade 7 students in the areas of geometry and dataanalysis and the three attitude components had a chi-square value of 0.0000. The exceptionwas the matrix dealing with the importance of geometry which had a chi-square value of0.0013. Thus, in five out of six possible cases, the relationship between attitude andperformance was similar for Grade 7 and Grade 10 students in that this relationship wasfound to be significant at the same level.The Grade 7 students generally held more favourable views towards geometry anddata analysis and achieved better results than did the students in Grade 10. The Grade 7and Grade 10 results differed in that the gap between the percentage of high and low74achievers who held favourable views towards geometry and data analysis was greater forstudents in Grade 10 than for students in Grade 7.However, an examination of the five matrices referred to previously indicated thatsome of the patterns that had emerged within the Grade 10 data were also present within theGrade 7 data. There were more than the expected number of students who consideredgeometry to be easy or enjoyable, or data analysis to be important, easy, or enjoyable, whoalso achieved good achievement scores in these areas Likewise, there were more than theexpected number achieving poor scores who also held unfavourable views in these fiveareas. The gap between expected value and actual value was lowest for students in themiddle performance range.Unlike the Grade 10 students, approximately 88% of the Grade 7 studentsconsidered geometry to be important. At all five achievement levels, at least 85% of thestudents held this view. These results are similar to those determined for Grade 10 anddata analysis. In that case, 83% of the students indicated that the topic was important.The previous discussion suggests that the relationship between achievement and the`easy/difficult' and like/dislike' components of attitude may be the same for students indifferent grade levels. These findings differ from those of other studies (Newman, 1984;Taylor & Robitaille, 1987) whose results linked the relationship between attitude andperformance to the grade level of the student. However, as was noted in the response toquestion 1, it appears that the 'importance' component of attitude may have a differentrelationship to attitude depending upon the topic or grade level under consideration.The final question posed by this study was as follows:4. What differences, if any, exist in the relationships in questions 1 and 2 amongstudents in the same grade who are enrolled in different mathematics courses?In order to answer this question, data relating to the views and performance ofstudents enrolled in Mathematics 10 and Mathematics 10A was examined. Mathematics 10students at all achievement levels held more favourable views towards data analysis than75did Mathematics 10A students. For geometry, the percentage of low-scoring students inboth courses who viewed geometry unfavourable was almost the same. As theachievement level rose, the gap between the views of the Mathematics 10 and Mathematics10A students grew larger.Similar relationships between attitudes towards geometry and data analysis andachievement in those areas were found for students enrolled in these courses in that, forfive out of six cases, the chi-square values for the matrices concerned with these data weresignificant at the 0.0000 level. The significance level for the remaining matrix was 0.0001.In general, for the students enrolled in Mathematics 10 and Mathematics 10A, the higherthe achievement level, the more they liked geometry and data analysis and the more theyconsidered them to be important and easy.These results suggest that, although students in different courses may have differentattitudes towards certain topics, the relationship between attitude and performance willfollow the same general patterns. However, these patterns will not necessarily be identicalfor each topic studied and may differ depending upon the achievement level of the student.ImplicationsThe results of this study affirm the need for mathematics educators to be aware ofthe impact of the affective domain on the learning process in mathematics. Although thefindings did not investigate the cause and effect relationship between performance andattitude, educators should recognize that there may be an interaction between these twovariables. As this relationship seems to exist for different domains of the curriculum aswell as for students of varying ability and in different grade levels, its existence needs to berecognized by educators working with students at all levels of the mathematics curriculum.The findings also emphasize the complex nature of the relationship between attitude andperformance. Teachers may always have assumed that students with favourable attitudestowards mathematics will generally perform well and those achieving low scores will have76negative attitudes. Teachers also should be aware that these relationships may vary withthe particular component of attitude expressed, with the individual topic under study, andwith the grade level of the student. Flexibility in dealing with students' performance andtheir attitudes is required.Of further interest is the observation that students may view mathematics, not as aglobal discipline, but rather as one composed of varying strands. Students appear todifferentiate among these strands and also to differentiate among their own perceptions ofthese strands. As students appear to be able to reflect metacognitively on these concepts,teachers should encourage students to do so.Limitations of the StudyThis study investigated the relationship between attitudes towards certain domainsof the mathematics curriculum and achievement in mathematics. The following conditionsare limitations on this work.The definitions of the terms geometry and data analysis are based on the implieddefinitions of these terms as used in the British Columbia mathematics curriculum guide.Student definitions of these terms, however, may differ from that of the curriculum. As anexample, one of the topics about which the Grade 10 sample group was asked to indicateits views was trigonometry, which suggests that trigonometry and geometry are twoseparate topics. The British Columbia mathematics curriculum, however, includestrigonometry under the heading of geometry.Another possible limitation is the reliability of student self-reporting of attitudes. Itis assumed that students have the ability to accurately evaluate their own attitudes. It is alsoassumed that students will choose to accurately report these attitudes.77Suggestions for Further ResearchThe results of this study have provided additional information regarding thecomplex relationship between attitudes towards mathematics and performance in thissubject. This work, however, suggests a variety of additional questions which may haveimplications for further research.The focus of this research was specifically on the geometry and data analysisdomains of mathematics. However, little research has been