EMPIRICAL AND CLINICAL MODELS FOR STUDENT PLACEMENT IN
THE COMMUNITY COLLEGE MATHEMATICS CURRICULUM
ROBERT NORMAN McLEOD
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
For Barbara, Charles, and Travis
Several individuals have contributed greatly to this work and
deserve recognition. First, the members of the doctoral committee,
especially the chairman and the cochairman, have been extremely
helpful and are due special thanks. Dr. James Wattenbarger, chairman,
has provided insight, guidance, and understanding from the very
beginning of the doctoral program. Dr. John Nickens, cochairman, has
added expertise, humor, and encouragement. Their assistance has been
invaluable and my genuine appreciation is extended to them.
I am also grateful to Dr. Ernie St. Jacques for his congenial and
helpful attitude and to Dr. Jim Pitts for his cooperation and
assistance. Their suggestions have contributed to the completion of
Individuals from two local community colleges have been very
helpful by sharing information and ideas. Dr. Tom Delaino of Santa Fe
Community College has offered suggestions and furnished data that have
greatly influenced this study. Associate Dean Betty Towry of Central
Florida Community College has provided background information and
material which were also beneficial. Many thanks are due both of
Finally, my family and friends have been an encouragement to me
throughout work in the doctoral program My wife, Barbara, has typed
numerous drafts of the dissertation as well as other papers. Her
support has been steadfast, and my love and appreciation are, as
always, due her. My father and his wife, my grandparents, my
brothers, and my sister have also shown much interest and have been
an encouragement. I am grateful to them all for their help and
TABLE OF CONTENTS
ACKNOWLEDGEMENTS.............................. .......... ..... iii
ABSTRACT ..................................................... vii
I INTRODUCTION......................................... 1
The Problem.......................................... 8
Methodology............ ............................... 9
Assumptions..... ..................................... . 10
Limitations.. ................................. 11
Justification for the Study............................ 11
Organization of the Research Report.................... 13
II REVIEW OF RELATED LITERATURE.......................... 15
Community College Students and Rationale for Placement. 15
Hierarchical Nature of Mathematics..................... 22
Effectiveness of Standardized Tests as Placement
Instruments..................................... .. 24
Alternative Means of Assessment for Placement.......... 32
Sumnary ............................................... 41
III METHODS AND PROCEDURES................................ 43
Sample Selection and Data Collection................... 45
Description of Data.................................... 47
Data Analysis......................................... 51
Student Interviews............................... .... 51
Development of the Clinical Model...................... 53
IV RESULTS................................................ 55
Discussion of Results to Question One.................. 59
Results With Respect to Question Two................... 60
Discussion of Results With Respect to Question Two..... 61
Results With Respect to Question Three................. 63
Discussion of Results to Question Three................ 73
V SUMMARY, CONCLUSIONS, RECOMMENDATIONS, AND
Summary ............................................ 75
Conclusions....................................... .. 76
Recommendations....................... .............. 76
Implications...................................... .. 79
A INTERVIEW GUIDE...................................... 82
B SUBJECT BY LEVEL MATRIX............................... 83
C SUGGESTED STEPS FOR CLINICAL EVALUATION................ 84
BIOGRAPHICAL SKETCH............................................ 91
Abstract of Dissertation Presented to the Graduate School of
the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy
EMPIRICAL AND CLINICAL MODELS FOR STUDENT PLACEMENT IN
THE COMMUNITY COLLEGE MATHEMATICS CURRICULUM
Robert Norman McLeod
Chairman: Dr. James L. Wattenbarger
Cochairman: Dr. John M. Nickens
Major Department: Educational Leadership
The problem of this study was to assess the appropriateness of
currently used placement tests in predicting student success in
mathematics courses and to identify additional factors which would
lead to appropriate placement of students into community college
mathematics courses. Placement test scores from the American College
Testing Program (ACT) and the Scholastic Aptitude Test (SAT) were
correlated with grades in entry-level mathematics courses for 605
first-time college students who had graduated from high school the
previous school year. Successful "high-risk" students, identified as
those with test scores below the cut-off points, were interviewed in
order to identify factors considered important to their successful
completion of college-level math courses. These and other factors
reported in the literature were utilized in a clinical evaluation
model which was reviewed by a panel of experts, composed of community
college instructors, counselors, administrators, and university
professors, for content and feasibility.
Correlations of test scores and grades in mathematics courses
were found to be low. Almost half of the correlations were negative
and only two of the positive correlations were greater than .50.
Three main factors emerged from the interviews as important. High
school preparation in mathematics, quality of student effort, and
clinical evaluation techniques were considered important by the
students interviewed. The panel of experts agreed that the concept of
clinical evaluation had merit. However, the panel disagreed over the
assumption that a hierarchy of subjects and levels in math could be
effectively used as a screening device in the model.
The study lends support to criticisms of the use of standardized
tests as a sole means of evaluation for mathematics placement. Other
factors which could be of use such as factors in a placement model
were identified, but not analyzed statistically. Recommendations for
further research of the validity of these factors in placement models
The concept of the "open door" as an admissions policy has been
viewed as fundamental to the realization of the mission and purpose of
the community college. Thornton (1966) noted that, historically, the
community college had attempted to provide educational opportunity to
the average student. In describing the open door philosophy of
community colleges, Thornton described any high school graduate, or
any person who seemed capable of profiting by the instruction offered,
as being eligible for admission. Cross (1971), however, maintained
that without proper consideration of the diversity of students that
enter colleges, the open door can turn into a "revolving door,"
wherein students exit the college unfulfilled.
Florida's community colleges require a high school diploma or its
equivalent for admission. However, many students possessing high
school diplomas are not properly prepared for college level work. For
example, Florida's high school graduation requirements currently
include a minimum of three credits in mathematics; however, no credit
in high school algebra is required. Because of this, many entering
students need a significant amount of additional preparation in math
to master college level courses. As a result, college preparatory
courses have been included in the curriculum of Florida's community
Consequently, appropriate placement of students into these
college preparatory courses becomes imperative. Roueche (1980)
commented that in order to give hope to students having learning
deficiencies who enroll, colleges were going to have to divorce
themselves from the "students have a right to fail" mind set, and
design instructional programs accordingly. Florida law requires
colleges to administer placement tests to all entering degree-seeking
students in order to assign them to courses commensurate with their
abilities. Because of the open door policy, entrance testing in
community colleges has been used as a means of placement rather than a
means of selection.
Four specific tests have been approved for placement purposes in
Florida's community colleges. The Florida legislature has mandated in
Rule 6A-10.315, FAC, that scores on one or more of the four approved
tests be used to place students in college preparatory communication
and computation courses, beginning the 1985 fall term. The four tests
are the American College Testing Program (ACT), Scholastic Aptitude
Test (SAT), Multiple Assessment Programs and Services (MAPS), and
Assessment Skills for Successful Entry and Transfer (ASSET). The cut-
off score for ACT has been set at 13, for SAT at 400, for MAPS at 206,
and at 12 for ASSET. Students who score within one standard deviation
of the cut-off scores may be exempted from or included in college
preparatory instruction on the basis of supplemental testing or
assessment documented by the institution. Identification of factors
in addition to standardized test scores that would provide information
concerning students' probability of success is therefore needed.
Students who score below the state-mandated cut-off scores on the
tests and are thus required to enroll in college preparatory courses
legitimately might ask how effective the tests and cut-off scores are
in predicting success in college level courses. Enrollment in college
preparatory courses will most likely increase the amount of time and
money spent for a college education and may lead to frustration and
lowered self-esteem. However, allowing students to enroll in courses
for which they are not properly prepared often diminishes the
probability of successfully passing the course. Since the goal of
effective placement is to assign students to courses in which they
have a reasonable probability of success, it is imperative to obtain
information as to what factors contribute to such success.
The necessity for some means of placement is widely recognized.
While some may support a "right-to-fail" concept that would allow
students to register for courses at any level, current practice
supports the use of placement methods to assure their right to
succeed. This has been accomplished not by restricting entry to
community colleges, but by identifying students' needs and then
providing the resources necessary to meet those specific needs.
The means of accomplishing effective placement, however, have
been elusive. Use of standardized testing has become pervasive in
admissions, placement, and curriculum decisions. A review of the
related literature revealed mixed results and only limited success,
when using standardized tests as predictors of academic success.
Other factors appear to be of equal or greater importance than test
scores when predicting academic success. Standardized tests persist,
however, as the most widely used type of evaluation for placement.
Hills (1971) stated "for some reason, though, there seems to be a halo
about test scores for placement that causes some faculty members to
believe that no other kind of data deserve consideration" (p. 707).
Evans (1975) cited the following statement by Winston Churchill
which helps explain some of the arguments for not using test scores as
a sole means of evaluation:
Examinations were a great trial to me. The
subjects which were dearest to the examiners were
almost invariably those I fancied least. . I
should have liked to be asked what I knew. They
always tried to ask what I did not know. When I
would willingly have displayed my knowledge, they
sought to expose my ignorance. This sort of
treatment had only one result: I did not do well
in examinations. (p. 270)
By virtually any other evaluative standard, Winston Churchill
would be considered successful. Yet, by his own admission, his
performance 'on examinations was poor. How many students entering the
community college system in this state and in this nation would
similarly do poorly on examinations but nevertheless have capabilities
and qualities that could lead to successful completion of college-
McClelland (1973) and others have questioned the use of
standardized tests as a sole means of evaluation. Depending on
certain factors such as the type of test (aptitude or achievement) and
method of scoring (norm-referenced or criterion-referenced),
interpretation of test scores has been subject to question. Testing
experts disagree on the amount of emphasis that should be placed on
test scores. A continuing controversy rests on the implicit
assumption that standardized tests do indeed measure what they purport
to measure (Haney, 1980). Until it can be demonstrated with certainty
that test scores provide information sufficiently accurate to justify
the decision under consideration, the controversy is likely to remain.
Clearly, there is no definitive answer to the question of how
much emphasis to place on test scores for various decisions. Bersoff
(1973) aptly noted that the validity of a measuring device must be
evaluated in terms of the purpose for which it is intended. The goal
of assessment has been seen as the acquisition of relevant information
that will contribute to decisions about desired changes in behavior.
With regard to placement, then, other factors of evaluating student
competence would add to our understanding of assessment techniques and
help in making important educational decisions.
The task facing educators is one of improving processes of
evaluating students. A desirable combination would, ideally,
incorporate the impartial, objective nature of tests with other, more
holistic, means. Alternative methods of assessment are seen by this
writer as useful for complementing and cross-validating standardized
tests. While objective tests can be of use in identifying which
students do not possess certain competencies, they add little (if any)
insight into specific deficiencies causing the low test results. A
more comprehensive model for placement, taking factors into
consideration other than test scores, is called for.
The strongest arguments in favor of using standardized tests for
placement purposes appear to be convenience and a lack of
standardization of grades or other measures. Many criteria other than
tests have been explored as predictors; however, problems with
quantification, objectivity, and standardization limit their usage.
Student opinion, teacher opinion, and clinical evaluation, among
others, have seemed to have promise for placement, only to be rejected
for various reasons.
Experts in mathematics have long known that use of standardized
tests as a sole means of evaluation is inadequate. The National
Council of Teachers of Mathematics (NCTM) Handbook on Evaluation in
Math (1961) concluded that many possible combinations of evaluative
techniques should be used. Studies comparing mathematics tests and
subsequent grades have demonstrated only moderate to limited
predictibility. However, a search of the literature revealed no
definitive methods that have been widely used to supplant the use of
standardized tests, which persist as one of the best single methods of
evaluation in mathematics. Some combination of previous grades in
math and test scores appears to be the best method of prediction
(Larson & Scontrino, 1976).
