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Work Sample & Commentary: The Never Ending Four

The task
The teacher asked students to complete a large scale mathematics project chosen from among five kinds. This student chose to do a “Pure Mathematics Investigation.” The teacher gave the student the written instructions illustrated here.

Circumstances of performance
This sample of student work was produced under the following conditions:
- alone in a group
in class - as homework
- with teacher feedback - with peer feedback
timed - opportunity for revision


In this classroom students were using the New Standards Elementary Mathematics Portfolio. The teacher intended students to be able to use their chosen projects to fulfill the requirements for the project exhibit in the portfolio.
This work sample illustrates a standard-setting performance for the following parts of the standards:
b Function and Algebra Concepts: Build iterations of simple non-linear patterns and recognize that these patterns are not linear.
a Statistics and Probability Concepts: Collect and organize data to answer a question.
a Problem Solving and Reasoning: Formulation.
g Mathematical Skills and Tools: Read, create, and represent data.
b Mathematical Communication: Show mathematical ideas in a variety of ways.
c Mathematical Communication: Explain solutions to problems clearly and logically.
d Mathematical Communication: Consider purpose and audience when communicating about mathematics.
e Putting Mathematics to Work: Pure mathematics investigation.

What the work shows
b Function and Algebra Concepts: The student builds iterations of simple non-linear patterns, including multiplicative and squaring patterns (e.g., “growing” patterns) with concrete materials, and recognizes that these patterns are not linear.
The student built (again and again) the non-linear pattern generated by this “trick” question (i.e., that the numbers appearing before a four turns up are always five or nine).

a Statistics and Probability Concepts: The student collects and organizes data to answer a question…by comparing sets of data.
The student represented data in a table in order to further explore the number relationships.

a Problem Solving and Reasoning: Formulation. Given the basic statement of a problem situation, the student:
• makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.
The student made all decisions regarding how to proceed with this entirely unformulated project.

g Mathematical Skills and Tools: The student reads, creates, and represents data on charts, tables, diagrams….

b Mathematical Communication: The student shows mathematical ideas in a variety of ways, including words, numbers, symbols,…charts, tables, diagrams….

c Mathematical Communication: The student explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.

The student used a narrative description and tables to illustrate the nature of the “problem” (or, in this case, “trick”).

d Mathematical Communication: The student considers purpose and audience when communicating about mathematics.
The student clearly explained the nature of the project as well as how it would be conveyed to the classroom audience.

e Putting Mathematics to Work: The student conducts a pure mathematics investigation, in which the student:
• decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns.

• describes a question or concept to investigate.

• decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models.

• carries out the investigation.

• writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.

The three errors in counting the letters (32, 13, and 30) and the spelling mistake (“ect” instead of “etc”) do not detract from the project.