Re: Sixth grade science teaching


Robert Grumbine wrote:
> In article <1127746688.253674.262870@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> widsith <widsith@xxxxxxxxx> wrote:
> >
> >Paul J Gans wrote:
> >> Grace Haliburton <kaosgrace@xxxxxxxxx> wrote:
>
> [snip]
>
> >Neither Grace, nor I, am arguing that memorization is not necessary,
> >but that memorization _alone_, is not the best learning method, and is
> >_fragile_.
>
> Nor are Paul are I suggesting that mathematics _ends_ with getting
> the tables in to memory. More in a moment.

I have actually not thought that you were. I am suspect about what
Paul is getting at with his insistance that memorizing the tables is
the _only_ way to know multiplication. As Grace and my experiences
show, is that this is simply not true.

> >If you are taught things as standalone facts, without
> >context, without relationship to eachother, once you forget, which you
> >will when you dont use it, its lost. If you are taught the item with
> >its context, then that context forms a web between the independent
> >facts and allows for far greater resilliency, even if a few of the data
> >points are lost.
>
> Context, principles (associative, distributive, commutative laws get
> you pretty far in re-developing the tables from a fragment), greater
> context, alternate ways of viewing the process (arrays of skittles,
> cuisinaire rods, areas of boxes, repeated addition, grouping of dots, ...)
> etc. are good additions and ways to provide multiple frameworks upon
> which to hang the facts. The multiple frameworks also make it more
> likely that the facts are remembered for longer and more meaninfully.
>
> The upshot, though, is that that, from where Paul and I are looking,
> these are all just ways of getting the things stuck into memory. It is
> important to get them stuck in to memory, we figure.

I think the context and structure is more important than simply a way
to get the elements stuck in memory, perhaps this is a difference in
expression and emphasis, perhaps it is a deeper disagreement. By
describing the relationships (whether this is done by dot grouping,
repeated addition, arrays of skittles, distributive and associative
laws or something else) one teaches the "table" and why its a table
rather than a list. I agree that one needs to "know" the
multiplication table to do multiplication, but I suspect we may
(actually that Paul and I, I rather suspect that you and I agree with
minor differences in emphasis) disagree as to what it means to "know"
that table. It seems to me that if you know the structure of the
table, you can deduce the elements and will eventually commit them to
memory, and indeed that is my experience. I do not see that knowledge
of the elements will naturally lead to knowledge of the structure, and
my experience with others indicates this. (Q: I forget, whats 6x9? Me:
Whats 6x10? Q: 60. So?) I also think that knowledge of the structure
allows greater scaling of the memorized portion. For example, I see
13x14 as 169+13, since at this point squares to 20 have been lodged in
my memory, rather than (12+10*3)+10*(4+10).

> The reason is
> as exemplified by Grace's calculus example. When you're learning
> the concepts of multiplication, getting 6x7 by adding 7 six times is
> ok. It's a correct principle and leads, albeit slowly, to a correct
> answer. But when you're out of multiplication class and in to, say,
> calculus, it becomes an obstacle to learning the calculus if you're
> taking that kind of time to get the coefficient in the derivative of
> 7x^6. By then, multiplication is assumed. It's an obstacle to further
> learning to not have it in hand.

Yes, multiplication is a required skill for either calculus or algebra.
I would argue though, that all that is required is "knowing" the
answer quickly. If you can count 7s fast enough, fine. (Youll also be
ahead of the game when a doctor or medic trys to test your orientation,
since the standard test involves counting backwards by 7.) In my
experience, students who dont memorize the full table start developing
shorcuts: counting by 14's, then 28+14, etc, as they fill in more of
the table, and so often can do the required math fairly quickly. (This
wasnt my case since I was almost never more than a single
addition/subtraction away from the answer I needed.)

> The speed is a related question. _some_ speed is a good idea. My
> second grade (no special school or anything, blue collar area and the
> town's school) had us learn the 9x9 addition and multiplication tables
> to get through 10 problems in 10 seconds. This was no challenge to most of the
> class, was made a game, was to progress at the student's own rate, etc.
> But it did mean that when algebra started several years later, we weren't
> still wrestling with the arithmetic aspect of doing algebra. We could
> focus on that mysterious (for most of the class) matter of having
> abstract symbols and to be carrying out inverse operations. Difficult
> enough on its own. But there was no purpose or inducement to getting
> speed beyond that point. A modest degree of speed is useful. A whole
> lot of speed, as in your year of 50 problem races, is just silly.

I thought there was a lot more emphasis on speed than was necessary.
Of course, there was a fair bit in the news around then about kids from
Japan or China who were lightning calculators, so that might have
influenced the fad du jour.

When I approached Algebra, I was still doing some extra calculating,
but was still one of the top of the class. (Given how I managed to
figure the parts of the table I didnt know, one might expect that.) By
Calculus, I was good on everything except 7x8=56 and 9x6=54. For some
reason I always mixed those two up. (I did get it right here didnt I?)
I dont think the difference in my computation ability caused any
noticable difference in my experience in those classes. (I admit
however that I might not be the best example.)

.

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