Test scores are commonly intended for use as part of a placement
program. Since test scores reflect past opportunities to learn, the
scores should not be used as the sole criterion in a placement
decision. Test scores provide a means of comparing students to a
common standard and represent a valuable criterion in placement
decisions. These decisions should be reviewed periodically, however,
and should be adjusted if classroom performance indicates students
have not been placed correctly (Florida MAPS: Technical Manual, 1984).
Placement is most systematic when based on the results of a
validity study. Hills (1971) has stated that utility should be of
primary consideration when deciding on placement methods and
procedures. If the standardized tests approved for the placement
process are appropriate, research would demonstrate the validity of
the tests for placement purposes. If not, perhaps other, more
utilitarian practices could be explored in order to minimize the
expense and maximize the effectiveness of placement procedures.
Research in the area of learning has indicated that the
probability of success is an important factor in subsequent learning.
Students who experience success are likely to demonstrate continued
success (Ferster & Perrott, 1968). The converse of this "success
breeds success" premise has also been demonstrated. Individuals who
are thwarted in initial attempts at accomplishing certain tasks have
developed the phenomenon referred to as "learned helplessness"
(Levine, 1977). Thus, students not having a reasonable chance of
success should be discouraged (or even prevented) from enrolling in
courses and programs for which the probability of failure is high.
For these reasons, community college administrators recognize the
necessity of assessment following admission in order to place students
in courses where they have a good chance to succeed. Students having
deficiencies have recently been required to take necessary
developmental work before proceeding to programs where the lack of
skill could cause failure (McCabe, as cited in Schinoff, 1982). The
success of this combination of assessment, placement, and instruction
has been documented (McCabe & Skidmore, 1983). The structured
approach resulted in increased completion rates and improved student
Since current practice is to think of assessment as a means of
sorting and screening students and is seen as a key to learning
(Justiz, 1985), effective placement techniques are needed. As a
result of objections to the use of standardized test scores for the
sole means of placement, the identification of factors that could be
used in addition to test scores is one of the purposes of this study.
Learning theorists conclude that individuals process information
differently (Cross, 1976; Ferguson, 1980; Roueche, 1980; Dunn, Dunn, &
Price, 1981). Admonitions to instructors to vary their teaching
techniques abound (Cross, 1976; Dunn & Dunn, 1979; Cohen & Brawer,
1982; Easton, Barshis, & Ginsberg, 1983-1984). Why, then, should
educators be locked into a one-dimensional method of assessment? If
teaching strategies should be flexible, it seems self-evident that
evaluation strategies should be equally free of stifling methodologies
that lock educators into a single type of evaluation for placement. A
study that analyzes the effectiveness of mathematics placement tests
used in Florida's community colleges and attempts to identify other
potential factors for use in placement strategies would be useful.
The necessity of appropriate placement in order to enhance
student success in community college mathematics courses has been
recognized. The factors that are currently used for placement are
certain state-mandated test scores which have been questioned when
used as the sole means of evaluation. The problem of this study was
to analyze the appropriateness of currently used placement tests in
predicting student success in mathematics courses and to identify
additional factors which would lead to appropriate placement of
students into community college mathematics courses.
Specifically, the following questions were addressed:
1. What is the relationship between selected placement test
scores and success in initial mathematics courses?
2. To what factors did high-risk students attribute their
successful completion of college-level mathematics courses?
3. What factors identified by the results of question number two
and in the literature may be used for components of a
clinical evaluation model which could augment the use of
standardized test scores for placement in mathematics
In order to investigate the appropriateness of the presently used
placement strategy, performance on the tests and actual performance in
the classroom were analyzed. Thus, to answer question number one,
test scores on the math portion of the ACT and SAT were correlated
with grades in entry-level mathematics courses from a community
college. The sample consisted of data from 605 students who had
enrolled at Santa Fe Community College in Gainesville, Florida. All
of the students were high school graduates from the 1983-84 school
year and were entering the college for the first time. The students
had entrance test scores on record (ACT or SAT were the only test
scores available) and had completed their first college math course
during either the summer or the fall term, 1984.
The scores and grades were correlated using the Pearson product-
moment correlation (Pearson's r). This technique enabled a comparison
of student test score to student grade in order to examine the
strength of the relationship between the variables. This was done for
each of eight math courses, for each of the two tests for a total of
16 comparisons. A correlation with grades from all eight courses was
In order to answer question number two, students were identified
who may have been considered "high-risk" because of low test scores,
but who had successfully completed a college-level math course on
their initial attempt. Specifically, students who scored 15 or below
on ACT or 430 or below on SAT and made C or better in a college
mathematics course other than MAT 1000 (Introductory Math Skills) or
MAT 1002 (Basic Mathematical Skills) were interviewed. Twenty
students were interviewed by telephone to determine the factors to
which they attributed their success in college-level math courses.
Question number three was answered by construction of a clinical
evaluation model which was examined by a panel of community college
math faculty members, counselors, administrators, and university
professors. The panel responded to questions concerning content,
feasibility, and utility of the model and made recommendations
concerning its use.
In selecting the sample of test scores, courses, and grades it
was assumed that the students and instructors in the particular
college which generated the data were representative of students and
instructors in community colleges throughout the state.
It was also assumed that the members of the panel of experts who
evaluated the model were representative of math instructors and
administrators in community colleges throughout the state.
Of the four tests that have been approved for placement purposes
in Florida's higher educational system, only ACT and SAT were
considered in this study. Furthermore, only the mathematics section
of the tests and mathematics courses were analyzed. Therefore, only
mathematics placement was studied.
The data for courses and grades earned were limited to summer
and fall semesters, 1984 only. Students who did not have a test score
on record, did not take a math course in those terms, or did not
complete their course (grade of "W" or "I") were not included in the
The sample was also limited to high school graduates from the
1983-84 school year. Since community college populations are composed
of large percentages of students who are not recent high school
graduates, limiting the sample to recent graduates eliminated other
elements of the community college population. However, this did serve
as a control for age of students.
This study was also limited to initial math courses only. No
data were collected on subsequent math courses to determine student
performance in the overall mathematics curriculum.
Justification for the Study
The community college student has been viewed as somewhat
different from the traditional college or university student (Koos,
1970). Primarily designed to alleviate financial, geographic, and
motivational barriers, the community college was established for
students that may have been hindered from attending college because of
these barriers (Wattenbarger, 1971).
As community colleges developed and grew in numbers across this
nation, this uniquely American educational institution began to
attract new and different types of students. Non-traditional
students--those with lower entrance scores, from different socio-
economic backgrounds, part-time, older, having a variety of goals and
interests--made up larger and larger percentages of the community
college student population (Cross, 1976). The concept of the open
door was seen as fundamental to the mission and purpose of the
community college. Open access, allowing virtually any student to
enroll having a high school diploma or its equivalent, was largely
responsible for altering the course of American higher education
(Medsker & Tillery, 1971).
This emerging wave of students brought challenges and problems
never before encountered. Accountability and standards of excellence
became issues of high priority. Quality became a key word in higher
education. Many felt that the large number of new, non-traditional
students had somehow limited the quality of higher education in the
United States (Thornton, 1966). Recently, certain trends have
indicated a gradual closing of the open door (Henderson, 1982).
Because of the concerns for excellence and quality, additional
means of evaluation have become necessary. Florida's mandatory
student placement procedure, and the College Level Academic Skills
Test (CLAST), provide a current example. Merely having an open door
policy, then, has not been sufficient in accomplishing the stated
goals of the community college. For various reasons, students not
able to complete course requirements (and more recently, standardized
test requirements) turn the open door into a revolving door (Cross,
1976). For these reasons, effective placement procedures are
Open access with a "right-to-fail" philosophy is not justifiable
given the mission and goals of the community college, which include
development of students' abilities to the fullest. Rather than
allowing students to enroll in college-level courses for which they
are not prepared, appropriate placement would ensure enrollment into
courses compatible with their abilities. Students would then have a
reasonable probability of success in the course. Placement and
remediation, then, should be viewed as affirmation of the mission and
goals of the community college.
Research to provide information concerning the appropriateness of
current placement tests would enable decision makers to evaluate the
usefulness of the tests. Identification of factors in addition to
standardized tests that could be used in placement strategies would,
it is hoped lead to their implementation in placement models. Further
research on these subsequent models could validate the effectiveness
of the additional factors. Therefore, the results of this study could
be used in a comprehensive evaluation of the placement process.
Ultimately, the students in Florida's community colleges should
benefit from improved placement procedures.
Organization of the Research Report
The report consists of four additional chapters following this
introduction. Chapter II contains a review of the related literature.
Chapter III is a description of the methods and procedures, including
selection of the sample, data collected, and description of the
statistical analysis. Chapter IV presents the results of the
research. Finally, Chapter V contains the summary, conclusions,
recommendations, and implications of the study.
REVIEW OF RELATED LITERATURE
Community College Students and Rationale for Placement
The population of students attending this nation's community
colleges is markedly different from the population of students
attending four-year institutions. The population of community college
students includes students from a wide range of ability, age, and
other factors (Koos, 1970). Roueche (1980) referred to this
population as diverse. He noted that more and more students with
serious learning deficiencies had enrolled in community colleges and
were expected to continue to do so.
Cross (1976) reported that the historical trends in college
access from aristocratic to meritorious to egalitarian had brought
increasing numbers of low-ability students into programs of post-
secondary education. Calling these low-ability students "new-
students," Cross operationally defined them as those scoring in the
lowest third among national samples of young people on traditional
tests of academic ability.
Thornton (1966) noted that, historically, the community college
had attempted to provide educational opportunity for the average
student. The "open door" philosophy was described by Thornton as
allowing any high school graduate, or any person who seemed capable of
profiting by the instruction offered as being eligible for admission.
In a statement summarizing entrance requirements, Thornton stated that
the responsibility for choice--success or failure--should rest with
the student, not with a standardized test nor with the decision of an
admissions counselor. Entrance testing in community colleges has not
been seen as a selection process as is common in four year
institutions, but rather as a means of placement.
This concept of an "open door" as an admissions policy has been
seen as fundamental to the realization of the mission and purpose of
the community college. Cross (1971), however, maintained that without
proper consideration of this diverse population of students, the open
door can turn into a "revolving door," wherein students exit the
college unfulfilled. She commented that community colleges encouraged
diversity, yet seemed unable to move away from the unproductive
preoccupation with wanting all to learn the same thing at the same
Roueche (1980) commented that in order to give hope to students
who enroll having learning deficiencies, colleges were going to have
to divorce themselves from the "students have a right to fail" mind-
set, and design instructional programs accordingly. Placement into
proper courses was seen as essential to success in completion of
Wiener (1985) reported that between 60 and 70 % of all
community college students must take remedial courses. The drop-out
rate among such students was reported at over half. Students who have
successfully completed remedial courses, however, demonstrated a much
greater probability of completing their college programs. Wiener
called for mandatory placement testing and subsequent assignment into
remedial programs when necessary.
Linthicum (1980) studied the procedures and instruments used to
place students in developmental programs at a community college. The
evaluation system was designed to identify levels of skills and
subsequently guide students into the appropriate program. Grades,
nationally normed tests, and certain institutionally developed tests
were used to assess levels of skills. Linthicum also looked at
measures of the affective domain as reported by means of an Affective
Measurement checklist and a Self-Assessment checklist.
In evaluating the general success of the program, Linthicum found
that student choice was not an effective method of placement and that
a mandatory placement program was essential for student success.
However, placement based solely on test scores was not effective.
