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This article was downloaded by: [Northeastern University] On: 11 November 2014, At: 20:22 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/upri20 High School Mathematics Curricula, University Mathematics Placement Recommendations, and Student University Mathematics Performance Ke Wu Norman , Amanuel G. Medhanie , Michael R. Harwell , Edwin Anderson & Thomas R. Post Published online: 30 Jun 2011. To cite this article: Ke Wu Norman , Amanuel G. Medhanie , Michael R. Harwell , Edwin Anderson & Thomas R. Post (2011) High School Mathematics Curricula, University Mathematics Placement Recommendations, and Student University Mathematics Performance, PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21:5, 434-455 To link to this article: http://dx.doi.org/10.1080/10511970903261902 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with

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Page 1: High School Mathematics Curricula, University Mathematics Placement Recommendations, and Student University Mathematics Performance

This article was downloaded by: [Northeastern University]On: 11 November 2014, At: 20:22Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH,UK

PRIMUS: Problems, Resources,and Issues in MathematicsUndergraduate StudiesPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/upri20

High School MathematicsCurricula, UniversityMathematics PlacementRecommendations, and StudentUniversity MathematicsPerformanceKe Wu Norman , Amanuel G. Medhanie , Michael R.Harwell , Edwin Anderson & Thomas R. PostPublished online: 30 Jun 2011.

To cite this article: Ke Wu Norman , Amanuel G. Medhanie , Michael R. Harwell ,Edwin Anderson & Thomas R. Post (2011) High School Mathematics Curricula,University Mathematics Placement Recommendations, and Student UniversityMathematics Performance, PRIMUS: Problems, Resources, and Issues in MathematicsUndergraduate Studies, 21:5, 434-455

To link to this article: http://dx.doi.org/10.1080/10511970903261902

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all theinformation (the “Content”) contained in the publications on our platform.However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness,or suitability for any purpose of the Content. Any opinions and viewsexpressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of theContent should not be relied upon and should be independently verified with

Page 2: High School Mathematics Curricula, University Mathematics Placement Recommendations, and Student University Mathematics Performance

primary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages,and other liabilities whatsoever or howsoever caused arising directly orindirectly in connection with, in relation to or arising out of the use of theContent.

This article may be used for research, teaching, and private study purposes.Any substantial or systematic reproduction, redistribution, reselling, loan,sub-licensing, systematic supply, or distribution in any form to anyone isexpressly forbidden. Terms & Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions

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PRIMUS, 21(5): 434–455, 2011Copyright © Taylor & Francis Group, LLCISSN: 1051-1970 print / 1935-4053 onlineDOI: 10.1080/10511970903261902

High School Mathematics Curricula, UniversityMathematics Placement Recommendations, andStudent University Mathematics Performance

Ke Wu Norman, Amanuel G. Medhanie, Michael R. Harwell,Edwin Anderson, and Thomas R. Post

Abstract: Recent “math wars” have drawn attention to how well various high schoolmathematics curricula prepare students for college-level mathematics. The purposeof this study was to investigate the relationship between the high school mathemat-ics curricula and students’ post-secondary mathematics placement recommendation,specifically how students responded to the mathematics placement recommendationsand the students’ performance in the first college mathematics class.

The results showed no relationship between students’ participation in a particularhigh school mathematics curriculum and mathematics placement recommendation, orbetween student high school mathematics curriculum and students’ responses to a uni-versity mathematics placement recommendation. However, students who took a more/less difficult class than what was recommended achieved significantly lower/highergrades than those who followed the recommendation. The findings have implicationsfor high school mathematics curricula selection, post-secondary student placement, andfuture research in this area.

Keywords: High school mathematics curricula, mathematics placement, post-secondary mathematics achievement.

1. INTRODUCTION

For most high school graduates in the United States, the bridge betweenhigh school and post-secondary mathematics coursework is the mathematics

Address correspondence to Ke Wu Norman, Mathematical Sciences Building,University of Montana, Room 01, 32 Campus Drive, Missoula, MT 59812, USA.E-mail: [email protected]

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Student University Math Performance 435

placement test. However, research on the relationships between high schoolmathematics curricula and

1. college mathematics recommendations,2. student response to the recommendation, and3. the impact of students’ responses on performance in their initial university

mathematics class,

has just started to appear in the literature.Mathematics placement tests are used at many post-secondary institutions

to assess students’ readiness for mathematics, especially their preparation forcalculus. Results of mathematics placement tests are used to guide decisionsabout the mathematics class students enroll in initially. Students who begintheir post-secondary mathematics study with a course appropriate for theirbackground increase the likelihood of succeeding in their initial class as wellas subsequent mathematics courses [7]. On the other hand, misplacement of astudent into an initial mathematics course can have negative effects on attitude,self-image, and college mathematics performance. College mathematics place-ment tests may also play a role in the enrollment of about one-quarter of allfreshmen in at least one post-secondary remedial mathematics course that theyshould have completed in high school.

