These work samples are drawn from two different
classrooms, but focus on the same concept of visualizing and representing
three dimensional objects in two dimensions.
In Sample 1, the teacher gave the following written instruction to
the students:
Andre built a shape with blocks. What would be the front view?
In Sample 2, the teacher gave the students Student Sheets 15 and 16
(from Seeing Solids and Silhouettes, a book in the series,
Investigations in Number, Data and Space, Dale Seymour Publications,
1995) which had these written instructions:
Choose two or three buildings. Draw their silhouettes from the front,
top, and right side. (Student Sheet 15)
Draw the front, top, and side silhouettes for three cube buildings.
Put the number of the building above its silhouettes. (Student Sheet
16)

Sample 1

These samples of student work
were produced under the following conditions: 
 alone 
in a group 
 in class 
as homework 
 with teacher feedback 
with peer feedback 
timed 
 opportunity for revision 
Both students were developing a New Standards Elementary Mathematics
Portfolio. Because of this, they knew the criteria in this system
for work that shows conceptual understanding. The criteria require
the students to use, represent, and explain the concept. The students
used the tasks as opportunities to show what they understood about
the concept of visualizing and representing three dimensional objects
in two dimensions.


Sample 1 was completed in Spanish in a
bilingual classroom. The translation was provided by the teacher.

Translation of Sample 1



Sample
2 includes a small mistake that does not detract from the evidence
of an overall understanding of this concept.
d
Geometry and Measurement Concepts:
The student uses many types of figures (…squares,…cubes…)
and identifies the figures by their properties, e.g., symmetry,
number of faces, two or three dimensionality, no right angles.

Sample 2

a
Mathematical Communication: The
student uses appropriate mathematical terms, vocabulary, and
language, based on prior conceptual work.
Sample
1 includes appropriate use of location words (e.g., “from
the right to the left”) and shape words (“cube”).
Sample
2 includes a good elementary definition of perspective, supported
by a clearly explained example.
Sample
2 includes some unnecessary and superfluous communication. While
this kind of communication is fairly common for students as
they develop their capacity to explain their thinking, reasoning,
and understanding, it does not add any mathematical communication
to the response. 


