Robert Hansen (RH) posted Dec 31, 2013 6:43 PM (http://mathforum.org/kb/message.jspa?messageID=9353461): <snip> > > 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.? > Indeed. Why not get together with some appropriate stakeholders (teachers; students; parents; others) and *design* some effective courses for them? > <snip> > > It follows from a simple principle of pedological > development that I will call Hansen?s law. > "pedological" (???!!!)
I'm more than 100% sure that, in this eagerly awaited "Hansen's Law" that you've proclaimed, you're NOT really discussing "pedological development" (whatever desperate thing THAT might mean!!) I do have serious conflicts with RH on a whole number of issues, but I am sure he is NOT interested in "pedological development".
Let's assume you really mean "pedagogical". In any case, your 'foundational theory' to the effect that "Children must be PUSHED to learn math" [Theory 'A'] (which I would guess underlies your "Hansen's Law") is just so much rubbish.
If "pedological" is just a slip of the typing-finger, your Theory 'A' is truly a serious error in understanding how humans (including children) learn. If you really wish to accomplish anything at all in education, you do need to get rid of this foolish idea. > > 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. > I personally believe that the *best* definition of "mathematical maturity" would come about through the development of appropriate math courses for various grade levels. Currently, the genuine Montessori approach has done that for primary school levels (MORE than 100 years ago, I believe - and our educational systems have not yet learned how to use that development!) I don't believe the Montessori system (or equivalent) has been developed for higher levels. A most challenging task awaits teachers, math educators and others! > > How do we progress from the repetitive and rote > instruction of arithmetic to the formal process of > theory building. > > Bob Hansen > Fundamentally, the way this could be done is quite simple to articulate (though it will not be quite so simple to do/ get done in practice):
Just put up the following 'Mission' (or 'sub-Mission'):
"To progress effectively from the process of 'teaching arithmetic' and enable learners to understand 'theory building' (and to actually build effective theories as needed in their own minds)" (This would probably be a minor 'sub-Mission' to the 'Mission': "To develop an effective math educational system").
- - and then learn how to put together (*integrate*) the great many good ideas that are already available out there amongst the stakeholders!
AND, of course, ALSO learn how to get rid of the bad ideas that are also floating around: some of these bad ideas we have seen floating around right here at Math-teach - details can be provided, if required.
Given some effective, practical tools needed to do all of the above-noted *integration* of good ideas (AND "getting rid of" bad ideas), this should really not take more than a couple of years to accomplish to the full satisfaction of all stakeholders! (This can be absolutely guaranteed; and it won't cost even a tiny fraction of all the HUGE time and money that are today being expended on improving 'math education', or STEM [or STEAM], or whatever).