A small college nurturing a large calculus clientele in a flexible calculus program recognizes the need for careful placement, and studies the effectiveness of its efforts with statistics to derive a formula for placement.
Background and Purpose
St. Olaf is a selective, church related, liberal arts college located in Northfield, Minnesota. The majority of the 2900 students are from the North Central states and nearly all of them come to college immediately after high school. Over the past 5 years, the median PSAT mathematics score of incoming St. Olaf students has ranged from 58 to 62 and approximately 60% were in the top 20% of their high school graduating class. In recent years, 8-10% of the St. Olaf students have graduated with a major in mathematics.
The St. Olaf Mathematics Department has 24 faculty members filling 18.9 F.T.E. positions. In addition to the major, the department also offers concentrations (minors) in statistics and computer science (there are no majors in these areas). The entry level mathematics courses include a two semester calculus with algebra sequence (covering one semester of calculus), calculus (honors and nonhonors sections using the same text), gateways to mathematics (a non-calculus topics course designed for calculus ready students), finite mathematics, elementary statistics, and principles of mathematics (intended for non-science majors). Each of these courses (with the exception of the first semester of the calculus with algebra course) can be used to fulfill the one-course mathematics requirement. This requirement, one of a new set of general education requirements instituted after approval of the college faculty, took effect with the incoming students in 1994. Prior to this time there was no formal mathematics course required, but most St. Olaf students took at least one of these courses to fulfill a two course science/mathematics requirement. In the last three years, the average total of fall semester calculus enrollments of first-year students has been 498. In addition, approximately 10 first-year students enrolled in sophomore level courses each fall.
|Fall Semester Calculus Enrollments|
|Calculus with Algebra||28|
|Calculus I (1st semester)||266|
|Math Analysis I (Honors Calculus I)||102|
|Calculus II (2nd semester)||9|
|Math Analysis II (Honors Calculus II)||93|
The St.Olaf mathematics placement program began in the late 1960's in response to the observation that students needed guidance in order to place themselves properly. One of the major realizations in the development of the program was the inadequacy of a single test for accurate placement. So in addition to creating a test (later replaced by MAA tests2), regression equations were developed to use admissions information and answers to subjective questions along with placement test scores to predict gradepoints as the one quantifiable measure of successful placement. Over the years, the one placement test evolved into three separate tests, the regression equations have been refined, and more sophisticated computer technology has been employed.
All new students are required to take one of the three mathematics placement exams, with exceptions made for non-degree international students and for students who have received a score of 4 or 5 on the College Board Calculus BC exam. The placement examinations are administered early during Week Onea week of orientation, department information sessions, placement testing and registration (there is no early registration for new students) which immediately precedes the beginning of fall semester. All three exams contain an initial list of subjective questions asking students for information about (1) their motivation for taking mathematics; (2) the number of terms of mathematics they plan to take; (3) the area in which they expect to major; (4) what their last mathematics course was, and their grade in this course; and (5) how extensively they have used calculators. Questions on the regular and advanced exams also ask for (6) how much trigonometry and calculus they have had; and (7) the mathematics course in which they think they should enroll. This latter question provides helpful information, but also indicates the need for providing guidance to the student since many would not place themselves into the proper course.
The exams all are timed exams with a ninety minute limit. Students taking the Advanced and Regular Exams are allowed to use calculators without a QWERTY keyboard. Currently, students are not allowed to use calculators on the Basic Exam. The three exams and their audience are listed below.
1. Advanced Exam: Designed for students who have had at least one semester of calculus and want to be considered for placement beyond first semester calculus at St. Olaf. This test consists of a locally written exam covering topics in first semester calculus (25 questions) together with a modified version of an MAA trigonometry and elementary functions exam (15 questions each). Approximately 220 students take this exam.
2. Regular Exam: Designed for students with standard mathematics backgrounds who intend to take calculus sometime during their college career. This test consists of a trigonometry and functions section (identical to that on the Advanced Exam) together with an MAA calculator based algebra exam (32 questions). Approximately 425 students take this exam.
3. Basic Exam: Designed for students with weaker mathematics backgrounds who have no plans to take calculus and who are hesitant about taking any mathematics. This test consists of an MAA exam over arithmetic and basic skills (32 questions) and an MAA algebra exam (32 questions). Approximately 120 students take this exam.
Following the exams, the questionnaire and test data are scanned into the computer, which grades the tests, then merges the test scores with the admissions data and other relevant information to predict a grade. Finally, the computer assigns a recommendation to each student using a cutoff program that is refined from year to year. Borderline and special cases (e.g., low class rank from a selective high school) are considered separately. Individual student recommendations are printed on labels which are then pasted on letters and distributed to student mail boxes the next morning. In addition, each academic advisor is sent an electronic message containing the results for their advisees.
The current St. Olaf mathematics placement program is administered by a member of the mathematics faculty who is given a half course teaching credit (out of a six-course standard load) for serving as director. As the present Director of Mathematics Placement, I send each new student a letter early in the summer describing the placement process and handle questions and concerns that arise. I also manage the details of giving the exams and reporting the results. Following the exams, I counsel students who have questions about their placement recommendation; notify instructors of students in their classes who did not take an examination or who did not follow the placement recommendation; and assist students with changes among calculus sections.
The placement recommendations are computed using a large number of regression equations. Dr. Richard Kleber, a mathematics faculty member and our senior statistician, has spent numerous years developing and refining these equations. Each summer he has run a series of regression studies in order to "fine tune" the regression equations. There are regression equations for each of 24 cases, depending on the information available for a particular student. These equations are used in five different sets as follows:
Use of Findings
As an example of these regression equations, the equation below predicts a grade (gr) on a four point scale using a normalized high school rank (based on a 20 to 80 scale), PSAT and ACT math scores, a self-reported math grade from the ACT exam and the algebra, trigonometry and function scores from the placement exam.
|gr =||-2.662 + 0.0385nrank + 0.00792psatm|
|+ 0.0470actm + 0.218mgrade + 0.0237ascore|
|+ 0.0277tscore + 0.0244fscore|
This particular equation, modified by information from the subjective questionnaire, is one of those used to make decisions about honors versus regular calculus placement for students who take the Regular Placement Exam. Typically, students with gr scores of 3.5 or above are given a recommendation for honors Calculus I; those with scores between 3.3 and 3.5 are told they may make their own choice between the honors and regular versions of Calculus I; and those with scores below 3.3 are given a recommendation for regular Calculus I.
The success and efficiency of this placement process is due to much "behind-the-scenes" effort by several people in the mathematics department and involves everyone in the department on the actual day of testing. This extensive effort is rewarded by the number of students who successfully complete their initial courses in mathematics. In the first semesters of the 1991-92 and 1992-93 academic years, 92% of the students who initially enrolled in a calculus course completed a semester of calculus; and of these, 92% received a grade of C- or above.
1 An earlier version of this article was published in the Fall 1994 issue of the Placement Test Newsletter, a publication of the MAA Committee on Testing.
2 For detailed information on the MAA Placement Test program, contact the MAA headquarters in Washington, DC (phone: