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The School of Mathematics in the College of Science & Engineering offers Bachelor of Science and Bachelor of Arts degrees in mathematics and includes specializations in actuarial science, mathematical biology, computer applications of mathematics, and math education. Sixty-five faculty members teach more than 350 undergraduate student majors. Mathematics majors choose from a rich range of courses, and the major curriculum is further supplemented by technical electives and coursework in physics. Some students broaden career options by taking classes in complementary disciplines like engineering, physics, or economics. Undergraduate students also have numerous opportunities to get involved in research through UROP, REUs, and other undergraduate research programs.
Writing in the School of Mathematics
The Mathematics faculty generated the following list in response to the question, “What characterizes academic and professional communication in this discipline?”
The School of Mathematics has identified communicating effectively as one of the central learning outcomes for its students. Communicating about mathematics in both spoken and written words involves a balance between the logical precision required to create new mathematical facts and the recognition of the informal language that reflects how we think about and assimilate new mathematics.
At the upper-division level, instructors expect their math major students to engage in deeper mathematics and communicate effectively about it. Undergraduate majors regularly participate in courses at the 5000 level and above depending upon their preparation and specialization. At present, the Mathematics department places a large burden on our Sequences, Series, and Foundations courses, MATH 2283 and 3283W, to train students in the techniques of mathematical proof and clear exposition of those proofs. While many students are successful in these so called “transition courses,” members of the department agree that student performance can be improved with more practice and feedback on writing and communication.
Fundamentally, writing in mathematics is characterized by logical arguments and deductions. In formal settings, this is codified in the notion of a proof – a logical argument that has its own stylized form – but even in informal communication the emphasis is on careful, internally consistent, deductive reasoning.Good mathematical writing is characterized by clarity and precision of explanation and the use of valid and sound logical arguments to establish results. Because mathematics relies on rigorous definitions of its terms, these definitions (and associated notation) must be clear and unambiguous. Arguments are often illustrated by (but not necessarily established with) well-chosen examples and counterexamples.
Stylistically, good mathematical writing is often concise, well-organized, and follows a focused and sequential development of the argument (without unnecessary detours or descriptions). Arguments should be well-motivated (often with explicit justifications) and, in longer forms, accompanied by an introductory roadmap, clearly marked transitions, and internal summaries. Ideally, mathematical writing is tailored to its audience, acknowledging the wealth of different backgrounds in the subject. A mathematically prepared reader should be able to understand and reconstruct the argument on their own, and mathematical operations and choices may need to be explained for lay readers.
At its heart, a proof is a logical argument designed to convince the reader of the truth of a claim. As such, it must clearly convey an idea, not merely move symbols about. In longer work, good writing highlights the most critical components of an argument. Finally, as with all writing in English, mathematical writing must employ correct grammar and sentence structures.
Writing Abilities Expected of Mathematics Majors
The Mathematics faculty generated the following list in response to the question, “With which writing abilities should students in this unit’s major(s) graduate?”
Minimum Requirements for Writing in the Major:
The demands of mathematical writing are substantial, being dictated by a high bar of rigor and logical precision. This is the foundation of mathematical writing, and perhaps its most important aspect. Secondary to that, exposition should be structured to illustrate the logical flow of an argument. Finally, mathematical writing addresses the needs of the intended audience. As such, a very well-prepared mathematics major should graduate with the following writing abilities, given in those three clusters:
Precision and Rigor:
- Assess whether an argument, whether logical or mathematical, is complete and correct.
- Explain and justify choices in method or approach when considering a problem or question. Be able to explain their choices and show their work.
- Know and follow conventions for mathematical exposition, including standard patterns of proof and English grammar and usage.
Exposition, Explanation, and Argumentation:
- Employ choices in language that illustrate the logical progression of the argument. In problems and shorter forms, organize writing in ways that illustrate the goal or main idea. In writing a mathematical paper or longer forms, establish a clear plan (roadmap) for writing.
- Draw attention to the critical components of a logical argument by highlighting themes and giving a sense of the big picture.
- Select illustrative examples and visualizations to amplify and clarify the argument being made.
- Write concisely, recognizing and eliminating extraneous information.
Audience and Context:
- Write mathematics (proofs, arguments, and exposition) that a reasonably prepared reader can understand and reconstruct.
- Consider an audience's needs and motivation when communicating mathematics and make effective choices about level of detail for non-technical audiences. The audience’s expertise and needs should guide the use of technical terms and level of detail.
Menu of Grading Criteria Used in Mathematics Courses
Precision and Rigor:
Assess whether an argument, whether logical or mathematical, is complete and correct.
- Addresses the necessary details, using valid logical statements. Points are deducted for erroneous reasoning and for omitting attention to crucial aspects of the conclusion.
Explain and justify choices in method or approach when considering a problem or question.
- Explains the choice of method in a way that justifies the chosen approach as the best one. Choices reflect both sound reasoning and mathematical taste.
- Defines terms, definitions, and notations near the beginning and used consistently throughout the document. No ambiguous, undefined, or sloppy terminology.
Know and follow conventions for mathematical exposition, including standard patterns of proof and English grammar and usage.
- Employs correct grammar and usage. Statements have a logical flow and are logically ordered (in paragraphs and in documents as a whole).
Exposition, Explanation, and Argumentation:
Employ choices in language that illustrate the logical progression of the argument. In problems and shorter forms, organize writing in ways that illustrate the goal or main idea. In writing a mathematical paper or longer forms, establish a clear plan (roadmap) for writing.
- Is organized in a way that makes it clear what the author is doing and makes it clear what conclusion is reached. Deductions are made for work that lacks a clear picture of what is assumed, lacks key information that helps the reader through the main part of the proof and/or lacks a clear conclusion which addresses the key issues.
Draw attention to the critical components of a logical argument by highlighting themes and giving a sense of the big picture.
- In detailed exposition, addresses a short list of topics/ideas/themes, which may be subdivided into smaller subsections. Deductions are made for work that doesn't summarize or synthesize key ideas.
Select illustrative examples and visualizations to amplify and clarify the argument being made.
- Clarifies the meaning of the mathematics through examples and illustrations. Examples contain requisite details to explain their meaning and value in context.
Write concisely, recognizing and eliminating extraneous information.
- Is concise. Contains enough detail to be complete and no more.
Audience and Context:
Write mathematics (proofs, arguments, and exposition) that a reasonably prepared reader can understand and reconstruct.
- States what the reader is expected to assume to be true and shows sufficient detail that a reader with that level of preparation can follow. Deductions are made for work that spends too much time on easy material that should be assumed. Deductions are made for gaps in the argument that assume too much.
Consider an audience's needs and motivation when communicating mathematics and make effective choices about level of detail for non- technical audiences. The audience’s expertise and needs should guide the use of technical terms and level of detail.
- Tailors to the audience (well-organized, includes the right amount of detail for an identified audience). The writing provides descriptions with enough detail that arguments can be reproduced by a peer.
The third component of Audience & Context, regarding presentation tools and software was stricken from the set of criteria. It was felt that learning to use such tools was advised in mathematics majors, but not an essential criterion of mathematical writing.
Highlights from the Writing Plan
In its second-edition Writing Plan, the School of Mathematics will continue to build from pilot interventions in MATH 5345 (Introduction to Topology) by expanding writing interventions in MATH 3283 (Sequences, Series, and Foundations) and Modern Algebra. Faculty are engaged in ongoing consultation with WEC staff and are integrating writing instruction into a professional development series for teaching assistants. Finally, the School of Mathematics will host a panel discussion on the uses of mathematical writing outside academia, drawing on the expertise of area professionals and recent alumni.