done concerning attitudes andindividual topics within the curriculum. A study of other mathematical topics, such asalgebra and number theory, would clearly be needed before any generalized conclusionsregarding attitudes and individual topics could be made. Likewise, the sample whichformed the basis of this work involved students from grades seven and ten. Further workinvolving students at different grade levels is needed. Research which encompassedcomponents of attitude other than those involved in this study would also broaden theframework of knowledge regarding attitudes towards particular domains of the curriculum.Attitude scales, such as the Mathematics Attitude Inventory and the Fennema-ShermanMathematics Attitude Scales, which examine a wide range of domains of attitude and whichhave statistical evidence supporting their validity would be useful for this purpose.Many of the findings of this study were based upon a descriptive analysis of thematrices designed to examine the data. A more empirical study of the data would allow formore specific information regarding the association between attitude and performance to bedetermined.One issue which has only been briefly referred to in the literature is that of thecultural context within which the learning of mathematics takes place (Kifer & Robitaille,1989; Hart, 1989). As societies become more multicultural in nature, it is important tounderstand the impact of culture on the relationship between attitude and performance in78mathematics. It cannot be assumed that this relationship will necessarily be the same for allsocietal groups.One further question which has only been minimally addressed is that of thecausality of the relationship between attitude and achievement (Cheung, 1988; Newman,1984). Although this may be a difficult question to precisely answer, it warrants,nonetheless, an examination.79REFERENCESAiken, L.R. (1970). Attitudes towards mathematics. Review of EducationalResearch, 40, 551-596.Aiken, L.R. (1976). Update on attitudes and other affective variables in learningmathematics. Review of Educational Research, 46, 293-311.Allport, G. (1967). Attitudes. In M. Fishbein (Ed.), Readings in attitude theory andmeasurement (pp. 3-13). New York: Wiley.Anderson, L.W. (1988). Attitudes and their measurement. In J.P. Keeves (Ed.),Educational research methodology and measurement: An internationalhandbook (pp.421-426). Oxford: Pergamon Press.Bassarear, T.J. (1987). The effect of attitudes and beliefs about learning, aboutmathematics, and about self on achievement in a college remedial mathematicsclass. Dissertation Abstracts International, 47, 2492A.Benbow, C. & Stanley, J. (1982). Consequences in high school and college of sexdifferences in mathematical reasoning ability: A longitudinal perspective. AmericanEducational Research Journal, 19, 598-622.Brassell, A., Petry, S., & Brooks, D. (1980). Ability grouping, mathematicsachievement, and pupil attitudes toward mathematics. Journal for Research inMathematics Education, 11, 22-28.Brown, C., Carpenter, T., Kouba, V., Lindquist, M., Silver, E., & Swafford, J. (1988a).Secondary school results for the fourth NAEP mathematics assessment: Algebra,geometry, mathematical methods and attitudes. Mathematics Teacher, 81, 337-347.Brown, C., Carpenter, T., Kouba, V., Lindquist, M., Silver, E., & Swafford, J.Secondary school results for the fourth NAEP mathematics assessment: Discretemathematics, data organization and interpretation, measurement, number andoperations. Mathematics Teacher, 81, 241-248.Burbank, I.K. (1970). Relationships between parental attitude toward mathematics andstudent attitude toward mathematics, and between student attitude towardmathematics and student achievement in mathematics. Dissertation AbstractsInternational, 30, 3359A-3360A.Caezza, J.F. (1970). A study of teacher experience, knowledge of and attitude towardmathematics and the relationship of these variables to elementary school pupils'attitudes toward and achievement in mathematics. Dissertation AbstractsInternational, 31, 921A-922A.Campbell, N. & Schoen, H. (1977). Relationships between selected teacher behaviors ofprealgebra teachers and selected characteristics of their students. Journal forResearch in Mathematics Education, 8, 369-375.80Cheung, K.C. (1988). Outcomes of schooling: Mathematics and attitudes towardsmathematics learning in Hong Kong. Educational Studies in Mathematics, 19, 209-219.Fennema, E. & Sherman, J. (1976). Fennema-Sherman Mathematics Attitude Scales:Instruments designed to measure attitudes toward the learning of mathematics byfemales and males. Journal for Research in Mathematics Education, 7, 324-326.Hart, L. E. (1989). Describing the affective domain: Saying what we mean. In D. B.McLeod & V.M. Adams (Eds.), Affect and mathematical problem solving: A newperspective (pp. 37-45). New York: Springer-Verlag.Hembree, R. (1990). The nature, effects, and relief of mathematics anxiety. Journal forResearch in Mathematics Education, 21, 33-46.Hogan, T.P. (1977). Students' interests in particular mathematics topics. Journal forResearch in Mathematics Education, 8, 115-122.Horn, E.A. & Walberg, N.J. (1984). Achievement and interest as functions of quantityand level of instruction. Journal of Educational Research, 77, 227-232.Jackson, P.W. (1990). Life in classrooms. (rev. ed.). New York: Teachers CollegePress.Kifer, E. (1990). Attitude measurement. Unpublished manuscript.Kifer, E., & Robitaille D. (1989). Attitudes, preferences and opinions. In D.F. Robitaille& R.A. Garden (Eds.), The IEA study of mathematics II: Contexts andoutcomes of school mathematics (pp. 178-208). Oxford: Pergamon Press.Lucock, R. (1989). Pupils learning mathematics: Beliefs and attitudes. DissertationAbstracts International, 49, 3294A.McLeod, D. B.(1989). Beliefs, attitudes, and emotions: New views of affect inmathematics education. In D. B. McLeod & V.M. Adams (Eds.), Affect andmathematical problem solving: A new perspective (pp. 245-258). New York:Springer-Verlag.Minato, S., & Yanase, S. (1984). On the relationship between students attitudes towardsschool mathematics and their levels of intelligence. Educational Studies inMathematics, 15, 313-320.Newman, R.S. (1984). Children's achievement and self-evaluations in mathematics: Alongitudinal study. Journal of Educational Psychology, 76, 857-873.Robitaille, D. (in press). 