When Five Plus Five Equals Eight

For the last several months, I have been privileged to work with hundreds of teachers as they unpack the Common Core State Standards. In each session, which have been held in grade level groups, we have worked through the meanings of the 8 Standards for Mathematical Practice and each time one point becomes extremely obvious.  These eight standards, or measures of student behavior, are inextricably linked.  It is no surprise really, as they were developed using information from the National Council of Teachers of Mathematics (NCTM) Process Standards and the National Research Council’s (NRC) Strands of Mathematical Proficiency.   The point is driven home, however, when teachers work through the process of identifying for themselves what each standard means and does not mean for their classroom.

NCTM’s five process standards include:

Problem Solving

  • Build new mathematical knowledge through problem solving
  • Solve problems that arise in mathematics and in other contexts
  • Apply and adapt a variety of appropriate strategies to solve problems
  • Monitor and reflect on the process of mathematical problem solving

Reasoning and Proof

  • Recognize reasoning and proof as fundamental aspects of mathematics
  • Make and investigate mathematical conjectures
  • Develop and evaluate mathematical arguments and proofs
  • Select and use various types of reasoning and methods of proof


  • Organize and consolidate their mathematical thinking through communication
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
  • Analyze and evaluate the mathematical thinking and strategies of others;
  • Use the language of mathematics to express mathematical ideas precisely.


  • Recognize and use connections among mathematical ideas
  • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
  • Recognize and apply mathematics in contexts outside of mathematics


  • Create and use representations to organize, record, and communicate mathematical ideas
  • Select, apply, and translate among mathematical representations to solve problems
  • Use representations to model and interpret physical, social, and mathematical phenomena

The five stands of mathematical proficiency are:

(1) Conceptual understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students] to learn new ideas by connecting those ideas to what they already know.” A few of the benefits of building conceptual understanding are that it supports retention, and prevents common errors.

(2) Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

(3) Strategic competence is the ability to formulate, represent, and solve mathematical problems.

(4) Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.

(5) Productive disposition is the inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p. 116)

When you combine the two you end up with these 8 Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

As teachers begin to discuss each of these eight standards they see how the process standards and mathematical proficiencies are interwoven throughout.  Examples of what that looks like for their classrooms and their practice can be found on videos at:  http://www.insidemathematics.org/index.php/common-core-standards

“Sum” thing to think about!

Hi, I am Renee' Smith and I began teaching math in, of all places, a maximum security prison. “I was 21 years old and 99 lbs. soaking wet. Some of my students were convicted murderers and as scary as it sounds, it was incredibly sad. Many of those men had failed in the school system and then turned to crime.” That early experience with public school dropouts, cemented my resolve to help future students understand and succeed in the area of mathematics. To that end I dedicated 16 years to the classroom, teaching 5th grade math through algebra. After obtaining my Masters in Education from Baker University, I joined the ESSDACK team in 2007. I have presented at the local, state, and national levels but my most worthwhile experience at ESSDACK has been working to take the fear out of math for teachers and students alike. Although my primary focus has been providing professional development in mathematics, I also devote time to podcasting and creating materials for teachers to use in their classrooms. My podcasts include, By the Numbers, Just Desserts and Math Snacks. Working at ESSDACK has provided me the opportunity to do many of the things I love. “I have the time to research current educational practices, I get to be creative, collaborative, and I meet tons of new people."

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