The common practice in universities and colleges is to gather informa-tion on students’ academic history to help make decisions on admission andmathematics course placement recommendations. This information includesstudents’ high school mathematics courses and grades as well as their mathe-matics scores from various aptitude exams such as the Scholastic AssessmentTest (SAT) and/or the ACT. These factors usually serve as reliable predictorsof university mathematics performance, and are used to place students intodifferent levels of initial mathematics courses.

Many post-secondary institutions also use mathematics placement examsdeveloped by the department of mathematics to provide course recommen-dations for incoming local or other freshmen (e.g., University of Wisconsinsystem, Iowa State University, University of Minnesota, Washington state post-secondary institution system, and the University of North Dakota). Theseuniversities are representative of large-sized research I universities in theUnited States. Available evidence suggests that mathematics placement examsused by colleges and universities generally focus on traditional topics such asalgebra, trigonometry, and calculus. For instance, the placement exam used bythe University of Wisconsin system contains three broad categories:

1. mathematics basics (e.g., arithmetic, intuitive geometry),2. algebra (e.g., linear, quadratic, exponential, and logarithm functions, and

geometry), and3. trigonometry [57].

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436 Norman et al.

Juxtaposed against the current use of college mathematics placement testsare the “math wars,” which have pitted advocates of various high schoolmathematics curricula against one another. Fueling the math wars are concernsthat students are not adequately prepared by their high school mathematicscurricula for college mathematics coursework [4, 32, 45].

The two extreme sides of the “math wars” are: the defenders of tra-ditional commercially developed (CD) mathematics curricula and advocatesof National Science Foundation-funded (NSFF) curricula [50]. Since the1990s many new NSFF curricula have been introduced into American K–12school systems. These curricula were designed to meet the National Councilof Teachers of Mathematics (NCTM) 1989 Standards [37]. In 2000, theNCTM published a revised document, Principles and Standards for SchoolMathematics [38], that expanded upon and updated the 1989 NCTM Standardsdocument [37].

Compared with CD curricula, the NSFF curricula cover broader topicsin mathematics including statistics, probability, and discrete mathematics [25,51]. In general, CD mathematics curricula stress traditional algorithms andprocedures, while NSFF curricula focus on developing student conceptual rea-soning abilities and skills in modeling-based problems situated in real lifecontexts with a commensurate de-emphasis on procedural skills [25, 51].

Advocates of CD curricula have expressed their concerns that using NSFFcurricula leads to poor college mathematics preparation [23, 59]. On the otherhand, proponents of NSFF curricula often point to the disappointing mathemat-ics achievement of all students, most of whom have completed a CD curriculum[22, 49]. They argue that changes in the development and implementation ofcurricula are needed. A number of studies of the impact of high school math-ematics curricula on students’ postsecondary mathematics achievement haveappeared since 2000 [17, 18, 23, 41, 48, 54]. However, it is important to addressthe role that post-secondary mathematics placement tests play in the impact ofhigh school mathematics curricula on college mathematics performance.

The purpose of this study is to investigate whether there is a relationshipbetween the high school mathematics curriculum a student completes, and

1. their college mathematics placement recommendation,2. students’ responses to the college mathematics placement recommendation,

and3. students’ performance in their first mathematics class as a function of

whether they followed the placement recommendation.

2. LITERATURE REVIEW

The literature on post-secondary mathematics placement tests is fairly recent.Researchers have generally found low predictive validity represented by

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modest to small correlations between placement test scores and the gradestudents earn in their initial mathematics course [9, 15, 27, 44].

Researchers have also considered other factors in examining placementrecommendations. These include students’ SAT or ACT math scores, highschool mathematics grades, percentile rank in high school graduating class,gender, ethnicity, and age. A common practice is to generate models thatproduce an overall (predicted) score that is the basis of the placement recom-mendation [13, 27, 30, 40, 46, 47, 52, 59].

Other research has focused on various features of the placement recom-mendation process, such as the format used to administer the placement test.Some post-secondary institutions use paper-and-pencil tests, which delay therecommendation, while others use computerized adaptive tests that give place-ment suggestions immediately after the test is completed [1, 6, 53]. While thereis substantial evidence in the psychometric literature that test format matters ina variety of ways [3, 29], there is no such evidence one way or the other in themathematics literature.

Studies of the relationship between students’ high school mathematicscurriculum experience, and college mathematics study experience started toappear in the literature in the 1990s [10, 33, 54]. There are conflicting findingsfrom different studies about how well NSFF students perform on college place-ment tests. Huntley, Rasmussen, Villarubi, Sangtong, and Fey [28] reportedthat students who completed three years of CORE-Plus (one of the NSFF cur-ricula) did not do as well as their CD counterparts on symbolic manipulationitems on the placement exam. In their study, students did not have access totechnology (calculators) when they took these test items.

