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DrDelMath

Reading Mathematics
by
DrDel

To learn mathematics in a traditional school environment, the student must be able to read mathematics.  Reading is not a natural phenomenon.  Reading mathematics certainly is not a natural phenomenon.  Reading mathematics is a skill that must be learned.  No one is born with the ability to read mathematics nor does one gain that ability in a traditional reading program.  Mathematics textbooks are written in a style completely different from other textbooks.  The student must learn to read that style of presentation. 
Some important elements of this style are:

In a non-mathematics textbook many paragraphs (or even pages) are used to present a single simple idea.
In a mathematics textbook a single simple sentence may present an enormous complex idea.
To read mathematics effectively one must develop a protocol which takes into account all of the peculiarities of mathematics writing.  This document is intended to help develop such a successful protocol.

Tip 1 from Dr. Del

Read and remember the title of your textbook as well as its authors.

In this College Algebra course at Meramec Community College, we use the fourth edition of a text named College Algebra written by James Stewart, Lothar Redlin, and Saleem Watson.  These men are from McMaster University, The Pennsylvania State University, and California State University at Long Beach.  The fact that these authors are at respected universities is a strong indication that they have good credentials as mathematicians and teachers.

You should observe that this title indicates the topic is algebra and it is at the college level.  Therefore, in this course we will study the subject of algebra.  We will not study arithmetic, trigonometry, geometry, or calculus.  We will study algebra!  We will not study it the same way you might have studied it in the past, because this is College Algebra not High School Algebra.

Tip 2 from Dr. Del

Read, think about, and remember each chapter title as you come to it.  At the same time read and think about each of the section titles for that chapter.

For example, when you get ready to study Chapter 1, look carefully at Page 72 where you find the chapter title; Equations and Inequalities.  That tells you the next 76 pages of this textbook are devoted to topics which are related in some way or another, probably very directly, to the study of two mathematical creatures; Equations and Inequalities.

If those two creatures are important enough to warrant 76 pages of instruction, they are probably pretty important.  It would seem natural to decide that one of the first things you want to discover is a PRECISE definition for each of these words.  That gives you at least two learning objectives when you turn to Page 72.

Now look at the seven section titles.  It appears that the concept of inequality will not be taken up until the last two sections, so your curiosity about that topic can be put on hold for a short time.  It is not at all clear what Section 4 has to do with equations.  You should just decide to let the authors tell you how Complex Numbers are related to equations when you get to Section 4.  When you get to Section 4 you should expect to learn the PRECISE definition of Complex Number.

The other section titles make it pretty clear that you will want to learn the PRECISE definitions of linear and quadratic.

You now have several learning objectives for your study of this chapter.

It is your responsibility to complete each of these learning objectives as you study the chapter.

Trust the authors and your teacher to reveal other learning objectives as you progress through each of the sections.

Tip 3 from Dr. Del

Read, think about, and remember each section title as you study the material in the section.  At the same time read and think about any subsection titles as you study that material.

For example, when you start studying Chapter 1, keep in mind that the first section deals with Basic Equations (the section title).  Next you should pay attention to the fact that the section is divided into subsections named:

As you study the material in the text you should always be aware of the Chapter, Section and subsection in which you are currently studying.

Attention to and actually doing the recommended activities in Tips 1, 2, and 3 have very real impact on how well you learn mathematics.  How those activities impact your learning is explained by Gagne:

“Semantic encoding is the process involved in moving information from short-term to long-term memory.  This process involves making the information meaningful by tying it to previously learned information structures (schemas) or establishing new structures.  Linkages of this sort would seem to be facilitated through the use of concept maps whereby the learner is enabled to see the structure of the material to be learned.” [Gagne p.68]

Without the use of a concept map some information in long-term memory is simply not available for retrieval.  It can’t be remembered.

Entries in this concept map are the course names, textbook titles, chapter titles, section titles, concept names, etc.  The student who pays attention to these various titles (and the associated hierarchy) is constructing a concept map which is essential for efficient retrieval of mathematics information from long-term memory which in turn facilitates (in fact, is essential for) learning.

Tip 4 from Dr. Del

Learn the vocabulary.
Why are definitions important?

If a reader does not know requisite definitions, mathematical statements become meaningless, textbook presentations are confusing at best, and lecture explanations have absolutely no value to the learner.  In addition, if the learner does not know requisite definitions his/her own written or spoken statements are incorrect and in many cases completely nonsensical.

Every mathematical term in the following sentence, taken from an elementary Algebra textbook, has been replaced by a randomly chosen word from a foreign language.

To natus one desenvolvemos by another, natus each consectetur of the first sagte by each geschiedensboek of the second liever and nesciunt dignissimos hálito.

