Levels 
Problem
Solving 
Reasoning
and Proof 
Communication 
Connections 
Representation 
Novice 
 No
strategy is chosen, or a strategy is chosen that
will not lead to a solution.

Little or no evidence of engagement in the task
is present.

 Arguments
are made with no mathematical basis.
 No
correct reasoning nor justification for reasoning
is present.

 No
awareness of audience or purpose is communicated.
 Little
or no communication of an approach is evident.
 Everyday,
familiar language is used to communicate ideas.


 No
attempt is made to construct mathematical representations.

Apprentice 
 A
partially correct strategy is chosen, or a correct
strategy for only solving part of the task is chosen.
 Evidence
of drawing on some relevant previous knowledge is
present, showing some relevant engagement in the
task.

 Arguments
are made with some mathematical basis.
 Some
correct reasoning or justification for reasoning
is present with trial and error, or unsystematic
trying of several cases.

 Some
awareness of audience or purpose is communicated,
and may take place in the form of paraphrasing of
the task.
 Some
communication of an approach is evident through
verbal/written accounts and explanations, use of
diagrams or objects, writing, and using mathematical
symbols.
 Some
formal math language is used, and examples are provided
to communicate ideas.

 Some
attempt to relate the task to other subjects or
to own interests and experiences is made.

 An
attempt is made to construct mathematical representations
to record and communicate problem solving.

Practitioner 
 A
correct strategy is chosen based on the mathematical
situation in the task.
 Planning
or monitoring of strategy is evident.
 Evidence
of solidifying prior knowledge and applying it to
the problemsolving situation is present.
Note:
The practitioner must achieve a correct answer.

 Arguments
are constructed with adequate mathematical basis.
 A
systematic approach and/or justification of correct
reasoning is present. This may lead to:
 Clarification
of the task.
 Exploration
of mathematical phenomenon.
 Noting
patterns, structures and regularities.

 A
sense of audience or purpose is communicated.
 Communication
of an approach is evident through a methodical,
organized, coherent, sequenced, and labeled response.
 Formal
math language is used throughout the solution to
share and clarify ideas.

 Mathematical
connections or observations are recognized.

 Appropriate
and accurate mathematical representations are constructed
and refined to solve problems or portray solutions.

Expert 
 An
efficient strategy is chosen and progress toward
a solution is evaluated.
 Adjustments
in strategy, if necessary, are made along the way,
and/or alternative strategies are considered.
 Evidence
of analyzing the situation in mathematical terms,
and extending prior knowledge is present.
Note:
The expert must achieve a correct answer.

 Deductive
arguments are used to justify decisions and may
result in more formal proofs.
 Evidence
is used to justify and support decisions made and
conclusions reached. This may lead to:
 Testing
and accepting or rejecting of a hypothesis or
conjecture.
 Explanation
of phenomenon.
 Generalizing
and extending the solution to other cases.

 A
sense of audience and purpose is communicated.
 Communication
at the practitioner level is achieved, and communication
of arguments is supported by mathematical properties
used.
 Precise
math language and symbolic notation are used to
consolidate math thinking and to communicate ideas.

 Mathematical
connections or observations are used to extend the
solution.

 Abstract
or symbolic mathematical representations are constructed
to analyze relationships, extend thinking, and clarify
or interpret phenomenon.