However, Schoen and Hirsch [48] found no significant differences on thealgebra sections of a college placement test among students with four years ofCORE-Plus, and individuals who completed four year of a CD curriculum. Forthe calculus readiness placement exam, Schoen and Hirsch did find significantdifferences favoring the CORE-Plus students. Students were allowed to usegraphing calculators on both placement exams in Schoen and Hirsh’s study. Itappears that the use of calculators may play a role on CORE-Plus students’performance on the placement exam.

Importantly, there appears to be no available research on the relationshipbetween high school mathematics curriculum and post-secondary mathematicsplacement recommendations, students’ responses to the recommendation, oron the impact on performance, in their initial mathematics class, of students’responses. The results of this study should add to our understanding of the roleof college mathematics placement tests as a moderating variable of the relation-ship between the high school mathematics curricula students complete, and thecollege mathematics courses students enroll in initially and their performancein those courses.

In this study, we posed the following three research questions:

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1. Is there a relationship between the high school mathematics curricu-lum students completed, and the university calculus placement testrecommendation, with various student background factors held constant,and, if so, what is its nature and magnitude?

2. Is there a relationship between the high school mathematics curriculum stu-dents completed, and student enrollment patterns after taking the universitycalculus placement recommendation, and, if so, what is its nature and mag-nitude? (i.e., are students from a particular curriculum more likely to take amore difficult than recommended course?)

3. Is there a relationship between students’ responses to the university calculusplacement recommendations, and performance in their initial mathematicscourses, with various students’ background factors held constant, and, if so,what is its nature and magnitude?

Examination of (1) should shed light on whether the high school mathe-matics preparation of college-bound students is related to the level of math-ematics classes in which they are advised to enroll initially. The results ofHarwell et al. [18] suggest that students from NSFF curricula tended to startwith less difficult mathematics classes than their CD and UCSMP counter-parts. Answering our first question may help to explain why this happened. Ourresults will also provide evidence relevant to a fundamental criticism of NSFFcurricula: that these curricula do an inadequate job of preparing students forsuccess in post-secondary mathematics [31]. The reason that we chose place-ment course recommendations as the key variable, instead of placement testscores, is that the critique is really on the course recommendations, not thetest score itself. In general, placement recommendations are made based onthe position of resulting score relative to certain cut-off scores. It is possiblethat students with difference placement test scores are placed into the samemathematics courses based on the break scores.

The study of (2) will compare student responses to the placement recom-mendations as a function of high school mathematics curricula (i.e., do studentsfollow the recommendation). We included this question because of anecdo-tal evidence that students who completed a NSFF curriculum in high schoolwere less likely to go along with the placement recommendation than were stu-dents completing a CD curriculum in high school. If corroborated, this wouldsuggest that there may be features of these high school mathematics curriculathat are related to motivation and expectations students have of themselves inmathematics.

The study of (3) will provide evidence of the appropriateness of the place-ment recommendation with respect to students’ performance in their initialmathematics classes. A finding that students who ignored a placement recom-mendation, and instead enrolled in a higher level course than recommended,had lower grades than those in the same course that followed the recommen-dation, thus providing support for the initial placement recommendation. On

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the other hand, a finding that students who chose to follow a recommenda-tion to begin with calculus, and who subsequently performed poorly in thisclass, would provide evidence that the initial recommendations were somehowflawed.

3. METHODOLOGY

This study employed a retrospective quasi-experimental design for cross-sectional data.

3.1. Population and Sampling

The target population consisted of university students in a Midwestern state.Students graduated from one of approximately 325 public high schools andcompleted at least three levels of high school mathematics in a traditional com-mercially developed (CD), National Science Foundation-funded (NSFF), orUniversity of Chicago School Mathematics Project (UCSMP) curricula, up to amaximum of five levels, with the fifth level including coursework beyond first-year calculus. A high school level was defined to be the content equivalent of ayear-long course taught one hour per day, and has been commonly referred toas a Carnegie unit.

The sample in this study consisted of 1,391 students from 100 high schoolswho enrolled at a single public research I university during the Fall 2002 or Fall2003 terms, and took the calculus readiness placement exam and at least onemathematics class in their first year of university study. Students who withdrewfrom the courses were not included in this study. Fall 2002 was chosen as thefirst term to study post-secondary performance because it allowed significantnumbers of students, who had completed at least three levels of a NSFF highschool mathematics curriculum, to be sampled.

3.2. Data

Archival data were collected from both high school and university sources.Data on student high school transcripts were obtained from the university inan electronic form. Variables extracted from this information included: GPA inhigh school mathematics classes, levels of high school mathematics completed(3, 4, or 5), gender, ethnicity (Native American, African American, Asian,Hispanic, or White), and type of high school mathematics curriculum (CD,UCSMP, NSFF).

