When: Before most whole group share outs
Why: 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.
How: Students have quiet time to think about a problem and work on it individually, and then time to share their response or their progress with a partner. Once these partner conversations have taken place, some students are selected to share their thoughts with the class.
When: After a task involving multiple representations
Why: 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.
How: Students are given a problem that can be approached and solved using multiple strategies, or a situation that can be modeled using multiple representations and create a “display” of their work. Students investigate each other's work by taking a tour of the visual displays and begin to compare work with the teacher providing questions for students to ask each other, pointing out important mathematical features, and facilitating comparisons. Students to refer to each other’s thinking by asking them to make connections between specific features of expressions, tables, graphs, diagrams, words, and other representations of the same mathematical situation
When: To introduce students to new representations
Why: To “think like mathematicians”, to use mathematical structure to match two different representations
How: Match descriptions to visuals by chunking and connecting to math you know. Connecting Representations has these five main parts: Launch; Connect Representations; Share and Study Connections; Create a Representation; and Meta-Reflection
When: Anytime you students are sharing their mathematical thinking and ideas with other students.
Why: Sentence frames can support student language production by providing a structure to communicate about a topic.
How: Helpful sentence frames are open-ended, so as to amplify language production, not constrain it.
| language function | sample sentence frames and question starters |
|---|---|
| describe |
|
| explain |
|
| justify |
|
| compare and contrast |
|
| question |
|
| describe |
|
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). Note: Notice and Wonder and I Notice/I Wonder are trademarks of NCTM and the Math Forum and used in these materials with permission.
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.
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.
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.
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.
Adapted from Illustrative Mathematics, Instructional Routines. https://im.kendallhunt.com/MS/teachers/3/instructional_routines.html