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Hover to see how factors connect to Operations. Then click connected factors to explore strategies related to multiple factors.

Operations specify how quantities can be combined and transformed. There are four arithmetic Operations: **addition**, **subtraction**, **multiplication**, and **division**. Throughout the school years, students develop an understanding of how Operations work with different types of numbers, such as fractions, decimals, and integers.

Students typically begin by thinking about Operations as telling them which computations to do (such as finding the sum or difference of two numbers). However, it is critical that students also develop a *conceptual* understanding of Operations. This conceptual foundation supports students in estimating approximate computations, alongside quickly and accurately calculating using procedures.

Students' skill with Operations is supported by these critical conceptual components:

- Understanding the
**algebraic properties**of the Operations involved in the problem: For instance, students should able to transform the harder sum 3+8+7 into the easier sum 3+7+8 (it's easy to see that 3+7 = 10, and then it's easy to add 8 to 10); - Understanding the
**Place Value**system and**decomposition**: For instance, a student could also transform 3+8+7 into 3+7+1+7 (making it easier to add up to 10, before adding 1 and 7); and - Understanding how
**relationships in a real world situation**can be expressed by Operations (i.e., modeling).

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.

Daily review strengthens previous learning and can lead to fluent recall.

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.

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.

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

Practicing until achieving several error-free attempts is critical for retention.

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

Math centers with math games, manipulatives, and activities support learner interests and promote the development of more complex math skills and social interactions.

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

Rhyming, alliteration, and other sound devices reinforce math skills development by activating the mental processes that promote memory.

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

By talking through their thinking at each step of a process, teachers can model what learning looks like.

Instruction in multiple formats allows students to activate different cognitive skills to understand and remember the steps they are to take in their math work.

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.

Having students teach their knowledge, skills, and understanding to their classmates strengthens learning.

When teachers connect math to the students' world, students see how math is relevant and applicable to their daily lives.

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

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.

When students engage in a dialogue with themselves, they are able to orient, organize, and focus their thinking.

When students monitor their comprehension, behavior, or use of strategies, they build their Metacognition.

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.

Providing visuals to introduce, support, or review instruction activates more cognitive processes to support learning.

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

Writing that encourages students to articulate their understanding of math concepts or explain math ideas helps deepen students' mathematical understanding.

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