Mathematical Thinking Core proposals

Course proposals for the Mathematical Thinking Core should describe how your course fits within your Core discipline, and how your Core discipline is situated within the purpose and values of liberal education.

Components of your proposal

Your proposal will include both a narrative description and a syllabus.
As you develop your proposal, you should not assume that the goals of your courses are obvious. It may be helpful to remember that the members of the Council on Liberal Education, like students in liberal education courses, come from units across the University. The council's aim is to ensure that liberal education courses meet the University's goals and that these goals are clear to students and to faculty members.

Narrative proposal

Your narrative proposal should explain how the course meets:

  1. The general requirements of liberal education.
  2. The common goals for all Core courses.
  3. The specific goals for the Mathematical Thinking Core.

Effective proposals will provide concrete examples from the course that illustrate how the course meets these goals, e.g., from the course syllabus, detailed outlines, course assignments, laboratory material, student projects, or other instructional materials or methods.

Your proposal should also include two brief statements that address:

  1. How your course addresses one or more of the University's Student Learning Outcomes.
  2. How the learning associated with this outcome will be assessed.


Because it is written for students, your syllabus should contain the following elements.

Language to help students understand what liberal education is and how this course fulfills its mission as a liberal education course. A course description at the head of the syllabus followed by a paragraph describing the precise aims according to the guidelines is one efficient way of doing this.

A clear explanation of how the particular course fulfills the Mathematical Thinking Core, so that students are aware of how and why the course meets LE requirements. This can be done through the stated course objectives, course topics, writing assignments, and required readings. You may also include supporting materials, such as lab manuals, sample assignments, or handouts.

Information about small group activities (small group discussion, debates, and so on) that will be employed in the course.

A brief paragraph describing the Student Learning Outcome(s) the course addresses, how it addresses these outcomes, and how the learning that is associated with the outcome will be assessed.

Additional syllabus guidelines:

  • For existing courses, the syllabus must be for a term within the past two years.
  • For courses under development, the syllabus may be provisional but still must document how the course will meet the LE requirement(s), as indicated above. A list of lecture topics or discussion topics should be included, with the understanding that dates, schedules, and readings may be tentative.
  • The syllabus needs to conform to the University Senate Syllabi Policy, approved December 6, 2001. It should be in English, or with an English translation provided.
  • Formatting is often lost when material is copied and pasted into the system. Try to keep formatting simple.


All liberal education courses must:

  • Explicitly help students understand what liberal education is, how the content and the substance of this course enhance a liberal education, and what this means for them as students and as citizens.
  • Meet one or more of the Student Learning Outcomes (SLO). In the syllabus you submit, specify which of the SLO(s) that the course meets, how it addresses the outcome(s), and how the learning that is associated with the outcome(s) will be assessed.
  • Be offered on a regular schedule.
  • Be taught by regular faculty or under exceptional circumstances by instructors on continuing appointments. Departments proposing instructors other than regular faculty must provide documentation of how such instructors will be trained and supervised to ensure consistency and continuity in courses.
  • Be at least 3 credits (or at least 4 credits for biological or physical sciences, which must include a lab or field experience component).

All Core courses must:

  • Employ teaching and learning strategies that engage students with doing the work of the field, not just reading about it.
  • Include small group experiences (such as discussion sections or labs) and use writing as appropriate to the discipline to help students learn and reflect on their learning.
  • Not (except in rare and clearly justified cases) have prerequisites beyond the University's entrance requirements.

To meet more than one requirement:

  • A course may be approved to meet one Core or one Theme or both a Core and a Theme. In the latter case, the Theme must be fully and meaningfully infused into the course (the old standard of "one-third of the course" will no longer be sufficient).
  • Courses may be submitted for both LE and WI designation.

Mathematical Thinking overview

Mathematics has a dual nature: It is a science and way of thinking, with its own language designed for logical discourse, and it also provides unique approaches to describing and understanding reality. Much of modern life rests on intellectual and scientific developments that are directed by mathematical equations and algorithms: space flight, computers, the Internet, weather modeling, security codes, and a host of others. To function as effective and responsible citizens, students need some understanding of the analytic processes that underlie these developments. Students should have some familiarity with two primary aspects of mathematical thinking.

The first aspect is mathematics as a body of knowledge. It is concerned with such issues as enumeration and computation, quantifying change, geometrical figures, shape, and symmetry. It deals with these topics via  precise, unambiguous symbolic language. Students need some facility in communication with these symbols to appreciate the power of its manner of expression. Students should understand some of the esthetically beautiful ideas and their history that have implications so powerful that science and technology would be impossible without  this underpinning—selected from  topics such as number theory, geometric analysis, calculus, probability and statistics, combinatorics,  and symbolic logic, among others. Students should appreciate that mathematical results are established by logical proofs or algorithms with rigorous methods for testing whether something in a symbolic language is an acceptable proof.

The second aspect of mathematical thinking is its broad applicability, its "unreasonable effectiveness" in the natural, biological and engineering sciences, as well in many of the social sciences and psychology. The essential concept  is "mathematical modeling." Using mathematical ideas many problems that arise in the everyday world can be abstracted and expressed as mathematical problems. The solutions, often obtained via scientific computation, are then applied to the original problem, and their conformance to reality checked. These elegant solutions to applied problems are necessary for a deeper understanding of the forces that continuously transform our world.

Mathematical Thinking Core objectives and criteria

There should be a variety of courses on mathematical thinking if the diverse needs of our students are to be met, and faculty from a variety of disciplines should participate. Responsibility for introducing students to mathematical thinking rests mainly with the courses in this part of the Core, but courses in the physical, biological, applied, and some of the social sciences will also properly address these issues. While courses should have applied dimensions, all should focus on the manipulation of mathematical or logical symbols. An appropriate course needs both to involve education in mathematical literacy, including communication with the special symbols of mathematics or logic (not prose only), and indication of how these concepts could be applied to analyze applied problems.

The Council urges the continued development of a different approach for those students for whom the traditional calculus route is inappropriate or not required for subsequent course work. Special courses dealing with "Great Ideas in Mathematics and its Applications" could be substantially more effective in providing these students with an understanding of diverse mathematical ways of thinking.

Acceptable tracks are: 1) courses dealing with "Great Ideas in Mathematics and its Applications," 2) calculus or other traditional math courses, 3) formal logic or applied courses that emphasize mathematical modes of thinking that go beyond rote computational skills. Courses on specific applications of mathematics, such as statistical methods, to a particular field are fine if there is emphasis on underlying mathematical ideas, rather than just recipes for the particular application.

To satisfy the Mathematical Thinking Core requirement a course must meet these criteria:

  • The course exhibits the dual nature of mathematics both as a body of knowledge and as a powerful tool for applications.
  • Students manipulate mathematical or logical symbols.
  • The prerequisite math requirements and mathematics used must be at least at levels that meet the standards for regular entry to the University.