Path: utzoo!utgpu!water!watmath!clyde!rutgers!tut.cis.ohio-state.edu!ut-sally!husc6!sri-unix!quintus!ok
From: ok@quintus.UUCP (Richard A. O'Keefe)
Newsgroups: comp.edu
Subject: Re: Learning arithmetic
Message-ID: <680@cresswell.quintus.UUCP>
Date: 22 Feb 88 10:28:50 GMT
References: <26@dogie.edu> <3340@killer.UUCP> <660@cresswell.quintus.UUCP> <8421@g.ms.uky.edu>
Organization: Quintus Computer Systems, Mountain View, CA
Lines: 72

In article <8421@g.ms.uky.edu>, mtbb34@ms.uky.edu (Becky McEllistrem) writes:
> >In article <660@cresswell.quintus.UUCP> ok@quintus.UUCP (Richard A. O'Keefe) 
> >writes:
> >>I think we need three things in elementary arithmetic teaching:
> >>	principles:	"This is WHY the addition algorithm works."
> 
>     And do you lecture that why or show them that why? Do you cut a square
>    to prove two triangles make a square or do you give them some big formula
>    that they won't remember?

If you cut a square in front of the class, it's STILL a lecture.
What I had in mind was something like this:
    a very important thing to get across is that addition is not something
    we made up, but something we discovered, that whether two bricks plus
    two bricks gives us four bricks or ten is not something we have the
    power to change, that whole numbers are defined by counting, that we
    can give whatever pictures we like to numbers and whatever names we
    like, but that doesn't change change the reality.

    Addition is based on counting.  I think it's important to get it
    across that someone who adds 3 to 5 by putting down three fingers
    and then putting down five and counting the result is going to get
    the right answer, because that's an instance of what addition MEANS.

    So now we have a well defined operation, but getting answers is
    incredibly tedious.  Finally getting to the point:  you explain
    WHY the (usual, base-10) addition algorithm works, by showing
    how it is connected to counting.  (What we'd be doing is giving
    an inductive proof that the diagram
	       encode
	(NxN)---------> (pairs of base-10 numerals)
	  | add by		| usual base-10
	  | counting		| addition algorithm
	  V    encode   	V
	  N ----------> (base-10 numeral)
    commutes, but we don't have to use the language of category
    theory to do it!)

    What this should get across, even if the explanation is a bit
    muddled, is that all this carrying stuff is not an arbitrary game
    that somebody made up, but that addition means something however
    we do it, and that this particular method does that.

    There are several steps in this which can be done by the pupils.
    
    The other mathematical operations can be explained the same way.

The really fundamentally important thing in this is really social:
it's the lesson that things like addition *can* be explained, and
that the teacher is telling you something that makes sense.  It
doesn't matter much if the pupils don't remember the explanation
of why the addition algorithm works:  the main thing is that they
remember that there IS an explanation and the teacher is willing
to show them.

In my original message, it was the "New Math" with terms like
"commutative" which are hard for a child to *say*, let alone
relate to its experience, that I claimed was boring and remote.
In my own schooling, which predated the adoption of "New Math"
by 1 year in NZ, we started with coloured rods, worked up to
play "shops" and recipes, and by the time I left high-school I
had a solid grounding in calculus and elementary statistics.
Now I have a 15-year-old cousin in Australia who was only taught
how to draw a straight-line graph last year, and finds word
problems a black mystery.  She's not thick, either.

Someone who *wants* to play the bagpipes will practice for hours,
driving the neighbours mad with endless repetitions of simple changes.
Someone who doesn't want to will find even picking the instrument up
boring and meaningless.  I don't believe that there's any problem with
drills as such; the problem is pupils who have never really seen/been
shown the *point*.