Level 
Understanding 
Strategies,
Reasoning, Procedures 
Communication 
Novice 

There is no solution, or the solution has no relationship
to the task.

Inappropriate concepts are applied and/or procedures
are used.
 The
solution addresses none of the mathematical components
presented in the task.


No evidence of a strategy or procedure, or uses
a strategy that does not help solve the problem.

No evidence of mathematical reasoning.

There were so many errors in mathematical procedures
that the problem could not be resolved.


There is no explanation of the solution, the explanation
cannot be understood or it is unrelated to the problem.

There is no use or inappropriate use of mathematical
representations (e.g. figures, diagrams, graphs,
tables, etc.).

There is no use, or mostly inappropriate use, of
mathematical terminology and notation.

Apprentice 

The solution is not complete indicating that parts
of the problem are not understood.
 The
solution addresses some, but not all of the mathematical
components presented in the task.


Uses a strategy that is partially useful, leading
some way toward a solution, but not to a full solution
of the problem.

Some evidence of mathematical reasoning.

Could not completely carry out mathematical procedures.
 Some
parts may be correct, but a correct answer is not
achieved.


There is an incomplete explanation, it may not be
clearly presented.

There is some use of appropriate mathematical representation.

There is some use of mathematical terminology and
notation appropriate of the problem.

Practitioner 

The solution shows that the student has a broad
understanding of the problem and the major concepts
necessary for its solution.
 The
solution addresses all of the components presented
in the task.


Uses a strategy that leads to a solution of the
problem.

Uses effective mathematical reasoning.

Mathematical procedures used.
 All
parts are correct and a correct answer is achieved.


There is a clear explanation.

There is appropriate use of accurate mathematical
representation.

There is effective use of mathematical terminology
and notation.

Expert 

The solution shows a deep understanding of the problem
including the ability to identify the appropriate
mathematical concepts and the information necessary
for its solution.
 The
solution completely addresses all mathematical components
presented int he task.
 The
solution puts to use the underlying mathematical
concepts upon which the task is designed.


Uses a very efficient and sophisticated strategy
leading directly to a solution.

Employs refined and complex reasoning

Applies procedures accurately to correctly solve
the problem and verify the results.
 Verifies
solution and/or evaluates the reasonableness of
the solution.
 Makes
mathematically relevant observations and/or connections.


There is a clear, effective explanation de tailing
how the problem is solved. All of the steps are
included so that the reader does not need to infer
how and why decisions were made.

Mathematical representation is actively used as
a means of communicating ideas related to the solution
of the problem.

There is precise and appropriate use of mathematical
terminology and notation.
