## 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.

To increase the quality of the reflections from your students, try these steps:

- Give the students
**one**reflection question; this should be written beforehand with the class period’s lesson content in mind. This question provides the students an opportunity to make a connection between concepts and procedures. (See sample reflection questions below.) - Give the students
**two**opportunities—during two different class periods—to reflect on the same content. - At least once a week, share a student reflection in class and ask students what parts of it represent a high-quality reflection and how we know. Then ask the students how they might improve the reflection to highlight mathematical connections.

Sample reflection questions to use:

- How did you use area models (perfect square diagrams) to apply the process of completing the square?
- Compare recursive and explicit formulas. What information do you need to use either type of formula? What are the advantages of each? What ideas about arithmetic sequences are highlighted in each?
- What did you learn about connecting the factored form of a quadratic to the graph?
- What are some features of a geometric sequence? Consider thinking about patterns, graphs, tables, and equations.
- What are some features of an arithmetic sequence? Consider thinking about patterns, graphs, tables, and equations.

Have you tried out this teaching idea in your classroom? We want to hear from you on how it went! Please share your experience with us.

#### 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.

To maximize the number of students actively engaged in writing about mathematics/explaining their solution, try these steps:

- Provide students with 2 examples of possible responses to the first part of the assignment: one quality response that includes justification and reference to context and the other that is brief, without explanation or reference to context.
- Provide students with 3-4 min. to study the responses and discuss with their partners.
- Ask students which explanation is better and why.
- Discuss positive qualities of each explanation, and details the student could have included to improve their explanation.
- Provide students with checklist of key components of a quality problem-solving response. (See sample checklist below.)
- Encourage students to provide each other with quality feedback when working on writing explanations – for example, by prompting student conversations when you circulate around the classroom.

Sample checklist students can use to help self-evaluate their work before submitting:

Algebra 1 Name:_______________

**Problem Solving Explanation Rubric**

Understand Question | "Mathematize" the Situation | Monitor Progress & Trying New Strategies | Accuracy | Make Sense of Solution |
---|---|---|---|---|

I used mathematics appropriate to the situation. | I have included mathematical reasoning, representation(s), and vocabulary in my work (show your work). | I tried new strategies when I was stuck. | My solution is correct. | I have included units and explained what my answer means in the context of the problem (used complete sentences). |

Have you tried out this teaching idea in your classroom? We want to hear from you on how it went! Please share your experience with us.

#### 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.