1990 mathematics assessment: Technical report. BritishColumbia: Ministry of Education.Robitaille, D., & Garden, R.A. (Eds.). (1989). The IEA study of matheamtics II:Contexts and outcomes of school mathematics. New York: Pergamon Press.Robitaille, D., & O'Shea, T. (Eds.). (1985). The 1985 British Columbia mathematicsassessment: General report. British Columbia: Ministry of Education.81Sandman, R.S. (1980) The Mathematics Attitude Inventory: Instrument and user'smanual. Journal for Research in Mathematics Education, 11, 148-149.Shaw M., & Wright, J. (1967). Scales for the measurement of attitudes. New York:McGraw-Hill.Taylor, A.R. & Robitaille, D.F. (in press). Instrumentation and sampling. In D.F.Robitaille (Ed.), 1990 mathematics assessment: Technical report. BritishColumbia: Ministry of Education.Taylor, A.R. & Robitaille, D.F. (1987). An analysis of factors related to students'opinions about and achievement in mathematics. British Columbia: Ministry ofEducation.Tsai, S. & Walberg, H. (1983). Mathematics achievement and attitude productivity injunior high school. Journal of Educational Research, 76, 267-272.82APPENDIX A1990 British Columbia Mathematics Assessment FormsGrade 10: Forms C and DGrade 7: Form A831990BRITISH COLUMBIAMATHEMATICS ASSESSMENTSTUDENT FORMGrade 10 Form C^ 1BACKGROUND INFORMATIONI ^ For each item, shade in the appropriate space on the answer sheet.1.^SexA) MaleB) Female2. AgeA) 14 or less D) 17B) 15 E) 18C) 16 F) 19 or more3. What program are you in?A) Regular program in EnglishB) French ImmersionC) Programme-cadre de francais4. What mathematics course are you currently taking (if you are nottaking one this semester, which course did you take last semester)?A) Math 8B) Math 9C) Math 9AD) Math 10E) Math 10AF) Introductory Math 11G) Math 11AH) Math 11I) Algebra 11J) Algebra 12K) A mathematics course not on this listL) I am not taking a mathematics course this year.5. Which of the following best describes the mathematicscourse you took or are taking this year?A) A ten-month courseB) A semestered course beginning in SeptemberC) A semestered course beginning in January or FebruaryD) I am not taking a mathematics course this year.E) Other1990 Mathematics Assessment2^Grade 10 Form C6. What made you decide to take the mathematics courseyou are currently taking (or the one you took lastsemester)? You many choose more than one response.A) The counsellor suggested it.B) My parent(s) or guardian(s) suggested it.C) My last year's mathematics teacher suggested it.D) I decided on my own.E) I had no choice because of my marks in previousmathematics courses.F) It is required for the next mathematics course Iwant to take.G) Most of my friends take this course.H) I took the course because I am good at mathematics.I)^Some other reason.7. Which mathematics course(s) do you intend to take in bothGrades 11 and 12? Mark all that apply.A) NoneB) Math 10C) Math 10AD) Math 11E) Introductory Math 11F) Math 11AG) Introductory Accounting 11H) Math 12I) Survey Math 12J) An enriched mathematics course (e.g. Advanced Placement,International Baccalaureate, Calculus, etc.)K) A mathematics course not on this list8. What do you plan to do after leaving secondary school? Choose one.A) Attend a business school or technical collegeB) Attend a vocational, art, or trade training schoolC) Attend a community college: university transfer programD) Attend a community college: career programE) Attend universityF) Look for a full-time jobG) Take a year off and then return to schoolH) Take a year off and then look for a jobI) Other plansJ) Undecided1990 Mathematics Assessment(g3Grade 10 Form C^ 3For the next three items, decide to what extent you agree or disagree.9. You have to be able to do mathematics to get a good job.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly Agree10. Most people use mathematics in their jobs.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly Agree11. When I leave school, I would like a job where I have to use mathematics.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly AgreeFor each of the next three items, three answers are needed.A) Tell how important you think the topic is.B) Tell how easy you think the topic is.C) Tell how much you like the topic.If you are not sure what a topic means, leave its three answers blank.12. GeometryA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult dislikeundecided undecided^undecidedimportant easy^likevery important^very easy^like a lot1990 Mathematics Assessment$84^Grade 10 Form C13. Data AnalysisA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lot14. TrigonometryA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lotFor the next three items, think of your mathematics classesduring a typical school week.15. We use computers in our mathematics class.A) Almost every periodB) OftenC) SometimesD) RarelyE) Never16. We have quizzes or tests in mathematics.A) Almost every periodB) OftenC) SometimesD) RarelyE) Never17. We work in small groups in our mathematics class.A) Almost every periodB) OftenC) SometimesD) RarelyE) Never1990 Mathematics AssessmentGrade 10 Form C^ 5ACHIEVEMENT SURVEY: Part 1, Items 1 - 20.These items are to be answered by all students.Mrs. Schmidt works in a local factory for $6.00 per hour, plustime and one-half after 40 hours. Last week she worked 45hours. How much did she earn?A) $240 .B) $270C) $285D) $405E).^I don't know.2. Simplify:^30 — 4 (8 — 2)A) 0B) 20C) 156D) 6E)^I don't know.3. Which number is largest?A) 324B) 5C)D)3458E)^I don't know.F1990 Mathematics Assessment06^ Grade 10 Form C4. Wendy bought 3 record albums on sale. The regular pricewas $7.24 each and the sale price was $1.50 off each record.If she paid 69¢ sales tax on her total purchase, how muchmoney did she spend?A) $17.22B) $17.91C) $21.72D) $22.91E)^I don't know.5. What is the mean of the following numbers?2, 2, 2, 3, 4, 5, 10A) 3B) 2C) 10D) 4E)^I don't know.1990 Mathematics AssessmentGrade 10 Form C6. From the graph below, the temperature at a depth of 2.5 km isclosest toA) 30°C 60B) 40°C ,c? 50C) 50°Cii40D) 60°CEeI-3020E) I don't know.100^1^2^3Depth in kilometres7. A bag contains 3 red marbles, 2 white marbles, and 20 blackmarbles. What is the probability of randomly choosing awhite marble?A) 0.08B) 0.5C) 0.92D) 2.0E)^I don't know.1990 Mathematics Assessmentci,8^ Grade 10 Form C8. Two dice are rolled. What is the probability of rolling a totalof 5?195361182D.-E)^I don't know.9. ABCD is a parallelogram. If Z A = 105° determine Z B.A) 75°B) 95°C) 105°D) 255°E)^I don't know.A)B)C)D)1990 Mathematics AssessmentGrade 10 Form Cq3910. Angles X and Y in the figure below are complementary. Ifthe measure of angle X is 24° less than the measure of angleY, then angle X isA) 48°B) 23°C) 57°D) 33°E)^I don't know.11. When Joe walks from his house to Kelly's house, he follows thepath through the open field. How far does he walk?A) 450 m^ Kelly'sHouseB) 500 mOpen FieldC) 550 m^HouseJoe's^ /D) 600 mE)^I don't know.CORNER400 mRIDGE ROAD/ PATHE1990 Mathematics Assessmentcr410^ Grade 10 Form C12. A ABC is a right triangle. Determine the length of side AC.A) 13B) 17C) 49D) 169E)^I don't know.13. Find the perimeter of the figure below which consists of anequilateral triangle and a semi-circle.