| 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.
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| 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.
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| 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 problem-solving 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.
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| 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.
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