Clearly, no one can learn mathematics from statements such as this.  However, that is precisely what a reader attempts when definitions are not learned.  After a few attempts it becomes clear that “the book doesn’t help at all” and the student stops reading the text and simply tries working problems. That strategy always leads to failure.

There are other reasons why definitions are important in the study of mathematics. 

"Preteaching vocabulary in the mathematics classroom removes cognitive barriers that prevent children from grasping new content."[Chard]

Here is what professor Stephen Maurer of Swarthmore college writes about the role of definitions in mathematics.

“Most disciplines don’t need to make definitions explicit nearly so often as mathematics does – they don’t need to be so precise nor do they deal so regularly with situations outside common experience.”[Maurer]

Definitions play a much more important role in mathematics than they do in any other area of study.  A word may (in fact probably will) have a different meaning in mathematics than in normal discourse.  One of the dictionary definitions for the word function is: “the action for which a person or thing is specially fitted or used or for which a thing exists”.  The definition of function in mathematics is very different.  That mathematical definition will be the subject of study in Chapter 2 of our College Algebra course.

Definitions in mathematics form a solid and completely adequate foundation upon which we base all our mathematical reasoning. 

A mathematical definition of a concept gives necessary and sufficient conditions for a creature to be an instance of that concept. 

For example, the definition of a prime number is:

A number is a prime number if and only if it is a natural number greater than 1 with exactly two divisors.

From this definition it is possible to conclude that 7.32 is not a prime number because it is not a natural number and therefore violates the necessary condition that a prime number be a natural number.  From this definition we can also conclude that 6 is not a prime number because it has four divisors and therefore violates the necessary condition that a prime number have only two divisors.  From this definition we can also conclude that 7 is a prime number because it is a natural number greater than 1 and it has only two divisors and therefore satisfies the necessary conditions stated in the definition.

Non-mathematical concepts on the other hand are frequently defined in a hazy and flexible manner.  Find any definition of prime real estate and note that it does not give necessary and sufficient conditions.  Rather it will be hazy, deliberately quite flexible, and open to personal interpretation.

In an article about teacher preparation, Professor H. Wu from the Mathematics Department at Berkeley claims that precise definitions form the basis of any mathematical explanation.  He correctly states:

“logical explanations – the essence of mathematics no matter how mathematics is defined – cannot be given without precise definitions.”[Wu].

Finally, I should point out that many/most of the exercises in College Algebra are simple one or two step logical consequences of a definition.  For example, a firm understanding of the term “graph” is the key to answering many questions. Just one example of this is the fact that an understanding of the word “graph” provides the basis for finding the points of intersection of two graphs.

Definitions are the first tool used when attempting to answer a mathematics question. 

Tip 5 from Dr. Del

Learn the vocabulary.  
How do you learn the vocabulary?

As you read the textbook:
As you read the textbook, insure that you know the meaning of every word.  In mathematics textbooks, the definition is usually provided when a new word is introduced.

If the textbook does not provide the definition or it is difficult to extract a formal definition, turn to the index to determine if this is the first occurrence of the word.  Turn to the first occurrence of the word in the book and try to find the definition.  If that fails turn to another source such as the chapter summary at the DrDelMath website or the Mathematics Glossary listed on the homepage of DrDelMath.

As you take lecture notes:
As you take lecture notes make sure you write all presented definitions precisely as they are presented in the lecture.

As part of your daily study plan:
As part of your daily study plan you should memorize all definitions.  To memorize a definition it is wise to start by writing it ten or more times.  Think about what you are writing.  After writing a definition ten times you should be able to recite or write it without reference to notes.  On a daily basis review the definitions by carefully reading(studying) them.  After committing a definition to memory continue your study of that definition.  Think about the necessary and sufficient conditions given in the definition.  Think about how the necessary and sufficient conditions determine whether a mathematical creature satisfies the definition.

As you prepare for a test: 
As you prepare for a test review all definitions --- think about how they help you to answer questions.

If you must still memorize definitions when you begin your review for a test, you have not been studying sufficiently or correctly or both.

As you use the DrDelMath website:
Insure that all the definitions provided in the chapter summary are committed to memory and that you understand every word in them.  Think about the necessary and sufficient conditions given in the definitions.  Think about how the necessary and sufficient conditions determine whether a mathematical creature satisfies the definition.

Insure that each of the definitions are included in the list that you review on a daily basis. 

A deck of flash cards may be the most convenient and efficient way to study definitions in mathematics.  A deck of flash cards makes it easy to review on a regular daily basis.

Cognitive Science as a Foundation for Tips 1 - 5

The Learning Process

There are two classifications of conditions for learning.  There are external conditions of learning which are controlled by the instructional developer or teacher.  There are internal conditions which derive from the stored memories of the learner. 

The diagram below is a widely accepted model of the processes involved in the act of learning.

Learning model from Gagne

The following is a simplified glimpse of some aspects of the internal conditions of learning.