(Use it = 3.14)A) 41.1 cmB) 33.1 cmC) 28.6 cmD) 20.6 cmE)^I don't know.1990 Mathematics AssessmentGrade 10 Form C^ 1114. The best estimate for the area of the circle shown below isA) 15 m2B) 75 m2C) 100 m2D) 5m2E)^I don't know.15. Towns A, B, and C are on the shore of a lake as shown in themap below. The distance from A to B is 7.8 km and thedistance from A to C is 2.4 km. Which one of the following isthe best estimate for the area of the lake?CA) 10 km 2B) 18 km 2C) 14 km 2D) 24 km 2E)^I don't know.1990 Mathematics AssessmentP9612^ Grade 10 Form C16. In the diagram below, the area of rectangle PQRS is 24 cm 2 .What is the area of the parallelogram QVRS?A) 48 cm 2B) 36 cm 2C) 24 cm2D) 18 cm 2^E)^I don't know.17. Simplify:^r + s— (r — s)A) 0B) 2r + 2sC) 2rD) 2sE)^I don't know.18. Evaluate : 2 — (2 — x) when x = —IA) 1B) —1C) 2D) —2E)^I don't know.1990 Mathematics AssessmentGrade 10 Form C^ 1319. Solve:^5.3x — 12 = 2.4x + 46A) x = 2B) x = 20340 C) x = 29D) x = 34770E)^I don't know.20. If you divide any positive number by a number greater than 2,then the answer will beA) less than half the original number.B) more than half the original number.C) a fraction.D) impossible to predict.E)^I don't know.This is the end of Part 1. Students who are currently taking Mathematics 10,or who did so last semester, go to Part 3 on page 24.All others complete Part 2 beginning on the next page: Items 21 - 40.1990 Mathematics Assessmentq 814^ Grade 10 Form CACHIEVEMENT SURVEY: PART 2, ITEMS 21 -4021. What is the value of 2 3 x 32 ?A) 72B) 36C) 54D) 48E)^I don't know.22. Evaluate:^(3-1)2A) 9B) —9C)— 1D) 9E) I don't know.23.^Evaluate:^- 3----2'A) 12B) 4C) 108D) 6E)^I don't know.1990 Mathematics Assessment99Grade 10 Form C^ 1524. Linda's new bike cost $159.99 and the sales tax was 5%.How much did she pay including tax?A) $164.99B) $167.99C) $172.98D) $177.99E) I don't know.125. Written as a fraction in lowest terms, ,,T% =A)14000 1C)40040D) 4E)^I don't know.1^226. Evaluate:^4 -5- + 3 —5B)3810C) 2D) 7 3E) I don't know.1990 Mathematics Assessment10016^ Grade 10 Form C27. At a party the ratio of boys to girls was 2 to 1. What percentof the people at the party were girls?2A) 66 —%3B)^50%1C) 33 3—%D) 200%E)^I don't know.1^228. Divide:^13÷ 2 -3A) 3B) 2C) 1 -3D) 932E)^I don't know.1990 Mathematics AssessmentGrade 10 Form C^17129. Multiply:^3 —2 x 2 -76A) 14B) 51-4C) 6 14D) 7-1E)^I don't know.30. A table and a graph of the same data are shown below. Whatis the value of x? Number of Cars Frequency0 or 12 or 34 or 56 or 72x730 -1 2 -3^4 -5^6 -7Number of CarsA) 3B) 4C) 5D) 6E)^I don't know.1990 Mathematics Assessmentlot18^ Grade 10 Form C31. In the graph below, rainfall in centimetres is plotted for 13weeks. The average weekly rainfall during the period isapproximatelyA0 54as3cc 211 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3WeekA)^1 cm8)^2 cmC) 3 cmD) 4 cmE)^I don't know.1990 Mathematics Assessment103Grade 10 Form C^ 1932. The table below shows scores for a class on a 10-point test.How many in the class made a score GREATER than 7?Test Score Tally Frequency4 I 15 III 36 14111 67 II 2^.,8 1111 49 III 310 I 1A) 2B) 8C) 10D) 12E)^I don't know.33. The coordinates of point Q areA) (-4, 3)B) (— 3, 4)C) (3, —4)D) (4, — 3)E)^I don't know. 1990 Mathematics Assessment20^ Grade 10 Form C34. The cosine of Z A in the figure is equal toA) 35B) 4C) 5D) 4E)^I don't know.35. A ABC is similar to A DEF. Determine the length of side BC.512D10A) 12B)245C) 5D) 10E)^I don't know.1990 Mathematics Assessment18m20 m11 mGrade 10 Form C^ 2136. A ABC is a right triangle. If L A = 40° and AC = 100, find AB.sin 40° = 0.6428cos 40° = 0.7660tan 40° = 0.8391A) 64.28B) 76.60C) 130.5D) 155.6E)^I don't know.37. Determine the perimeter of this figure.A) 49 mB) 58 mC) 76 mD) 87mE)^I don't know.1990 Mathematics Assessmenti (X,22^Grade 10 Form C38. What is the approximate total surface area of the solidcylinder below? Use the formulaSurface Area = 2itr 2 + bah. A) 880 cm2B) 1187 cm2C) 308 cm 2D) 934 cm2E)^I don't know.20 cm39. If x = 4, y = 2, and z = 0.5, the value of 2xy 2z isA) 16B) 32C) 64D) 128E)^I don't know.1990 Mathematics AssessmenttoGrade 10 Form C^ 2340. Solve for x: 3x + 7 = 5x + 4A) x =B) x=C) x =D) x =11- 23- 2,32112E)^I don't know.This is the end of Part 2. Check your workand hand in your booklet and answer sheet.1990 Mathematics Assessment24^Grade 10 Form CACHIEVEMENT SURVEY: PART 3, ITEMS 41-60This section is for students who are taking Math 10 now,or who took Math 10 last semester.41. Find the sum in simplest radical form.A) 5JB)C) 15D) 3 .4E)^I don't know.42. Find the difference in simplest radical form.26 —475A) '415B) -\17s-C) 3TD) 2E)^I don't know.IOS1990 Mathematics Assessmenttog25Grade 10 Form C43. Points A, B, and C are plotted in the diagram below. If thesepoints are supposed to lie on the same line, which one of thefollowing statements can we conclude is true?A) Point A is incorrectly plotted.B) Point B is incorrectly plotted.A• •BC) Point C is incorrectly plotted.D) Any one of points A, B, or C isincorrectly plotted.E) I don't know.44. A collection of coins consists of 5 quarters, 2 dimes,6 pennies, 3 nickels, and-4 one-dollar coins. What is thelikelihood that if one is drawn at random, it will be a nickel?A) 15%B) 12%C) 25%D) 10%E)^I don't know.45. If the lines y = mix + b1, and y m2x + b2 are parallel, thenA) bi = b2B) y = x1C) mi.= — m2D) ml = m2E)^I don't know.c1990 Mathematics Assessment►tO26^ Grade 10 Form C46. Which one of the following is the graph of x + y = — 3?A) Y B) YC) Y D) 4 IIHIIII ^ XE)^I don't know.1990 Mathematics AssessmentGrade 10 Form C^27^tit47. In the figure AB // DG and AC // FE. If Z1 = 130° andL2 = 100°, find L AGH.A) 80°B) 100°C) 130°D) 150°E)^I don't^know.48. In the figure below, line NQ is parallel to line OP, NQ = 4 CM,OP = 6 cm, and MQ = 8 cm. Find the length of MP.A) 10 cmB) 12 cmC) 14 cmD) 16 cmE)^I don't know.49. The figure below illustrates a water canal and a method ofmeasuring its width. If PS = 24 m, PR = 2 m, and RT = 5 m,how wide is the canal?A) 24 mB) 32 mC) 40 mD) 60 mE)^I don't know.1990 Mathematics Assessmentt t a28^ Grade 10 Form C50. Which triangle can you be sure is similar to AMNP?12320° 4A)12B)12C) 918D)12E) I don't know.1990 Mathematics AssessmentipliI I 3E Grade 10 Form C^ 2951. In triangle ABC, BD is the median to AC. What additionalinformation is required to show that triangle ABD iscongruent to triangle CBD?A) AB = CDB) AB = BCC) BD = ABD) AC = BCE)^I don't know.A52. In order for ml to be parallel to m2, which one of thefollowing must be true?m 12 1rn 2 ,s^3 4A) L 3 and Z 4 must both be right angles.B) Z 1 must have the same measure as Z 4.C) L 1 + Z 4 must equal 180°.D) L 3 must have the same measure as L 2.E)^I don't know.os,1990 Mathematics Assessment30^ Grade 10 Form C53. Expand: (a — 7)(a — 2)A) a2 — 5a + 14B) a2 — 9a — 14C) a2 — 9a + 14D) a2 — 14a — 9E)^I don't know.54. Factor completely over the rational numbers: 4a 2 — 8A) (2a + 4) (2a — 2)B) 2(2a + 2) (a — 2)C) 2 2(a2 — 2)D) 22(a + 2) (a — 1)E)^I don't know.55. Factor completely: a2 — 36b2A) (a — 12b)(a + 3b)B) (a — 6b)(a — 6b)C) (a — 6b)(a + 6b)D) (a — 4b)(a + 9b)E)^I don't know.1990 Mathematics AssessmentISGrade 10 Form C^ 3156. A = 3n 2 — 2n + 5. If n = — 2, then the value of A isA) —5B) 10C) 15D) 21E)^I don't know.57. Which one of the following is the graph of 2x — 3 5?0^4E)^I don't know.58. Which one of the following statements about the equation2(x — 7) =2x +5 is true?A) The equation has no solution.B) The equation has infinitely manysolutions.C) x =0D) x =19E)^I don't know.1990 Mathematics