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Students are amazed at how they can lift their levels of achievement by following the rubric and knowing what is necessary for all levels.

- A Title One Teacher
 
 

Exemplars NCTM Standard Rubric

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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 connections are made.
  • 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 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.
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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