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Topic: Summer '96 Institute: Middle School Working Group
Replies: 29   Last Post: Jul 19, 1996 3:55 AM

 Messages: [ Previous | Next ]
 Allen Adler Posts: 272 Registered: 12/3/04
math and language arts
Posted: Jul 16, 1996 12:13 AM

One poster has mentioned his interest in language arts and has
apparently taken an interest in the crossword called the Sator
Magic Square in this connection. In a separate discussion, the
difficulties of translating mathematics written in English into
mathematics written in Spanish has come up.

I would like to elaborate on my comment, in the latter discussion,
about mathematics being a language and see where it leads from the
standpoint of language arts.

Topic 1: Mathematical idiom

If you show someone an open bag of potato chips which still has some
potato chips in it, there are various circumlocutions to describe
it. "You want some chips? There's some left."

A mathematician will simply remark that there is a nonempty bag of
potato chips. The word "nonempty" is typical of mathematical idiom.
It doesn't normally arise in the parlance of nonmathematicians.

There are other key phrases and usages which, even if not unique
to mathematics, are standard mathematical usage which one finds
less often in common parlance. One is "if and only if". Someone
might say, "I'll give you a quarter if you let me copy your
homework." From a logical point of view, it is conceivable that
the quarter will be given even if permission is not granted to
copy the homework, but most likely the student intended the
word if to mean "if and only if".

Another is the tendency of some people to write "We have" before
displaying a formula or a conclusion. E.g., "Combining these two
identities and clearing denominators, we have x=7." or
"After a similar line of reasoning, which we omit here, we
have that the line opposite the largest angle is the largest side."

Another example involves words that mean one thing in English
and something entirely different in mathematics. An advanced
example is "limit" or "compact". But more elementary notions
such as "group" or even "relation" are already different.

One activity is to try to come up with amusing equivocations
that play on the different usages. E.g.,
"A group is a collection of individuals who associate and there
is usually an individual, called the identity element, with the
property that any individual who associates with the identity element
is no longer a member of the group."

I have not even scratched the surface. Anyway, exploring this aspect
of mathematics also involves the language arts.

One source to look at for some informal and formal discussions of
this, look at the discussion of the Dog Walking Ordinance, reprinted
in the Word Of Mathematics from a work of Robert Graves and Alan
Hodge.

Topic 2: Language Decipherment

Take an English speaking student and show him/her some mathematics
written in another language, preferably one with a lot of cognates
with English. There is a tremendous amount of information in whatever
tables or formulas the writing contains. The notation is likely to
be common to speakers of either language and that fact serves
to give the reader some idea of the context of the discussion even
if he/she doesn't know the language in question. Under such
conditions, one can often decipher enough of the language to make
sense of what is going on, even if one can't make a detailed
translation.

One of the pleasant features of decipherment is that one can form
hunches as to what the text actually says, but then one has to
check whether the proposed translation leads to correct mathematics.

Another is realizing that one is staring at a familiar usage
(e.g. if and only if) in a language one has never studied and
now knowing how to say that. In some cases, one might never
know how to look it up.

Topic 3: wff and proof

There is a game which was available when I was in high school
and which I still see advertised sometimes. It involves tossing
dice which have logical symbols on the faces and trying to
arrange them so as to arrive at the longest well formed formula
one can make with those symbols. It was called wff and proof.
This game can be used to illustrate another aspect of mathematics
and language arts.

One can also go further using predicate calculus. (BTW, calculus is another
good example of a word that has different meanings in mathematics
than in common parlance. Sometimes when people ask me what calculus
is, I like to tell them it is a substance found on the teeth of
mathematicians.)

Topic 4: Productions and natural language

I'm sure some very simple examples can be given to students to
illustrate some of the concepts of formal languages. Then one
can explain that similar ideas are being applied to study
natural languages.

One way to introduce the examples might be
as puzzles in which the students, after seeing that the teacher
accepts some strings of symbols and not others, have to guess
the pattern in the acceptable strings. Or a computer, with infinite
time and patience, can do the accepting and rejecting.

It can be pointed out that a computer program is a string of
symbols which the machine will accept or reject and that learning
to program in that language in part involves learning to distinguish
acceptable programs from unacceptable ones.

Similarly, learning a natural language, such as one's native language,
involves learning to recognize which strings of sounds or symbols
are acceptable in that language. Rules of grammar and usage amount
to trying to approach a description of the acceptable strings,
just as in the exercises one did at the beginning.

Maybe another variant would be to introduce "pig latin" or various
other modifications of ordinary speech and to specify or guess
the rules of formation.

Topic 5: Secret codes

Students can make up secret codes and pass notes using them.
Other students can try to decode them. The easiest would
be substitution codes.

This is to some extent a language art. It also involves some
statistical knowledge of the language. It wouldn't hurt to bring
to their attention the existence of such puzzles in the newspapers.

Incidentally, one can also use magic squares to encode secret
messages, although the messages might take up a lot of space.

Date Subject Author
7/15/96 Sandy Sherman
7/15/96 Suzanne Alejandre
7/15/96 Eric Sasson
7/15/96 Donald W. Smith
7/15/96 Eric Sasson
7/15/96 Sarah Seastone
7/15/96 Anne Wheelock
7/15/96 Cynthia Lanius
7/15/96 Denise Miller
7/15/96 Suzanne Alejandre
7/15/96 Suzanne Alejandre
7/15/96 Suzanne Alejandre
7/15/96 Lou Talman
7/15/96 Anne D. Sandler
7/16/96 Donald W. Smith
7/16/96 Suzanne Alejandre
7/16/96 Eric Sasson
7/16/96 Suzanne Alejandre
7/16/96 Eric Sasson
7/17/96 Jean Abel
7/17/96 Suzanne Alejandre
7/17/96 David Grant
7/18/96 Donald W. Smith
7/18/96 Suzanne Alejandre
7/18/96 Suzanne Alejandre
7/18/96 Eric Sasson
7/18/96 Sarah Seastone
7/19/96 Denise Miller