What Time Is It?

Lillian R. Lieber

Always fascinating to find out that it's 1959.

POSTULATES are, as you know,
the RULES for playing some "game".
Surely anyone in his right mind
would not even try to
play a game without knowing its
BASIC RULES!
And yet,
some people, young and old,
try to play the games of
arithmetic and algebra
WITHOUT EVER realizing
that these HAVE basic rules!
Now, mind you
it is NOT BECAUSE these rules
are difficult,
NOT BECAUSE there are too many of them!
On the contrary, they are very simple
and very few,
as you will soon see.
Why is it then that
youngsters, in their study of
these subjects,
are usually NOT given
the POSTULATES?
But, instead, are given
various directions for
doing various little things ------
thousands of them! -----
just as if a beginner in football
were told
"now grab the ball",
"now run this way",
"now run the other way",
etc., etc.,
without ever telling him
about the "goal",
or what he is really supposed to accomplish
or what he is allowed to do
or not allowed,
in short just pushing him around
in ways that may be clear enough to
the "pusher" or "teacher"
but which
the learner does not understand at all,
for he does not know what it is that
he is trying to do,
and gets quite bewildered by
the enormous number of details
with which he is overwhelmed!
Surely no one would ever think of
teaching football this way,
and yet this is the way
mathematics is often taught!
No wonder so many people
think they "hate math."

Now why is this so?

Is it because a certain psychologist
once emphasized the idea that
there are millions of
"S - R" bonds"
(Stimulus-Response bonds),
in arithmetic, for instance,
each of which is
something separate and distinct
and must be individually learned -----
thus the Stimulus "1 + 1"
must bring the Response "2",
"2 + 1" must bring "3",
etc., etc.,
ad infinitum.

Now I do not presume to
criticize this,
but surely it must be good psychology
to get a BROAD view of a game
and be aware of
the set of rules which govern it!

Another objection someone raised to
this "postulational" approach
is that words like
"commutative",
"associative",
"distributive",
etc.,
(which describe some of the postulates,
as you will see)
are just too hard for teenagers!
To which I can only say -----
let us not underrate teenagers!
If we do not believe in them
and in their great drive to achieve,
we may turn them aside altogether
from good, hard pursuits,
and they may then use their strength
in other ways -----
not necessarily good ways -----
for strength they HAVE and
MUST use it.
And to think that
the above-mentioned words are
"too hard" for them
is just arrant nonsense!

Lillian Rosanoff Lieber
(LATTICE THEORY, pp. 34 - 36)