On 2015-12-21,
malcolm...@btinternet.com <
malcolm...@btinternet.com> wrote:
> On Monday, December 21, 2015 at 12:55:57 AM UTC, Herman Rubin wrote:
>> Throughout history, too much em;phasis has been placed on memorization
>> rather than understanding. Memorization is easy to test, and rhis is
>> one reason why the educationists and politicians like it. It does not
>> require thinking, but I object to the objectives of objective tests.
> Historically that's probably been the case. Although before 1600
> there were no printed books and it was necessary to have a more
> oral culture. The druids had a 20 year training period, and wrote
> nothing down. The Jewish oral law was also not committed to writing
> until Judah haNasi ordered transcription as an emergency measure.
The Egyptians had a rather widespread, at least among merchants,
language (Demotic) in the second millennium BCE. So did the
Babylonians with their cuneiform writing, and also the various
Eemitic people in the Fertile Crescent; most of this has vanished.
However, much Ugaritic writing, with a necessarily cuneiform
alphabet, has been found, and it casts quite a bit of light on
God's name and the Tanakh, mostly on the kh part; many of the
Psalms are monotheized versions of Ugaritic psalms.
Aramaic was a widely used language in the Western part of the
Fertile Crescent, and the present Hebres script comes from that,
so it was certainly written. That writing has disappeared; it
was very hard for writing to survive then. In the sixth century
BCE, Athenian citizens were required to be literate.
> But nowadays there has been a movement in the opposite direction,
> to eliminate all rote learning. But it's not sensible to teach
> children to work out times table from first principles, then
> expect them to use the paper and pencil algorithm for long
> multiplication.
On the contrary, I propose to teach the structure of the
non-negastive integers, and to construct the addition and
multiplication tables, and to do it for other bases as well.
I have also seen a claim of good performance only using
multiplication by 1, 2, 5, and 10, and using the distributive
law. This is an old Korean? method known as Chisenbop.
If the child knows how to construct sums and products,
the memorization is likely to come, and as a mathemawtician
I can tell you that it is rare that a mathematician needs to
do arithmetic. I am quick at it, and keep in practice, but
I did not do hours of drill.
> It simply takes too long. Either require them
> to commit the times table to memory, or accept that we don't do
> pencil and paper any more. The latter has implications that need to
> be thought about, for example if you don't know that 7 x 7 is 49,
> you can't "see" that (x -7)(x +7) is going to be x^2 - 49. You
> have to teach quadratics differently.
You would know that it is x^2 - 7^2. I can immediately see that
(x- 73)(x+73) = x^2 - 5329; so what? The ability to do arithmetic
quickly and accurately is useful, but is not basic to understanding
Understanding the structure, and how to prove results, is far
more important.
> Unfortunately, as you say, times tables become a fetish. Parents
> drill children in the car. They mean well, but it's likely to
> turn children off mathematics. Teaching the table by rote is
> just a job. It should take about a term ((8 x 8) - 8)/2 + 8
> entries = 36 things to learn. 11 weeks of term, of which we write
> off week 1 to get the children settled, and allow them to play games
> in the last week. So nine weeks. Four maths lessons a week =
> (let's hear it) 36 lessons. So one lesson a table entry. If you
> can't teach the table by rote in that time, your technique is
> very poor.
Just let them understand, and have the necessary amount of practice.
Algebraic notation should be used in the beginning; this is the
important part of algebra.
> However the child who does best on the rote learning exercise
> probably won't be the best mathematician. That's not really a
> problem, just a quirk, unless teacher attaches insane value to
> current class position.
Memorization will do nothing to teach how to use numbers, and
one can learn that without all the memorization. As I said,
it is useful but not necessary. They will realize that they
can do reasonable word problems more quickly if they use
algebra and quicker addition and multiplication.
Also, when having them do word problems, do not restrict the
number of variables used. Eliminating variables is cheap.