Re: At what point (age) does learning a new language become futile?


Brian M. Scott wrote:

Read it again: I said that it *can* be difficult under those
circumstances. Of course it isn't always, or there would be
very little undergraduate mathematics that I could teach
effectively, since I had no trouble with any of it. In fact
I've been teaching a wide range of undergraduate mathematics
for over 30 years, and one of my stronger suits is working
out what a student's misunderstandings are and finding
explanations that work for that student. Many potential
trouble spots are easy to see, whether one had difficulties
there or not. None the less, there are some very basic
topics with which I cannot do this nearly so effectively,
because I sometimes -- not always -- simply cannot work out
what the student's misconception is.

Hmm. That's where empathy comes in, I think - all those
soft, squishy emotional things that are completely
impossible to quantify (or to teach).

I specifically picked an example in which empathy is not the
determining factor. The issue is specifically the area in
which the student is having trouble.

Yes, but how can you tell, without empathy, exactly *what* part of the
concept the student is having trouble with? I've had some students who
had astonishingly weird - and wrong - ideas about what the concept was
supposed to be; being emotionally attuned to the student (hmm....
intellectually attuned to the student? That's probably more accurate)
is what helps you find those weird misconceptions. In many cases, the
student is completely unable to tell you just what the trouble is, or
what he's misunderstanding; without the ability to see things through
his eyes, as it were, there's no way you can teach him anything.

Also, I think that experience matters quite a lot. I've
been a tutor for more than 10 years now, 3 of those years
full-time. I've had just about every kind of question
one can ask about grade school math by now. It's a lot
easier to work out what the student's difficulty is if
one has seen 10 other students with similar problems.

And having seen the same difficulty umpteen times before
doesn't do a damn' bit of good if on this occasion none of
the techniques that you've developed to deal with it works.
When you've tried every way that you can think of to get an
idea across, and you've run out of ideas, and nothing's
worked, you're well and truly stuck.

Yes, but having more ideas and more explanations always helps.

The best teachers I have encountered
are the ones who have a truly deep understanding of the
subject they're teaching; those people are almost always
the ones who were good at the subject in question as
kids.

Of course. That group also includes some of the worst,
however.

You know - I'm not sure of that at all as far as
schoolteaching is concerned. You teach college math,
right? You see a totally different set of instructors
than I see, as a tutor on (mostly) the school level.

Oh, I see some of them too -- before they get into the
schools!

You have my sincere sympathy.

It's true that the phenomenon is much more common at the
college and university level, but I've seen it at the high
school level as well. Indeed, I can think of a near example
from my own high school: he was uncommonly knowledgeable for
a high school math teacher, and a very nice, hard-working
man, but while he certainly was not a horrible teacher, he
also wasn't extremely effective with weaker students. He
just didn't have the knack. (I'm ignoring elementary school
simply because elementary school teachers with a deep
understanding of mathematics make hen's teeth look common.)

I suppose; I guess I was thinking more about the elementary-school
level. I've been fortunate enough to know two really really good
elementary school math teachers - people who possess a truly deep
understanding of mathematics, and not just at the elementary school
level. You don't go into elementary school teaching if you don't know
how to teach and don't love teaching - and if you also have a profound
knowledge of the subject you're teaching, you will do wonderfully.

And I do agree with you - the reason that most of my clients are
elementary-school or middle-school students is because there are almost
no elementary school math teachers who actually know math. One of my
students told me that her math teacher counts on her fingers - and
sometimes, miscounts and gets the answer wrong. The mind boggles.

(And of course almost the whole of
his knowledge of mathematics was completely irrelevant to
the arithmetic of fractions.)

Sure; but arithmetic is still part of mathematics.

This is almost wholly irrelevant to the question.

Not entirely; surely a mathematician should at least have
a vague knowledge of how to add/subtract/multiply/divide
fractions? At the very least, to understand how the
process works?

The point is that one doesn't have to be a mathematician to
understand how the process works; a very large number of
non-mathematicians are at least as well qualified to explain
the arithmetic of fractions, conceptually as well as
mechanically.

Yes, yes, yes - my statement was more along the lines of "Every
mathematician should know how to explain fractions", rather than "There
exist non-mathematicians who know how to explain fractions". In other
words, the set of all mathematicians is a sub-set of the set of people
who understand fractions - or at least (I fervently hope), it should
be.

LM

.

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