Assessing Allegheny College’s introductory
calculus and precalculus courses
Ronald Harrell and Tamara Lakins
December 30, 2003
Abstract: With the goal of improving student learning, Allegheny College
assesses the effectiveness of its introductory calculus and precalculus courses
by analyses of grade data, conversations with client departments, and information
regarding such courses at similar institutions. The initial assessment led
to substantial revisions in its offerings.
Background and goals: What did we hope to accomplish?
Allegheny College, a national liberal arts college located in Meadville, PA,
has an enrollment of approximately 1800 students. The mathematics department
teaches approximately 550 students per year in its introductory calculus and
precalculus courses, which from 1990 to 2003 consisted of Math A (Intermediate
Algebra), a one-year sequence Math 155/156 (Calculus/Precalculus), and a traditional
calculus course Math 160 (Calculus I).
The original intent of Math A was to prepare students for the sequence Math
155/156. It also became a specific prerequisite for the following courses in
client departments: introductory chemistry, introductory computer science,
and the research design and statistics course in the psychology department.
The original intent of the sequence Math 155/156 was to provide an alternate
entry point into the regular calculus sequence for students with weaker precalculus
backgrounds. In particular, the sequence covered selected precalculus topics
in addition to the calculus topics traditionally covered in Calculus I and was
designed to prepare students for Calculus II.
At the beginning of the fall semester, entering first-year students are placed
in a mathematics course based on an algebra-trigonometry-precalculus-based placement
exam and/or consultation with a member of the mathematics department. The placement
is a non-binding recommendation.
The goals of our assessment project were to determine whether the intermediate
algebra course and precalculus/calculus sequence were addressing the needs of
our students and to make any needed changes to the courses, which in any case
would include the addition of a regular assessment program. The intermediate
algebra course had existed for over 20 years, and the precalculus/calculus sequence
had been used for 13 years. In neither case had we ever assessed their effectiveness,
and we had anecdotal evidence from members of the mathematics department that
the courses were not preparing students at the level we expected for subsequent
courses. It was time to take a hard look at both.
Description: What did we do?
During the 2001-02 academic year the mathematics department assessed the effectiveness
of Math A (Intermediate Algebra) and Math 155/156 (Precalculus/Calculus) by
reviewing data regarding student performance in these and subsequent courses,
by conducting conversations with faculty in client departments, and by reviewing
information about precalculus and introductory calculus offerings at the 26
colleges and universities identified by Allegheny College as in our comparison
Data on the distribution of grades were examined for the 359 entering first-year
students who enrolled in Math A during a fall semester from 1997 to 2000 and
who went on to take introductory chemistry, Math 155, introductory computer
science, or the psychology statistics course. We omit the discussion of students
who took the introductory computer science course, since their number was too
small to draw viable conclusions. In addition, a second comparison of grade
distributions was made for students taking these subsequent courses, based on
mathematics placement level and regardless of whether the students took Math
A. For this second comparison, grades in Fall 1997 through Spring 2001 were
considered, and the sample consisted of students who entered Allegheny in a
fall semester from 1997 to 2000.
Data on the distribution of grades were examined for the 310 entering first-year
students who enrolled in Math 155 in a fall semester from 1997 to 1999 and who
subsequently took Math 156 (the second course in the precalculus/calculus sequence)
and Math 170 (Calculus II) at Allegheny College. These latter students were
compared to the 224 entering first-year students who began in Math 160 (Calculus
I) in a fall semester from 1997 to 1999 and who subsequently took Math 170 at
All grade data considered were obtained from the Allegheny College Registrar
in electronic format. Student names were removed from the data, and fictitious
identification numbers were used in order to ensure student anonymity.
We consulted 1-2 faculty members (typically the chair or
faculty who teach courses requiring quantitative or mathematical skills) in
our client departments regarding Math A and our calculus offerings. Client
departments include biology, chemistry, computer science, economics, environmental
science, geology, mathematics, physics, and psychology. In the case of Math
A, we wished to specifically learn why Math A was a prerequisite for the four
earlier mentioned courses, and which skills currently taught in Math A were
considered essential for these courses. In the case of calculus and those departments
that require it for their major, we wanted to know specifically why calculus
was required and which skills were considered essential.
Data from the colleges and universities in our comparison group were obtained
by researching the college and university web sites; when necessary, details
were clarified by email and phone calls to the appropriate department chair,
or another faculty member designated by a department chair. In particular,
we wished to determine the following information about each school in our comparison
group: its precalculus and introductory calculus course offerings and the mechanisms
by which it enables students with deficiencies in their mathematics preparation
to prepare for calculus.
Initially, we planned to review the high school backgrounds of a sample of
our students. This would have required reading folders of individual students
to determine which math courses had been taken and which grades had been obtained.