Qualitative factors such as motivation and self-concept were suggested
as important factors in the learning process. Linthicum emphasized
the necessity of flexibility in placement. She reported that reading
tests were better predictors of academic success than were math tests
and even recommended that math courses not be taken during the first
term by low ability students.
A study by Cordrey (1984) examined the effectiveness of a "skills
prerequisite program" used at a community college. The program
included mandatory placement testing and a curriculum of prescribed
remedial courses as a result of placement. The study examined
placement patterns, drop out rates, effects of remediation on
subsequent grade point average (GPA), and academic persistence.
Results of the study indicated that withdrawal from courses was
reduced as a result of the placement program. An institutionally
developed test was used for placement in math. The study also showed
that remediation did have a positive effect on future success in
academic courses. Other writers have questioned the effectiveness of
Haase and Caffrey (1983) reported information concerning the
assessment and placement process that had recently been instituted at
a community college. The Stanford Test of Academic Skills (TASK) was
used in addition to institutionally developed diagnostic assessment
techniques. An increase in retention of students as a result of the
placement program was reported. They concluded by commenting on the
necessity of continuously monitoring placement procedures and changing
assessment techniques when necessary. It was recommended that methods
beyond mere testing were needed and that flexibility was imperative.
Reap (1979) reviewed the American College Testing (ACT)
Assessment Program in terms of its use at a community college. The
purpose of the evaluation was to determine how effective ACT was in
helping the college reach its educational goals. Specifically the
study sought to answer two questions: (1) Did ACT provide an accurate
description of the entering freshmen? (2) Did ACT operate as an
effective predictor of student success? Reap concluded that the first
question could be answered in the affirmative, but the latter question
in the negative.
The review in Buros (as cited in Reap, 1979) reported that the
predictive validity of ACT was as satisfactory as the state of the
measurement art then permitted. Since ACT was used as a placement
instrument, Reap's study examined its effectiveness at that
institution. The correlation of math grades with ACT scores was
reported as .19. Reap further reported that when high school grade
averages were included, the effectiveness of prediction was increased,
and suggested that possibly high school transcripts should be
considered for predictive purposes. While it did appear that the
effectiveness of ACT in predicting grades increased as scores
increased, Reap concluded that the ACT did not appear to be successful
as an effective predictor of student success.
Clark (1980) explored six factors in attempts at determining
which variables were significantly related to student success in four
different math courses. These were
Placement test scores
High school GPA
Prior college units taken
Prior college GPA
High school math grades
Prior college math grades
Grades in high school and college math courses were found to be
significantly related to success (defined as grade "C" or better)
using the Chi-square technique at the .05 level of confidence.
Allen (1981) in describing instructional techniques in a
Fundamental Algebra course at a community college emphasized the
necessity of proper placement. "A 40 question 'Co-op Test'" was
administered to all entering students at the college. Initially,
students had the option of taking their desired course in mathematics;
however, the college agreed that waivers of the suggested placement
were counter-productive. Students were placed into the Fundamentals
of Algebra course on the basis of test scores, with no waivers. This
decision was based on a prior study by Allen which found that when a
student was given the appropriate placement test and placed into the
recommended course according to math department guidelines there was a
positive correlation between placement scores and grades in initial
math courses. Allen recommended correct placement as the first step
for success in the initial math course.
Palow (1979) advocated that assessment and placement were an
"integral part of a comprehensive program of instruction in
mathematics" (p. 1). Citing the SAT scores from the previous year
which indicated that the freshman class was the "least academically
prepared" group in the history of the examination, Palow emphasized
the necessity of a system of assessment and placement to "match and
funnel" students into a compatible system of instruction.
The assessment and placement system described by Palow consisted
of combining the results of two paper and pencil inventories. The
first inventory was a multiple choice mathematics test geared to the
course which the students had indicated they wanted to pursue. The
second inventory was the Canfield-Lafferty Learning Styles Inventory,
designed to determine an individual's preferred way of learning.
Through a set of decision rules programmed into a computer the student
was assigned to one of four modes in individualized instruction.
Results of the placement program were not available; however, the
rationale for the necessity of placement was consistent with other
writers' recommendations. It was also significant in that it
represented one of the only programs that included students' preferred
learning style as a factor in placement.
Wood (1980) noted that the assumption that algebra should be the
first course in mathematics for an entering college student was
unwarranted. Since open-door colleges have been faced with
ever-larger numbers of entering freshmen who have studied very little
or no mathematics, Wood made a strong case for effective placement
programs in mathematics at the community college level.
Wood suggested three probable causes for failures among entering
freshmen: (1) students that had not completed two years of high
school algebra; (2) students that had completed two years of high
school algebra, but had been out of school for three or more years;
and (3) students that had completed two years of high school algebra,
but with minimal grades of low "C" or "D."
A testing program recommended by Wood included two placement
testing instruments--ACT and institutionally developed mathematics
tests composed by the faculty. Six different entry-level courses
ranging from a math review course (practical arithmetic) through the
first course in calculus were offered.
Wood's recommendations for an appropriate placement program in
math were summarized as follows:
1. A majority of junior college freshmen have deficiencies in
mathematics that range from partial to total.
2. Records show that these deficiencies do not necessarily imply
a lack of ability. They frequently spring from insufficient
high school training and/or a time lapse between high school
3. For students of normal or above-normal ability, these
deficiencies can be effectively removed by a review course of
one or two semesters. Our experience leads us to believe
that the two-semester plan is the better one for any college
with an open-door policy.
4. Presidents and academic deans of colleges need to be aware
that short of returning to high school, students with serious
mathematical deficiencies have no way to improve without such
a review course or courses. This is especially true of
mathematics because of its cumulative nature.
5. Placement tests for entering freshmen as well as advanced-
standing examinations (in college algebra and trigonometry)
for well-prepared students provide an efficient way to
achieve accurate student placement.
6. The results of our investigation support the philosophy that
any junior college that maintains an open-door policy to all
high school graduates accepts responsibility for providing
students with courses in which they have a reasonable chance
to succeed. (Wood, 1980, p. 64)
Hierarchical Nature of Mathematics
Wilson (1971) discussed and illustrated testing for evaluation
purposes in secondary school mathematics. In the process of this
discussion, a framework or model of the secondary school mathematics
curriculum was constructed. Based on a taxonomy of educational
objectives developed by Bloom (1956), the model described the
mathematics curriculum in terms of content and behaviors. The content
consisted of number systems, algebra, and geometry while the behaviors
included the cognitive and affective domains.
Mathematics content was described as being progressive or
sequential in nature. For instance, the concepts involved in number
systems were arranged in order from simple to complex. Many of the
concepts that were described required proficiency in prior content.
Number systems preceded algebra and geometry. However, it was
emphasized that much of the content was incorporated throughout the
curriculum rather than in a specific course. In describing the
sequential aspects of the mathematics curriculum, Wilson said that
although there was a sequential nature to the mathematics curriculum,
a given topic may be presented at increasing levels of sophistication.
While some topics logically precede others, the dividing line between
content areas was described as unimportant.
The levels of behavior were described as being both hierarchical
and ordered. The levels were subdivided into the cognitive and the
affective domain. The cognitive domain included computation,
comprehension, application, and analysis. The affective domain
included interests and attitudes, and appreciation.
Computation items were designed to require recall of basic facts
and terminology. Emphasis was upon knowing and performing operations
and not upon deciding which operations were appropriate.
Comprehension related to recall of concepts and generalizations. The
emphasis was upon demonstrating understanding of concepts and their
relationships, not upon using concepts to produce a solution.
Application items required recall of relevant knowledge, selection of
appropriate operations, and performance of the operations. Analysis
items required a nonroutine application of concepts such as the
detection of relationships, the finding of patterns, and the
organization and use of concepts and operations in a nonpracticed
Wilson commented on the hierarchical and ordered nature of the
levels of behavior:
It is ordered in the sense that analysis is more cognitively
complex than application, which is in turn more cognitively
complex than comprehension, and the computation level
includes those items which are the least cognitively
complex. It is hierarchical in that, for example, an item
at the application level may require both comprehension
level skills (selection of appropriate operations) and
computation level skills (performance of an operation).
The affective domain was not described as hierarchical. It was
emphasized, however, that the affective domain must not be discounted
when considering instruction and evaluation of mathematics.
Effectiveness of Standardized Tests as Placement Instruments
While several sources in the literature demonstrated the
necessity for placement in community colleges, the means of achieving
effective placement have been somewhat elusive. As cited, most
methods of placement involve the use of a standardized testing
instrument, either as a sole means of placement or used in conjunction
with other criteria. One of the purposes of placement is the
prediction of success in the course or courses into which the students
are placed. Standardized tests have demonstrated only limited
predictive value. Apparently other factors are involved which have
been difficult to detect.
McClelland (1973) questioned the use of standardized tests,
as predictors of academic success and also as predictors of "success
in life" as defined by certain accomplishments such as job status,
earnings, satisfactions, etc. McClelland noted that researchers have
had difficulty in demonstrating that grades in school are related to
any other behaviors of importance--other than doing well on aptitude
tests. Making a case that standardized tests and grades only
correlate highly with one another, McClelland urged that a wider array
of talents should be assessed for college entrance. While the
argument applies mainly to the use of tests as selection instruments,
their value as placement instruments has also been subject to
Haney (1980) examined the use of standardized tests in a broad
context and related considerable controversy over their use. Sharp
disagreement was reported among testing experts on issues of test bias
and validity. Haney maintained that while a wide variety of
inferences may be drawn from any test score, one acid test of what
inferences were drawn was how the scores were used by social
Haney commented on the differences between aptitude tests which
were reported as being used for screening or selection purposes and
minimum competency tests. The competency testing movement generally
represented a government or institutional effort to regulate and
improve schooling. Haney cited Jensen who commented that the only
justification for competency testing for placement purposes was
evidence that the alternative treatments were more beneficial to the
individuals assigned to them than would be the case if everyone got
the same treatment. Jensen concluded that minimum competency testing
would not contribute to the solution of the problem of test bias and
validity since it appeared to be an unnecessary stigmatizing practice
with no redeeming benefits to individual pupils or to society.
Jensen apparently felt that alternative treatments, i.e.,
remediation, or low level classes, were not effective. This view is
consistent with information reported by Hills (1971), who cited
research that indicated evaluation of remedial courses was not
effective. Popham (1975), however, felt that competence testing for
placement was useful because it allowed isolation and remediation of
instructional deficiencies. Apparently, Popham assumed that
remediation was possible and effective.
A distinction has been made in the literature on testing between
two types of tests--norm referenced and criterion referenced.
Fundamentally, the difference is based on comparison of performance to
other individuals (norm referenced) or comparison to certain pre-
determined standards (criterion referenced).
Glaser (1963) and Popham and Husek (1969) were the first to
introduce and popularize the field of criterion-referenced tests (CRT)
which promised to be a significant breakthrough in education. The
CRTs were seen as a means of maximizing the potential of each student.
Cross (1976) discussed individualized, competency-based or mastery
learning and pointed out similar advantages of CRTs.
Hambleton, Swaminathan, Algina, and Coulson (1978) noted that the
introduction of CRTs was intended to meet the testing and measurement
requirements in objectives-based instructional programs. Problems
arose regarding a precise, acceptable definition of criterion-
referenced test, the central issue being the use of the word
"criterion." "Criterion" is best defined for these purposes as a
domain of behaviors, not a performance standard, minimum proficiency
or cut-off score. Popham (1975) provided the best workable definition
of CRT: "A criterion referenced test is used to ascertain an
individual's status (referred to as a domain score) with respect to a
well-defined behavior domain" (p. 2).
