Scoring Tutorial

About Exemplars Math Rubric

There are 4 levels of performance

The Novice level

demonstrates little or no evidence of understanding the mathematical concepts of the task. The student does not have enough knowledge to successfully engage with the problem and requires re-teaching.

This student work does not meet the standard.

The Apprentice level

has the broadest range, from a student who is just beginning to demonstrate mathematical understanding to a student who almost meets the standard. The student is able to begin the problem, but not reach a successful solution. This may be because they have a conceptual misunderstanding or because they make a simple calculation error. There may or may not be an attempt to make a mathematical connection or representation.

This student work does not meet the standard.

The Practitioner level

meets the standard and is proficient in understanding the underlying mathematics of the task. The student has a suitable strategy leading to a correct answer and a full solution. There is an appropriate mathematical connection and representation.

This student work meets the standard.

The Expert level

goes beyond the standard and shows a deeper understanding of the underlying mathematics. The achievement level of Expert is not a common occurrence when assessing student work. At least some of the mathematical language or notations used exceed grade level. Mathematical connections extend the solution to other mathematics or to a deeper understanding of the mathematics in the task. A representation analyzes relationships and interprets phenomenon.

This student work meets and exceeds the standard.

Based on 5 Mathematical CriteriaDerived from the NCTM Process Standards

Problem Solving (P/S)

Problem solving means engaging in a task for which the solution method is not known in advance.

Reasoning & Proof (R/P)

Mathematical reasoning and proof offer powerful ways of developing and expressing insights about a wide range of phenomena.

Communication (Com)

Communication is a way of sharing ideas and clarifying understanding.

Connections (Con)

Connections are mathematically relevant observations that students make about their problem-solving solution.

Representation (Rep)

The ways in which mathematical ideas are represented are fundamental to how people can understand and use those ideas.

How Exemplars Scoring Works

The Exemplars assessment rubric allows teachers to examine student work against a set of analytic criteria. Each criterion is taken into consideration individually and assessed as Novice, Apprentice, Practitioner (meets the standard), or Expert.

The work is then given an Overall Achievement Level score. In coming to the overall assessment (Overall Achievement Level), a paper cannot receive a score higher than the lowest score on any of the five criteria. Thus, if a student does not have any representation on her or his work, the “Representation” score would be Novice and the Overall Achievement Level would be assessed at Novice. If a student has an Apprentice score in “Communication” and all other scores are Practitioner, the student’s Overall Achievement Level would be assessed at Apprentice. In order to meet the standard, a student has to achieve the Practitioner level or above for each of the five criteria. Because the Exemplars rubric is performance based, it is not possible to take a mode or mean “grade” from the assessed criteria.

While many schools and districts require an overall Overall Achievement Level for a task, others do not. What is important is to know where the student stands on each criterion and what the next instructional steps are for that student.

Exemplars Exception to the rule

The National Council for the Teachers of Mathematics has suggested that the “Connections” criterion can be demanding for students because it requires more cognitive thinking and reflection. (For more information and tips on this subject refer to the write up on “Understanding Mathematical Connections” under the “Getting Started” section on the Library.) Therefore, there is one exception to the Overall Achievement Level score. If a student has all Apprentice scores or above but a Novice in “Connections,” the student may receive an Overall Achievement Level score of Apprentice. The student cannot be a Practitioner (or Expert) because not all of the criteria scores meet the standard.

The rationale behind this decision is that if a student has correct problem solving and reasoning as well as communication and a correct representation but did not make a mathematical connection, it would be very difficult to assign the student an Overall Achievement Level of Novice, because the thinking and the solution are correct. This “exception” to the rule is well received by many schools that are looking for a way to give an overall assessment score to a student’s problem-solving piece.

Review the Complete Exemplars Math Rubric

Novice
• Problem Solving(P/S)

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 present.

• Reasoning & Proof(R/P)

Arguments are made with no mathematical basis.

No correct reasoning nor justification for reasoning is present.

• Communication(Com)

No awareness of audience or purpose is communicated.

No formal mathematical terms or symbolic notations are evident.

• Connections(Con)

No connections are made or connections are mathematically or contextually irrelevant.

• Representation(Rep)

No attempt is made to construct a mathematical representation.

Apprentice
• Problem Solving(P/S)

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.

• Reasoning & Proof(R/P)

Arguments are made with some mathematical basis.

Some correct reasoning or justification for reasoning is present.

• Communication(Com)

Some awareness of audience or purpose is communicated.

Some communication of an approach is evident through verbal/written accounts and explanations.

An attempt is made to use formal math language. One formal math term or symbolic notation is evident.

• Connections(Con)

A mathematical connection is attempted but is partially incorrect or lacks contextual relevance.

• Representation(Rep)

An attempt is made to construct a mathematical representation to record and communicate problem solving but is not accurate.

Practitioner
• Problem Solving(P/S)

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.

• Reasoning & Proof(R/P)

Arguments are constructed with adequate mathematical basis.

A systematic approach and/or justification of correct reasoning is present.

• Communication(Com)

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 to share and clarify ideas. At least two formal math terms or symbolic notations are evident, in any combination.

• Connections(Con)

A mathematical connection is made. Proper contexts are identified that link both the mathematics and the situation in the task.

Some examples may include one or more of the following:

• clarification of the mathematical or situational context of the task
• exploration of mathematical phenomenon in the context of the broader topic in which the task is situated
• noting patterns, structures and regularities
• Representation(Rep)

An appropriate and accurate mathematical representation is constructed and refined to solve problems or portray solutions.

Expert
• Problem Solving(P/S)

An efficient strategy is chosen and progress towards 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.

• Reasoning & Proof(R/P)

Deductive arguments are used to justify decisions and may result in formal proofs.

Evidence is used to justify and support decisions made and conclusions reached.

• Communication(Com)

A sense of audience and purpose is communicated.

Communication of an approach is evident through a methodical, organize, coherent, sequenced and labeled response. Communication of an argument is supported by mathematical properties.

Formal math language and symbolic notation is used to consolidate math thinking and to communicate ideas. At least one of the two formal math terms or symbolic notations is beyond grade level.

• Connections(Con)

Mathematical connections are used to extend the solution to other mathematics or to a deeper understanding of the mathematics in the task.

Some examples may include one or more of the following:

• testing and accepting or rejecting of a hypothesis or conjecture
• explanation of phenomenon
• generalizing and extending the solution to other cases
• Representation(Rep)

An appropriate mathematical representation is constructed to analyze relationships, extend thinking and clarify or interpret phenomenon.