Writing Mathematics — Part I

Learn to communicate mathematically.

Read, write, and orally communicate mathematical concepts.

Introduction

College Algebra students should be advised early in the semester that their grade will be largely dependent on the following :

1. Ability to correctly state mathematical concepts.
2. Ability to correctly use mathematical terms and symbols when writing.
3. Ability to correctly use mathematical concepts to solve problems.
4. Correctness of the method for solving a problem.
5. Written presentation of a process for solving a problem.
6. Understanding of mathematical concepts as exemplified by written work.
7. Ability to recognize and use connections within mathematics.
8. Ability to formulate and use generalizations.

Even a casual reading of the above eight items reveals considerable emphasis on writing mathematics correctly.  A student’s previous experience in mathematics may well lead him/her to believe that such emphasis on writing in mathematics is an aberration. This brief essay is designed to remove all doubt about the need and value of writing in mathematics. It is based on quotations from respected authorities and educators as well as my personal observations during more than 30 years of teaching.

Students should be advised that they will not receive satisfactory grades for writing down some final “answer”. Professor Kevin Lee from Purdue University Calumet sums it up well with:

“There is good reason why Herman Melville wrote Moby Dick as a novel and not as the single sentence:

The whale wins.

For the same reason, just writing down your final conclusions in an assignment will not be sufficient in a college math class.” [Lee]

Professor Lee also correctly claims;

“The ideas are the mathematics. So a page of computations without any writing or explanations contains no mathematics.” [Lee]

Establishing the Need for Good Writing

Included in the Missouri State Level Goals for General Education is the following statement about mathematics.

“Students should develop a level of quantitative literacy that would enable them to make decisions and solve problems and which could serve as a basis for continued learning.”

The following statements are in part the intended implementation, by STLCC, of the State Level General Education Goals.

• “Represent mathematical information graphically, symbolically, numerically, and verbally with clarity, accuracy, and precision.”
• “Formulate and use generalizations based upon pattern recognition.”
• “Recognize and use the connections within mathematics.”

In 1992, the Tennessee State Board of Education set forth a number of goals related to mathematics for all high school students. Included in that list of five goals was:

“Learn to communicate mathematically.”

The following year that same board made a number of specific recommendations to be implemented in the mathematics curriculum. First in that list of goals was:

“Read, write, and orally communicate mathematical concepts.”

Many other states have similar policies regarding communication and mathematics.

We may conclude from the above and numerous other sources that:

Educators at the state level are in agreement.
Mathematical communication is important.

Individual educators share this view. Dr. Kevin P. Lee provides a good example: [Lee]

“The mathematics learned in college will include concepts which cannot be expressed using just equations and formulas.”

“…being able to write clearly is as important a mathematical skill as being able to solve equations.”

Why is Writing Important in Mathematics?

In a statement of his teaching philosophy, Professor Maurer of Swarthmore College states:

“Writing is an essential form of communication, especially for subtle material like mathematics. Some people think writing and mathematics are disjoint activities, but far from it. In mathematics you use all the tools of ordinary language plus the additional conventions of mathematical symbolism – solutions consist of both words and symbols. So writing plays an important role in my courses.” [Maurer2]

In the first paragraph of a 1996 essay, E. Berry and J. Lawson state:

“In any discipline, the successful communication of ideas is at least as important as the ideas themselves. Most disciplines develop standard usages and restrictions that differ from everyday English. Mathematics is not an exception.” [Berry]

There are two important aspects to writing mathematics correctly:

• The mathematics must be correct.
• The writing must be grammatically correct.

There is widespread agreement among educators that writing mathematics helps students learn the concepts. There is also almost universal agreement that communication in the discipline is essential to utilizing any discipline in everyday life, and that good communication skills are important to career advancement.

Methods to Improve Mathematics Writing

The role of definitions in mathematical writing and the proper form for writing definitions should be emphasized through a number of assignment activities which require the student to write important mathematical definitions.  Absolute perfection should be demanded for these assignments.

A number of assignment activities should contain a model for writing a particular type of process. The student should then be expected to adhere to that model to write responses to several questions.  The model provided should emphasize the concepts involved rather than the numerical computations.

The statement of some Quiz and Test questions should contain the final “answer” and request the student to write a proper argument that leads to the given conclusion.

Examples and discussions in the textbook usually illustrate proper mathematics writing. In those instances where poor writing is used in a textbook, the instructor should point out the correct style.

Examples and discussions presented by the instructor should always illustrate proper mathematics writing.

Instructors should use the language of mathematics and should encourage/insist that their students do the same.  Writing by the instructor and by students should use fundamental words such as:

Addend, sum, minuend, subtrahend, difference, factor, product, dividend, divisor, quotient, set, element, subset, union, intersection,  numerator, denominator, natural number, whole number, rational number, irrational number, real number, complex number, etc.