The Mathematics Educator 2004, Vol. 14, No. 2, 27–33 Mathematics Placement Test: Helping Students Succeed Norma G. Rueda & Carole Sokolowski A study was conducted at Merrimack College in Massachusetts to compare the grades of students who took the recommended course as determined by their mathematics placement exam score and those who did not follow this recommendation. The goal was to decide whether the mathematics placement exam used at Merrimack College was effective in placing students in the appropriate mathematics class. During five years, first-year students who took a mathematics course in the fall semester were categorized into four groups: those who took the recommended course, those who took an easier course than recommended, those who took a course more difficult than recommended, and those who did not take the placement test. Chi-square tests showed a statistically significant relationship between course grade (getting a C– or higher grade) and placement advice. The results indicate that students who take the recommended course or an easier one do much better than those who take a higher-level course or do not take the placement exam. With achievement in coursework as the measure of success, we concluded that the placement test is an effective tool for making recommendations to students about which courses they should take. There is a widespread recognition of the need for She explained that the placement recommendations appropriate placement in the mathematics courses for were based on a large number of regression equations undergraduate freshmen. Many colleges and that required considerable expertise in development universities around the nation have used the and periodic redefinition. The placement test also Mathematical Association of America (MAA) required the coordination of numerous categories of Placement Test; others have designed their own exams student data used in the equations. Approximately 85% or used a combination of placement exams and other of students who enrolled in a calculus course based on measurements, such as ACT or SAT mathematics the recommendations from the placement test at St. scores and high school GPAs. Since the MAA has Olaf College received a grade of at least C–. discontinued its placement test program, the Cohen, Friedlander, Kelemen-Lohnas and Elmore responsibility has been put on individual institutions to (1989) recommended a placement procedure that was develop their own placement exam. The purpose of our less technically sophisticated than St. Olaf’s, but still study was to determine the effectiveness of Merrimack required considerable background data about students. College’s placement test by examining the connection They recommended multiple criteria methods, which between students enrolling in recommended courses included a placement test customized to an institution’s and their success in those courses. curriculum. They started with sixty variables, and found the eight best predictors: high school graduation Literature Review status, number of hours employed, units planned, age, In this section, we investigate some of the specific high school grade point average, mathematics methods reported in the literature for placing placement test score, reading placement test score, and undergraduates in their first mathematics course. We English placement test score. Cohen et al.’s study was point to some of the assumptions in these reports and based on (a) thousands of surveys completed by suggest some of the drawbacks to the method of students and faculty members at eight California placement. community colleges, (b) a comparison between student Cederberg (1999) reported on the three placement scores on assessment tests and grades in different tests administered at St. Olaf College in Minnesota. courses, and (c) the relationship between student characteristics and their grades. Krawczyk and Toubassi (1999) described a simpler Norma G. Rueda is a professor in the Department of placement procedure used by the University of Mathematics at Merrimack College, North Andover, MA. Her research interests include mathematical programming and Arizona. The University of Arizona used two applied statistics. placement tests adapted from the 1993 California Carole Sokolowski is an Assistant Professor in the Department Mathematics Diagnostic Testing Project (see of Mathematics at Merrimack College, North Andover, MA. http://mdtp.ucsd.edu/). Students chose which test they Her research area is undergraduate mathematics education. felt was most appropriate for their ability and choice of Rueda & Sokolowski 27 major. One test covered intermediate algebra skills the mathematics/science requirement. During data