Several months ago I wrote to a number of mathematicians to gather samples from various campuses of present practice concerning mathematics or quantitative literacy requirements for all students. This report summarizes highlights from the responses I received and documents the great variety of such requirements. Since neither my solicitation nor the responses are proper random samples of higher education, readers should be cautious about drawing general inferences from this data. However, the respondents do include institutions of sufficient diversity to provide an appropriate portrait of present practice in higher education.

*Arithmetic.* In this report, "arithmetic" refers to the topics
taught primarily in grades 6-8, including percentages, fractions, square
roots, decimals, scientific notation.

*Algebra I.* The traditional content of ninth grade algebra: linear
equations, algebraic manipulations, evaluation and simplification of
algebraic expressions, elementary graphing.

*Elementary Geometry.* In this report, "elementary geometry" refers
not to the traditional sophomore course in Euclidean (axiomatic) geometry,
but to measurement of lengths, angles, areas, and volumes that normally
appear in the mathematics textbooks in grades 6-9. Examples include the
area of a triangle and a circle, volume of boxes and cylinders,
applications of the Pythagorean theorem and of the angle sum in triangles,
and recognition that congruent triangles have identical measurements.

*Euclidean Geometry.* The traditional, classic course, often
watered-down, covering the axioms, theorems, proofs, and constructions of
Euclid. Usually taught at the sophomore level in U. S. high schools,
following Algebra I.

*Algebra II.* The traditional high school junior course, including
factoring, quadratic formula, logarithms, triangle trigonometry, and
graphs and formulas for simple conic sections (ellipses, parabolas,
hyperbolas).

*College Algebra.* The fourth year of a traditional high school
curriculum emphasizing treatment of elementary functions (polynomial,
exponential, logarithmic, and trigonometric) in preparation for the study
of calculus. This course (or one roughly comparable in purpose and
sophistication) is often called "Elementary Functions," "Precalculus,"
or "Analytic Geometry and Trigonometry."

*Calculus.* Traditionally, the first university course for those
maintaining normal pace towards scientific studies. Currently, about 20%
of high school students take calculus in some form as seniors. Often
offered in several versions known in the trade as full, short,
hard, or soft.

*Computing.* Use of a computer (PC or campus time-sharing system) and
elementary programming (usually in BASIC). These topics are commonly
taught in high school or at colleges as part of non-credit short courses.
They sometimes are part of quantitative literacy requirements.

*Logic.* Propositional calculus, truth tables, and translations
between English sentences and symbolic representation of logical
implications. Often taught as part of various mathematics courses, or in
college courses offered by Departments of Philosophy.

*Data Analysis.* Often called "Exploratory Data Analysis," this
course emphasizes the graphical representation of data (histograms,
scatter-plots, box plots), interpretation of graphs and of data, and
simple summary statistics (mean, median, mid-range, correlation). It uses
few algebraic formulas and relies heavily on real data from everyday
examples.

*Liberal Arts Mathematics.* This course is offered in some form on
virtually all campuses, but there is little consistent pattern to course
titles or content. Many are historical and philosophical in nature;
others emphasize practical topics (e.g., statistics, matrices, computer
programming); still others use topics from number theory, geometry, and
graph theory to introduce the process of mathematical discovery and proof.
All have in common a student clientele with high mathematics anxiety and
weak algebraic skills.

Regardless of where or when courses are taught, the implicit logic of mathematics provides an intrinsic order to the sophistication and difficulty of these various courses. In my judgement, based on descriptions in the various documents sent by the several respondents to this survey, this order is approximately as follows:

A. Arithmetic

B. Algebra I, Computing, Elementary Geometry

C. Algebra II, Euclidean Geometry, Logic, Data
Analysis, Liberal Arts Mathematics

D. College Algebra

E. Calculus

**University of California at Los Angeles.** Entrance requirement:
Algebra I and II, and Euclidean Geometry. College of Letters and Sciences
has a "Quantitative Reasoning" requirement which may be satisfied by a
variety of mathematics or statistics (data analysis) courses in other
departments--especially in the social sciences. College Algebra does not
meet the requirement; nothing lower is offered for credit. The
Mathematics Department "strongly resists any attempt to have a purely
mathematical graduation requirement." [Phil Curtis]

