Sample A.39
Mathematics Assessment
California Department of Education
As described in detail in the California Mathematics Framework (1992), mathematically powerful students are those who can draw on mathematical ideas, tools, and techniques to think and communicate. In responding to an open-ended question and accomplishing its task, a student demonstrates mathematical power. As the student engages in and responds to the task, he or she draws from his or her thinking capacity, understanding of mathematical ideas, ability to use tools and techniques, communication skills, and ability to shape a coherent and focused response.
Teachers use a rubric to score and evaluate students' responses to open-ended questions or investigations. The scoring rubric, based on the goals articulated in the Mathematics Framework, describes how well the student work meets the expected standard of performance. Teachers should study the general rubric as well as the evaluation process to understand how the mathematical power is assessed through open-ended mathematical problems.
Table 1 shows a general rubric used jot evaluate responses to open-ended questions in mathematics. The rubric articulates, at six levels, the extent to which student work accomplishes the purpose of the task and demonstrates mathematical understanding, reasoning, thinking, communicating, and use of tools and techniques. Level 6 represents the highest quality of work, and level 1, the lower quality of performance.
This rubric can be applied to any open-ended task. Therefore, before applying this rubric to mathematics assessment, the scorer must explore all possible ways in which it relates to a particular problem. In other words, the scorer looks for the mathematical ideas, thinking, communication, tools and techniques that a student can use to solve a particular problem.
GENERAL RUBRIC
Table 1
Level 6
Solid work that may go beyond the requirements of the task(s), showing for example:
- complete understanding of the task's mathematical concepts and processes.
- clear identification of all of the important elements of the task(s).
- where appropriate, clear evidence of doing purposeful mathematics, including investigating, experimenting, modeling, designing, interpreting, analyzing, or solving.
- excellent prose and mathematical supporting arguments that may include examples or counter-examples.
- creativity and thoughtfulness in communicating the results and the interpretations of those results, to an identified audience, using dynamic and diverse means.
- multiple solutions based upon different assumptions about or interpretations of the task(s).
- unusual insights into the nature of and the resolution of problems encountered in the task(s).
- a high level of mathematical thinking that includes, where appropriate, making comparisons, conjectures, interpretations, predictions, or generalizations.
- exceptional skill in choosing appropriate mathematical tools and techniques in the resolution of problems in task(s).
Level 5
Fully achieves the requirements of the task(s), showing for example:
- good understanding of the task's mathematical concepts and processes.
- identification of most, if not all, of the important elements of the task(s).
- evidence of doing purposeful mathematics, including where appropriate, investigating, experimenting, modeling, designing, interpreting, analyzing, or solving.
- clear, successful communications with an identified audience.
- one solution and interpretation of those results.
- evidence of mathematical thinking that includes, where appropriate, making comparisons, conjectures, interpretations, predictions, or generalizations.
- use of variety of tools and techniques appropriate to the form of the task(s) and the requirements of the task.
Level 4
Substantially completes the requirements of the task(s), showing for example:
- an understanding of most of the task's mathematical concepts and processes.
- identification of the important elements of the task(s), but some less important ideas are missing.
- some aspects of investigations, experiments, model building, designs, interpretations, analysis, solutions required by the task(s) may be missing, but most of the parts are included.
- adequate communication with an identified audience, but with limited clarity and variety.
- occasional evidence of mathematical thinking involving comparisons, conjectures, interpretation, predictions, or generalizations.
- a limited variety of tools and techniques used to resolve the situation presented in the task(s).
Level 3
Limited completion of the requirements of the task(s), showing for example:
- an understanding of some of the task's mathematical concepts and processes, but with evidence of gaps in those understandings.
- identification of some of the important elements of the task(s), but assumptions about some of the elements may be flawed.
- communication of some ideas, but generally makes inadequate attempts to communicate, often failing to address the identified audience, and difficulty in expressing mathematical ideas.
- inadequate mathematical thinking that includes ineffective analysis procedures, limited solution strategies, unclear mathematical arguments, and inappropriate interpretation of results.
- a selection of some inappropriate tools and techniques used to resolve the situation presented in the task(s).
Level 2
Requirements of the task(s) not completed, showing for example:
- only fragmented understanding of the task's mathematical concepts and processes, accompanied by disorganized, incomplete results.
- Identification of only a few, usually superficial, elements of the task(s).
- attepts to address the intended audience that may be incoherent, muddled, or incomplete.
- attempts to explain or justify results that are convoluted, illogical, circular, or unrelated to the results shown.
Level 1
Does not achieve any requirements of the task(s), showing for example:
- an irrelevant, nonsensical, or illegible response that has no valid relationship to the task(s).
- no understanding of the task's mathematical concepts and processes.
- unsuccessful attempt, if any, to communicate with the intended audience. Usually communication is not attempted.
- no attempt to explain or justify results. If attempt is made, it is often unrelated to the task, illegible, or incoherent.
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STUDENT EXAMPLES
Score: 6
The response shows:
- completion of requirements of the task, including the mathematical processes by which the choice was made.
- comparison of mean, median, and mode, all correctly computed, and the judgment that the differences between these measures for the scores of the two bowlers are not significant.
- two line graphs which indicate clearly the lack of consistency by Bill, which is then chosen as the deciding factor.
- a clear, coherent discussion of these characteristics in a creative and thoughtful way.
Score: 2
The response shows:
- lack of understanding of mean in reporting that the same total of scores and number of games resulted in different "averages".
- incorrect total, which could have been more accurately computed with a calculator.
- a carefully drawn bar graph, but no justification for its inclusion as the graph was not referred to in the explanation.
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