Date: Feb 5, 2013 6:20 AM
Author: Robert Hansen
Subject: Re: Proving a definition of multiplication (wrong) by induction
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. 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.