| Routine | Description | Example |
|---|---|---|
| Think, Pair, Share | This is a teaching routine useful in many contexts whose purpose is to give all students enough time to think about a prompt and form a response before they are expected to try to verbalize their thinking. First they have an opportunity to share their thinking in a low-stakes way with one partner, so that when they share with the class they can feel calm and confident, as well as say something meaningful that might advance everyone’s understanding. Additionally, the teacher has an opportunity to eavesdrop on the partner conversations so that they can purposefully select students to share with the class. | Give 2 minutes of quiet work time and then invite students to share their sentences with their partner, followed by whole-class discussion. |
| Stop & Jot | This processing activity gives students the opportunity to respond to questions in writing at different points throughout the lesson. |
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| Notice and Wonder |
When: This routine can appear as a warm-up or in the launch or synthesis of a classroom activity. Why: The purpose is to make a mathematical task accessible to all students with these two approachable questions. By thinking about them and responding, students gain entry into the context and might get their curiosity piqued. Taking steps to become familiar with a context and the mathematics that might be involved is making sense of problems (MP1). How: Students are shown some media or a mathematical representation. The prompt to students is “What do you notice? What do you wonder?” Students are given a few minutes to think of things they notice and things they wonder, and share them with a partner. Then, the teacher asks several students to share things they noticed and things they wondered; these are recorded by the teacher for all to see. Sometimes, the teacher steers the conversation to wondering about something mathematical that the class is about to focus on. |
Here is a table that shows how many rolls of paper towels a store receives when they order different numbers of cases. What do you notice? What do you wonder? |
| Which One Doesn’t Belong |
When: Warm-ups Why: Which One Doesn’t Belong fosters a need to define terms carefully and use words precisely (MP6) in order to compare and contrast a group of geometric figures or other mathematical representations. How: Students are presented with four figures, diagrams, graphs, or expressions with the prompt “Which one doesn’t belong?” Typically, each of the four options “doesn’t belong” for a different reason, and the similarities and differences are mathematically significant. Students are prompted to explain their rationale for deciding that one option doesn’t belong using their existing ideas and language and given opportunities to make their rationale more precise. |
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| Would You Rather |
When: depending on task, anywhere in the lesson Why: When students are asked to communicate their thinking when presenting and justifying solutions to problems they solve mentally, students develop more accurate, efficient, and flexible strategies. How: In this routine students are given two scenarios or choices and must apply their number sense and mathematical reasoning to make a choice. Students are then asked to communicate and represent mathematical thinking in many ways while explaining and justifying their preference. |
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| Open Middle |
When: warm ups Why: Students engage in tasks that have multiple ways of solving and include optimization such that it is easy to get an answer but more challenging to get the best or optimal answer. How: Routine that requires student to share their thinking with tasks the include the following: a “closed beginning” meaning that they all start with the same initial problem. a “closed end” meaning that they all end with the same answer. an “open middle” meaning that there are multiple ways to approach and ultimately solve the problem. |
Directions: Given the point (3,5), use digits 1-9, at most one time, to find a point (__, __) that minimizes the slope of the line that passes through the two points. The slope cannot be undefined. |
| Compare and Connect | To foster students’ meta-awareness as they identify, compare, and contrast different mathematical approaches, representations, and language. Teachers should demonstrate thinking out loud (e.g., exploring why one might do or say it this way, questioning an idea, wondering how an idea compares or connects to other ideas or language), and students should be prompted to reflect and respond. This routine supports meta-cognitive and meta-linguistic awareness, and also supports mathematical conversation. | “Focus discussion on different approaches to the second question. If any students with less-efficient methods were selected, have them go first in the sequence, or present one of these representations yourself. As students are presenting their work, encourage them to explain the meaning of any numbers used and the reason they decided to use particular operations. For example, if a student multiplies 80 by 5/8, ask them to explain what ⅝ means in this context and why they decided to multiply 80 by it. It can be handy to have representations like double number lines or tables displayed to facilitate these explanations.” |
| Connection Representations | Connecting Representations is an instructional routine that positions students to think structurally as they connect two representations by articulating the underlying mathematics. An the essential goal of this routine is expanding students’ repertoire of structural noticings, MP7. |  |
| Stronger and Clearer |
When: Stronger and clearer can be used to support students in revising and refining their mathematical thinking. Why: To provide a structured and interactive opportunity for students to revise and refine both their ideas and their verbal and written output. This routine provides a purpose for student conversation as well as fortifies output. The main idea is to have students think or write individually about a response, use a structured pairing strategy to have multiple opportunities to refine and clarify the response through conversation, and then finally revise their original written response. Throughout this process, students should be pressed for details, and encouraged to press each other for details. How: Students have an opportunity to draft a response; opportunity to share with at least 2 people, and have an opportunity to revise their response and reflect on their process. Teacher facilitates conversation, affirming student responses. |
Response – First Draft The first draft of a response is a student’s opportunity to write or draw their initial thoughts in response to the Prompt. Pair Meetings (2-3 times) Each meeting gives each partner an opportunity to be the speaker and an opportunity to be the listener. As the speaker, each student shares their ideas (without looking at their first draft, when possible). As a listener, each student should (a) ask questions for clarity and reasoning, (b) press for details and examples, and (c) give feedback that is relevant for the language goal. (1–2 min each meeting) Response – Second Draft Finally, after meeting with 2–3 different partners, students write a second draft. This draft should naturally reflect borrowed ideas from partners, as well as refinement of initial ideas through repeated communication with partners. |
| Critique, Correct, and Clarify |
When: Anytime students are sharing their thinking in writing Why: To give students a piece of mathematical writing that is not their own to analyze, reflect on, and develop. The intent is to prompt student reflection with an incorrect, incomplete, or ambiguous written argument or explanation, and for students to improve upon the written work by correcting errors and clarifying meaning. Teachers can model how to effectively and respectfully critique the work of others with meta-think-alouds and press for details when necessary. This routine fortifies output and engages students in meta awareness. |
PRESENT: Present a partial argument, explanation, or solution method. Teacher can play the role of the student who produced the response, and ask for help in fixing it.
PROMPT: Prompt students to identify the error(s) or ambiguity, analyze the response in light of their own understanding of the problem, and work both individually and in pairs to propose an improved response. SHARE: Pairs share out draft improved response. REFINE: Students refine their own draft response. |
Routines adapted from Illustrative mathematics and Stanford Center for Assessment, Learning, and Equity’s Principles for the Design of Mathematics Curricula: Promoting Language and Content Development.
