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Mathematical Flexibility is the ability to maintain and shift among multiple representations of numbers and between problem-solving strategies in an adaptive manner. Students can use Mathematical Flexibility to better understand mathematical concepts and more adaptively use strategies to find solutions to math problems.

There are two key types of Mathematical Flexibility:

**Representational flexibility**supports students in thinking about number and relationships in different ways. For example, a fraction may be represented as a numerical ratio, a place on a number line, or a slice of a pie, while a simple function may be represented using a table, a verbal statement, or an equation. Being able to represent mathematical concepts in multiple ways supports conceptual understanding and allows students to see the deeper similarities among math problems.**Procedural flexibility**supports students in considering different strategies when solving a problem. Not only do students need to be able to solve problems quickly and accurately, they also need to make strategy choices in an adaptive manner. Procedural flexibility allows students to shift away from more cumbersome procedures to use more efficient problem-solving strategies given the content or context (e.g., dividing both sides of an equation by a constant as a first vs. a last step in the problem), which improves speed and supports greater accuracy in mathematical thinking and problem solving.

Providing math tasks with high cognitive demand conveys high expectations for all students by challenging them to engage in higher-order thinking.

As students solve problems in a group, they learn new strategies and practice communicating their mathematical thinking.

Students activate more cognitive processes by exploring and representing their understandings in visual form.

Continual use of foundational skills with different problems reinforces a conceptual understanding of math skills.

Thinking of and about patterns encourages learners to look for and understand the rules and relationships that are critical components of mathematical reasoning.

Teaching students to recognize the structures of algebraic representations helps them transfer solution methods from familiar to unfamiliar problems.

Discussing strategies for solving mathematics problems after initially letting students attempt to problem solve on their own helps them understand how to organize their Algebraic Thinking and intentionally tackle problems.

Analyzing incorrect worked examples is especially beneficial for helping students develop a conceptual understanding of mathematical processes.

When students explain their thinking process aloud with guidance in response to questions or prompts, they recognize the strategies they use and solidify their understanding.

The flipped classroom has two parts: cooperative group activities in class and digitally-based individual instruction out of class.

As students walk through stations working in small groups, the social and physical nature of the learning supports deeper understanding.

Adding motions to complement learning activates more cognitive processes for recall and understanding.

In guided inquiry, teachers help students use their own language for constructing knowledge by active listening and questioning.

As students work with and process information by discussing, organizing, and sharing it together, they deepen their understanding.

Math games allow students to practice many math skills in a fun, applied context.

When students have meaningful discussions about math and use math vocabulary, they develop the thinking, questioning, and explanation skills needed to master mathematical concepts.

Visualizing how ideas fit together helps students construct meaning and strengthens recall.

Providing physical and virtual representations of numbers and math concepts helps activate mental processes.

Visual representations help students understand what a number represents as well as recognize relationships between numbers.

Students deepen their understanding and gain confidence in their learning when they explain to and receive feedback from others.

Providing space and time for students to reflect is critical for moving what they have learned into Long-term Memory.

Math games and manipulatives for vision differences support math development for learners with visual needs.

Children's literature can be a welcoming way to help students learn math vocabulary and concepts.

Providing students a voice in their learning is critical for making learning meaningful.

When students create their own number and word problems, they connect math concepts to their background knowledge and lived experiences.

Students deepen their math understanding as they use and hear others use specific math language in informal ways.

Untimed tests provide students the opportunity to flexibly and productively work with numbers, further developing their problem-solving abilities.

Analyzing and discussing solved problems helps students develop a deeper understanding of abstract mathematical processes.

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