
Re: A question about straight lines
Posted:
Feb 1, 2014 11:14 AM


Bob
On Feb 1, 2014, at 9:28 AM, Joe Niederberger <niederberger@comcast.net> wrote:
> R Hansen says: >> 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. >> You can only get all of those details through experience, not a perfect explanation. > > Naive set theory is exactly what is needed to cut through the fog here.
I am not sure we are talking about the same ?fog?. You seem to be after a logical explanation. Before you can explain the fog doesn?t the student need to sense the fog? I am after more sophisticated senses which require time, maturity and experience.
> (Plugs are just a bit of a silly diversion I'll agree with you there. And I meant to say earlier I was *not* creating a lesson plan.) So let's get on with the naive set theory and see if it helps. Explanations, coupled with our understandings of basic concepts like sets, unions, intersections, and combined with following the below, step by step, plus pondering, and asking questions to clarify things that aren?t clear, and working more problems, etc. etc. is what you mean by ?experience?. I agree. > > I'm talking now to Neighbor, he'll have to read up on sets, subsets, unions, intersections, etc. to follow along. I may be asking far too much of him below. He might try reading the Wikipedia page on naïve set theory, and never make it to the exercise below. Oh well, he?s not in my class! > > What?s been at issue here is what happens to the cut point, when the original line is ?to be taken separately?, as two. > > 1. We are going to view a line segment now as a set of points; ?made up of points?, if you prefer. > > 2. Lets pick one point though (our cut point.) We can generate a property that applies to all other points by reference to that one. > Grant me that all other points lie on one side, or the other side, of our reference cut point. > > 3. We will express this mathematically as follows: > Let C be our cut point, our reference point. If A is any point on the line whatsoever, either: > (1) A = C (they are the same point in fact.) > (2) A is on one side of C, which I will write as A < C > (3) A is on the other side of C, and I?ll notate this as A > C > > 4. Now I'm going to reinterpret your question about bisecting a line in set terms: you want: > a. to create two subsets L2 and L3, of L, > b. such that L1 and L2 still qualify as lines, and > c. L1 union L2 = L, and > d. L1 intersect L2 = {} (empty set).
When young students split a segment into two segments, they don?t meet condition (d). They create two complete (closed) segments which meet conditions a, b, and c, but not d. Both segments have an endpoint in common. When you remove that endpoint, I don?t think they know what is left. Because they didn?t really know what was there to begin with.
> 5. Here are some subsets of L, I?m going to expect you to ?translate? from what I write into a more or less English form. I?ll do the first one and third one: > [1] { x in L  x > C } Translation: this is the subset of L that consists of all points, x, in L, such that x > C. Its the line segment you get by ?taking? all points that lie on one side of C. > [2] { x in L  x < C } > [3] { x in L  x > C *or* x = C } This is like [1], except I also include the point C itself in this subset. > [4] { x in L  x < C or x = C } > > I?ll claim without proof that all 4 of these are still line segments of some variety.
(1) and (2) are not ?segments? that I am aware of (and you say that later). Geometry seems to require a sufficient number of particular points (remember our previous discussion of ?particular?). I understand the sets you are referring to, but as Lou pointed out, based on my understanding of real numbers, not simple geometry. I think you are being overly optimistic with respect to how young a child can be and really know real numbers.
> > 6. Think about those 4 subsets listed above, and what they mean *as sets of points*! > > 7. Pick 2 of those that meet the conditions a, c, d, listed in bullet 4. We?ll forget b for later, and just assume they all are line segments in fact. > > Can you do it?
And how does this teach children about points, lines and infinity? All I am saying is that what you wrote here, even after it is translated into english, won?t make any difference to a 12 year old. They can?t put any of this to use. Not until they have a feel for what a line actually is, which doesn?t occur until some real familiarity with real numbers. This doesn?t even start occurring till algebra. And only for a few it seems.
If Neighbor understood real numbers and applied that to points, he would then be able to use what you wrote. He is still struggling with an incorrect sense of what points are. He still has a tendency (as all of us do) to think that two points can be beside each other, with no other point in between. The only way I know of getting over that tendency is to study the real numbers and what we mean by ?point? and ?line?.
Bob Hansen

