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.
Students decided to use diagrams, lists, and equations.
• uses previously learned strategies,
skills, knowledge, and concepts to make decisions.
All of the samples demonstrate use of simple addition, as well as
skill in drawing diagrams and making lists.
• 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.
In Samples 1, 2, and 3 the students used diagrams, lists, and equations.
• makes connections among concepts in
order to solve problems.
All students
made connections between simple number concepts and patterns 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:…
• makes the solution into a general
rule that applies to other circumstances.
In Samples 1 and 2 the students went beyond the problem by finding
a rule that would work for any number of people. Although the students
of Samples 3 and 4 accomplished the task assigned, they did not
state the general rule.
b
Mathematical Communication: The student
shows mathematical ideas in a variety of ways, including words,
numbers,…pictures,…diagrams….
c
Mathematical Communication: The student explains
solutions to problems clearly and logically, and supports solutions
with evidence, in both written and oral work.
There are some grammatical and spelling errors
in these work samples. For example, in Sample 1, the student misspelled
“drawed” and “egzact.” In Sample 4, the
student made some grammatical mistakes (e.g., the capitalization
of “kantite” in the middle of a sentence) and misspelled
a few words (e.g., “Jwen” instead of “Jwenn,”
and “sink” instead of “senk”). However,
these errors do not detract from the mathematics in the work. This
was a class assignment and was not edited for spelling and grammar.