and placed students in one of three levels of algebra or collection for this study, business administration a liberal arts mathematics course. The second test majors were required to take Applied College Algebra, covered college algebra and trigonometry skills and Calculus for Business, and one other mathematics placed students in courses from finite mathematics course. Students majoring in science or engineering through Calculus I. In the fall of 1996, their data generally were required to take more mathematics; for indicated that approximately 17% of freshmen placed example, engineering students were required to take in College Algebra through Calculus I failed or three calculus courses and one course in differential withdrew from their respective courses, compared with equations. They also took Precalculus if they did not a 50% attrition rate in the early 1980’s before the place out of this course on their placement exam. mandatory testing and placement. Apart from a test, Since some incoming freshmen are not prepared to they also considered other factors, such as high school take a college-level mathematics course, a non-credit GPA. developmental mathematics course, Math I, has been A number of studies have investigated the use of offered at Merrimack College since the fall of 1994. standardized tests, such as ACT and SAT. Bridgeman Before 1994 we administered a mathematics diagnostic and Wendler (1989) found that the mathematics SAT exam to incoming students, but were unable to score was a relatively poor predictor of grades accommodate students who were not ready to take a compared to placement exams. Their results were college-level mathematics course. Instead, they based on grades from freshman mathematics courses at enrolled in a mathematics course at a higher level than ten colleges. Odell and Schumacher (1995) showed their exam score indicated they should. There was a that a placement test used in conjunction with high failure rate among these students. mathematics SAT scores could be a better predictor Because students at Merrimack College often have than SAT scores alone. Their conclusion was based on difficulty in other courses if they have not completed data from a private business college in Rhode Island. Math I, proper mathematics placement has become Callahan (1993) studied the criteria followed at Cottey important to all of our departments. For example, the College in Missouri to place students in the appropriate Chemistry Department now requires the students who level course, as well as their placement success rates. place into Math I to complete the course before they As with the studies mentioned above, Cottey College take several of their chemistry courses. Krawczyk and used several variables to achieve their results – the Toubassi (1999) have found similar results at the MAA Placement Test, ACT and SAT mathematics University of Arizona in which all chemistry students scores, and years of high school mathematics taken. and 90% of biology students—whose placement test Each of these studies assumed that the success rates scores indicated they should be placed in intermediate were based on students following the placement algebra or lower—received grades below C– in their advice. Mercer (1995), on the other hand, conducted a chemistry or biology courses. study to compare pass rates in a college-level This interest in correct mathematics placement mathematics class for mathematically unprepared extends beyond the chemistry department as evidenced students who enrolled in developmental courses and by the many questions from the business and liberal those who did not. The results of this study showed a arts faculty, as well as science and engineering faculty statistically significant relationship between passing at the meetings in preparation for orientation advising. the course and following placement advice. A major reason for our concern about student success is that successful students are more likely to remain in Background their studies. A high level of student retention is not Merrimack College, located in North Andover, only academically and socially desirable at a school, Massachusetts, is a small four-year Catholic college but it makes sense economically. offering programs in the liberal arts, business, the We do not place students according to their prior sciences, and engineering. Among the college’s high school GPA or whether they have taken calculus. distribution requirements, students must complete three We do not assume that these factors indicate whether mathematics or science courses, with no more than two or not they need algebra. In fact, many students placed courses from the same department. Most of the into Math I have had four years of high school students take at least one mathematics course. Most mathematics, including precalculus (and occasionally liberal arts majors usually choose Basic Statistics, calculus), but according to our placement test they do Finite Mathematics, or Discrete Mathematics to satisfy not appear to understand the basic concepts of algebra. 28 Mathematics Placement Test Mathematics Placement Test Business: Score below 9 in Part I ⇒ Math I All incoming freshmen at Merrimack College are For those who score above 8 in Part I: expected to take a mathematics placement test • Score below 30 in Parts I–II ⇒ Applied College Algebra developed by members of the Mathematics • Score of 30 or higher in Parts I–II ⇒ Calculus for Department. There are two versions of the exam, one Business Liberal Arts: for students who will major in Business Administration Score below 9 in Part I ⇒ Math I or Liberal Arts, and one for students in Science and Any other score ⇒ Enroll in a mathematics elective course Engineering. The version for Business Administration and Liberal Arts consists of two parts, elementary and Data Analysis intermediate algebra. The version for Science and We have performed statistical analyses since 1997 Engineering students contains a third part that tests to study whether there was a relationship between the students’ understanding of functions and graphs. score on the placement test and how well first year Students are instructed not to use calculators. From students did on the first mathematics course taken at 1994 until 1999 the placement exam was taken at Merrimack College. In order to determine this Merrimack College in June during orientation or at the relationship, we first examined the correlation between beginning of the academic year. Since 2000 the exam total score on the placement test and students’ grades. has been mailed to students at home. A Scantron form A preliminary study with n = 372 showed that the with the students’ answers is mailed back to correlation between the grade earned in a mathematics Merrimack College and graded. Students are informed course and the total score on the placement test was that using outside resources or calculators may result in 0.466. For the same study, the correlation between their being placed in a course for which they are grades earned and SAT scores was 0.334. A multiple unprepared and may result in their failing or regression model to estimate the final grade based on withdrawing from the course. There is not a difference the SAT score and the placement exam score gave the between these mail-in results and the previous equation: Final Grade = 0.80 + 0.00122(SAT) + monitored exam results with respect to the percentages 0.0641(Placement Exam). In addition, a t test for each of students who place into the various mathematics of the variables in the model indicated that the SAT courses. Thus, we believe that most students heed our had a t value of 1.01 (p = 0.314) and the Placement warnings. The recommendations are available to Exam had a t-value of 6.58 (p < 0.0005). Based on the t students and advisors during June orientation. See test values and the p-values, we removed the SAT Appendix A for some problems similar to those given variable from the model and concluded that the in the mathematics placement test. placement exam was a better predictor of student Part I of the placement test consists of seventeen success. questions on elementary algebra. If a student does not Although we first used t-tests to compare the total answer at least fifty percent of these questions score on the placement test with SAT scores as correctly, then that student must take Math I. For predictors of students’ final grades, we ultimately students who score above fifty percent on Part I, liberal decided against them as a means to further examine the arts majors may take any mathematics elective course; effectiveness of our placement test as a placement tool business administration majors are placed into a for two reasons. First, there was not a true linear college algebra or a business calculus course, relationship between the total score on the placement depending on their overall score; and science and test and placement. It was more like a step-function, engineering students may be placed into a college with a range of scores in each part of the test being algebra, Precalculus or Calculus I course, depending on considered for placement. Second, there was a their total score. The specific recommendations relatively weak relationship between total score on the resulting from the mathematics placement exam are as placement test and final grades because students are follows: placed into so many different levels of courses. For Science and Engineering: example, a student may have a very low placement test Score below 9 in Part I ⇒ Math I score, be properly placed into an elementary algebra For those who score above 8 in Part I: class, and earn a high grade in that course. Thus, a • Score 20 or lower in Parts I–III ⇒ Applied College simple correlation between total score on the Algebra placement test and grades earned was considered • Score between 21 and 34 in Parts I–III ⇒ Precalculus inappropriate to determine the effectiveness of the • Score of 35 or higher in Parts I–III ⇒ Calculus I Rueda & Sokolowski 29 placement test, and therefore we decided to categorize or not the student followed the placement test result the data. recommendation. Each first-year student was categorized according Results to the level of mathematics course taken: (1) the course was easier than that recommended by the placement We wanted to know whether there was an test score; (2) the course was the recommended course; association between the level of the course taken and (3) the course was more difficult than that the grade earned. Table 1 shows the number and recommended (higher-level); or (4) the placement test percentage of students who did well in the class (C– or was not taken. Although the test was required, some higher) and the number and percentage of students who students—usually transfer students—were allowed to did poorly (D+ or lower, or withdrew from the class) take a mathematics course based on their mathematics from 1997 through 2001. It was generally accepted that grade(s) at a previous institution. This policy has not grades of D and F were unsatisfactory, as evidenced by worked well and is being changed to require all the fact that almost all comparable studies used cut-off incoming students to complete the placement test. grades of C or C–. Those that used the C cut-off often Within these categories, students were counted had a minimum grade requirement of C for a student to according to whether they (a) did well (received a move on to a subsequent course. Our department does grade of C– or better) or (b) did poorly (received a not have such a requirement, and thus the choice of C– grade below C–). Chi-square tests were performed to for this study was somewhat arbitrary. We felt that our determine whether there was a relationship between the professors might be less likely to slightly inflate the level of the course taken and the grade received in the grade of C– to C than would those at schools with the class. Given the eight possible categories previously minimum requirement. As described above, students described in this study, the chi-square test indicated were classified according to the level of the course whether the percentage of students, say, who did well taken: easier than the one recommended, the one in each category of course level taken was significantly recommended, or a higher-level course than the one different from that in any of the other categories. The recommended. A fourth category was used in order to use of the chi-square test assumes that a random include students who did not take the placement exam, sample, representative of the population, was taken. In but took a mathematics course. this study, we used the entire population of freshmen The data were analyzed using the chi-square test students who took mathematics in their entering fall (see Table 2). We found that there was a relationship semester at our college for the years from 1997 to 2001 between the two variables for first year students who (n = 1710). The null hypothesis stated that there was no took a mathematics course in the fall of 1997 [χ2 (3, n association between the two variables. The alternative = 369) = 24.66, p < 0.0005]. The same conclusion held hypothesis stated that the grade depended on whether for the data corresponding to the fall of 1998 Table 1 Number and Percentage of Students Each Year Disaggregated by Grade and Course Category 1997 1998 1999 2000 2001 Course < D+ > C– < D+ > C– < D+ > C– < D+ > C– < D+ > C– 6 23 2 17 1 12 5 21 2 8 Easier 21% 79% 11% 89% 8% 92% 19% 81% 20% 80% 47 168 57 168 55 220 57 205 50 200 Recom. 22% 78% 25% 75% 20% 80% 22% 78% 20% 80% 30 30 33 34 6 16 4 12 7 15 Higher- Level 50% 50% 49% 51% 27% 73% 25% 75% 32% 68% 28 37 22 28 4 18 12 16 18 16 No Exam 43% 57% 44% 56% 18% 82% 43% 57% 53% 47% n 111 258 114 247 66 266 78 254 77 239 30 Mathematics Placement Test Table 2 Contingency Table and Chi-Square Test (Expected counts are printed below observed counts; shaded cells indicate expected counts less than 5.) 