President's Message

Catherine Martin

President, Colorado Council of Teachers of Mathematics

This issue of the Colorado Mathematics Teacher

continues the focus on the Standards for Mathematical

Practice by taking a closer look at Math Practice 2: Reason

abstractly and quantitatively. Quantitative reasoning refers

to the ability to apply mathematical concepts and skills to

solve real problems, while abstract reasoning refers to the

process of representing a problem abstractly (e.g., with

an equation) and then manipulating the abstractions to

solve the problem. This practice, then, challenges students

to solve contextual problems by moving back and forth

between reasoning abstractly and reasoning quantitatively

as they make sense of the problem and solution in light of

the context.

When engaged in MP2, students begin by

carefully examining the problem and focusing on the

quantities and the units of those quantities within

the context. Once students develop an understanding

of the problem, they move to creating an abstract

representation of it. Thus, they decontextualize the

problem by representing it using numbers, symbols,

expressions, equations, graphs, and/or tables that

they can then manipulate. As they continue their work

with the representations, they pause when needed

to contextualize the problem and make sense of

their work as related to the original context. Math

Practice 2 calls for students to zoom in and out and

to decontextualize and re-contextualize as they move

back and forth between the problem context and its

mathematical representation.

To further understand MP2 as it plays out in

K–5 and 6–8, refer to “Standards for Mathematical

Practice: Commentary and Elaborations for K–5” and

“Standards for Mathematical Practice: Commentary and

Elaborations for 6–8”; both documents are available at

http://commoncoretools.me/author/wgmccallum/.

Teachers can support students’ engagement in and

development of their expertise with Math Practice 2 in

a variety of ways. First, we need to provide challenging

real-world problems and time for students to grapple

with and engage in discourse around them. These

problems, too, need to be ones that require students to

carefully examine the units of the quantities within, as

they seek to make sense of and understand the problem.

Secondly, we need to support students in their ability

to abstract the quantities in the problem using, as

mentioned previously, numbers, symbols, expressions,

equations, graphs, and/or tables. Additionally, we need

to encourage students to continually move back and

forth in connecting their mathematical representations

with the original context of the problem. To accomplish

this, students will need to explain and communicate

their thinking within their classroom community. And,

finally, we should help them understand that the ability

to reason abstractly and quantitatively is an essential

skill to their success in using mathematics to make sense

of our world.

As you begin the school year and engage your

students in the Standards for Mathematical Practice,

be sure to look for opportunities to deepen your own

understanding of them; save the date for the fall CCTM

Conference on September 25 (afternoon) and September

26 (all day). A conference focus will be on the new NCTM

publication, Principles to Actions: Ensuring Mathematical

Success for All. As you know, the Common Core State

Standards describe what students should know and be

able to do in mathematics AND the varieties of expertise

that students should develop and bring to bear on their

content learning. The eight Mathematics Teaching

Practices in Principles to Actions provide us with the

framework for supporting students in their endeavors

to meet the expectations of the Common Core. We are

delighted to welcome Diane Briars, President of NCTM,

and Jo Boaler to our conference and hope to see you

there!