[QUOTE=R.D. Silverman;285527]And would someone [i]please[/i] confine sm88 to posting his "stuff" to the misc. math. thread?? He isn't helping at all, and will confuse the accolyte.[/QUOTE]It seems to me that sm88 is also an acolyte who has been exposed to the Socratic method both here and in other threads. He is a self-confessed acolyte who is now beginning to understand how Socratic teaching works:[quote=sm88]thanks for this comment it spurred me to look up those preliminaries again on my new computer. I think I get a little more of the puzzle of functions because the notation looks similar to sets as it's performed on sets and it also now looks familiar to the notation of equivalence classes which kinda helps I think. [/quote]

Yes, he may be blundering and he may be confusing other acolytes. However, that is what acolytes do. It's all part of the process of becoming a master in turn.

Paul

[QUOTE=xilman;285538]Herein lies a problem, at least as far as I see it. You, I, Bob, Tom and doubtless others reading this thread recognize the Socratic style when presented and we use it when attempting to educate others. Unfortunately, Socratic teaching is not that widespread and it is almost unused in primary and secondary educational establishments. At least, that is my observation.

A novice unused to this technique seems to interpret questions which are intended to indicate a course of enquiry and self-education as if they are belittling the novice in the eyes of on-lookers. The impedance mismatch then tends to cause more heat than illumination.

Perhaps some of us --- Bob and myself, amongst others --- should try to keep this phenomenon in mind and to tailor our use of the Socratic method of education to our likely audience. That's not to say we should avoid it --- certainly not --- but to recognize its limitations when the subtlety is beyond the intended audience's present level of sophistication.

Paul[/QUOTE]

The Socratic method gets the student to [i]think[/i] about what is
going on. As Kingsfeld said: "Questions and answers. Questions and
answers". Good teachers get the students to [i]think[/i] as opposed
to memorize. Indeed. There is in on-going discussion now in sci.math where
some college level teachers are asking: "how do you handle students
who merely memorize proofs, as opposed to learning how to construct them".

When the student is required to answer questions, he has to think about
what is going on. This is in contrast to a pure lecture style. Students
tend to learn by rote from the latter. The result of this is that it becomes
impossible for them to solve problems that they have not seen before.

One can not acquire [b]reasoning ability[/b] by memorization.

[QUOTE=R.D. Silverman;285542]The Socratic method gets the student to [i]think[/i] about what is
going on. As Kingsfeld said: "Questions and answers. Questions and
answers". Good teachers get the students to [i]think[/i] as opposed
to memorize. Indeed. There is in on-going discussion now in sci.math where
some college level teachers are asking: "how do you handle students
who merely memorize proofs, as opposed to learning how to construct them".

When the student is required to answer questions, he has to think about
what is going on. This is in contrast to a pure lecture style. Students
tend to learn by rote from the latter. The result of this is that it becomes
impossible for them to solve problems that they have not seen before.

One can not acquire [b]reasoning ability[/b] by memorization.[/QUOTE]I agree with everything you wrote there.

I'm suggesting that we continue to persuade students to think. I'm arguing that because Socratic reasoning isn't widely recognized, much less practiced, we should sometimes be more blatant as to [b]why[/b] we are acting the way we do.

Lest I be accused of hypocrisy, I'm well aware that my own style is often too oblique for the intended audience. Mea culpa. All I can do is attempt to improve.


Paul

[QUOTE=xilman;285551]I agree with everything you wrote there.

I'm suggesting that we continue to persuade students to think. I'm arguing that because Socratic reasoning isn't widely recognized, much less practiced, we should sometimes be more blatant as to [b]why[/b] we are acting the way we do.

Lest I be accused of hypocrisy, I'm well aware that my own style is often too oblique for the intended audience. Mea culpa. All I can do is attempt to improve.


Paul[/QUOTE]

Did we scare off the O.P.? Was the scope of the required effort too much?

[QUOTE=R.D. Silverman;285574]Did we scare off the O.P.? Was the scope of the required effort too much?[/QUOTE]I don't know. In an ideal world the O.P. would post a reaction to the subsequent discussion. Have you tried a PM to make that suggestion?

[QUOTE=CRGreathouse;285400]x^(-29) is a rational function, not a polynomial. Polynomials don't allow negative (or fractional) exponents.[/QUOTE]

In general, yes. But (I'm not telling you, but the O.P.) there are instances
when they do allow negative exponents. When they are elements of a
finite field, for example. Which is why I said that the domain matters.

[QUOTE=xilman;285538]Herein lies a problem, at least as far as I see it. You, I, Bob, Tom and doubtless others reading this thread recognize the Socratic style when presented and we use it when attempting to educate others. Unfortunately, Socratic teaching is not that widespread and it is almost unused in primary and secondary educational establishments. At least, that is my observation.

A novice unused to this technique seems to interpret questions which are intended to indicate a course of enquiry and self-education as if they are belittling the novice in the eyes of on-lookers. The impedance mismatch then tends to cause more heat than illumination.

Perhaps some of us --- Bob and myself, amongst others --- should try to keep this phenomenon in mind and to tailor our use of the Socratic method of education to our likely audience. That's not to say we should avoid it --- certainly not --- but to recognize its limitations when the subtlety is beyond the intended audience's present level of sophistication.[/QUOTE]I recommend that Socratic questions should be accompanied by a short explanation that the question is being posed only for the purpose of socratically blah, blah and blah.

[QUOTE=xilman;285575]I don't know. In an ideal world the O.P. would post a reaction to the subsequent discussion. Have you tried a PM to make that suggestion?[/QUOTE]

I sent a PM. No reply.