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

The Common Core State Standards for mathematics focus on **mathematical understanding**.

"[T]he development of these Standards began with research-based learning progressions detailing what is known today about how students' mathematical knowledge, skill, andunderstandingdevelop over time.... These standards define what students shouldunderstandand be able to do in their study of mathematics. Asking a student tounderstandsomething means asking a teacher to assess whether the student has understood it. But what does mathematicalunderstandinglook like? One hallmark of mathematicalunderstandingis the ability to justify, in a way appropriate to the student's mathematical maturity,whya particular mathematical statement is true or where a mathematical rule comes from."

We are fortunate that the CCSS mathematics content standards provide clarity about *what* students are expected to know and understand and, from the mathematical practice standards, *how* they are expected to work in mathematics classrooms. We know that mathematical understanding is key. Now what?

An integral part of preparing to teach the CCSS is studying the underlying meaning of the mathematics in content standards and planning how to engage students in making sense of these ideas.

What might you work on when you come together with colleagues to discuss the underlying meaning of the mathematics? Imagine you are discussing a grade 3 Operations and Algebraic Thinking standard.

3.0A.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (*Commutative property of multiplication*.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (*Associative property of multiplication*.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (*Distributive property*.)

Before we proceed, take a moment to note that this standard appears under the heading, "**Understand** properties of multiplication and the relationship between multiplication and division". Your time collaborating with colleagues will be best spent if you make sense of and link your practice to the understanding aspects of the standards. Here are suggestions about how you might proceed.

One thing you might do in your collaborative study of 3.OA.5 is to explore the meaning of the associative property of multiplication. You might arrive at this description of the associative property:

Solving for the total number of items (the product) in "a groups with b items" (a x b) and "c groups of these groups" is the same as thinking about “a groups” of (b x c). Both ways of putting the groups of items together result in the same product because, regardless of how the groups are put together, the same number of items are being combined.

Your group of colleagues might identify a variety of contexts and discuss what each illuminates about the property of associativity for students. For example, each of the following situations can be modeled by an expression using the associative property of multiplication.

- Three shelves in a grocery store contain 5 boxes of cereal, 3 boxes deep.
- There are three classrooms and each classroom has 8 boys and 8 girls.
- A roller coaster car seats 3 people in each seat, and there are 4 seats in each car. Each roller coaster train has 7 cars.

Exploring associativity with a variety of tools can also help to shed light on the meaning of the concept. Working with colleagues, you might create three-dimensional figures and discuss what the figures help to illuminate about the associative property of multiplication.

The three dimensional figures below are great representations of the three expressions:

Imagine what students might say if they understand the relationship between the three dimensional figures and the expressions.

When asked, "What does the associative property help us think about?" Students might say, "The associative property tells about equal groups of equal groups."

You and your colleagues need not "go it alone." The Thinking Through a Lesson Protocol is an important teacher tool for developing lessons that use students’ mathematical thinking as the critical ingredient in developing their understanding of key disciplinary ideas. (Smith, M.S., Bill, V.L. & Hughes, E.K. (2008, October). Thinking through a lesson: Successfully implementing high level tasks. *Mathematics Teaching in the Middle School, 14*(3), 132-138.)