Controversy existed regarding terminology for tests of this sort,
with the terms criterion-referenced, domain-referenced, and objectives
referenced being the three discussed most. Popham advocated the term
"criteria-referenced" because of considerable public support and the
ill-advisedness of beginning a new campaign for "domain-referenced"
tests, even though the latter term is probably most descriptive.
According to McClelland (1973) competency-based testing should
1. be criterion referenced (not norm referenced)
2. be designed to reflect changes in what the individual has
learned (not measure "native intelligence")
3. provide public and explicit information on how to improve on
the characteristics) tested (not keep answers secreted away
from the public)
4. assess competencies involved in clusters of life outcomes
(not test esoteric qualities that are of little use in the
a. communication skills
c. moderate goal setting
d. ego development
5. involve operant as well as respondent behavior (not require
only pre-determined responses that may be unfairly limited,
i.e., all true-false or multiple choice)
6. sample operant thought processes to get maximum
generalizability to various action outcomes (so that students
can see the relevance of the skill, its application, and
ramifications in life situations)
The rapid acceptance of criterion-referenced testing in general
was not without problems, however. An urgent need for establishment
of standards, both for the development of and for demonstrating
validity of CRTs was noted by Evans (1975).
Novick and Lewis (1974) dealt with these problems as well as a
problem concerning the length of CRTs for a specified objective,
The minimum acceptable length depends on the
manner in which test information is used to make
decisions about individual students, the level of
functioning required for defining mastery of an
objective, the relative losses incurred in making
false positive and false negative decisions, the
background information available on the student
and on the instruction process, and the premium on
testing time within the instructional process.
Novick and Lewis adopted guidelines which effectively said that
test lengths of 12 items or fewer for a specified objective were very
desirable. Lengths above this and up to 20 were tolerable. Tests
that were longer than this for a single objective were described as
discomforting. Tables of test lengths, taking the above mentioned
factors into account, were published. They concluded that "mastery
must be confirmed by a test that permits demonstration of non-mastery"
(1974, p. 158).
Special categories of CRTs were identified by Keesling (1974), in
particular, the specific type of mastery learning in which order of
presentation was crucial. Objectives that were subject to a priority
ordering based upon task analysis or theories of instruction were
distinguished from objectives that required no ordering of
presentation or transfer of training. Keesling concluded that the
validity of the proposed structure of relationships among objectives
was very important. This hierarchy is not always detectable, however.
Hambleton et al. (1978) commented that when it is possible for a set
of learning objectives to be arranged into a learning hierarchy, the
strategy of branch-testing would seem to offer considerable potential
for decreasing the amount of testing while improving its quality.
The problems associated with criterion-referenced tests,
particularly such issues as test score validity, determination of cut-
off scores, and complicated legal actions by the courts have not been
totally resolved. However, Hambleton et al. (1978) reported that
sufficient theory and practical guidelines were available for
construction of at least adequate CRTs and criteria-referenced testing
Numerous studies have examined specific standardized tests, most
notably the SAT and the ACT and their correlation with grades. The
results have been inconclusive. Schade (1977) reported a study
carried out at a community college to determine the predictive
validity of various parts of two standardized tests toward academic
achievement as measured by the first semester GPA. The tests used in
the study were the ACT and the Missouri College Placement Test (MCPT).
Each of the ACT segments showed a significant correlation with first
semester GPA. The ACT mathematics segment had the lowest correlation,
r2 = .125. Schade described the predictive power of the tests as
"poor to moderate" and indicated that the values of the correlation
coefficients were comparable to results that had been reported from a
variety of institutions. None of the MCPT segments showed a
Schade theorized as to why his study and several others which he
cited failed to demonstrate high correlations between grades and test
scores. One possibility mentioned was a change in student motivation,
either positive or negative. College-level work could possibly
stimulate previously low achievers with lower test scores into better
performance or could stifle earlier motivation. A second possibility
noted was that the tests could be suspect. A low score was actually
impossible to interpret, possibly indicating insufficient native
capacity or inadequate training in the skills and abilities tested.
Test bias may have also been a factor. Thirdly, grades and a lack of
standardization of grading systems could have contributed to the poor
Nolan (1976) also conducted a correlational study using ACT sub-
test scores and grades earned in corresponding subject areas to
determine the predictive value of ACT. The correlation analysis
yielded coefficientis of such low magnitude as to make him conclude
that there was no significant relationship between ACT scores and
academic performance. The r2 for mathematics grades and the ACT math
sub-test was reported at .07. Nolan stated that "it appears that high
school grades alone are the best predictors considering the negligible
amount of variance accounted for by ACT scores" (p. 4).
Larson and Scontrino (1976) examined the validity of high school
grade point average and of the verbal and mathematical portions of the
SAT as predictors of college performance over an eight year period.
They reported multiple-correlation coefficients that were
"consistently high." Interestingly, the mean proportion of variance
accounted for in the eight-combined samples by using all three
predictor variables in combination was only 4.7% greater than the mean
proportion of variance accounted for by using the high school GPA as a
Fincher (1974) examined SAT scores in a state university system
over a 13 year period. He noted that the value of SAT scores is
their use in conjunction with high school grades. The SAT scores
alone (the zero-order) correlated with college performance was not
considered a sufficient indicator. However, used in addition to the
high school record, SAT was judged to improve predictive efficiency.
Fincier's comment regarding the mathematics portion of SAT was
noteworthy. He commented that the mathematics scale contributed with
less consistency than the verbal section of the test and would not
appear to be highly useful in one-half of the situations where it was
Other findings of interest by Fincher were confirmation of
previous findings that females were more predictable in academic
performance than males (when comparing SAT scores, high school grades,
and college grades) and the loss of information from combining SAT
math and verbal scores. Each portion of the test taken separately was
a better predictor than the combined score. These results were
consistent with previous findings and Fincher recommended continued
use of SAT as a selection instrument.
Price and Kim (1976) compared high school grades and entrance
test scores with performance in college. The ACT scores were used as
the standardized test scores. College GPA was mostly determined (75%)
by both high school grades and ACT scores. However, Price and Kim
concluded that it appeared reasonable to believe that ACT scores were
more significant and important in predicting a person's ability to
perform in college than were high school grades because the beta
coefficients of four specific fields of the ACT program were
relatively larger than those of high school grades.
Alternative Means of Assessment for Placement
Grades have traditionally been a basic means of evaluating,
recording, and reporting students' progress. Several sources in the
literature have expressed concern with using grades for objective
measurement purposes since grades are often determined subjectively.
Haase and Caffrey (1983) stated that grades were not good measures of
performance because of grade inflation and lack of standardization.
Schade (1977) commented similarly about the lack of
standardization of grades:
Among the many teachers, areas of study, and
institutions, there are a plethora of grading
standards. A student who achieves at a certain
level in one class should be expected to achieve
at approximately the same level in other classes.
Yet this does not always happen. If students were
to take the same course from different teachers,
whether at the same or at a different institution,
they would not necessarily make the same grades.
Different academic areas will tend to have
different standards. Some disciplines are
notoriously stringent and demanding while others
are the opposite. Compounding this difficulty is
the fact that lower ability students will tend to
gravitate towards those areas that are less
taxing. (p. 19)
Goldman and Slaughter (1976) questioned the use of the grade
point average (GPA) as a validation criterion. Most studies
attempting to validate standardized tests compare test scores to GPA.
However, because grades appear to be more explainable by unmeasured
traits than by test scores or previous grades, their use as a
validation criterion becomes suspect. Goldman and Slaughter further
maintained that composite GPA was a poorer predictor than single class
grades. Since composite GPA is made up of decidedly nonequivalent
components it is less reliable and hence less predictable than grades
from a single class.
Longstreet (1975) questioned the use of grades as fair and
objective measures of performance, maintaining that grades are used
for convenience in administration and for tradition. Longstreet noted
the important difference between grading based upon knowledge of
subject matter and grading based on comparative scores. Longstreet
called for alternatives to traditional grading such as mastery
learning, contract grading, self-assigned grading and conferences with
students, commenting that "criteria truly significant to the
development of an intellectually independent and creative individual
cannot be reduced to . a few letters or percentage points, however
convenient these may be bureaucratically" (p. 246).
McClelland (1973) questioned the validity of grades as
predictors, asserting that while grades and test scores correlate
highly with one another, neither can accurately predict future
measures of success in life. He noted that researchers have had
difficulty demonstrating that grades in school are related to any
other behaviors of importance other than doing well on aptitude tests.
He stated that while grade level attained seemed related to future
measures of success in life, performance within grade was related only
slightly. Results of several studies indicate that superior on-the-
job performance is related in no way to better grades in college.
Though grades have been the subject of criticism, their use is
widespread, practically universal. As previously noted, standardized
test scores have also been subject to criticism for many of the same
reasons as grades. Alternative means of assessment have been reported
in the literature. However, there seems to be no agreement on what,
if any, measures should be used as alternatives. Grades and
standardized tests persist as the most common means of measurement.
Clinical and holistic methods of evaluation have been suggested as
alternatives. The concept of clinical evaluation, an analytical
assessment of competencies and deficiencies, and the prescription of
treatment based on direct communication and observation by a
practicing professional, was referred to by several writers.
Holistic evaluation, based on the theory that reality is made up
of unified wholes that are different from the simple sum of their
parts, has also been proposed as a viable addition to measurement and
evaluation methods dominated by traditional grades and standardized
tests. Oral exams, performance tests, situational tests,
observations, and checklists were suggested as complements and
alternatives (Roueche, 1980). A holistic approach emphasizes the
importance of the whole and the interdependence of its parts
Clinical and holistic evaluation have been referenced numerous
times pertaining to mathematics; however, the specific terms have not
been used. Many writers who proposed alternatives to standardized
testing made reference to clinical and holistic approaches to
Wilson (1971) concluded that standardized testing alone was not
sufficient for proper evaluation in mathematics:
Mathematics learning is a many-component task. It
should be measured or evaluated over a broad range
of criteria. The evaluation of mathematics
learning in terms of a single measure leads in
incomplete or even erroneous information . .
The use of standardized tests in the evaluation of
classroom learning is of limited value. They are
inappropriate for formative evaluation. For
summative evaluation, standardized tests tend to
concentrate on one level of behavior (and hence
limit the range of outcomes to be considered) or
combine scores, levels of behavior or content (and
hence limit the information that may be available
on the test). (p. 264)
Sueltz (1961) in the National Council of Teachers of Mathematics
(NCTM) Handbook on Evaluation on Math commented that to determine the
level of sophistication of a student's work and the depth of
understanding of a major topic requires a much more refined procedure
than mere standardized testing. The writers of the Handbook further
conclude that evaluation of the thinking and procedures employed by
students usually is better done by careful observation and interview
than by objective testing.
Various writers have called for holistic and clinical approaches
to evaluation as alternatives to standardized tests and traditional
grading. Quinto and McKenna (1977), in an NEA published monograph,
suggested several alternatives to standardized testing as means of
evaluation. The alternatives included contract grading, conferences
with students, and checklists based on observations. Kopfstein (1980)
commented on community college reading and study skills programs,
proposing certain methods of evaluation that would be considered
clinical, although the term is not specifically used. Student
interviews were one of the primary methods suggested. O'Reilly,
Vogler, and Asche (1980) suggested several options that could be used
as alternatives to standardized tests, all of which would be
considered clinical or part of a holistic evaluation, although again
the terms are not used. Ginsburg (1975) recommended evaluating
student progress through "direct oral communication." Interviews and
conversations with students were essential to this type of evaluation.
