On Dec 30, 2013, at 2:11 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> You can easily find that the usual distributive law no longer holds. But so what? If we took the distributive law as a given (axiom) then we cannot use this new sign law, but if we are investigating a possible new system, we cannot assume all the results that follow from the starting points (not detailed here) of the old system. > (But, are there slightly different but similar laws that do hold?) > > In short, your example of "inconsistency" is too narrow, and amounts to a kind of circular reasoning. Of course one cannot introduce new laws or identities willy-nilly into a already established system without possibly introducing inconsistency, but the usual system of integers in arithmetic is not the only possible consistent system hat uses the signs +, -, x, /. > > So the question becomes -- is there a scheme for introducing "/" with signs that is compatible with the above sign laws for "x"? Obviously order of operands will now be important, although we are not used to thinking about that too much in ordinary arithmetic. > > That is mathematical thinking, and explanation such matters ("what is consistency?") should be part of an introduction to mathematical logic.
But we are talking about 8th graders, not Euclid. I didn't mean formal consistency and my statements were pedagogical, not mathematical. I mean whatever informal consistency is available to an 8th grader. Sometimes that takes the form of just ?It seems to work.?
I hear your argument. My flimsy informal definition of consistency isn?t formal. But it suffices for high school aged students.
It follows from a simple principle of pedological development that I will call Hansen?s law.
In the course of the development of mathematical awareness, do not ascribe levels of understanding to activities when those activities can be performed routinely without said levels of understanding.
And I am talking about the progression of mathematical awareness in students, not what math looks like to middle aged adults.
Walking high school students through another "consistent system? is like playing the same song in a different key. To you or I this activity may be an insightful example of what makes a song a song. But you and I have a lot more experience. To immature students, all you did was raise all the notes a fixed number of steps. And while we are on this point, and I don?t know how to say it other than to just say it, there are reasons as to why the song was written in the key it was written. Reasons that are inherent to the creative process of song writing, and inherent to the creative process of mathematical theory building. Yes, if you had originally heard the song in a different key first, you wouldn?t know the difference. But that notion is only applicable to listening to other people?s songs. When you create a song, it has a key, and it isn?t arbitrary. Suffice it to say that part of understanding a body of work in higher mathematics is recognizing its key. Then you can recognize its theme.
All I am doing here is trying to better define "mathematical maturity? and the nature of its progression.
How do we progress from the repetitive and rote instruction of arithmetic to the formal process of theory building.