Using a Rubric to Score Student Responses
Once you decide to use openended assessments in your mathematics teaching, one of the first questions you will likely ask is, How do I score the responses? A rubric is, for the most part, a new name for what teachers have been doing for a long time. Sometimes we use a very simple rubric that only has two indicators, namely, right or wrong, full credit or no credit. Sometimes our rubrics are more complexright, wrong, or sort of. Many of us use a "stickynote system": when we grade a set of tests and we see an error that we have encountered before, we think to ourselves, How much did I count off for that? and we look back and then make a sticky note that says minus 2 for sign error to keep our grading consistent.
In other words, a rubric is a list of indicators that helps us rank responses based on some criteria. A rubric can be analytic or holistic. An analytic rubric is divided into several dimensions. Suppose you want to grade a student on the dimensions of communication, mathematical correctness, and completeness. You would use an analytic rubric with indicators for each of those three dimensions. Holistic rubrics help teachers assess the whole task on one scale. For openended questions, a holistic rubric is the most effective and user friendly.
A scoring rubric is helpful in several ways.
First, it helps us focus on what students know and can do rather than on what they do not know and cannot do. Suppose you ask students to write two mixed numbers whose sum is and to explain how they know that their two numbers satisfy the condition. Now suppose one student provides two mixed numbers whose sum is not and another student provides two improper fractions whose sum is not . Even though both students' answers are incorrect, the first student knows something the second student does not: by providing mixed numbers, he shows that he knows what mixed numbers are. A rubric can help focus your attention on what mathematical knowledge is apparent from the response.
Second, a rubric helps us keep grading consistent. Suppose a teacher asks students to explain why multiplication is the appropriate operation to solve a particular problem. Student A responds, "To get the right answer." Student B responds, "If you don't multiply, it is going to be wrong." Student C responds, "It is what you have to do in order to get the answer right." The teacher who graded these responses scored student A's response with 0 points, student B's response with 2 points and student C's response with 3 points (scale: 03), a wide range for responses that are essentially the same. Using a scoring rubric can help prevent these kinds of scoring inconsistencies.
Finally, we can ask students to evaluate previous students' responses to a question using the same scoring rubric we will use. This helps students better understand our expectations and learn what differentiates highlevel responses from lowlevel responses. Assessment thus becomes less mysterious.
The following rubric is useful in scoring responses to openended assessment items:
Scoring Rubric 
0 

Response indicates no appropriate mathematical reasoning 

1 

Response indicates some mathematical reasoning but fails to address the item's main mathematical ideas 

2 

Response indicates substantial and appropriate mathematical reasoning but is lacking in some minor way(s) 

3 

Response is correct and the underlying reasoning process is appropriate and clearly communicated 
Now let's use this rubric to score the following assessment item:
Draw a parallelogram on the dot paper below that is similar but not congruent to the parallelogram above. Explain how you know the two parallelograms are similar but not congruent.
Response A:


Response B:

Response C:
In response A, the student showed some evidence of appropriate mathematical reasoning in that she drew a parallelogram. However, she missed the item's main mathematical point that the sides should be proportional. This response would receive a score of 1. Student B drew a parallelogram that was similar and not congruent to the original parallelogram, but her explanation did not make it clear that the sides had to be proportional. She pointed out that the sides should have different lengths (so that the new parallelogram would not be congruent to the original), and her sides were actually proportional, but her explanation did not explicitly state that the sides needed to be proportional. Response B would be scored 2. Student C drew a parallelogram that was similar but not congruent and pointed out that similar means relatively the same shape (that is, proportional) and not the same size. The word "relatively" makes the difference, warranting a score of 3. A score of 0 would be given to a response that is a shape other than a parallelogram or a parallelogram that is not similar to the original shape, because there would be no evidence of appropriate mathematical reasoning.
As with all grading, different teachers will grade openended items differently, because a particular teacher focuses on some aspect of the response more than another teacher does. It is less important that all teachers use the rubric in the same way and more important that a particular teacher uses it the same way for all of her or his students. Once your students become familiar with the rubric and how you use it, they will learn what kind of information they need to include in their responses in order to earn full credit.
Talking with your colleagues about student responses to openended questions and how to score the responses using the rubric is very helpful. Below is an openended question followed by student responses. We encourage you to look at these responses with a few of your colleagues and discuss how you would score the responses using the rubric. When you see that other teachers value many of the same criteria you do, you will feel more comfortable using the rubric.
Draw and give the dimensions of a rectangle and a triangle that have the same area. Show that the two figures have the same area.
Response A:
Response B:
Response C:
Response D: