Issues surrounding placement of students in math courses at the  post-secondary level have implications for mathematical practices of teachers and students at the secondary and post-secondary levels.  Presently these matters are receiving considerable attention at Memorial University of Newfoundland with the increasing prominence of the Mathematics Skills Inventory (MSI) as a screening device for entry into first year math courses. Chris Brown's article provides an organized expository piece on the topic of placement testing and remedial mathematics.  The references and additional titles in the appended bibliography provide a helpful resource to those who may wish to further examine the research in this field.  The article itself represents a revision of a paper initially prepared this past semester for a graduate course: (ED6630) Critical Issues in Mathematics Education.  It has been informative assisting Chris with these revisions in an effort to prepare the paper for publication in Morning Watch.  Previously unpublished research of my own also appears in this issue.  The research was conducted in a math help centre specifically designed to assist students who had been placed into remedial mathematics courses at a large university.  Both Chris and I feel that the simultaneous publication of the two papers will enrich the potential discussion that may ensue.  Comments on the papers are welcomed. 
John Grant McLoughlin
 
 

 It is widely recognized that many students entering post-secondary institutions - community colleges and universities - are deficient in the mathematics background that is presumed necessary for the successful completion of post-secondary level mathematics courses.  Various reasons may be proposed to explain this situation:
 

 • Recent high school graduates enter with math grades that give a false impression of satisfactory backgrounds either because the grades tend to be inflated, or because the actual content coverage differs from the official curriculum.

 • High school math courses may not include the particular content needed to prepare students for post-secondary math courses.

 • Mature students who have been out of school for a number of years have lost their math skills or may never have acquired them at all.  (This applies especially to entering students at community and technical colleges, but to a lesser degree, at universities as well.)

 Whatever the reasons may be, a common approach to dealing with this problem is to assess all students' math skills at entry with a placement test, and then place underprepared students in a remedial math course (or series of courses) that is intended to bring the students' skills up to the necessary level. The student who successfully completes the remedial program should reasonably expect to succeed in first-year post-secondary level mathematics. By this means, post-secondary institutions hope to extend the opportunities of higher education to a larger number of people and enhance the enrolment and retention rates in mathematics.

 Of course this process is not as simple or foolproof as it seems.  One of the most important questions that need to be asked is:  How effective is the partnership of placement tests and remedial mathematics in improving the success of underprepared students?  This article will offer insight into this question by providing a brief overview of widely used placement tests in mathematics and of typical methods of their administration and use.  Evidence will be presented from selected longitudinal studies that purport to show that students did benefit from placement testing and remediation.  Opposing views and general criticisms of this process will also be considered.  The paper concludes with the author's personal view on the use of mathematics  placement testing and remedial mathematics.
 
 

 Tests are widely used to assess the mathematics skills of students entering the first year of studies at post-secondary institutions.  For example, a survey of 1,297 such institutions in the United States found that over 90% used math assessment tests (Lederman, Ribaudo, and Ryzewic, 1985).  In the local context, Memorial University of Newfoundland and the College of the North Atlantic both use math assessment tests, though in different ways and degrees.  Broadly speaking, placement tests take two forms:  standardized scholastic tests developed by national or state/provincial bodies, and locally developed tests created for the purpose by an institution.  Tests commonly used in the U.S. include:
 

 1. Scholastic Aptitude Test - Math (SAT-M);

 2. American College Testing Program (ACT);

 3. Assessment of Skills for Successful Entry and Transfer (ASSET);

 4. New Jersey College Basic Skills Placement Test (NJCBSPT);

 5. Mathematical Association of America Placement Test Program (MAA);

 6. Descriptive Tests of Mathematical Skills (DTMS).

 Most of these are actually sets of tests that include one or more components on mathematics.  A Canadian diagnostic/placement test is the Mathematics Skills Inventory (MSI) developed by Rudolph Zimmer at Fanshawe College in Ontario.  It is in use at Memorial University of Newfoundland.

 Placement tests may be administered to students in a variety of ways. They may be taken by high school students in the last month or two before graduation.  They may be given to students during the first week of university/college attendance or they may be available to take at any time during a six-month (or longer) period before a student is scheduled to start a program.  Students may take the tests in a variety of settings ranging from a large hall with hundreds of others, to a small room with a few others, or a computer terminal all alone.

 Of course it is the results of the placement tests, the students' scores, that are considered meaningful.  Ranges of scores, referred to as cut scores, are selected by the standard test developers or the institution's math faculty to assign students to categories, and thus courses.

 These categories may be loosely described as prepared, underprepared, and very underprepared (Three categories are most common, though sometimes just two are used).  For a test with a maximum score of 100, say, students with 60 or more would be assigned to a college-level math course, students with scores from 50-59 would be assigned to a developmental course (elementary algebra), and the rest would be assigned to a basic math course (arithmetic only).

 With placement decided, the next step in the process is the remedial mathematics program.  What do these programs teach?  A survey of 79 post-secondary institutions in the U.S. produced a broad categorization of 4 typical courses (McDonald, 1988).  A quick look over the content briefly described there (see Table 1) shows that most mathematics from kindergarten to high school is included.

 Adapted from:  "Developmental Mathematics Instruction:  Results of a National Survey".  Anita D. McDonald (1988).  79 institutions with 'exemplary' programs responded.

 Most institutions offered two remedial courses:  basic math and beginning algebra.  McDonald also noted that the content of "Intermediate Algebra" was often included in a college-level course at many institutions, making it remedial for some but college-level for others.

 The assignments resulting from placement tests may be mandatory, recommended, or just guidelines for a student to consider in their math course selection.  Students may or may not have the option of appealing a placement or taking a retest at a later time with the aim of achieving a higher course placement.  These variations can make it difficult to sort out the exact effects of course assignments on students' future success.  In the next section, two studies which provide support for placement testing and remediation are reviewed in some detail.  Course assignments for both of these situations were nominally mandatory but some flexibility in terms of appeals and retesting was allowed.
 
 

A Standardized Placement Test Example

 A longitudinal study of the cohort of students entering a Connecticut  community-technical college in Fall 1990 covered the period from Fall 1990 to Spring 1993 (Sturtz and McCarroll, 1993).  All students took the New Jersey College Basic Skills Placement Test (NJCBSPT) which included two mathematics sections - mathematical computation and elementary algebra. Students' scores resulted in recommendations for placement in one of three courses - Basic Math I, Basic Math II, or a college-level math course.  Details of the cut scores and numbers of students recommended for each are presented in tables 2 and 3 which have been adapted from Sturtz, Alan J. & McCarroll, Judith A. (1993).

 Placement was nominally mandatory.  However, some students followed recommendations; others appealed placements and enrolled in higher courses; others still opted to take a lower than recommended course, while some did not enroll in any math course.  In fact, about 29% of the students originally tested never enrolled in any math course over the six terms of the study.  Most of these students had discontinued studies at the institution.

 What happened to the students who did enroll?  Table 4 shows that success rates in math were good for those who followed recommendations, but slightly lower for those who took a course higher than originally assigned.

(*Success = Grade A, B, or C.)
 

 What exactly does this mean in terms of the assignments by the placement tests?  Considering the group of students recommended for Basic Math I we see that 63% of 322, or 203 were successful.  But, 60% of 35, or 21 students that appealed and took Basic Math II were also successful at the higher level.  Thus, it might be argued that the cut scores alone had misplaced these 21 students out of 357 in the Basic Math I group for an error rate of about 6%.  A similar review of the Basic Math II group shows that 67% of 15, or 10 students were successful at a higher course.  Again, based on cut scores alone, 10 out of 121 or about 8% of the students were misplaced.  Would such error rates be acceptable in a mandatory assignment process?

 Sturtz and McCarroll suggest another way to assess the success of the process.  The underprepared students who followed recommendations were compared to those who did not in terms of persistence (mean number of terms attended) and quality point average (similar to GPA) over the six terms of the study.

They conclude:  "Students who were successful in their recommended basic-level courses tended to continue enrollment ... for a slightly greater number of terms (Sturtz and McCarroll, 1993, p. 17)."  This conclusion really does not say much as it is probable that those who failed in the remedial math courses would be more likely to withdraw from college and would obviously have a lower mean attendance as a group.  With regard to QPA, it also seems obvious that the successful groups must show the higher QPA's whether or not they followed recommendations.  In fact, it is the students who challenged assignments and took higher level courses who show the highest QPA and persistence.  Does this show that self-selecting a higher than recommended course is the best route to success?

 Finally, the study looked at how the remedial cohort fared with college-level math.  While asserting that overall the evidence supports the placement process, they note:  "Data for completion of college-level math courses are inconclusive (Sturtz and McCarroll, p. 17)."  Only 28% of students who enrolled in Basic Math I, and 55% of students who enrolled in Basic Math I

 This study reveals a number of the difficulties in evaluating a mathematics course placement process.  It is relatively straightforward to collect data on test scores, grades, and attendance; it is not so straightforward to interpret the data and determine if it is reasonable to say that the placement process is significantly beneficial to underprepared students.

An Institutionally Developed Placement Test

 At Pembroke State University in North Carolina, the Mathematics Department created its own test to assess math skills and place students in remedial or first-year university math courses (Truman, 1992).  Development and piloting of the process took about two years.  A large part of the paper deals with how the test was constructed, and how the content, validity, reliability and other statistical aspects of the test were assessed and improved.  The author then reviews the success of the placement and remediation process over the most recent three-year period.

 The placement test was administered to all entering students during summer freshmen orientation.  Based on scores, students were assigned to one of two remedial courses, and one of three first-year math courses.  Students who felt they were misplaced could apply for a retest; only 2% of 1375 students have done so in the three-year period reported on in the paper (Truman, p. 62). To evaluate the effectiveness of following recommendations, the final grades of first-year students in the Fall term were compared with their choice of course level for three consecutive years.

(adapted from Truman, 1992, p. 63).
 

 Reviewing the figures in this table, we see that a large proportion of students - 262/775 or 34% - took courses below the placement level.  As might be expected, this group shows a relatively high average grade.  The 461 students who followed recommendations include 53 who failed, or 11.5%.  This can be described as an 11.5% false-positive error for the placement test, i.e., the test placed that portion too high.  Finally, 52 out of 775, or 6.7% took higher courses than recommended and, not surprisingly, show a low mean grade. However, 24 of these students were successful as defined in the previous study, meaning that 24/775 or 3% were placed too low by the initial test.  (The reader might compare this to the 6-8% low placement error for the test in the first study.)  As far as Truman is concerned the evidence from these results is clear:
 

 After 5 years of mathematics placement testing at Pembroke, the mathematics department is convinced that this program provides an efficient, practical, and workable method of placing students in mathematics courses which give them the best chance for educational success (Truman, p. 64).

 Is this conclusion reasonable?  Certainly, the rate of failures is relatively low, as is the rate of dissatisfaction with placements.  But, something does not seem quite right with some of the numbers.  For example, if 1,375 students took the math placement test in the period described above why are there only 775 students in the table of grades?  What happened to the remaining 600?  And if just 2% of 1,375, or 23 students (Truman, p. 62) asked for a retest, how is it that 52 students took courses higher than placement level?  Of these 52 students, 24 of them were successful.  This calls to question the claim that only 3% were placed too low.  The fact that 46.2% of the students who took the initiative to get placed in a higher than initially recommended course were successful suggests that the potential for inappropriate placement in this direction merits further attention.  This false-negative error rate is unacceptably high.  In addition, there is no follow-up data presented to indicate if successful underprepared students are enrolling in college-level math and what their success rates in those courses might be. Omission of this type of data in studies evaluating remedial math programs is unfortunately relatively common (Akst, 1986).

 The papers by Sturtz and McCarroll (1993) and Truman (1992) clearly support the use of math placement testing and remediation in post-secondary institutions.  They provide evidence to show that the process is beneficial to students.  These papers also present, either implicitly or explicitly, indications that a score from a placement test alone does not tell the whole story.  For example, though placement was nominally mandatory at both institutions, each one also provided a way for students to challenge and alter placements.  Many students who had been tested never took any math courses at all.  Did placement testing scare them away from math courses, confirming one more time that they can't do it?  There must be other factors that affect a student's success in college-level mathematics.  In the next section, criticisms of the math placement process and some views opposed to placement testing will be explored.
 
 

 The critique of the mathematics placement process may be described in terms of two broad camps - those who are fundamentally opposed to the use of tests for assigning people to particular courses or programs and those who accept the placement process but suggest it needs to be refined to include the effects of other factors beside math test scores alone.  Perhaps we could call these groups Rejectors and Revisors respectively.
 

Rejectors' Views

 In the field of assessment, placement testing is seen as a subset of selection testing.  According to Glaser and Silver (1994), "Selection testing attempts to measure human abilities prior to a course of instruction so that individuals can be appropriately placed, diagnosed, certified, included or excluded."  (p. 395)  The last word in that quote signals the main point in some of the opposition to placement testing.  Placement tests may function to exclude people from post-secondary education rather than aid access because they may be seen as just one more hurdle.  Assigning people to ability groups is seen to be a kind of academic tracking, and may actually serve to reproduce or entrench inequities rather than help eliminate them (Kingan and Alfred, 1994).  For example, Glaser and Silver, summarizing Oakes (1985), note:  "In studies of the academic tracking of students for mathematics instruction, data regarding instructional practices suggest that students assigned to the lower tracks of many high schools tend to receive less actual mathematics instruction, less homework, and more drill-and-practice of low-level factual knowledge and computational skill than students assigned to middle and higher tracks (p. 398)."

 Another aspect of the exclusion or barrier view is the notion that remedial courses deter enrollment due to the extra time and money needed to complete a program (Morante, 1989), or that placement in remedial classes stigmatizes students with respect to their peers and may lead them to become demoralized and drop out (Kingan and Alfred, 1994).  This kind of grouping may also have serious implications when visible minorities are "over-represented" in remedial classes.

 Some opposition to placement testing and remediation derives from a financial argument combined with a touch of what might be called higher education snobbery.  In this view, underprepared students and remedial courses just do not belong in college or university as their presence tends to lower standards.  The time and money needed for testing and remediation is better spent on the students who are prepared and the resources they need (Almeida,1986).  Aligned with this view is the notion that underpreparedness is the result of poor content or instruction in high school math courses, so the problem should be fixed there (Platt, 1987).
 

Revisors' Criticisms

 A significant amount of the criticism directed at mathematics placement  testing is focused on the research which suggests that many other factors, particularly noncognitive or psychosocial factors, are important in determining a student's success in mathematics (Bridgeman and Wendler, 1991; House, 1995; Penny and White, 1998; Ting and Robinson, 1998).  These factors may  include:  self-confidence, commitment, attendance, gender, ethnic background, age or maturity, financial circumstances, self-rating of math ability, parent's education level, motivation, teacher's attitudes, mode of instruction, and teacher's gender (Ting and Robinson, 1998).  Critics propose that some of these factors should be assessed in specially designed questionnaires or interviews and used in conjunction with math test scores to make better placement decisions.

 Another area of the criticism focuses on the math placement test itself.  The tests may need improvements in terms of content and predictive validity, discrimination, reliability and the choice of cut scores (Morante, 1989; Truman, 1992).  Within the same frame also lies the debate over using general achievement tests (such as SAT) versus content-specific basic skills placement tests.  After an extensive review of assessment and placement, Jerry Weber (1986) concludes:
 

 Content-specific placement tests in combination with other student data will yield effective assessment forming a basis for placement decisions.  Performance on general achievement tests (ACT or SAT) or a subsection of one achievement test should not determine basic skills course placement (p. 28).

 Similarly, Wattenbarger and McLeod (1989) note that studies conducted on Florida colleges show that "...standardized entrance examinations do not provide information of sufficient accuracy to justify placement into the mathematics curriculum based solely on the math portion of the tests (SAT and ACT were used in one study) (p. 18)."  Most community colleges and a high proportion of universities do use institutionally created content-specific tests, though about half the universities are likely to rely on SAT or ACT scores (Lederman, et al., 1985; McDonald, 1988).

 Finally, the remedial courses offered to upgrade students' math skills are subject to criticism from a number of perspectives.  Courses vary widely in content, duration, and mode of delivery.  This may simply reflect different needs in different contexts and an effort to be flexible on behalf of students' needs.  More significant is the observation that many do not use any special instructional strategies directed at the characteristics of underprepared students (Laughbaum, 1992).  The faculty who teach the remedial program may be temporary, less qualified, and not well integrated into the post-secondary mathematics departments to the detriment of their students (Penny and White, 1998).
 
 

 Clearly the problem of how best to deal with underprepared math students at the post-secondary level is as thorny and long standing as many problems in education.  The combination of placement tests and remedial mathematics courses is a well-established process that many post-secondary institutions provide as a solution to this problem.  But is it an effective one?  The research evidence reviewed in this paper, and similar studies, indicate that it may provide a reasonable level of success for some students in some contexts.  However, if placement is based solely on the results of math skills tests without considering significant noncognitive factors, research suggests that a high number of unsatisfactory placements may occur.  Further there is concern that this process perpetuates social, gender and ethnic biases, and discourages enrollment.  Then, how should we proceed?

 It is my view that retaining all first-year students, regardless of background in the first-year math course, is not an option.  The pace and level of these courses quickly frustrates and discourages underprepared students. Attempting to remediate these students in regular class time is not possible for most, and extra help in off-class hours usually suffices for only a few of those who need help.  Under these circumstances, I think the only suitable option is to employ a mathematics placement test that identifies those who are underprepared, and to offer suitable remedial courses which provide the necessary mathematics upgrading.  But this process should be careful, considerate and flexible.  Careful - in the selection, evaluation and ongoing review of the placement test(s) itself, and in selection and training of faculty to teach remedial courses.  Considerate - of the other factors in the students' situations.  Flexible - in the timing and mode of delivery of both the placement test and the remedial courses.  Students should have the option to take the test before the regular program begins and be able to do a retest if they wish.  Remedial courses should be available in the summer and at night at a reasonable number of sites.  They should be delivered in ways that reflect the  nature of adult underprepared students.  A process like this should have the qualities needed to make it a prescription for success. 

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