In fields other than education, especially the medical field,
writers have called for additional methods of assessment. Clinical
observation procedures have been proposed. Kuliecke, Lloyd, and
Mathis (1982) identified problems in evaluation of medical students
and emphasized the necessity of process evaluation and not mere
product evaluation. Factors were cited that were responsible for poor
performance on medical examinations by medical students. Lack of
complexity in data analysis procedures, quality of training, mental
set at time of examination, experience with taking exams, effort put
into training, and barriers to assimilating training (especially
foreign language barriers) were cited. Kuliecke et al. called for
models of evaluation that could take these factors into account.
Shugars, May, and Vann (1981) discussed evaluation of dental
students. They maintained that cumulative data from several faculty
members should be used as a basis for grades. In addition, means for
assessing professionality in students was stressed. A clinical
judgment examination, wherein faculty members carefully observed
students, was integral to the dental evaluation model.
Halpin, Halpin, and Schaer (1981) compared objective measures of
writing to holistically scored essays. The holistic method of scoring
was based upon a generalized impression or global quality of the
essay. Halpin et al. found that 26% of the variance in the essay
scores was explained by the objective measures. Clemson (1978)
reported a somewhat higher correlation, 49% for r2, in a similarly
conducted study. Objective measures were not sufficient in explaining
the total range of variation in holistically scored samples. The
holistic method of evaluation appeared to be at least as useful as
objective measures in assessing competence.
Attitude and the Affective Domain
Corcoran and Gibb (1961) discussed attitude appraisal in the
learning of mathematics, noting that suitable instruments were not
widely available. Attitude towards math involves both cognitive and
non-cognitive aspects. They noted that a student's attitude toward
mathematics is a composite of intellectual appreciation of the subject
and emotional reactions to it. Corcoran and Gibb examined attitude
according to direction (attraction or repulsion) and intensity (strong
or weak). Other important aspects of attitude toward mathematics were
noted as consistency, salience, reaction to difficulty, interest, and
Various methods of assessing attitudes toward math were reported,
basically separated into self-reports wherein students reported their
own attitudes, and observer reports, based on interviews with the
students. Thurstone scaling methods, Likert scales, Guttman-type
scales, and Hoyt-MacEachern scales were used in self-reporting
attitude appraisal. The Minnesota National Laboratory instrument was
suggested for observer reporting. Fouche (1961) summarized Corcoran
and Gibb's chapter by stating that ignoring attitude evaluations
completely would be almost to treat students as mere learning
machines, devoid of feelings and emotions, and such higher, complex
behavior as creativity and discovery. Fouche indirectly referred to
holistic evaluation in math:
A little thought will show that what is generally
meant by "testing" is really uniform testing of a
number of students. The conscientious, skillful,
.perceptive mathematics teacher is constantly
making evaluations of a student's verbal answers,
of his blackboard work, of his facial expressions
and other non-verbal behavior, but tests are
required nonetheless in order to have uniform and
easily comparable information about the behavior
of all students in the same situation. (p. 172)
Since no two students have the same learning style, Roueche
(1980) maintained that no single methodology of evaluation would fit
all students. Some students are "right-hemisphere preferenced; that
is, they excel at holistic and spatial functions. Traditional methods
of instruction and evaluation, designed around left-hemisphere
strengths (verbal and analytic) of the middle class, will not reach
them according to Roueche.
Within the last decade research having far-reaching implications
for evaluation and placement has been done regarding learning styles.
Dunn, Dunn, and Price (1977) stated that how a student learns is
perhaps the most important factor in his academic achievement. Dunn
and Dunn (1979) developed a "Learning Style Inventory" (LSI) based on
research data that yielded 18 categories which suggest that learners
are affected by the following elements:
1. immediate environment (sound, temperature, light, and
2. emotionality (motivation, responsibility, persistence, and
3. sociological needs (self, pairs, peers, teams, adult, and/or
4. physical needs (perceptual strengths and/or weaknesses, time
of day, intake of foods and fluids, and mobility).
Dunn and Dunn (1979) reported that teachers were able to
recognize some learning style elements with considerable accuracy.
Certain other elements were admittedly more difficult to assess. They
further commented that people "learn in ways that differ dramatically,
but certain students achieve only through selected methods--methods
that frequently fail to produce academic results for others" (p. 238).
The Productivity Environmental Preference Scale pepsS), an adult
version of the LSI, has also been developed by Dunn and Dunn. The
PEPS would be suitable for use in community college curriculum,
instruction, and evaluation.
Farr (1971) confirmed in a study of learning styles that
individuals could accurately predict the modality in which they could
demonstrate superior learning performance. The data revealed that "it
is advantageous to learn and be tested in the same modality and that
such an advantage is reduced when learning and testing are both
conducted in an individual's non-preferred modality" (p. 242). The
most desirable conditions existed when learning and testing were both
in the student's preferred modality.
Domino (1970) reported that students who had been exposed to a
teaching style consonant with the ways they believed they learned
scored higher on tests, fact knowledge, attitude, and efficiency of
work than those who had been taught in a manner dissonant with their
orientation. Hunt (1981) asserted that learning style described
students in terms of those educational conditions under which they are
most likely to learn. To say that a student differs in learning style
means that certain educational approaches are more effective than
others, said Hunt. This differed somewhat from Dunn and Dunn in that
Hunt viewed learning style as a malleable trait.
Hunt (1981) further stressed the reciprocal relation between
psychological theory and educational practice, concluding that
reciprocity is a central feature in matching of styles. Taking such a
reciprocal view of matching, Hunt declared, more reasonably accounts
for the continuously changing nature of the teaching/learning
Davidman (1981) was critical of some of the assertions made
regarding learning styles. In particular, the validity of the LSI was
questioned. Davidman argued that, although the LSI provided
interesting information, it should not be taken as a clear and
irrefutable indication of a student's pattern of learning. He
concluded by stating his belief that brief, teacher-made instruments
would initiate a more useful diagnostic process.
Dunn, Dunn, and Price (1981) countered Davidman's criticism by
maintaining that the LSI offered a reliable and practical alternative
to "soft evaluation." They stated that "in conjunction with teachers'
insights and experiences, it could provide the foundation for building
learning environments designed to meet the needs of individuals"
Palow (1979), as previously cited, listed learning-style
preference as a major consideration for placement of students.
However, no other sources were found that used learning style
preference as a placement strategy.
Placement has been a continuing problem for community colleges in
particular, because of the great diversity that characterizes their
clientele. Since entering students vary greatly in academic skills,
some means of assigning students to courses commensurate with their
skills has been considered imperative. Standardized tests have been
the most commonly used means of assessment for student placement.
High school grades have also been used, with slightly better success
than tests. Alternative means of assessment, including clinical and
holistic evaluation, assessment of the affective domain, and learning-
style preference have been proposed. Because of the difficulty in
reliably measuring these alternatives, the use of traditional grades
and standardized tests persist as the most widely used placement
Numerous studies have considered the relationship of standardized
tests and grades. Most of these, however, examine the predictive
validity of the tests for use as selection instruments. Since
mandatory testing for placement is relatively recent in community
colleges, information regarding the effectiveness of certain commonly
used tests as placement instruments was limited. Only a small amount
of information was found in the literature regarding appropriate
placement strategies in community colleges. Most of the studies had
a similar rationale and underscored the necessity for placement.
However, it appears that an effective, agreed-upon system of placement
has not been reported.
METHODS AND PROCEDURES
Research Design and Selection of Variables
In order to determine the relationship between placement tests
and student performance in mathematics, and thus address the first
question of this study, several decisions were required in order to
design a suitable research model. As previously noted, four tests
have been approved by the Florida legislature for placement purposes
in higher education. While information on all four tests would have
been desirable, data on two of the tests, MAPS and ASSET, were not
available from the college used in this study. Both ACT and SAT have
long been used as admissions tests and data were available. The MAPS
and ASSET tests are relatively new and data concerning these two
tests, when available, should be used in subsequent studies.
High school grades have been commonly reported in the literature
as effective predictors of college grades. Several writers
recommended using high school grades in conjunction with placement
tests for the most effective prediction of college grades. Prior
grades typically have proven to be the best predictors of subsequent
grades. Therefore, the possibility of including high school grades as
a second independent variable was considered. However, since previous
research has been adequate in demonstrating the usefulness of high
school grades, they were not used as a second independent variable.
Since test scores were the only state-mandated measures for math
placement at the time of this study, standardized test scores were
used as the sole independent variable.
Several alternatives were available for selection as measures of
student performance to use as the dependent variable. Various
measures of performance and combinations thereof have been reported in
the literature in the assessment of student performance. Student
opinion, teacher opinion, clinical evaluation methods, other
standardized tests, grade point average, student retention, and other
factors have been used to measure relative success. Convincing
arguments have been made for the use of measures other than grades in
evaluating student performance. Grades have been criticized as
lacking objectivity and standardization. These same criticisms apply,
however, to the previously mentioned alternative means of assessment.
Grades have continued to be accepted as the most commonly used measure
of student performance. Therefore, grades in math courses were chosen
as the measure of student performance for this study.
Relationships between test score and grade would be most
meaningful when considered for each course individually. Grades in
different mathematics courses would obviously have different meanings
since math courses are sequentially arranged in the curriculum. For
instance, a grade of "A" in a college preparatory course would have a
much different meaning than a grade of "A" in calculus, since calculus
is near the end of the sequential mathematics curriculum, whereas
college preparatory math is a simpler course.
For these reasons, ACT and SAT scores and grades were correlated
for each of the eight selected math courses. A composite correlation
of test score and grades was also performed for each test.
Factors found in the literature and thought to be of potential
use in the placement process were studied by interviewing students who
had enrolled in college mathematics courses and were successful, i.e.,
achieved grades of C or better in courses other than MAT 1000
(Introductory Math Skills) or MAT 1002 (Basic Mathematical Skills) but
had test scores below or slightly above the cut-off scores.
Sample Selection and Data Collection
Given the variables which were to be used in the study--placement
test scores and grades--a source of data containing these variables
was located. Data from all students that had enrolled for the first
time in the summer or fall semester of 1984 who had entrance test
scores on record as well as a math course and grade were obtained
through the Management Information Services division of Santa Fe
Community College, one of the 28 community colleges in Florida's
higher educational system. The college is located in Gainesville, in
north-central Florida, and draws students mainly from its local
district, as well as from throughout the state and from other states
The demographics of the selected cases obtained from the college
were found to be similar to the statewide population of community
college students. Table 3.1 shows the similarity.
The breakdowns by race and sex for recent statewide high school
graduates was quite similar as well. Statewide data represent
students in the 15-19 age group, while data from the college in the
study represent 1984 high school graduates (Table 3.2).
Demographics of First-Time Community College Students
Santa Fe Community College Students Statewide
Sex Male 44% Sex Male 41.3%
Female 56% Female 58.7%
Race Black 11% Race Black 10.8%
White 79% White 77.3%
Other 10% Other 11.8%
Demographics of Recent High School Graduates Entering the
Community College System
Santa Fe Community College Students Statewide
Sex Male 43.4% Sex Male 43.7%
Female 56.5% Female 56.3%
Race Black 13.6% Race Black 11.4%
White 79.3% White 72.4%
Other 7.0% Other 16.0%
The data for students actually used in the sample, i.e., those
who had a test score and passed a course, were also similar. Table
3.3 depicts this.