The university maintained electronic records of the high school coursestaken by each applicant and the associated grades, but we quickly detecteddiscrepancies in how particular mathematics courses were coded in univer-sity records and the correct descriptors. For example, some high schools

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440 Norman et al.

reported “Integrated 1, 2, 3, and 4” as their high school mathematics courses,information which the university then recorded. We use the descriptor “inte-grated” to denote detailed study of algebra, geometry, probability, and statisticseach year. This is sometimes used interchangeably with “Standards-based,” theterm sometimes used to denote NSFF curricula. In some cases, the coursesrecorded as “integrated” were actually a traditional publisher’s text series thathad a subtitle with the word “integrated” in it. In other cases, districts usingNSFF (CORE-Plus) curricula recorded students’ course titles as Algebra I,Geometry, and Algebra II because of parental pressure.

These experiences led us to contact each of the 100 high schools to inde-pendently obtain descriptions of their mathematics programs—including themathematics courses offered and the textbooks used—to allow accurate cate-gorizations of students enrolling in the university as having completed a CD,UCSMP, or NSFF mathematics curriculum in high school. Contacts at the highschools were usually a mathematics curriculum coordinator, senior or mastermathematics teacher, principal, or assistant principal. After categorizing highschool mathematics curricula, four NSFF curricula were represented in oursample: Interactive Mathematics Program (IMP), Mathematics: Modeling OurWorld (MMOW), Contemporary Mathematics in Context (CORE-Plus), andMathematics Connections. These curricula were not differentiated in the dataanalysis.

University variables collected included: when a student enrolled at the uni-versity (Fall 2002, Fall 2003), course or placement recommendation based onthe calculus placement test score, ACT mathematics score, and name and gradeof mathematics courses in the first year of university study. An important vari-able was the academic unit a student was admitted to, which we categorized asthe following:

1. Science-, Technology-, Engineering-, and Mathematics-related colleges ordepartments (STEM),

2. Humanities (e.g., sociology, literature, political science), and3. General College.

General College admits students who were academically not ready to be admit-ted by any other academic units at the university. These three academic unitstend to have different mathematics course-taking requirements that need to betaken into account. For example, STEM students are generally required to takemore, and more difficult, mathematics classes than students in Humanities.This implies differences among students in the kind of starting mathematicsclass recommended.

Another university variable was a student’s mathematics grade, whichwas based on a 4-point grade scale—ranging from A (scale value of 4.0)to F (scale value of 0)—that was treated as showing a course interval scaleof measurement. Under university rules, students have the grading option of

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receiving a Satisfactory/Unsatisfactory or, at the discretion of the instructor,an Incomplete, but these options appeared in only a few student records in oursample and were not examined.

The university calculus readiness exam was developed by the Departmentof Mathematics. The criterion-related validity coefficient of the exam (corre-lation between placement test scores and ACT mathematics scores) is 0.546,and the predictive validity coefficient of the exam (correlation between place-ment test scores and mathematics course grades earned) is 0.603[39], which areamong the typical range of validity statistics of placement tests [9, 15, 27, 44,59]. The test is offered online, and students are not allowed to use calculators.

Among the 30 multi-choice items on the university calculus place-ment exam, 23 (76.7%) test knowledge in Algebra/Functions while theremaining items test content in Geometry/ Trigonometry [39]. The cate-gory of Algebra/Functions includes knowledge in linear, exponential, power,polynomial, logarithmic, rational, and periodic functions. The category ofGeometry/Trigonometry includes patterns with regard to shape, size, location;representational patterns with drawings, coordinates, or vectors; organizationof geometric facts, and relationships through deductive reasoning.

There are two other placement exams offered by the university: an algebrareadiness exam and the General College placement exam. In this study, we lim-ited our focus to recommendations provided by the university placement testin calculus for two reasons. First, calculus and its immediate antecedents areviewed by many in the mathematics community as critical to post-secondarymathematics learning, understanding, and success, and by some as a keydifference in the preparation of students from CD and NSFF high schoolmathematics curricula. Second, narrowing the focus to students who took thecalculus readiness placement test imposes a homogeneity on the sample thatshould enhance inferences by removing groups of students with wildly dif-ferent mathematics backgrounds (e.g., students who need significant remedialmathematics work, and those who are ready to begin their post-secondarymathematics studies with advanced classes such as Calculus II or III).

