Re: Sixth grade science teaching

Robert Grumbine wrote:
> In article <1127770961.697789.52400@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
> widsith <widsith@xxxxxxxxx> wrote:
> >


> >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.
>
> Mental frameworks are important. With the framework, the elements
> have a place to 'hang', so are retained better. Even after the elements
> are lost, with a framework, they're easier to re-hang. On the other
> hand, a pure framework can't stand. If the only thing you have is,
> say, principles of what multiplication means, but never hang facts
> on it (say that 8*7 = 56), that framework collapses as soon as you
> turn away.

Indeed. I am not arguing that the framework should be taught without
the facts. Rather, it is my position that they should be taught
together. That some portion of the theory, enough to provide that
framework, should be provided _simultaneous_ with the facts. In the
question of a multiplication table, that might be as simple as
exploring with the class the diagonal symmetry and the the fact that if
8*5=40, 8*6=48=40+8=8*5+8. Give them some understanding of how the
elements of the table tie together to make a table along with the
elements themselves. This is in contrast to the claim by others that
"they can learn nice theory later."

I am not suggesting that we should start with Peano Axioms in first
grade. Nor am I suggesting that we need to provide the concepts of a
Group or Field when introducing multiplication. I am concerned that if
the elements are delivered as independent facts, completly divorced
from the theory then when the theory is introduced later on, the
integration of the theory with those facts will be that much harder. I
am suggesting that the student should be exposed to some of the ideas
behind multiplication at the same time as the elements of the table.
Even if it is a fairly small portion of those ideas, it gives them the
understanding that there is something tying these items together and
when that something is examined in more detail, they have a better idea
of where it fits with what they have already learned.

<snip>

> >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).
>
> Or 196 - 14 (14 being the square root of 200, almost, is useful, so
> I hang on to that better than 13 squared, though I get that too), or
> 13*14 = 130 + 13*4. One trick of rapid mental arithmetic is to
> start with the most significant digits, rather than, as we're taught in
> class, the least. Has the advantage that your partial answers are
> increasingly accurate, and start out on the right order of magnitude.
> Truncate/round/hedge at liesure.

Exactly. But if you have learned multiplication as a mechanical
process, reversing that order will be very difficult. Even if you have
learned theory later.

> There are entire books, some still in press (or at least in bookstores),
> devoted to 'shortcuts' in mental arithmetic. At various times, such
> things were taught, though perhaps not the entire book level, in
> math classes. You'll notice that we're not recommending that. (Memorizing
> huge tables and families of things to speed mental arithmetic ... ouch.
> I've redeveloped a number of them, it turns out, because I often
> do mental computation. But that's me, not a skill for most people
> in most of life.)
>
> We're agreed that it is a results matter -- sufficient math facts
> and principles to develop the answer to elementary addition and
> multiplication problems with adequate speed. For some people, the
> most effective way of getting there is to put the entire table into
> memory. For others, a handful of pairs and principles are sufficient.

True. And Im not arguing that a handful of pairs and priciples are all
that should be taught. The elements of the table should be memorized.
(Some of us take longer to do so than others.) But I believe so should
some of the structure of the table, and they should be learned at the
same time.

The only reason I have brought up my _personal_ methods was to disprove
by counterexample the claim (made by others) that, "The *ONLY* way to
to multiplication is by memorizing the tables." There exist other
ways, and some of us have used them functionally for years.

> A key, though, is that at any given point, some people are going to
> find it easier to memorize a full table. The younger you're looking
> at, the greater that fraction.

I have taken some pains during this post to use the term "elements of
the table" to refer to specific items such as "2x7=14" and to refer to
the "structure of the table" seperately. (And by this I mean such
things as the diagonal symmetry, the fact that the fours column is a
skip count of the 2s, etc.) As I said above, I am not suggesting that
students shouldnt be encouraged to memorize all the elements of the
table, but I am suggesting that teaching the "full table" requires
something more than simply the elements of the table.

There is also something of a pet-peeve of mine here: If one is to
teach only the elements and mechanical methods of addition and
multiplication, can we please call the subject something else? Perhaps
"Computation"? It certainly has no bearing on _Mathematics_, any more
than Home Ec has on Chemistry.

> On the other hand, it can run quite late in education. When I
> was taking physical chemistry, a class with mostly juniors in it and
> a prerequisite of multivariate calculus, I was shocked (a few times
> really) at one point. The professor announced that he would be
> showing us the master equations, a set of 4 equations that, armed
> also with an understanding of physical principles, would enable us
> to do all of thermodynamics. 'Now isn't that better than having to
> memorize a separate equation for every single possible situation?'
> he continued. I was shocked by the overwhelming majority of the
> class (pre-meds) chorusing 'NO!'. They seriously did want a massive
> list of things to memorize, rather than to memorize 4 equations and
> understand a handful of principles. Fortunately for me, they did
> not get their wish.

I can only shake my head. Having attended Johns Hopkins some years
ago, I had my own encounters with pre-meds and I am not surprised.
Diappointed, but not surprised. To an extent though, I cannot help but
suspect that such a preference for memorization is an artifact of how
these students have been taught in the past.

.

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