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Re: CL # 4, Some comments
Posted:
Nov 11, 1996 5:06 PM
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Ralph
I think the distinction that you and Gary are discussing is between the
actions that can be carried out on real objects (boys and striped shirts)
on the one hand; and the numerals that I suspect Gary would say are
constructed by each of us to represent lots of actions we have already
carried out with real objects on the other hand. Once someone drops the use
of real objects as nouns with number names as adjectives and starts using
number names as nouns, that person has already constructed an abstraction.
That is, numbers and operations on numbers form a system in which 5 + 3 = 8
in all cases.
Of course, in constructing that knowledge, each of us has numerous
opportunities to test our knowledge against that of others and of the real
situation itself. I think the recognition that numbers (as nouns) take on
properties that are not always those of the real situations from which
number knowledge is constructed is one of the reasons that young children
often find operations with numbers difficult; it is also why hastening
young children into the use of symbols may have sad outcomes for some
children.
May I request that you ensure that your posted remarks are gender-inclusive?
Helen
>On Mon, 11 Nov 1996, W. Gary Martin wrote:
>
>[Raimi] > > I believe it is useful. Children should always be able to
>> >distinguish the behavior of numbers from the behavior of apples, and to
>> >realize that one is a model for the other, and can therefore sometimes be
>> >misleading. I don't believe you are in any doubt about what the
>> >properties of numbers are, with addition, multiplication, etc., and how
>> >addition sometimes models the mixing of two fluids and sometimes not. The
>> >failure of the model in the case of "adding" alcohol to water does not
>> >imply any ambiguity in the problem "Find 3+5", and children should learn
>> >this early.
>
>[Gary Martin:]
>
>> I think children MUST understand that for 3+5 to equal 8 one is dealing
>> with certain conditions
>
>[RAR] Not at all. 3+5 is always equal to 8.
>
>
>> -- sets of comparable objects, and the sets are
>> joinable in a certain way. For example, (3 boys) + (5 kids wearing stripes)
>> may not be 8.
>
>[RAR] I'll say.
>
>> Otherwise, we're just manipulating symbols. They need to be
>> aware of possible sources of ambiguity in working with 3+5.
>
>[RAR] No; there are no ambiguities in "3+5=8". "Working with" things might
>introduce ambiguities, though, as in the case of the boys and the stripes.
>
>> Further, which is a model for which?
>
>[RAR] The numbers, which were invented by men, are the model for the boys
>and whatever, which were invented by God.
>
>
>> Is it that the situation is
>> misleading when 3+5 not= 8? Or is it that the students don't really
>> understand what 3+5 means if they say the answer is 8?
>
>[RAR] This is hard for me to understand. My view is that 3+5=8, and
>that when a three and a five seem to describe something, and 8 fails to
>describe some related thing, then "3+5=8" is not a model for that
>situation.
>
>
>Ralph A. Raimi Tel. 716 275 4429, or (home) 716 244 9368
>University of Rochester Fax 716 244 6631
>Rochester, NY 14627 Homepage: http://www.math.rochester.edu/u/rarm
==================================================
Helen Mansfield
Curriculum Research and Development Group
University of Hawaii
1776 University Ave
Honolulu, HI 96822
Phone: +1 808 956 9956 (W)
+1 808 593 0920 (H)
Fax: +1 808 956 4984
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