On 10/17/2003 05:10 AM, Tom Wayburn wrote:
> What sayeth you-all? Is not '7 + 8' = '15' by
> *definition*?
No, not by definition. That's not the definition
of 15, not the definition of 8, not the definition
of 7, not the definition of +, and not the definition
of =.
You can't have more than one definition of the
same thing. 9+6 is also 15.
The following are definitely not equivalent:
-- true
-- true and known by rote
-- true by definition
-- etc.
I consider it appropriate for kids in second grade
to learn the addition table by rote (for all one-digit
addends at least).
The definition of 15, if a definition is really needed,
comes from the rules of Hindu numerals: we have
1 in the tens place and 5 in the ones place.
> Or, shall we assume every integer has an immediate
> successor (IS) and argue that 8 = IS(7) according to our god-given
> counting numbers (Axiom 0); and, knowing 2 X 7 = 14, conclude ....
Where is that coming from? Is that mockery? In
this form, it is too sophisticated for second grade
but not sophisticated enough to be correct.
That looks like a travesty of the proof that 1 + 1 = 2.
I know the actual proof, but I suspect few 2nd-grade
teachers know it, and I suspect few 2nd-graders are
interested.
It is appropriate for 2nd-graders to learn various
properties of addition, properties of equality, etc.,
such as commutativity and associativity of addition.
I believe this is non-controversial.
The approach that identifies a few facts as axiomatic,
such that all other facts are derived therefrom, has
been fashionable in mathematics for well over 2000
years.
Whether the axiomatic approach is a suitable model of
"reasoning in general" is gravely suspect. People who
know a thing or two about psychology (e.g. William James)
and how to do science and teach science (e.g. Richard
Feynman) have made a pretty strong case for concentrating
on what's *known* (as opposed to what's axiomatic) and
learning how the various known things are *connected*.
These connected facts generally form an undirected cyclic
graph, in contrast to the directed acyclic graph implied
by the axiomatic approach.
If the axiomatic approach is suboptimal for adults,
it is reeeeeally suboptimal for 2nd-graders.