The tasks 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

 Circumstances of performance 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.

 These work samples illustrate standard-setting performances for the following parts of the standards: a Geometry and Measurement Concepts: Give and respond to directions about location. b Geometry and Measurement Concepts: Visualize and represent two dimensional views of simple rectangular three dimensional shapes. d Geometry and Measurement Concepts: Use many types of figures. a Mathematical Communication: Use appropriate mathematical terms, vocabulary, and language.

What the work shows
a Geometry and Measurement Concepts: The student gives and responds to directions about location, e.g., by using words such as “in front of,” “right,” and “above.”
Both students used location words in their explanations of their thinking and understanding.

b Geometry and Measurement Concepts: The student visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes.
Both students produced two dimensional views to represent three dimensional objects.

 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.