Beyond Understanding in Math

Promote success in math with the Common Core Standards for Mathematical Practice

The Common Core State Standards for Mathematics begin with eight Standards for Mathematical Practice. Unlike more common content standards, these standards describe behaviors, expertise, and habits of mind necessary for successful work with math. Great math instruction pairs content standards with these standards of practice to ensure student success with math.

The Standards for Mathematical Practice are based on “processes and proficiencies” described by the National Council of Teachers of Mathematics in their process standards and the Strands of Proficiency described in the National Research Council’s Adding It Up report.

In short, the Standards for Mathematical Practice describe students who:

1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique 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.

These standards do not replace, or supersede, content standards. Without understanding of math concepts like number, value, function, and so on, there cannot be math proficiency! That being said, they are the place the Common Core State Standards for math begin. Without the productive dispositions the standards describe, student success with math in “the real world” beyond memorizing or completing formulas will be limited.

The CCSS for Mathematical Practice provide specific targets to help guide us as we develop math instruction and determine our classroom learning environment. Here are examples of what successful student math practice can look like and ways educators can build mathematics expertise in the classroom.

1. Make sense of problems and persevere in solving them

Problem solving is not simply the ability to solve an equation. Computation or application of known or expected formulas is not the same as “engaging in a task for which the solution method is NOT known in advance.” (NCTM Standards for School Mathematics, p.52)

When practicing this standard, students demonstrate:

• the ability to question, plan, and reflect.
• habits of persistence and curiosity.
• confidence in unfamiliar situations.

Effective problem solvers evaluate progress, reflect, and adjust what they are doing. Teachers can foster this by creating a classroom culture where it’s okay to ask questions, take risks, and make mistakes; an environment where curiosity and persistence are rewarded and reflection is promoted. They can also explicitly teach students problem-solving skills such as looking for patterns, listing possibilities, organizing ideas, finding similar or equivalent problems or cases, and working backward.

2. Reason abstractly and quantitatively

Reasoning involves making sense of quantities and how they relate to each other, as well as the capacity to follow a rational set of steps when approaching or solving a problem. Young students may need to use physical objects, while older students can abstract and use symbols, but both groups of students are capable of making logical distinctions and decisions.

When practicing this standard, students demonstrate:

• competence representing a problem with objects or symbols.
• the ability to apply properties of objects and operations.

To reason effectively, students must follow a rational, systematic series of steps based on sound mathematical procedures. Teachers can foster this by creating a classroom culture where students are rewarded for discovery, where it’s okay to play and experiment, to guess and try along the way. They can also explicitly teach students reasoning skills such as looking for patterns, rules, and generalities, considering alternatives, and using and manipulating symbols to represent a problem.

3. Construct viable arguments and critique reasoning of others

To reason effectively students, must discover the argument (rule) then analyze and evaluate it. As they make their own arguments and evaluate the arguments of others, they apply analytical skills.

When practicing this standard, students demonstrate:

• clear explanations and conclusions.
• sound argument.
• open sharing of thought process.
• analytical thinking.

Justifying and explaining ideas improves students' reasoning skills and their conceptual understanding. Teachers can foster this by creating a classroom culture that provides interesting problems, engages in comparing, and values listening, multiple perspectives, and thinking out loud. They can also explicitly teach students analytical skills such as formulating good questions, breaking into components, seeing logic, justifying conclusions, comparing, and evaluating.

4. Model with mathematics

When students create math models, they apply what they know about math to solve problems. Effective modeling with mathematics isn’t just creating graphs and representing data. It’s the ability to represent problems in different ways.

When practicing this standard, students demonstrate:

• ability to see quantities and relationships.
• application of knowledge and formulas.
• analysis with and of charts, graphs, and formulas.

Justifying and explaining ideas improves students' reasoning skills and their conceptual understanding. Teachers can foster this by creating a classroom culture that uses math manipulatives and encourages students to state assumptions, find generalizations, and make predictions. Working with data (or even playing with numbers) in the forms of diagrams, tables, charts, and graphs can also help students indentify number relationships and make predictions.

5. Use appropriate tools strategically

Students should be familiar with a variety of age-appropriate tools and choose the best one for the task at hand. Tools can be as simple as pencil and paper or as complex as today’s powerful graphing calculators. Concrete models and symbolic representations are also mathematical tools.

When practicing this standard, students demonstrate:

• knowledge of a variety of tools.
• ability to use a tool correctly.
• capacity to see limitations of a tool.
• effective choice of tools for a task.

While they obviously need to teach students how to use tools such as calculators, protractors, and spreadsheets, teachers must also give students opportunities to make decisions about which tool to use, even if this the lesson learned is that the tool students chose wasn’t the best choice.

6. Attend to precision

Students should be concerned with precise language when communicating and accuracy when working. Building vocabulary and using precise language leads to more effective communication and is essential to mathematical communication.

When practicing this standard, students demonstrate:

• attention to detail.
• precise communication.
• accurate calculations.

Teachers can foster this by creating a classroom culture in which quality is highly prized. Rewarding students when they take their time to do work carefully and encouraging them to check their work also results in improved accuracy and lets them know that you value effort.

7. Look for and make use of structure

Students use patterns and structure to help them break down complex problems into smaller, manageable components.

When practicing this standard, students demonstrate:

• ability to shift perspective.
• comfort with complexity.
• capacity to discern patterns or structure.

Teachers can foster this by creating a classroom culture that engages students with complex questions and problems, encourages play with patterns and models, and rewards testing and trying. They can help students cope with complex problems by teaching them to see structure and patterns, to break things down into component parts, and to take a different perspective.

8. Look for and express regularity in repeated reasoning

Students are able to see repetition and use that understanding to define mathematical rules, make predictions, and develop formulas.

When practicing this standard, students demonstrate:

• attention to detail.
• discerning of patterns or structure.
• ability to identify processes.

Teachers can foster this by creating a classroom culture that promotes comparisons, is filled with music, and egages students with sequencing and creating patterns. Teachers can help students find and express repetition by asking them to describe what is happening, identify similarities and differences, make predictions based on their observations of patterns, and combine all of this knowledge into rules and formulas.

It's the process!

The Common Core Standards for Mathematical Practice don’t so much describe “what” to teach as “how” to teach. Modeling, questioning, routines, discussions, and task selection help create an environment that gives students an opportunity to practice and develop these behaviors, skills, and habits of mind. All of these standards can be practiced in a project-based learning environment.

These standards can be addressed throughout the curriculum and aren’t limited to how we instruct in math. I would even argue that these standards provide an obvious entry point to pushing math across the curriculum and should be part of an intentional focus in all areas of learning.