Reading Mathematics — Reading the Narrative

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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