The teacher gave students the prompt (the
diagrams were the size of tangram pieces) illustrated here.
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 


Students had previously worked with tangrams. They
had begun to study area relationships and fractions.
This student was preparing a New Standards Elementary Mathematics Portfolio
and was aware of the requirements for the portfolio. In providing evidence
of problem solving work, students using New Standards Mathematics Portfolios
must show that they can go beyond the problem and make connections, extensions
or generalizations (
c).
This work sample illustrates a standardsetting
performance for the following parts of the standards:

d 
Arithmetic
and Number Concepts: Describe and compare quantities by using
simple fractions. 
e 
Geometry and
Measurement Concepts: Solve problems by showing relationships
between and among figures. 
a 
Problem Solving
and Reasoning: Formulation. 
b 
Problem Solving
and Reasoning: Implementation. 
c 
Problem Solving
and Reasoning: Conclusion. 
e 
Mathematical
Skills and Tools: Refer to geometric shapes and terms correctly. 
b 
Mathematical
Communication: Show mathematical ideas in a variety of ways. 
c 
Mathematical
Communication: Explain solutions to problems clearly and logically. 

d
Arithmetic and Number Concepts: The student
describes and compares quantities by using concrete and real world
models of simple fractions; that is:
• finds simple parts of wholes.
The student
explained and showed in a model that there can be four equal (parallelogram)
parts within the whole (parallelogram).
• uses drawings, diagrams, or models
to show what the numerator and denominator mean, including when adding
like fractions, e.g.,
+ , or when showing that
is more than .
The student used two models to show that one small parallelogram is
equivalent to one out of four equal parts of the whole parallelogram,
i.e., that “Tracy’s shape was
not a
of Terri’s shape.” 

e
Geometry and Measurement Concepts: The student
solves problems by showing relationships between and among figures,
e.g., using congruence and similarity….
The student
explained and modeled that “…one big shape can equal (i.e.,
be congruent to) four little of the same (i.e., a similar) shape.”
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 prespecified manipulative, or go through
a predetermined set of steps.
Given a
problem situation with many possible approaches, the student decided
what materials to use (pattern blocks and tiles) and what approach
to follow.
• uses previously learned strategies,
skills, knowledge, and concepts to make decisions.

The student used knowledge about the relationship between simple
parts and wholes to decide who was correct in the dispute.
• uses strategies, such as using manipulatives
or drawing sketches, to model problems.
b
Problem Solving and Reasoning: Implementation.
The student makes the basic choices involved in planning and carrying
out a solution; that is, the student:
• makes up and uses a variety of strategies
and approaches to solving problems and uses or learns approaches
that other people use, as appropriate.
The
student used fractional and geometric strategies to solve the problem.
• makes connections among concepts
in order to solve problems.
The student
made connections between number and geometry concepts to solve the
problem.
• solves problems in ways that make
sense and explains why these ways make sense, e.g., defends the
reasoning, explains the solution.
c
Problem Solving and Reasoning: Conclusion.
The student moves beyond a particular problem by making connections,
extensions, and/or generalizations; for example, the student:…
• explains how the problem is
similar to other problems he or she has solved….
The student
went beyond the problem by illustrating that the same approach and
concepts can be used in a different situation.

e
Mathematical Skills and Tools: The student
refers to geometric shapes and terms correctly with concrete objects
or drawings, including triangle,…parallelogram….
The student
referred correctly to the parallelogram, which is not mentioned
by name in the task.
b
Mathematical Communication: The student
shows mathematical ideas in a variety of ways, including words,
numbers,…pictures….
c
Mathematical Communication: The student
explains solutions to problems clearly and logically, and supports
solutions with evidence, in both written and oral work.
The solution
is explained clearly.
The conclusion is supported by diagrammatic evidence.


