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The Relationship Between Attitudes and Performance inBusiness CalculusConcetta Depaolo, Constance H. Mclaren,

To cite this article:Concetta Depaolo, Constance H. Mclaren, (2006) The Relationship Between Attitudes and Performance in Business Calculus.INFORMS Transactions on Education 6(2):8-22. http://dx.doi.org/10.1287/ited.6.2.8

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The Relationship Between Attitudes andPerformance in Business Calculus

Concetta Depaolo And Constance H. MclarenIndiana State University

College of BusinessOperations Management & Analysis Analytical Department

Terre Haute, IN [email protected]@indstate.edu

Abstract

Undergraduate business curricula include coursework in both statistics and calculus, subjects that can bedaunting to even capable students. This study examines undergraduate business students' attitudes towardand performance in both business statistics and calculus, and determines that after controlling for ability, attitudesplay a significant role in performance. Although self-reported attitudes become more positive over the courseof the semester, attitudes toward calculus are less positive than those toward statistics, and negative attitudesare related to lower exam scores. For students with no prior calculus background, this relationship betweennegative attitudes and poor exam performance appears to be particularly strong. The conclusions have impli-cations for increasing learning by improving attitudes.

Editor's note: This is a pdf copy of an html document which resides at http://ite.pubs.informs.org/ Vol6No2/DePaoloMcLaren

1. Introduction

In the mid 1990s, curriculum planners in the businessschool at a medium-sized U.S. comprehensive univer-sity recommended that a proof-based, one-semestercalculus course taught in the math department nolonger be required for business majors. Instead, instruc-tion in basic calculus topics with more relevance tobusiness applications would be provided by businessfaculty as an independent unit within an existingcourse.

Concurrent curriculum review led to a reorganizationof the topics in the required business statistics, man-agement science, and operations management classes.The result was the creation of three required one-semester (three credit hour) classes: a Business Statis-tics I class (covering descriptive statistics, probability,inference, and quality concepts), a Business StatisticsII class (including regression analysis, forecasting, anddecision analysis), and an operations managementclass (incorporating optimization in addition to typicalOM topics). Staffing and curriculum considerationsled to the decision to place calculus instruction within

the second business statistics course; however, statisticsinstruction is not calculus-based.

The independent calculus unit within Business Statis-tics II occupies approximately four weeks, or twelve50-minute class sessions, and comes at or near the endof the fourteen-week semester. Courses required laterin the curriculum, such as finance and operationsmanagement, use basic calculus concepts and theBusiness Statistics II class is a prerequisite for them.

Multiple sections of both required statistics coursesare offered each semester, are usually taught bytenured/tenure track faculty, and typically consist of35-40 students each. Non-business majors, includingmath and economics majors, rarely enroll in thecourses. Before enrolling in the first business statisticsclass, a student must have completed prerequisitecourses in algebra and computer tools. All sectionsuse the same textbook for both statistics classes, mostrecently Statistics for Business and Economics, 9e, byAnderson, Sweeney, and Williams.

Institutional research data (see First Year ExperienceSurveys) gathered from these business students indi-

© INFORMS ISSN: 1532-05458INFORMS Transactions on Education 6:2(8-22)

DEPAOLO & MCLARENThe Relationship Between Attitudes and Performance in Business Calculus

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cate that most are of traditional college age (over 90%entered college within one year of high school gradu-ation) and many are first generation college students(fewer than 25% of them come from families whereboth parents have a college degree). Over 35% reportedthat they did well in high school without studying andanother 12% reported not studying at all. When askedwhich academic area would be the hardest, 30-40% ofthem listed "math." This apprehension about mathseems to confirm what instructors have observed andappears consistent with conclusions from the CRAFTYCurriculum Foundations Project of the MAA (2000):

Business students, although able, are often math phobic. Courses should strive to lessenmath phobia, enable students to be more comfortable with mathematics, and help studentsappreciate the relevance of mathematics.

Consistent with this recommendation, the placementof calculus within the business curriculum rather thanin the math department allows instructors to offer amore appropriate level of instruction (applicationsrather than proofs) and to provide more meaningfuland relevant examples. Teaching materials focus onapplications and the inexpensive and friendly text (Forgotten Calculus, 3e, by Bleau) includes a variety ofexamples and the solutions to all exercises. The calcu-lus topics covered include functions, differentiation,unconstrained optimization of one variable, and simpleintegration. Yet despite the basic level of the materialand the authors' conscious efforts to provide activelearning within a supportive environment, some stu-dents who have performed well on the course's earliermultiple regression and business forecasting examsexpress apprehension about the study of calculus andscore poorly on the calculus exam. This behavior hasconsistently frustrated instructors and has raisedquestions about the role of attitudes and math phobiain performance.

