Re: Promath

[Robert Hopcroft <hopcroft@uswest.net>]

"Odile Bénassy" wrote:
> 1) grade :
> any grade
> my own work's scope is part of our "classes de 2nde et 1ere" (15-17
> years old
> in France)
> but I've thought also of very simple computation for very young children
> as I already said : "only interface should be different", the mechanics
> is still the same

> 2) general areas :
> many
> everything that need training in a rather systematic way
> for  example :
> numerical computing - formal computing - algebra - logic - trigonometry
> - everything that's similar to algebra in functions, like elementary
> differential and integral computation -
> also : reasoning, every kind of job where student has to produce a
> well-formed and meaningful sentence to express a mathematical idea, this
> situation happens at any grade, already when you ask a 7 years old child
> to describe simple situations

Remember, I have nothing to do with education, but here is how I would go
about it. I got out a clean sheet of paper and said, "Let's start at the
beginning". Every number has a successor, ZERO is a number. First task,
grasping the meaning of zero. This leads to sets and in particular the empty
set. A set is a collection of objects such as a set of dishes or a set of
books. It would be nice if things were integrated and in the process of
teaching words, the concept of set was introduced in a variety of ways
setting up the child for being comfortable with the idea. Present the
following, "Here is a set of books and they are going to be moved". To move
them, "Here is an empty box". How many books are in the box? Zero! Put a book
into the box. How many books are in the box. One! Put another book into the
box. How many books are in the box. Two! Continue until 15 books are in the
box. Now write down the numbers one has just said (0,1,2.....14,15). How many
numbers were spoken. Sixteen because zero is a number. What has one learned?
When one counts, one starts with the empty set. Now write down the numbers 0
... 15. Beside them write the binary equivalent. This can be done as a game.
In the first column write 0,1 0,1 etc. In the second column write 0,0,1,1,
0,0,1,1 etc. In the third column write four zeroes followed by four ones etc.
Finally in the fourth column write eight zeroes followed by eight ones. By
now the pattern should be apparent. In each succeeding column double the
number of zeroes and ones. Forever!  Not only does every number have a
successor, but here is a way to generate them. Now look at the set (0...15).
How many numbers have no ones? One! How many numbers have a single one. Four!
What do these numbers have in common. They are the powers of two. How many
numbers have two ones. Six! Three ones. Four! Four ones. One! What do the
numbers (1,4,6,4,1) represent. The binomial coefficients. What does this
mean? It's the possible outcomes of flipping a coin heads/tails or 0/1, four
times. Now lets take a regular tetrahedron and number the corners with the
numbers which contain one one, namely (1,2,4,8). One could get into the two
possibilities (mirror images). Now we have six lines with the endpoints
numbered. Let's label the lines by adding the endpoint numbers. The numbers
are unique, namely the six numbers with two ones. Why? Because the numbers
generating them have a single one in a unique location. Now let's label the
four faces by adding the three numbers at the corners of the triangle. Again
we get the four unique numbers with three ones. Finally let's label the
tetrahedron by adding the four corner numbers to get 15. What happened to the
number with no ones. This is the label of the empty set of no tetrahedrons.
Thus ZERO is a number. One could present just the triangle to younger
children and the tetrahedron to older ones. Thus presenting the ideas twice
during the childs schooling. This was the point where I was pondering the
presentation of geometric figures. Continuing, a regular tetrahedron has
seven axis of symmetry. There are four passing through a corner and the
opposite face. What is the sum of these labels. Fifteen! The remaining three
axis of symmetry pass through the midpoints of opposite edges. What is the
sum of these labels? How about fifteen!  One could continue on with group of
the 12 rotations of the regular tetrahedron. I have more but it gets sketchy
as I still have to fill in the details. I think I'll stop here and take a
break and back later.

Bob