Monday, January 16, 2006

Profession Hair Color Sold On Line

mathematics education reform and lifestyles

loci: Who cares?


well known is the locus of a point moving always remaining the same distance from two separate fixed points A and B. That is, the point X moves in the plane such that AX = XB or, equivalently, AX-XB = 0. Well, I mean ... well known for one who has ever seen and used many times. This is the bisector of the segment AB, ie perpendicular to AB at its midpoint.

The analytical form of view this result is placed in the Cartesian coordinates of points A and B in the simplest way possible: A = (a, 0) and B = (b, 0). So if X = (x, y), we apply the distance formula between two points for

(xa) ^ 2 + y ^ 2 = (xb) ^ 2 + y ^ 2, where we get

xa = xb or xa =- x + b.

From the first equation gives a = b there is no segment AB because both points coincide (and nothing can be concluded.)

From the second you get x = (a + b) / 2. And this is the result we want.

But this requires analytical result a "translation." First you have to "read" him that if the abscissa (x-point moves) remains constant, then the point X describes a line perpendicular to the axis x (moving parallel to the axis and then
always stays the same distance ( a + b) / 2 of it). Second must be "read" that (a + b) / 2 is the midpoint between A and B. Close

But, right now! this speech is raised from the standpoint of the teacher. Let's look now from the standpoint of the boy of 16 who is taking his first course in analytic geometry. What do you know and what does not? Assuming

understand natural language English, are in any way some terms you may not know:

locus "?
fucking "fixed?
point "mean?
"Cartesian plane?

Professor reflect on these possible unknowns can be paralyzed and conclude that mathematics education is impossible. Also because the current educational reform could be demanding not only learn these concepts
but learns them significantly.

But "significantly" is an adjective with a thousand interpretations ...
and the parent seems to be 1) team building, 2) engage in any activity that creates appropriate, 3) discussion and 4) conclusion ...

And the key to this interpretation of "activity", so that the learning of relevant content (in terms of discipline) has been replaced in practice by implementation of significant activities for students (item of view of experts in education). Neither good nor bad just a trend of contemporary education. Opening 2

But look at this other locus. Details: segment AB constant k, the point X moves so that ^ 2-XB AX ^ 2 = k.

Riddle: What describes locus X?

Development 2 Solution: (for extreme cases)

If k = AB ^ 2 then AX = XB ^ 2 ^ 2 + AB ^ 2 and is (recalling the Pythagorean theorem) that the locus is a perpendicular to AB and B.

If AB =- k ^ 2 then AX ^ 2 + AB ^ 2 = XB ^ 2 and is (again by Pythagoras) the locus is a perpendicular to segment AB but now by A.

If k = 0 then there is the bisector as locus described by the point X.

Of these three extreme cases can develop the assumption that the locus is a perpendicular searched the segment AB. And then there's another idea: k depends on the cross (and the crossing depends on k) of the intersection of the perpendicular to the segment (with the line, rather) AB. (Assume that crosses X ', then k = AX' ^ 2-XB '^ 2.)

is left as an exercise for the reader the analytical demonstration with X = (x, y), A = (a, 0) , B = (b, 0) and k either, where you should get - after doing some algebra - 2 (ab) x = a ^ 2-b ^ 2 + k. As an exercise also aims to "read" here
the geometric interpretation in two parts as in the case of the perpendicular: how do we know that the locus is perpendicular to segment AB? How know where it intersects the line AB? Close


I would like to stress here, as a closing comment, that the activity of problem solving school mathematics there are three well-defined moments: a formulation (analytical or synthetic) of the problem - using data to define a solution plan - a plan monitoring, and interpretation of results should answer the question posed in the title.

And to the question of education expert "what applies to this?", Would respond with "is a workout." And if the experts say: A training and what for Why? Well, this is a cognitive skills training, to
while the trainee is being trained to show you the potential of symbolic reasoning in mathematics.

And if you insist: And all this will serve you in your adult life? Well, it all depends on your lifestyle and what specific practices are given in it ... You do what you have been served not have developed those skills? JMD in VL

greets (and have presented them this picture when he went to eat squash blossom quesadillas to the Faculty of Sciences UNAM - October 2005)

Thursday, January 12, 2006

Feeling Weak Headache

Effect flashback in mathematics education

Effect flashback in mathematics education

For some reason (documented in the literature of cognitive psychology by Daniel Kahneman), many educators believe that the solution of a mathematical problem (school mathematics) can be discovered by any trainee. But it is easy to see that even the teacher can be difficult to discover (or rediscover). These days

was designing a training module on the use of complex numbers to solve geometric problems. At one point she needed to justify (prove) that multiplication by the complex z = r (cost + Isent) executes two actions on another complex z '(z times): the longer times and tour r t degrees.

And needed proof of this result, in turn, the formulas of sine and cosine of the angle sum. A widely used formulas without anyone wondering why they are valid. Are some formulas in some way and "natural" is nauralizado use in solving problems. But considering that the audience targeted by the training module that concerns me is that of teens interested in math contest, then I myself felt the need to demonstrate these formulas. (Here comes the personal opinion, it is difficult to decide in a situation Teaching what to say and what to keep ... but ...)

And I said, "cakewalk." This should be easy ... But no. The key idea of \u200b\u200bthe show did not come, although he was convinced that it should be easy. So I had to resort to a book. Found appropriate by the Dolciani (Modern Introductory Analysis), a very good school math book despite being seventies (I mean the time of axiomatic fashion.)

I had seen and proved the theorem once, so to see the Dolciani experienced a rediscovery and I felt a little embarrassed with myself. Because the key idea is extremely simple! (It is the classic, "expresses the amount of two different modes, retainers and clear") And yes, it is difficult not to conclude - in these cases - that "anyone can come up with."

The reader may consider the following figure as a test "almost speechless" in the formula of the cosine of the angle sum. Just "see" that PQ = P'Q ', applying the formula of distance between two points and clear.

But I wish to emphasize here is that this feeling of "anyone can come up with is a kind of reverse conclusion:" Now I saw the solution seems trivial, so it was always trivial ".

Let me conclude these reflections with a moral teaching. What seems to be the case, trying to escape the "optical illusions" retrospective effect is that an idea, however simple it may seem in retrospect, may be inaccessible to the learner's cognition without the help of a suggestion from the instructor. But you can also say that such ideas are exemplary and should remain accessible in the memory of cognizable interest in the math contest. How? Well, that would be a topic for another post, but it might help to start with an inventory of key ideas for solving problems of competition - the way a chess player tirelessly studying openings, finishes, and middle game tactics. JMD in VL

wish you a happy (or not so unhappy lost) 2006