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Mathematical Flexibility is the ability to maintain and flexibly shift among multiple representations of numbers and problem-solving strategies. Students can use Mathematical Flexibility when they are problem solving to remember and shift among these multiple representations and strategies to help them better understand mathematical concepts and find solutions to math problems.

There are two key types of Mathematical Flexibility:

**Representational Flexibility**supports students in thinking about numbers in different ways. For example, a number may be represented as a point on a number line or a grouping of dots. Being able to represent numbers in multiple ways supports conceptual understanding of numbers overall 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 learned early on to more efficient strategies, which improves speed and supports greater accuracy in mathematical thinking and problem solving.

Building with blocks is ideal for promoting early geometric and Spatial Skills.

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

CRA is a sequential instructional approach during which students move from working with concrete materials to creating representational drawings to using abstract symbols.

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.

Knowing the language of math is critical because students must use this language to understand math concepts and determine calculations needed.

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.

Dot cards build number sense and promote early math skills, particularly Spatial Skills and Non-symbolic Number knowledge.

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.

Free choice supports learner interests and promotes the development of more complex social interactions.

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.

Teaching students through guided play encourages them to take an active role in their learning and supports the development of a broad array of cognitive skills.

Spending time with new content helps move concepts and ideas into Long-term Memory.

Math games use numbers and Spatial Skills, allowing students to practice many math skills in a fun, applied context.

When students have meaningful conversations 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 strengthen recall.

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

Easy access to seeing the relationships between numbers promotes number sense as students see these connections repeatedly.

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.

Students develop their skills by listening to and speaking with others in informal ways.

Three-phase lesson format is a problem-solving structure to promote meaningful math learning by activating prior knowledge, letting students explore mathematical thinking, and promoting a math community of learners.

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

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