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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 basic arithmetic operations: **addition**, **subtraction**, **multiplication**, and **division**. Students typically begin by learning addition and subtraction before learning the more complex procedures of multiplication and division around third grade.

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).

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.

10 minutes in each math session devoted to building fluent retrieval of basic arithmetic facts sets the foundation for learning new concepts.

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

In explicit number naming, the structure of the number name labels the number in Place Value order and clearly states the quantity.

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.

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.

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 use numbers and Spatial Skills, allowing 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 conversations 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 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.

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

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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