On Tue, Jun 4, 2013 at 1:39 AM, Robert Hansen <firstname.lastname@example.org> wrote:
> > On Jun 3, 2013, at 7:32 PM, kirby urner <email@example.com> wrote: > > Basic finding an unknown, rules of equality, we can call those "algebra > skills" and we can keep weaving in that same material. > > > That isn't algebra. That is the dumbed down version that it became after > decades of mass trying. That is algebra after you strip it of its art and > sense. Algebra is making sense of one or more mathematical relationships > and using that sense as the means to some mathematical end. It isn't any > particular result or some finite set of instructions like the recipe for > banana bread. That is why I don't care how many computers you have. Without > that sense and art you can't apply those computers to "algebra" any more > than a chimpanzee can write Shakespeare using a typewriter. > >
No that's not algebra at all. Al Jabber was a set of methods and techniques developed in Baghdad and filtered into European civilization in such forms as Liber Abaci. People learned how to do what we call "algorithms" around addition and multiplication and so on -- basic arithmetic. Combined with double entry bookkeeping, the stage was set for a nouveau riche to arise, independently of the church and landed nobility. Venice played a big role, as the money was in global trade, shipping. Of course there's more to it: Pisa was a place where Italians competed to factor polynomials as a kind of public sport, with sponsors, like cock fighting, with mathematicians as cocks. This pushed factoring pretty hard and eventually gave us Galois Theory, more of group theory, and abstract algebra.
I think abstract algebra and boolean algebra are great subjects to draw upon for high schoolers and do so a lot. Yes, it's way better than what's in the text books today. I'm a better math teacher than 99% of high school math teachers, when it comes to content. Actually I'll claim 99.99%. I'm the undisputed champion of forward-thinking curriculum writing in many circles, and deservedly so. I'm pretty much the only one with both 'Divided Spheres' and 'Math for the Digital Age' in my syllabus.
I test my ideas, write them up, get feedback, and reiterate. Few have done as much for math reform as I have. I'm very proud of my achievements but of course (obligatory award acceptance line): I've stood on the shoulders of giants.
What "algebra" means is best answered historically, taking the long view. I very much encourage this. Mathematics with no historical perspective, as taught today is ridiculously dumbing down, which is why the average American adult is pig-ignorant unless they've been exposed to lots of stuff besides what they got in formal schooling -- which many of them have, which is why I think there's some hope for them, backward though they may be in global terms.
"Algebra is making sense of one or more mathematical relationships and using that sense as the means to some mathematical end." -- that's just vague preaching, fluff, no content. You should tighten up your thinking. A lot.
> This happens to many teachers. Day in and day out they are going through > the motions of teaching mathematics to brand new faces and they forget the > point of the process. They forget the pedagogy and development or they > never understood it in the first place. Or I suppose they are charged with > really difficult cases. They moan "Why am I teaching kids to add numbers? A > calculator can add numbers!" And I tell them "You are not teaching them > just to add numbers. You are teaching them the sense of ADDITION, and if > you don't use numbers what the hell are you going to use?" We don't need to > teach children that calculators can add numbers. We need to teach children > what "add numbers" means. There is no other way to do that with sufficient > payback (acquisition of senses) than to have children add numbers. The same > applies to algebra. You cannot show students how to solve with tools > (computers) before they have developed the personal sense of what "solve" > means. They have to experience it and the nuances of it. Mentally. >
Right, they need to know (a) positional notation (which is best taught in connection with the physical abacus, or an emulated one) (b) bases, because when you shift over a column, it may not be powers of 10 you're dealing with (c) algorithms, for using these positional representations to do the basic operations.
If your curriculum has no base 2 or base 16, it is garbage, utter garbage. Of course a given resource may skip that or talk about something else, but when you add it all up and find no hexadecimal by the end of high school, you know you're dealing with a basket case school that has no real purpose other than to pay some adult prison guard day care people so mom and dad can both work while junior flounders in the "care" of know-nothing incompetents.
> > You are in a state of euphoria about all the things a middle aged > Princeton educated man can talk about but your chief problem is that you > seem to have no recollection or sense of how you got to where you are. You > want to start these kids at the end of that journey, rather than at the > beginning. Yet, in previous discussions you are quick to defend older texts > like Dolciani. If I were you, I would get a copy of a textbook you studied > as a child and go through it start to finish and try to put yourself back > in that time. Try to remember the discussions and exercises and your > transition from not knowing to knowing. >
Unlike yourself, I go into schools, teach rooms full of teenagers, was a full time math teacher. Your real experience pales to insignificance compared to mine. You make a good dramatic foil though, thanks for the contrast.
> > There is no easy button to all of this. There are more computers in this > world and they are more powerful than you or I would have ever imagined as > children. And everyone has one. There has been no math revolution because > it has nothing to do with computers. It has to do with thinking and being > smart and the technology of thinking and being smart hasn't changed in the > last 50,000 years and will not change in the next 50,000 years. > > Bob Hansen >
I do not agree with this assessment. The artifacts make a huge difference to the civilization. Navigation and spherical trig go together. Google Earth and similar assets change everything. There have been many "math revolutions" over time, and then there's been the idea of "public education" which was another revolution. The saga continues, yes, but it hasn't been some static background with algebra having a fixed meaning. When it comes to thinking and being smart, yes, we need to encourage that, always will need to. I think that's what I'm doing. That may mean pointing out the weaknesses in your analysis, which is parochial and relatively uninformed compared to mine. Sorry, but that's objectively a fact. You should learn from me.