**Colgate University.** No current graduation requirement, but under
study is a proposed requirement of one quantitative course, either a
mathematics course at or above the level of calculus, or a computer
science course at the level of the standard Pascal-based Introduction to
Computer Science, or a special "quantitative reasoning" course blending
components of practical mathematical topics like data analysis and logic
with computing. [Tom Tucker]

**Towson State University.** Present requirement of three courses in
science or mathematics; emerging state-mandated requirement in
quantitative literacy. Analysis of present student expertise reveals
frequent lack of number sense, confusion about percentages and graphs
("ignored by students since they can't read them"), confusion caused by
different notational conventions among different disciplines, and split
opinion on the value of computing (a good thing for statistics, great
skepticism in other areas). [Martha Siegel]

**Ohio State University.** Computational skills examination (at level
of Algebra I with bits of Algebra II) followed by two courses selected
from a variety of departments. One of these courses must emphasize
mathematical or logical analysis, the other data analysis.

**California State University at Northridge.** State mandated Entry
Level Mathematics (ELM) examination covering Arithmetic, Algebra I, and
Elementary Geometry, followed by one of seven mathematics courses
(including, for example, calculus, statistics, college algebra, and
liberal arts mathematics). Slightly more than half of the students who
take the ELM exam fail it and must then enroll in various remedial courses
called "Arithmetic and Measurement Geometry" and "Elementary Algebra"
which cover the curriculum of grades 1-9. [Elena Marchisotto]

**Bellevue Community College.** Requirement: one mathematics course
beyond Algebra II. Typical courses used to satisfy the requirement are
College Algebra, Liberal Arts Mathematics, Logic, and Finite Mathematics
(for business and social science students). All but Logic require use of
computers.

**Mississippi State University.** Requirement: College Algebra plus
one or two additional courses. Students who begin with calculus
automatically meet the requirement upon completion of this course. The
College Algebra requirement part of the requirement is mandated at the
state level and applied to all public institution in Mississippi. Since
this rule took effect, many courses with the title of "College Algebra"
have been introduced with a curriculum that more closely matches lower
courses (such as Algebra II). "There seems to be a widespread opinion in
the mathematics community that College Algebra is inappropriate as the
core mathematics course that all students should take." [Gordon E.
Jones]

**Rhode Island College.** Requirement: Demonstrating mathematical
competency by passing an exam or taking a course covering Arithmetic,
Algebra I, and simple parts of Data Analysis; in addition, one or two
courses in mathematics (calculus, finite mathematics, probability, etc.)
computer science, or logic. [Henry Guillotte]

**Macalester College.** No requirement, "despite gentle pushing of John
Davis during his eight years as President." [Wayne Roberts]

**State University of New York at Stony Brook.** A quantitative
proficiency examination covering Algebra II and Data Analysis. Can be met
in various ways, ranging from a score of 600 on the SAT Advanced
Mathematics Achievement test, to a score of 85% on the 11th grade New York
Regents examination, to passing pre-calculus (college algebra) with a
grade of C- or higher. Certain courses in other departments also meet
this requirement. [Alan Tucker & A. W. Knapp]

**Tulane University.** One four-hour course is required, either in
mathematics (finite mathematics, statistics, calculus) or in computer
science, or symbolic logic (in philosophy). There is some faculty
dissatisfaction with the level of work that students are able or willing
to exert, so they are trying to raise the passing standards in these
courses. [Jerome Goldstein]

**University of California at Davis.** An attempt to introduce a
quantitative literacy requirement in the College of Letters and Science
failed for three reasons: (1) A large number of students already take
sufficient mathematics as part of requirements for science and
engineering; (2) The only students who would be affected by a requirement
are those in the arts and humanities; the faculty in those areas see no
need for the requirement and can't agree on what courses might be part of
a requirement if there were one; (3) There is little enthusiasm in the
mathematics department for such a requirement, especially since it will
draw into the department students who are likely to be antagonistic toward
mathematics. [Henry Alder]