Assessment32^ Grade 10 Form C59. Find the value of x such that: 5x 2 + 15 = 95A) 4B) — 4C) ±4D) All real numbersE)^I don't know.60. In the rectangular solid below, the length of the diagonal PS isA) 21 cmB) 17 cmC) 20 cmD)4-20") cmE)^I don't know. R8 cm9 cmThis the end of Part 3. Check your workand hand in your booklet and answer sheet.1990 Mathematics Assessment1990BRITISH COLUMBIAMATHEMATICS ASSESSMENTSTUDENT FORMGrade 10 Form D^ 1BACKGROUND INFORMATIONFor each item, shade in the appropriate space on the answer sheet.SexA) MaleB) Female2. AgeA) 14 or less^D) 17B) 15^E)^18C) 16 F) 19 or more3. What program are you in?A) Regular program in EnglishB) French ImmersionC) Programme-cadre de francais4. What mathematics course are you currently taking (if you are nottaking one this semester, which course did you take last semester)?A) Math 8B) Math 9C) Math 9AD) Math 10E) Math 10AF) Introductory Math 11G) Math 11AH) Math 11I) Algebra 11J) Algebra 12K) A mathematics course not on this listL) I am not taking a mathematics course this year.5. Which of the following best describes the mathematicscourse you took or are taking this year?A) A ten-month courseB) A semestered course beginning in SeptemberC) A semestered course beginning in January or FebruaryD) I am not taking a mathematics course this year.E) Other1181990 Mathematics Assessmentits2^ Grade 10 Form D6. What made you decide to take the mathematics courseyou are currently taking (or the one you took lastsemester)? You may choose more than one response.A) The counsellor suggested it.B) My parent(s) or guardian(s) suggested it.C) My last year's mathematics teacher suggested it.D) I decided on my own.E) I had no choice because of my marks in previousmathematics courses.F) It is required for the next mathematics course Iwant to take.G) Most of my friends take this course.H) I took the course because I am good at mathematics.I)^Some other reason.7. Which mathematics course(s) do you intend to take in bothGrades 11 and 12? Mark all that apply.A) NoneB) Math 10C) Math 10AD) Math 11E) Introductory Math 11F) Math 11AG) Introductory Accounting 11H) Math 12I) Survey Math 12J) An enriched mathematics course (e.g. Advanced Placement,International Baccalaureate, Calculus, etc.)K) A mathematics course not on this list8. What do you plan to do after leaving secondary school? Choose one.A) Attend a business school or technical collegeB) Attend a vocational, art, or trade training schoolC) Attend a community college: university transfer programD) Attend a community college: career programE) Attend universityF) Look for a full-time jobG) Take a year off and then return to schoolH) Take a year off and then look for a jobI) Other plansJ) Undecided1990 Mathematics AssessmentGrade 10 Form DFor the next three items, decide to what extent you agree or disagree.9. You have to be able to do mathematics to get a good job.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly Agree10. Most people use mathematics in their jobs.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly Agree11. When I leave school, I would like a job where I have to use mathematics.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly AgreeFor each of the next three items, three answers are needed.A) Tell how important you think the topic is.B) Tell how easy you think the topic is.C) Tell how much you like the topic.If you are not sure what a topic means, leave its three answers blank.12. GeometryA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult dislikeundecided undecided^undecidedimportant easy^likevery important^very easy^like a lot1990 Mathematics Assessmentiao3tat4^Grade 10 Form D13. Data AnalysisA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lot14. Working with decimals, fractions, and percentA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lotFor the next three items, think of your mathematics classesduring a typical school week.15. We use computers in our mathematics class.A) Almost every periodB) OftenC) SometimesD) RarelyE) Never16. We have quizzes or tests in mathematics.A) Almost every periodB) OftenC) SometimesD) RarelyE) Never17. We work in small groups in our mathematics class.A) Almost every periodB) OftenC) SometimesD) RarelyE) Never1990 Mathematics AssessmentGrade 10 Form D^5ACHIEVEMENT SURVEY: Part 1, Items 1 - 20.These items are to be answered by all students.1. A used automobile can be bought for cash for $2850, or on credit with adown payment of $400 and $80 a month for three years. How muchmore would a person pay by buying on credit than by buying the car forcash?A) $ 3280B) $ 640C) $ 430D) $ 400E)^I don't know.2. How many white squares will there be in the 10th figure in thefollowing pattern?- • • •A) 45B) 46C) 55D) 512E)^I don't know.1990 Mathematics Assessment6^ Grade 10 Form D3. Which number is largest?A)^3453458E)^I don't know.4. In a school election with three candidates, Mike received 120 votes,Lawrence received 30 votes, and Lesley received 50 votes. Whatpercent of the total vote did Mike receive?A) 30%B) 40%C) 60%D) 120%E) I don't know.5. What is the mean of the following numbers?2, 2, 2, 3, 4, 5, 10A) 3B) 2C) 10D) 4E)^I don't know.B)C)D)1990 Mathematics Assessment)ay-Grade 10 Form D^ 76. How many passengers used the airport in June?AIRLINE PASSENGERS FOR FIRST SIX MONTHS OF THE YEARAirports Hundreds of Passengers per MonthJan.^Feb.^Mar.^Apr.^May^JuneTotalBay City 9 3 5 7 2 4 30Camden 6 8 1 5 8 2 30Dover 8 5 9 6 6 3 37Fiske 5 6 6 1 3 7 28Grange 1 2 3 6 7 10 29TOTAL 29 24 24 25 26 26 154A) 7B) 26C) 700D) 2600E)^I don't know.7. A bag contains 3 red marbles, 2 white marbles, and 20 black marbles.What is the probability of randomly choosing a white marble?A) 0 08B) 0 5C) 0.92D) 2.0E)^I don't know.1990 Mathematics AssessmentGrade 10 Form D8. Two dice are rolled. What is the probability of rolling a total of 5?A)190^5D)^361C)D)2E)^I don't know.9. Which two angles are each supplementary to Z 4?A) Z1 and Z2B) Z2 and Z 3C) L5andZ1D) L1 and Z 3E) I don't know.10. A ABC is a right triangle and AABD is equilateral. Z BDC =A) 90°B) 120°C) 135°D) between 120° and 135°E)^I don't know.1990 Mathematics AssessmentJoe'sHouseKelly'sHouseti/Open Field/^E woPATH400 mRIDGE ROADGrade 10 Form D^ 911. When Joe walks from his house to Kelly's house, he follows the paththrough the open field. How far does he walk?A) 450 mB) 500 mC) 550 mD) 600 mE) I don't know.CORNER12. A RST and A RXY are right triangles. If RY = 4, XY = 3, and ST = 6,find RT.A) 7B) 8C) 10D) 12E) I don't know.13. The best estimate for the area of the circle shown below isA) 15 m 2B) 75 m2C) 100 m 2D) 5 m 2E)^I don't know.1990 Mathematics Assessment1 0^ Grade 10 Form D14. Find the surface area of the square pyramid shown below.Surface Area = b 2 + 4(lbh), where b is the length of the base and h isthe slant height.A) 192 cm 2B) 144 cm 2C) 336 cm2D) 384 cm 2E)^I don't know.12 cm15. What is the surface area of the.rectangular prism shown below?Surface Area = 2(1w + lh + wh)A) 30 m 2B) 31 m 2C) 60 m 2D) 62 m 2E)^I don't know.1990 Mathematics Assessmentla.Grade 10 Form D^ 1116. The