abstractmath.org
help with abstract math
Produced by Charles Wells. Home. Website Contents Website Index
Posted 2
June 2008
ATTITUDE incomplete
This page explains some of the ideas that behind abstractmath.org.
Abstractmath.org is a website with attitude(s). I have a definite point of view about what is included here and how it is presented. This incomplete section is a summary of some of the more important ideas behind this website. Most of these ideas are discussed in the Handbook with references to the math ed and cognitive science research literature.
Contents
What abstractmath.org does NOT do
Math language is a living language
Math English and ordinary English are different
Background
Abstractmath.org is based on my text A Handbook of Mathematical Discourse and on class notes I wrote during 35 years of teaching mathematics at Case Western Reserve University. The material is drawn from
¨ The work of many scholars in linguistics, mathematical education, philosophy, and cognitive science. Their works are cited in detail in the Handbook.
¨ My own observations of students.
¨ My lexicographical research described in the Handbook.
What abstractmath.org does NOT do
¨ Abstractmath.org is not a
source of extended treatments of particular math subjects. There are many such sources on the web.
¨ Abstractmath.org does not
provide a dictionary of technical terms in math. MathWorld and Wikipedia do this well.
¨ Abstractmath.org does not go into depth about problem solving and proof techniques. It is certainly a good idea to teach both these things, but one website can’t do everything. Links.
¨ Abstractmath.org does not attempt to present a balanced view of math education, cognitive science or anything else, although it draws on research in these areas.
Covert
curriculum
My
intent in creating abstractmath.org is to bring out as much as I can
those
aspects of understanding, doing and communicating math
that many mathematicians and students are not always aware of.
Abstract math, like any other academic discipline, contains
explicit ideas that are taught to the student and also hidden ideas and assumptions and
methods that are not communicated to the student. This puts the student in the position of an
anthropologist trying to understand the culture of the fearsome tribe of Mathematicians. As anyone who has dealt with more than one
culture can tell you:
¨ There are things the natives know about themselves and will tell you. (American Southerners like to eat grits and will tell you.)
¨ There are things the natives think they know about
themselves and will tell you, but they are wrong. (Americans never think they are class-conscious.)
¨ There are things the natives know about themselves and won’t tell you. (To give an example about Americans is logically impossible and to give an example about another nationality would be rude.)
¨ There are lots and lots of things the natives don’t know about
themselves. (Americans
don’t like to tell people what to do but they are not very aware of the fact.)
The languages of math
I tell it the way it is
Most of the discussion on this website of both the symbolic language and math English
is descriptive, not prescriptive.
The presentation here is aimed at describing how math is written and spoken, not how it should be written and spoken. I do not often talk about “right” and “wrong” usage. After all, you have to put up with it the way it is!
Math language is a living language
¨ Both math English and the symbolic language are living languages, just like English or Spanish. (But the symbolic language is mostly written rather than spoken.)
¨
New words or symbols appear in any language and gradually replace old ones.
· Several hundred years ago “you” gradually replaced “thou” and “thee” in English.
·
Around thirty years ago (a fact that can be checked, but I
have not done so) some mathematicians starting writing “ ” to mean “let x = 42”. This usage seems to be spreading slowly.
¨ New usages appear and old usages are discarded.
· “Between you and I” is apparently replacing “between you and me” in the language of young educated Americans. Whether this wins out or not is yet to be seen.
· In
math English a hundred years ago the plural of “formula” was “formulae”. Now it is almost always “formulas”. (More here.)
· Some textbook authors, for
example Epp, have started using “if and only if” instead of
“if” in definitions (more here).
Math English and ordinary English are different
Math English is a special form of English with
differences in vocabulary and
in usage.
¨ Math
English uses ordinary words with special meanings. For
example, in math, a group is a very specific type of
structure, it is not just a bunch of things.
¨ It
uses the structural words of English such as “if” and “or”
that don’t mean the same thing in math English as they do
in ordinary English.
¨ Math
English has rules that change the meaning of words
depending on context. For example, “if”
means “if and only if” in a definition.
How
language changes
How
language changes:
The
older people who object to new usages die.
Younger
people continue talking the way they are used to.
That
is how language changes.
When
a new word or usage begins to replace an old one, many people who think they
know what they are talking about object strongly and irrationally. I expect some people who read my remark above about “between you and I” will flame me with remarks
like:
“They are not educated if they say ‘between you
and I’.”
“You are contributing to the dumbing down of American
culture.”
Remarks such as these are made
mostly by older people. Older people generally die before younger people, so
sooner or later the younger people “win”.
Presentation
Abstractmath.org is written in a style that
¨ is informal.
¨ has lots of bulleted lists
¨ has sidebars
¨ uses different font colors and weights
There is no excuse for the kind of heavy handed academic writing that was prevalent in the past in academic books, and I am glad that we have moved away from that in the last thirty years.
Still, my style here is experimental and not always consistent.
Suggestions are welcome.
Metaphors and images
Exciting but dangerous
We should reveal the
metaphors and images we use when thinking about math
but we should also explain the dangers and pitfalls of
using them
Many of us who teach math have an ambivalent attitude towards the use of images and metaphors in math. They are exciting but dangerous.
¨ They help us to understand math objects new to us.
¨ They help us to understand applications of theorems about the objects.
¨
But they can make you think a math objects has properties that it does not.
An example is the "next real number" idea (thinking of the real line
as a row of points or locations, so next to one point there must be a "next
one" not true.) That is suggested by the metaphor "line" applied to the set
of reals.
Rich and rigorous
When we think about and do math, we jump back and forth between the rich
mode and the rigorous mode of thinking.
¨ In rich mode we use images and metaphors a lot because they suggest how to think about the objects and what the applications might be. They also make math a rich and interesting subject (a point often neglected).
¨ Then when we set out to prove something, we adopt an entirely different, impoverished mental image of mathematical objects (the rigorous mode). They are inert, they don't change, they don't affect anything. In other words, they are dead. (Elsewhere I have called it rigor mortis mode.) That view fits with the properties of the logic used in mathematical reasoning. For example, thinking as the objects as inert helps lessen the confusion caused by "if...then" since it removes thoughts of causality and time order. It is very important to think in this mode during proof construction, but it is also like going from a color picture to a black and white picture.