To this end, a study was devised to determine if theauthors' observations regarding the relationship be-tween attitudes and performance were accurate. Theunusual nature of the course provided a unique oppor-tunity to differentiate, in an unchanging environment,attitudes toward and performance on exams in calcu-lus as compared to other quantitative topics includingbusiness statistics. Before discussing the researchquestions, methods, and results of the study, a briefoverview of theoretical background and relevant liter-ature is given.

2. Theoretical Background

2.1. Ingredients of Learning

Before investigating how instructors can positivelyaffect learning, one must understand the ingredientsthat contribute to learning, including those related tocognitive functions, environment, and behavior. Assummarized by Kirton (2003), three cognitive functionsare involved in an individual's learning or problemsolving capacity: cognitive effect, cognitive resource,and cognitive affect. Cognitive effect includes a per-son's ability and preferred learning style. Cognitiveresource refers to a person's knowledge, skills andexperience. Cognitive affect involves an individual'sattitudes, beliefs, values, needs and wants; it serves todirect a person's motivation or energy to complete atask or achieve a goal. All three cognitive functionsoperate within an environment, called the social effect,that is external to the individual. Environment refersto culture, climate, and interaction with others. Inacademia, it includes the class setting, teaching tech-niques, and any feedback provided to the student(Kirton, 2003).

The actions that constitute an individual's behaviorare the product of many operating variables, includingcognitive functions and the environment (Kirton, 2003).For example, cognitive functions, including ability,motivation, and attitudes, will determine whethersomeone is willing and able to perform a task. Theenvironment has a reciprocal relationship with behav-ior. An individual's behavior influences the environ-ment, and behavior is adjusted as feedback from theenvironment is processed (Kirton, 2003).

Although ability is recognized as a major contributorto student learning, it is not the only important consid-eration. Many studies measure the effects of factorsother than ability on learning in statistics and mathe-matics courses. Some focus on attitudes, while othersinclude environmental influences. A brief overviewof several studies follows.

2.2. Factors Associated with Achievement

Many studies have investigated the effects of factorssuch as attitudes, anxiety, and math background andability on achievement in statistics courses (e.g. Robertsand Saxe, 1982; Harvey, Plake and Wise, 1985; Sutarso,1992; Feinburg and Halperin, 1978; Adams and Hol-



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comb, 1986). Lalonde and Gardner (1993) investigatedmath background and ability, math and statisticsanxiety, motivational intensity, and attitudes to see ifthey were related to achievement, and found that mathaptitude lessens anxiety, which improves attitudesand motivation, which leads to greater effort, whichleads to increased achievement. Tremblay, Gardner,and Heipel (2000) built upon Lalonde and Gardner 'smodel by adding motivational variables including in-terest in math and grades in previous courses. Theyfound that low levels of aptitude, lack of interest inmath, and unfavorable attitudes toward the courseresult in high levels of anxiety, which lead to poorperformance.

2.3. Attitudes and Anxiety

Authors and instructors have observed and document-ed negative attitudes toward statistics, math and/orother quantitative courses, confirming the relationshipbetween cognitive affect and learning. For example,Dillon (1982) reports that students felt " panicky, " "sick, " " worried, " " frustrated, " " doomed, " and "overwhelmed " about having to take statistics. Someauthors (e.g. Gal and Ginsburg, 1994; Gordon, 1995)suggest that these negative attitudes relate to highschool math experiences, and that perceptions ofstatistics as not useful for jobs or future study maylead to feelings of dislike or disinterest in the subject.Widespread observations by instructors of statisticsno doubt led to the creation of assessment instrumentssuch as the SAS − Statistics Attitude Survey (Robertsand Bilderback, 1980) and the ATS − Attitudes TowardStatistics (Wise, 1985) that measure student compe-tence, attitudes, interest, and beliefs about statistics.

Studies that have investigated statistics anxiety includeOnwuegbuzie and Wilson (2003), who cite level ofbasic mathematical skill, extent of and success withprior mathematics courses, and mathematics anxietyas explanatory factors, and Schacht and Stewart (1992),who reference factors contributing to anxiety amongstudents and suggest class activities to reduce it.

Some authors have suggested a link between statisticsanxiety and math anxiety (e.g. Zimmer and Fuller,1996). Tobias (1991) discusses steps in reducing mathanxiety, concluding " My research with six hundredundergraduates proved, instead, that most averagestudents have all the cognitive equipment they needto do advanced algebra, intermediate-level statistics,

and college calculus. The problem is that they don'tbelieve they do. " (p. 1)

2.4. The Effects of Environment on Learning

Many authors have investigated the role of environ-ment on learning. Moos (1979) discusses how learningenvironments affect student satisfaction, learning, andpersonal growth. The author reports more success ina climate with high student involvement, personalstudent-teacher relationships, and innovative teachingmethods.