Organizing that information would have been complicated by the variety of math
courses now taught in high schools. So given our resources, we decided to drop
this part of the study.
Insights: What did we learn?
The analysis of the data, on student performance in Math A (Intermediate Algebra)
and the courses for which it became a prerequisite, indicated that
Math A did not seem to adequately prepare students for any of the
subsequent courses, except possibly the psychology statistics course.
(Year-by-year details of all the grade information discussed here
can be found in Appendix A.)
In particular, of the 359 entering first-year students who enrolled
in Math A during their first semester at Allegheny College, 246 (69%)
earned a successful grade of C or higher. Of these 246 students,
112 went on to take introductory chemistry with 74 (66%) earning a
grade of C or higher, 183 took Math 155 with 111 (61%) earning a grade
of C or higher, and 54 took the psychology statistics course with
46 (85%) earning a grade of C or higher.
Even more striking was that our second comparison showed that, of all students
who enrolled in introductory chemistry or Math 155 from Fall 1997 to Spring
2001, students who placed in Math A and who did not take it prior to enrolling
in one of these courses often fared as well as, or better than, students who
placed in Math A and took it first. In particular, 49 of the 74 students (66%)
who placed in Math A but did not take it or Math 155 prior to enrolling in introductory
chemistry earned a grade of C or higher in chemistry, while 85 of the 141 students
(60%) who placed in Math A and took it or Math 155 prior to taking introductory
chemistry earned a grade of C or higher in chemistry. In the case of Math 155,
67 of the 107 students (63%) who placed in Math A but did not take it prior
to taking Math 155 earned a grade of C or higher in the course, while 124 of
the 231 students (54%) who placed in Math A and took it prior to taking Math
155 earned a grade of C or higher in the course. Thus, it appeared that Math
A was ineffective in preparing students for subsequent courses.
The analysis of the data on student performance in the one-year
precalculus/calculus sequence (Math 155/156) indicated that a smaller number
of these students went on to take Calculus II (Math 170) than one might expect.
Of the 310 entering first year students who enrolled in Math 155 during a fall
semester from 1997 to 1999 only 52 (17%) went on to take Calculus II (either
at Allegheny or elsewhere) by Spring 2001, while of the 446 entering first year
students who enrolled in Math 160 during the same semesters, 248 (56%) went
on to take Calculus II (either at Allegheny or elsewhere) by Spring 2001. Furthermore,
students in Allegheny’s Math 170 who began in Math 155/156 were generally not
as successful as those who began in Math 160. Of the 47 first-year students
who began in Math 155 during a fall semester from 1997 to 1999 and who went
on to take Math 170 at Allegheny by Spring 2001, 29 (62%) earned a grade of
C or higher. On the other hand, of the 224 students who began in Math 160 and
went on to take Math 170 at Allegheny by Spring 2001, 168 (75%) earned a grade
of C or higher. Thus the Math 155/156 sequence was not serving students as
well as we had hoped.
The conversations with faculty in client departments revealed
that Math A was serving more than one purpose. The chemistry and mathematics
departments required Math A, or placement out of Math A, as a prerequisite for
entry level courses which require students to possess traditional algebra skills.
However, faculty in the biology, computer science, environmental science, and
psychology departments emphasized wanting their students to possess general
quantitative skills (such as good problem solving skills, the ability to translate
word problems into an appropriate mathematical model, being able to work with
and interpret data, and having good number sense), rather than specific kinds
of algebra skills. Clearly at the level below calculus, something more than
a course that only reviews algebra was needed.
In the case of Math 155/156 the chemistry, mathematics, and
physics departments expected a thorough treatment of the concepts of calculus
and a good knowledge of computational skills. The other client departments
required at least some calculus for their major programs and expected students
to be able to understand and apply the concepts of calculus. Often these same
departments wanted calculus courses to be less theoretical and were primarily
interested in having their students learn how to do only the more elementary
computations. Thus there were two kinds of clientele for calculus courses.
Finally, an examination of course offerings at the 26 schools
in our comparison group indicated that we were only one of 2 institutions that
offer a course at the level of Math A, and only 2 institutions offer a course
at the level of college algebra. On the other hand, a total of 15 colleges
offer precalculus, 7 offer a combined precalculus and calculus sequence similar
to our Math 155/156, and 5 colleges offer no course which directly prepares
students for calculus. Finally, 8 institutions offer a one-semester alternative
to the traditional first calculus course (usually a course which emphasizes
applications from the social and/or life sciences).
Redesigning: What did we do?
