## Refined Routines

The teaching ideas here are instructional routines teachers can implement in their classrooms to **help students become more deeply and actively engaged in understanding algebra.**

These ideas focus on how teachers can help students better engage, which we define as making deep mathematical connections, justifying and critiquing mathematical thinking, and solving challenging problems – or **Connect**,** Justify**, and **Solve.** We view the classroom as an under-utilized source for testing and refining instructional routines that are continuously informed by what teachers see day-by-day, class-by-class. These ideas have been tested and refined by our network members and may be promising strategies for your classroom, so take a look!

## Connect

Making connections among mathematical algorithms, concepts, and application to real-world contexts, where appropriate.

#### Connect Making Connections between Procedures and Concepts with Student-Structured Math Talks

This routine harnesses the power of group work to help students — rather than the teacher — make connections between procedures and concepts when new material is introduced.

#### Connect Introducing New Material with Open-Ended Problems

Rather than telling students what they will be studying next, what if you presented them with an open-ended, novel problem instead? You can use productive struggle to help your students connect with the mathematics.

#### Connect Making Connections with Exit Tickets

Many teachers use exit tickets to assess understanding of a skill or concept that has been taught. Instead, let’s use them to examine the connections students make between mathematics and real-life applications.

#### Connect End-of-Class Reflections

Do you teach right up to the bell? Use this routine to help students reflect on their own learning so you can use that to adjust your teaching.

#### Connect Using Exit Tickets for Skills or Concept Connections

How do you assess student retention of learning? This routine uses exit tickets as a catalyst for discussing connections between concepts and procedures.

## Justify

Communicating and justifying mathematical thinking as well as critiquing the reasoning of others.

#### Justify Infusing Justification into Problem Solving

Many students lack confidence in their problem-solving skills. Learn how non-traditional problems and scaffolding can help your students become more confident problem solvers.

#### Justify Student Math Talk in the Classroom Routine — Entry Tickets

Use entry tickets to help your students reflect on their prior learning before moving forward with a topic.

#### Justify Teaching Descriptive Statistics through Claims

The process of claim, evidence, and reasoning is regularly used in humanities and science classes. Use this process to both introduce and practice concepts in the math classroom.

#### Justify Improving Student Justifications through Questions and Prompts

Students often struggle to provide quality verbal or written justifications without significant teacher prompting. Use small-group work with targeted prompts to help your students improve.

#### Justify Improving Student Justifications through Sense Making

Choosing high-quality tasks and pairing them with a reasoning routine can create a more student-centered classroom.

#### Justify Sharing Errors and Stuck Points

Your students can move beyond looking at errors procedurally to critiquing the work of others using this routine to provide feedback to one another.

#### Justify Using Formative Assessment Tickets to Support Justification

It’s important for students to leave class with a sound understanding of the essential question addressed that day. With this routine, students build confidence by summarizing their learning.

#### Justify Changing the Structure of Structured Math Talk

By combining private reasoning time with structured math talk, students learn to communicate their mathematical understandings to others.

## Solve

Making sense of and solving challenging math problems that extend beyond rote application of algorithm.

#### Solve Using Written Examples to Help Students Explain Thinking

Math is about more than just getting a correct answer. By studying exemplars, students learn how to write quality explanations of their mathematical thinking.

#### Solve Inserting Non-rote Problems into Instruction

Break the cycle of students just repeating your actions by presenting them with structures for approaching non-rote problems.

#### Solve Assessing Homework

Homework review can eat up a significant chunk of class time. Use this routine to help students see mistakes as opportunities for improvement.

#### Solve Self-Monitoring Checklist

You can’t be everywhere! Help your students learn the components of quality math discussions by using a self-monitoring checklist.

#### Solve Incorporating Problem Solving and Related Perseverance into Instruction

Perseverance is difficult to teach and reinforce. Use this routine to help students stick with solving unfamiliar and challenging problems.