After taking the calculus readiness placement test, enrollment in one of thefollowing is recommended to students:

1. Precalculus I (review of high school algebra),2. Precalculus II (refresher course in preparation for calculus which includes

trigonometric functions, DeMoivre’s Theorem, and solutions of linear andnonlinear systems of equations),

3. Short Calculus (differential and integral calculus in one variable, anddifferential calculus in two variables), or

4. Calculus I.

On the basis of the descriptions of these courses we decided to group therecommendations into two levels: Precalculus I and II formed one level, and

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Short Calculus and Calculus I the other level. We treated Precalculus I andII recommendations as similar, and the Short Calculus and Calculus I recom-mendations as similar, but treated the differences in mathematics preparationbetween Precalculus (I, II) and Calculus (Short, Calculus I) as sufficient todetect the effects of interest.

There were three possible student responses to the calculus placementrecommendation at the university:

1. followed the recommendation,2. did not follow the recommendation and took a more difficult mathematics

course, and3. did not follow the recommendation and took a less difficult mathematics

course.

University precalculus and calculus classes are offered in various academicunits but are always taught by instructors in the Department of Mathematics.Variation across sections of precalculus and calculus classes was curtailedsomewhat by the Department of Mathematics’ policy of standardizing coursesyllabi, using common student assignments and examinations, and using asingle course-specific distribution of scores for assigning grades.

3.3. Data Analyses

We began with various descriptive analyses to detect important patterns inthe data. Then we addressed the first research question (Is there a relation-ship between high school mathematics curricula and university placementtest recommendations?) using a binary logistic regression model in whichhigh school mathematics curriculum was the key independent variable, andthe calculus placement recommendation served as the dependent variable. Toanswer the second research question (Is there any relationship between highschool mathematics curricula and student response to the university place-ment recommendation?) we used a multinomial logistic regression modelin which high school mathematics curriculum was the independent vari-able of interest and student response to the recommendation the dependentvariable.

The third research question (Is there any relationship between students’responses to the placement recommendations and their grades earned in thefirst university mathematics course?) was answered by using linear regressionanalyses. Here the key independent variable was student response to the cal-culus placement recommendation (followed; did not follow and took a moredifficult mathematics course; did not follow and took a less difficult mathe-matics course) and grade in a student’s first mathematics class, the dependentvariable. All statistical hypotheses were tested using α = 0.05.

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Student University Math Performance 443

3.4. Limitations of the Study

Data in this study were collected from a single large Midwestern publicuniversity. Variation in how placement information is used by academic coun-selors is of particular concern since it is poorly understood. For example, it isunclear whether academic counselors in Humanities and STEM departmentsand programs make the same placement recommendations based on the sameinformation, or whether the nature of the counseling is similar (i.e., supportiveand interactive versus prescribed and abrupt). There is some anecdotal evidencethat different units at this university (Humanities, STEM, and General College)handle placement decisions differently. One of the reasons we think this maymatter is that the students are, generally, just out of high school and unfamiliarwith post-secondary education and, in many cases, unsure about their mathe-matical preparation. This may make them more likely to accept the “official”placement recommendation even if they disagree with it.

4. RESULTS

4.1. Descriptive Analyses

A total of 1,391 of the students in our sample took the calculus readinessplacement test and received a course recommendation. Based on the coursenames recommended by the placement test compared with the actual coursesstudents took, among those who took the calculus readiness placement test,63.3% (N = 880) followed the placement recommendation, 11% did not fol-low the recommendation and took a less difficult mathematics course, while25.7% did not follow the placement recommendation and took a more difficultmathematics class.

Next, we examined the relationships between when a student enrolled (Fall2002, Fall 2003) and the high school variables ACT mathematics, high schoolmathematics GPA, ethnicity, and gender, which were either not significant orsignificant, but quite weak in magnitude, with less than 1% of the varianceexplained. Thus, students who enrolled in Fall 2002 (50.1%) or Fall 2003(49.9%) appeared to have similar backgrounds, and were combined and treatedas a single group for the remaining analyses.

4.2. Binary Logistic Regression Analysis

A binary logistic regression with calculus placement (precalculus, calculus) asthe dependent variable was used to help answer the first research question: Isthere a relationship between high school mathematics curricula (CD, UCSMP,NSF) and the mathematics course placement recommendation generated by

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444 Norman et al.

the university calculus placement exam. The variables included in the logis-tic regression equation were: gender, high school curriculum (UCSMP, CD,and NSF, with the latter as the reference group), academic units (STEM,Humanities, and General College, with the latter as the reference group), ACTmathematics scores, high school mathematics GPA, high school mathematicslevels, and ethnic groups (African American, Asian, Hispanic, and White, withthe latter as the reference group).