1997 1998 1999 2000 2001 Course < D+ > C– < D+ > C– < D+ > C– < D+ > C– < D+ > C– 6 23 2 17 1 12 5 21 2 8 Easier 8.72 20.28 6.00 13.00 2.58 10.42 6.11 19.89 2.44 7.56 47 168 57 168 55 220 57 205 50 200 Recom. 64.67 150.33 71.05 153.95 54.67 220.33 61.55 200.45 60.92 189.08 Higher- 30 30 33 34 6 16 4 12 7 15 Level 18.05 41.95 21.16 45.84 4.37 17.63 3.76 12.24 5.36 16.64 28 37 22 28 4 18 12 16 18 16 No Exam 19.55 45.45 15.79 34.21 4.37 17.63 6.58 21.42 8.28 25.72 n 111 258 114 247 66 266 78 254 77 239 Chi-Square test 24.66 21.22 N/A 6.56 18.42 p value <0.0005 <0.0005 N/A 0.087 <0.0005 [χ2 (3, n = 361) = 21.22, p < 0.0005]. We did not In addition, students who withdrew from a course were perform a chi-square test for the data corresponding to counted and were assigned a number grade of 0 for this 1999 because there were 3 cells with expected counts study. less than 5 (see shaded cells in Table 2) and Moore Discussion and Conclusion (2001) does not recommend the use of the chi-square From Table 1 we saw that students who took the test when more than 20% of the expected counts are recommended course or an easier one did much better less than 5. The relationship was not significant for the than those who took a higher-level course or did not fall of 2000 at an alpha level of 0.05 [χ2 (3, n = 332) = take the placement exam. The same conclusion can be 6.56, p = 0.087] when taking p < 0.05 to be statistically drawn from Table 3, with the exception of the year significant. The relationship was again significant for 2000. In 2000 there was no significant difference the data corresponding to the fall of 2001 [χ2 (3, n = among the average grades received by those students 316) = 18.42, p < 0.0005]. In sum, the percentage of who took the recommended course and those who took students who did well in their first undergraduate a higher-level course. An explanation for this may be mathematics course was higher for those students who that the proportion of students who took a higher-level followed the advice or took an easier course than the course dropped from 19% in 1998 to 7% in 1999, and one recommended based on their placement test score. to 5% in 2000. The few students who took a higher- The chi-square test only showed evidence of some level course seemed to know that they would be able to association between the variables. We then looked at succeed. In addition, the percentage of students placed the tables to determine the nature of the relationship or into our developmental course, Math I, has been association (Moore 2001). Table 3 shows the number decreasing. Twenty five percent of the 521 who took of students who took the recommended course, an the mathematics placement exam in 1998 were easier one, a higher-level course, as well as the number recommended to take Math I. Twenty percent of the of students who did not take the placement exam, and 532 who took the mathematics placement exam in the mean and median grades on a 4.0 scale for each 1999 were in that category. Those figures went down group from 1997 to 2001. The following to 14% out of 525 students in 2000 and 16% out of 517 correspondence between letter-grades and number- in 2001. One of the reasons for this decrease was that grades was used at Merrimack: students were allowed to retake the placement exam and to place out of our developmental mathematics A 4.0 B 3.0 C 2.0 D 1.0 course. Even though that possibility was available to A– 3.7 B– 2.7 C– 1.7 D– 0.7 students before, more effort has been made in the last B+ 3.3 C+ 2.3 D+ 1.3 F 0.0 Rueda & Sokolowski 31 Table 3 Mean and Median Grade (on a 4.0 Scale) by Year and Course Category 1997 1998 1999 2000 2001 n n n n n n ean dia n ean dia n ean dia n ean dia n ean dia M e M e M e M e M e Course M M M M M (%) (%) (%) (%) (%) 29 2.8 3.0 19 2.7 2.7 13 3.1 3.3 26 2.4 2.5 10 2.4 2.7 Easier 8% 5% 4% 8% 3% 215 2.4 2.7 225 2.4 3.0 275 2.5 2.7 262 2.5 2.7 250 2.6 3.0 Recom. 58% 62% 83% 79% 79% 60 1.7 1.5 67 1.6 1.7 22 2.2 2.0 16 2.4 2.5 22 1.9 1.7 Higher- Level 16% 19% 7% 5% 7% 65 1.8 2.0 50 1.7 2.0 22 2.4 2.2 28 1.8 2.2 34 1.6 1.2 No Exam 18% 14% 7% 8% 11% n 369 361 332 332 316 few years to avoid improperly placing students in a adequate records and analyzing them with regard to the non-credit course (Math I). placement test’s effectiveness, as we have done in this With the exception of 1999, there is no significant study, is a key component in maintaining the validity difference, using z tests, between the proportions of and reliability of the test itself. A number of years ago, students who did well or poorly among those students the placement test score was viewed as the basis for a who did not take the mathematics placement exam. “recommended” mathematics course for each student, It is not surprising that students who enrolled