Several authors have discussed or studied the effectof environment within college classes, and specificallyin statistics courses. For example, Fraser and Fisher(1982) concluded that there were many significantcorrelations between outcomes and perceptions ofenvironment. Gal and Ginsburg (1994) assert thatlearning statistics requires a supportive atmospherewhere students feel safe, are motivated, and are com-fortable with experimentation and inconclusive results.Tremblay, Gardner and Heipel (2000) concluded thatattitude toward the instructor has a direct effect onmotivation, which has positive effect on achievement.A similar finding was reported by Onwuegbuzie andWilson (2003). Some authors suggest that the use ofreal-world situations, hands-on activities, or smallgroup exercises can reduce anxiety, which should, inturn, increase satisfaction and achievement (e.g. On-wuegbuzie and Wilson, 2003; Mvududu, 2003; Schachtand Stewart, 1992; Garfield, 1993).

3. Purpose of This Study

Although there are multiple ingredients of learning,including ability (cognitive effect), preparation (cogni-tive resource), and environment (social effect), the fo-cus of this study is on attitudinal factors (cognitiveaffect) that influence learning. In consideration ofcommon sense and a large body of literature, it isconceded that ability, preparation, and environmenthave an effect on achievement. This study seeks toexplore the existence and nature of the relationshipbetween attitudes about calculus/quantitative coursesand performance, after controlling for ability, prepara-tion, and environment.

The instructors involved in this study strive to createa non-intimidating environment for the Business



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Statistics II class. Students are encouraged to askquestions, are involved in hands-on activities, and aregiven real-world examples to motivate learning. Classsections of this course are small enough for instructorsto know students' names and for students to becomecomfortable talking with their instructors both in andoutside of the classroom. This atmosphere is intendedto increase mutual trust and to encourage students toshare their reactions about what they are learning.Based on conversations with students, the instructors'assessment of student engagement, and performanceon exams, the authors have developed five informalobservations about the relationship between attitudesand performance that this study seeks to validate:

• Observation 1: Student attitudes regarding calculusand quantitative classes become more positive overthe course of the semester as students become morecomfortable with the instructor and the curriculum;that is, at the onset students tend to anticipate thatthe material will be more difficult than it turns outto be.

• Observation 2: Students appear to have differentattitudes toward calculus than they do aboutstatistics and other quantitative classes. Some stu-dents who are performing well going into the cal-culus unit still express an inordinate amount ofapprehension about calculus, feeling that it willbe much more difficult than the statistics portionof the course.

• Observation 3: Even capable students may havenegative attitudes that detract from their perfor-mance on exams. Frequent comments such as " I'mjust not good at math " are sometimes a reflectionof ability; however they appear to become a self-fulfilling prophecy when spoken by students whohave appeared to follow along in class, to correctlycomplete assignments, and to thoroughly preparefor the exam,.

• Observation 4: Expressions of anxiety and negativeattitudes are more prevalent throughout the calcu-lus unit than during the statistics portion of thecourse. Furthermore, these attitudes appear to co-incide with lower average scores on the calculusexam compared to the statistics exams. This de-crease in performance is widespread, includingeven capable students who were quite successfulwith the earlier statistics material. For example,almost 20% of students earning an A or B average

on the statistics exams earn a D or an F on the cal-culus exam.

• Observation 5: Students who have never takencalculus before appear to have negative attitudesabout the subject and are more likely to exhibitsigns of " math anxiety " that impedes their perfor-mance on exams. They appear to anticipate thatthe study of calculus will be much more difficultthan it is, even when the instructors try to allaytheir fears. To these students, the fear of the un-known appears to be great.

To determine the validity of these observations, severalresearch questions were developed:

1. Do student attitudes improve significantly overthe course of the semester?

2. Are student attitudes toward calculus distinct fromtheir attitudes about statistics and other quantita-tive classes?

3. Are attitudes significant in predicting exam scoresafter controlling for ability, preparation, and envi-ronmental factors?

4. Is the relationship between attitudes and perfor-mance different and/or stronger for calculus thanit is for statistics?

5. Is the relationship between attitudes and perfor-mance different for those who have not taken cal-culus than it is for those who have?

Methods used to investigate these questions are de-scribed in the next section.

4. Methods

This study, approved by the university's institutionalreview board, involved classes taught by the authorsover three spring semesters. Data collected consistedof student records, performance on in-class exams,and a paper and pencil survey. Students present inclass when the surveys were distributed overwhelm-ingly agreed to participate in the study, with only ahandful of exceptions. After giving informed consentand agreeing to let the instructors access their studentrecords and use their exam scores, these students filledout the survey during class time. Students who wereabsent from class or who subsequently dropped thecourse (approximately 14% and 2%, respectively, of



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the students originally enrolled in the targeted classes),were not included in the study. Consistent with poorattendance, these students earned lower exam grades,on average, than those who did participate, and weremore likely to be upperclassmen who were taking theclass later than is customary.

