Reading Mathematics Tips 1, 2 & 3

Introduction

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:

  • Organization with titles for chapters, sections, and subsections
  • Use of page layout, fonts, formatting, and color to organize and emphasize
  • Dependence on the language (including symbols and conventions) of mathematics
  • Extensive use of technical terms as the basis for every concept
  • Logical arguments which lead inarguably from some assumption to a conclusion
  • Exclusive use of deductive reasoning
  • Use of precise, concise, well defined presentations
  • Every presentation is reduced to barest essentials
  • Use of examples and exercises

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.  This document consists of 11 tips from DrDel.  To facilitate writing these tips, I reference a textbook which I used recently.  Since there is very little variation in College Algebra textbooks, this textbook will work just fine as a universal example textbook.

References for all the tips:

[Bruff] Derek Bruff, Tips for Reading Your Mathematics Textbook,
  http://derekbruff.org/?p=2465
[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/documents/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   Kinds of Concept Maps

Copyright 2007 by Delano P. Wegener, Ph.D.
All Rights Reserved. Use of text, images and other content on this website are subject to the terms and conditions specified on our Copyright and Fair Use page.

Tip 1 from DrDel

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

A text which I recently used for a College Algebra course  is 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 DrDel

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.

  • Learn the definition of equation.
  • Learn the definition of inequality.
  • Learn the definition of linear.
  • Learn the definition of quadratic.
  • Learn the definition of complex number.

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 DrDel

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:

  • Linear Equations
  • Solving Equations using Radicals
  • Solving for One Variable in Terms of Others

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.  Concept maps will be discussed more in future tips and post.

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