best estimate for the volume of the solid shown below isA) 29B) 56C) 550D) 720^E)^I don't know.8.2^17. Simplify:^r + s — (r —s)A) 0B) 2r + 2sC) 2rD) 2sE)^I don't know.18. Simplify:^(3p + 2q) — (p + q)A) 2p+qB) 2p + 3qC) 4p — 3qD) 4p— qE)^I don't know.1990 Mathematics Assessment12^Grade 10 Form D19. If five is added to a certain number and the sum is multiplied by three,the result is —17. Find the number.A) — 32B) — 4C) — 7-32D) — 10-3E)^I don't know.20. How high would a stack of one million pennies be?A) 2 mB) 200 mC) 2 000 mD) 20 000 mE)^I don't know.This is the end of Part 1. Students who are currently taking Mathematics 10, or who didso last semester, go to Part 3 on page 21.All others complete Part 2 beginning on the next page: Items 21 - 40.1990 Mathematics AssessmentI B oGrade 10 Form D^ 13ACHIEVEMENT SURVEY: PART 2, ITEMS 21-4021. What is the value of 2 3 x 32 ?A) 72B) 36C) 54D) 48E)^I don't know.22. Simplify:^10 + 35 ÷ 5 + 2A) 19B) 11C) 15D) 6-7E)^I don't know.23. Evaluate:^(2 2)-2A) —16B) 0C)116D) 16E) I don't know.1990 Mathematics Assessment13114^ Grade 10 Form D24. Written as a decimal, 20% equalsA) 0.2B) 0.02C) 2.0D) 20.0E)^I don't know.325. Write —8 as a decimal.A) 0.3B) 0.24C) 0.375D) 2.666E)^I don't know.26. The scientific notation for 634.78 isA) 0.634 78 x 10 -3B) 6.3478 x 10 -2C) 63.478 x 10D) 6.3478 x 102E)^I don't know.1990 Mathematics AssessmentIGrade 10 Form D^ 15127. A marathon runner covers 42 km in 2-2 hours. His average speed isA) 8.4 km/hB) 16.8 km/hC) 25.2 km/hD) 33.6 km/hE)^I don't know.28. The sales tax is 5%. How much would the sales tax be on a new carthat costs $6750.00?A) $675.00B) $337.50C) $67.50D) $33.75E)^I don't know.129. Find the missing term:^2—a^1 33283C) 3D) 6E)^I don't know.A)B)1990 Mathematics Assessment80604020043A) 2B) 10C) 15D) 20E)^I don't know.111•11111111111•111111111111111111111111111111 1M111•11111•1111111111111111111•MIll■11111■1111111M2inall••ln■111111111111•111■1EMI■RCEIM■111•11111111111/MIIMMEIIIII■INIPENIMENI1111111111111111■Imn-vemisamamminminir.:iii-amornamounium1^2Time in hours)1 6^ Grade 10 Form D30. A team scored an average of 3 points per game for 5 games. How manypoints altogether were scored in the 5 games?A) 5B) 15C) 25D) 75E)^I don't know.31. The distance travelled by two cars during a period of four hours isshown in the graph below. Three hours after starting, how manykilometres is car A ahead of car B?1990 Mathematics Assessment1 34Grade 10 Form D^ 1732. The graph shows the distance travelled by a tractor during a period offour hours. How fast is the tractor moving?AA) 1 km/hB) 2 km/hC) 4 km/hD) 8 km/hE)^I don't know.EN.—a)UCas4c7)E8765432133. The coordinates of point Q areA) (— 4, 3)B) (— 3, 4)C) (3, — 4)D) (4, — 3)E) I don't know./ 0^1^2 3 4Time (hours)YIIIIIIIIIIIIIII MEM.MIMI 1111111•1111111MIMI MIMEMIMI MEMEMMEN MEMNum mumMUM MEM■1111M11 MIME1111111111MIN 01111111•1111iMEM 1111111111M11■111■■111111=111■MEW 4M=go-0It1990 Mathematics AssessmentA 3:5-18^ Grade 10 Form D34. A DEF is similar to A XYZ. Find the length of YZ.35A) 26.25B) 28C) 37.3D) 43.75E)^I don't know.35. The tangent of L A in the figure is equal toA)B)C)455434D)43E) I don't know.1990 Mathematics Assessment3(0Grade 10 Form D^ 1936. A ABC is a right triangle. If L A = 40° and AC = 10, find BC.sin 40° = 0.6428cos 40° = 0.7660tan 40° = 0.8391A) 11.917B) 6.428C) 8.391D) 0.08391E)^I don't know.37. The area of this figure isA) 39 cm 2B) 44 cm 2C) 90 cm 2D) 120 cm 2E)^I don't know. 6 cm10 cm5 cm38. The volume of the rectangular prism below is 576 cm 3 . What is itsheight?A) 4 cmB) 8 cmC) 16 cmD) 32 cmE)^I don't^know.1990 Mathematics Assessment131-20^ Grade 10 Form D39. Evaluate:^— 4a(a — 3b) when a = 2 and b = —1A) —40B) —8C) 8D) 40E)^I don't know.40. Solve for n:^4(n — 3) — 5 = 7nA) n =B) n =C) n =D) n =—17317373—73 E)^I don't know.This is the end of Part 2. Check your workand hand in your booklet and answer sheet.1990 Mathematics AssessmentGrade 10 Form D^21ACHIEVEMENT SURVEY: PART 3, ITEMS 41 -60This section is for students who are taking Math 10 now,or who took Math 10 last semester.41. Write in simplest radical form.3 -NrA) 124TB)C) 6 -V 12D)E)^I don't know.42. Find the quotient in simplest radical form.6-\F2T2V 10A) 4ArldB) 3'C)D) INTIT)E)^I don't know.131990 Mathematics Assessment$1500 -$1200-Profit $900 -inDollars $600 _$300 _110 20 310^1^1■3922^ Grade 10 Form D43. If the relationship shown in the graph below continues between profitand the number of suits sold, what will be the approximate profit for thesale of 35 suits?Profit for a B.C. Suit ManufacturerNumber of Suits SoldA) $600B) $900C) $1200D) $1500E)^I don't^know.44. A collection of coins consists of 5 quarters, 2 dimes, 6 pennies,3 nickels, and 4 one-dollar coins. What is the likelihood that if one isdrawn at random, it will be a nickel?A) 15%B) 12%C) 25%D) 10%E)^I don't know.1990 Mathematics AssessmentYAD^B^A-•---' ^.4,-*---CAO I• XV1 14-0Grade 10 Form D^ 2345. Which line is the graph of the equation 2x - y = - 4?A) AB) BC)D) DE)^I don't know.46. Which one of the following equations is satisfied by both of the orderedpairs (3, -1) and (10, - 4)?A) 3x - 7y = 2B) - 3x + 7y = 2-^ 2C) Y= -Tx +D) Y= - 7 x + —23^3E)^I don't know.1990 Mathematics Assessment14124^ Grade 10 Form D47. Quadrilateral ABDC is made up of 2 equilateral triangles ABC and BCD.The measure of L ABD isA) 60°B) 90°C) 120°D) 150°E)^I don't^know.48. If A ABC is similar to A PRQ, which one of the following is true?AA^AB PQtv BC = QPAB_B) PQ ACPR ACC) QR = BCAB PRD) AC—PQE) I don't know.1990 Mathematics AssessmentGrade 10 Form D^ 2549. In the right triangle below L C = 40° and BC = 20.0. Use the followinginformation to find AB.sin 40° = 0.6428cos 40° = 0.7660tan 40° = 0.8391A) 16.8B) 15.3C) 12.9D) 23.8E)^I don't know.A50. If x > 0 and y < 0, then the point (x, y) is located in quadrantA) IB) IIC) IllD) I VE)^I don't know.1990 Mathematics Assessment14 326^ Grade 10 Form D51.^In triangle PQR, ST will be parallel to RQ ifA) Ll 4-Z2-1-Z3=180°B) Z 2 + 5 = 180°C) L3=L4D) L4=L2E)^I don't know.T52. In the figures below AB = XZ, AC = XY, and L CAB and L YXZ eachmeasure 110°. Are the triangles congruent? If so, choose the answerthat tells you this. 1 CA) Yes (S-S-S)B) Yes (A-S-A)C) NoD) Yes (S-A S)E)^I don't know.1990 Mathematics AssessmentGrade 10 Form D^ 2753. Simplify:^15p6, 3p2A)^5p35p4C) 12p3D) 12p4E)^I don't know.54. If .Nra— = 0.9z4, then a =A) 0.3z2B) 0.81z2C) 0.32z6D) 0.81z8E)^I don't know.,V0.0049x855. Simplify:Y6A) 0.7x4y3B) 0.07x4y30.07x4 C) Y3D)0.007x4Y3E)^I don't know.1990 Mathematics Assessment1 46-28^ Grade 10 Form D56. R = + —1 . If x = 2 and y = 3, then the value of R isX y1652556E)^I don't know.57. Solve for x: 5x— 15x + 6 <16A) x <1B) x >1C) x <-1D) x > —1E)^I don't know.158. Solve for x: x + 3 = —2 x — 1A) x= — 8B) x= — —8C) x= 4D) x= 8E)^I don't know.A)C)D)1990 Mathematics Assessment0 214(0Grade 10 Form D^ 2959. Solve the following equation for m: y = mx + bA) m=y— b—xB) m =y b C) m=y+b— xD) m = 1 (b — y)E)^I don't know.60. The graph of x 0 or x >2 isA) ^0 2C )^I I I I I I I I I 11+• I I I I I ID)E) I don't know.This is the end of Part 3. Check your work and handin your booklet and answer sheet.1990 Mathematics Assessmenta 15I^1-'^ --" ^- ^ • -4^.