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College-Algebra.Com

Writing Mathematics Correctly

Attention

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

Ability to correctly state mathematical concepts.
Ability to correctly use mathematical terms and symbols when writing.
Ability to correctly use mathematical concepts to solve problems.
Correctness of the method for solving a problem.
Written presentation of a process for solving a problem.
Understanding of mathematical concepts as exemplified by written work.
Ability to recognize and use connections within mathematics.
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 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 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.

Basic Rules for Writing Mathematics

"Mathematics writing is different from ordinary writing and harder - in addition to all the requirements of ordinary good writing, there are additional constraints and conventions in mathematics. An additional constraint is that mathematics follows much more demanding rules of logic than ordinary discourse, and you must make your logic clear. Some of the additional conventions are those for defining new concepts and those for organizing the material .... "[Maurer, p.3]

There are two important aspects to writing mathematics correctly:

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

Some basic rules for writing in general and writing mathematics in particular are presented here. More rules and conventions may be added at some future time. No significance should be attached to the order in which these rules are presented.

"The fact that some mathematics conventions have been universally adopted around the world suggests that they accomplish something important." [Maurer]

For that reason it is unwise for a novice to vary from standard conventions, even though they are not absolute rules.

Write with a pencil so that errors may be erased and replaced with correct work.
Use complete sentences, correct grammar, and correct spelling.
Symbols (like the + symbol) that have a specific mathematical meaning are reserved for mathematical use.
Don't submit a first draft.
Many mathematical adjectives and nouns have precise mathematical meanings, and an English synonym will not serve as a replacement. For example, "element" and "part" are not interchangeable when referring to an element of a set.
Look at examples of writing in the textbook, and try to emulate the style.
Start each sentence with a word, not a mathematical symbol.
Two mathematical expressions or formulas in a sentence should be separated by more than just a space or by punctuation; use at least one word.
Words have meanings: be aware of them. For example, an equation has an equal sign in it.
Don't use abbreviations.
Don't end a line with an equal sign or an inequality sign.
Insure that every statement is mathematically correct.
Strive for a good balance between words and symbols.
Honor the equal sign.
Use different letters for different things.
Remember mathematics is case sensitive.
Define all terms or variables.
Once a variable has been assigned a meaning, do not re-use it with a different meaning in the same context.
There is a distinction between the definite article ( "the" ) and the indefinite articles ( "a" and "an" ).
Avoid the use of imprecise terms.
Be sure that the use of a term agrees with the definition of that term.
Include transitional phrases to help guide the reader.
Conclude the solution of a problem with a clear and complete statement of the conclusion.

Some of the above come from: "Writing Mathematics"[Berry] and some from "Course Description for Math 248 at The University of Illinois" [Grayson].

Common Errors to Avoid When Writing Mathematics

The most common errors fall into the following categories:

Not answering the question.
Incorrect English.
Incorrect mathematics.
Hard-to-read format.
Use of pronouns such as it.
Using the same symbol for different quantities.
Too many words and too few symbols, or vice versa.
Leaving too little space for possible corrections.

Listed here are some of the common errors which should be avoided. This list is certainly not complete. Additions will be made to it in future semesters and a supplement or duplicate list will be maintained on the DrDelMath website. An excellent collection of errors and math facts is maintained by Professor Russell Blyth at St. Louis University. That site is MathMistakes.Info and the algebra section is HERE.

Don't write f = x + 1	when you mean	f(x) = x + 1.
Don't write n = even = 2n	when you mean	If n is even, then n = 2k for some k.
Don't write n² = 16 = n = ±4	when you mean	n² = 16 implies n = ±4.
Don't write k = k + 1	when you mean	Replace k by k + 1.
Do not write (3, 4, 8)	when you mean	{3, 4, 8}.
Do not write a ⊂ A	when you mean	a ∈ A.
Don't write length + area	when you mean	length and area.
Don't write 1.4	when you mean	√2.

Do not confuse the words equation, expression, and function.

An expression is an algebraic combination of terms containing no verb.
An equation is a mathematical statement which contains an equal sign.
A function consists of a domain, a range, and a rule.

Writing About Division
It is correct to write:	The quotient of a divided by b.
It is correct to write:	Divide both sides of the equation by 10.
It is incorrect to write:	The quotient of a and b.
It is incorrect to write:	Divide 10 to both sides of the equation.
It is incorrect to write:	Divide 10 by both sides of the equation.
It is incorrect to write:	Divide the right side of the equation by 10.
Faced with 10x = 20 it is incorrect to claim that dividing 20 by 10 yields x = 2.
Statements involving division must make it clear which is the divisor and which is the dividend.

Writing About Subtraction
It is correct to write:	The difference of a subtracted from b.
It is correct to write:	Subtract 5x from both sides of the equation.
It is incorrect to write:	The difference of a and b.
It is incorrect to write:	Subtract 5x to both sides of the equation.
It is incorrect to write:	Subtract both sides by 5x.
It is incorrect to write:	Minus 5x from both sides of the equation.
It is incorrect to write:	- 5x from both sides of the equation.
Statements involving subtraction must make it clear which is the subtrahend and which is the minuend.

Writing With Arithmetic Operations
It is incorrect to write 3 + - 4.	Correct syntax is 3 + (- 4).
It is incorrect to write 3 ÷ -4.	Correct syntax is 3 ÷ (- 4).
It is incorrect to write 3 - - 4.	Correct syntax is 3 - (- 4).
It is incorrect to write two operation symbols next to each other.
It is incorrect to write of moving variables or numbers from one side of an equation to the other. There is no mathematical operation called "move".

References:

[Lee] "A Guide to Writing Mathematics" LINK
(http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingman.pdf)

[Lee2] "Tips for Reading Mathematics" LINK
(http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/readingtips.pdf)

[Lee3] "A Mathematical Writing Checklist" LINK
(http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingcheck.pdf)

[Maurer] "Advice for Undergraduates on Special Aspects of Writing Mathematics" LINK
(http://www.swarthmore.edu/library/cornell/WRITE_PRIMUS.pdf)

[Berry] "Writing Mathematics" LINK
(http://www.math.ualberta.ca/~orivasplata/writing_aids/purdue2_write.pdf)

[Grayson] "Course Description for Math 248 at The University of Illinois" LINK
(http://www.math.uiuc.edu/~dan/Courses/2003/Fall/248/)

[McCain Library - Agnes Scott College] "Writing in Math is Integral" LINK
(http://writing_center.agnesscott.edu/handouts/034_math.html)

[DrDel] "Common Errors" LINK
(http://www.drdelmath.com/special_topics/common_errors.htm)

[Blyth] "MathMistakes.Info" LINK
(http://mathmistakes.info/)