The CCSS and the Importance of Assessing Students' Mathematical Practices

Posted on June 21st, 2012 by Victoria Bill and Pam Goldman

We are glad to hear that the new CCSS math assessment tasks will require students to make use of mathematical practices while working with mathematical content. We are not happy to hear the possibility that test reports will contain information about student performance with regard to the content standards but not information about performance with regard to standards for mathematical practice. If the rumor proves true, it will result in a lost opportunity to provide teachers with vital information to support the kind of teaching and learning advocated by the CCSS for Mathematics.

Only by reporting results about the standards for mathematical practice will we know if students can use the practices independently and in high-stakes situations. Only by reporting results about the standards for mathematical practice are we likely to impact practice in most classrooms—for, as we know, when something is assessed, it is worked on.

Reports on the mathematical practices would provide teachers insights into ways in which the items will measure the practices, give them feedback on their students' use of the practices, and shape the pedagogy they use in supporting student use of the practices.

Let's look at an example of how standards for mathematical practices can be measured and what we can learn by measuring them. Consider the following task:

The Box of Candies Task

Now examine the work one student did in response to the task.

The Box of Candies Task

First, think about what we would consider if we were to measure the student's work against the content standards. The student's repeated addition equation is evidence of her understanding of the factors in the multiplication equation (3.OA1—use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities). The student claims that 6 x 3 and 3 x 6 are the "same thing." A convincing element of the work is that the student backs up her claim by showing the same product for each expression (she shows that the product of each is 18). The student also writes both a multiplication equation and a related repeated addition equation for each problem indicating that, regardless of the order of the factors, the product remains the same (3.OA5—apply properties of operations as strategies to multiple and divide).

Now, think about what we would consider and what we might learn if we were to measure the same work against the standards for mathematical practice. This work demonstrates the use of four of the eight mathematical practices. The student attempts to explain the relationship between the two expressions in two ways, though the two ways are not equally effective. The student work can be interpreted as showing that the student attempts to make sense of the task and to persevere and complete all parts of the task (MP#1). Some people may believe that the student persevered when solving the task because all parts are completed and two different methods are attempted. The first attempt, use of fact families—two multiplication and two division problems that make use of the same factors and products—does very little to convince us that the two expressions are equivalent. More effective is the viable argument the student constructed (MP#3) with the repeated addition equations written in relationship to the respective multiplication expressions to convince us that 3 sixes are the same as 6 threes because both are equal to 18 candies. The repeated addition and multiplication equations are accurate and the relationships made between the two operations are correct. Note that the student's division facts are not accurate. In terms of MP#4 (model with mathematics), we would not deduct points for the computational error because the student includes other work of value and the division facts are not pertinent to the task. Thus, the mathematical practice "attend to precision" (MP#6) does not come into play.

The student would also receive credit for demonstrating an understanding of mathematical structure (MP#7—looking for and making use of structure) (for more on MP#7, see our previous blog). Based on the two repeated addition equations that are related to the visual model and the accurate indication of the products of each, we can see that the student understands that the structure of an array problem can be described by the repeating rows or by the repeating columns and that we can refer to equal groups of rows with three cookies in each row or equal columns of six cookies in each column and still arrive at a product of 18 (the meaning of the commutative property). The student would also receive credit for demonstrating an understanding of mathematical structure.

From this one simple example, you can see that student understanding and use of mathematical practices can and should be assessed. The practices are important. Knowing that they will be tested and that data will be provided about how well students can use them in a high stakes environment can have a positive impact on classroom practice.