Posted on June 13th, 2012 by Victoria Bill and Pam Goldman

The CCSS for Mathematics include both content standards and standards for mathematical practice. The content standards define "what students should understand and be able to do." The standards for mathematical practice describe "varieties of expertise that mathematics educators...should seek to develop in their students."

Some practice standards may have been familiar to mathematics educators in the past as "best practices." Others may be new. Some educators think of the content standards as the "what" and the practice standards as the "how." According to the CCSS, "The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students." How can we do this?

Some educators argue that we should just give students as many tasks as possible that draw on or require them to use as many mathematical practices as possible when determining the solution path for a task that targets a mathematical concept. Some educators argue that by continually giving students high-level tasks to work on, they will eventually engage with all the math practices. We argue that we must design strategically for engagement of students with all the mathematics practices regularly.

Consider Mathematical Practice Standard #7, "Look for and make use of structure." This practice may be new to educators. Don't overlook its potential value for both student learning and for formative assessment. Unless we give students opportunities to work on tasks that target the standards for mathematical content *and* require students to explain their reasoning with diagrams, equations, or written explanations of the structure of the mathematics of the task, we might find ourselves with a limited or false sense of student understanding.

Let's think this through using a third grade task:

Samantha has 2 bags of candy with 6 pieces of candy in each bag. How many pieces of candy does Samantha have?

Jerome has 4 bags of candy with 3 pieces of candy in each bag. How many pieces of candy does Jerome have?

Compare Samantha and Jerome's bags of candy. Who has more and how do you know? Make a diagram or write an equation that explains how you know who has the most and why?

Jerome has 4 bags of candy with 3 pieces of candy in each bag. How many pieces of candy does Jerome have?

Compare Samantha and Jerome's bags of candy. Who has more and how do you know? Make a diagram or write an equation that explains how you know who has the most and why?

This simple task asks students to work with several *content standards*:

- Standard 3.OA1 (Interpret products of whole numbers). Students wrestle with the meaning of the factors 2x6 and 4x3 in a multiplication problem.
- Standard 3.OA3 (Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities). Students compare how much candy Samantha and Jerome have.

This simple task asks students to use a variety of *mathematical practices*:

- MP#1 (Make sense of problems and persevere in solving them). Students must make sense of the problem and persevere as they attempt to determine the way to represent both students’ amounts of candy with a diagram and equations.
- MP#2 (Reason abstractly and quantitatively). Students must reason abstractly and quantitatively because the task is a contextual situation and students are required to write an equation to represent each student's candies. When students re-contextualize the algorithms in the context of the situation and explain the meaning of the expressions, they will demonstrate if they can work quantitatively.
- MP#3 (Construct viable arguments and critique the reasoning of others). Students are likely to reason that “Jerome and Samantha have the same amount of candy because both have 12 pieces of candy, thus constructing a viable argument.
- MP#4 (Model with Mathematics). Students' equations, diagrams of the bags of candies and their written explanation will let us know if they have a means of modeling with mathematics.

Students engage in four practice standards, pretty good! Now consider what more occurs when a final prompt is added to the task:

Some students claim Samantha and Jerome have the same amount of candy even though one of them has 2 bags and the other has 4 bags. Explain with a diagram or an equation why you agree or disagree.

Students are asked to use an additional *mathematical practice*:

MP#7 (Look for and make use of structure) comes into play. Without the final prompt, we may fail to find out if students really understand the reason why both expressions equal 12 candies. The focus on 2 bags versus 4 bags draws students' attention to the number of equal groups in relationship to the number of items in each of the groups. Ideally, students will explain that the two students’ amounts are equivalent because although Jerome has more bags he has fewer candies in each whereas Samantha has fewer bags but each bag has more candies so, in the end, the total number of candies are the same.

So, yes, tasks must address a content standard and as many of the mathematical practices as possible. But at some point during a unit of study it is crucial for students to **look for and make use of a structure**, to engage with the mathematical practice that requires them to reason about the underlying mathematical concepts which are articulated in many of the content standards. **Remember to keep your eye on and design for engagement with Mathematical Practice #7, look for and make use of structure.**