Date: May 31, 2001 4:34 AM
Author: Osher Doctorow
Subject: Re: Two Column Proof (was: Algebra/Geometry Text Suggestions)
From: Osher Doctorow osher@ix.netcom.com, Thurs May 31, 2001 12:30AM
Oops! I made a typo (it's past my bedtime). It's r! that makes the
difference, not (n-r)!. The idea is the same. When r is bigger than 1, say
r = 2 or more, combinations are always fewer in number than permutations
provided that n is at least one larger than r.
Now that I'm here again, I will say a few more words. When you expand the
key or core idea of a proof logically, you don't memorize sequences of
steps, because of the above combination versus permutation argument
(including my previous contribution). What you do is, you ask what you need
to do next to get from the premise to the conclusion using the core idea.
Once you start, you keep retrieving relevant theorems from your
knowledge/memory which are triggered by either the words or symbols of the
previous step. If you retrieve too many theorems, you can usually rapidly
cut them down because all but one or two will refer to extraneous knowledge
beyond what the theorem mentions. Usually you get down to one theorem to
use in the particular step of the proof, or one definition or premise
(*given*) or axiom or postulate or corollary. If you really have two, try
each and see what happens (there is usually a way to determine by looking
which is most relevant, however). At no time are you memorizing sequences
in two-column proofs.
Osher
----- Original Message -----
From: "Osher Doctorow" <osher@ix.netcom.com>
To: "Joshua Zucker" <joshua.zucker@stanfordalumni.org>; "Haim"
<hpipik@netzero.net>; <math-teach@forum.swarthmore.edu>
Sent: Thursday, May 31, 2001 1:11 AM
Subject: Re: Two Column Proof (was: Algebra/Geometry Text Suggestions)
> From: Osher Doctorow osher@ix.netcom.com Wed. May 30, 2001 11:40PM
>
> Aha! Just as I thought! Another bicycle rider trying to teach us car
> drivers!
>
> Actually, two column proofs involve an organization of logic. Proofs are
> deductive logic from definitions and axioms and postulates and premises to
> conclusions. When written in two columns, this organizes the proofs.
Some
> students may benefit from different ways of organizing the proofs, but
among
> the key things are (1) searching for the key idea of the proof and
> expressing it at least to oneself succinctly and concisely, (2) citing the
> reason for each step in the proof, (3) writing the steps in the order in
> which the logic is used. Actually, if you know your previous courses
> (prerequisites) well and know what the previous theorems and definitions
and
> axioms and postulates and corollaries say, and if you have looked
carefully
> at examples of each and worked problems in each, you can sometimes find
the
> key idea of the proof by searching in your memory for the words expressed
in
> the premise and conclusion, which may bring up the relevant prior theorem
or
> corollary by which to prove the theorem in question. If, in addition, you
> actually read the proof, and when reading it consciously search for the
key
> idea and try to express it in a few words - the idea which, when expanded,
> leads to the other results - then you would be wise to make a list of key
> ideas associated with theorems, and if a teacher flunked you for being
able
> to associate key ideas with proofs of particular theorems, he or she would
> probably be making a bad mistake. Once you get practice in isolating key
> ideas, you should try to logically expand them. You can ask yourself -
what
> do I need to go from the key idea to the conclusion of the theorem? Write
> out what you think you need to know. Try saying it in English sentences
and
> translating it into mathematics. If you get stuck at some sentence or
step,
> look at the words in English or math and try to retrieve from your
knowledge
> what theorems relate to the words - triangle, circle, square,
parallelogram,
> polygon, plane, sphere, etc. Don't memorize sequences of steps except in
> algorithms (one of the most common errors of people who do poorly in
> mathematics), but do memorize definitions, axioms, postulates, statements
of
> theorems, corollaries. The reason for not memorizing sequences of steps
> except in algorithms is that a permutation (ordered list) contains more
> elements than a combination (unordered list or set). So when you memorize
> things in a particular order, you put an extra load on your memory. In
> fact, let n! = 1 times 2 times 3 times...times n. Then the number of
> combinations of r things selected from n > = r things is n!/[r!(n-r)!],
> wheras the number of permutations of r things selected from n > = r things
> is n!/(n-r)!. Notice the (n-r)! in the denominator of combinations but
not
> in the denominator of permutations. When n - r is greater than 1 (for
> example, choose all sets of r = 3 objects from n = 5 objects), then the
> number of combinations is less than the number of permutations because you
> are dividing the former by (n - r)! which is greater than 1, and the
bigger
> n - r is, the bigger the difference.
>
> Now, then. I recommend staying off your bicycles altogether in automobile
> traffic in large urban regions. If Abraham Lincoln could walk, so can
you
> unless you are disabled. Bicycles can be used in non-car areas, and some
> cities use them in bicycle paths which do not intersect car paths. When a
> bicycle path runs in the street parallel to car traffic, I recommend
flying
> and parachute and falling lessons. A bicycle is a perfect target for an
> angry driver, a sleepy driver, a driver who doesn't know the rules, a very
> old driver who can barely see outside the Motor Vehicle Department, and
very
> young drivers eager to test out their accelerators, not to mention drunk
> drivers, etc. I also know about horses and herds of cows. Do not follow
a
> herd of cows over a cliff, whether or not you have practiced falling.
>
> Osher
>
> ----- Original Message -----
> From: "Joshua Zucker" <joshua.zucker@stanfordalumni.org>
> To: "Haim" <hpipik@netzero.net>; <math-teach@forum.swarthmore.edu>
> Sent: Wednesday, May 30, 2001 7:58 PM
> Subject: Re: Two Column Proof (was: Algebra/Geometry Text Suggestions)
>
>
> > At 7:33 PM -0700 5/30/01, Haim wrote:
> > >Dear Friends,
> > >
> > > As with the discussion on long division, some months ago, I find
> > >the longer the discussion goes on, the less I understand.
> > >
> > > I always understood the two column proof to be a pedagogical
> > >device, much like training wheels on a bicycle. Some children need
> > >training wheels longer than others, a few children don't need them at
> > >all, most children benefit from having them for some period of time
> > >early in their bicycle riding experience.
> >
> > If you read the literature from the professionals at teaching
> > bicycle riding, they recommend that you eliminate the training
> > wheels, which teach bad habits of balance. Instead, start by
> > coasting downhill on a gentle grassy slope where it won't hurt
> > too much when you fall down. After some experience, start
> > pedaling a bit as you go down the hill. Then, once you're not
> > falling too much any more, head out to a bike path or a cul-de-sac
> > and then finally go get some experience with the real roads!
> >
> > And no, I'm not making that up just for the analogy.
> >
> > > Is someone asserting something else about the two column proof?
> >
> > Is my analogy transparent enough?
> >
> > --Joshua Zucker
> >
> >
>