help with abstract math
Produced by Charles Wells Revised 2017-01-28
Introduction to this website website TOC website index blog
BACKGROUND AND ATTITUDE
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.
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:
- 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.
Abstractmath.org is written in a style that
- is informal
- has lots of bulleted lists
- uses different font colors and
- uses short and snappy sentences (or so I wish -- unfortunately I am no Hemingway)
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
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.
- Links. I can provide an instant link to a site that explains an expression the reader might not be familiar with. In abstractmath.org, these links mostly go to other places in abmath or to Wikipedia, sometimes to other sites on the web. I have avoided providing references to sites that exist only on paper, particularly if they are behind a paywall.
- Space is free. I can refer in one location to a diagram that occurs elsewhere in the article by giving a link, but it is better simply to paste the diagram into the current location. It doesn't add cost or weight to do this! I have not done this as much as I should have in abstractmath.org, but I am working on it.
- Constant updating I don't have to stay humiliated for ten years because I made a mistake in a book (such as the one you can find very early in the Handbook). I can change it!
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:
- There are things the natives know
about themselves and will tell you.(American
Southerners like to eat grits/drink sweet tea 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
don’t like to tell people what to do but they are not very aware of the fact.)
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
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!
- 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 "$x:=42$" to mean "let $x = 42$". This usage seems to be spreading slowly. See colon equals.
- 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
math English a hundred years ago the plural of “formula” was “formulae”. Now it is usually “formulas”. (More here.)
- Mathematicians frequently deliberately change the meaning of a technical word. This happens far more often then it does in natural languages.
- In the nineteenth century, some mathematicians included $1$ as a prime number. They abandoned this deliberately because it made it hard to state the fundamental theorem of arithmetic.
- In the mid-twentieth century, because of the needs of topology and category theory, mathematicians began requiring that a function have a specified codomain. In my mathematical lifetime I have seen this gradually taking over as standard.
- Some textbook authors have started using “if and only if” instead of
“if” in definitions (more here). It remains to be seen if this will win.
Math English is a special form of English with
differences in vocabulary and
English uses ordinary words with special meanings. For
example, in math, a group is a very specific type of
mathematical object. It is not just a bunch of things.
uses the structural words of English such as “if” and “or”
that don’t mean the same thing in math English that they mean
in ordinary English.
English has rules that change the meaning of words
depending on context. For example, “if”
means “if and only if” in a definition. This use of context confuses many beginning students who don't know why they are confused, and too often the teacher does not point this characteristic of math prose.
older people who object to new usages die.
people continue talking the way they are used to.
is how language changes.
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
“They are not educated if they say ‘between you
“You are contributing to the dumbing down of American
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 and very often, perhaps even usually, incomplete.
Metaphors and images
We should reveal the
metaphors and images
we use when thinking about math
but we should also explain
the dangers and pitfalls of
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.
- Example: 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
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.
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.
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, PlanetMath and Wikipedia do this well. The abstractmath.org Glossary includes mostly terms that cause beginners problems.
- 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.
- 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.
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