The significant results in Table 1 indicate that students with stronger priormathematics background are more likely to be placed into university calculus.For example, with other variables being held constant, students who com-pleted more high school mathematics levels were 1.64 times more likely tobe recommended to start in calculus relative to precalculus, compared to stu-dents who completed fewer high school mathematics levels. Similarly, studentswith higher ACT mathematics scores and high school mathematics GPAs were

Table 1. Binary Logistic Regression Predicting Calculus Placement Recommendation

Effect B SE B Exp (B)

Intercept −17.1 1.21Gender −0.28 0.15UCSMP 0.47 0.27CD 0.26 0.23STEM 0.61 0.42Humanities 0.04 0.43ACT Math∗ 0.43 0.03 1.54HS Math GPA∗ 0.92 0.18 2.51HS Math levels∗ 0.50 0.11 1.64African American 0.84 0.59Asian 0.43 0.22Hispanic 0.37 0.55

Gender (0 = male, 1 = female), CD = Commercially Developed, UCSMP =University of Chicago School Mathematics Project, STEM = Academic units in sci-ence, technology, engineering or mathematics, HS Math levels = high school mathlevels completed, HS Math GPA = high school mathematics GPA, African American(0 = no, 1 = yes), Asian (0 = no, 1 = yes), Hispanic (0 = no, 1 = yes). B = Beta is theunstandardized coefficient, SE B = standard error of Beta, Exp(B) = exponentiatedBeta.

∗Chi-square tests were used for the tests of significance and the results were signif-icant at P < 0.05. ACT Math, HS MATH GPA, and HS Math levels were significantlyrelated to the likelihood of being recommended to a calculus course as opposed to a pre-calculus course. For instance, with other variables held constant, one unit higher on theACT Math score was associated with a student being approximately one and one-halftimes (1.54) more likely to be placed into a calculus course as opposed to a precalculuscourse.

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more likely (by factors of 1.54 and 2.51, respectively) to be recommendedfor calculus as opposed to precalculus, compared to students with lower ACTmathematics scores or high school mathematics GPAs. There was no signifi-cant difference between male and female students on the likelihood of beingplaced into university calculus as opposed to precalculus.

The key result in Table 1 is the absence of a significant relationshipbetween high school mathematics curricula and the calculus placement recom-mendation (precalculus, calculus), after controlling all the other variables. Thismeans that a student’s high school mathematics curriculum was not relatedto his or her mathematics placement recommendation at the university. Inother words, none of the high school mathematics curricula (CD, UCSMP,and NSFF) sufficiently or insufficiently prepared students for placement inuniversity calculus courses.

4.3. Multinomial Logistic Regression Analysis

To investigate whether there was a relationship between students’ high schoolmathematics curricula and their responses to the placement recommendation(followed, did not follow and took a more difficult mathematics course, did notfollow and took a less difficult mathematics course), with other factors heldconstant, we used a multinomial logistic regression model. Covariates in thismodel were the same as those in the binary logistic model with one excep-tion. Hispanic students were omitted because only one Hispanic student tooka mathematics course of lower than recommended difficulty. The dependentvariable was student response to the recommendation.

The results presented in Table 2 indicate that female, non-STEM students,and students who took fewer high school mathematics classes are likely to takean initial mathematics classes of difficulty lower than that recommended bythe calculus placement test. For instance, females were one and one-half timesmore likely (B = 0.42, Exp (B) = 1.53, p = 0.04) to take a lower difficulty thanrecommended course than male students.

The results for the model predicting the likelihood of a student takinga mathematics class of higher than recommended difficulty level, relative tofollowing the recommendation, indicated that being in a Humanities programmade the student a third as likely (B = –1.25, p = 0.00) to take a mathemat-ics class of higher than recommended difficulty, compared to those not inHumanities. Students with a stronger prior mathematics background (higheraverage ACT mathematics scores, high school mathematics GPA, and highschool mathematics levels) were more likely to take a mathematics classmore difficult than that recommended by the calculus placement exam. Therewere no significant differences between male and female students, and amongdifferent ethnic and curricula groups.

Students’ responses to the placement recommendation (follow, not followand took a more difficult course, not follow and took a less difficult course)

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Table 2. Multinomial Logistic Regression Predicting Response to Calculus PlacementTest Recommendation

Effect B SE B Exp (B)

Took lowerIntercept 1.72 1.08Gender∗ 0.42 0.21 1.53UCSMP −0.44 0.36CD −0.14 0.30STEM∗ −1.37 0.42 0.25Humanities −0.60 0.42ACT Math −0.01 0.03HS Math GPA 0.02 0.23HS Math Levels∗ −0.55 0.15 0.58African American 0.63 0.54Asian −0.10 0.29

Took higherIntercept∗ −3.69 0.91Gender −0.26 0.16UCSMP −0.01 0.27CD 0.24 0.24STEM −0.21 0.41Humanities∗ −1.25 0.43 0.29ACT Math∗ 0.06 0.02 1.06HS Math GPA∗ 0.70 0.19 2.02HS Math levels∗ −0.25 0.11 0.78African American −0.34 0.60Asian 0.16 0.20

Analysis does not include Hispanic students due to small n for analysis oflog odds of taking a lower difficulty course. Gender (0 = male, 1 = female), CD =Commercially Developed, UCSMP = University of Chicago School MathematicsProject, STEM = Academic units in science, technology, engineering or mathemat-ics (0 = non-STEM major, 1 = STEM major), HS Math levels = high school mathlevels completed, HS Math GPA = high school mathematics GPA, African American(0 = no, 1 = yes), Asian (0 = no, 1 = yes). B = Beta is the unstandardized coefficient,SE.B = standard error of Beta, Exp (B) = exponentiated Beta.

