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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.

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 common problem structures 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 mathematical thinking and intentionally tackle problems.

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

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

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

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

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

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.

When students explain their thinking process aloud, they recognize the strategies they or others use and solidify their understanding.

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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