# WRITING AND REASONING

I think writing is more a matter of clear thinking. And if students can think clearly enough to write these proofs correctly and, where possible, write simply, then that's going to help them in other kinds of writing as well. This is a good exercise in ordering your thoughts, eliminating unnecessary baggage, and getting things down in a clear and concise way. — Professor Christopher Allday

Formal mathematical proofs . . . teach you to structure your thinking process into a stream of relevant data, ignoring whatever is irrelevant. — Student

### COURSE GOALS

The major objective is to foster student understanding of abstract algebra and the ability to use the method of reasoning required to prove theorems and explain solutions to abstract, mathematical problems. As students become increasingly skilled in thinking clearly and ordering their thoughts, they should gain greater aptitude in writing clearly and concisely.

### WRITING ACTIVITY

FIVE MATH PROOF ASSIGNMENTS

Every week, students are assigned homework exercises in which they write proofs of various math principles. For each assignment, students must write out proofs to answer a set of mathematical problems such as the following:

Writing was the ONLY way to learn the material. . . Neither writing nor mathematics is a "spectator sport." Moreover, in writing proofs, you learn and practice not only mathematics, but also the principles of logical argument; and the discipline of these communications skills is widely APPLICABLE. --Student
Proofs of the solutions to such problems are to be logical arguments written in the English language, much of which is later replaced by mathematical symbols. Here is an example of part of a proof:
Students have one week to complete the set of proofs for each assignment. The instructor does not collect the first two assignments, but goes over them in class, writing answers in proof-form on the chalkboard. Students compare those answers to their own, asking for clarification when needed. The next assignment is graded. All the assignments require logical organization. I would encourage students to form study groups to help one another on non-graded assignments. This exchange of ideas, I think, would reinforce the instruction. [To encourage more class participation] I might suggest that from time to time a portion of a lecture period be devoted to a brief assignment of current material and the class be divided into groups, each tackling the same or different problems and then sharing results with the entire class.--Student
PURPOSE: Writing proofs of theorems or other statements allows students the opportunity to practice logical thinking and document logical arguments. Students also gain proficiency in the language of abstract mathematical proofs and gain greater understanding of the methods and level of abstraction necessary in abstract algebra Aside from the core material, I strengthened my skills at analytic thought and logical argument. I also practiced research and review skills in order to complete the assignments.--Student

### RELATED ACTIVITY

FOUR TAKE-HOME EXAMS

Take-home exams require students to answer approximately ten problems in written proof form, which results in six or more pages for each exam.

PURPOSE: Take-home exams allow students enough time to thoroughly contemplate algebraic problems and write them out in neat and logical form. Students thus demonstrate that they understand the concepts, format, and reasoning used in abstract algebra and that they are proficient in logical analytical thinking.

## Professor Allday comments on his class (excerpts from an interview):

The first learning goal is to understand the material: abstract algebra, the method of reasoning, the style of proving things in a logical and orderly fashion. This is all very new to the students. There's very little computation involved compared to reasoning.

I think the more advanced the material gets, the more likely it is that there are different approaches. Usually in the more elementary stages there is the simplest approach; maybe there is also a more circuitous approach to the problem. I encourage the students to find the simplest one, and sometimes I even take points off if they've presented a solution that is roundabout but not necessarily wrong. And if the reasoning is correct, that's good; but if the reasoning is correct and the simplest possible, that's even better. And I might give ten out of ten for a correct and simple solution and reasoning, and eight out of ten for correct and unnecessarily long reasoning.

By example, I try to teach them a writing strategy [and the writing style specific to math]. I write a tremendous number of proofs on the board and they also read proofs in the book, and I hope the combination of the two will give them an idea of how to do it.

To some extent it seems odd that a math course is a writing-intensive course. In a sense, I would encourage them to write as little as possible, tell them that it's a thinking-intensive course rather than a writing-intensive course, and emphasize finding the shortest, most concise proof. On the other hand, ordering their thoughts, getting their thoughts and reasons clear in their heads, and putting them on paper are good experiences for many other kinds of expository writing or reasoned writing.