∗Chi-square tests were used for the tests of significance and the results were sig-nificant at P < 0.05. Gender, STEM major, and HS Math levels were significantlyrelated to the likelihood of a student who did not follow the course recommendationbased on the placement test and took a course less difficult than recommended. Forinstance, with other variables held constant, female students were approximately oneand one-half times (1.53) more likely to take less difficult courses than male students;and STEMmajored students were one-fourth (0.25)more likely to take less difficultcourses than non-STEM students. Humanities, ACT Math, HS GPA, and HS Math lev-els were significantly related to the likelihood of a student who did not follow the courserecommendation based on the placement test and took a course more difficult than rec-ommended. For instance, Humanities-majored students were approximately one-third(0.29) more likely to take more difficult courses than non-Humanities students; and oneunit higher on HS GPA was associated with twice (2.02) the likelihood of taking moredifficult courses.

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may be related to students’ motivation/expectation toward mathematics andtheir selfconfidence in mathematics. Table 2 shows that students who com-pleted different high school mathematics curricula (CD, UCSMP, NSFF) didnot respond to the placement recommendations differently, meaning that noneof the curricula had more specific features that related to students’ motivationsand expectations of themselves in mathematics. However, male students andstudents with stronger prior mathematics knowledge were more likely not tofollow the placement recommendations.

4.4. Linear Regression Analysis

The third research question, “is there a relationship between students’responses to the university calculus placement recommendation (followed, didnot follow and took a more difficult mathematics course, did not follow andtook a less difficult mathematics course) and students’ performance (grade)in their initial mathematics course, with various background factors held con-stant,” was tested using linear regression. Covariates in this model include thosein the logistic regression models, and the difficulty level of the initial universitymathematics class a student took, as well as the student’s response to place-ment recommendation, which was represented with dummy-coded predictorsthat used followed-the-placement recommendation as the reference group. Thedependent variable was grade earned in a student’s first university mathematicscourse. The results are presented in Table 3.

The key findings in this table are the significant slopes for students whotook less difficult or more difficult mathematics classes than recommended.For the former, the slope of 0.28 indicates that, with other predictors held con-stant, students who took less difficult mathematics classes than recommendedearned, on average, a grade that was 0.28 units (0–4 point scale) above thatearned by students who followed the recommendation or took a more difficultmathematics class than recommended.

5. DISCUSSION

The result of no gender difference on mathematics placement (calculus, pre-calculus) is similar to the findings of some studies of gender differences onmathematics achievement [5, 35], but contradicts other studies which foundthat, in general, males outperformed females in mathematics [2, 19, 42]. Thisstudy showed that female students were more likely to take mathematicscourses that were less difficult than those recommended by the placement test.This may be due to societal influences that lead to female students’ negativeattitudes toward mathematics and a lower self-confidence in subject [14, 16,43, 56].

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Table 3. Linear Regression Analysis for Grade in First UniversityMathematics Class

Effect B SE B

Intercept −1.40 0.334Gender 0.01 0.06Difficulty 0.05 0.07UCSMP −0.09 0.10CD 0.07 0.09STEM −0.07 0.14Humanities 0.01 0.14ACT Math∗ 0.06 0.01HS Math GPA∗ 0.56 0.07HS Math levels∗ 0.13 0.04African American 0.51 0.19Asian 0.08 0.08Hispanic 0.12 0.21Took Lower∗ 0.28 0.10Took Higher∗ −0.17 0.07

Model R2 = 0.15, Gender (0 = male, 1 = female), CD = CommerciallyDeveloped, UCSMP = University of Chicago School MathematicsProject, STEM = Academic units in science, technology, engineering, ormathematics; HS Math levels = high school math levels completed, HSMath GPA = high school mathematics GPA, African American (0 = no,1 = yes), Asian (0 = no, 1 = yes), Hispanic (0 = no, 1 = yes), Took Lower(1= took less difficult mathematics class than recommended, 0 = didnot), Took Higher (1 = took more difficult mathematics class than rec-ommended, 0 = did not). B = Beta is the unstandardized coefficient, SEB = standard error of Beta.

∗Chi-square tests were used for the tests of significance and the resultswere significant at P < 0.05. For instance, with other variables held con-stant, one unit increase on the ACT Math score was associated, on average,with 56 units (on scale of 0–4 points) of higher grade in the first universitymathematics class.

