Green Sheet: CEPC 05/06-7
To: St. Olaf College Faculty
Re: Proposed Revisions to General Education Requirement for “Mathematical Reasoning”
At the March 2, 2006 Faculty Meeting, CEPC will move the adoption of new statements to define the general education requirement for “Abstract and Quantitative Reasoning” (AQR). The motion includes (1) description, (2) guidelines, (3) comments, (4) intended learning outcomes, and (5) rationale for the requirement as part of St. Olaf College general education. A rationale for the motion follows.
CEPC will further move that if these changes are approved, the new AQR requirement will become one element of a set of revised GE requirements that will be implemented as a group rather than one by one.
Abstract and Quantitative Reasoning (AQR) (Foundation Studies)
Abstract and Quantitative Reasoning: A course that develops analytic thinking skills through systematic focus on abstract and quantitative reasoning.
- Abstract reasoning is the study of structures and patterns that arise in quantitative or computational settings. Quantitative reasoning is the use of formal structures and methods to model and analyze phenomena in the natural and human-made worlds. An AQR course should include elements of both of these reasoning activities.
- An AQR course should develop students' problem-solving proficiency through analytic thinking, not merely routine calculation. An AQR course should develop skills and ideas beyond what is typically attained in secondary school.
- An AQR course should incorporate multiple elements of abstract or quantitative reasoning (e.g., symbolic, geometric, and numerical perspectives; data analysis and statistical inference; visualization; algorithms and formal models).
- AQR courses may attach varying relative weights to abstract and quantitative reasoning. However, most courses will include significant elements of both modes of thought.
The structures, patterns, and phenomena modeled might come from almost any area, including the natural sciences, the social sciences, the arts and humanities, and mathematics itself. The main focus, however, should be on abstract and quantitative reasoning themselves, rather than solely on particular applications. Concrete data, if employed, are best collected in a variety of disciplines and settings, as this approach illustrates the power and transportability of the methods under study.
Examples. Courses in calculus, gateways to mathematics, computer science, formal logic, game theory, and statistics are among (but not necessarily exhaustive among) those that would probably address AQR goals. In calculus, for example, mathematical functions are used to represent and study phenomena of motion, including acceleration, velocity, and displacement. Data structures, formal operations, quantification, and methods of inference are studied in computer science and formal logic. In statistics, students collect, represent, structure, and draw inferences from data.
- Students should learn to solve novel problems in novel ways, rather than simply to perform routine procedures. AQR courses should be more than “technical cookbooks”; they should help students build flexible tools for thinking and solving problems. Courses that are mainly remedial at the college level (such as college algebra) should not receive AQR credit.
- Treating a variety of perspectives helps students build transferrable problem-solving skills. The processes of modeling and disciplined inference from results should not only be practiced but also addressed as subjects in their own right.
Examples. Biomedical data in a statistics course, position and velocity functions in a calculus course, and games in the economic sense can all be described symbolically, numerically, and graphically; seeing objects from several perspectives offers depth and mastery. Algorithms, data structures, and formal models in computer science and logic help students see structures and methods that underlie particular applications.
Intended Learning Outcomes:
Students will demonstrate
- an ability to recognize and employ patterns, structures, and models appropriate to particular theoretical or applied problems, as well as derive and understand properties of patterns, structures, and models themselves;
- an ability to apply abstract and quantitative reasoning to solve problems in novel contexts.
- an ability to approach problems from multiple perspectives, employing a variety of strategies.
Rationale for the AQR requirement:
Abstract and quantitative reasoning help us detect, describe, interpret, and employ structures and patterns that surround us in the natural and human-made worlds. These ways of thought are closely intertwined: Pattern and structure in quantitative information give rise to abstract models, which in turn inform effective quantitative reasoning.
Abstract and quantitative reasoning are valuable components of a liberal education, transportable across many fields of study. Abstract reasoning lends organization and focus to observation and experimentation, while quantitative comparison and prediction are crucial in scientific applications and in informing public policy debates.
Students in the natural and social sciences employ abstract and quantitative reasoning as essential intellectual tools. The study of abstract and quantitative reasoning as subjects in their own right, by contrast, is characteristic of (but not restricted to) work in mathematics, statistics, and computer science.
As a Foundation Study, the AQR requirement emphasizes methods of thought more than specific content areas.
Rationale for the motion:
Dean May’s task force on General Education called for the reinvigoration of campus conversation about GE. It called for even more public statements of the convictions that underlie the curriculum. And it called for specific review of a short list of requirements. In response to these recommendations, CEPC has overseen a process of review of the guidelines and descriptions for several GE requirements (PHA, MAR, NST, MCS, BTS-T, BTS-B).
The present statements were drafted by CEPC on the basis of recommendations from a working group on MAR (Associate Dean Van Wylen, NSM Chairs and former Chairs: Gross, Huff, Jacobel, Pearson, Walczak, A. Walter and Zorn, and Professors Grenberg, Judge and Nichol). Responding to suggestions from the Dean’s task force, CEPC asked this working group in particular to address the question of whether the general education of our students must include a course in “mathematical reasoning” (or some other phrase that suggests the distinctive approach to problems that characterizes the work of our colleagues in the MSCS department), or whether general education is better served by requiring a certain level of “quantitative literacy” (or some other phrase that suggests the need for some mastery of quantitative methods in their applications to other kinds of problems).
The proposed AQR requirement falls clearly on the “reasoning” side of this divide. Abstraction and quantitative reasoning focus explicitly on ideas and ways of thinking that unify and cut across particular applications. Quantitative literacy (QL), by contrast, is linked strongly to specific applications, which arise naturally in many and different courses. By focusing on reasoning rather than on QL itself, an AQR course makes abstract and quantitative reasoning tools available for applications that arise across the curriculum -- and thus, indirectly, advances the QL cause better than a QL-focused requirement would do.
Changes from MAR. The MAR requirement was originally seen as principally “mathematical” – hence the M in the acronym. Over time, courses such as Statistics 110 and Computer Science 121 were given the MAR credential. Although these courses probably changed somewhat in their own right, it is also true that the MAR category has evolved to become more comprehensive over the years.
The AQR requirement as newly described would not only better reflect what has become actual practice but also focus attention more clearly on the characteristic modes of thought that AQR courses seek to teach. The AQR might also somewhat broaden the range of courses that meet the requirement – but would continue to require explicit focus on analytic thinking, not routine calculation.