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Topic: Is this an exceptionally hard set of questions to answer?
Replies: 68   Last Post: Nov 11, 2002 7:54 PM

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 Alberto C Moreira Posts: 266 Registered: 12/6/04
Re: Is this an exceptionally hard set of questions to answer?
Posted: Oct 31, 2002 9:06 AM

Kevin Foltinek <foltinek@math.utexas.edu> said:

>Alberto Moreira <junkmail@moreira.mv.com> writes:
>

>> I would postpone the concept of
>> associativity to senior high or so, when they first need to face it in
>> its formal embodiment.

>
>Associativity (and commutativity and distributivity) are needed and
>used much sooner than senior high. For example, distributivity is
>used extensively in the "long multiplication" algorithm.

You don't need to bother with the property, which is merely a
formalization of the intuition. At this level, the intuition of
addition, through counting, is enough. Long division is a shorthand
for repeated subtraction, which, again, is intuitive. For example, if
you divide 7321 by 16 using the algorithm, you single out "73" which
is actually 7300. Dividing 73 by 64 gives you 4x16=64 and remainder 9,
that's actually 6400 + 900. What you are actually doing is subtracting
16 hundreds from 7321 4 times:

7321-1600-1600-1600-1600 = 921

The associativity you need to worry about operates at an intuitive
level, no need to bother about formalizing it. If a student has number
sense, it's jolly easy to point out what the algorithm is doing, but
if he or she doesn't, it's nearly impossible to justify it,
associativity or not.

>There's a little trap there: "1+2+3" does not even make sense until
>you know that addition is associative.

In intuitive terms, 1+2+3 makes all the sense in the world: count one,
than count two more, than count three more. Counting to fifteen is
nothing but adding 1+1+1+1+1+1+1+1+1+1+1+1 +1+1+1. The fact that later
on mathematics models addition as a function that takes two arguments
and barfs out a result is a consequence of the mathematical MODEL of

So, I come back to my point: before the restricted model is taught, we
must sediment the intuition.

>The trouble with intuition is that it easily can be, and often is,
>wrong. Indeed, it is intuitive to many students that (1/2) + (2/3) =
>(1+2)/(2+3).

It is not intuitive, and it's a consequence of the perversity of the
archaic notation we use to represent arithmetic. If you write
(1/2)+(2/3), you are saying "divide 1 by 2, take the result, put in
your pocket; divide 2/3, and add the result to the value you put
inside your pocket. That is what the NOTATION is saying, it has
nothing to do with the intuition of the operations themselves. I could
put it in postfix notation and state it as "1 2 / 2 3 / +", or better,
something like "1 2 / @a 2 3 / @b a b +" and that would possibly
capture the intent of the notation way better: take 1, take 2, divide,
put the result in a, take 2, take 3, divide, put the result in b, take
the value in a, take the value in b, add. No doubt whatsoever about
what's the sequence the operations will be performed.

The point is, we use a CONVENTION that division has higher precedence
than addition. So, 1/2+3 means (1/2)+3 and not 1/(2+3), and that's a
mere notational convention. The intuition is, what the sequence of the
operations ? If I divide before I add, I get one result; if I add
before I divide, I get another result. The notation establishes a
the semantics of the operations themselves. Now, if I have a solid
number sense, I will note that 1/2+3/4 can be ambiguous, unless we
establish the notion of precedence among operations. And I do believe
that confusing notation with concept is a serious problem that many
teachers of elementary mathematics don't seem to be able to rid
themselves of.

>The student who understands the properties of the
>operators is (in my experience) less likely to make this sort of
>error, while the student who merely develops an intuition that
>operations "behave nicely", without understanding exactly what
>"nicely" means, expects that the addition of fractions can be
>accomplished using the "nice" and "intuitive" method.

The student who understands the way the operations themselves work
will have little problem fathoming the ins and outs of notation. The
issue here isn't the understanding of how the operations work, but the
understanding of what the notation is trying to say. Two different
issues !

>I have heard comments that arithmetic and basic algebra appear as
>"magic" to students, that there is some mysterious force involved that
>makes things work. Things are only magic if you don't know what's
>going on, if you don't know the rules.

Arithmetic is intuitive. Arithmetic notation is contrived. There are
two problems here, one is to understand arithmetic, the other is to
grasp our notation of arithmetic. We mustn't confuse the two issues !

