abstractmath.org 2.0
help with abstract math

Produced by Charles Wells     Revised 2017-04-23
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Covert curriculum

The languages of math

Metaphors and images

What abstractmath.org does not do


Abstractmath.org is a website with attitude. I have a definite point of view about what is included here and how it is presented.   This section is a summary of some of the more important ideas behind this website.   Many of these ideas are discussed in the Handbook with references to the math ed and cognitive science research literature.


Abstractmath.org is based on my text A Handbook of Mathematical Discourse and on my discrete math class notes (and others)  I wrote during 35 years of teaching mathematics at Case Western Reserve University. The material is drawn from:



Abstractmath.org is written in a style that

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.

Designed for the web, not for paper

Abstractmath.org is intended to be read on a computer, tablet or smartphone. It is not designed to be published on paper, because being delivered on the web makes many things easier for the reader.

Covert curriculum

My intent in creating abstractmath.org is to bring out 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:

Unfortunately many teachers don't tell the students some behaviors they should know because they aren't consciously aware of the behaviors themselves, or because they think some things they do are "obvious". Abstractmath.org tells you about some of these things, based on my experience as a teacher and on the math ed literature.

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

Math English and ordinary English are different

Math English is a special form of English
with differences in vocabulary and in usage.

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”.

Remarks about usage in abstractmath.org

The Handbook of Mathematical Discourse has 428 citations for usages in the mathematical research literature. After finishing the Handbook, I started abstractmath.org and decided that I would quote the Handbook for usages when I could but would not spend any more time looking for citations myself, which is very time consuming. Instead, in abmath I have given only my opinion about usage. A systematic, well funded project for doing lexicographical research in the math literature would undoubtedly show that my remarks were sometimes incorrect or incomplete.

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 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.

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.  

What abstractmath.org does NOT do


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