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Colorado Council Teachers of Mathematics

Professional Organization of Educators

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
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