Quantitative Literacy for College Students

A Memorandum to CUPM

Lynn Arthur Steen, July 1, l989


A report on college requirements for quantitative literacy prepared in July 1989 for the Committee on the Undergraduate Program in Mathematics (CUPM) of the Mathematical Association of America. A summary of a 1979 CUPM report on QL appears at the end of this memo. The most recent reportfrom the CUPM Subcommittee on Quantitative Literacy was issued in 1996. (QL Home Page)


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.


Since each of the reports I received are expressed in terms whose meanings depend on local context, I have attempted in this summary to translate each response into a common traditional vocabulary:

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.

Implicit Sequencing

Arithmetic, Algebra I, Elementary Geometry, Algebra II, College Algebra, and Calculus represent the "pipeline" courses of mathematics, from elementary school to college. Euclidean geometry continues to be taught in most high schools, and is required for entrance by many colleges, but it is almost never referred to again in college-level graduation requirements. Computing is commonly taught in high school; Logic and Data Analysis sometimes, and Liberal Arts Mathematics rarely.

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

Comments from Campuses

What follows are summaries of material I received, expressed in the categories defined above. Unless otherwise noted, all comments about requirements (for entrance and for graduation) apply to all students. (Although I have tried to summarize faithfully what individuals wrote, both the interpretations and possible misinterpretations are my responsibility, not those of the cited correspondents.)

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]

1979 CUPM Minimal Competencies Survey

Don Bushaw, who was chair of CUPM ten year ago, sent me a copy of the report of a random survey of 1000 mathematicians from AMS, SIAM, and MAA concerning graduation requirements and minimal competencies for all college students. The report is based on 335 returns from this survey, now exactly ten years old. Here are some highlights:

Tentative Conclusions

  1. 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.

  2. 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.

  3. Many institutions require for graduation less than other institutions (even of comparable type) require for admission.

  4. Concern about core curricula and assessment of student learning is gaining momentum at all levels of society.

  5. 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.

  6. 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.

  7. 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.

  8. 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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Last Update: 25 June 1999
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