Because of the unique placement of a calculus unitwithin a statistics course and the focus on calculusperformance, it was decided that instruments specificto this course should be developed. The first cohort,consisting of 91 students enrolled in Spring 2002,completed three surveys over the course of thesemester one at the start of the term, one after thestatistics portion of the class but before the calculusunit, and one at the end of the semester when the cal-culus unit was complete. The first survey asked back-ground questions about the students' high school mathexperiences in addition to questions about current at-titudes toward math, quantitative classes, and calculus.Subsequent surveys for the first cohort tracked onlythe attitudinal questions.

Additional cohorts were enrolled in the Springsemesters of 2003 and 2005. These students gave in-

formed consent to access their student records andwere administered only the first survey near the begin-ning of the semester. In each of these terms, 69 studentsparticipated, resulting in a total sample size of n = 229.

In the surveys, all attitudinal questions were measuredon a 5-point Likert scale, with students marking oneof five multiple-choice responses, labeled from (1)Disagree Strongly to (5) Agree Strongly. Students wereasked to self-report usual math grades from highschool (5 = As, 4 = Bs, 3 = Cs, 2 = Ds, 1 = Fs). In addition,student records were accessed to ascertain each stu-dent's SAT Math score (or ACT score converted to anSAT equivalent according to admissions practices) andgrades in the prerequisite algebra class (measured ona standard 4-point scale). Each student's gender andclass standing (sophomore, junior, or senior) were alsorecorded, and exam scores were provided by the in-structor. Student scores on the calculus exam providedthe measure of performance in calculus; performancein statistics was measured by the mean of the scoreson the two midterm exams covering regression andforecasting. The variables included in the study arelisted in Table 1.

Table 1:

Student records did not contain SAT Math and mathgrade data for some of the students. Of the 229 partic-ipants, 31 did not have SAT (nor converted ACT) in-formation in their student records. An inspection ofthese student records indicated that these students

were typically transfer, nontraditional, or internationalstudents. In addition, four students did not have agrade for the prerequisite math class. Three of thesestudents were transfer students who were also missing



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SAT scores. In all, 32 students (14%) were missing ei-ther SAT or a math grade or both.

An investigation of the differences between thosemissing and not missing data, which is discussed indetail in the Analysis and Results section, revealedonly slight differences between the groups. It was de-sired to include the cases with missing values in theanalysis so that the sample would more accuratelyreflect the population that includes such transfer,nontraditional, and international students. Using amethod discussed in Cohen & Cohen (1975), missingvalues were replaced with the mean and a binaryvariable was created to indicate that these cases weremissing data. This indicator variable was then includedas an independent variable in analyses such as the re-gression models described below.

To explore research question 1, data from the first co-hort that tracked changes in attitudes over the courseof the semester were analyzed using paired t-tests andthe Bowker test for symmetry (Bowker, 1948). To de-termine if attitudes toward calculus were distinct fromattitudes toward other quantitative subjects (researchquestion 2), a Principal Components Analysis was

conducted on data from the first survey common toall three cohorts. These same data were used to devel-op multiple regression and Principal Components re-gression models to investigate the remaining questionsregarding the effects of attitudes on performance. Theresults of the analyses are discussed below, and all arebased on a 5% significance level unless otherwisespecified.

5. Analysis and Results5.1. Descriptive Statistics

Of the 229 students in the study, 51% were female and38% had upperclass standing. The mean SAT Mathscore was 500, and 36% had earned an A in the prereq-uisite algebra class. Slightly more than 50% of thesample indicated that they liked their math classes inhigh school, and 28% reported taking a calculus coursein high school. More than half of respondents (53%)indicated that they thought math was hard in highschool. A summary of the descriptive statistics for thedata set is shown in Table 2. In cases where relativefrequencies are reported, " 5 " always corresponds tothe most positive response.

Table 2:

5.2. Identification of Control Variables

When the differences by gender and upperclass (junioror senior) status were investigated using independentsamples t-tests and chi-square tests for independence,women were found to have significantly higher mathgrades and exam scores than men, and exhibited morepositive attitudes on several of the attitudinal surveyquestions (AttMathHS, AttMathNow, CalcAtt, Cal-cVal). These results are consistent with instructor ob-

servations that women have better attendance andstudy habits, and tend to be more conscientious aboutcompleting assignments than men.