in the to be followed or not, as the student chose. Today, the recommended course or an easier course performed entire Merrimack community appears to view the test better than did students who enrolled in a higher-level score with increased respect because of the results course than the one recommended. What is important presented in this paper. These ongoing statistical here is that approximately 80% of these students who validations of the connections between proper took the recommended or easier course succeeded with placement and successful achievement have served to a grade of C– or higher. legitimize the placement test as part of a larger effort to We have found the mathematics placement exam increase retention on our campus. to be a useful tool to place students in the appropriate mathematics course, and we have been successful in REFERENCES convincing most of our students to follow our advice Bridgeman, B., & Wendler, C. (1989). Prediction of grades in with respect to which courses to take. A challenge for college mathematics courses as a component of the placement us has been persuading students to take Math I, the validity of SAT-mathematics scores. (College Board Report No. 89-9). New York, NY: College Entrance Examination non-credit class, when they are not ready for a college Board. level course. While there is no perfect placement Callahan, S. (1993, November). Mathematics placement at Cottey method, we have found that our test is better than SAT College. Paper presented at the Annual Conference of the scores in placing students into the appropriate course. American Mathematical Association of Two-Year Colleges, In addition, our multiple-choice test is easier to Boston. (ERIC Document Reproduction Service No. ED administer than methods used at other schools 373813) mentioned in this paper. Cederberg, J. N. (1999). Administering a placement test: St. Olaf College. In B. Gold, S. Keith, & W. Marion (Eds.), From our experience, a well-designed in-house Assessment practices in undergraduate mathematics (pp. placement test geared towards our curriculum is a 178−180). Washington, DC: Mathematics Association of simple and powerful tool for placing incoming students America. in an appropriate mathematics course. Keeping 32 Mathematics Placement Cohen, E., Friedlander, J., Kelemen-Lohnas, E., & Elmore, R. Mercer, B. (1995). A comparison of students who followed (1989). Approaches to predicting student success: Findings mathematics advisement recommendations and students who and recommendations from a study of California Community did not at Rochester Community College. Practicum prepared Colleges. Santa Barbara, CA: Chancellor’s Office of the for Nova Southeastern University, Ft. Lauderdale, FL. (ERIC California Community Colleges. (ERIC Document Document Reproduction Service No. ED 400014) Reproduction Service No. ED 310808) Moore, D. (2001). Statistics: Concepts and controversies (5th ed.). Krawczyk, D., & Toubassi, E. (1999). A mathematics placement New York: W. H. Freeman and Company. and advising program. In B. Gold, S. Keith, & W. Marion Odell, P., & Schumacher, P. (1995). Placement in first-year college (Eds.), Assessment practices in undergraduate mathematics mathematics courses using scores on SAT-math and a test of (pp. 181−183). Washington, DC: Mathematics Association of algebra skills. PRIMUS, 5, 61−72. America. Appendix A: Sample Problems The following problems are similar to the ones given on the actual exam, but the format is different. These sample problems are free response. The actual exam has a multiple choice format, in which several answers are provided to each problem, and only one of them is correct. 1. Without a calculator, evaluate: The following problems are representative of the additional section of the Placement Test for Science and Engineering majors. 3.42 a. |5 – 9| b. 3 −8 c. d. 8 5 .02 9−12 6. Let f(x)= 1 . Find the domain of f. x−1 2. Simplify the following: € 7. Find the zeros of the function 2x−3 f(x)= . a. 12x2+3x b. −2x3 c. (25x4y8)−1/2 d. a + 4 € x+1 3x (−2x)3 (3x3)2 a+2 a−3 8. Find the inverse function, f−1(x), if f(x)=3 x+2. e. 5x+18−4(x+7) f. log(x2−1)−log(x+2)+logx 9. Which of the following points is not on the graph of y=ex2−1? 3. Solve the following for x: € (0, e−1), (1, 0), (2, e3), (3, e8) a. ax−b=5 b. |2x+1| = 5 c. x= 2x+15 10. Convert 135° to radians. d. 1 =64 e. x+2y=5 f. x2 +x<6 4x x+4y=7 11. Simplify in terms of sinθ:1−cos2θ=? 4. Solve, then simplify the radical: x2 −2=−4x 12. Which of the following is greatest? sin 30°, sin 45°, sin 90°, sin 180° 5. Find an equation for the line through the points (–1, –2) and (1, 4). Give the slope, m, and the y-intercept, b. Rueda & Sokolowski 33