Although many research studies have shown that African-American andHispanic students are more likely to have lower standardized test scores thanWhite students [11, 12, 34, 36, 55], it is important to note that the achievementgap has been closing over the past 25 years [8, 20, 21]. In this study, the resultsshowed no significant differences on the mathematics placement recommen-dations among different ethnic (African-American, Asian, Hispanic, White)groups.

This study contributes to the literature in the area of connecting students’high school mathematics experiences and their initial college mathematicscourse placement, their responses to the course recommendation, and their

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performance in the course. It was motivated by an ongoing and impor-tant discussion within the mathematics community on the ability of varioushigh school mathematics curricula to prepare students to succeed in collegemathematics coursework.

The main finding of the study is that students’ high school mathemat-ics curricula and their calculus readiness placement test recommendations(precalculus, calculus) were unrelated. This implies that, insofar as the uni-versity calculus readiness placement test is concerned, the high school math-ematics programs completed by college-bound students are not related to theinitial level of mathematics classes recommended to them. Our data suggestthat students are not disadvantaged by any of the three high school mathematicscurricula (CD, UCSMP, NSFF) studied with respect to university mathematicsplacement recommendation. Advocates of particular high school mathematicscurricula are likely to be disappointed because these results do not provide clearevidence of an advantage for any one curriculum over another [4, 23, 25, 32,45, 50, 51, 59, 60].

Along the same lines, these findings provide evidence that any advantagethat CD students may have on the university placement test, or the disadvan-tage that NSFF students may have on the placement test because of content,appear to be minimized. That is, neither NSFF, CD, nor UCSMP studentshave any advantage with respect to the university placement recommendations.However, it remains an open question as to what course recommendation NSFFstudents would receive if the placement test, which emphasizes standard algo-rithms and repetitive procedures consistent with CD curricula, reflected a morebalanced mixture of items illustrating an expanded domain of topics and moreclearly paralleling NSFF curricula.

In other words, considering the disadvantage that NSFF students havedue to less content alignment between NSFF curricula and the placement test,the lack of a difference in placement recommendations, compared to CD andUCSMP students, is important. Similarly, it is unclear how NSFF studentswould perform in university mathematics precalculus and calculus coursesstructured to be more consistent with an NSFF high school curriculum, asopposed to their current alignment with commercially developed high schoolcurricula and instruction. The learning environment, pedagogy, and technol-ogy used in college mathematics courses may be very different from a typicalhigh school classroom using a NSFF mathematics curriculum. For example,LaBerge, Sons, and Zollman [33] interviewed 30 mathematics faculty membersfrom 11 universities and colleges in the Midwest, and found that 28 intervie-wees used lecture most of the time or always; 23 stated that they never orseldom asked students to write about mathematics in their “own language,” and19 faculty members mentioned that they rarely have students work in groups.This contrasts greatly with the teaching style recommended by the NSFF highschool mathematics curricula [24, 58].

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Future work could include investigation of the following:

1. students’ longitudinal mathematics achievement and course-taking patternsat college as a function of their high school mathematics curricula,

2. the way that various academic counselors at universities use placementinformation and the nature of the counseling process, and

3. identifying psychometric properties of placement tests.

These are all areas of possible study that would add significantly toour knowledge of college mathematics placement tests and their variousramifications.

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BIOGRAPHICAL SKETCHES

Ke Wu Norman is an assistant professor in the Department of MathematicalSciences at the University of Montana. Her research focuses on teacher edu-cation and professional development, and student achievement in science,technology, engineering, and mathematics, at post-secondary institutions, as afunction of their mathematics experiences in high school. She enjoys teach-ing content and methods courses to preservice teachers and professionaldevelopment courses for in-service teachers.

Amanuel G. Medhanie is a doctoral student in the Department of EducationalPsychology at the University of Minnesota. The area of emphasis in his studiesis Quantitative Methods in education.

Michael R. Harwell is a professor in the Department of EducationalPsychology at the University of Minnesota. His research focuses on moderators

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and mediators of the relationship between SES and educational achievement,missing data methods, and the impact of high school mathematics curricula oncollege mathematics performance.

Edwin Anderson is a retired mathematics teacher who taught at Minneapolispublic schools for 39 years. He received his Ph.D. in 1976 from the Universityof Minnesota Twin Cities. He has been the Co-PI and project coordinator forseveral National Science Foundation-funded research projects at the Universityof Minnesota.

Thomas R. Post is a professor in the Department of Curriculum and Instructionat the University of Minnesota. He has conducted research dealing with math-ematical learning and concept development, teacher development, and therelationship between theoretically based instructional activities and the natureof the resulting student conceptual development. His recent research projectsfocus on the impact of reform mathematics curricula on student mathematicsachievement on standardized tests at the K-16 grades.

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