Students with upperclass standing were found to havesignificantly lower math grades, exam scores, and SATscores, and had indicated significantly more negativeattitudes on survey questions relating to attitudes to-ward quantitative classes (AttMathHS, AttMathNow,EasyNow). Students who are on-track take the coursein their sophomore year, so those who enroll later may



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have avoided the class, have had trouble meeting theprerequisites, or have been unsuccessful in a previousattempt. Thus, lower grades and more negative atti-tudes are not surprising.

Average exam scores of one instructor were significant-ly lower than exam scores of the other. A review ofexams indicates that one instructor tended to createlonger and more difficult exams. No other differencesbetween instructors were found in any of the surveyquestions, math grades, or SAT scores.

When students enrolled in different terms are com-pared, ANOVAs show no significant differences inexam scores or SAT scores and chi-square tests for in-dependence indicate no significant differences ingrades, and only two significant differences in atti-tudes. Specifically, students enrolled in Spring 2002felt that calculus would be significantly more difficultthan students enrolled in Spring 2005 (as measuredby the variables EasyCalc and CalcComp).

Chi-square tests and t-tests were used to comparestudents who had taken calculus to students who hadnot taken calculus. T-tests revealed that those who hadtaken calculus before had significantly higher examscores and SAT scores, while chi-square tests showedthey earned significantly higher grades in high schooland in the prerequisite math course. Additional chi-square tests suggested that these students were morelikely to believe that the calculus material would beeasier; however, they did not feel that high school orcurrent math classes were easier, nor did they havesignificantly different attitudes toward calculus thanthe students who had not taken it.

SAT scores and/or grades in the prerequisite algebraclass were missing for some students. Chi-square andt-tests show no differences in attitudes, exam scores,and most survey questions between these studentsand others. The only difference identified was thatthose missing data were less likely to report earningAs in high school (HSGrades).

Because of the differences found by gender, upperclassstatus, instructor, term, missing, and whether the stu-dents had taken calculus, these variables were includedas control variables in the regressions used to investi-gate the research questions.

5.3. Research Question 1

To determine if student attitudes regarding quantita-tive classes and calculus improve significantly overthe course of the semester, responses to survey ques-tions were compared at various points during thesemester. These data were gathered only for the 91students participating in the Spring 2002 semester.Because a small number of these students did not takeor fully complete either the second or third surveys,the sample sizes in the following analyses will differ.

When responses to current attitudes questions at thebeginning of the semester are compared to thosemeasured before the calculus unit using paired t-tests,only one significant difference was found: studentshave more positive attitudes about math/quantitativeclasses in the middle of the term (mean = 3.51) thanthey do at the beginning (mean = 3.21, T = -3.69, p <0.001, n = 90, where a 5 indicates the most positive at-titude).

On the other hand, several changes were found in at-titudes when survey responses before the calculus unitare compared to those measured after the calculus unit(see Table 3, where 5 indicates the most positive atti-tude). In particular, significant positive changes werefound in students' attitude toward calculus (CalcAtt)and how easy they felt calculus to be (EasyCalc, Calc-Comp). To alleviate any concerns about the use of atest for means of ordinal data, even though the samplesize is large, each of these significant differences wasconfirmed using a non-parametric test, the Bowkertest of symmetry, which tests for non-random changesin subject responses to categorical data over time (seeBowker, 1948).

Table 3:


To determine if student attitudes toward calculus aredistinct from their attitudes about other quantitativeclasses, a Principal Components Analysis was per-



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formed on the independent variables in the study(excluding binary indicator variables). The analyseswere first performed without 32 (of 229) cases withmissing SATMath or MathGrade, and then again usingall 229 cases with missing values replaced by themeans. The results of the PCAs were nearly identical:the number of components extracted is the same, thepercentage of variation explained is within 1%, andall of the individual items load most strongly on thesame components with most factor loadings within.04 of their previous value. Therefore it was judged

appropriate to replace the missing values by theirmeans so that the data set would be as complete aspossible.

Three components with eigenvalues over one wereextracted, and a varimax rotation was applied. As Ta-ble 4 shows, there appear to be two components relat-ed to attitudes, one more strongly related to calculusattitudes, and one more strongly related to attitudestoward math/quantitative classes.

Table 4:


To determine if attitudes are related to performancein calculus and statistics, regression analyses wereused. First, two ordinary least squares (OLS) multipleregression models were run in which calculus examscore and the mean statistics exam score were the de-pendent variables. Independent variables includedattitudinal and ability/preparation measures. Binarycontrol variables for female, upperclass status, priorstudy of calculus, instructor, and the academic termwere also included as independent variables. An addi-tional binary variable was included to indicate whethermissing data had been replaced by the mean.

