Date: Nov 30, 2012 1:38 PM
Author: Joe Niederberger
Subject: Re: Some important demonstrations on negative numbers
>First, with no guidance whatsoever, you can try to find a route up the mountain. That would be like mathematics prior to the discovery/invention of negative numbers (as a small example).
You are not getting it - negative numbers were used for a long time while people regularly struggled trying to make sense of "quantities less than nothing". There's absolutely no point in putting anyone through that.
Then some glib showmen came up with terminology like "abstract quantities" -- as if that helps! I view that a lot like your "formal reasoning" - tastes great, ...
I'm saying present it as clearly as we know at this point - negative numbers are a combined concept - magnitude plus a direction. Why? We can discuss conceptual mappings, formal considerations in kid's terms (keeping distributive law etc.), history, etc.
"Quantities less than nothing" = emperor has no clothes.
>But even though we know the path the student must still make the journey and the path is still as fraught with peril as it ever was. It is still full of "old confusions" and the student must develop formal thinking in order to overcome those confusions.
Nonsense - the old confusions such as "quantities less than nothing" can be avoided, or briefly mentioned and discarded, from the get go.