Demographics of Sample of 1984 Hiah School Graduates
Sex Male 280 46.2
Female 325 53.8
Total 605 100.0
Race Black 67 11.0
White 484 80.0
Other 54 9.0
Total 605 100.0
The data were collected from a computer print-out from the
college which contained information on all of the entering students
for the 1984-85 academic year (including both summer and fall
semesters) who had graduated from high school the previous academic
year, 1983-84. Only data from those students who had a test score
on record and had enrolled in and completed a math course were
collected. A total of 605 complete sets of data were used in the
Description of Data
The two placement tests used in this study were the ACT and
SAT examinations. As noted previously, these tests were designed
for predictive purposes and have been commonly used as selection
instruments. However, these tests were widely used for placement
purposes in most of the community colleges in Florida at the time of
Kline (1972), in Buros' Seventh Mental Measurement Yearbook,
explained that the ACT Mathematics Placement Examination was developed
to assist colleges in placing entering students in the mathematics
classes most appropriate for their ability and preparation. The test
contains four categories: intermediate algebra, college algebra,
trigonometry, and "special topics." Kline suggested that the test may
be most useful for predicting success in highest level freshman
mathematics, analytic geometry, and calculus. The reliability, KR-21,
was reported at .81 for the total score.
Dubois (1972) and Wallace (1972) also reported in the Buros
yearbook. The purpose of SAT was described as an aid in assessing
students' competence for satisfactory achievement in college. The SAT
was designed to be effective over the full range of abilities of
students. The test had been found to have "reasonably good"
validities for predicting college achievement. Internal consistency
reliability was reported at .90 for SAT-mathematics. Alternate-form
coefficients of correlation were reported at .88. It was also noted
that multiple correlation studies with high-school grades yielded
better predictive results for college grades than SAT alone.
Scores on the ACT in this study range from 1 to 32. The mean
score on ACT was 14.262, median score was 14.5, and the mode was 10.
One standard deviation was 6.288. The mean SAT score was 433.029,
with standard deviation of 92.529. Median score was 429.333 and the
mode was 450. Of the 605 student scores used in the study, 421 were
ACT scores and 184 of the scores were SAT. Scores on the SAT ranged
from 210 to 710.
Eight math courses were chosen for use in this study, ranging in
content from fundamental skills in arithmetic to the first course in
calculus. These courses were by far the ones selected most often. A
small percentage of students chose other courses, such as business
math or the second course in calculus, but were not included in the
study. Twenty-three (23) students were enrolled in MAT 1000,
Introductory Math Skills. This course was designed for students who
needed to develop basic computational skills and improve accuracy with
basic arithmetic facts. Included in the course was a math lab, where
students worked individually to improve skills.
One hundred ninety-eight (198) students enrolled in MAT 1002,
Basic Mathematical Skills, which included the arithmetic of whole
numbers, fractions, decimals, and percent. This course also had a
lab component where students worked individually to develop and
One hundred ten (110) students enrolled in MAT 1024, Elementary
Algebra. This course represents the first of three algebra courses
offered by the college and included the study of algebraic notation
and terminology and the addition, subtraction, multiplication, and
division of general algebraic expressions, among other topics.
One hundred fifty-one (151) students enrolled in MAC 1102,
Intermediate Algebra. This course was the second in the algebra
series and included the study of complex rational expressions,
exponents, roots, radicals, and other algebraic topics.
One hundred twenty-one (121) students enrolled in MAC 1104,
College Algebra, the third course in the algebra series. This course
included the study of relations, functions and conic sections, theory
of equations, systems of equations, exponential and logarithmic
functions, and other related topics.
The sixth course of the eight courses used in this study was MGF
1113, Principles of Mathematics. Only 13 students enrolled in this
course which presented an overview of the various branches of
mathematics and their development. Sets, logic, introduction to
algebra, statistics, probability, and geometry were among the topics
Trigonometry, MAC 1114, was the choice of 22 students. This
course covered the study of the six trigonometric functions, their
interrelationships, and their application to both right and oblique
Finally, 29 students enrolled in MAC 2311, Calculus I with
Analytic Geometry. This course was the first course offered in the
calculus series and included the study of limits, the derivative and
its geometric interpretation, continuity, integration of algebraic and
trigonometric functions, and applications of integration to area.
Students who received grades of "W" (withdrawal) or "I"
(incomplete) were not included in the study. Sixty-two (62) students,
approximately 9%, had grades of "W" or "I" and were omitted.
Test scores, course, and grade were entered and analyzed using
the Statistical Program for the Social Sciences (SPSS) package.
Initially, the data were separated by test score, ACT or SAT, yielding
two groups. Since no student had scores on both the ACT and SAT, no
direct comparison of tests was made. The relationship of test score
to grade was determined separately for each test.
The data were then categorized by course. The eight courses were
analyzed for each test, yielding 16 tables and graphs (summarized in
Chapter IV). The Pearson-product moment correlation (Perason's r) was
computed for each course, correlating test score with grade. Finally,
composite correlations were done for each test across all courses,
comparing test score to grade.
The literature suggested that other factors should be used in
addition to test scores when evaluating students for placement or
other purposes. However, identifying, defining, quantifying, and
using these other factors has been rather difficult and, as yet, no
strategies have been implemented on a statewide basis that make use of
any factors other than test scores.
In an attempt to identify additional factors that may be of use
in placement of students, interviews were conducted. Students who
scored below or slightly above the state-mandated cut-off scores
(effective August, 1985) were identified. Those students who
successfully completed a college-level mathematics course ("C" or
better) were asked what factors they considered important to their
success. The purpose of these interviews was not to draw any
conclusions or generate any statistical evidence on other factors for
placement. Rather, the identification of potential factors for use in
a placement model was the intent of the interviews.
The cut-off scores on ACT and SAT (math) have been mandated as 13
and 400, respectively. Students who scored below the cut-off or
slightly above were interviewed. Specifically, students whose score
on ACT was 15 or less, or 440 or less on SAT, were interviewed, but
only if they achieved a "C" or better in a college-level math course
(MAT 1024 or above).
Travers (1969) discussed the effects of an interviewer on the
results. As much uniformity as possible was recommended in order to
maintain consistency in the conditions under which the information was
collected. Interviews were suggested that were not too highly
structured, allowing the interviewee to respond freely but still in
accord with the subject at hand. The same introductory and concluding
remarks were recommended. Therefore, each interview began with an
introduction of the caller, a brief explanation of the purpose of the
interview, and a question as to whether the student would be willing
to participate. Following the interview, the interviewer thanked the
student for cooperating. The interview process was very similar,
then, for each student contacted.
Difficulties in quantification of results of interviews were also
reported by Travers. However, the purpose of the interviews in this
study was not to generate empirical data, but rather to identify
potential factors for use in placement strategies. Therefore,
quantification of responses was not part of the interview process,
except to report in a general way what the students said.
An interview guide is located in the appendix of this report.
The guide represents a framework from which the interviewer (this
writer) conducted the interviews. Twenty students were interviewed,
all by telephone. Results of the interviews are reported in Chapter
Development of the Clinical Model
As a result of the literature search in developing a rationale
for this study, several alternatives to standardized testing emerged.
Using the term "clinical evaluation," the writer attempted to combine
several of these aspects into a model which could be used to evaluate
student competence in mathematics. The intended use of the model was
to provide additional information beyond test scores, that could be
useful in decisions regarding placement, remediation, and curriculum.
The model was grounded in theoretical concepts that seemingly
would be of value when applied to practical problems in community
colleges, such as placement, design of college-preparatory courses,
and evaluation of student competence. In an attempt to validate the
effectiveness of the model, a "panel of experts" was consulted and
asked to respond to the model with respect to content and feasibility.
The panel consisted of knowledgeable persons in the field of
community college mathematics. Specifically, eight persons were
contacted. Two of the individuals were university professors familiar
with the problems of mathematics placement in the community colleges.
One of the panel members was a community college administrator,
likewise familiar with mathematics and related problems. One was a
counselor at a community college who was experienced in problems
of mathematics evaluation and placement. Four of the panel members
were mathematics instructors at community colleges, one of whom was
the department head.
The panel members were in agreement with the concept of such a
model. All of the members supported the idea that information in
addition to test scores would be useful. At various stages of the
development of the model various panel members made suggestions to
enhance the development of the model. The model was revised several
times and eventually a final version was submitted to the panel for
their comments, criticism, and suggestions as to the feasibility and
content of the model. A description of the model and a summary of the
panel's reaction to the model are included in Chapter IV.
Results are presented in this chapter in response to the three
questions outlined in Chapter I. Results of the data analysis to
determine the relationship between placement test scores and success
in initial math courses (Question one) are presented in Table 4.1.
This table summarizes the correlations of ACT and SAT scores and
grades earned in the eight entry level mathematics courses described
in Chapter III. The correlations are presented separately for ACT and
SAT. The Pearson product-moment correlation (r) is given for each
course. The final two rows represent the correlation for a composite
of grades across all courses, for ACT and for SAT.
A further analysis of the comparison of test scores and grades is
provided in Tables 4.2 and 4.3. The correlations reported in Table
4.1 which were significant at the .95 level are included in these
tables. Test scores are grouped into four categories which are
indicated in a horizontal arrangement. Grades are listed vertically
from A down through F. A cross-tabular effect is thus presented,
showing the number (n) and percentage of students who earned a
specific grade (A through F) and whose test score was in a certain
category. Table 4.2 presents ACT scores and grades, cross-tabulated
for the three courses which had significant r levels, and the
composite of all ACT scores and grades. Table 4.3 presents the same
information for SAT scores.
Correlation of Test Score and Grade for ACT and SAT
for Entry Level Mathematics Courses
Course Test n r
Introductory Math ACT 20 -.01
SAT 3 -.50
Basic Math ACT 160 .31*
SAT 27 -.04
Elementary Algebra ACT 71 .04
SAT 32 .19
Intermediate Albegra ACT 77 -.07
SAT 54 .22*
College Algebra ACT 62 .21*
SAT 42 .47*
Principles of Mathematics ACT 7 -.42
SAT 5 -.24
Trigonometry ACT 11 .78*
SAT 10 .69*
Calculus ACT 13 .25
SAT 12 -.16
Composite ACT 423 .16
Composite SAT 188 .12
Note: Composite score included 4 scores on SAT and 2 scores on ACT
which were subsequently dropped because the courses were not
used in the study.
*Significant at the 95% level.
Cross-Tabulation of ACT Scores and Grades
0-7 8-15 16-23 24-32
Course Grades N % N % N % N %
Basic Math A 6 4% 29 18% 0 0% 0 0%
B 15 9% 26 16% 0 0% 0 0%
C 9 6% 19 12% 2 1% 0 0%
D 0 0% 7 4% 1 0% 0 0%
F 27 17% 19 12% 0 0% 0 0%
College A 0 0% 2 3% 5 8% 8 13%
Algebra B 0 0% 2 3% 10 16% 5 8%
C 0 0% 1 2% 10 16% 0 0%
0 0 0% 2 3% 2 3% 1 2%
F 0 0% 1 2% 11 18% 2 3%
Trigonometry A 0 0% 0 0% 2 18% 3 27%
B 0 0% 0 0% 1 9% 0 0%
C 0 0% 0 0% 3 27% 0 0%
0 0 0% 1 9% 1 9% 0 0%
F 0 0% 0 0 0 % 0% 0 0%
Composite A 8 2% 42 10% 20 5% 18 4%
B 19 5% 37 9% 29 7% 5 1%
C 14 3% 34 8% 37 9% 3 1%
D 1 0% 19 5% 17 4% 2 1%
F 36 9% 33 8% 40 10% 3 1%
Cross-Tabulation of SAT Scores and Grades
210-330 340-450 460-570 580-710
Course Grades N % N % N % N %
Intermediate A 0 0% 2 4% 2 4% 1 2%
Algebra B 0 0% 7 13% 7 13% 1 2%
C 0 0% 15 26% 1 2% 1 2%
0 1 2% 5 9% 1 2% 0 0%
F 1 2% 6 11% 2 4% 1 2%
College A 0 0% 2 5% 5 12% 4 10%
Algebra B 0 0% 3 6% 3 6% 1 2%
C 0 0% 5 12% 2 5% 0 0%
D 0 0% 5 12% 2 5% 0 0%
F 0 0% 5 12% 5 12% 0 0%
Trigonometry A 0 0% 0 0% 2 20% 2 20%
B 0 0% 0 0% 1 10% 0 0%
C 0 0% 3 30% 0 0% 0 0%
0 0 0% 1 10% 1 10% 0 0%
F 0 0% 0 0% 0 0% 0 0%
Composite A 9 5% 10 5% 12 6% 7 4%
B 9 5% 18 10% 17 9% 5 3%
C 1 0% 30 16% 7 4% 3 2%
D 2 1% 18 10% 4 2% 0 0%
F 6 3% 20 11% 9 5% 2 1%
Discussion of Results to Question One
Question one, which addressed the relationship between placement
test scores and performance in initial college mathematics courses,
was answered by correlating test scores and grades. As indicated in
Table 4.1, the Pearson product-moment coefficient of correlation (r)
was computed for each course, comparing test score to grade. The r
statistic is a numerical descriptive measure of the correlation
between two variables which measures the strength of the linear
relationship between them (McClave, 1982).