As Table 5 indicates, some of the attitudinal variableswere significant in predicting exam scores in bothmodels. The variable representing high school mathattitudes (AttMathHS) was significant in the statisticsmodel, and the variable representing how easy mathwas in high school (EasyHS) was significant in thecalculus model. Current attitudes toward quantitativeclasses (AttMathNow) was significant in both models.Also significant in predicting both types of exam scores

were SAT Math scores, grades in the prerequisite mathclass (MathGrade) and in high school (HSGrades),while gender and instructor were only significant inthe statistics model. The variable indicating whetherthe student had taken calculus before was not signifi-cant in either model, nor was the variable indicatinga missing SAT score or math grade.

The models shown in Table 5 were examined for evi-dence of multicollinearity. The largest Variance Infla-tion Factor is 2.455 (tolerance = 0.407), which indicatessome multicollinearity. Some of the coefficients appearto have opposite sign than one might expect. For ex-ample, in the calculus model, the negative coefficientof EasyHS indicates that the easier students foundmath in high school, the lower the expected calculusexam grade. In the statistics model, AttMathHS has anegative coefficient, indicating better attitudes in highschool lead to lower expected exam grades. Additionalevidence of multicollinearity results from the examina-tion of the criteria put forth by Belsely, Kuh & Welsch(1980). A conditioning index in excess of 30 coupledwith two variance proportions greater than 0.5 indi-cates that some multicollinarity is present.



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Table 5:

In light of this finding, Principal Components Regres-sion was used to test for the effects of the independentvariables on calculus and statistics exam scores. Basedon the Principal Components shown in Table 4, factorscores were created automatically by the statisticalsoftware using the regression approach (see e.g.Tabachniak & Fidell, 2001). For the purposes of theregression model, the components were named on thebasis of the variables most strongly loading on eachas follows:

• Component 1: Calculus Attitudes

• Component 2: Quantitative Attitudes

• Component 3: Ability/Preparation

Note that the final component, consisting of SAT Mathscore, grade in the prerequisite algebra class, andgrades in high school, was named Ability/Preparationbecause it is difficult to separate the effects of abilityand preparation within these measures. These threecomponents were used for the attitudinal and abili-ty/preparation variables in the Principal ComponentsRegression model, along with the seven binary controlvariables. Missing data values were replaced with themeans, giving a total of n = 229 records. The results ofthis analysis are shown in Table 6. The b i and betavalues represent, respectively, the unstandardized andstandardized regression coefficients.

In both the calculus and statistics models, both thecalculus attitudes and the quantitative attitudes com-ponents were significant. As expected, the abili-

ty/preparation component was significant, and showeda large effect on scores compared to the other variables.The coefficients of the (standardized) factors indicatethat as the ability/preparation factor score increasesby one standard deviation, then scores on the calculusexam could be expected to increase by 8.2 points outof 100, while statistics scores increase by an estimated6.5 points. Increases of approximately 2 to 3 points canbe expected in both models as the attitude factor scoresincrease by one standard deviation. Those who hadtaken calculus before did not score significantly higherthan others when other variables are taken into ac-count.

Also significant in both models were gender and in-structor. The female variable had a significant positiveeffect in both models. An examination of gradebookdata reveals that women enrolled in this course tendto have fewer absences, and are more consistent incompleting and earning higher grades on assignments,which might shed light on the relationship betweengender and exam scores.

The models also indicate that one instructor had sig-nificantly lower calculus and statistics exam scores.Possible reasons for this result include that this instruc-tor's exams may also be more difficult or may be scoredless generously. Another possibility is that studentsof this instructor are more likely to be upperclassmen(39% vs. only 21% of the other instructor's students)who might have delayed taking the course or are re-peating it.



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Table 6:

The Variance Inflation Factors for these models are allunder 1.40 (tolerances above .71), so there is little evi-dence of multicollinearity. The regression assumptionsappear to be met for these models except for someheteroscedasticity, which is likely to be caused bystudents with lower ability/preparation factor scoresshowing a wide range of exam scores, while studentswith high ability/preparation scores generally earnhigh exam scores with little variation.

To determine whether this heteroscedasticity has aneffect on which variables are significant in the models,generalized least squares (GLS) regressions were run.In the calculus model, each of the variables foundsignificant in the original model were also significantin the GLS model. With respect to the statistics model,the only substantive change in significance was thatthe quantitative attitudes component appeared onlymarginally significant (p = 0.081) in the GLS model;however, the significance of the calculus attitudescomponent remained.

Finally, to alleviate any concerns that replacing missingSAT Math scores with the mean has unduly affectedthe results, all analyses were rerun without the SATMath variable. The Principal Components Analysisextracted slightly different components; however, thePrincipal Components regressions indicated that, al-though the R-squared values decreased substantially,a combined grades/attitudinal component and a per-ceived easiness component were both significant inpredicting both calculus and statistics exam scores.This result also supports the result that attitudes arerelated to performance.


