Goetz (2005) believes cooperation plays an important role in learning, and that, we should assess what we value

(p. 12). Ergo, his assessments in his precalculus classes contain cooperative activities in which students' grades are partially dependent on the interactivity and communication between the students. His tasks are posed as open-ended word problems with real-world applications. In one example, given a data set the students were expected to construct a mathematical equation that models that data. There was a diverse range of answers varying from polynomial to rational to exponential functions.

The students' grades were based on a grading rubric, which the students were aware of before the exam. One of the elements on the rubric took group participation into account, and the students were expected to explain the specific roles each member played in the cooperative activity. That way, each student would get the deserved amount of credit. Goetz believes in using assessment as a tool for learning, such that students will turn an exam into a learning experience. This corresponds with NCTM's view that assessment should enhance student learning.

Grading rubrics are essential for assessment. In my personal experience and from advice from my educators, I can report that students' scores on assessments without rubrics can become subjective and open to interpretation, and are a potential source of conflict between teacher and parents. In order to provide an explicit and objective grade, a grading rubric must be used. The higher resolution a rubric has, the more accurate students' score will be on that particular task. Sometimes, though, teachers create rubrics that are hard for students to understand, thus the students might score more poorly than if they had a clear indication of what is expected of them.

Brown-Herbst (1999) had her middle-school students construct their own grading rubric. Her class's rubric was based on a final draft submitted by teachers from twelve schools participating in a statewide project in Alaska. While constructing the rubric, her students had to interpret the language used by teachers to gain an understanding of the spectrums of performance. After three days of debate and discussion among the middle-schoolers, they finalized a rubric that was to be used on not only their end of year exam, but also on that project itself. In other words, the students were being assessed by their own criteria.

A project such as the one implemented by Brown-Herbst (1999) takes much time and planning, but the knowledge and skills gained by the students are worth it. Students reflected NCTM's (2000) Communication process standard: they translated mathematical teacher language into mathematical student language, and conveyed concepts and ideas to one another and refined them. Even previously implicit ideas have been made explicit by students who asked each other to clarify meaning; e.g., [a] seventh-grade girl spoke up:

(p. 453).I need to know exactly what the

math thing

is

A final example as another form of assessment is offered by Bailey and Chen (2005). They introduced the graphing portfolio, in which students are expected to *trace out* a picture or graphic by using functions (either cartesian or a combination of cartesian and polar) to illustrate the lines in the graphic. This method is slightly related to Goetz's (2005) example, in that students are working backwards with functions: given a function's graph (or a curve of best fit), they need to find the equation. Graphing portfolios are useful for an artistic and creative touch in a mathematics course.

References

- Bailey, E. C., & Chen, F. (). Graphing portfolios in calculus: Reinforcing concepts and inviting creativity.
*Mathematics Teacher, 98*(6), 404–407. - Brown-Herbst, K. (). So math isn't just answers.
*Mathematics Teaching in the Middle School, 4*(7), 448–455. - Goetz, A. (). Using open-ended problems for assessment.
*Mathematics Teacher, 99*(1), 12–17.