Date: Feb 5, 2013 9:10 AM
Author: GS Chandy
Subject: Re: Proving a definition of multiplication (wrong) by induction
Robert Hansen (RH) posted Feb 5, 2013 4:50 PM (GSC's remark interspersed):
> On Feb 4, 2013, at 9:21 PM, Jonathan Crabtree
> <firstname.lastname@example.org> wrote:
> > Surely a definition is not a 'word problem' given
> by teachers designed to be ambiguous in order to
> force parents and students to work out what it means?
> Actually, that is "the process" in the context of
> "teaching". The student must figure things out.
> Without context, natural language is ambiguous. It is
> that way because it can't be any other way and be
> compatible with how the human mind works.
With or without context, 'natural language' is ambiguous (which is also one of the reasons for its extraordinary power and poetic beauty). 'Natural language' is ambiguous because it leaves largely unclarified and therefore ambiguous the inter-relations between the factors of the system(s) under discussion. Of course, 'providing context' does help to remove some of that ambiguity - but by no means all of it or enough of it to enable us design systems. Which is the main reason for the extension of our 'natural language' that I call 'prose + structural graphics' (p+sg). To a significant extent, models developed using p+sg help reduce the ambiguities that otherwise often plague our understanding of the complex systems we need to deal with.
> To Dave's
> point, no teacher would teach multiplication, in any
> manner, without providing the associated context by
> writing on the board a+a+a+...+a. There are only two
> important elements in that context related to
> multiplication, a and b times. For the life of me, I
> do not understand why you think introducing a third
> element at this point, (b - 1), is helping the notion
> of multiplication in any way.
> Also, I can't find any reference in my son's math
> book of "added to itself". I'm not even sure that
> this is even a popular expression. At this point, it
> is looking more like a curio than an established
> practice. What is established however is that a x b
> is a added b times or b added a times. This is shown
> symbolically with repeated addition like 4 x 3 = 3 +
> 3 + 3 + 3, or visually with arrays and groups of
> objects. Nowhere in any of these contexts is it
> easily represented (to students first tackling the
> operation of multiplication) that (b - 1) or (a - 1)
> is involved.
> Wouldn't this (linguistic) discussion of this
> purported figure of speech ("added to itself") be
> better dealt with later (in algebra) after the
> student has sufficient experience? How can involving
> "b - 1" at the beginning of teaching multiplication
> be "better"? Why not simply drop the words "to
> itself" and just say "a added b times"? This seems to
> be the popular way of putting it.
> Bob Hansen
Message was edited by: GS Chandy