The next installment in our series explores how Math Practices 5 and 6 help students approach problems with confidence and express their conceptual understanding with precision.

*In this four-part series, the EdReports mathematics team explores the Standards for Mathematical Practice and why they're essential for every student to learn and grow. Read the rest of the series here:*

*How Math Practices 1–3 Help All Students Access Math Learning and Build Skills for the Future**Mathematics for All: How Modeling Transforms Student Learning**How Math Practices 7 and 8 Power Student “Lightbulb” Moments*

Through elementary and middle school, I was good at math—but it never really lit a spark for me. That all changed in ninth grade, thanks to Ms. Franklin.

I always remember the class when Ms. Franklin challenged us to use the quadratic formula to create a water-glass xylophone. We calculated and measured out the correct liquid volumes to set all the glasses to different musical pitches, then performed a flawless rendition of “Mary Had a Little Lamb.”

Ms. Franklin brought creativity into the classroom in a way I’d never seen before, making math fun and relevant to help us grow our mathematical proficiency. When I became a teacher, I knew I wanted to do the same for my students.

To inspire a love for math and to build the skills that students need for college and careers, the 8 Standards for Mathematical Practice are a must. Practices 5 and 6 in particular build confidence and agency, encouraging students to think critically and show deep understanding of mathematical concepts through precise expression. Let’s take a closer look!

Example of MP5 in a 6th Grade ClassroomStudents engage with the following word problem: A runner took 300 minutes to run 40 miles. If she runs at a constant speed:A) How long would she take to run 12 miles? B) What distance would she run in 30 minutes? C) What’s her speed in miles per hour? D) What’s her pace in minutes per mile? The teacher elicits solutions from the whole class, modeling the use of several different tools and strategies. These could include finding the unit rate of minutes per mile, using a double number line, using a tape diagram, and compiling a table to compare minutes against miles like the one below. |

Minutes: | 300 | 30 | 15 | 7.5 | 60 | 90 |

Miles: | 40 | 4 | 2 | 1 | 8 | 12 |

**Why this Practice is important**:

Students engaging in MP5 understand the range of available tools, including physical tools (e.g. a ruler or protractor), cognitive strategies (e.g. formulas or methods), and software tools (e.g. a spreadsheet or graphing calculator). Students can also understand the benefits and limits of applying a given tool to a particular problem or context.

MP5 is a great confidence builder. When I taught sixth grade, my students were finding their feet in the transition to middle school, grappling with new and daunting responsibilities outside of class: finding their lockers, getting to class on time, turning in homework. They were similarly daunted by math problems and often didn’t know where to begin.

But when students know what tools are available, they're energized to get started, make mistakes, and be creative. There’s little more satisfying in math than looking at a problem and feeling confident that you know how to approach it. Math Practice 5 also promotes student agency, putting the onus on learners to understand a problem conceptually and figure out the best method or tool to apply to it.

Example of MP6 in a High School ClassroomStudents solve quadratic equations using two different methods: graphing and the quadratic formula. The class splits into two debate groups—each is assigned one of the methods and prepares a presentation to defend it. The teams give their presentations and debate, engaging with MP6 by using precise language, giving clear definitions, and explaining their reasoning. The precision required by the debate format encourages students to collectively demonstrate and deepen their understanding of the underlying mathematical concepts. |

**Why this practice is important:**

As students engage in MP6, they use precision when giving definitions, when applying the correct symbols and units of measurement, and when calculating accurately and efficiently. In doing so, they demonstrate conceptual understanding and procedural skills—and the better you understand the math, the easier it is to be precise when applying skills.

Math is a language of its own—replacing the decimal with a comma in the number 3.142 makes a big difference! Conjugating verbs as a language learner can seem arbitrary if you don’t know that the verb endings reference different subjects; the same goes for understanding the concepts behind the language of math.

As a new teacher, I saw my students’ frustration and confusion around MP6: expressing volume with inches squared, mixing up x and y axes. My eureka moment came when I realized they weren’t attending to precision because they didn’t understand the “why.”

A student who’s taken the real-world measurements of a cuboid’s length, width, and height stands a better chance of connecting those three values with the concept of cubic inches than a learner focusing solely on the algorithmic procedure of calculating volume. That conceptual understanding positions students to express their answers precisely using the correct unit.

High-quality instructional materials help students develop mathematical knowledge and skills via intentional development of all the Math Practices. That’s why EdReports’ mathematics review tools address how materials elicit the Practices in student tasks and guide teachers to support that engagement.

In the case of MP5, our educator reviewers consider factors such as how well materials require students to choose and use tools and strategies that deepen their mathematical understanding and to recognize tools’ benefits and limitations.

For MP6, reviewers look for evidence including whether materials require students to communicate using grade-level appropriate vocabulary, to calculate accurately and efficiently, and to formulate clear explanations of their thinking. They also look at how well materials guide teachers to model all eight Practices and to monitor and give feedback on students’ engagement with them.

Making the most of the Math Practices means having high expectations of every student—enabling them to learn, grow, and prepare for college and careers. Math Practices 5 and 6 inspire learners to dive into challenges confidently, express themselves and justify their reasoning precisely, and apply their conceptual understanding and procedural skills to tackle real-world problems.

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