Re: Multiplication Tables

Tina (
Wed, 08 Jan 1997 23:23:35 +0000

Charles wrote:
> On Tue, 7 Jan 1997, Tina wrote:
> > I, myself, spent my 6th grade in a Math class in a German Grammar school
> > with the most memorable teacher. We spent the whole year expressing
> > ourselves mathematically without using any numbers. It was the
> > best Math I ever learned.
> This sounds great. Could you elaborate on what you did?
> Charles

Hi, Charles and Kathy.
I'll try to describe it, but it has been a long time.

In 6th grade we were not allowed to use any numbers, i.e. 1, 2, 3, etc.
or first, second, etc.
You can actually try that on your own.
When we discussed homework, for example, we couldn't just say:
the first, second, or so question.
We learned quickly to use a reference point instead. If we
wanted to give the answer to the first question we would have to
say, "the question in regard to..."
The second question would be "the one after the question in regard to,
What was fun for the students in this case was the individuality and
the innovativeness with which students would try to express themselves.
To say, the first question, the second question, etc. is not too
but trying to say something different really called on the students'
resources and imagination.
And that is what it was all about.
Numbers were "amounts", or "quantity", adjectives would narrow
down whether it was very large, large, even, or uneven. Further
points would help to understand whether it was in the tens, hundreds or
larger numbers. For example, in order to say 5 + 6, one could say:
A small, family-sized uneven amount added to a second
just a little larger, even amount
adds up to somewhat more than double of the first amount and
is also uneven.
The object was not so much to guess what number one might have in mind
but to see numbers as having characteristics and individualities and
which one of those characteristics would continue when brought together
with another number.
For example uneven numbers, no matter which one, are quite unique.
Alone they can cause trouble. An uneven amount of something can
stir up a lot of problems.
When brought together with a (different, less trouble causing)
even number, they like to dominate.
When brought together with another uneven number however they have
a quite different result in changing into something even.
(Any philosophy in that?)
Is it better to cut up a cake into even numbers or uneven numbers?
Other words that we used a lot were "more than", "less than" and
equal, to express relationships of "quantities" or "amounts".

I hope I am making sense, it is quite difficult to explain
and it has been a long time.
The major point of "not using numbers" was using our brains
and the individuality with which one could express something as simple
as 2 + 2. or 10 x 5, etc.

20 years later, as an English High School teacher, I would get
tired of saying,
Question number 1, 2, etc. and at times I would use
my 6th grade skills and the High School kids loved it.
I would use anything but numbers to refer to questions and
toss in at times some SAT words.

I doubt anyone teaches Math anymore the way I was taught in 6th grade.

In regard to the German Mathbook, I want to do a little bit of research
first to find out if it is available here. I have it send to me from
I'll let you know about Publisher, name, etc. ASAP.