```Date: Dec 1, 2012 5:31 AM
Author: Robert Hansen
Subject: Re: Some important demonstrations on negative numbers

On Nov 30, 2012, at 10:12 PM, "Clyde Greeno" <clydegreeno@cox.net> wrote:> "6 + 4i" means the (6,4) long-lat location in the coordinate plane or on the globe ... or the full-planar vector that carries (0,0) to (6,4). "153" means means a specific point on the (Peano) line of Arabic numerals. The "power analysis" of that numeral is a 3-dimensional vector whose base-8 value is 107.Now you are getting it Clyde. You can apply all sorts of meanings and interpretations to numbers. Wouldn't that alone suggest to you that numbers themselves don't have any particular preference other than that they retain their mathematical (field) properties.> Numbers are devoid of meanings only for persons who were so-educated (or blindly or mis-educated)Well, they have their mathematical meanings, like 1 comes before 2, 2 + 3 = 5, and so on, but, unless you apply them to something, they don't have any specific physical meaning or interpretation.>  ... to deal with numeric formalities, without actually knowing what those formaities are talking about.Don't you mean something more like "... to deal only with numeric formalities without knowing how to apply those formalities..." I have never said that application is unimportant nor have I said that it isn't part of learning mathematics. But application by itself?> We are scholastically taught that "applying" mathematics to the real world is an aspect of "mathematics."I don't think students are explicitly taught this distinction. The mathy ones pick it up. The others fail algebra. What is missing in a lot of today's curriculums (because of the damnation we call "conceptualization") is the main sequence, 1 + 1 is 2, 1 + 2 is 3, ...> It is even more true (though untaught)  that *abstracting* mathematics from the real world is a crucial aspect of mathematics. In fact, that is what enables us to "apply" mathematics ... and to grasp the common-sensibility of its formalities.Well, I don't know how I would approach a 6 year old and start teaching them to count and add without referencing the real world almost entirely, but I know eventually we will be doing the times tables and when we get to fractions we will know how to add and multiply them, as numbers. Around 3rd or 4th grade, the disentanglement starts in earnest but math and its application don't become fully disentangled till around algebra 2.Bob Hansen
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