The implicit assumption in such a comparison of test scores to
grades would be that if a positive linear relationship existed
between test score and grades, that relationship would be apparent
from the r statistic. Correlations of a highly positive nature would
indicate such a relationship. However, the results of this study
indicate quite low correlations, with many of them actually
Only two of the positive correlations were at the .50 magnitude
or higher. Grades in MAC 1114, Trigonometry, were correlated with ACT
scores at .78 and with SAT scores at .69. Other moderately positive
correlations were .47 for grades in MAC 1004, College Algebra, and SAT
scores, .31 for grades in MAT 1002, Basic Mathematical Skills, and ACT
scores and .25 for grades in MAC 2311, Calculus, and ACT scores. The
most striking aspect of the correlations appears to be the seven
negative values of r, and the modest to low nature of the correlations
in general. High positive correlations would have indicated large
proportions of low test scores corresponding to low grades and high
test scores corresponding to high grades. Low to moderate
correlations are seen as indicating weak relationships, with high test
scores sometimes associated with high grades, but also occurring with
low grades. Thus, the results from the correlations performed in
response to question one indicate that there is only a moderate to
weak relationship between test scores and grades. Standardized test
scores appear to be of only minimal value in predicting success in
initial mathematics courses. Apparently, other factors must be used
in appropriate placement.
Results With Respect to Question Two
Question two was an attempt to determine additional factors which
could be of use in placement strategies. The question asked what
factors "high-risk" students attributed to their successful completion
of college mathematics courses. As previously described, students
that may have been expected to do poorly in college-level courses, or
be excluded altogether, because of low test scores were interviewed.
These students had successfully completed a college-level course (MAT
1024 or above) in their first attempt. The interviews focused on why
they felt they were able to perform as well as they did.
The students were all willing to talk about their experiences in
the math courses. Using the interview guide (in Appendix A) as a
framework, the students' responses were tabulated. Results appear in
Factors Considered Important by Students for Successful
Completion of College-Level Math Courses
Instructor Background Tutor Effort/Time Math Lab
80% 100% 20% 90% 40%
Since there was no uniform manner of responding, each student's
responses were unique. There were, however, many similarities in the
comments the students made. The percentages given in Table 4.4 are
based on generalizations made by the interviewer in order to place
responses into one of the categories on the checklist. When
clarification was necessary, the interviewer asked a direct question
in order to determine the student's intended meaning.
Discussion of Results With Respect to Question Two
The intent of the survey of "high-risk" students was to identify
possible factors that could be of use in the placement process. By
speaking directly with students, the interviewer was able to get a
more personal feeling for which factors were apparently important.
The literature had identified many of the factors mentioned by the
students. The results of the interviews verified many of the factors
mentioned in the literature such as motivation/effort, high school
performance (grades), and "clinical evaluation" in the form of direct
personal assistance by the instructor, a tutor and/or use of the math
lab. Two factors not mentioned by the students that were found in the
literature were attitude toward math (affective domain) and learning-
It would appear from the results of the interviews that three
main factors emerged as important. High school performance or
background in math, motivation and effort, and "clinical evaluation"
during the course were all mentioned prominently in the interviews.
It seems very likely that these factors, given adequate means of
defining and quantifying them, could be of great value in the placing
Every one of the students interviewed indicated that they had
completed high school mathematics courses beyond the first course in
algebra. Two of the students had taken calculus, 11 had taken
trigonometry/analytic geometry, 5 had taken the second course in
algebra, and 2 had taken geometry. This was unexpected since they
each were below the present cut-off score for placement in college
A second important factor as identified by the interviews and in
the literature related to student effort. In general the students'
responses concerning effort and time spent preparing for the course
were related to high school background. For instance, the two
students who had taken calculus reported minimal time and effort
studying. Other students considered their time and effort to have
been extremely important. As reported, 40% made use of the math lab.
Quality of student effort is seen as an important factor in
student achievement. Pace (1980) maintained that pronouncements
should not be made concerning college impact without taking quality of
effort into consideration. Not only must the offerings of the
institution be considered but what the students do with those
offerings as well. The special value of measuring quality of effort
was demonstrated to be the most influential single variable in
accounting for students' attainment in the study by Pace.
The third factor suggested in the interviews and having
considerable support in the literature was "clinical evaluation."
Direct personal assistance by the instructor or a tutor and use of the
math lab pertain to this type of assessment. Methods of identifying
students' deficiencies and prescribing remedial work were seen as
another important factor which should be used in the placement and
Results With Respect to Question Three
The third question of this study pertains to the construction of
a clinical evaluation model which makes use of factors identified in
the literature and in the student interviews described in Question
two. Such a model was constructed as a result of this study and was
evaluated by the panel of experts described in Chapter III.
A description of the model is presented herein. The panel's
reactions to the model and the discussion thereof follows.
Description of the Clinical Evaluation Model
Clinical evaluation is defined as an analytical assessment of
deficiencies and the prescribing of treatment based on direct
observation by a practicing professional. Clinical evaluation allows
for a holistic approach to evaluation that is not possible using
standardized testing alone. Use of the model should enable the
evaluator and the student to understand more fully the nature of the
deficiency in a given area than would be possible using standardized
The model presumed a hierarchy of levels of skills in
mathematics, wherein certain skills cannot be mastered without
previous mastery of more fundamental skills. Use of the model
depended greatly on the professional capabilities of the evaluator--
ideally a community college mathematics instructor. The professional
judgment and skill of the evaluator were seen as extremely important.
The model was divided into subject areas--Arithmetic, Algebra,
Geometry, Statistics--and levels--Algorithms, Concepts,
Generalizations, and Problem Solving. Figure 4.1 is a flowchart of
the clinical evaluation process. Individual flowcharts for each
subject area are presented as Figures 4.2 through 4.5. Each level on
the model presumed mastery of certain prerequisite levels. These
levels are shown in relation to one another as they would occur in the
evaluation process. Arithmetic was seen as a prerequisite subject for
all other subjects on the model (see Appendix B).
Using the Model
The clinical evaluation would begin with Algebra. The flowchart
(Fig. 4.1) represents the sequences that would follow. If the student
demonstrates mastery of the required skills in Algebra, it is
theorized that the student could also perform the required skills in
Arithmetic and thus the evaluation would proceed to Geometry and then
NO NO NO
STOP STOP STOP
Figure 4.1. Flowchart for Clinical Evaluation
to Statistics. If the evaluator determines that the student has not
demonstrated at least 70% mastery of the skills in Algebra, the
evaluation must then also include Arithmetic. If Algebra and
Arithmetic were evaluated at less than 70% competence, the evaluation
Figures 4.2 through 4.5 represent the sequences of the clinical
evaluation for each subject area. The sequencing provides for rapid
progress through the flowchart, specifically if the student
demonstrates competence of higher levels, by presuming mastery of the
lowest levels (Algorithms). If, however, the student did not
demonstrate mastery of the higher levels (concepts, or generalizations
and problem solving) the Algorithm level must also be evaluated.
Mastery is defined at 70% or greater for each subject area.
By following the sequence indicated by the flowcharts, the
evaluator would consult a checklist to determine what specific skills
comprise that particular subject and level. The evaluator would then
formulate a problem pertaining to the first skill or select a sample
problem from a data bank. The student would attempt to solve the
problem, listing or explaining the steps taken and the result. The
evaluator must then determine whether the student's result is correct.
If so, the evaluator would proceed to the next enumerated skill for
that subject/level. If not, the evaluator would formulate another
problem, ask a related line of questions or anything further that may
help determine whether or not the student could perform the designated
skill. Since clinical evaluation calls for direct oral communication
between the student and evaluator, the evaluator would initiate
dialogue by asking appropriate, related questions. If, in the
evaluator's judgment, the student could not perform that skill, the
evaluator would note the skill and continue to the next skill. When
all the skills had been evaluated, the flow chart would be consulted
to determine the next step in the evaluation process (see Appendix C).
Algebra. The evaluation for Algebra would begin with the
Concepts level. If the student demonstrated mastery at the 70% level
or above, the evaluation would proceed to the Generalization and
Problem Solving questions. If not, the evaluation would proceed to
the Algorithm level. If that level is evaluated at less than 70%, the
evaluation of Algebra would terminate and proceed to Arithmetic.
Similarly, if the evaluation of Generalizations and Problem Solving
were evaluated at less than 70%, the evaluation would proceed to
Algorithms. In order to proceed to Geometry, the student would
demonstrate proficiency in 70% of the skills. If not, the evaluation
would proceed to Arithmetic.
Arithmetic. If Arithmetic was evaluated, the starting point
would be Concepts. Mastery would be demonstrated by performance at
70% or above. If the evaluator determined that the student had not
demonstrated mastery at the concepts level, the evaluation would
proceed to Algorithms. If Algorithm skills in Arithmetic were
determined as deficient, the evaluation would terminate. If, however,
the student demonstrated mastery of Arithmetic Algorithms, the
evaluation would proceed to Generalizations and Problem Solving. If
the student demonstrated mastery at 70% or above the evaluation would
proceed to Geometry. If not (and thus the overall performance in
Arithmetic is less than 70%) the clinical evaluation would terminate.
Geometry. The evaluation of Geometry would begin with the
Concepts level. The student would demonstrate mastery of at least 70%
or the evaluation would proceed to Algorithms. The student would
demonstrate mastery of Algorithm skills or the Geometry component
would terminate. If the student did demonstrate mastery of Concepts
the evaluation would proceed to Generalizations and Problem Solving.
The demonstration of mastery would be at least 70% or the Algorithm
level would be evaluated as well. Overall mastery of over 70% should
be demonstrated for Geometry.
Statistics. The evaluation of students' competence in Statistics
would begin with the Concepts level. If mastery was demonstrated at
70% the evaluation would proceed to Generalizations and Problem
Solving. If not, Algorithms would be evaluated. All of the
Statistics algorithms should be demonstrated successfully or the
Statistics component would terminate. The Generalizations and Problem
Solving skills should be demonstrated at least 70% or the Algorithms
level would be evaluated as well. Overall mastery of Statistics would
be demonstrated at 70%.
