Mathematics includes a set of knowledge and skills used to access science, technology, engineering, and mathematics (STEM) applications and careers as well as a tool for improving reasoning and analytic thinking. Effective **math instructional strategies** include using manipulatives to introduce mathematics concepts and applying systematic strategies for solving word problems.

### Overview of Math Principles and Practices

**Principle 1: Establish schoolwide practices for enhancing mathematics understanding within relevant content area instruction.**

- Practice 1: Encourage students to apply their understanding of mathematics concepts and procedures to draw conclusions and propose solutions about history, science, social studies, economics, and other content areas.
- Practice 2: Ask students to analyze events and phenomena from a quantitative perspective and use their analyses to develop arguments and provide just cations.

**Principle 2: Use a universal screener to identify students at risk for mathematics difficulties and to determine interventions to provide these at-risk students. Monitor the development of mathematics knowledge and skills of identified students.**

- Practice 1: Identify a system for screening and progress monitoring that prioritizes content and skills necessary for subsequent mathematics development.
- Practice 2: Select a cut score for screening that balances the need to help the most at-risk students with the resources available.

**Principle 3: Help students recognize number systems and expand their understanding beyond whole numbers to integers and rational numbers. Use number lines as a central representational tool in teaching this and other rational number concepts.**

- Practice 1: Use measurement activities and number lines to help students understand that fractions and decimals are numbers and share number properties.
- Practice 2: Provide opportunities for students to locate and compare fractions and decimals on number lines.
- Practice 3: Use number lines to improve students’ understanding of fraction equivalence, fraction density (the concept that there are an in nite number of fractions between any two fractions), and negative fractions.
- Practice 4: Explain that fractions can be represented as common fractions, decimals, and percentages, and develop students’ ability to translate among these forms.

**Principle 4: Develop students’ conceptual understanding of mathematics and provide ample opportunities to improve procedural fluency.**

- Practice 1: Use area models, number lines, and other visual representations to improve students’ understanding of formal computational procedures.
- Practice 2: Use meaningful fact practice activities for students lacking a strong foundation in math facts.
- Practice 3: Address common misconceptions regarding computational procedures. Practice 4: Present real-world contexts with plausible numbers for problems.

**Principle 5: Provide explicit and systematic instruction during instruction and intervention.**

- Practice 1: Include explicit teacher or peer modeling and demonstrate key concepts and skills.
- Practice 2: Include worked examples of key concepts and skills.
- Practice 3: Gradually transition from teacher-modeled problem solving to student- directed problem solving.
- Practice 4: Include opportunities for students to talk aloud about the skills, knowledge, or problem-solving procedures they are learning.
- Practice 5: Provide immediate and corrective feedback with opportunities for students to correct errors.
- Practice 6: Include sufficient, distributed, and cumulative practice and review.

**Principle 6: Instruction should include strategies for solving word and algebra problems that are based on common underlying structures.**

- Practice 1: Include systematic instruction on the structural connections between known, familiar, and novel word problems.
- Practice 2: Teach common problem types and their structures, as well as how to categorize and select appropriate solution methods for each problem type.

**Principle 7: For students who struggle in mathematics, instruction and intervention materials should include opportunities to work with representations of mathematical ideas. Teachers should be proficient in the use of these representations.**

- Practice 1: Employ visual representations to model mathematical concepts.
- Practice 2: Explicitly link a visual representation or model with the abstract mathematical symbol or concept.
- Practice 3: Use consistent language across similar representations.

**Principle 8: Establish a schoolwide plan to identify and improve teachers’ mathematical and pedagogical content knowledge.**

- Practice 1: Assess teachers’ needs in relation to mathematics content knowledge and mathematics pedagogical content knowledge across content areas.
- Practice 2: Select and implement high-quality professional development that acknowledges different teachers’ needs.
- Practice 3: Improve teachers’ knowledge and understanding of making practice decisions based on research evidence and student data.

**Principle 9: Discontinue using practices that are NOT associated with improved outcomes for students and teachers.**

- Practice 1: Examine the evidentiary bases of practices currently used in teaching mathematics and identify and eliminate practices that are contra- indicated by existing evidence.
- Practice 2: Monitor student learning formally and informally and use trend data to determine whether and how to adjust current practices.