Information stays in short-term memory for a very short time (measured in seconds) except during an activity called rehearsal.  Information which is to be remembered for use at a later time must be semantically encoded and stored in long-term memory.

During the learning process, information is retrieved from long-term memory into short-term memory where it combines with other items in short-term memory to bring about new kinds of learning.

Suppose the learner has previously learned the meaning of the two terms “mathematical expression” and “equal sign”.  To say he has learned this information means it has been semantically encoded and stored in long-term memory and is ready for retrieval into short-term memory.

When this student is presented with the definition “An equation is a mathematical statement which contains an equal sign”, the two previously learned bits of information are retrieved from long-term memory into short-term memory where they are combined with the new definition to be encoded and stored in long-term memory.  At this point the student has learned the meaning of  the word equation as it is used in mathematics.  I should point out that the process is actually a bit more complex but this example illustrates what must happen during the learning process.

What can go wrong in the above process to prevent learning from happening?  There are many potential problems.   We have absolutely no control over some but the learner, the instructional developer,  and instructor can prevent some problems. 

In the above example, it is clear that the learner must retrieve two items from long-term memory.  If that retrieval does not take place, learning does not happen.  Retrieval might fail because the two necessary items are not in long-term memory or they are in long-term memory but not available for retrieval.

Careful sequencing of courses and topics in mathematics education is an attempt to insure that the learner has previously learned requisite material and has it stored in long-term memory ready for retrieval.

Availability for Retrieval

“Semantic encoding is the process involved in moving information from short-term to long-term memory.  This process involves making the information meaningful by tying it to previously learned information structures (schemas) or establishing new structures.  Linkages of this sort would seem to be facilitated through the use of concept maps whereby the learner is enabled to see the structure of the material to be learned.” [Gagne p.68]

“As in the case of individual facts, the learning and storage of larger units of organized verbal information occurs within the context of a network of interconnected and organized propositions previously stored in the learner’s memory.”[Gagne p.84]

Without the use of concept maps some information in long-term memory is simply not available for retrieval.  It can’t be remembered.

There are four common kinds of concept maps with a few other specialized maps that help in certain situations.  The concept map which is most commonly associated with mathematics is called the Hierarchy Concept Map. 

Here is a simplified  picture of a Hierarchy Concept Map which might be used for mathematics.

Sample mind map for mathematics

The entries in this concept map are the course names, textbook titles, chapter titles, section titles, concept names, etc. The student who pays attention to these various titles (and the associated  hierarchy) is constructing a concept map which is essential for efficient retrieval of mathematics information from long-term memory.

That is why the previous five tips for reading mathematics encourage you to pay attention to the organizational titles.   They help build the very essential concept map.

Another View

Another way of visualizing how your long-term memory organizes information might be the following.

There is a “room” reserved in your LTM (long-term memory) for mathematics information.  The room is divided into “sections” labeled “Algebra”, “Geometry”, “Analysis”, and so on for each of the major segments of mathematics. 

Each of these sections of the room contain numerous “file cabinets”, each reserved for a subset of the room section.  So in the “Algebra” section of the room, among others there will be a file cabinet for Functions.  Inside this file cabinet are folders for topics such as Zeros of functions, Linear functions, Quadratic functions.

This organization permits efficient recovery of information from any one of the folders.

When the learner uses no concept map, it is comparable to having all the millions of mathematics facts strewn about on the floor of the mathematics room.  Obviously in such disarray, virtually nothing is available for retrieval and therefore very little learning can  take place.

Six Tips for Reading the Narrative in a Mathematics Textbook

As you sit down to study your 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 an infinite number of types of mathematics problems, so there is no way to learn every single problem-solving technique. 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.

References:


[Bruff] Derek Bruff, Tips for Reading Your Mathematics Textbook,
           http://www.derekbruff.com/talks/tips_reading.pdf
[Chard] Dr. David Chard, Vocabulary Strategies for the Mathematics Classroom,
           http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf
[Gagne] R. M. Gagne, L. J. Briggs, W. W. Wager, Principles of Instructional Design, 19
[Maurer] Advice for Undergraduates on Special Aspect of Writing Mathematics
           http://www.swarthmore.edu/library/cornell/WRITE_PRIMUS.pdf
[Simonson] Shai Simonson and Fernando Gouvea, How to Read Mathematics
          http://www.stonehill.edu/compsci/History_Math/math-read.htm
[Wu]  What Is So Difficult About the Preparation of Mathematics Teachers?
          http://math.berkeley.edu/~wu/pspd3d.pdf
[Zucker] Steven Zucker, Teaching at the University Level, Notices of the AMS, August 1996, p.863
          http://www.ams.org/notices/199608/comm-zucker.pdf
Kinds of Concept Maps   http://classes.aces.uiuc.edu/ACES100/Mind/c-m2.html

 

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