-"f^ ,7-BRITISH COLUMBIAMATHEMATICS ASSESSMENT(GI JE-D1NSTUDENT FORMGrade 7 Form ABACKGROUND INFORMATIONFor each item, shade in the appropriate space on the answer sheet.1. SexA) MaleB) Female2. AgeA) 10 or less^D) 13B) 11^E) 14 or moreC) 123. What program are you in?A) Regular Grade 7 program in EnglishB) Early French ImmersionC) Late French ImmersionD) Programme-cadre de francais4. In this class, mathematics is taught inA) English.B) French.For the next three items, decide to what extent you agree or disagree.5. You have to be able to do mathematics to get a good job when you growup.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly Agree6. Most people use mathematics in their jobs.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly Agree1990 Mathematics Assessment14-92^Grade 7 Form A7. When I leave school, I would like a job where I have to use mathematics.A) Strongly DisagreeB) DisagreeC) Do not knowD) AgreeE)^Strongly AgreeFor each of the next four items, three answers are needed.A) Tell how important you think the topic is.B) Tell how easy you think the topic is.C)^Tell how much you like the topic.If you are not sure what a topic means, leave its three answers blank.8. Learning geometryA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult dislikeundecided undecided^undecidedimportant easy^likevery important^very easy^like a lot9. Working with data and graphsA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lot10. Learning to use calculatorsA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant ea-,y likevery important^very easy^like a lot1990 Mathematics AssessmentI SOGrade 7 Form A^ 311. Learning strategies for problem solving like lookingfor patterns and making modelsA^ B^ Cnot at all important^very difficult^dislike a lotnot important^difficult^dislikeundecided undecided undecidedimportant easy likevery important^very easy^like a lotFor the next three items, think of your mathematics classesduring a typical school week.12. We use computers in our mathematics class.A) Almost every dayB) OftenC) SometimesD) RarelyE) Never13. The teacher helps individual students.A) Almost every dayB) OftenC) SometimesD) RarelyE) Never14. We review our homework and discuss solutions.A) Almost every dayB) OftenC) SometimesD) RarelyE) Never1990 Mathematics Assessment4Use this page for your rough work.1990 Mathematics AssessmentAsaGrade 7 Form A^ 5ACHIEVEMENT SURVEY1. Subtract: 2008— 189 A) 819B) 1181C) 1819D) 2181E)^I don't know.2.^As of June 1, 1976, the population of Canada was 22 589 416.Round off 22 589 416 to the nearest ten thousand.A) 22 580 000B) 23 000 000C) 22 600 000D) 22 590 000E) I don't know.3. Meg wants to mail party invitations to 36 friends. Envelopesare only sold in packets of 15 and cost 75¢ per packet. Howmuch will she have to spend for envelopes?A) $1.50B) $1.80C) $2.25D) $2.70E)^I don't know.1990 Mathematics Assessment►536^ Grade 7 Form A4. The value of 3 + 4(5 + 2) isA) 25B) 26C) 31D) 49E)^I don't know.5. Estimate the product: 9.785 x 11.134 x 2.9065 x 8.910A) 3000B) 2000C) 300D) 200E)^I don't know.6. Write —18 2 in lowest terms.A)^3 42B) 3C) 11D) 1 1 -E)^I don't know.1990 Mathematics AssessmentGrade 7 Form A7. Which one of the following numbers is largest?A) 0.694B) 0.07C) 0.76D) 0.0816E)^I don't know.8. Dividing by 1000 is the same as multiplying by which one ofthe following?A) 0.01B) 0.001C) 0.000 1D) 0.000 01E)^I don't know.19. John had 12 baseball cards. He gave —3 of them to Jim. Howmany does John have left?A) 4B) 6C) 8D) 9E)^I don't know.1990 Mathematics Assessment155"8^Grade 7 Form A210. Each of the students in the drama club ate —3 of a pizza at theyear-end party. If they ate 12 pizzas in total, how manystudents are there in the club?A) 8B) 18C) 24D) 36E)^I don't know.11.^Which one of the following is equivalent to 2 : 3 ?A) 3 : 4B) 5 : 12C) 3 : 2D) 4 : 6E) I don't know.12. A machine seals 225 boxes in 3 hours. There are 1000 boxesto seal. How many will be left unsealed after an 8-hour shift?A) 400B) 600C) 800D) 925E)^I don't know.1990 Mathematics AssessmentGrade 7 Form A^ 913. There are 140 players in a tournament. The ratio of girls toboys is 3 : 4. How many girls are there?A) 40B) 60C) 80D) 105E)^I don't know.14. Write 45% as a fraction in lowest terms.1A) 454B) 51C) 59D) 20E)^I don't know.15. Which one of the following shows a discount of 10%?A) 30e off $3.00B) 35e off $3.00C) 40e off $3.00D) 45e off $3.00E)^I don't know.1990 Mathematics Assessment10^ Grade 7 Form A16. What is the opposite of —2?A)B)—122C) 0D) 2E)^I don't know.17. When a positive number is divided by a negative number, theanswer isA) positive.B) negative.C) zero.D) You can't tell without knowing what thenumbers are.E)^I don't know.18. Which of the following statements is false?A) Zero is smaller than any positive number.B) All positive numbers are larger than zero.C) All positive numbers are larger than all negativenumbers.D) Zero is smaller than any negative number.E)^I don't know.1990 Mathematics Assessment1 1 ^Grade 7 Form A19. The table shows the numbers of various coins found in a box.Coin Number found$1^(dollar coin) 250¢ (fifty-cent piece) 625¢ (quarter) 110¢ (dime) 35¢ (nickel) 81¢ (penny) 3Which one of the following graphs shows this?A) 86cc1.1J'`n 4Z 20B) 86ccLu420 I^1¢ 5¢ 10e 25e 50e $1COIN1e 5e 10e 25e 50e $1COINC) 8ce 6420 D) 8cc 64201e 5c 10e 25e 50e $1COIN1e 5e 10e 25¢ 50¢ $1COINE) I don't know.1990 Mathematics AssessmentIfi12^ Grade 7 Form A20. About how many is a million?A) The number of hairs on your headB) The number of grains of sand on a beachC) The number of people that could be packed ontoa soccer field standing upD) The number of tennis balls needed to fill aclassroomE)^I don't know.21. For a party game each number shown below was painted on adifferent ping pong ball, and the balls were thoroughly mixedup in a bowl. If a ball is picked from the bowl by a blindfoldedperson, what is the probability that the ball will have a 4 onit?2, 3, 4, 4 , 5 , 6, 8, 8, 9, 101A) 21B) 41C) 51D) 10E)^I don't know.1990 Mathematics AssessmentE) I don't know.Grade 7 Form A22. What spinner would you use to conduct a probabilityexperiment on your friends' favourite colours if half of themprefer blue, a third of them prefer red, and the rest like green?23. Two plastic discs are tossed in the air and, when they land,the numbers that show are added together. One of the discshas 1 on one side and 2 on the other. The second disc has 3on one side and 4 on the other. What sums are possible?A) 5 onlyB) 1, 2, 3, and 4C) 4, 5, and 6D) 1, 2, 3, 4, 5, and 6E)^I don't know.24. To find out how much time Grade 7 students spend watchingTV, whom should you ask?A) Your friendsB) The parents of Grade 7 studentsC) Grade 7 studentsD) Students in the schoolE)^I don't know.1990 Mathematics AssessmentIbi14^ Grade 7 Form A25. Which one of the following names does not read the same ifwritten on a card „and held up to a