Archive for January, 2013

Writing Mathematics – Part II

January 25, 2013

Learn to Communicate Mathematically.

Read, write, and orally communicate mathematical concepts.

A Few Basic Rules

No significance should be attached to the order in which these basic 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.
  • Do not submit your first draft.
  • Many mathematical adjectives and nouns have precise mathematical meanings,
  • Emulate the writing style found in the textbook and/or lectures.
  •  Start each sentence with a word, not a mathematical symbol.
  • 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.
  • Honor the equal sign.
  • Avoid the use of imprecise terms.
  • 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.
  • Do not use abbreviations.
  • Do not 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.
  • Use different letters for different things.
  • Remember mathematics is case sensitive.
  • Define any terms or variables which you use.
  • 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”).
  • Do not start a sentence with a formula.

 Examples of Incorrect Mathematics Writing

  • It is incorrect to speak or write of the quotient of a and b. When speaking or writing about division, statements must make it clear which is the divisor and which is the dividend.
  • It is incorrect to speak or write of the difference of a and b. When speaking or writing about subtraction, statements must make it clear which is the subtrahend  and which is the minuend.
  • It is incorrect to connect several different equations with equal signs, where the intermediate equal signs are intended to convey “equivalent to”.  For example, x = y – 3 = x + 3 = y is very confusing and altogether wrong.
  • It is incorrect to speak of “moving a number or variable”; there is no mathematical operation called “move”.
  • Equations can be equivalent but cannot be equal.
  • Writing “solve” when the action is “compute” or “evaluate”.
  • Writing f = x + 1 instead of  f(x) = x + 1.
  • Writing  n = even = 2n instead of “If n is even, then n = 2k for some k”.
  • Writing n2 = 16 = n = ±4 instead of n2 = 16 implies n = ±4
  • Writing k = k+ 1 instead of “replace k by k + 1”.
  • Writing (2, 3, 8) instead of {2, 3, 8}.
  • Writing  x\subset A instead of x\in A.
  • Writing “length + area” instead of “length and area”.
  • Writing a decimal approximation instead of an irrational number.
  • Confusing 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, range, and rule.
  • Confusing the terms polynomial, polynomial equation, and polynomial function.

Common Syntax Errors to Avoid

  • 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.

 Never Violate Long-Standing Word Usage

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

  • It is correct to write: divide both sides of the equation by 10
  • 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 correct to write: subtract 5x from both sides of the equation
  • 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

References:

[Lee] “A Guide to Writing Mathematics”

http://www.cs.ucdavis.edu/~amenta/w10/writingman.pdf

[Lee] “Tips for Reading Mathematics”

http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/readingtips.pdf

[Lee] “A Mathematical Writing Checklist”

http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingcheck.pdf

[Maurer] “Advice for Undergraduates on Special Aspects of Writing Mathematics”

http://www.swarthmore.edu/documents/WRITE_PRIMUS.pdf

[Berry] “Writing Mathematics”

http://agora.cs.wcu.edu/~jlawson/teaching/writing/writing.pdf

[Grayson] “Course Description for Math 248 at The University of Illinois”

http://www.math.uiuc.edu/~dan/Courses/2003/Fall/248/

[McCain Library – Agnes Scott College] “Writing in Math is Integral”

http://writing_center.agnesscott.edu/handouts/34_math.html .

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Writing Mathematics — Part I

January 20, 2013

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.

Reading Mathematics — Reading the Narrative

January 17, 2013

Six Tips for Reading the Narrative in a Mathematics Textbook

As you sit down to study a mathematics textbook and after you have duly noted the textbook title, chapter title, section title, and subsection titles as appropriate to the part you wish to study, you must heed the following tips.  These tips (6 – 11) are modifications, adaptations, and copies of tips for reading a mathematics textbook as assembled and justified by Derek Bruff while at Harvard University [Bruff].  These tips are certainly not original with Dr. Bruff.  Every serious student or teacher of mathematics has long been aware of the protocol for reading a mathematics book.  These tips are meant to encourage the beginning student to follow that protocol.

Tip 6 from Dr. Del

Read the preface of your textbook, scan any appendices, become familiar with the table of contents, index, and any other listings provided in the textbook.

The preface usually addresses special features of the textbook. Knowing the special features and special symbols used in the book will make the textbook more useful and less confusing. Becoming familiar with the various indexes will be helpful later when you want to look up the definition of a word or review a concept.

“Reading Mathematics is not at all a linear experience …Understanding the text requires cross references, scanning, pausing and revisiting” [Simonson]

Tip 7 from Dr. Del

Read the narrative of each section.

