
Re: How teaching factors rather than multiplicand & multiplier confuses kids!
Posted:
Nov 11, 2012 6:00 PM


On Nov 10, 2012, at 5:44 PM, kirby urner <kirby.urner@gmail.com> wrote:
> What do you mean by formal reasoning? > > Is that with some special notation, or are we being treated to a > display of it here? > > Reasoning is often taken up in speech and debate, when coaches go over > the structure of a rational argument. > > Is that what you're talking about? Rhetoric?
No, not rhetoric. Even though some rhetoric is based on formal reasoning, formal reasoning is not a required element of rhetoric, persuasion is. Formal reasoning is the ability to understand and work with an abstract theory. It is pedantic (for lack of a better word). Persuasion is not an element of formal reasoning.
Note: All theories are abstract so "abstract theory" is redundant.
>> Fragments of group theory fit easily in Algebra 1 and make >> both better, but because Group Theory as a whole is quite >> sophisticated it gets thrown out until after Calculus. >> >> >> I am good with the extra for experts inserts, but you have to get the >> algebra first before you can teach the student the why behind it, otherwise >> it is just pretend. Besides, Clyde will tell you that since they are doing >> algebra, and since algebra is controlled by group theory, they understand >> group theory. So, no reason to teach group theory twice.:) >> > > This is reinforcing the stereotype that you need to be especially > gifted to get it in a different sequence.
No, I am only saying that you have to get algebra before you attempt to get the theory of algebra. In the context of my statement, expert simply means that the kid got algebra fairly well.
> The segment I'm talking about involves distilling the totatives of a > number N, using the GCD algorithm and then showing how totatives > multiplied modulo N have > > (a) Closure > (b) Associativity > (c) Inverse elements > (d) an Neutral element > > (CAIN  plus this group is also Abelian (Biblical pun)). > > Explain what each of those properties of a group means. Play around > with more of them. It's simple stuff, easy peasy. Amenable to > "gamefication". > > This is what's called "spiraling" by the way, where you go into > something a little, from one angle, and then get into it more later, > from another angle. John Saxon was emphatic about "spiraling".
Yeah, well that isn't spiraling. Spiraling starts with a foundational treatment and the "spiraling" occurs after that, not before. What you wrote would be called enrichment or extra for experts.
>> Junior is saddled with the same >> linear sequence is grandparents had. Is that a good thing? By >> definition? >> >> >> Well, Junior is just human, like his grandparents. Naturally, the >> progression would be the same, right? >> > > Is this what you call "formal reasoning" then? > > You seem to consider yourself a good example of what your favored > curriculum would turn out.
I would like more students to think rationally.
> > I assume we're being treated to an example of what "reasoning" means, > am I safe to assume that?
Yes.
While a forum like this would be classified as rhetoric, my rhetoric is based mostly on formal reasoning and yours mostly on ideology and word play.
I have this theory that there is a natural progression to all of this. When we lay down the ideas and theory of mathematics in a student, that process inevitably follows the same pattern in which it was discovered by humankind. I am not saying that we must visit every success and failure of the past, but the progression of sophistication is the same. This isn't a unique theory, it is known fairly well in art and music. When you study art through history, the topics like form, shadow, foreground, perspective, appear in the same order as they do when we teach art. This is because children begin with the same primitive understanding of art as did primitive adults in the past. And they advance through the layers of sophistication in art in the same way that the generations of artists did, in the past. Whether it is one child learning art, or the whole human race discovering it, the progression is the same. It is quicker for the modern child (it doesn't take 100's or 1000's of ye! ars) because they do not have to discover it as did humankind, we are able to teach it to them. Otherwise we would stay forever at square one. Irregardless of the fact that we can teach, we are still bound by the same natural progression.
Let me explain it another way...
Suppose that a calamity befalls the earth and one million people manage to escape and settle on a distant planet like earth. Assume that this new earth is as abundant in natural resources as the old earth was, and also assume that these settlers have brought with them all of the knowledge we currently possess. They have brought books, computers, iPads and DVDs filled with all of our knowledge. But that is all. They were unable to bring our infrastructure with them, just the knowledge of our infrastructure.
How long would it take these people to produce an iPad?
Keep in mind that they brought numerous iPads with them, so they not only know how they are made they have actual examples of them. BUT it will take them many decades if not 100 years to manufacture the first iPad on the new earth. In fact, the iPads (and the computers) they brought with them will probably be dead and gone by that time, unless they take some extraordinary steps to preserve them.
The reason it will take this long is because of the nature of technology. Even though we have the blueprint we will still have to build all this technology up again and to do so we will have to retrace the steps in roughly the same order. First we will have to meet our basic needs, water, food, and shelter. And this doesn't mean we will start by building a Publix and a Walmart. We will be primitive. We will have to craft tools. Wood, then stone and then iron. This will go on and on, each iteration providing something to build upon in the next. And we haven't even reached the industrial age yet on this new earth. We will have to locate iron ore and reinstitute the primitive skills to process it. Internal combustion engines? I seriously doubt it. We don't even have the tools to drill for oil, let alone the refinery in which to distill it. In the beginning we will rely on animal power (if we have animals). Eventually, over time, iteration after iteration, the technological infr! astructure on the new earth will grow. Eventually, we will get our steam engine. Things will start to pick up speed now, but we are still a very long way from building a fabrication plant capable of making the chips that go into an iPad. We will get there, eventually, but we will have to retrace the steps because each step was only possible from the step before it.
When a child begins the journey of acquiring the theory of mathematics they are very much like our settlers on the new earth starting from scratch. In fact, as the child advances in mathematics aren't they advancing the state of the art of their own thinking? They are building a sort of technology, a technology of mind. Just like are settlers, they have to go through the progression and just like our settlers they have to linger a bit at each iteration to ensure that the infrastructure they are building will support the next iteration.
The settlers already had the blueprint for technology. They knew the journey. When teaching a chid however, it is the teacher that holds that information. They have been there and done that. It is their job to guide the child on that journey.
Bob Hansen

