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A Diamond Is Forever Unclear, as A Term -- and How to Teach from That

Date: 12/17/2010 at 05:28:29
From: lily
Subject: Is the term 'diamond' appropriate in a mathematical context

Many people use the term "diamond" to describe certain mathematical
shapes. I have researched but cannot find an authoritative definition. How
is this word appropriate or not appropriate in a mathematical context?

I have always thought that a diamond was a "shape," but it does not appear
to be used in primary school mathematics. Not sure what to tell students
if it is appropriate to call it a diamond or not?



Date: 12/17/2010 at 09:55:04
From: Doctor Ian
Subject: Re: Is the term 'diamond' appropriate in a mathematical context

Hi Lily,

I guess how you would approach this depends on what it is you're trying to
teach.

If the goal is for students to be able to go out and identify a diamond
when they see one, or know what is meant by that word when they read or
hear it, then it seems like a good idea to teach kids what a "diamond" is.
But note that "diamond" is an informal term, without a precise
mathematical definition. For example, the "diamonds" on playing cards
sometimes have slightly curved sides, and so aren't even polygons.

But if the goal is to help the students get a feel for what it's like to
"do mathematics," then the relevant concept here is "subsets." Children
deal with subsets outside of mathematics; for example, every Cocker
Spaniel is a dog, but not every dog is a Cocker Spaniel; every dog is a
mammal, but not every mammal is a dog.... So with this plane geometry
vocabulary, all you're really doing is showing that the same kind of
relationship sometimes holds among mathematical objects. For example,

   Every rectangle is a parallelogram, but not every parallelogram is a
   rectangle.

   Every square is a rectangle (which means every square is also a
   parallelogram), but not every rectangle is a square.

   Every square is a rhombus, but not every rhombus is a square.
   
   ... and so on.

And there are reasons why subset classifications like this are
interesting. For example, if you know something is true of any
parallelogram, then to show that it's also true of a rhombus, all you need
to do is note that a rhombus IS a parallelogram, and you're done. This can
save a lot of work!

"Diamond," like "oval," doesn't really fit into this context, because
those terms don't have precise definitions. (If someone says that
something is "oval," they might mean sort of egg-shaped, or they might
mean elliptical, or they might mean it's a rectangle with semi-circles on
the end, or just that it's kind of curvy and longer in one dimension than
the other. You can't really know for sure.)

These words are part of everyday language, and rely on a certain amount of
"you-know-what-I-mean" to be useful in communication. Whereas in math,
we're interested in eliminating as much "you-know-what-I-mean" as
possible, so we can be sure that if we're talking about something, we both
understand it in exactly the same way.

One interesting point of discussion, though, is that many people will
think that a square rotated 45 degrees "becomes" a rhombus (or a
"diamond"), as though shape somehow depends on orientation. Having
students discuss whether this is reasonable can provide a nice
introduction to ideas about symmetry.

Does this help?

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 12/17/2010 at 15:54:26
From: Doctor Rick
Subject: Re: Is the term 'diamond' appropriate in a mathematical context

Hi, Lily. 

I'd like to add to what Dr. Ian has said.

He mentioned that sometimes people take a square, rotate it 45 degrees,
and call it a diamond. That reminded me of the following exchange that I
had a few years ago:

Question: Teacher wants Kindergarten students to differentiate between a
rhombus and a diamond. (It's part of the Kindergarten learning assessment
in February to distinguish between the two!) Some Internet sites indicate
they are the same, so I'm wondering how to tell the difference. They look
the same to both daughter and Mom. And at the Kindergarten level, I
couldn't see a shape summary that showed both rhombus and diamond. Most
basic websites on shapes do not include a rhombus; the few that do are
written beyond the reading level of a 5 year-old.

Answer: This does seem strange. For a mathematical viewpoint, see the
following well respected site:

    http://mathworld.wolfram.com/Diamond.html 

It identifies several ways in which the word "diamond" is used. The first
is synonymous with rhombus. The second is a square rotated so its sides
are at 45 degrees to the vertical; this usage is not really a shape, as
shapes are generally viewed as independent of orientation. However, this
usage is familiar to anyone who knows anything about baseball.

I searched a bit further on the Internet for education sites that refer to
both the rhombus and the diamond, and found this lesson plan, written for
other teachers:

"Make six ovals (ellipses), six diamonds, and six rhombi from construction
paper.... Draw an oval (ellipse), a diamond, and a rhombus. Point to each
shape and say its name. Ask the students to name each shape. Explain that
an oval has no sides. Explain that a diamond and rhombus each have four
sides. Ask the students if they have ever seen anything with these
shapes.... Give each student a different shape. The students may be
sitting at their seats or in a circle on the floor. Call out 'ovals, stand
up,' then 'ovals, sit down; rhombi, stand up.'"

Evidently, the author of this lesson considers the diamond and the rhombus
to be different shapes. The problem, as with a lot of educational sites I
have seen, is that it doesn't tell us what a diamond and a rhombus are!
When I downloaded the accompanying documentation, I saw two rhombi, one
with two sides horizontal, the other with the diagonals horizontal and
vertical. They weren't labeled, so I can only guess that the former is
supposed to be a rhombus and the latter a diamond, differing only in their
orientations.

However, notice that in one part of the activity, the shapes have been cut
out of paper and given to the students. Orientation is lost when this is
done.

The "standard" supposedly being assessed by this activity is: "Identifies
circle, square, triangle, oval (ellipse), diamond, rhombus, and rectangle
in various orientations/positions." To me, this suggests that the shapes,
including a diamond and a rhombus, can be distinguished regardless of
their orientation. If a diamond and a rhombus "shape" are distinguished
only on the basis of "orientation," the standard contradicts itself. But
what else could distinguish them? I have no clue.

I don't like this at all! I'd like to know what the teacher has to say,
and also the author of the lesson I found. Please fill me in when you
learn what the distinction is supposed to be.

End of quote. Lily, I never heard back from this parent, or from a teacher
who raised the same issue. I'm glad to hear you at least raising the
question, and I hope it leads to a good discussion.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Elementary Definitions
Elementary Triangles and Other Polygons

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