>Nonsense. With two rows on an abacus, you can count to 100. With two
>pairs of hands, you can count to 20 (assuming you use the usual simple
>form of finger-counting). If you really think that an abacus is an
>extension of our use of our fingers as counting tools, you don't
>understand our digits representation of numbers (or you don't know how
>to use an abacus).

Counting in tens is enough for elementary school. For example, say we
want to add 74+28. I count 8,9-10-11-12, so, I jot down 2, and carry
one. Now I count, 7, 8-9 (for the tens), 10 (for the carry), and I jot
down the 10 next to the 2: result, 102. Man, I used that when I
learned arithmetic with my father even before I went to school. I
still remember my little first and second grade colleagues counting in
their fingers to figure out the result of an addition, because that
was intuitive enough - and my teachers would bother them nonstop to
get them to stop using their fingers instead of leveraging on that
intuitive impulse.

The important thing is to develop the concept of carry, and do things
one column at a time.

>If you refuse to recognize that the length of a rod is a perfectly
>valid representation of a number once you have agreed on what has
>length 1, fine. Make little cuts on the rods so that it looks like a
>bunch of cubes glued together, then you can count the cubes. Many
>children have no difficulty imagining these cuts (or similarly
>imagining the rod made up of cubes). Apparently you do.

It is obvious, to me at least, that digital feel is a lot more exact
than analog feel. So, length is not a good thing to use, unit is.
Don't glue anything, just count - and here, cubes or rods are as good
as one's own fingers. The difference is, our fingers are at the tips
of our hands all the time.

And if nothing else, in this age of computers, digital intuition is a
whole lot more useful than its analog counterpart.

Alberto.

Date Subject Author
9/28/02 Karl M. Bunday
9/30/02 Alberto C Moreira
9/30/02 Shmuel (Seymour J.) Metz
10/5/02 Moufang Loop
10/7/02 Shmuel (Seymour J.) Metz
9/30/02 Stephen Herschkorn
9/30/02 Magi D. Shepley
10/1/02 Karl M. Bunday
10/2/02 Kevin Foltinek
10/2/02 Karl M. Bunday
10/3/02 Alberto C Moreira
10/3/02 Kevin Foltinek
10/3/02 Jim Hunter
10/4/02 Herman Rubin
10/4/02 Alberto C Moreira
10/5/02 Herman Rubin
10/4/02 Alberto C Moreira
10/4/02 Kevin Foltinek
10/5/02 Alberto C Moreira
10/6/02 Virgil
10/6/02 Herman Rubin
10/6/02 Jim Hunter
10/6/02 Virgil
10/7/02 Kevin Foltinek
10/8/02 Alberto C Moreira
10/8/02 Kevin Foltinek
10/9/02 Alberto C Moreira
10/10/02 Kevin Foltinek
10/11/02 Alberto C Moreira
10/14/02 Kevin Foltinek
10/15/02 Alberto C Moreira
10/15/02 Kevin Foltinek
10/16/02 Alberto C Moreira
10/16/02 Kevin Foltinek
10/14/02 Kevin Foltinek
10/16/02 Alberto C Moreira
10/16/02 Kevin Foltinek
10/12/02 Shmuel (Seymour J.) Metz
10/14/02 Kevin Foltinek
10/25/02 Van Bagnol
10/25/02 Alberto C Moreira
10/26/02 Van Bagnol
10/27/02 Alberto C Moreira
10/27/02 Herman Rubin
10/28/02 Kevin Foltinek
10/29/02 Alberto C Moreira
10/24/02 Van Bagnol
10/25/02 Van Bagnol
10/26/02 Alberto C Moreira
10/28/02 Kevin Foltinek
10/29/02 Alberto C Moreira
10/29/02 Kevin Foltinek
10/31/02 Alberto C Moreira
10/31/02 Kevin Foltinek
11/2/02 Alberto C Moreira
11/2/02 David Redmond
11/3/02 Alberto C Moreira
11/3/02 Alberto C Moreira
11/4/02 Kevin Foltinek
11/2/02 Virgil
11/4/02 Kevin Foltinek
11/5/02 Alberto C Moreira
11/5/02 Kevin Foltinek
11/6/02 Alberto C Moreira
11/7/02 Kevin Foltinek
11/9/02 Alberto C Moreira
11/11/02 Kevin Foltinek
10/3/02 Kevin Foltinek
10/5/02 Magi D. Shepley