The above findings indicated that our lower level course offerings were not
diverse enough to meet students’ needs, and some courses did not accomplish
their intended purpose. After much discussion, the mathematics department
replaced Math A (Intermediate Algebra) and Math 155/156 (Precalculus/Calculus),
which were hierarchically designed to prepare students for the regular
calculus sequence, with four courses that provide students with three
options for beginning the study of mathematics, depending on their
individual goals. The new courses are briefly described below; more
detailed descriptions and course goals are given in Appendix
B. Important in the design of the new courses was meeting the
needs expressed by client departments.
The first option, Math 110 (Elementary Mathematical
Models), replaced Math A. Math 110 is an elementary algebra-based modeling
course that emphasizes the study of real world problems and models, and rates
of change. Algebra is reviewed as needed. The course is for those students
who need a mathematics course but not a calculus course. The intended audience
consists of humanities and social science students, who take it to fulfill a
graduation requirement or who find it useful in a major field, such as economics,
environmental studies, political science, or psychology.
The second option, Math 150 (Precalculus), is a standard
college level course on the subject, intended only for those students who need
to take the regular calculus sequence, but who also need to brush up on precalculus
topics before doing so. Topics covered in the course were formerly taught in
the sequence Math 155/156.
The third option is the sequence Math 157/158 (Calculus
I and II for Social/Life Sciences), which replaces the Math 155/156 sequence.
The sequence is for those students who need calculus, but not the thorough and
more rigorous treatment presented in the regular calculus sequence. The emphasis
is on the concepts of calculus and how they occur in problems from the life
and social sciences. Topics in both single and multivariate calculus are covered.
This option serves primarily biology, economics, and environmental science students.
The mathematics placement exam, which was previously used
to determine placement in either Math A, Math 155, or Math 160, is still used
by the department. The department now requires a particular score on that exam
in order to recommend placement into Math 160. Students who do not achieve
the target score may enroll in Math 110, Math 150, or Math 157, depending on
high school background, confidence, and intended major. These latter three
courses have no formal prerequisite. While Math 110 is a terminal course and
is not intended to prepare students for Math 157, some students may opt to take
it before attempting Math 157.
Students who have already received college credit for a
calculus course may not take Math 110, 150, or 157 for credit. Furthermore,
students who begin in the Math 157-158 sequence and later change to a major
requiring the ordinary calculus sequence Math 160-170 are treated on an individual
basis. A student may take Math 160 for credit after receiving credit for Math
157 but not after receiving credit for Math 158. (Such a student may still
take Math 160, but will not receive credit toward graduation for the course.)
Students who wish to take Math 170 after Math 158 receive individual advising.
In addition to creating these new courses and options,
we also created a way to have an ongoing assessment of each in order to monitor
their effectiveness. Content goals for each course are assessed using selected
questions on final exams to gauge how well students have mastered the material.
For the goal
which pertains to each of the new courses, as well as the goal
- students will be able to communicate mathematical information
in written form,
- students will be able to choose, implement, refine, and interpret appropriate
mathematical models for various real-world problems,
which pertains to Math 110 and the sequence Math 157/158, the assessment consists
of short writing assignments, projects, or appropriate homework where writing
is emphasized. Thus the assessment data consists of scores on selected final
exam questions and instructors’ impressions of the writing assignments and/or
projects during the semester.
At the end of each semester the instructors for each course meet briefly to
review and discuss the assessment data for that semester. They then submit
a report of their findings to the department chair, who makes the contents of
the report available to the entire department. It is hoped that, by reviewing
a small amount of assessment data each year, the department will be able to
maintain an ongoing and accurate picture of the effectiveness of the courses
and our assessment methods, while at the same time not placing too great a burden
on the faculty. Periodically, perhaps every three to five years, we will do
a wider assessment, similar to the one reported here, that determines how well
these courses prepare students for subsequent courses, not only in mathematics,
but in other areas.
Other Comments and Recommendations. After Lakins compiled
and analyzed the data, the department spent several weeks of intense discussion
creating the above four replacement courses. The two major sticking points
were the exact nature of the replacement courses and finding a reasonable ongoing
assessment plan for each course. Some faculty questioned whether ongoing assessment
plans were needed, and getting them to see the usefulness and benefits of such
plans was a hard sell. To help students and advisors the department also made
up documents explaining the new courses in detail and indicating which courses
would benefit which students.
The authors wish to acknowledge the support and assistance of Allegheny College’s
Associate Dean Richard Holmgren in planning and implementing this assessment
project. Funds from Dean Holmgren’s Arthur Vining Davis Foundation grant supported
the 2001-02 salary of Tamara Lakins, as well as the travel costs of both authors
to three SAUM Assessment in Undergraduate Mathematics workshops. We also acknowledge
the assistance and advice of our workshop team leader Bonnie Gold.