mirror?A) AVAB) EVEC) MOMD) OTTOE)^I don't know.26. Which one of the following diagrams showsthe reflection of the face in the line n? nn nn nE) I don't know.1990 Mathematics AssessmentGrade 7 Form A27. Angle A and Angle B are congruent. If Angle A has a measureof 35°, what is the measure of Angle B?A) 35°B) 55°C) 145°D) 325°E)^I don't know.28. The two triangles shown below are similar. What is themissing length on the large triangle?3A) 3 —47 1B) 2C) 24D) 30E)^I don't^know.1990 Mathematics AssessmentP • • Q• •R^S0^1^2 3 4 ■ X43211(D316^ Grade 7 Form A29. With 4 toothpicks you can make 1 small square. With 7toothpicks you can make 2 small squares, and with 10toothpicks you can make 3 small squares. What is the largestnumber of small squares that you can construct with 34toothpicks?4A) 10B) 12C) 13D) 16E)^I don't know.4^710IIIY1230. Which point has the coordinates (2, 3)?A)B)C)D)E)^I don't know.1990 Mathematics AssessmentI b4Grade 7 Form A^ 1731. A graduated cylinder contains 500 mL of water. A rock isplaced in the cylinder and the water level rises to 683 mL.What is the volume of the rock?A) 183 cm 3B) 317 cm 3C) 683 cm 3D) 1183 cm 3E)^I don't know.32. The excavation for a swimming pool is a rectangular hole thatis 10 m long, 3 m wide, and 3 m deep.• A dumptruck can carry12 m 3 of fill. How many truckloads did it take to remove thefill from the excavation?A) 3B) 7C) 8D) 12E)^I don't know.33. 250 g is how many kilograms?A) 25B) 250C) 0.25D) 2.5E)^I don't know.1990 Mathematics Assessment18^ Grade 7 Form A34. Daley's Fruit Stand is on the highway 400 m west of AshStreet. Poplar Street is 1.2 km east of Ash Street along thehighway. How far is Daley's Fruit Stand from Poplar Street?A) 401.2 mB) 520 mC) 1.6 kmD) 5.2 kmE)^I don't know.35. A small car has a fuel tank that holds 35 L of gas. The carconsumes 7.5 L for each 100 km driven. If a trip is 250 km,how much gas remains if the trip was started with a full tank?A) 16.25 LB) 18.75 LC) 53.75 LD) 1840 LE)^I don't know.36. Which one of the following stands for the product of a numberand 6?A) y+ 6B) y-6C) 6yD) 6E)^I don't know.1990 Mathematics AssessmentGrade 7 Form A37. Of the following expressions, which one represents a numbern increased by 5?A) 5 — nB) n + 5C) 5 < nD)5nE)^I don't know.38. Gary works G hours for $7 per hour and Andy works A hours for$5 per hour. What is the total amount of money that they arepaid?A) 7A + 5GB) 12(A + G)C) 12AGD) 7G + 5AE)^I don't know.39. When the input is x the output isA) 19B) 2x — 1C) 2x + 1D)E)^I don't know.INPUT OUTPUT3 74 95 116 137 158 17. .. •• •x1990 Mathematics Assessment20^ Grade 7 Form A40. Solve: 8 = 16A) 2B) 8C) 24D) 128E)^I don't know.1990 Mathematics Assessment
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The relationship between attitudes towards specific mathematics topics and achievement in those domains Walsh, Carmel Frances 1991
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Title | The relationship between attitudes towards specific mathematics topics and achievement in those domains |
Creator |
Walsh, Carmel Frances |
Date Issued | 1991 |
Description | The purpose of this study was to investigate the nature of the relationship between high school students’ attitudes towards particular mathematics topics and their achievement in those areas. In order to examine this relationship, data collected by the 1990 British Columbia Mathematics Assessment concerning students in Grade 7 and Grade 10 were analyzed. This assessment involved over 37 000 students at the Grade 7 level and over 31 000 Grade 10 students. Data concerning students’ perceptions as to the importance, difficulty, and likeability of various mathematics topics were collected by the assessment. Achievement scores based on student performance on a 40-item, multiple-choice test of mathematical ability were also obtained. Each of the domains of the British Columbia mathematics curriculum were represented on the achievement test. Achievement items were also constructed at three cognitive behaviour levels: computation, comprehension, and application/problem solving. Geometry and data analysis were the two domains of the mathematics curriculum which were the focus of this work. The data relating to these topics were analyzed through the use of chi-square analysis. Matrices were designed which compared students’ perceptions of geometry and data analysis with their achievement in those domains. Each of the three components of attitude—difficulty, importance, and likeability—were treated as independent variables. Chi-square values were determined for each matrix and an analysis of the patterns exhibited by the cells was also undertaken. With one exception, each matrix had a chi-square value which was significant at the 0.0001 level. The remaining matrix was significant at the 0.001 level. An examination of the Grade 10 data indicated that a significant relationship between students’ attitudes towards geometry and data analysis and students’ achievement in those domains existed. The number of students who considered geometry or data analysis to be important, easy, or likeable and who also obtained good scores on the achievement portion of the assessment was greater than the expected value for those cells of the matrices. Likewise, the number of students who indicated that geometry or data analysis was not important, was difficult, or disliked the topic and who also obtained low achievement scores was greater than the expected value. Similar patterns were observed when students’ overall achievement in mathematics was compared with their attitudes towards data analysis and geometry. Students in Grade 7 generally achieved higher scores in the mathematics assessment and held more favourable views towards data analysis and geometry than did students in Grade 10. However, results showed that the relationship between each of the components of attitude and achievement in geometry and data analysis followed trends similar to, but not as strong as, those found for students in Grade 10. For the purposes of this study, the Grade 10 data were also separated into two groups. The data concerning students enrolled in the more challenging Mathematics 10course were compared with the data relating to students enrolled in the less difficult Mathematics 10A course. Students enrolled in the Mathematics 10 course achieved higher scores and held more positive views towards data analysis and geometry than did the students enrolled in Mathematics 10A. The relationships between each of the components of attitude studied and achievement within each of the domains, however, were similar for both groups of students. |
Extent | 5852164 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-10 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0086400 |
URI | http://hdl.handle.net/2429/1784 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of Curriculum and Pedagogy (EDCP), Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1992-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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