On Nov 30, 2012, at 1:38 PM, Joe Niederberger <niederberger@comcast.net> wrote:

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.

R.H says:
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.

Joe, numbers are not quantities. Quantities are physical, they include a number, a unit, a direction, if needed, and for goodness sake, a freaking context. Mathematics deals only with the number part of all that. The rest is physics. I am somewhat confused by your comments above. If you are saying that we must fib in the beginning, like saying "If you owe someone 5 dollars that is like having -5 dollars", then I am way ok with it. I understand the side show going on, where we try to find applications of mathematics to the real world. That is all healthy. However, you won't have this formal system to apply to the real world (or anything else) if you don't develop it. I mean, you can tell kids for years that negative numbers are like owing people money and then ask them what (-1)^{some odd integer} is.

My point was, it isn't like we are asking the students to invent negative numbers (find a route up Mt Everest). The route has already been found. But they still need to make the journey and understanding negative numbers is quite the perilous journey. Is it not?

How long does it take before the student understands that quantities and numbers are entirely different things? That there isn't any such thing as a quantity less than nothing? That numbers, including negative numbers, are entirely a mathematical concept and are only valid in a mathematical model and if that model be a model of something physical, you must interpret the results of that model carefully and within the limits of the model?

How long does it take for the student to understand that? Not to explain it, just to understand it? To be honest, I doubt I could have ever explained it without the fact that I majored in physics. But I definitely studied numbers just for the sake of numbers since I can remember. I liked how they added and subtracted and multiplied etc.

Bob Hansen