triangle, circle, square!

Mrs. Hernandez,

Thank you for all of your hard work. You have no doubt touched the lives
of many children, and they will always remember your efforts as I
remember those of my first grade teacher. Please don't let this email
give you the impression that I think otherwise. I'm just doing my best
to help my son makes it through some tough times. D* is showing
aptitude in mathematics including an intuition, technical savy, and a
persistent memory. I would like for him take advantage of his aptitudes
despite the rough sailing.

All about diamonds:

At our conference today, you explained that D* would not accept
that a square rotated did not become a diamond. You also corrected me on
my definition of 'diamond'. I have a pretty strong math education, and
have worked as a professional mathematician. I have publications in the
area, so I looked it up.

In mathematics we would say something more precise such as “the shape
name of a Euclidean solid does not change under translation, rotation,
or scale transformations.” This is true for those shapes; however,
Euclid's Elements (plane geometry) does not define a diamond as a
fundamental shape. It speaks only of line segments, triangles,
quadrilaterals (squares, rectangles, rhombuses, and trapezoids),
polygons, and circles. To talk about diamonds, we must look a little
deeper.

The shape 'diamond' is defined in the mathematical area that studies
tiling and tessellation. Mathematically, a diamond is defined as any
shape created by two triangles that share a hypotenuse. This definition
includes squares, rhombuses, and rectangles – leaving only predilection
when one decides to call a tile a diamond rather than one of the other
quadrilaterals.

In the colloquial, as described in Webster's Third New International
Dictionary (the big unabridged monster), a diamond is defined as “a
square or rhombus-shaped configuration usually having a **distinctive
orientation**”. This is why, for example, the highway department is not
incorrect when referring to 'diamond shaped' warning signs such as those
now on McNeil; although they are rotated squares. The diamonds on
playing cards happen to be sheared, which is ok by the definition too,
as they can be made from two triangles that share their long sides.

In other words, according to mathematical definition from the last
couple thousand of years, and from modern colloquial usage, my six year
old was correct in both his understanding of the shape's definition and
the sensitivity of the use of the term 'diamond' based on orientation of
the plane shape. D* knew this because we had discussed this very
subject in one of our math lessons. I hope you will clear up the issue
with him, as he obviously thinks it is important. In the future he will
not penalize for having had advanced lessons.

Ironically, this was brought up in the bigger context of why D*
sticks on things even when they are obviously wrong. I suspect that one
reason D* emphasizes such points is that he is dealing with
something that in the past most six years olds didn't have to: He is
having to sifting out whether directions from an authority figure (like
a teacher, mom, or dad) should be followed or not. This is no doubt due
to his being exposed to conflicting feelings and directions from his
mother and father. Chances are he is not alone in this at Deepwood, as 1
in 3 first marriages are ending in divorce, and many spouses insist on
being bitter.

According to this interpretation, it is important to D* that a
rotated square is a diamond, because that information was given by his
father. Thus he is using the argument with you to determine if he should
trust information from his father (against the advice of his mother),
and by extension trust his father in general. I understand from a
psychologist that it is normal for children D*' age to think in
such concrete black and white terms. D* is simply doing his best
to use a six year old's tool set to deal with what was in the past only
a problem for older children.
In my opinion D* is persevering with an amazing amount of courage,
determination, and good will. For these reason, beyond the existence
extracurricular lessons, I hope you will give weight to his concerns,
and embrace them, rather than participating in making a black and white
issue out of a technical detail. We have a lot of time to master plane
geometry, but not so much time to make D* feel comfortable with
lessons at school. Of course, I also recognize that we need to work on
his acceptance of the fact that it isn't bad for two people to give
slightly conflicting information. And I think it is wonderful that you
all are teaching first graders about invarient properties.

I would like to learn more about the math curriculum used at Deepwood,
and I would like to inform you about D*' extracurricular math
lessons. In this way we can reduce the likelihood that his advanced
study will be misinterpreted and suppressed. Please let me know when we
can do this.

An additional question: is the school familiar that D* is
currently visiting a psychologist? This has not been mentioned. If so,
how is the school involved? Also, D*' mother informed me the other
day that she has been acting as though she had exclusive medical rights.
This is wrong, as she admitted when it was pointed out in email. H*
and I have *joint* medical, and H* is under court order to involve
me with any medical decisions (i.e. not to act solely). This includes
the use of any medication, or the use of counseling. Please let me know
if the school needs to see copies of documents in order to establish
this.

Sincerely Yours,

Tom Lynch

P.S, see you next Wednesday. If you don't do computer lab, I would like
to observe class.
Received on Tue Sep 21 18:21:04 2004