**Harvard University.** A Quantitative Reasoning Requirement (QRR) was
introduced ten years ago as part of the new core requirements. The
requirement, which must be completed during the first year, is in two
parts: Data Analysis and Computing. The Data Analysis component--the
harder of the two--includes realistic problems involving presentation
(graphing) of data, uncertainty, and growth. The Computing part is as
described above in the list of definitions. Students meet the requirement
by passing exams in each part--each exam is offered several times per
year--or by taking a special course designed to cover these topics (and
more). [Deborah Hughes Hallett]

**Florida State University.** Florida requires all students to pass the
College Level Academic Skills Test (CLAST) before becoming juniors. The
mathematics part of this test--the larger part--covers Arithmetic, Algebra
I, Elementary Geometry, and topics from Data Analysis and Logic. The test
is multiple choice, and includes questions ranging from fifth grade
arithmetic to challenging multi-step problems involving quantitative
reasoning. Reports indicate a high failure rate on this exam. (Ed. Note:
The CLAST is significantly more challenging at a conceptual and
problem-solving level than the California ELM exam--which is intended for
student two years younger.) [Bettye Anne Case]

- 40% of the respondents who are in academic institutions reported that
their institution had a graduation requirement in mathematics for all
students. Almost all of these requirements were met by course
requirements rather than examinations. The average number of credits in
these mathematics requirements was 5.6 semester hours.
- Respondents reported the course titles used to meet the requirement.
"The variety of nomenclature used was astonishing." The most common
courses mentioned, in order, were calculus, probability and statistics,
college algebra, "basic college mathematics" (probably Algebra I and II),
elementary algebra, and computer programming or appreciation.
- 70% of the respondents at institutions without a requirement favored
having one. (Of course, by responding to the survey, they signalled a
special interest in this issue.)
- Respondents were asked to rank topics suitable for a college
mathematics requirement. The topics that received strongest (majority)
support were, in effect, those covered in Arithmetic, Algebra I,
Elementary Geometry, and Data Analysis as defined above. Topics from
College Algebra, Logic, and Computer Programming--courses commonly used to
meet a general mathematics requirement--ranked rather low in this
survey.

- There is no focused consensus in higher education about the
appropriate level of mathematics (or quantitative skills) to be required
of all college graduates. Examples in this survey encompass a four
year range of high school study--from Algebra I to College Algebra.
- All institutions that have quantitative literacy requirements appear
to want something beyond what is normally taught in elementary school.
However, none of the reporting institutions require subjects that are
distinctly at the college level (e.g., calculus or its equivalent). What
colleges do seem to agree on is that college graduates should know some
high school mathematics.
- Many institutions require for graduation less than other institutions
(even of comparable type) require for admission.
- Concern about core curricula and assessment of student learning is
gaining momentum at all levels of society.
- Requirements defined and imposed external to the college or university
rarely reflect the broad, modern aspects of applicable mathematics; most
often they are expressed in terms of courses that emphasize mechanical
manipulations.
- Special courses designed as part of a core curriculum to meet a
quantitative literacy requirement serve only those students who are
ill-prepared in mathematics. Students who major or minor in the sciences
never take them.
- Mathematics stands out from other disciplines in having a
college-level graduation requirement defined in terms of high school-level
accomplishment. This posture, for a discipline know for its rigor and
high standards, makes everyone associated with these requirements
uncomfortable.
- Most mathematics departments have sufficient enrollment pressure from
students who need mathematics for their majors to not need the enrollment
support of a college-wide requirement. Most mathematicians, moreover,
have little interest in teaching students who reach college still in need
of the type of remedial work normally defined as constituting the
college-wide requirement. As a consequence, mathematics departments
usually are not strong advocates for mathematics requirements.

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