To determine if the relationship between attitudes andperformance is different and/or more pronounced forcalculus than it is for statistics, the multiple and Prin-cipal Components regression models were examined.The differences in the OLS regressions (see Table 5)on the calculus and statistics scores were small:AttMathHS (attitude toward math in high school) wassignificant in the statistics model but not in the calculusmodel, whereas EasyMath (how easy math was in highschool) was significant in the calculus model but notthe statistics model. The Principal Components Regres-sions (see Table 6) indicated that both calculus atti-tudes and quantitative attitudes were significant inpredicting calculus and statistics exam scores. Theseresults still hold even when outliers (representing threestudents with extremely low calculus exam scores andtwo with extremely low statistics scores) were re-moved.

The results of Table 6 indicate that both the unstan-dardized and standardized regression coefficients forthe attitudinal factors are larger in the calculus modelthan in the statistics model. Additional analyses revealthat both the zero-order correlations and the squaredsemi-partial correlations associated with the attitudefactors are also larger in the calculus model (see Table7). Although these differences are not large, they doindicate that the attitude factors have slightly moreimportance in the calculus model. Therefore, althoughrelationship between attitudes and exam scores ap-pears to be similar in nature for both calculus andstatistics, attitudes do appear to play a slightly largerrole in calculus performance.



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Table 7:


The final question to be investigated is whether therelationship between attitudes and performance isdifferent for those who have and have not taken calcu-lus before. In all of the regression models run, the in-dicator variable TookCalc was not significant, suggest-

ing that when all other variables are controlled for,prior background in calculus does not have an effecton performance. To further investigate if attitudeshave a different effect based on prior experience, thedata set was split into two groups (TookCalc = 1 andTookCalc = 0) and the Principal Components Regres-sions were run separately for each group. The resultsfor calculus scores are shown in Table 8.

Table 8:

For those students who had not taken calculus before,results indicate that in addition to ability and gender,both calculus and quantitative attitudes appear to havesignificant relationships with calculus exam scores.This model is almost identical to the overall modelshown in Table 6, with the exception that the instructorvariable is not significant here. One possible interpre-tation of this difference may be that students who havenot taken calculus before find the exam to be difficult,regardless of the instructor.

For those students who had no previous calculus ex-perience, the same variables found significant in thecalculus model also appeared significant in the statis-tics model (not shown). Again, the statistics model isconsistent with the model for all students (refer toTable 6), with the exception of the non-significance ofthe instructor variable.

For those students who had previous experience withcalculus, neither attitudinal factor had a significantrelationship with calculus exam scores; however, theability factor was significant (see Table 8). These resultsmay suggest that previous exposure to calculus canreduce anxiety and negative attitudes so that they donot detract from performance. As was found in theoverall model, the instructor with more " upperclass" students appears to have lower scores on calculusexams among those students with previous calculusexperience. Exam difficulty or scoring practices couldbe a factor in this result. The presence of a significantrelationship between the " missing " variable and cal-culus exam scores might be explained by the fact thatthere were only 9 students who had taken calculusand had missing data. Upon closer examination, it wasdetermined that the majority of these students wereinternational students. It seems reasonable to infer thatthey may have had a different level of experience with



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calculus than domestic students had, which may con-tribute to their higher scores.

The fact that gender appeared significant in the " allstudent " model for calculus exam scores (refer to Table6) and is not significant for students with previouscalculus experience might be related to effort. As dis-cussed previously, when all students are considered,there were significant differences between men andwomen with respect to the number of assignmentsmissed, scores on homework assignments, and ab-sences. However, when only students who had takencalculus before were compared by gender, no suchdifferences were found.

When the statistics model for those who had takencalculus (not shown) and the overall statistics model(see Table 6) are compared, it was found that the abil-ity factor and the gender and instructor variables werestill significant, but that the two attitudinal factorswere not. This result suggests that, for students withstronger quantitative backgrounds that include calcu-lus, negative attitudes, if present, do not appear to berelated to performance on statistics exams.

6. Discussion and Conclusions

The results of the statistical analyses indicated thatoverall, attitudes toward quantitative classes and cal-culus can improve over the course of the semester.There was a significant increase in positive attitudesabout quantitative classes throughout the first part ofthe semester while the statistics modules are taught.This result may be explained, in part, by the supportiveenvironment created by the instructors, who are acces-sible, provide real-world examples, and include hands-on activities. These improved attitudes may also bedue to success early in the course, leading students tohave increased confidence in their abilities.