Most of these tips are about reading the narrative in the textbook. The most important material in a mathematics textbook is the narrative — the presentation of concepts. Set aside time to read the textbook when you have no intention of working on exercises. This will enable you to truly focus on the mathematical concepts at hand. If in the past, you have opened your textbook only when doing exercises (looking at the rest of the book only for examples), you must rid yourself of this bad habit now.

“Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics.  Students need to learn how to read mathematics, in the same way they learn how to read a novel or a poem, listen to music, or view a painting.”[Simonson]

All of the tips presented here are an attempt to help you learn that protocol.  How you read the narrative is an important part of that protocol.

Tip 8 from Dr. Del

Read the narrative several times.

The first reading should be to scan for major ideas.
During this first reading you should be interested in extending your mind map to include the new major concepts. If the narrative concerns topics already in your mind map, then the narrative should correct, refine, or extend your mind map.

The second reading should be to identify and learn important definitions.
To maximize the benefit from the lecture, this (and the first) reading of a section and memorization of necessary vocabulary should be done before the lecture about the section.

Make a list of the mathematics terms encountered in the section. Some of these will be old familiar terms and some will be new. You must know (flawlessly) the precise definitions of all these terms. The DrDelMath website will have precise definitions of all new terms introduced in the section. It is your responsibility to look up and review any definitions which you have forgotten.

An excellent way to begin the process of learning the definitions is to memorize them and the best way to begin the memorization process is to write the definition ten or more times.  Flash cards are a good mechanism for studying and reviewing definitions and important properties.

The third reading should be the first attempt to understand the details.
Don’t be in a hurry! To be sure this third reading involves the decoding of the words found on the page, but that is the least important and time consuming activity. The third reading should involve a great deal of reflection, contemplation, questioning, as well as construction of examples and non-examples. The textbook examples and pictures are designed to illustrate with less abstraction new abstract concepts presented in the section. Read them for that purpose. They are intended to increase you understanding — not to present templates for problem solutions.

“Reading mathematics too quickly, results in frustration. A half hour of concentration in a novel buys you 20-60 pages with full comprehension (depending on how experienced you are at reading novels). The same half hour in a mathematics textbook buys you 0-3 lines (depending on how experienced you are at reading mathematics). There is no substitute for work and time. “[Simonson]

The fourth reading should be to understand the topic as a unified whole. Mathematics is very logical and unified. Not only should the material in a section fit together, but those concepts must fit logically with previously learned topics. Look for similarities and differences between the current concept and previously learned concepts.

Subsequent readings should be for a better understanding or for review. Simultaneously reading your lecture notes and the text narrative are necessary to fit them together as a unified whole. If you have trouble with an exercise, you need to re-read the narrative looking for a better understanding of the concepts as it applies to the particular exercise. Review is a regular part of the learning process, so re-reading the narrative should be a natural and regular part of your study activities.

Tip 9 from Dr. Del

Focus on Concepts.

There are infinitely many types of mathematics problems, so there is no way to learn every single problem-solving technique.  It might be said the only important problems are the ones that do not appear in textbooks.  Mathematics is about ideas. The mathematics problems which you are assigned are expressions of these ideas. If you can learn the key concepts, you will be able to solve any type of problem (including ones you have never seen before) involving those concepts.  In support of the contention that ideas are the important mathematics, Dr. Steven Zucker of John Hopkins University states:

“One of my basic tenets is that the students have no right to know what an upcoming exam is going to look like.[Zucker]

Tip 10 from Dr. Del

Don’t bother highlighting.

Unlike most other textbooks, mathematics textbooks use chapter titles, section titles, and sub-section headings to organize material and provide the basis for the necessary mind map. Mathematics textbooks also use page layout, fonts, and colors very well to organize information and make it easily visible. Words used as headings such as Definition, Theorem, Axiom, Property, Proof, and Example serve to identify and classify certain segments of the text.  There’s usually little use in highlighting or underlining in a mathematics textbook although it is sometimes helpful to mark something that you might want to find quickly at a future time.  An attempt to underline or highlight everything that is important will result in the entire narrative being highlighted.

Tip 11 from Dr. Del

Read with pencil, paper, and eraser.

As you read the text, you should write notes.  Check calculations. Write your own examples.  Believe your textbook, but check the work you see there anyway — insure that you can supply all the missing details.  You don’t learn difficult material just by reading a nice presentation of the material – you need to break out pencil and paper and convince yourself that you follow the reasoning and computations. It is also important that you be able to produce a similar argument on your own.  That is much more difficult than following a nicely presented argument.  You might try to work out examples before looking at their solutions in the textbook.  Make up your own examples to illustrate the concepts and do the necessary computations to insure that your example illustrates what you want it to.  Combine your lecture notes and material from the DrDelMath website with the text material by writing your own “mini textbook” about the subject.


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