On Jan 31, 2014, at 4:20 PM, Neighbor <email@example.com> wrote:
>> You probably intended this as a game of analogies >> between two adults already blessed with a sophisticated >> understanding of points, lines and infinity. The >> following is directed at using such accidental >> analogies in teaching? > > Actually, I don't have a sophisticated understanding of points and lines, for if I would, why should I have started this very discussion? I only want to understand.
You are confusing sophistication with formality. You have been studying this situation of points and lines for some time. Long enough to start asking sophisticated questions. You couldn?t answer them because you lacked formal thinking skills, which also involves better reasoning (logic).
> >> How does this help a student? I can?t see how this in >> any way leads to any understanding, sophisticated or >> not, of points, lines and infinity. When I reach such >> silliness in my own pedagogy I STOP. You are beyond >> the student?s thinking. > > Then I will try to explain to you how this can lead to any understanding, and you can agree or refuse what I'm going to say: > Well, first of all, I dare say, and you will agree, that when you want to learn something, and have understanding in that, you first ought to start from the bases of the subject you're going to study, and analyze the bases very carefully, spending a lot of time in their contemplation, for if you misunderstand the bases totally or partially, when you proceed in further inquiries of the subject, you will reach a lot of false conclusions, based on your misunderstandings of the bases. But to properly understand the bases, you need to understand why they are as they are and not different, and to understand this you need rational arguments. My meaning is that he who wants to learn well anything, with proper understanding of the matter, has to understand everything he does in the process of learning, at least having reason in what he does, for if not, if he accepts anything as a doctrine, without the understanding of why that thing is as it is, by reason that he's lazy or by any ! ot! > her reason, I dare say, he will never properly understand the subject, 'cause his knowledge is based on things that he accepted as "doctrines", and he didn't try to understand why they are as they are with rational arguments, and therefore his very knowledge will be a fake knowledge. Would you not agree with me?
What you said here is all generic. Go back and specifically tell me how plugs and glue can lead a student to gain a sophisticated understanding of points, lines and infinity? And assume that the student agrees 100% with your explanation that the point is like glue and connects the two open segments or like a double ended male plug and connects the two segments. What does that have to do with the details of points, lines and segments. You can only get all of those details through experience, not a perfect explanation. And this all takes time.
>> You are beyond the student?s thinking. > > Actually I am myself a student, and I don't think this to be beyond my thinking, of course, if I want to have a rational thinking.
You misunderstood what I said (and chopped it up strangely). I was talking about elementary aged students. Time has a lot to do with all of this (not Crabtree?s version of time). I mean age and maturity. You have been thinking about points, lines and infinity for some time. When I say ?beyond their thinking? I mean beyond their maturity and sophistication.
> >> Just stick with the mechanical definitions of open and > closed segments of real numbers. > > What you mean, if I'm not mistaken, is to stick with the mechanical definitions without the understanding of them?
You are mistaken. I mean that in the early stages, until the student has matured and acquired sophisticated modes of thinking, most of their mathematical experience is mechanical. Later, they keep revisiting that experience with more sophisticated modes of thinking. As you are doing.
> That is to say, take this definitions as doctrines, and accept them without any problem? But don't you see that your future knowledge will be based on doctrines? On something you don't understand, for you only accepted the definitions as true and therefore you don't know why they are as they are and not different. > >> They will get the sophistication in time, and with use. > > Of course they will. But don't you agree that with the use and time they will learn to do everything mechanically, and their wonder of why the definitions they have accepted are true will vanish with time.
Accepted as true? The student should be intelligent enough to recognize that these definitions have stood for hundreds of years and that they have been thoroughly studied. I don?t think that is what you meant to say. I think you mean that the student is confused or has dilemmas. That is a natural part of learning mathematics and the teacher helps them through it. Like we are trying to do here with you.
> > >> Also, would a young student even understand the >> meaning or the source of the meaning for male and >> female plugs?:) >> >> Bob Hansen > > Maybe not, but we need something simple to start with.
Yes, but something mathematically simple, not something else entirely different. There is no way to teach an elementary student naive set theory and it is just silliness to try. You will understand what I mean when you try to answer my specific question. What insight into points, lines and infinity, at all, would a student get from glue and plugs? I suspect everything you offer will be from other exercises and not that analogy. You will realize that such an approach is nothing but an illusion.
Mathematics is not about definitions. Definitions are just part of the language. Mathematics is about experience and thinking about the experience. The language is important, but not in the way you are hoping. That is why mathematical texts start with the definitions and then, through usage and context, explain what those definitions mean. It seems backwards, I know, but it will seem perfectly natural once you get the hang of it.