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Discussion: Traffic Jam Applet tool
Topic: Fractions, concept and calculations
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Subject:   RE: Fractions, concept and calculations
Author: Mathman
Date: Nov 27 2004
On Nov 27 2004, mgre wrote:

Just a couple of point if I may, the jist of which is that you are neither right
nor wrong, but must deal with whatever situation you are involved in as unique,
allowing that other situations are equally unique.  It's not a matter of
avoiding theory, but realising the depth to which it is needed in different
situations.  I can talk a good deal more about our experience vs theirs, and how
to balance that, later.

> This is the first year I've been asked to teach math

I taught it for 30 years or so, and studied, understood [as best I could], and
applied it long before then.  Like yourself, I'm self-taught in computer
science/programming, and went on to teach that and help others as well ...and
still do.

> and am teaching a special ed class of grade 9 students who lack
> understanding in many areas.

Two points here, if I may:  They are special ed, not you.  You can not
necessarily pass along your own learning experiences, or manner in which you
learned to them.  I also taught students who had great learning difficulty [and
enjoyed them thoroughly.]  An example: A boy asked how to do certain problems
using the calculator.  I showed him. He asked for another demo, and I showed him
again.  Got it!  He sat down and happily worked through the whole period.  The
next day:  "Sir, would you show me how to do that again?"  No problem, and he
went to the task again.

We had already gone through the "theory" several times as well in different
situations, and htey had been studying, or had at least been exposed to the
material over a period of some years.

> I've been working a lot with fractions.
> >I'm with you all
> the way until you got to memorizing with or
> without understanding.

I've been told I might have some capability in the subject [rightly or wrongly].
In unverstiy, stuck on one particular item, I could have stayed there until I
thoroughly understood it and beat it. [I DO like to beat it,and not have it beat
me.]  But time was passing.  I had to move on, or lose track.  Cutting it short,
by the end of the year I did understand that which had been missing, as it fit
into place.

The point:  Do we wait until they DO understand, or move on?  The point is not
to simply let it go, but to come back to it the form of private tutoring,
not just class-time... from time to time.  It is not to be ignored, but
neither should it slow down progress of learning at that time.

>Of course, I keep my mind open for understanding and after
> using the model I'll often have an "aha" experience - "Oh, I get it!
> That's what's happening!"

>If I don't memorize or use a model I
> don't really understand, I can't move on. I'm mired in my non-
> understanding. So I think sometimes we have to trust that
> understanding will catch up.

That is precisely the point I am trying to make here.  We all go through that at
one point or another, and the argument [discussion] is to say that sometimes we
*must* move along, or lose all for the want of one.  And so we allow the
students the same privilege out of necessaity times.

>All this is not to say that we don't
> try for understanding.

Of course we do, but we don't beat our heads against the wall.  I tried forever
with one adult student with learning problems to get across the idea that a
diagonal of a rectangle is longer than the other sides.  He could understand
only when several were measured that this might be so.  The problem is that not
all things can be measured.  Some need to be calculated to advance beyond grade
2, and so there is some larger, deeper understanding necessary.  But some of
that comes with experience over the years...before the "Aha!".

I worked also in the trades [industry], and taught those who would.  There is no
way, shape or form that they would begin to understand, or want to understand
the mathematical theory of conic sections.  But that didn't stop the carpenter
from learning how to draw an ellipse using a piece of string, or with two
concentric circles.  Nor did it stop him from building spiral staircases using
the carpenter's square [take a really good look at that one some day.]  Neither
did it stop the tinsmith from cutting pieces to fit, nor the draftsman form
doing layout drawings.

I know a girl who has brain
> damage and needs to be taught over and over again before she "gets"
> something.

She is not alone.  We are all there at one point or another.

We don't need to be a mechanic to learn to drive a car.  We do need to learn all
about mechanics if we want to be a mechanic.  Not all are mechanics, nor need to
be.  We need to recognise what might be the most useful, and for some, the
theory of mathematial processes can be nothing less than frustrating.  So don't
expect all to learn equally, or equally well.  We all have our gifts, and we all
have our limitations.  That is not to deny effort, but to avoid sheer

From a LONG time back:  An Australian friend talked of a university prof who
*almost never* had a failure.  His method was to teach how to do things, and
then follow that up with the why.  The truth is, how many of us examine on the
"why", rather than the "how"?  it is the "how" that we use daily.

Just for fun, and to indicate depth to which we should go or not to teach some
aspects of fractions, how would you approach this one do it yourself,
...and how would you expect young learners to approach the same?  This is not a
"trick question", but a matter of balancing our wider experience against theirs,
and this is a trivial example to make the point.

(2 1/2) / (3 1/2) =


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