Levels | Descriptions |

1 | Responses completely lack precision, coherence, and purposefulness. For instance, responses are too vague, irrelevant, incomplete, fragmented, inaccurate, or incorrect. |

2 | Responses do not meet the criteria of precision, coherence, and purposefulness. However, responses address the questions and have minimal mathematical errors. Mathematical understanding tends to focus knowledge at the surface, or mechanical level. |

3 | Responses show some elements of precision, coherence, and purposefulness. For instance, there is evidence of an attempt or effort to emphasize the key mathematical idea, its rationale, the logical progression of mathematical concepts, and the connectedness among different mathematical concepts, procedures, and ideas. In addition, responses show an attempt to scaffold mathematical ideas for students. |

4 | Responses exemplify precision, coherence, and purposefulness. There is consistent (or substantial) evidence of an attempt or effort to emphasize the key mathematical idea, its rationale, the logical progression of mathematical concepts, and the connectedness among different mathematical concepts, procedures, and ideas. In addition, responses show attention to how to scaffold mathematical ideas to students (e.g., from simple to complex; from specific to general). |