Following the clinical evaluation, appropriate remedial work
for each deficient category should be prescribed. Math labs in the
community colleges would be equipped with remediation techniques
such as programmed instruction units (perhaps computerized),
workbooks, cassette tapes, flash cards, etc., that address the
The model, therefore, would be used to allow the evaluator and
the student to identify deficiencies and plan learning activities
which could strengthen those deficiencies. The model could be used as
GO TO GO TO
Figure 4.2. Algebra Flowchart
Figure 4.3. Arithmetic Flowchart
GO TO GO TO
Figure 4.4. Geometry Flowchart
GO TO GO TO
Figure 4.5. Statistics Flowchart
a component of a well planned placement and instructional process in
mathematics. As noted previously, expert professional personnel would
be required in addition to a well-equipped "math lab" for the entire
developmental process to succeed. The extent to which these resources
and personnel are available determine the effectiveness and
feasibility of implementing a model such as this.
Discussion of Results to Question Three
The eight member panel which was consulted for their expert
opinion concerning the model was in agreement on certain aspects but
disagreed on others. Each member of the panel agreed that such a
model, if possible to construct, would be of great use in placement
and curriculum decisions. The panel split four to four over the
theoretical assumptions on which the model was based.
The model was based on the assumption that since mathematics was
of a hierarchical nature, the hierarchy could be used to evaluate
mathematical competencies. Wilson (1971) based an instructional model
on the hierarchical nature of mathematics. However, even though math
curriculum is structured according to such a hierarchy, four of the
panel members believed students do not reliably recall skills
according to that hierarchy. For this reason, the four panel members
doubted that the hierarchical nature of math could be used in the
evaluation model. Interestingly and probably significantly, the four
panel members to express such doubt were all community college
mathematics instructors who were familiar with the abilities and
skills of students. The counselor, administrator, and the two
university professors accepted the assumption that mathematical
hierarchy could be so used.
Finally, all of the panel members expressed concern over the
feasibility of implementing the model. Those who assumed that the
hierarchy could screen students were more optimistic. The assumption
was that fewer skills would actually be evaluated since many of the
skills would have been presumed mastered if students demonstrated
proficiency of the "higher" skills. Those who rejected the
hierarchical concept contended that nearly all skills would need to be
evaluated individually, thus greatly increasing the length of a
"clinical evaluation." The large amounts of time, resources, and
personnel necessary to implement a nonhierarchical model in the
colleges rendered it unfeasible in the opinion of the panel.
SUMMARY, CONCLUSIONS, RECOMMENDATIONS,
Placement for community college students in math has been
recognized as imperative. Effective, widely accepted placement
techniques have not been reported in the literature. This study has
analyzed the relationship between tests used for placement purposes
and subsequent grades in mathematics courses. In order to identify
and verify other factors which could be used in the placement process,
interviews were conducted with students who had successfully completed
their initial college level course, but had been identified as "high-
risk" as a result of low test scores. Three main factors emerged from
the interviews as important. High school preparation in mathematics,
quality of student effort, and clinical evaluation techniques were
considered important by the students in leading to their successful
completion of college-level courses.
A clinical evaluation model was developed which was based on
concepts found in the literature and confirmed by students in the
interviews. A panel of experts reacted'to the model. The model
was seen as lacking validity by four of eight experts and problems
with feasibility of implementing a variation of the model were
The correlations for ACT and SAT scores and grades in initial
mathematics courses were found to be generally low. It is concluded
that the relationship between test scores and grades was weak for the
data under consideration in this study. The results of this study
lend support to criticism of using standardized test scores as a sole
means of evaluation in the placement process.
Other factors that could be used in placement strategies were
sought. Based on a review of related literature and interviews with
"high-risk" students, factors necessary for successful completion of
college-level courses were identified. High school math background,
student effort, and availability of clinical evaluation techniques
were the three most promising for use.
A clinical evaluation model based on theoretical concepts from
the literature was constructed. A panel of experts concluded that
an acceptable variation of the model would not be useful given
constraints of time, resources, and personnel.
Heretofore, no standardized test or other single means of
evaluation has emerged as an effective placement instrument.
Mandatory placement is a relatively recent practice in Florida's
community colleges and suitable methods of accomplishing it are still
being sought. Research designed to study possible methods of
placement is needed. A combination of two or more factors would seem
to offer the most promise for effective placement, particularly in
light of the lack of a single reliable instrument.
The following variables seem to be worthy of further study:
1. high school math grades
2. other standardized tests
3. student effort/motivation
4. clinical evaluation methods
5. learning-style preference
6. affective domain/attitude toward math
Problems have been identified for each of the possible variables
listed above. However, since no acceptable, widely used placement
techniques are available, their investigation would appear to be
Grades have been criticized as a variable for use in research
studies because of lack of standardization. This characteristic does
not seem likely to change. Even though there are problems using
grades, other studies have shown them to be useful in predicting
academic success. Particularly in mathematics, where high school
courses are sequentially arranged, would consideration of grades be
appropriate because information on the level of mathematics in
students' backgrounds should be beneficial.
Standardized tests other than ACT and SAT may prove to be better
placement instruments. Certainly MAPS and ASSET should be studied to
determine their effectiveness as placement instruments. Other tests,
particularly competency-based, content exams should be considered as
possible placement instruments.
Quality of student effort has been identified as an important
factor. Outcomes have been predicted quite well using quality of
student effort as a variable. Further investigation in this area
should prove valuable.
Clinical evaluation techniques, in theory, seem to offer great
promise for appropriate placement. The analytical assessment of
competencies and deficiencies based on direct observation by a
competent observer would have advantages that standardized tests could
not provide. Specifically, the identification of exact areas of
deficiency in math would be possible. Considering the hierarchical
nature of mathematics, knowledge of certain deficiencies would be most
useful in the placement process.
Unfortunately, such a model for clinical evaluation in mathematics
would not be accepted by many practitioners. Specifically, the model
developed in this study was not acceptable based on opinions of
knowledgeable persons in the field. Various problems were identified
in the construction of such a model. Much of the evaluation would
greatly depend on the observer, and it would be very difficult to
maintain consistency from one observer to another. Problems also were
identified with quantification of the results of the observations.
Difficulty in defining a true hierarchy in math skills was also
Nevertheless, if some or all of these problems could be
minimized, clinical evaluation of math skills would appear to have
possibilities for placement of students. More research in the area
As previously noted, learning-style preference has been
considered in only a small number of the studies reported in the
literature. This area also holds potential for placement and should
be researched more fully.
The affective domain and the student's attitude toward math have
been difficult to study. Emphasis has too often been placed on the
cognitive domain. The affective domain represents a vital component
of how students learn, yet has often been neglected in instruction and
learning paradigms. Possibly because of the lack of reliable
instruments to measure it, the attitudes and emotions are not assessed
and considered in placement strategies.
One rather obvious implication of this study is to confirm the
opposition of several writers to the use of standardized tests as the
sole means of evaluation. Indeed, the composers of standardized tests
caution against their use as the sole criterion in a placement
decision. Even so, the recently mandated cut-off scores require the
use of standardized test scores, although provisions were made for the
use of additional means of evaluation for placement. Therefore, this
study clearly implies the need for other factors which can be used for
The concept of clinical evaluation based on the hierarchical
nature of mathematics formed the theoretical basis for the
construction of the clinical evaluation model. The hierarchical
nature of mathematics has long been recognized and utilized in
curriculum design. The sequential nature of mathematics courses in
general follows a logical progression. However, the assumption that
this hierarchy would be useful for evaluation purposes has not been
demonstrated by this study. Even though mathematics must be presented
in the proper sequence for the purpose of sound instruction, students
do not reliably recall skills in the order presented. Therefore, an
evaluation model which is based on strict application of the hierarchy
would lack validity. It is an oversimplification to presume that
because a student demonstrates proficiency in certain areas of the
mathematics curriculum that student could also perform at lower
levels. Similarly, it cannot be assumed that proficiency in higher
levels of the cognitive domain (applications, for example) necessarily
imply proficiency in lower levels such as algorithms.
The implication, then, is that for purposes of summative
evaluation, the assumed use of the hierarchy is insufficient. This
implies that for a valid surmative evaluation each skill at each level
should be evaluated separately. There is evidence that a clinical
means of evaluation would be more accurate than using standardized
tests, but in either case mastery of lower levels cannot be presumed
because of demonstration of proficiency at higher levels.
This is not to say that the hierarchy cannot be used in the
placement process. If it can be demonstrated that skills presumed to
be prerequisite are indeed such, a test or clinical evaluation to
determine mastery of these prerequisite skills would be most useful
for placing students into the proper course. The narrower the range
of skills, the more detectable the prerequisites would be. For these
reasons, use of the hierarchical nature of math would be helpful in
formative evaluation such as for placement. For example, if a series
of questions that have been demonstrated to be necessary prerequisites
for, say, an algebra course were available, evaluation of students'
mastery of those skills could be used to determine readiness for the
course. This evaluation could be a standardized test or a clinical
evaluation, although again the clinical evaluation is recommended
given adequate resources and personnel.
In conclusion, then, use of the hierarchy for assumption of
proficiency of lower skills because of proficiency in higher skills
has not been demonstrated to be valid. Therefore use of the
mathematical hierarchy for summative evaluation (such as CLAST) is not
recommended. However, using the hierarchical and sequential nature of
mathematics shows promise for use in formative evaluation such as
1. Identification of caller.
2. Would you be willing to answer a few questions regarding the math
course you took in the fall (or summer) of 1984?
3. You've been identified as doing well in the course (name course
specifically). The questions relate to why you feel you were able
to successfully complete the course. What reasons would you give
for your success?
4. What other factors would you consider important?
5. Any others? What about ? (refer to checklist)
6. About how many hours per week did you study for this course?
7. Any other reasons that you can think of?
8. Thank you for your help.
Reasons or factors given by students for successful completion.
HIGH SCHOOL PREPARATION
WEEKLY HOURS OF STUDY
SUBJECT BY LEVEL MATRIX
ALGORITHMS CONCEPTS GENERALIZATIONS PROBLEM-SOLVING
SUGGESTED STEPS FOR CLINICAL EVALUATION
1. The evaluator presents a problem pertaining to the skill that
is to be evaluated. The students should solve the problem, listing
the steps taken and the answer. The evaluator examines each step as
well as the final answer.
2. The evaluator presents a solution and answer to a problem
that pertains to the skill; however, the solution steps and answer
contain an error. The student is to identify the error or errors and
3. The student is asked to formulate a problem/question that
requires use of the skill under consideration, then list the solution
steps and the answer.
Some or all of these recommended steps may be used to initiate
discussion and thus provide information concerning the student's
understanding and competence. Other techniques may be used, such as a
related line of questioning, particularly if the evaluator senses the
need for such. Creativity on the part of the evaluator would be
It is here that the holistic approach to evaluation is necessary
as well as use of the hierarchical nature of mathematics.
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Robert Norman McLeod was born August 5, 1950, in Jacksonville,
Florida. The first of four children of H.F. and the late Norma B.
McLeod, he moved with his family to Miami where he completed
elementary and secondary education in the public schools of Dade
He attended the University of Florida from 1968 through 1972,
receiving the B.A. with honors, majoring in psychology. After
returning to Dade county he taught psychology, mathematics, and
reading there for six years, during which time he earned the Master's
and Ed.S. degrees from Nova University.
In 1980 he was married to Jackie Barbara Polly. They moved to
Ocala in 1981 as work began on the Ph.D. Two sons, Charles Robert and
Travis Gordon, were born in 1982 and 1984, respectively. He has
taught mathematics in the Marion county school system since August,
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
es L. attenoarger, airman
ofessor of Education 1 Leadership
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
ohn M.' Nickens,' C6chairman
Professor of Educational Leadership
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is
fully adequate, in scope and quality, as a dissertation for the degree
of Doctor of Philosophy.
Ernest H. St. Jacques
Associate Professor of Educational