Friday, September 10, 2010

How to Read Mathematics

"Mathematics is “a language that can neither be read nor understood without initiation.” 

    A reading protocol is a set of strategies that a reader must use in order to benefit fully from reading the text. Poetry calls for a different set of strategies than fiction, and fiction a different set than non-fiction. It would be ridiculous to read fiction and ask oneself what is the author's source for the assertion that the hero is blond and tanned; it would be wrong to read non-fiction and not ask such a question.  This reading protocol extends to a viewing or listening protocol in art and music. Indeed, much of the introductory course material in literature, music and art is spent teaching these protocols.
  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.  Ed Rothstein’s book, Emblems of Mind, a fascinating book emphasizing the relationship between mathematics and music, touches implicitly on the reading protocols for mathematics.
   When we read a novel we become absorbed in the plot and characters.  We try to follow the various plot lines and how each affects the development of the characters.  We make sure that the characters become real people to us, both those we admire and those we despise.  We do not stop at every word, but imagine the words as brushstrokes in a painting.  Even if we are not familiar with a particular word, we can still see the whole picture.  We rarely stop to think about individual phrases and sentences. Instead, we let the novel sweep us along with its flow and carry us swiftly to the end.  The experience is rewarding, relaxing and thought provoking.
    Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space.  Both a mathematics article and a novel are telling a story and developing complex ideas, but a math article does the job with a tiny fraction of the words and symbols of those used in a novel.  The beauty in a novel is in the aesthetic way it uses language to evoke emotions and present themes which defy precise definition.  The beauty in a mathematics article is in the elegant efficient way it concisely describes precise ideas of great complexity.

What are the common mistakes people make in trying to read mathematics?  How can these mistakes be corrected?

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

Don’t be a Passive Reader

“A three-line proof of a subtle theorem is the distillation of years of activity.  Reading mathematics… involves a return to the thinking that went into the writing”

 Mathematics says a lot with a little.  The reader must participate.  At every stage, he/she must decide whether or not the idea being presented is clear.  Ask yourself these questions:

Why is this idea true?
Do I really believe it?
Could I convince someone else that it is true?
Why didn't the author use a different argument?
Do I have a better argument or method of explaining the idea?
Why didn't the author explain it the way that I understand it?
Is my way wrong?
Do I really get the idea?
Am I missing some subtlety?
Did this author miss a subtlety?
If I can't understand the point, perhaps I can understand a similar but simpler idea?
Which simpler idea?
Is it really necessary to understand this idea?
Can I accept this point without understanding the details of why it is true?
Will my understanding of the whole story suffer from not understanding why the point is true?
Putting too little effort into this participation is like reading a novel without concentrating.  After half an hour, you wake up to realize the pages have turned, but you have been daydreaming and don’t remember a thing you read.

Don’t Read Too Fast

Make the Idea your Own

An Example of Mathematical Writing

  ---example will be presented in the next article                                          
                                                                                                                             --Shai Simpson 

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