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All information come from "5 Practices for Orchestrating Productive Mathematics Discussions" By Margaret S. Smith and Mary Kay Stein.

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A goal of the lesson could be to have students be able to show two ways to solving the problem with mastery. This is a very basic example.

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This practice is specifically just for you as a teacher to have a sense of what you want to produce  or the mathematical goal you have in mind for the lesson. This helps to orchestrate discussion throughout the class period.

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Here we could connect the place values in the manipulatives to vertical addition for the students who may be struggling of seeing how vertical addition is working.

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The sequence in which you choose to go through your solutions is based on teacher preference and may be based on your goals of your lesson and the main solutions you may be wanting to cover.

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This is the same image shown as in Selecting stage and can be combined in the right column.

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In the middle column it demonstrates a few ways that a teacher may record the solutions that he/she has seen her students using.

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These here are some examples that a teacher may come up with for the problem 82+32

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While monitoring, you will want to have your anticipation sheet with you, marking solutions that students have used or not used to keep record in helping with your selecting process. These solutions may be correct or incorrect solutions (teacher preference.

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The right column combines selecting and sequencing together. You are select the student solutions that you want to cover as a class and in which order you will do so. This is where they may be correct or incorrect.

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For this sheet, you may lay it out as displayed or it can be helpful to leave room for your own personal notes, new strategies or incorrect strategies.

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While monitoring is happening, as the teacher is making her way around the classroom observing what students are using as their solutions, he/she may ask questions to what or why the students are getting at while solving.

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Connecting is tying strategies together from other students work to one another to help make connections they may understand better than others. These connections may help guide them to make progress in their mathematics or the classes mathematical goals.

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Sequencing is the order in which the selected students and their solutions will present their thoughts and solutions to the class.

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The teacher hand selects students or groups to share the strategies students used when solving and seeing during your monitoring process. These selected students later share their work for discussion with the rest of the class.

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The teacher wanders the classroom while the students are working in small groups, pairs, or individually, and observes the solution students are using for a problem.

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Anticipating is how the teacher suspects the students will solve a problem(s) that he/she will give to the class to solve.