After the calculus unit, a significant positive shift inthe attitudes about calculus appeared (see Table 3).The most significant changes occured for the EasyCalcand CalcComp variables, indicating that students an-ticipated that calculus would be more difficult than itactually was, supporting the idea of a calculus " pho-bia. "

Results that do not support this finding include thefact that there was no significant change from the be-

ginning to the end of the semester in how easy stu-dents felt quantitative classes were. It is interestingthat, although students indicated liking quantitativeclasses significantly more before calculus than at thestart of the term, there was a slight, marginally signif-icant decrease is these attitudes after calculus (p-value= 0.067). This result appears to suggest that anyprogress in improving attitudes made in the beginningof the term while covering statistics might be partiallynegated by the calculus unit. Even though attitudestoward calculus improved, students expressed muchless affinity for it than they did for quantitative classesin general.

The Principal Components Analysis supported theobservation that students make a distinction betweencalculus and other quantitative classes. There appearedto be two components, one more strongly related tocalculus attitudes, and one more strongly related toattitudes toward math/quantitative classes. Althoughsome of the items loaded on both factors (e.g.AttMathNow and EasyNow), clearly not all of theitems loaded on a single large attitudinal factor, indi-cating some difference in how the subjects are viewedby students.

When trying to determine the effect of attitudes onperformance, it is clear that attitudinal variables aresignificant predictors, whether one relies on the OLSregressions or the Principal Components regressions.These results hold when three outliers identified inthe calculus models and two in the statistics modelswere removed, with only slight changes in the p-val-ues. They also hold when a GLS regression model wasused to address heteroscedasticity, and when SATscores were omitted from the analyses, although theattitudinal components took slightly different form.Thus, it is safe to assume that attitudes do have animpact on performance, although that effect is smallcompared to the effects of ability/preparation. Thisfinding supports the instructors' observations thateven capable students may be hindered by apprehen-sion.

The observation that attitudes have a larger effect oncalculus performance than on statistics performancewas supported, though not strongly, by these results.The differences in the OLS regressions were slight,and the same attitudinal components were significantin both the calculus and statistics Principal Compo-nents regressions. What can be said is that the attitudi-



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nal factors have slightly more importance in the calcu-lus model than in the statistics model, as measured bythe standardized and unstandardized regression coef-ficients and the squared semi-partial and zero-ordercorrelations.

Lastly, students who had not taken calculus beforeappeared to be more strongly affected by attitudesthan students who had. For those without prior calcu-lus experience, both the calculus and quantitative atti-tudes components were significant in the PrincipalComponents regressions, in addition to the abili-ty/preparation component. For those who had calculusbefore, only the ability/preparation factor appearedsignificant. Students who have already taken a calculuscourse are likely to be more talented and their perfor-mance appears to be unaffected by their attitudes,unlike their peers without prior calculus experience.

There are several limitations to this study. First of all,the study is limited by the availability of data to mea-sure ability and preparation. Ideally, some sort of IQscore would be used to measure ability and a thoroughunderstanding of each student's high school and col-lege math experiences would be represented in vari-ables used for preparation. However, measures suchas these are not readily available; instead, the SATmath score and grades were used to represent abilityand preparation. These are admittedly imprecise andeach is likely to reflect some combination of abilityand preparation, as evidenced by the Principal Com-ponents Analysis. In addition, the use of grades intro-duces an unknown amount of variation into the study,as grades assigned by different instructors may be in-consistent.

Because the focus of this study is on performance, onlystudents who completed the course are included andthe results are not generalizable to all students whoenroll. Future research may focus on the role of atti-tudes for those students who withdraw before com-pleting the course.

While 36% to 44% of the variation in exam scores hasbeen explained, there are other measures that couldbe expected to contribute to the prediction of perfor-mance on exams that were not included in this analy-sis. For example, measures of effort, motivation, ortime spent preparing were not used. Future researchshould focus on the collection of these data, some ofwhich might be obtained through student surveys,

and how these variables contribute to the predictionof exam scores.

Despite these limitations, this study has been valuableto instructors by validating the relationship betweenattitudes and performance. Of the ingredients oflearning, instructors have no influence over ability,very little over preparation (unless remedial help canbe provided), but do have influence on attitudes andthe environment. Within the environment created bythese instructors, attitudes improved over the courseof the semester, and improved attitudes are related toincreased performance. Therefore, if instructors areaware of and address negative attitudes, especiallyamong those with limited mathematical backgrounds,students may have more success with the course.

This research shows that negative attitudes and appre-hension are negatively related to success, particularlyfor those with no prior calculus experience, lendingcredence to the CRAFTY observations about the pres-ence of math anxiety in business students. Participantsin this study were exposed to a brief, independentcalculus module taught within a familiar environment;it seems likely that business students who take a morerigorous, full-semester calculus class with a new in-structor would exhibit no less apprehension. Therefore,the conclusions of this study should be of interest tothose who teach business calculus and statisticscourses, no matter how those courses are structured.

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